[{"data":1,"prerenderedAt":67053},["ShallowReactive",2],{"post-\u002Fblog\u002F2026\u002F0007-dirichlet-distribution-and-evidential-deep-learning":3,"i-ci:tag":64987,"i-mingcute:time-line":64987},{"post":4,"nextPost":64998,"prevPost":65339},{"id":5,"title":6,"body":7,"description":64983,"draft":64984,"enableCodeHighlight":64984,"enableCodeHightlight":64984,"enableComment":64985,"enableD2":64984,"enableKatex":64985,"extension":64986,"image":64941,"important":64984,"location":64987,"meta":64988,"navigation":64985,"ogImage":64987,"path":64989,"seo":64990,"stem":64991,"summary":64992,"tags":64993,"time":64996,"__hash__":64997},"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",[882,883,885],"theorem",{"title":884},"定义 1.1（概率单纯形）",[11,886,887,888,941,942,994,995,1527,1528,867],{},"对整数 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2",[15,911,913,932],{"className":912,"ariaHidden":48},[47],[15,914,916,920,923,926,929],{"className":915},[52],[15,917],{"className":918,"style":919},[56],"height:0.8193em;vertical-align:-0.136em;",[15,921,102],{"className":922,"style":174},[61,122],[15,924],{"className":925,"style":128},[127],[15,927,275],{"className":928},[132],[15,930],{"className":931,"style":128},[127],[15,933,935,938],{"className":934},[52],[15,936],{"className":937,"style":859},[56],[15,939,906],{"className":940},[61],"，",[15,943,945,963],{"className":944,"translate":19},[18],[15,946,948],{"className":947},[23],[25,949,950],{"xmlns":27},[29,951,952,960],{},[32,953,954,956,958],{},[35,955,102],{},[81,957,239],{},[89,959,91],{},[40,961,962],{"encoding":42},"K-1",[15,964,966,985],{"className":965,"ariaHidden":48},[47],[15,967,969,973,976,979,982],{"className":968},[52],[15,970],{"className":971,"style":972},[56],"height:0.7667em;vertical-align:-0.0833em;",[15,974,102],{"className":975,"style":174},[61,122],[15,977],{"className":978,"style":772},[127],[15,980,239],{"className":981},[375],[15,983],{"className":984,"style":772},[127],[15,986,988,991],{"className":987},[52],[15,989],{"className":990,"style":859},[56],[15,992,91],{"className":993},[61]," 维标准单纯形定义为\n",[15,996,998,1108],{"className":997,"translate":19},[18],[15,999,1001],{"className":1000},[23],[25,1002,1003],{"xmlns":27},[29,1004,1005,1105],{},[32,1006,1007,1019,1021],{},[227,1008,1009,1011],{},[35,1010,232],{"mathvariant":231},[32,1012,1013,1015,1017],{},[35,1014,102],{},[81,1016,239],{},[89,1018,91],{},[81,1020,244],{},[32,1022,1023,1025,1028,1030,1032,1038,1040,1042,1044,1050,1052,1054,1060,1062,1069,1071,1073,1075,1077,1103],{},[81,1024,87],{"fence":48},[35,1026,1027],{"mathvariant":251},"x",[81,1029,244],{},[81,1031,741],{"stretchy":86},[266,1033,1034,1036],{},[35,1035,1027],{},[89,1037,91],{},[81,1039,94],{"separator":48},[81,1041,97],{},[81,1043,94],{"separator":48},[266,1045,1046,1048],{},[35,1047,1027],{},[35,1049,102],{},[81,1051,750],{"stretchy":86},[81,1053,83],{},[227,1055,1056,1058],{},[35,1057,259],{"mathvariant":258},[35,1059,102],{},[81,1061,264],{},[266,1063,1064,1066],{},[35,1065,1027],{},[35,1067,1068],{},"i",[81,1070,275],{},[89,1072,278],{},[81,1074,94],{"separator":48},[282,1076,284],{},[286,1078,1079,1093,1099,1101],{"scriptlevel":278,"displaystyle":86},[289,1080,1081,1083,1091],{},[81,1082,293],{},[32,1084,1085,1087,1089],{},[35,1086,1068],{},[81,1088,244],{},[89,1090,91],{},[35,1092,102],{},[266,1094,1095,1097],{},[35,1096,1027],{},[35,1098,1068],{},[81,1100,244],{},[89,1102,91],{},[81,1104,105],{"fence":48},[40,1106,1107],{"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\\}",[15,1109,1111,1164],{"className":1110,"ariaHidden":48},[47],[15,1112,1114,1117,1155,1158,1161],{"className":1113},[52],[15,1115],{"className":1116,"style":695},[56],[15,1118,1120,1123],{"className":1119},[61],[15,1121,232],{"className":1122},[61],[15,1124,1126],{"className":1125},[338],[15,1127,1129],{"className":1128},[342],[15,1130,1132],{"className":1131},[346],[15,1133,1135],{"className":1134,"style":695},[350],[15,1136,1137,1140],{"style":716},[15,1138],{"className":1139,"style":358},[357],[15,1141,1143],{"className":1142},[362,363,364,365],[15,1144,1146,1149,1152],{"className":1145},[61,365],[15,1147,102],{"className":1148,"style":174},[61,122,365],[15,1150,239],{"className":1151},[375,365],[15,1153,91],{"className":1154},[61,365],[15,1156],{"className":1157,"style":128},[127],[15,1159,244],{"className":1160},[132],[15,1162],{"className":1163,"style":128},[127],[15,1165,1167,1170],{"className":1166},[52],[15,1168],{"className":1169,"style":394},[56],[15,1171,1173,1179,1182,1185,1188,1191,1194,1235,1238,1241,1244,1247,1250,1253,1294,1297,1300,1303,1306,1335,1338,1341,1344,1385,1388,1391,1394,1397,1400,1403,1406,1466,1469,1509,1512,1515,1518,1521],{"className":1172},[161],[15,1174,1176],{"className":1175,"style":402},[146,401],[15,1177,87],{"className":1178},[406,407],[15,1180,1027],{"className":1181},[61,411],[15,1183],{"className":1184,"style":128},[127],[15,1186,244],{"className":1187},[132],[15,1189],{"className":1190,"style":128},[127],[15,1192,741],{"className":1193},[146],[15,1195,1197,1200],{"className":1196},[61],[15,1198,1027],{"className":1199},[61,122],[15,1201,1203],{"className":1202},[338],[15,1204,1206,1227],{"className":1205},[342,472],[15,1207,1209,1224],{"className":1208},[346],[15,1210,1213],{"className":1211,"style":1212},[350],"height:0.3011em;",[15,1214,1215,1218],{"style":482},[15,1216],{"className":1217,"style":358},[357],[15,1219,1221],{"className":1220},[362,363,364,365],[15,1222,91],{"className":1223},[61,365],[15,1225,497],{"className":1226},[496],[15,1228,1230],{"className":1229},[346],[15,1231,1233],{"className":1232,"style":504},[350],[15,1234],{},[15,1236,94],{"className":1237},[153],[15,1239],{"className":1240,"style":157},[127],[15,1242,97],{"className":1243},[161],[15,1245],{"className":1246,"style":157},[127],[15,1248,94],{"className":1249},[153],[15,1251],{"className":1252,"style":157},[127],[15,1254,1256,1259],{"className":1255},[61],[15,1257,1027],{"className":1258},[61,122],[15,1260,1262],{"className":1261},[338],[15,1263,1265,1286],{"className":1264},[342,472],[15,1266,1268,1283],{"className":1267},[346],[15,1269,1272],{"className":1270,"style":1271},[350],"height:0.3283em;",[15,1273,1274,1277],{"style":482},[15,1275],{"className":1276,"style":358},[357],[15,1278,1280],{"className":1279},[362,363,364,365],[15,1281,102],{"className":1282,"style":174},[61,122,365],[15,1284,497],{"className":1285},[496],[15,1287,1289],{"className":1288},[346],[15,1290,1292],{"className":1291,"style":504},[350],[15,1293],{},[15,1295,750],{"className":1296},[178],[15,1298],{"className":1299,"style":128},[127],[15,1301,83],{"className":1302},[132],[15,1304],{"className":1305,"style":128},[127],[15,1307,1309,1312],{"className":1308},[61],[15,1310,259],{"className":1311},[61,427],[15,1313,1315],{"className":1314},[338],[15,1316,1318],{"className":1317},[342],[15,1319,1321],{"className":1320},[346],[15,1322,1324],{"className":1323,"style":695},[350],[15,1325,1326,1329],{"style":716},[15,1327],{"className":1328,"style":358},[357],[15,1330,1332],{"className":1331},[362,363,364,365],[15,1333,102],{"className":1334,"style":174},[61,122,365],[15,1336],{"className":1337,"style":128},[127],[15,1339,264],{"className":1340},[132],[15,1342],{"className":1343,"style":128},[127],[15,1345,1347,1350],{"className":1346},[61],[15,1348,1027],{"className":1349},[61,122],[15,1351,1353],{"className":1352},[338],[15,1354,1356,1377],{"className":1355},[342,472],[15,1357,1359,1374],{"className":1358},[346],[15,1360,1363],{"className":1361,"style":1362},[350],"height:0.3117em;",[15,1364,1365,1368],{"style":482},[15,1366],{"className":1367,"style":358},[357],[15,1369,1371],{"className":1370},[362,363,364,365],[15,1372,1068],{"className":1373},[61,122,365],[15,1375,497],{"className":1376},[496],[15,1378,1380],{"className":1379},[346],[15,1381,1383],{"className":1382,"style":504},[350],[15,1384],{},[15,1386],{"className":1387,"style":128},[127],[15,1389,275],{"className":1390},[132],[15,1392],{"className":1393,"style":128},[127],[15,1395,278],{"className":1396},[61],[15,1398,94],{"className":1399},[153],[15,1401,284],{"className":1402},[127],[15,1404],{"className":1405,"style":157},[127],[15,1407,1409,1412],{"className":1408},[531],[15,1410,293],{"className":1411,"style":537},[531,535,536],[15,1413,1415],{"className":1414},[338],[15,1416,1418,1458],{"className":1417},[342,472],[15,1419,1421,1455],{"className":1420},[346],[15,1422,1424,1444],{"className":1423,"style":550},[350],[15,1425,1426,1429],{"style":553},[15,1427],{"className":1428,"style":358},[357],[15,1430,1432],{"className":1431},[362,363,364,365],[15,1433,1435,1438,1441],{"className":1434},[61,365],[15,1436,1068],{"className":1437},[61,122,365],[15,1439,244],{"className":1440},[132,365],[15,1442,91],{"className":1443},[61,365],[15,1445,1446,1449],{"style":574},[15,1447],{"className":1448,"style":358},[357],[15,1450,1452],{"className":1451},[362,363,364,365],[15,1453,102],{"className":1454,"style":174},[61,122,365],[15,1456,497],{"className":1457},[496],[15,1459,1461],{"className":1460},[346],[15,1462,1464],{"className":1463,"style":593},[350],[15,1465],{},[15,1467],{"className":1468,"style":157},[127],[15,1470,1472,1475],{"className":1471},[61],[15,1473,1027],{"className":1474},[61,122],[15,1476,1478],{"className":1477},[338],[15,1479,1481,1501],{"className":1480},[342,472],[15,1482,1484,1498],{"className":1483},[346],[15,1485,1487],{"className":1486,"style":1362},[350],[15,1488,1489,1492],{"style":482},[15,1490],{"className":1491,"style":358},[357],[15,1493,1495],{"className":1494},[362,363,364,365],[15,1496,1068],{"className":1497},[61,122,365],[15,1499,497],{"className":1500},[496],[15,1502,1504],{"className":1503},[346],[15,1505,1507],{"className":1506,"style":504},[350],[15,1508],{},[15,1510],{"className":1511,"style":128},[127],[15,1513,244],{"className":1514},[132],[15,1516],{"className":1517,"style":128},[127],[15,1519,91],{"className":1520},[61],[15,1522,1524],{"className":1523,"style":402},[178,401],[15,1525,105],{"className":1526},[406,407],"\n其内部记作 ",[15,1529,1531,1605],{"className":1530,"translate":19},[18],[15,1532,1534],{"className":1533},[23],[25,1535,1536],{"xmlns":27},[29,1537,1538,1602],{},[32,1539,1540,1543,1546,1558,1560,1562,1564,1566,1578,1580,1586,1589,1591,1593,1595,1598,1600],{},[35,1541,1542],{"mathvariant":231},"int",[81,1544,1545],{},"⁡",[227,1547,1548,1550],{},[35,1549,232],{"mathvariant":231},[32,1551,1552,1554,1556],{},[35,1553,102],{},[81,1555,239],{},[89,1557,91],{},[81,1559,244],{},[81,1561,87],{"stretchy":86},[35,1563,1027],{"mathvariant":251},[81,1565,83],{},[227,1567,1568,1570],{},[35,1569,232],{"mathvariant":231},[32,1571,1572,1574,1576],{},[35,1573,102],{},[81,1575,239],{},[89,1577,91],{},[81,1579,264],{},[266,1581,1582,1584],{},[35,1583,1027],{},[35,1585,1068],{},[81,1587,1588],{},">",[89,1590,278],{},[81,1592,94],{"separator":48},[282,1594,284],{},[35,1596,1597],{"mathvariant":231},"∀",[35,1599,1068],{},[81,1601,105],{"stretchy":86},[40,1603,1604],{"encoding":42},"\\operatorname{int}\\Delta^{K-1} = \\{\\mathbf{x} \\in \\Delta^{K-1} : x_i > 0,\\ \\forall i\\}",[15,1606,1608,1671,1692,1745,1801],{"className":1607,"ariaHidden":48},[47],[15,1609,1611,1614,1621,1624,1662,1665,1668],{"className":1610},[52],[15,1612],{"className":1613,"style":695},[56],[15,1615,1617],{"className":1616},[531],[15,1618,1542],{"className":1619},[61,1620],"mathrm",[15,1622],{"className":1623,"style":157},[127],[15,1625,1627,1630],{"className":1626},[61],[15,1628,232],{"className":1629},[61],[15,1631,1633],{"className":1632},[338],[15,1634,1636],{"className":1635},[342],[15,1637,1639],{"className":1638},[346],[15,1640,1642],{"className":1641,"style":695},[350],[15,1643,1644,1647],{"style":716},[15,1645],{"className":1646,"style":358},[357],[15,1648,1650],{"className":1649},[362,363,364,365],[15,1651,1653,1656,1659],{"className":1652},[61,365],[15,1654,102],{"className":1655,"style":174},[61,122,365],[15,1657,239],{"className":1658},[375,365],[15,1660,91],{"className":1661},[61,365],[15,1663],{"className":1664,"style":128},[127],[15,1666,244],{"className":1667},[132],[15,1669],{"className":1670,"style":128},[127],[15,1672,1674,1677,1680,1683,1686,1689],{"className":1673},[52],[15,1675],{"className":1676,"style":142},[56],[15,1678,87],{"className":1679},[146],[15,1681,1027],{"className":1682},[61,411],[15,1684],{"className":1685,"style":128},[127],[15,1687,83],{"className":1688},[132],[15,1690],{"className":1691,"style":128},[127],[15,1693,1695,1698,1736,1739,1742],{"className":1694},[52],[15,1696],{"className":1697,"style":695},[56],[15,1699,1701,1704],{"className":1700},[61],[15,1702,232],{"className":1703},[61],[15,1705,1707],{"className":1706},[338],[15,1708,1710],{"className":1709},[342],[15,1711,1713],{"className":1712},[346],[15,1714,1716],{"className":1715,"style":695},[350],[15,1717,1718,1721],{"style":716},[15,1719],{"className":1720,"style":358},[357],[15,1722,1724],{"className":1723},[362,363,364,365],[15,1725,1727,1730,1733],{"className":1726},[61,365],[15,1728,102],{"className":1729,"style":174},[61,122,365],[15,1731,239],{"className":1732},[375,365],[15,1734,91],{"className":1735},[61,365],[15,1737],{"className":1738,"style":128},[127],[15,1740,264],{"className":1741},[132],[15,1743],{"className":1744,"style":128},[127],[15,1746,1748,1752,1792,1795,1798],{"className":1747},[52],[15,1749],{"className":1750,"style":1751},[56],"height:0.6891em;vertical-align:-0.15em;",[15,1753,1755,1758],{"className":1754},[61],[15,1756,1027],{"className":1757},[61,122],[15,1759,1761],{"className":1760},[338],[15,1762,1764,1784],{"className":1763},[342,472],[15,1765,1767,1781],{"className":1766},[346],[15,1768,1770],{"className":1769,"style":1362},[350],[15,1771,1772,1775],{"style":482},[15,1773],{"className":1774,"style":358},[357],[15,1776,1778],{"className":1777},[362,363,364,365],[15,1779,1068],{"className":1780},[61,122,365],[15,1782,497],{"className":1783},[496],[15,1785,1787],{"className":1786},[346],[15,1788,1790],{"className":1789,"style":504},[350],[15,1791],{},[15,1793],{"className":1794,"style":128},[127],[15,1796,1588],{"className":1797},[132],[15,1799],{"className":1800,"style":128},[127],[15,1802,1804,1807,1810,1813,1816,1819,1822,1825],{"className":1803},[52],[15,1805],{"className":1806,"style":142},[56],[15,1808,278],{"className":1809},[61],[15,1811,94],{"className":1812},[153],[15,1814,284],{"className":1815},[127],[15,1817],{"className":1818,"style":157},[127],[15,1820,1597],{"className":1821},[61],[15,1823,1068],{"className":1824},[61,122],[15,1826,105],{"className":1827},[178],[11,1829,1830,1904,1905,1963,1964,2105,2106,2157,2158,2210,2211,2263,2264,2292],{},[15,1831,1833,1857],{"className":1832,"translate":19},[18],[15,1834,1836],{"className":1835},[23],[25,1837,1838],{"xmlns":27},[29,1839,1840,1854],{},[32,1841,1842],{},[227,1843,1844,1846],{},[35,1845,232],{"mathvariant":231},[32,1847,1848,1850,1852],{},[35,1849,102],{},[81,1851,239],{},[89,1853,91],{},[40,1855,1856],{"encoding":42},"\\Delta^{K-1}",[15,1858,1860],{"className":1859,"ariaHidden":48},[47],[15,1861,1863,1866],{"className":1862},[52],[15,1864],{"className":1865,"style":695},[56],[15,1867,1869,1872],{"className":1868},[61],[15,1870,232],{"className":1871},[61],[15,1873,1875],{"className":1874},[338],[15,1876,1878],{"className":1877},[342],[15,1879,1881],{"className":1880},[346],[15,1882,1884],{"className":1883,"style":695},[350],[15,1885,1886,1889],{"style":716},[15,1887],{"className":1888,"style":358},[357],[15,1890,1892],{"className":1891},[362,363,364,365],[15,1893,1895,1898,1901],{"className":1894},[61,365],[15,1896,102],{"className":1897,"style":174},[61,122,365],[15,1899,239],{"className":1900},[375,365],[15,1902,91],{"className":1903},[61,365]," 是 ",[15,1906,1908,1925],{"className":1907,"translate":19},[18],[15,1909,1911],{"className":1910},[23],[25,1912,1913],{"xmlns":27},[29,1914,1915,1923],{},[32,1916,1917],{},[227,1918,1919,1921],{},[35,1920,259],{"mathvariant":258},[35,1922,102],{},[40,1924,685],{"encoding":42},[15,1926,1928],{"className":1927,"ariaHidden":48},[47],[15,1929,1931,1934],{"className":1930},[52],[15,1932],{"className":1933,"style":695},[56],[15,1935,1937,1940],{"className":1936},[61],[15,1938,259],{"className":1939},[61,427],[15,1941,1943],{"className":1942},[338],[15,1944,1946],{"className":1945},[342],[15,1947,1949],{"className":1948},[346],[15,1950,1952],{"className":1951,"style":695},[350],[15,1953,1954,1957],{"style":716},[15,1955],{"className":1956,"style":358},[357],[15,1958,1960],{"className":1959},[362,363,364,365],[15,1961,102],{"className":1962,"style":174},[61,122,365]," 中的紧凸集，其顶点为坐标向量 ",[15,1965,1967,1998],{"className":1966,"translate":19},[18],[15,1968,1970],{"className":1969},[23],[25,1971,1972],{"xmlns":27},[29,1973,1974,1995],{},[32,1975,1976,1983,1985,1987,1989],{},[266,1977,1978,1981],{},[35,1979,1980],{"mathvariant":251},"e",[89,1982,91],{},[81,1984,94],{"separator":48},[81,1986,97],{},[81,1988,94],{"separator":48},[266,1990,1991,1993],{},[35,1992,1980],{"mathvariant":251},[35,1994,102],{},[40,1996,1997],{"encoding":42},"\\mathbf{e}_1, \\dots, \\mathbf{e}_K",[15,1999,2001],{"className":2000,"ariaHidden":48},[47],[15,2002,2004,2007,2047,2050,2053,2056,2059,2062,2065],{"className":2003},[52],[15,2005],{"className":2006,"style":829},[56],[15,2008,2010,2013],{"className":2009},[61],[15,2011,1980],{"className":2012},[61,411],[15,2014,2016],{"className":2015},[338],[15,2017,2019,2039],{"className":2018},[342,472],[15,2020,2022,2036],{"className":2021},[346],[15,2023,2025],{"className":2024,"style":1212},[350],[15,2026,2027,2030],{"style":482},[15,2028],{"className":2029,"style":358},[357],[15,2031,2033],{"className":2032},[362,363,364,365],[15,2034,91],{"className":2035},[61,365],[15,2037,497],{"className":2038},[496],[15,2040,2042],{"className":2041},[346],[15,2043,2045],{"className":2044,"style":504},[350],[15,2046],{},[15,2048,94],{"className":2049},[153],[15,2051],{"className":2052,"style":157},[127],[15,2054,97],{"className":2055},[161],[15,2057],{"className":2058,"style":157},[127],[15,2060,94],{"className":2061},[153],[15,2063],{"className":2064,"style":157},[127],[15,2066,2068,2071],{"className":2067},[61],[15,2069,1980],{"className":2070},[61,411],[15,2072,2074],{"className":2073},[338],[15,2075,2077,2097],{"className":2076},[342,472],[15,2078,2080,2094],{"className":2079},[346],[15,2081,2083],{"className":2082,"style":1271},[350],[15,2084,2085,2088],{"style":482},[15,2086],{"className":2087,"style":358},[357],[15,2089,2091],{"className":2090},[362,363,364,365],[15,2092,102],{"className":2093,"style":174},[61,122,365],[15,2095,497],{"className":2096},[496],[15,2098,2100],{"className":2099},[346],[15,2101,2103],{"className":2102,"style":504},[350],[15,2104],{},"。低维特例给出了全部几何直观：",[15,2107,2109,2127],{"className":2108,"translate":19},[18],[15,2110,2112],{"className":2111},[23],[25,2113,2114],{"xmlns":27},[29,2115,2116,2124],{},[32,2117,2118,2120,2122],{},[35,2119,102],{},[81,2121,244],{},[89,2123,906],{},[40,2125,2126],{"encoding":42},"K = 2",[15,2128,2130,2148],{"className":2129,"ariaHidden":48},[47],[15,2131,2133,2136,2139,2142,2145],{"className":2132},[52],[15,2134],{"className":2135,"style":57},[56],[15,2137,102],{"className":2138,"style":174},[61,122],[15,2140],{"className":2141,"style":128},[127],[15,2143,244],{"className":2144},[132],[15,2146],{"className":2147,"style":128},[127],[15,2149,2151,2154],{"className":2150},[52],[15,2152],{"className":2153,"style":859},[56],[15,2155,906],{"className":2156},[61]," 时它是一维线段，",[15,2159,2161,2180],{"className":2160,"translate":19},[18],[15,2162,2164],{"className":2163},[23],[25,2165,2166],{"xmlns":27},[29,2167,2168,2177],{},[32,2169,2170,2172,2174],{},[35,2171,102],{},[81,2173,244],{},[89,2175,2176],{},"3",[40,2178,2179],{"encoding":42},"K = 3",[15,2181,2183,2201],{"className":2182,"ariaHidden":48},[47],[15,2184,2186,2189,2192,2195,2198],{"className":2185},[52],[15,2187],{"className":2188,"style":57},[56],[15,2190,102],{"className":2191,"style":174},[61,122],[15,2193],{"className":2194,"style":128},[127],[15,2196,244],{"className":2197},[132],[15,2199],{"className":2200,"style":128},[127],[15,2202,2204,2207],{"className":2203},[52],[15,2205],{"className":2206,"style":859},[56],[15,2208,2176],{"className":2209},[61]," 时是二维三角形，",[15,2212,2214,2233],{"className":2213,"translate":19},[18],[15,2215,2217],{"className":2216},[23],[25,2218,2219],{"xmlns":27},[29,2220,2221,2230],{},[32,2222,2223,2225,2227],{},[35,2224,102],{},[81,2226,244],{},[89,2228,2229],{},"4",[40,2231,2232],{"encoding":42},"K = 4",[15,2234,2236,2254],{"className":2235,"ariaHidden":48},[47],[15,2237,2239,2242,2245,2248,2251],{"className":2238},[52],[15,2240],{"className":2241,"style":57},[56],[15,2243,102],{"className":2244,"style":174},[61,122],[15,2246],{"className":2247,"style":128},[127],[15,2249,244],{"className":2250},[132],[15,2252],{"className":2253,"style":128},[127],[15,2255,2257,2260],{"className":2256},[52],[15,2258],{"className":2259,"style":859},[56],[15,2261,2229],{"className":2262},[61]," 时是三维四面体。它与 ",[15,2265,2267,2280],{"className":2266,"translate":19},[18],[15,2268,2270],{"className":2269},[23],[25,2271,2272],{"xmlns":27},[29,2273,2274,2278],{},[32,2275,2276],{},[35,2277,102],{},[40,2279,102],{"encoding":42},[15,2281,2283],{"className":2282,"ariaHidden":48},[47],[15,2284,2286,2289],{"className":2285},[52],[15,2287],{"className":2288,"style":57},[56],[15,2290,102],{"className":2291,"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。",[11,2294,2295,2296,2453,2454,2679,2680,2710,2711,2771,2772,2927,2928,3001,3002,3076],{},"本文采用如下测度约定：以 ",[15,2297,2299,2335],{"className":2298,"translate":19},[18],[15,2300,2302],{"className":2301},[23],[25,2303,2304],{"xmlns":27},[29,2305,2306,2332],{},[32,2307,2308,2314,2316,2318,2320],{},[266,2309,2310,2312],{},[35,2311,1027],{},[89,2313,91],{},[81,2315,94],{"separator":48},[81,2317,97],{},[81,2319,94],{"separator":48},[266,2321,2322,2324],{},[35,2323,1027],{},[32,2325,2326,2328,2330],{},[35,2327,102],{},[81,2329,239],{},[89,2331,91],{},[40,2333,2334],{"encoding":42},"x_1, \\dots, x_{K-1}",[15,2336,2338],{"className":2337,"ariaHidden":48},[47],[15,2339,2341,2345,2385,2388,2391,2394,2397,2400,2403],{"className":2340},[52],[15,2342],{"className":2343,"style":2344},[56],"height:0.6389em;vertical-align:-0.2083em;",[15,2346,2348,2351],{"className":2347},[61],[15,2349,1027],{"className":2350},[61,122],[15,2352,2354],{"className":2353},[338],[15,2355,2357,2377],{"className":2356},[342,472],[15,2358,2360,2374],{"className":2359},[346],[15,2361,2363],{"className":2362,"style":1212},[350],[15,2364,2365,2368],{"style":482},[15,2366],{"className":2367,"style":358},[357],[15,2369,2371],{"className":2370},[362,363,364,365],[15,2372,91],{"className":2373},[61,365],[15,2375,497],{"className":2376},[496],[15,2378,2380],{"className":2379},[346],[15,2381,2383],{"className":2382,"style":504},[350],[15,2384],{},[15,2386,94],{"className":2387},[153],[15,2389],{"className":2390,"style":157},[127],[15,2392,97],{"className":2393},[161],[15,2395],{"className":2396,"style":157},[127],[15,2398,94],{"className":2399},[153],[15,2401],{"className":2402,"style":157},[127],[15,2404,2406,2409],{"className":2405},[61],[15,2407,1027],{"className":2408},[61,122],[15,2410,2412],{"className":2411},[338],[15,2413,2415,2444],{"className":2414},[342,472],[15,2416,2418,2441],{"className":2417},[346],[15,2419,2421],{"className":2420,"style":1271},[350],[15,2422,2423,2426],{"style":482},[15,2424],{"className":2425,"style":358},[357],[15,2427,2429],{"className":2428},[362,363,364,365],[15,2430,2432,2435,2438],{"className":2431},[61,365],[15,2433,102],{"className":2434,"style":174},[61,122,365],[15,2436,239],{"className":2437},[375,365],[15,2439,91],{"className":2440},[61,365],[15,2442,497],{"className":2443},[496],[15,2445,2447],{"className":2446},[346],[15,2448,2451],{"className":2449,"style":2450},[350],"height:0.2083em;",[15,2452],{}," 为自由坐标、",[15,2455,2457,2500],{"className":2456,"translate":19},[18],[15,2458,2460],{"className":2459},[23],[25,2461,2462],{"xmlns":27},[29,2463,2464,2497],{},[32,2465,2466,2472,2474,2476,2478,2491],{},[266,2467,2468,2470],{},[35,2469,1027],{},[35,2471,102],{},[81,2473,244],{},[89,2475,91],{},[81,2477,239],{},[266,2479,2480,2482],{},[81,2481,293],{},[32,2483,2484,2486,2489],{},[35,2485,1068],{},[81,2487,2488],{},"\u003C",[35,2490,102],{},[266,2492,2493,2495],{},[35,2494,1027],{},[35,2496,1068],{},[40,2498,2499],{"encoding":42},"x_K = 1 - \\sum_{i\u003CK} x_i",[15,2501,2503,2559,2578],{"className":2502,"ariaHidden":48},[47],[15,2504,2506,2510,2550,2553,2556],{"className":2505},[52],[15,2507],{"className":2508,"style":2509},[56],"height:0.5806em;vertical-align:-0.15em;",[15,2511,2513,2516],{"className":2512},[61],[15,2514,1027],{"className":2515},[61,122],[15,2517,2519],{"className":2518},[338],[15,2520,2522,2542],{"className":2521},[342,472],[15,2523,2525,2539],{"className":2524},[346],[15,2526,2528],{"className":2527,"style":1271},[350],[15,2529,2530,2533],{"style":482},[15,2531],{"className":2532,"style":358},[357],[15,2534,2536],{"className":2535},[362,363,364,365],[15,2537,102],{"className":2538,"style":174},[61,122,365],[15,2540,497],{"className":2541},[496],[15,2543,2545],{"className":2544},[346],[15,2546,2548],{"className":2547,"style":504},[350],[15,2549],{},[15,2551],{"className":2552,"style":128},[127],[15,2554,244],{"className":2555},[132],[15,2557],{"className":2558,"style":128},[127],[15,2560,2562,2566,2569,2572,2575],{"className":2561},[52],[15,2563],{"className":2564,"style":2565},[56],"height:0.7278em;vertical-align:-0.0833em;",[15,2567,91],{"className":2568},[61],[15,2570],{"className":2571,"style":772},[127],[15,2573,239],{"className":2574},[375],[15,2576],{"className":2577,"style":772},[127],[15,2579,2581,2585,2636,2639],{"className":2580},[52],[15,2582],{"className":2583,"style":2584},[56],"height:1.0771em;vertical-align:-0.3271em;",[15,2586,2588,2591],{"className":2587},[531],[15,2589,293],{"className":2590,"style":537},[531,535,536],[15,2592,2594],{"className":2593},[338],[15,2595,2597,2627],{"className":2596},[342,472],[15,2598,2600,2624],{"className":2599},[346],[15,2601,2604],{"className":2602,"style":2603},[350],"height:0.1786em;",[15,2605,2606,2609],{"style":553},[15,2607],{"className":2608,"style":358},[357],[15,2610,2612],{"className":2611},[362,363,364,365],[15,2613,2615,2618,2621],{"className":2614},[61,365],[15,2616,1068],{"className":2617},[61,122,365],[15,2619,2488],{"className":2620},[132,365],[15,2622,102],{"className":2623,"style":174},[61,122,365],[15,2625,497],{"className":2626},[496],[15,2628,2630],{"className":2629},[346],[15,2631,2634],{"className":2632,"style":2633},[350],"height:0.3271em;",[15,2635],{},[15,2637],{"className":2638,"style":157},[127],[15,2640,2642,2645],{"className":2641},[61],[15,2643,1027],{"className":2644},[61,122],[15,2646,2648],{"className":2647},[338],[15,2649,2651,2671],{"className":2650},[342,472],[15,2652,2654,2668],{"className":2653},[346],[15,2655,2657],{"className":2656,"style":1362},[350],[15,2658,2659,2662],{"style":482},[15,2660],{"className":2661,"style":358},[357],[15,2663,2665],{"className":2664},[362,363,364,365],[15,2666,1068],{"className":2667},[61,122,365],[15,2669,497],{"className":2670},[496],[15,2672,2674],{"className":2673},[346],[15,2675,2677],{"className":2676,"style":504},[350],[15,2678],{},"，并以 ",[15,2681,2683,2697],{"className":2682,"translate":19},[18],[15,2684,2686],{"className":2685},[23],[25,2687,2688],{"xmlns":27},[29,2689,2690,2694],{},[32,2691,2692],{},[35,2693,1027],{"mathvariant":251},[40,2695,2696],{"encoding":42},"\\mathbf{x}",[15,2698,2700],{"className":2699,"ariaHidden":48},[47],[15,2701,2703,2707],{"className":2702},[52],[15,2704],{"className":2705,"style":2706},[56],"height:0.4444em;",[15,2708,1027],{"className":2709},[61,411]," 上的 ",[15,2712,2714,2735],{"className":2713,"translate":19},[18],[15,2715,2717],{"className":2716},[23],[25,2718,2719],{"xmlns":27},[29,2720,2721,2733],{},[32,2722,2723,2725,2727,2729,2731],{},[81,2724,741],{"stretchy":86},[35,2726,102],{},[81,2728,239],{},[89,2730,91],{},[81,2732,750],{"stretchy":86},[40,2734,753],{"encoding":42},[15,2736,2738,2759],{"className":2737,"ariaHidden":48},[47],[15,2739,2741,2744,2747,2750,2753,2756],{"className":2740},[52],[15,2742],{"className":2743,"style":142},[56],[15,2745,741],{"className":2746},[146],[15,2748,102],{"className":2749,"style":174},[61,122],[15,2751],{"className":2752,"style":772},[127],[15,2754,239],{"className":2755},[375],[15,2757],{"className":2758,"style":772},[127],[15,2760,2762,2765,2768],{"className":2761},[52],[15,2763],{"className":2764,"style":142},[56],[15,2766,91],{"className":2767},[61],[15,2769,750],{"className":2770},[178]," 维 Lebesgue 测度 ",[15,2773,2775,2813],{"className":2774,"translate":19},[18],[15,2776,2778],{"className":2777},[23],[25,2779,2780],{"xmlns":27},[29,2781,2782,2810],{},[32,2783,2784,2787,2793,2796,2798],{},[35,2785,2786],{"mathvariant":231},"d",[266,2788,2789,2791],{},[35,2790,1027],{},[89,2792,91],{},[81,2794,2795],{},"⋯",[35,2797,2786],{"mathvariant":231},[266,2799,2800,2802],{},[35,2801,1027],{},[32,2803,2804,2806,2808],{},[35,2805,102],{},[81,2807,239],{},[89,2809,91],{},[40,2811,2812],{"encoding":42},"\\mathrm{d}x_1 \\cdots \\mathrm{d}x_{K-1}",[15,2814,2816],{"className":2815,"ariaHidden":48},[47],[15,2817,2819,2823,2826,2866,2869,2872,2875,2878],{"className":2818},[52],[15,2820],{"className":2821,"style":2822},[56],"height:0.9028em;vertical-align:-0.2083em;",[15,2824,2786],{"className":2825},[61,1620],[15,2827,2829,2832],{"className":2828},[61],[15,2830,1027],{"className":2831},[61,122],[15,2833,2835],{"className":2834},[338],[15,2836,2838,2858],{"className":2837},[342,472],[15,2839,2841,2855],{"className":2840},[346],[15,2842,2844],{"className":2843,"style":1212},[350],[15,2845,2846,2849],{"style":482},[15,2847],{"className":2848,"style":358},[357],[15,2850,2852],{"className":2851},[362,363,364,365],[15,2853,91],{"className":2854},[61,365],[15,2856,497],{"className":2857},[496],[15,2859,2861],{"className":2860},[346],[15,2862,2864],{"className":2863,"style":504},[350],[15,2865],{},[15,2867],{"className":2868,"style":157},[127],[15,2870,2795],{"className":2871},[161],[15,2873],{"className":2874,"style":157},[127],[15,2876,2786],{"className":2877},[61,1620],[15,2879,2881,2884],{"className":2880},[61],[15,2882,1027],{"className":2883},[61,122],[15,2885,2887],{"className":2886},[338],[15,2888,2890,2919],{"className":2889},[342,472],[15,2891,2893,2916],{"className":2892},[346],[15,2894,2896],{"className":2895,"style":1271},[350],[15,2897,2898,2901],{"style":482},[15,2899],{"className":2900,"style":358},[357],[15,2902,2904],{"className":2903},[362,363,364,365],[15,2905,2907,2910,2913],{"className":2906},[61,365],[15,2908,102],{"className":2909,"style":174},[61,122,365],[15,2911,239],{"className":2912},[375,365],[15,2914,91],{"className":2915},[61,365],[15,2917,497],{"className":2918},[496],[15,2920,2922],{"className":2921},[346],[15,2923,2925],{"className":2924,"style":2450},[350],[15,2926],{}," 为参考测度。在该约定下 ",[15,2929,2931,2954],{"className":2930,"translate":19},[18],[15,2932,2934],{"className":2933},[23],[25,2935,2936],{"xmlns":27},[29,2937,2938,2952],{},[32,2939,2940],{},[227,2941,2942,2944],{},[35,2943,232],{"mathvariant":231},[32,2945,2946,2948,2950],{},[35,2947,102],{},[81,2949,239],{},[89,2951,91],{},[40,2953,1856],{"encoding":42},[15,2955,2957],{"className":2956,"ariaHidden":48},[47],[15,2958,2960,2963],{"className":2959},[52],[15,2961],{"className":2962,"style":695},[56],[15,2964,2966,2969],{"className":2965},[61],[15,2967,232],{"className":2968},[61],[15,2970,2972],{"className":2971},[338],[15,2973,2975],{"className":2974},[342],[15,2976,2978],{"className":2977},[346],[15,2979,2981],{"className":2980,"style":695},[350],[15,2982,2983,2986],{"style":716},[15,2984],{"className":2985,"style":358},[357],[15,2987,2989],{"className":2988},[362,363,364,365],[15,2990,2992,2995,2998],{"className":2991},[61,365],[15,2993,102],{"className":2994,"style":174},[61,122,365],[15,2996,239],{"className":2997},[375,365],[15,2999,91],{"className":3000},[61,365]," 的体积为 ",[15,3003,3005,3035],{"className":3004,"translate":19},[18],[15,3006,3008],{"className":3007},[23],[25,3009,3010],{"xmlns":27},[29,3011,3012,3032],{},[32,3013,3014,3016,3019,3021,3023,3025,3027,3029],{},[89,3015,91],{},[35,3017,3018],{"mathvariant":231},"\u002F",[81,3020,741],{"stretchy":86},[35,3022,102],{},[81,3024,239],{},[89,3026,91],{},[81,3028,750],{"stretchy":86},[81,3030,3031],{"stretchy":86},"!",[40,3033,3034],{"encoding":42},"1\u002F(K-1)!",[15,3036,3038,3063],{"className":3037,"ariaHidden":48},[47],[15,3039,3041,3044,3048,3051,3054,3057,3060],{"className":3040},[52],[15,3042],{"className":3043,"style":142},[56],[15,3045,3047],{"className":3046},[61],"1\u002F",[15,3049,741],{"className":3050},[146],[15,3052,102],{"className":3053,"style":174},[61,122],[15,3055],{"className":3056,"style":772},[127],[15,3058,239],{"className":3059},[375],[15,3061],{"className":3062,"style":772},[127],[15,3064,3066,3069,3072],{"className":3065},[52],[15,3067],{"className":3068,"style":142},[56],[15,3070,91],{"className":3071},[61],[15,3073,3075],{"className":3074},[178],")!","，这一点在后文讨论熵时会重新出现。",[877,3078,3080],{"id":3079},"_12-形式化定义","1.2 形式化定义",[882,3082,3084],{"title":3083},"定义 1.2（Dirichlet 分布）",[11,3085,3086,3087,3359,3360,3571,3572,3600,3601,3688,3689,4097,4098,4126,4127,4162,4163,4242,4243,4246],{},"设参数向量 ",[15,3088,3090,3144],{"className":3089,"translate":19},[18],[15,3091,3093],{"className":3092},[23],[25,3094,3095],{"xmlns":27},[29,3096,3097,3141],{},[32,3098,3099,3103,3105,3107,3113,3115,3117,3119,3125,3127,3129],{},[35,3100,3102],{"mathvariant":3101},"bold-italic","α",[81,3104,244],{},[81,3106,741],{"stretchy":86},[266,3108,3109,3111],{},[35,3110,3102],{},[89,3112,91],{},[81,3114,94],{"separator":48},[81,3116,97],{},[81,3118,94],{"separator":48},[266,3120,3121,3123],{},[35,3122,3102],{},[35,3124,102],{},[81,3126,750],{"stretchy":86},[81,3128,83],{},[289,3130,3131,3133,3139],{},[35,3132,259],{"mathvariant":258},[32,3134,3135,3137],{},[81,3136,1588],{},[89,3138,278],{},[35,3140,102],{},[40,3142,3143],{"encoding":42},"\\boldsymbol{\\alpha} = (\\alpha_1, \\dots, \\alpha_K) \\in \\mathbb{R}_{>0}^K",[15,3145,3147,3172,3293],{"className":3146,"ariaHidden":48},[47],[15,3148,3150,3153,3163,3166,3169],{"className":3149},[52],[15,3151],{"className":3152,"style":2706},[56],[15,3154,3156],{"className":3155},[61],[15,3157,3159],{"className":3158},[61],[15,3160,3102],{"className":3161},[61,3162],"boldsymbol",[15,3164],{"className":3165,"style":128},[127],[15,3167,244],{"className":3168},[132],[15,3170],{"className":3171,"style":128},[127],[15,3173,3175,3178,3181,3223,3226,3229,3232,3235,3238,3241,3281,3284,3287,3290],{"className":3174},[52],[15,3176],{"className":3177,"style":142},[56],[15,3179,741],{"className":3180},[146],[15,3182,3184,3188],{"className":3183},[61],[15,3185,3102],{"className":3186,"style":3187},[61,122],"margin-right:0.0037em;",[15,3189,3191],{"className":3190},[338],[15,3192,3194,3215],{"className":3193},[342,472],[15,3195,3197,3212],{"className":3196},[346],[15,3198,3200],{"className":3199,"style":1212},[350],[15,3201,3203,3206],{"style":3202},"top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;",[15,3204],{"className":3205,"style":358},[357],[15,3207,3209],{"className":3208},[362,363,364,365],[15,3210,91],{"className":3211},[61,365],[15,3213,497],{"className":3214},[496],[15,3216,3218],{"className":3217},[346],[15,3219,3221],{"className":3220,"style":504},[350],[15,3222],{},[15,3224,94],{"className":3225},[153],[15,3227],{"className":3228,"style":157},[127],[15,3230,97],{"className":3231},[161],[15,3233],{"className":3234,"style":157},[127],[15,3236,94],{"className":3237},[153],[15,3239],{"className":3240,"style":157},[127],[15,3242,3244,3247],{"className":3243},[61],[15,3245,3102],{"className":3246,"style":3187},[61,122],[15,3248,3250],{"className":3249},[338],[15,3251,3253,3273],{"className":3252},[342,472],[15,3254,3256,3270],{"className":3255},[346],[15,3257,3259],{"className":3258,"style":1271},[350],[15,3260,3261,3264],{"style":3202},[15,3262],{"className":3263,"style":358},[357],[15,3265,3267],{"className":3266},[362,363,364,365],[15,3268,102],{"className":3269,"style":174},[61,122,365],[15,3271,497],{"className":3272},[496],[15,3274,3276],{"className":3275},[346],[15,3277,3279],{"className":3278,"style":504},[350],[15,3280],{},[15,3282,750],{"className":3283},[178],[15,3285],{"className":3286,"style":128},[127],[15,3288,83],{"className":3289},[132],[15,3291],{"className":3292,"style":128},[127],[15,3294,3296,3300],{"className":3295},[52],[15,3297],{"className":3298,"style":3299},[56],"height:1.1168em;vertical-align:-0.2755em;",[15,3301,3303,3306],{"className":3302},[61],[15,3304,259],{"className":3305},[61,427],[15,3307,3309],{"className":3308},[338],[15,3310,3312,3350],{"className":3311},[342,472],[15,3313,3315,3347],{"className":3314},[346],[15,3316,3318,3336],{"className":3317,"style":695},[350],[15,3319,3321,3324],{"style":3320},"top:-2.4519em;margin-left:0em;margin-right:0.05em;",[15,3322],{"className":3323,"style":358},[357],[15,3325,3327],{"className":3326},[362,363,364,365],[15,3328,3330,3333],{"className":3329},[61,365],[15,3331,1588],{"className":3332},[132,365],[15,3334,278],{"className":3335},[61,365],[15,3337,3338,3341],{"style":716},[15,3339],{"className":3340,"style":358},[357],[15,3342,3344],{"className":3343},[362,363,364,365],[15,3345,102],{"className":3346,"style":174},[61,122,365],[15,3348,497],{"className":3349},[496],[15,3351,3353],{"className":3352},[346],[15,3354,3357],{"className":3355,"style":3356},[350],"height:0.2755em;",[15,3358],{},"，记 ",[15,3361,3363,3403],{"className":3362,"translate":19},[18],[15,3364,3366],{"className":3365},[23],[25,3367,3368],{"xmlns":27},[29,3369,3370,3400],{},[32,3371,3372,3378,3380,3394],{},[266,3373,3374,3376],{},[35,3375,3102],{},[89,3377,278],{},[81,3379,244],{},[289,3381,3382,3384,3392],{},[81,3383,293],{},[32,3385,3386,3388,3390],{},[35,3387,1068],{},[81,3389,244],{},[89,3391,91],{},[35,3393,102],{},[266,3395,3396,3398],{},[35,3397,3102],{},[35,3399,1068],{},[40,3401,3402],{"encoding":42},"\\alpha_0 = \\sum_{i=1}^K \\alpha_i",[15,3404,3406,3461],{"className":3405,"ariaHidden":48},[47],[15,3407,3409,3412,3452,3455,3458],{"className":3408},[52],[15,3410],{"className":3411,"style":2509},[56],[15,3413,3415,3418],{"className":3414},[61],[15,3416,3102],{"className":3417,"style":3187},[61,122],[15,3419,3421],{"className":3420},[338],[15,3422,3424,3444],{"className":3423},[342,472],[15,3425,3427,3441],{"className":3426},[346],[15,3428,3430],{"className":3429,"style":1212},[350],[15,3431,3432,3435],{"style":3202},[15,3433],{"className":3434,"style":358},[357],[15,3436,3438],{"className":3437},[362,363,364,365],[15,3439,278],{"className":3440},[61,365],[15,3442,497],{"className":3443},[496],[15,3445,3447],{"className":3446},[346],[15,3448,3450],{"className":3449,"style":504},[350],[15,3451],{},[15,3453],{"className":3454,"style":128},[127],[15,3456,244],{"className":3457},[132],[15,3459],{"className":3460,"style":128},[127],[15,3462,3464,3468,3528,3531],{"className":3463},[52],[15,3465],{"className":3466,"style":3467},[56],"height:1.2809em;vertical-align:-0.2997em;",[15,3469,3471,3474],{"className":3470},[531],[15,3472,293],{"className":3473,"style":537},[531,535,536],[15,3475,3477],{"className":3476},[338],[15,3478,3480,3520],{"className":3479},[342,472],[15,3481,3483,3517],{"className":3482},[346],[15,3484,3486,3506],{"className":3485,"style":550},[350],[15,3487,3488,3491],{"style":553},[15,3489],{"className":3490,"style":358},[357],[15,3492,3494],{"className":3493},[362,363,364,365],[15,3495,3497,3500,3503],{"className":3496},[61,365],[15,3498,1068],{"className":3499},[61,122,365],[15,3501,244],{"className":3502},[132,365],[15,3504,91],{"className":3505},[61,365],[15,3507,3508,3511],{"style":574},[15,3509],{"className":3510,"style":358},[357],[15,3512,3514],{"className":3513},[362,363,364,365],[15,3515,102],{"className":3516,"style":174},[61,122,365],[15,3518,497],{"className":3519},[496],[15,3521,3523],{"className":3522},[346],[15,3524,3526],{"className":3525,"style":593},[350],[15,3527],{},[15,3529],{"className":3530,"style":157},[127],[15,3532,3534,3537],{"className":3533},[61],[15,3535,3102],{"className":3536,"style":3187},[61,122],[15,3538,3540],{"className":3539},[338],[15,3541,3543,3563],{"className":3542},[342,472],[15,3544,3546,3560],{"className":3545},[346],[15,3547,3549],{"className":3548,"style":1362},[350],[15,3550,3551,3554],{"style":3202},[15,3552],{"className":3553,"style":358},[357],[15,3555,3557],{"className":3556},[362,363,364,365],[15,3558,1068],{"className":3559},[61,122,365],[15,3561,497],{"className":3562},[496],[15,3564,3566],{"className":3565},[346],[15,3567,3569],{"className":3568,"style":504},[350],[15,3570],{},"。若随机向量 ",[15,3573,3575,3588],{"className":3574,"translate":19},[18],[15,3576,3578],{"className":3577},[23],[25,3579,3580],{"xmlns":27},[29,3581,3582,3586],{},[32,3583,3584],{},[35,3585,1027],{"mathvariant":251},[40,3587,2696],{"encoding":42},[15,3589,3591],{"className":3590,"ariaHidden":48},[47],[15,3592,3594,3597],{"className":3593},[52],[15,3595],{"className":3596,"style":2706},[56],[15,3598,1027],{"className":3599},[61,411]," 取值于 ",[15,3602,3604,3632],{"className":3603,"translate":19},[18],[15,3605,3607],{"className":3606},[23],[25,3608,3609],{"xmlns":27},[29,3610,3611,3629],{},[32,3612,3613,3615,3617],{},[35,3614,1542],{"mathvariant":231},[81,3616,1545],{},[227,3618,3619,3621],{},[35,3620,232],{"mathvariant":231},[32,3622,3623,3625,3627],{},[35,3624,102],{},[81,3626,239],{},[89,3628,91],{},[40,3630,3631],{"encoding":42},"\\operatorname{int}\\Delta^{K-1}",[15,3633,3635],{"className":3634,"ariaHidden":48},[47],[15,3636,3638,3641,3647,3650],{"className":3637},[52],[15,3639],{"className":3640,"style":695},[56],[15,3642,3644],{"className":3643},[531],[15,3645,1542],{"className":3646},[61,1620],[15,3648],{"className":3649,"style":157},[127],[15,3651,3653,3656],{"className":3652},[61],[15,3654,232],{"className":3655},[61],[15,3657,3659],{"className":3658},[338],[15,3660,3662],{"className":3661},[342],[15,3663,3665],{"className":3664},[346],[15,3666,3668],{"className":3667,"style":695},[350],[15,3669,3670,3673],{"style":716},[15,3671],{"className":3672,"style":358},[357],[15,3674,3676],{"className":3675},[362,363,364,365],[15,3677,3679,3682,3685],{"className":3678},[61,365],[15,3680,102],{"className":3681,"style":174},[61,122,365],[15,3683,239],{"className":3684},[375,365],[15,3686,91],{"className":3687},[61,365],"，且关于上述参考测度的密度为\n",[15,3690,3692,3769],{"className":3691,"translate":19},[18],[15,3693,3695],{"className":3694},[23],[25,3696,3697],{"xmlns":27},[29,3698,3699,3766],{},[32,3700,3701,3704,3706,3708,3711,3713,3715,3717,3733,3748],{},[35,3702,3703],{},"f",[81,3705,741],{"stretchy":86},[35,3707,1027],{"mathvariant":251},[81,3709,3710],{},"∣",[35,3712,3102],{"mathvariant":3101},[81,3714,750],{"stretchy":86},[81,3716,244],{},[3718,3719,3720,3722],"mfrac",{},[89,3721,91],{},[32,3723,3724,3727,3729,3731],{},[35,3725,3726],{},"B",[81,3728,741],{"stretchy":86},[35,3730,3102],{"mathvariant":3101},[81,3732,750],{"stretchy":86},[289,3734,3735,3738,3746],{},[81,3736,3737],{},"∏",[32,3739,3740,3742,3744],{},[35,3741,1068],{},[81,3743,244],{},[89,3745,91],{},[35,3747,102],{},[289,3749,3750,3752,3754],{},[35,3751,1027],{},[35,3753,1068],{},[32,3755,3756,3762,3764],{},[266,3757,3758,3760],{},[35,3759,3102],{},[35,3761,1068],{},[81,3763,239],{},[89,3765,91],{},[40,3767,3768],{"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":4075,"style":4076},[350],"height:0.143em;",[15,4078],{},[15,4080,239],{"className":4081},[375,365],[15,4083,91],{"className":4084},[61,365],[15,4086,497],{"className":4087},[496],[15,4089,4091],{"className":4090},[346],[15,4092,4095],{"className":4093,"style":4094},[350],"height:0.2769em;",[15,4096],{},"\n则称 ",[15,4099,4101,4114],{"className":4100,"translate":19},[18],[15,4102,4104],{"className":4103},[23],[25,4105,4106],{"xmlns":27},[29,4107,4108,4112],{},[32,4109,4110],{},[35,4111,1027],{"mathvariant":251},[40,4113,2696],{"encoding":42},[15,4115,4117],{"className":4116,"ariaHidden":48},[47],[15,4118,4120,4123],{"className":4119},[52],[15,4121],{"className":4122,"style":2706},[56],[15,4124,1027],{"className":4125},[61,411]," 服从参数为 ",[15,4128,4130,4144],{"className":4129,"translate":19},[18],[15,4131,4133],{"className":4132},[23],[25,4134,4135],{"xmlns":27},[29,4136,4137,4141],{},[32,4138,4139],{},[35,4140,3102],{"mathvariant":3101},[40,4142,4143],{"encoding":42},"\\boldsymbol{\\alpha}",[15,4145,4147],{"className":4146,"ariaHidden":48},[47],[15,4148,4150,4153],{"className":4149},[52],[15,4151],{"className":4152,"style":2706},[56],[15,4154,4156],{"className":4155},[61],[15,4157,4159],{"className":4158},[61],[15,4160,3102],{"className":4161},[61,3162]," 的迪利克雷分布，记作 ",[15,4164,4166,4194],{"className":4165,"translate":19},[18],[15,4167,4169],{"className":4168},[23],[25,4170,4171],{"xmlns":27},[29,4172,4173,4191],{},[32,4174,4175,4177,4180,4183,4185,4187,4189],{},[35,4176,1027],{"mathvariant":251},[81,4178,4179],{},"∼",[35,4181,4182],{"mathvariant":231},"Dir",[81,4184,1545],{},[81,4186,741],{"stretchy":86},[35,4188,3102],{"mathvariant":3101},[81,4190,750],{"stretchy":86},[40,4192,4193],{"encoding":42},"\\mathbf{x} \\sim \\operatorname{Dir}(\\boldsymbol{\\alpha})",[15,4195,4197,4215],{"className":4196,"ariaHidden":48},[47],[15,4198,4200,4203,4206,4209,4212],{"className":4199},[52],[15,4201],{"className":4202,"style":2706},[56],[15,4204,1027],{"className":4205},[61,411],[15,4207],{"className":4208,"style":128},[127],[15,4210,4179],{"className":4211},[132],[15,4213],{"className":4214,"style":128},[127],[15,4216,4218,4221,4227,4230,4239],{"className":4217},[52],[15,4219],{"className":4220,"style":142},[56],[15,4222,4224],{"className":4223},[531],[15,4225,4182],{"className":4226},[61,1620],[15,4228,741],{"className":4229},[146],[15,4231,4233],{"className":4232},[61],[15,4234,4236],{"className":4235},[61],[15,4237,3102],{"className":4238},[61,3162],[15,4240,750],{"className":4241},[178],"。其中归一化常数为",[661,4244,4245],{},"多元 Beta 函数",[15,4247,4249,4318],{"className":4248,"translate":19},[18],[15,4250,4252],{"className":4251},[23],[25,4253,4254],{"xmlns":27},[29,4255,4256,4315],{},[32,4257,4258,4260,4262,4264,4266,4268],{},[35,4259,3726],{},[81,4261,741],{"stretchy":86},[35,4263,3102],{"mathvariant":3101},[81,4265,750],{"stretchy":86},[81,4267,244],{},[3718,4269,4270,4299],{},[32,4271,4272,4286,4289,4291,4297],{},[289,4273,4274,4276,4284],{},[81,4275,3737],{},[32,4277,4278,4280,4282],{},[35,4279,1068],{},[81,4281,244],{},[89,4283,91],{},[35,4285,102],{},[35,4287,4288],{"mathvariant":231},"Γ",[81,4290,741],{"stretchy":86},[266,4292,4293,4295],{},[35,4294,3102],{},[35,4296,1068],{},[81,4298,750],{"stretchy":86},[32,4300,4301,4303],{},[35,4302,4288],{"mathvariant":231},[32,4304,4305,4307,4313],{},[81,4306,741],{"fence":48},[266,4308,4309,4311],{},[35,4310,3102],{},[89,4312,278],{},[81,4314,750],{"fence":48},[40,4316,4317],{"encoding":42},"B(\\boldsymbol{\\alpha}) = \\frac{\\prod_{i=1}^K \\Gamma(\\alpha_i)}{\\Gamma\\left(\\alpha_0\\right)}",[15,4319,4321,4354],{"className":4320,"ariaHidden":48},[47],[15,4322,4324,4327,4330,4333,4342,4345,4348,4351],{"className":4323},[52],[15,4325],{"className":4326,"style":142},[56],[15,4328,3726],{"className":4329,"style":3867},[61,122],[15,4331,741],{"className":4332},[146],[15,4334,4336],{"className":4335},[61],[15,4337,4339],{"className":4338},[61],[15,4340,3102],{"className":4341},[61,3162],[15,4343,750],{"className":4344},[178],[15,4346],{"className":4347,"style":128},[127],[15,4349,244],{"className":4350},[132],[15,4352],{"className":4353,"style":128},[127],[15,4355,4357,4361],{"className":4356},[52],[15,4358],{"className":4359,"style":4360},[56],"height:1.6984em;vertical-align:-0.52em;",[15,4362,4364,4367,4598],{"className":4363},[61],[15,4365],{"className":4366},[146,3837],[15,4368,4370],{"className":4369},[3718],[15,4371,4373,4590],{"className":4372},[342,472],[15,4374,4376,4587],{"className":4375},[346],[15,4377,4380,4450,4458],{"className":4378,"style":4379},[350],"height:1.1784em;",[15,4381,4382,4385],{"style":3853},[15,4383],{"className":4384,"style":3857},[357],[15,4386,4388],{"className":4387},[362,363,364,365],[15,4389,4391,4394],{"className":4390},[61,365],[15,4392,4288],{"className":4393},[61,365],[15,4395,4397,4403,4444],{"className":4396},[161,365],[15,4398,4400],{"className":4399,"style":402},[146,365,401],[15,4401,741],{"className":4402},[365],[15,4404,4406,4409],{"className":4405},[61,365],[15,4407,3102],{"className":4408,"style":3187},[61,122,365],[15,4410,4412],{"className":4411},[338],[15,4413,4415,4436],{"className":4414},[342,472],[15,4416,4418,4433],{"className":4417},[346],[15,4419,4422],{"className":4420,"style":4421},[350],"height:0.3173em;",[15,4423,4424,4427],{"style":4054},[15,4425],{"className":4426,"style":4058},[357],[15,4428,4430],{"className":4429},[362,4062,4063,365],[15,4431,278],{"className":4432},[61,365],[15,4434,497],{"className":4435},[496],[15,4437,4439],{"className":4438},[346],[15,4440,4442],{"className":4441,"style":4076},[350],[15,4443],{},[15,4445,4447],{"className":4446,"style":402},[178,365,401],[15,4448,750],{"className":4449},[365],[15,4451,4452,4455],{"style":3885},[15,4453],{"className":4454,"style":3857},[357],[15,4456],{"className":4457,"style":3893},[3892],[15,4459,4461,4464],{"style":4460},"top:-3.535em;",[15,4462],{"className":4463,"style":3857},[357],[15,4465,4467],{"className":4466},[362,363,364,365],[15,4468,4470,4534,4538,4541,4544,4584],{"className":4469},[61,365],[15,4471,4473,4476],{"className":4472},[531,365],[15,4474,3737],{"className":4475,"style":537},[531,535,536,365],[15,4477,4479],{"className":4478},[338],[15,4480,4482,4525],{"className":4481},[342,472],[15,4483,4485,4522],{"className":4484},[346],[15,4486,4489,4510],{"className":4487,"style":4488},[350],"height:0.9191em;",[15,4490,4492,4495],{"style":4491},"top:-2.1786em;margin-left:0em;margin-right:0.0714em;",[15,4493],{"className":4494,"style":4058},[357],[15,4496,4498],{"className":4497},[362,4062,4063,365],[15,4499,4501,4504,4507],{"className":4500},[61,365],[15,4502,1068],{"className":4503},[61,122,365],[15,4505,244],{"className":4506},[132,365],[15,4508,91],{"className":4509},[61,365],[15,4511,4513,4516],{"style":4512},"top:-2.931em;margin-right:0.0714em;",[15,4514],{"className":4515,"style":4058},[357],[15,4517,4519],{"className":4518},[362,4062,4063,365],[15,4520,102],{"className":4521,"style":174},[61,122,365],[15,4523,497],{"className":4524},[496],[15,4526,4528],{"className":4527},[346],[15,4529,4532],{"className":4530,"style":4531},[350],"height:0.3214em;",[15,4533],{},[15,4535],{"className":4536,"style":4537},[127,365],"margin-right:0.1952em;",[15,4539,4288],{"className":4540},[61,365],[15,4542,741],{"className":4543},[146,365],[15,4545,4547,4550],{"className":4546},[61,365],[15,4548,3102],{"className":4549,"style":3187},[61,122,365],[15,4551,4553],{"className":4552},[338],[15,4554,4556,4576],{"className":4555},[342,472],[15,4557,4559,4573],{"className":4558},[346],[15,4560,4562],{"className":4561,"style":4051},[350],[15,4563,4564,4567],{"style":4054},[15,4565],{"className":4566,"style":4058},[357],[15,4568,4570],{"className":4569},[362,4062,4063,365],[15,4571,1068],{"className":4572},[61,122,365],[15,4574,497],{"className":4575},[496],[15,4577,4579],{"className":4578},[346],[15,4580,4582],{"className":4581,"style":4076},[350],[15,4583],{},[15,4585,750],{"className":4586},[178,365],[15,4588,497],{"className":4589},[496],[15,4591,4593],{"className":4592},[346],[15,4594,4596],{"className":4595,"style":3918},[350],[15,4597],{},[15,4599],{"className":4600},[178,3837],[11,4602,4603],{},"该定义之所以合法，依赖于下述恒等式。它是整个理论的算术基础，值得单独给出证明。",[882,4605,4607],{"title":4606},"命题 1.3（归一化恒等式）",[11,4608,4609,4610,4732,4733],{},"对任意 ",[15,4611,4613,4641],{"className":4612,"translate":19},[18],[15,4614,4616],{"className":4615},[23],[25,4617,4618],{"xmlns":27},[29,4619,4620,4638],{},[32,4621,4622,4624,4626],{},[35,4623,3102],{"mathvariant":3101},[81,4625,83],{},[289,4627,4628,4630,4636],{},[35,4629,259],{"mathvariant":258},[32,4631,4632,4634],{},[81,4633,1588],{},[89,4635,278],{},[35,4637,102],{},[40,4639,4640],{"encoding":42},"\\boldsymbol{\\alpha} \\in \\mathbb{R}_{>0}^K",[15,4642,4644,4669],{"className":4643,"ariaHidden":48},[47],[15,4645,4647,4651,4660,4663,4666],{"className":4646},[52],[15,4648],{"className":4649,"style":4650},[56],"height:0.5782em;vertical-align:-0.0391em;",[15,4652,4654],{"className":4653},[61],[15,4655,4657],{"className":4656},[61],[15,4658,3102],{"className":4659},[61,3162],[15,4661],{"className":4662,"style":128},[127],[15,4664,83],{"className":4665},[132],[15,4667],{"className":4668,"style":128},[127],[15,4670,4672,4675],{"className":4671},[52],[15,4673],{"className":4674,"style":3299},[56],[15,4676,4678,4681],{"className":4677},[61],[15,4679,259],{"className":4680},[61,427],[15,4682,4684],{"className":4683},[338],[15,4685,4687,4724],{"className":4686},[342,472],[15,4688,4690,4721],{"className":4689},[346],[15,4691,4693,4710],{"className":4692,"style":695},[350],[15,4694,4695,4698],{"style":3320},[15,4696],{"className":4697,"style":358},[357],[15,4699,4701],{"className":4700},[362,363,364,365],[15,4702,4704,4707],{"className":4703},[61,365],[15,4705,1588],{"className":4706},[132,365],[15,4708,278],{"className":4709},[61,365],[15,4711,4712,4715],{"style":716},[15,4713],{"className":4714,"style":358},[357],[15,4716,4718],{"className":4717},[362,363,364,365],[15,4719,102],{"className":4720,"style":174},[61,122,365],[15,4722,497],{"className":4723},[496],[15,4725,4727],{"className":4726},[346],[15,4728,4730],{"className":4729,"style":3356},[350],[15,4731],{},"，\n",[15,4734,4736,4814],{"className":4735,"translate":19},[18],[15,4737,4739],{"className":4738},[23],[25,4740,4741],{"xmlns":27},[29,4742,4743,4811],{},[32,4744,4745,4762,4776,4794,4797,4799,4801,4803,4805,4807,4809],{},[266,4746,4747,4750],{},[81,4748,4749],{},"∫",[227,4751,4752,4754],{},[35,4753,232],{"mathvariant":231},[32,4755,4756,4758,4760],{},[35,4757,102],{},[81,4759,239],{},[89,4761,91],{},[289,4763,4764,4766,4774],{},[81,4765,3737],{},[32,4767,4768,4770,4772],{},[35,4769,1068],{},[81,4771,244],{},[89,4773,91],{},[35,4775,102],{},[289,4777,4778,4780,4782],{},[35,4779,1027],{},[35,4781,1068],{},[32,4783,4784,4790,4792],{},[266,4785,4786,4788],{},[35,4787,3102],{},[35,4789,1068],{},[81,4791,239],{},[89,4793,91],{},[282,4795,4796],{}," ",[35,4798,2786],{"mathvariant":231},[35,4800,1027],{"mathvariant":251},[81,4802,244],{},[35,4804,3726],{},[81,4806,741],{"stretchy":86},[35,4808,3102],{"mathvariant":3101},[81,4810,750],{"stretchy":86},[40,4812,4813],{"encoding":42},"\\int_{\\Delta^{K-1}} \\prod_{i=1}^K x_i^{\\alpha_i - 1} \\,\\mathrm{d}\\mathbf{x} = B(\\boldsymbol{\\alpha})",[15,4815,4817,5089],{"className":4816,"ariaHidden":48},[47],[15,4818,4820,4824,4908,4911,4971,4974,5071,5074,5077,5080,5083,5086],{"className":4819},[52],[15,4821],{"className":4822,"style":4823},[56],"height:1.3371em;vertical-align:-0.3558em;",[15,4825,4827,4831],{"className":4826},[531],[15,4828,4749],{"className":4829,"style":4830},[531,535,536],"margin-right:0.1945em;position:relative;top:-0.0006em;",[15,4832,4834],{"className":4833},[338],[15,4835,4837,4899],{"className":4836},[342,472],[15,4838,4840,4896],{"className":4839},[346],[15,4841,4844],{"className":4842,"style":4843},[350],"height:0.186em;",[15,4845,4847,4850],{"style":4846},"top:-2.3442em;margin-left:-0.1945em;margin-right:0.05em;",[15,4848],{"className":4849,"style":358},[357],[15,4851,4853],{"className":4852},[362,363,364,365],[15,4854,4856],{"className":4855},[61,365],[15,4857,4859,4862],{"className":4858},[61,365],[15,4860,232],{"className":4861},[61,365],[15,4863,4865],{"className":4864},[338],[15,4866,4868],{"className":4867},[342],[15,4869,4871],{"className":4870},[346],[15,4872,4875],{"className":4873,"style":4874},[350],"height:0.7741em;",[15,4876,4878,4881],{"style":4877},"top:-2.786em;margin-right:0.0714em;",[15,4879],{"className":4880,"style":4058},[357],[15,4882,4884],{"className":4883},[362,4062,4063,365],[15,4885,4887,4890,4893],{"className":4886},[61,365],[15,4888,102],{"className":4889,"style":174},[61,122,365],[15,4891,239],{"className":4892},[375,365],[15,4894,91],{"className":4895},[61,365],[15,4897,497],{"className":4898},[496],[15,4900,4902],{"className":4901},[346],[15,4903,4906],{"className":4904,"style":4905},[350],"height:0.3558em;",[15,4907],{},[15,4909],{"className":4910,"style":157},[127],[15,4912,4914,4917],{"className":4913},[531],[15,4915,3737],{"className":4916,"style":537},[531,535,536],[15,4918,4920],{"className":4919},[338],[15,4921,4923,4963],{"className":4922},[342,472],[15,4924,4926,4960],{"className":4925},[346],[15,4927,4929,4949],{"className":4928,"style":550},[350],[15,4930,4931,4934],{"style":553},[15,4932],{"className":4933,"style":358},[357],[15,4935,4937],{"className":4936},[362,363,364,365],[15,4938,4940,4943,4946],{"className":4939},[61,365],[15,4941,1068],{"className":4942},[61,122,365],[15,4944,244],{"className":4945},[132,365],[15,4947,91],{"className":4948},[61,365],[15,4950,4951,4954],{"style":574},[15,4952],{"className":4953,"style":358},[357],[15,4955,4957],{"className":4956},[362,363,364,365],[15,4958,102],{"className":4959,"style":174},[61,122,365],[15,4961,497],{"className":4962},[496],[15,4964,4966],{"className":4965},[346],[15,4967,4969],{"className":4968,"style":593},[350],[15,4970],{},[15,4972],{"className":4973,"style":157},[127],[15,4975,4977,4980],{"className":4976},[61],[15,4978,1027],{"className":4979},[61,122],[15,4981,4983],{"className":4982},[338],[15,4984,4986,5063],{"className":4985},[342,472],[15,4987,4989,5060],{"className":4988},[346],[15,4990,4992,5003],{"className":4991,"style":4008},[350],[15,4993,4994,4997],{"style":4011},[15,4995],{"className":4996,"style":358},[357],[15,4998,5000],{"className":4999},[362,363,364,365],[15,5001,1068],{"className":5002},[61,122,365],[15,5004,5005,5008],{"style":4023},[15,5006],{"className":5007,"style":358},[357],[15,5009,5011],{"className":5010},[362,363,364,365],[15,5012,5014,5054,5057],{"className":5013},[61,365],[15,5015,5017,5020],{"className":5016},[61,365],[15,5018,3102],{"className":5019,"style":3187},[61,122,365],[15,5021,5023],{"className":5022},[338],[15,5024,5026,5046],{"className":5025},[342,472],[15,5027,5029,5043],{"className":5028},[346],[15,5030,5032],{"className":5031,"style":4051},[350],[15,5033,5034,5037],{"style":4054},[15,5035],{"className":5036,"style":4058},[357],[15,5038,5040],{"className":5039},[362,4062,4063,365],[15,5041,1068],{"className":5042},[61,122,365],[15,5044,497],{"className":5045},[496],[15,5047,5049],{"className":5048},[346],[15,5050,5052],{"className":5051,"style":4076},[350],[15,5053],{},[15,5055,239],{"className":5056},[375,365],[15,5058,91],{"className":5059},[61,365],[15,5061,497],{"className":5062},[496],[15,5064,5066],{"className":5065},[346],[15,5067,5069],{"className":5068,"style":4094},[350],[15,5070],{},[15,5072],{"className":5073,"style":157},[127],[15,5075,2786],{"className":5076},[61,1620],[15,5078,1027],{"className":5079},[61,411],[15,5081],{"className":5082,"style":128},[127],[15,5084,244],{"className":5085},[132],[15,5087],{"className":5088,"style":128},[127],[15,5090,5092,5095,5098,5101,5110],{"className":5091},[52],[15,5093],{"className":5094,"style":142},[56],[15,5096,3726],{"className":5097,"style":3867},[61,122],[15,5099,741],{"className":5100},[146],[15,5102,5104],{"className":5103},[61],[15,5105,5107],{"className":5106},[61],[15,5108,3102],{"className":5109},[61,3162],[15,5111,750],{"className":5112},[178],[11,5114,5115,5118,5119,5212],{},[661,5116,5117],{},"证明"," 考虑 ",[15,5120,5122,5146],{"className":5121,"translate":19},[18],[15,5123,5125],{"className":5124},[23],[25,5126,5127],{"xmlns":27},[29,5128,5129,5143],{},[32,5130,5131],{},[289,5132,5133,5135,5141],{},[35,5134,259],{"mathvariant":258},[32,5136,5137,5139],{},[81,5138,1588],{},[89,5140,278],{},[35,5142,102],{},[40,5144,5145],{"encoding":42},"\\mathbb{R}_{>0}^K",[15,5147,5149],{"className":5148,"ariaHidden":48},[47],[15,5150,5152,5155],{"className":5151},[52],[15,5153],{"className":5154,"style":3299},[56],[15,5156,5158,5161],{"className":5157},[61],[15,5159,259],{"className":5160},[61,427],[15,5162,5164],{"className":5163},[338],[15,5165,5167,5204],{"className":5166},[342,472],[15,5168,5170,5201],{"className":5169},[346],[15,5171,5173,5190],{"className":5172,"style":695},[350],[15,5174,5175,5178],{"style":3320},[15,5176],{"className":5177,"style":358},[357],[15,5179,5181],{"className":5180},[362,363,364,365],[15,5182,5184,5187],{"className":5183},[61,365],[15,5185,1588],{"className":5186},[132,365],[15,5188,278],{"className":5189},[61,365],[15,5191,5192,5195],{"style":716},[15,5193],{"className":5194,"style":358},[357],[15,5196,5198],{"className":5197},[362,363,364,365],[15,5199,102],{"className":5200,"style":174},[61,122,365],[15,5202,497],{"className":5203},[496],[15,5205,5207],{"className":5206},[346],[15,5208,5210],{"className":5209,"style":3356},[350],[15,5211],{}," 上的 Gamma 积分之积：",[15,5214,5216],{"className":5215,"translate":19},[212],[15,5217,5219,5329],{"className":5218,"translate":19},[18],[15,5220,5222],{"className":5221},[23],[25,5223,5224],{"xmlns":27,"display":221},[29,5225,5226,5326],{},[32,5227,5228,5243,5245,5247,5253,5255,5257,5273,5287,5306,5320,5322,5324],{},[5229,5230,5231,5233,5241],"munderover",{},[81,5232,3737],{},[32,5234,5235,5237,5239],{},[35,5236,1068],{},[81,5238,244],{},[89,5240,91],{},[35,5242,102],{},[35,5244,4288],{"mathvariant":231},[81,5246,741],{"stretchy":86},[266,5248,5249,5251],{},[35,5250,3102],{},[35,5252,1068],{},[81,5254,750],{"stretchy":86},[81,5256,244],{},[266,5258,5259,5261],{},[81,5260,4749],{},[289,5262,5263,5265,5271],{},[35,5264,259],{"mathvariant":258},[32,5266,5267,5269],{},[81,5268,1588],{},[89,5270,278],{},[35,5272,102],{},[5229,5274,5275,5277,5285],{},[81,5276,3737],{},[32,5278,5279,5281,5283],{},[35,5280,1068],{},[81,5282,244],{},[89,5284,91],{},[35,5286,102],{},[289,5288,5289,5292,5294],{},[35,5290,5291],{},"t",[35,5293,1068],{},[32,5295,5296,5302,5304],{},[266,5297,5298,5300],{},[35,5299,3102],{},[35,5301,1068],{},[81,5303,239],{},[89,5305,91],{},[227,5307,5308,5310],{},[35,5309,1980],{},[32,5311,5312,5314],{},[81,5313,239],{},[266,5315,5316,5318],{},[35,5317,5291],{},[35,5319,1068],{},[282,5321,4796],{},[35,5323,2786],{"mathvariant":231},[35,5325,5291],{"mathvariant":251},[40,5327,5328],{"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：",[15,8077,8079],{"className":8078,"translate":19},[212],[15,8080,8082,8299],{"className":8081,"translate":19},[18],[15,8083,8085],{"className":8084},[23],[25,8086,8087],{"xmlns":27,"display":221},[29,8088,8089,8296],{},[32,8090,8091,8105,8107,8109,8115,8117,8119,8165,8225,8227,8230,8232,8234,8240,8242,8258,8272,8290,8292,8294],{},[5229,8092,8093,8095,8103],{},[81,8094,3737],{},[32,8096,8097,8099,8101],{},[35,8098,1068],{},[81,8100,244],{},[89,8102,91],{},[35,8104,102],{},[35,8106,4288],{"mathvariant":231},[81,8108,741],{"stretchy":86},[266,8110,8111,8113],{},[35,8112,3102],{},[35,8114,1068],{},[81,8116,750],{"stretchy":86},[81,8118,244],{},[32,8120,8121,8123,8131,8147,8157,8159,8161,8163],{},[81,8122,741],{"fence":48},[289,8124,8125,8127,8129],{},[81,8126,4749],{},[89,8128,278],{},[35,8130,6790],{"mathvariant":231},[227,8132,8133,8135],{},[35,8134,5858],{},[32,8136,8137,8143,8145],{},[266,8138,8139,8141],{},[35,8140,3102],{},[89,8142,278],{},[81,8144,239],{},[89,8146,91],{},[227,8148,8149,8151],{},[35,8150,1980],{},[32,8152,8153,8155],{},[81,8154,239],{},[35,8156,5858],{},[282,8158,4796],{},[35,8160,2786],{"mathvariant":231},[35,8162,5858],{},[81,8164,750],{"fence":48},[32,8166,8167,8169,8185,8199,8217,8219,8221,8223],{},[81,8168,741],{"fence":48},[266,8170,8171,8173],{},[81,8172,4749],{},[227,8174,8175,8177],{},[35,8176,232],{"mathvariant":231},[32,8178,8179,8181,8183],{},[35,8180,102],{},[81,8182,239],{},[89,8184,91],{},[5229,8186,8187,8189,8197],{},[81,8188,3737],{},[32,8190,8191,8193,8195],{},[35,8192,1068],{},[81,8194,244],{},[89,8196,91],{},[35,8198,102],{},[289,8200,8201,8203,8205],{},[35,8202,1027],{},[35,8204,1068],{},[32,8206,8207,8213,8215],{},[266,8208,8209,8211],{},[35,8210,3102],{},[35,8212,1068],{},[81,8214,239],{},[89,8216,91],{},[282,8218,4796],{},[35,8220,2786],{"mathvariant":231},[35,8222,1027],{"mathvariant":251},[81,8224,750],{"fence":48},[81,8226,244],{},[127,8228],{"linebreak":8229},"newline",[35,8231,4288],{"mathvariant":231},[81,8233,741],{"stretchy":86},[266,8235,8236,8238],{},[35,8237,3102],{},[89,8239,278],{},[81,8241,750],{"stretchy":86},[266,8243,8244,8246],{},[81,8245,4749],{},[227,8247,8248,8250],{},[35,8249,232],{"mathvariant":231},[32,8251,8252,8254,8256],{},[35,8253,102],{},[81,8255,239],{},[89,8257,91],{},[5229,8259,8260,8262,8270],{},[81,8261,3737],{},[32,8263,8264,8266,8268],{},[35,8265,1068],{},[81,8267,244],{},[89,8269,91],{},[35,8271,102],{},[289,8273,8274,8276,8278],{},[35,8275,1027],{},[35,8277,1068],{},[32,8279,8280,8286,8288],{},[266,8281,8282,8284],{},[35,8283,3102],{},[35,8285,1068],{},[81,8287,239],{},[89,8289,91],{},[282,8291,4796],{},[35,8293,2786],{"mathvariant":231},[35,8295,1027],{"mathvariant":251},[40,8297,8298],{"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。",[11,9312,9313,9314,9342,9343,9371,9372,9375],{},"命题 1.3 的证明过程本身包含了比结论更多的信息：",[15,9315,9317,9330],{"className":9316,"translate":19},[18],[15,9318,9320],{"className":9319},[23],[25,9321,9322],{"xmlns":27},[29,9323,9324,9328],{},[32,9325,9326],{},[35,9327,5858],{},[40,9329,5858],{"encoding":42},[15,9331,9333],{"className":9332,"ariaHidden":48},[47],[15,9334,9336,9339],{"className":9335},[52],[15,9337],{"className":9338,"style":6026},[56],[15,9340,5858],{"className":9341},[61,122]," 与 ",[15,9344,9346,9359],{"className":9345,"translate":19},[18],[15,9347,9349],{"className":9348},[23],[25,9350,9351],{"xmlns":27},[29,9352,9353,9357],{},[32,9354,9355],{},[35,9356,1027],{"mathvariant":251},[40,9358,2696],{"encoding":42},[15,9360,9362],{"className":9361,"ariaHidden":48},[47],[15,9363,9365,9368],{"className":9364},[52],[15,9366],{"className":9367,"style":2706},[56],[15,9369,1027],{"className":9370},[61,411]," 的",[661,9373,9374],{},"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。",[877,9377,9379],{"id":9378},"_13-生成性刻画gamma-商表示","1.3 生成性刻画：Gamma 商表示",[882,9381,9383],{"title":9382},"命题 1.4（Gamma 商表示）",[11,9384,13,9385,9529,9530,9697,9698,9768,9769,9797,9798,10326,10327,10403,10404,5981,10483,867],{},[15,9386,9388,9419],{"className":9387,"translate":19},[18],[15,9389,9391],{"className":9390},[23],[25,9392,9393],{"xmlns":27},[29,9394,9395,9416],{},[32,9396,9397,9404,9406,9408,9410],{},[266,9398,9399,9402],{},[35,9400,9401],{},"γ",[89,9403,91],{},[81,9405,94],{"separator":48},[81,9407,97],{},[81,9409,94],{"separator":48},[266,9411,9412,9414],{},[35,9413,9401],{},[35,9415,102],{},[40,9417,9418],{"encoding":42},"\\gamma_1, \\dots, \\gamma_K",[15,9420,9422],{"className":9421,"ariaHidden":48},[47],[15,9423,9425,9429,9471,9474,9477,9480,9483,9486,9489],{"className":9424},[52],[15,9426],{"className":9427,"style":9428},[56],"height:0.625em;vertical-align:-0.1944em;",[15,9430,9432,9436],{"className":9431},[61],[15,9433,9401],{"className":9434,"style":9435},[61,122],"margin-right:0.0556em;",[15,9437,9439],{"className":9438},[338],[15,9440,9442,9463],{"className":9441},[342,472],[15,9443,9445,9460],{"className":9444},[346],[15,9446,9448],{"className":9447,"style":1212},[350],[15,9449,9451,9454],{"style":9450},"top:-2.55em;margin-left:-0.0556em;margin-right:0.05em;",[15,9452],{"className":9453,"style":358},[357],[15,9455,9457],{"className":9456},[362,363,364,365],[15,9458,91],{"className":9459},[61,365],[15,9461,497],{"className":9462},[496],[15,9464,9466],{"className":9465},[346],[15,9467,9469],{"className":9468,"style":504},[350],[15,9470],{},[15,9472,94],{"className":9473},[153],[15,9475],{"className":9476,"style":157},[127],[15,9478,97],{"className":9479},[161],[15,9481],{"className":9482,"style":157},[127],[15,9484,94],{"className":9485},[153],[15,9487],{"className":9488,"style":157},[127],[15,9490,9492,9495],{"className":9491},[61],[15,9493,9401],{"className":9494,"style":9435},[61,122],[15,9496,9498],{"className":9497},[338],[15,9499,9501,9521],{"className":9500},[342,472],[15,9502,9504,9518],{"className":9503},[346],[15,9505,9507],{"className":9506,"style":1271},[350],[15,9508,9509,9512],{"style":9450},[15,9510],{"className":9511,"style":358},[357],[15,9513,9515],{"className":9514},[362,363,364,365],[15,9516,102],{"className":9517,"style":174},[61,122,365],[15,9519,497],{"className":9520},[496],[15,9522,9524],{"className":9523},[346],[15,9525,9527],{"className":9526,"style":504},[350],[15,9528],{}," 相互独立且 ",[15,9531,9533,9572],{"className":9532,"translate":19},[18],[15,9534,9536],{"className":9535},[23],[25,9537,9538],{"xmlns":27},[29,9539,9540,9569],{},[32,9541,9542,9548,9550,9553,9555,9557,9563,9565,9567],{},[266,9543,9544,9546],{},[35,9545,9401],{},[35,9547,1068],{},[81,9549,4179],{},[35,9551,9552],{"mathvariant":231},"Gamma",[81,9554,1545],{},[81,9556,741],{"stretchy":86},[266,9558,9559,9561],{},[35,9560,3102],{},[35,9562,1068],{},[81,9564,94],{"separator":48},[89,9566,91],{},[81,9568,750],{"stretchy":86},[40,9570,9571],{"encoding":42},"\\gamma_i \\sim \\operatorname{Gamma}(\\alpha_i, 1)",[15,9573,9575,9630],{"className":9574,"ariaHidden":48},[47],[15,9576,9578,9581,9621,9624,9627],{"className":9577},[52],[15,9579],{"className":9580,"style":9428},[56],[15,9582,9584,9587],{"className":9583},[61],[15,9585,9401],{"className":9586,"style":9435},[61,122],[15,9588,9590],{"className":9589},[338],[15,9591,9593,9613],{"className":9592},[342,472],[15,9594,9596,9610],{"className":9595},[346],[15,9597,9599],{"className":9598,"style":1362},[350],[15,9600,9601,9604],{"style":9450},[15,9602],{"className":9603,"style":358},[357],[15,9605,9607],{"className":9606},[362,363,364,365],[15,9608,1068],{"className":9609},[61,122,365],[15,9611,497],{"className":9612},[496],[15,9614,9616],{"className":9615},[346],[15,9617,9619],{"className":9618,"style":504},[350],[15,9620],{},[15,9622],{"className":9623,"style":128},[127],[15,9625,4179],{"className":9626},[132],[15,9628],{"className":9629,"style":128},[127],[15,9631,9633,9636,9642,9645,9685,9688,9691,9694],{"className":9632},[52],[15,9634],{"className":9635,"style":142},[56],[15,9637,9639],{"className":9638},[531],[15,9640,9552],{"className":9641},[61,1620],[15,9643,741],{"className":9644},[146],[15,9646,9648,9651],{"className":9647},[61],[15,9649,3102],{"className":9650,"style":3187},[61,122],[15,9652,9654],{"className":9653},[338],[15,9655,9657,9677],{"className":9656},[342,472],[15,9658,9660,9674],{"className":9659},[346],[15,9661,9663],{"className":9662,"style":1362},[350],[15,9664,9665,9668],{"style":3202},[15,9666],{"className":9667,"style":358},[357],[15,9669,9671],{"className":9670},[362,363,364,365],[15,9672,1068],{"className":9673},[61,122,365],[15,9675,497],{"className":9676},[496],[15,9678,9680],{"className":9679},[346],[15,9681,9683],{"className":9682,"style":504},[350],[15,9684],{},[15,9686,94],{"className":9687},[153],[15,9689],{"className":9690,"style":157},[127],[15,9692,91],{"className":9693},[61],[15,9695,750],{"className":9696},[178],"（形状 ",[15,9699,9701,9719],{"className":9700,"translate":19},[18],[15,9702,9704],{"className":9703},[23],[25,9705,9706],{"xmlns":27},[29,9707,9708,9716],{},[32,9709,9710],{},[266,9711,9712,9714],{},[35,9713,3102],{},[35,9715,1068],{},[40,9717,9718],{"encoding":42},"\\alpha_i",[15,9720,9722],{"className":9721,"ariaHidden":48},[47],[15,9723,9725,9728],{"className":9724},[52],[15,9726],{"className":9727,"style":2509},[56],[15,9729,9731,9734],{"className":9730},[61],[15,9732,3102],{"className":9733,"style":3187},[61,122],[15,9735,9737],{"className":9736},[338],[15,9738,9740,9760],{"className":9739},[342,472],[15,9741,9743,9757],{"className":9742},[346],[15,9744,9746],{"className":9745,"style":1362},[350],[15,9747,9748,9751],{"style":3202},[15,9749],{"className":9750,"style":358},[357],[15,9752,9754],{"className":9753},[362,363,364,365],[15,9755,1068],{"className":9756},[61,122,365],[15,9758,497],{"className":9759},[496],[15,9761,9763],{"className":9762},[346],[15,9764,9766],{"className":9765,"style":504},[350],[15,9767],{},"、率 ",[15,9770,9772,9785],{"className":9771,"translate":19},[18],[15,9773,9775],{"className":9774},[23],[25,9776,9777],{"xmlns":27},[29,9778,9779,9783],{},[32,9780,9781],{},[89,9782,91],{},[40,9784,91],{"encoding":42},[15,9786,9788],{"className":9787,"ariaHidden":48},[47],[15,9789,9791,9794],{"className":9790},[52],[15,9792],{"className":9793,"style":859},[56],[15,9795,91],{"className":9796},[61],"）。令\n",[15,9799,9801,9890],{"className":9800,"translate":19},[18],[15,9802,9804],{"className":9803},[23],[25,9805,9806],{"xmlns":27},[29,9807,9808,9887],{},[32,9809,9810,9812,9814,9828,9834,9836,9839,9845,9847,9857,9859,9861,9863,9865,9867,9873,9875,9877,9879,9885],{},[35,9811,5858],{},[81,9813,244],{},[289,9815,9816,9818,9826],{},[81,9817,293],{},[32,9819,9820,9822,9824],{},[35,9821,1068],{},[81,9823,244],{},[89,9825,91],{},[35,9827,102],{},[266,9829,9830,9832],{},[35,9831,9401],{},[35,9833,1068],{},[81,9835,94],{"separator":48},[127,9837],{"width":9838},"2em",[266,9840,9841,9843],{},[35,9842,1027],{},[35,9844,1068],{},[81,9846,244],{},[3718,9848,9849,9855],{},[266,9850,9851,9853],{},[35,9852,9401],{},[35,9854,1068],{},[35,9856,5858],{},[81,9858,94],{"separator":48},[127,9860],{"width":9838},[35,9862,1027],{"mathvariant":251},[81,9864,244],{},[81,9866,741],{"stretchy":86},[266,9868,9869,9871],{},[35,9870,1027],{},[89,9872,91],{},[81,9874,94],{"separator":48},[81,9876,97],{},[81,9878,94],{"separator":48},[266,9880,9881,9883],{},[35,9882,1027],{},[35,9884,102],{},[81,9886,750],{"stretchy":86},[40,9888,9889],{"encoding":42},"s = \\sum_{i=1}^K \\gamma_i, \\qquad x_i = \\frac{\\gamma_i}{s}, \\qquad \\mathbf{x} = (x_1, \\dots, x_K)",[15,9891,9893,9911,10079,10216],{"className":9892,"ariaHidden":48},[47],[15,9894,9896,9899,9902,9905,9908],{"className":9895},[52],[15,9897],{"className":9898,"style":6026},[56],[15,9900,5858],{"className":9901},[61,122],[15,9903],{"className":9904,"style":128},[127],[15,9906,244],{"className":9907},[132],[15,9909],{"className":9910,"style":128},[127],[15,9912,9914,9917,9977,9980,10020,10023,10027,10030,10070,10073,10076],{"className":9913},[52],[15,9915],{"className":9916,"style":3467},[56],[15,9918,9920,9923],{"className":9919},[531],[15,9921,293],{"className":9922,"style":537},[531,535,536],[15,9924,9926],{"className":9925},[338],[15,9927,9929,9969],{"className":9928},[342,472],[15,9930,9932,9966],{"className":9931},[346],[15,9933,9935,9955],{"className":9934,"style":550},[350],[15,9936,9937,9940],{"style":553},[15,9938],{"className":9939,"style":358},[357],[15,9941,9943],{"className":9942},[362,363,364,365],[15,9944,9946,9949,9952],{"className":9945},[61,365],[15,9947,1068],{"className":9948},[61,122,365],[15,9950,244],{"className":9951},[132,365],[15,9953,91],{"className":9954},[61,365],[15,9956,9957,9960],{"style":574},[15,9958],{"className":9959,"style":358},[357],[15,9961,9963],{"className":9962},[362,363,364,365],[15,9964,102],{"className":9965,"style":174},[61,122,365],[15,9967,497],{"className":9968},[496],[15,9970,9972],{"className":9971},[346],[15,9973,9975],{"className":9974,"style":593},[350],[15,9976],{},[15,9978],{"className":9979,"style":157},[127],[15,9981,9983,9986],{"className":9982},[61],[15,9984,9401],{"className":9985,"style":9435},[61,122],[15,9987,9989],{"className":9988},[338],[15,9990,9992,10012],{"className":9991},[342,472],[15,9993,9995,10009],{"className":9994},[346],[15,9996,9998],{"className":9997,"style":1362},[350],[15,9999,10000,10003],{"style":9450},[15,10001],{"className":10002,"style":358},[357],[15,10004,10006],{"className":10005},[362,363,364,365],[15,10007,1068],{"className":10008},[61,122,365],[15,10010,497],{"className":10011},[496],[15,10013,10015],{"className":10014},[346],[15,10016,10018],{"className":10017,"style":504},[350],[15,10019],{},[15,10021,94],{"className":10022},[153],[15,10024],{"className":10025,"style":10026},[127],"margin-right:2em;",[15,10028],{"className":10029,"style":157},[127],[15,10031,10033,10036],{"className":10032},[61],[15,10034,1027],{"className":10035},[61,122],[15,10037,10039],{"className":10038},[338],[15,10040,10042,10062],{"className":10041},[342,472],[15,10043,10045,10059],{"className":10044},[346],[15,10046,10048],{"className":10047,"style":1362},[350],[15,10049,10050,10053],{"style":482},[15,10051],{"className":10052,"style":358},[357],[15,10054,10056],{"className":10055},[362,363,364,365],[15,10057,1068],{"className":10058},[61,122,365],[15,10060,497],{"className":10061},[496],[15,10063,10065],{"className":10064},[346],[15,10066,10068],{"className":10067,"style":504},[350],[15,10069],{},[15,10071],{"className":10072,"style":128},[127],[15,10074,244],{"className":10075},[132],[15,10077],{"className":10078,"style":128},[127],[15,10080,10082,10086,10195,10198,10201,10204,10207,10210,10213],{"className":10081},[52],[15,10083],{"className":10084,"style":10085},[56],"height:1.0925em;vertical-align:-0.345em;",[15,10087,10089,10092,10192],{"className":10088},[61],[15,10090],{"className":10091},[146,3837],[15,10093,10095],{"className":10094},[3718],[15,10096,10098,10183],{"className":10097},[342,472],[15,10099,10101,10180],{"className":10100},[346],[15,10102,10105,10119,10127],{"className":10103,"style":10104},[350],"height:0.7475em;",[15,10106,10107,10110],{"style":3853},[15,10108],{"className":10109,"style":3857},[357],[15,10111,10113],{"className":10112},[362,363,364,365],[15,10114,10116],{"className":10115},[61,365],[15,10117,5858],{"className":10118},[61,122,365],[15,10120,10121,10124],{"style":3885},[15,10122],{"className":10123,"style":3857},[357],[15,10125],{"className":10126,"style":3893},[3892],[15,10128,10130,10133],{"style":10129},"top:-3.4461em;",[15,10131],{"className":10132,"style":3857},[357],[15,10134,10136],{"className":10135},[362,363,364,365],[15,10137,10139],{"className":10138},[61,365],[15,10140,10142,10145],{"className":10141},[61,365],[15,10143,9401],{"className":10144,"style":9435},[61,122,365],[15,10146,10148],{"className":10147},[338],[15,10149,10151,10172],{"className":10150},[342,472],[15,10152,10154,10169],{"className":10153},[346],[15,10155,10157],{"className":10156,"style":4051},[350],[15,10158,10160,10163],{"style":10159},"top:-2.357em;margin-left:-0.0556em;margin-right:0.0714em;",[15,10161],{"className":10162,"style":4058},[357],[15,10164,10166],{"className":10165},[362,4062,4063,365],[15,10167,1068],{"className":10168},[61,122,365],[15,10170,497],{"className":10171},[496],[15,10173,10175],{"className":10174},[346],[15,10176,10178],{"className":10177,"style":4076},[350],[15,10179],{},[15,10181,497],{"className":10182},[496],[15,10184,10186],{"className":10185},[346],[15,10187,10190],{"className":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",[15,10328,10330,10355],{"className":10329,"translate":19},[18],[15,10331,10333],{"className":10332},[23],[25,10334,10335],{"xmlns":27},[29,10336,10337,10353],{},[32,10338,10339,10341,10343,10345,10347,10349,10351],{},[35,10340,1027],{"mathvariant":251},[81,10342,4179],{},[35,10344,4182],{"mathvariant":231},[81,10346,1545],{},[81,10348,741],{"stretchy":86},[35,10350,3102],{"mathvariant":3101},[81,10352,750],{"stretchy":86},[40,10354,4193],{"encoding":42},[15,10356,10358,10376],{"className":10357,"ariaHidden":48},[47],[15,10359,10361,10364,10367,10370,10373],{"className":10360},[52],[15,10362],{"className":10363,"style":2706},[56],[15,10365,1027],{"className":10366},[61,411],[15,10368],{"className":10369,"style":128},[127],[15,10371,4179],{"className":10372},[132],[15,10374],{"className":10375,"style":128},[127],[15,10377,10379,10382,10388,10391,10400],{"className":10378},[52],[15,10380],{"className":10381,"style":142},[56],[15,10383,10385],{"className":10384},[531],[15,10386,4182],{"className":10387},[61,1620],[15,10389,741],{"className":10390},[146],[15,10392,10394],{"className":10393},[61],[15,10395,10397],{"className":10396},[61],[15,10398,3102],{"className":10399},[61,3162],[15,10401,750],{"className":10402},[178],"，且 ",[15,10405,10407,10434],{"className":10406,"translate":19},[18],[15,10408,10410],{"className":10409},[23],[25,10411,10412],{"xmlns":27},[29,10413,10414,10431],{},[32,10415,10416,10418,10421,10423,10425,10427,10429],{},[35,10417,1027],{"mathvariant":251},[81,10419,10420],{},"⊥",[282,10422,6793],{},[282,10424,6793],{},[282,10426,6793],{},[81,10428,10420],{},[35,10430,5858],{},[40,10432,10433],{"encoding":42},"\\mathbf{x} \\perp\\!\\!\\!\\perp s",[15,10435,10437,10462,10474],{"className":10436,"ariaHidden":48},[47],[15,10438,10440,10444,10447,10450,10453,10456,10459],{"className":10439},[52],[15,10441],{"className":10442,"style":10443},[56],"height:0.6944em;",[15,10445,1027],{"className":10446},[61,411],[15,10448],{"className":10449,"style":128},[127],[15,10451,10420],{"className":10452},[132],[15,10454],{"className":10455,"style":7099},[127],[15,10457],{"className":10458,"style":7099},[127],[15,10460],{"className":10461,"style":7099},[127],[15,10463,10465,10468,10471],{"className":10464},[52],[15,10466],{"className":10467,"style":10443},[56],[15,10469,10420],{"className":10470},[132],[15,10472],{"className":10473,"style":128},[127],[15,10475,10477,10480],{"className":10476},[52],[15,10478],{"className":10479,"style":6026},[56],[15,10481,5858],{"className":10482},[61,122],[15,10484,10486,10520],{"className":10485,"translate":19},[18],[15,10487,10489],{"className":10488},[23],[25,10490,10491],{"xmlns":27},[29,10492,10493,10517],{},[32,10494,10495,10497,10499,10501,10503,10505,10511,10513,10515],{},[35,10496,5858],{},[81,10498,4179],{},[35,10500,9552],{"mathvariant":231},[81,10502,1545],{},[81,10504,741],{"stretchy":86},[266,10506,10507,10509],{},[35,10508,3102],{},[89,10510,278],{},[81,10512,94],{"separator":48},[89,10514,91],{},[81,10516,750],{"stretchy":86},[40,10518,10519],{"encoding":42},"s \\sim \\operatorname{Gamma}(\\alpha_0, 1)",[15,10521,10523,10541],{"className":10522,"ariaHidden":48},[47],[15,10524,10526,10529,10532,10535,10538],{"className":10525},[52],[15,10527],{"className":10528,"style":6026},[56],[15,10530,5858],{"className":10531},[61,122],[15,10533],{"className":10534,"style":128},[127],[15,10536,4179],{"className":10537},[132],[15,10539],{"className":10540,"style":128},[127],[15,10542,10544,10547,10553,10556,10596,10599,10602,10605],{"className":10543},[52],[15,10545],{"className":10546,"style":142},[56],[15,10548,10550],{"className":10549},[531],[15,10551,9552],{"className":10552},[61,1620],[15,10554,741],{"className":10555},[146],[15,10557,10559,10562],{"className":10558},[61],[15,10560,3102],{"className":10561,"style":3187},[61,122],[15,10563,10565],{"className":10564},[338],[15,10566,10568,10588],{"className":10567},[342,472],[15,10569,10571,10585],{"className":10570},[346],[15,10572,10574],{"className":10573,"style":1212},[350],[15,10575,10576,10579],{"style":3202},[15,10577],{"className":10578,"style":358},[357],[15,10580,10582],{"className":10581},[362,363,364,365],[15,10583,278],{"className":10584},[61,365],[15,10586,497],{"className":10587},[496],[15,10589,10591],{"className":10590},[346],[15,10592,10594],{"className":10593,"style":504},[350],[15,10595],{},[15,10597,94],{"className":10598},[153],[15,10600],{"className":10601,"style":157},[127],[15,10603,91],{"className":10604},[61],[15,10606,750],{"className":10607},[178],[11,10609,10610,10612,10613,10649,10650,10990,10991,11129],{},[661,10611,5117],{}," 由独立性，",[15,10614,10616,10630],{"className":10615,"translate":19},[18],[15,10617,10619],{"className":10618},[23],[25,10620,10621],{"xmlns":27},[29,10622,10623,10627],{},[32,10624,10625],{},[35,10626,9401],{"mathvariant":3101},[40,10628,10629],{"encoding":42},"\\boldsymbol{\\gamma}",[15,10631,10633],{"className":10632,"ariaHidden":48},[47],[15,10634,10636,10639],{"className":10635},[52],[15,10637],{"className":10638,"style":829},[56],[15,10640,10642],{"className":10641},[61],[15,10643,10645],{"className":10644},[61],[15,10646,9401],{"className":10647,"style":10648},[61,3162],"margin-right:0.0639em;"," 的联合密度为 ",[15,10651,10653,10717],{"className":10652,"translate":19},[18],[15,10654,10656],{"className":10655},[23],[25,10657,10658],{"xmlns":27},[29,10659,10660,10714],{},[32,10661,10662,10668,10686,10700,10702,10704,10706,10712],{},[266,10663,10664,10666],{},[81,10665,3737],{},[35,10667,1068],{},[289,10669,10670,10672,10674],{},[35,10671,9401],{},[35,10673,1068],{},[32,10675,10676,10682,10684],{},[266,10677,10678,10680],{},[35,10679,3102],{},[35,10681,1068],{},[81,10683,239],{},[89,10685,91],{},[227,10687,10688,10690],{},[35,10689,1980],{},[32,10691,10692,10694],{},[81,10693,239],{},[266,10695,10696,10698],{},[35,10697,9401],{},[35,10699,1068],{},[35,10701,3018],{"mathvariant":231},[35,10703,4288],{"mathvariant":231},[81,10705,741],{"stretchy":86},[266,10707,10708,10710],{},[35,10709,3102],{},[35,10711,1068],{},[81,10713,750],{"stretchy":86},[40,10715,10716],{"encoding":42},"\\prod_i \\gamma_i^{\\alpha_i - 1} e^{-\\gamma_i} \u002F \\Gamma(\\alpha_i)",[15,10718,10720],{"className":10719,"ariaHidden":48},[47],[15,10721,10723,10727,10767,10770,10868,10940,10944,10947,10987],{"className":10722},[52],[15,10724],{"className":10725,"style":10726},[56],"height:1.1957em;vertical-align:-0.2997em;",[15,10728,10730,10733],{"className":10729},[531],[15,10731,3737],{"className":10732,"style":537},[531,535,536],[15,10734,10736],{"className":10735},[338],[15,10737,10739,10759],{"className":10738},[342,472],[15,10740,10742,10756],{"className":10741},[346],[15,10743,10745],{"className":10744,"style":6064},[350],[15,10746,10747,10750],{"style":553},[15,10748],{"className":10749,"style":358},[357],[15,10751,10753],{"className":10752},[362,363,364,365],[15,10754,1068],{"className":10755},[61,122,365],[15,10757,497],{"className":10758},[496],[15,10760,10762],{"className":10761},[346],[15,10763,10765],{"className":10764,"style":593},[350],[15,10766],{},[15,10768],{"className":10769,"style":157},[127],[15,10771,10773,10776],{"className":10772},[61],[15,10774,9401],{"className":10775,"style":9435},[61,122],[15,10777,10779],{"className":10778},[338],[15,10780,10782,10860],{"className":10781},[342,472],[15,10783,10785,10857],{"className":10784},[346],[15,10786,10788,10800],{"className":10787,"style":4008},[350],[15,10789,10791,10794],{"style":10790},"top:-2.4231em;margin-left:-0.0556em;margin-right:0.05em;",[15,10792],{"className":10793,"style":358},[357],[15,10795,10797],{"className":10796},[362,363,364,365],[15,10798,1068],{"className":10799},[61,122,365],[15,10801,10802,10805],{"style":4023},[15,10803],{"className":10804,"style":358},[357],[15,10806,10808],{"className":10807},[362,363,364,365],[15,10809,10811,10851,10854],{"className":10810},[61,365],[15,10812,10814,10817],{"className":10813},[61,365],[15,10815,3102],{"className":10816,"style":3187},[61,122,365],[15,10818,10820],{"className":10819},[338],[15,10821,10823,10843],{"className":10822},[342,472],[15,10824,10826,10840],{"className":10825},[346],[15,10827,10829],{"className":10828,"style":4051},[350],[15,10830,10831,10834],{"style":4054},[15,10832],{"className":10833,"style":4058},[357],[15,10835,10837],{"className":10836},[362,4062,4063,365],[15,10838,1068],{"className":10839},[61,122,365],[15,10841,497],{"className":10842},[496],[15,10844,10846],{"className":10845},[346],[15,10847,10849],{"className":10848,"style":4076},[350],[15,10850],{},[15,10852,239],{"className":10853},[375,365],[15,10855,91],{"className":10856},[61,365],[15,10858,497],{"className":10859},[496],[15,10861,10863],{"className":10862},[346],[15,10864,10866],{"className":10865,"style":4094},[350],[15,10867],{},[15,10869,10871,10874],{"className":10870},[61],[15,10872,1980],{"className":10873},[61,122],[15,10875,10877],{"className":10876},[338],[15,10878,10880],{"className":10879},[342],[15,10881,10883],{"className":10882},[346],[15,10884,10886],{"className":10885,"style":8010},[350],[15,10887,10888,10891],{"style":716},[15,10889],{"className":10890,"style":358},[357],[15,10892,10894],{"className":10893},[362,363,364,365],[15,10895,10897,10900],{"className":10896},[61,365],[15,10898,239],{"className":10899},[61,365],[15,10901,10903,10906],{"className":10902},[61,365],[15,10904,9401],{"className":10905,"style":9435},[61,122,365],[15,10907,10909],{"className":10908},[338],[15,10910,10912,10932],{"className":10911},[342,472],[15,10913,10915,10929],{"className":10914},[346],[15,10916,10918],{"className":10917,"style":4051},[350],[15,10919,10920,10923],{"style":10159},[15,10921],{"className":10922,"style":4058},[357],[15,10924,10926],{"className":10925},[362,4062,4063,365],[15,10927,1068],{"className":10928},[61,122,365],[15,10930,497],{"className":10931},[496],[15,10933,10935],{"className":10934},[346],[15,10936,10938],{"className":10937,"style":4076},[350],[15,10939],{},[15,10941,10943],{"className":10942},[61],"\u002FΓ",[15,10945,741],{"className":10946},[146],[15,10948,10950,10953],{"className":10949},[61],[15,10951,3102],{"className":10952,"style":3187},[61,122],[15,10954,10956],{"className":10955},[338],[15,10957,10959,10979],{"className":10958},[342,472],[15,10960,10962,10976],{"className":10961},[346],[15,10963,10965],{"className":10964,"style":1362},[350],[15,10966,10967,10970],{"style":3202},[15,10968],{"className":10969,"style":358},[357],[15,10971,10973],{"className":10972},[362,363,364,365],[15,10974,1068],{"className":10975},[61,122,365],[15,10977,497],{"className":10978},[496],[15,10980,10982],{"className":10981},[346],[15,10983,10985],{"className":10984,"style":504},[350],[15,10986],{},[15,10988,750],{"className":10989},[178],"。施行与命题 1.3 相同的替换 ",[15,10992,10994,11022],{"className":10993,"translate":19},[18],[15,10995,10997],{"className":10996},[23],[25,10998,10999],{"xmlns":27},[29,11000,11001,11019],{},[32,11002,11003,11009,11011,11013],{},[266,11004,11005,11007],{},[35,11006,9401],{},[35,11008,1068],{},[81,11010,244],{},[35,11012,5858],{},[266,11014,11015,11017],{},[35,11016,1027],{},[35,11018,1068],{},[40,11020,11021],{"encoding":42},"\\gamma_i = s x_i",[15,11023,11025,11080],{"className":11024,"ariaHidden":48},[47],[15,11026,11028,11031,11071,11074,11077],{"className":11027},[52],[15,11029],{"className":11030,"style":9428},[56],[15,11032,11034,11037],{"className":11033},[61],[15,11035,9401],{"className":11036,"style":9435},[61,122],[15,11038,11040],{"className":11039},[338],[15,11041,11043,11063],{"className":11042},[342,472],[15,11044,11046,11060],{"className":11045},[346],[15,11047,11049],{"className":11048,"style":1362},[350],[15,11050,11051,11054],{"style":9450},[15,11052],{"className":11053,"style":358},[357],[15,11055,11057],{"className":11056},[362,363,364,365],[15,11058,1068],{"className":11059},[61,122,365],[15,11061,497],{"className":11062},[496],[15,11064,11066],{"className":11065},[346],[15,11067,11069],{"className":11068,"style":504},[350],[15,11070],{},[15,11072],{"className":11073,"style":128},[127],[15,11075,244],{"className":11076},[132],[15,11078],{"className":11079,"style":128},[127],[15,11081,11083,11086,11089],{"className":11082},[52],[15,11084],{"className":11085,"style":2509},[56],[15,11087,5858],{"className":11088},[61,122],[15,11090,11092,11095],{"className":11091},[61],[15,11093,1027],{"className":11094},[61,122],[15,11096,11098],{"className":11097},[338],[15,11099,11101,11121],{"className":11100},[342,472],[15,11102,11104,11118],{"className":11103},[346],[15,11105,11107],{"className":11106,"style":1362},[350],[15,11108,11109,11112],{"style":482},[15,11110],{"className":11111,"style":358},[357],[15,11113,11115],{"className":11114},[362,363,364,365],[15,11116,1068],{"className":11117},[61,122,365],[15,11119,497],{"className":11120},[496],[15,11122,11124],{"className":11123},[346],[15,11125,11127],{"className":11126,"style":504},[350],[15,11128],{},"，联合密度变换为",[15,11131,11133],{"className":11132,"translate":19},[212],[15,11134,11136,11326],{"className":11135,"translate":19},[18],[15,11137,11139],{"className":11138},[23],[25,11140,11141],{"xmlns":27,"display":221},[29,11142,11143,11323],{},[32,11144,11145,11147,11149,11151,11153,11155,11157,11159,11203,11206,11282,11284,11286,11288,11290,11292,11295,11301,11303,11305,11307,11309,11311,11313,11315,11317,11319,11321],{},[35,11146,11],{},[81,11148,741],{"stretchy":86},[35,11150,5858],{},[81,11152,94],{"separator":48},[35,11154,1027],{"mathvariant":251},[81,11156,750],{"stretchy":86},[81,11158,244],{},[3718,11160,11161,11189],{},[32,11162,11163,11179],{},[227,11164,11165,11167],{},[35,11166,5858],{},[32,11168,11169,11175,11177],{},[266,11170,11171,11173],{},[35,11172,3102],{},[89,11174,278],{},[81,11176,239],{},[89,11178,91],{},[227,11180,11181,11183],{},[35,11182,1980],{},[32,11184,11185,11187],{},[81,11186,239],{},[35,11188,5858],{},[32,11190,11191,11193,11195,11201],{},[35,11192,4288],{"mathvariant":231},[81,11194,741],{"stretchy":86},[266,11196,11197,11199],{},[35,11198,3102],{},[89,11200,278],{},[81,11202,750],{"stretchy":86},[81,11204,11205],{},"⋅",[3718,11207,11208,11254],{},[32,11209,11210,11212,11214,11220,11222,11236],{},[35,11211,4288],{"mathvariant":231},[81,11213,741],{"stretchy":86},[266,11215,11216,11218],{},[35,11217,3102],{},[89,11219,278],{},[81,11221,750],{"stretchy":86},[5229,11223,11224,11226,11234],{},[81,11225,3737],{},[32,11227,11228,11230,11232],{},[35,11229,1068],{},[81,11231,244],{},[89,11233,91],{},[35,11235,102],{},[289,11237,11238,11240,11242],{},[35,11239,1027],{},[35,11241,1068],{},[32,11243,11244,11250,11252],{},[266,11245,11246,11248],{},[35,11247,3102],{},[35,11249,1068],{},[81,11251,239],{},[89,11253,91],{},[32,11255,11256,11270,11272,11274,11280],{},[5229,11257,11258,11260,11268],{},[81,11259,3737],{},[32,11261,11262,11264,11266],{},[35,11263,1068],{},[81,11265,244],{},[89,11267,91],{},[35,11269,102],{},[35,11271,4288],{"mathvariant":231},[81,11273,741],{"stretchy":86},[266,11275,11276,11278],{},[35,11277,3102],{},[35,11279,1068],{},[81,11281,750],{"stretchy":86},[81,11283,244],{},[35,11285,9552],{"mathvariant":231},[81,11287,1545],{},[81,11289,741],{"stretchy":86},[35,11291,5858],{},[81,11293,11294],{"separator":48},";",[266,11296,11297,11299],{},[35,11298,3102],{},[89,11300,278],{},[81,11302,94],{"separator":48},[89,11304,91],{},[81,11306,750],{"stretchy":86},[81,11308,11205],{},[35,11310,4182],{"mathvariant":231},[81,11312,1545],{},[81,11314,741],{"stretchy":86},[35,11316,1027],{"mathvariant":251},[81,11318,11294],{"separator":48},[35,11320,3102],{"mathvariant":3101},[81,11322,750],{"stretchy":86},[40,11324,11325],{"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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,[15,11923,3102],{"className":11924,"style":3187},[61,122,365],[15,11926,11928],{"className":11927},[338],[15,11929,11931,11951],{"className":11930},[342,472],[15,11932,11934,11948],{"className":11933},[346],[15,11935,11937],{"className":11936,"style":4051},[350],[15,11938,11939,11942],{"style":4054},[15,11940],{"className":11941,"style":4058},[357],[15,11943,11945],{"className":11944},[362,4062,4063,365],[15,11946,1068],{"className":11947},[61,122,365],[15,11949,497],{"className":11950},[496],[15,11952,11954],{"className":11953},[346],[15,11955,11957],{"className":11956,"style":4076},[350],[15,11958],{},[15,11960,239],{"className":11961},[375,365],[15,11963,91],{"className":11964},[61,365],[15,11966,497],{"className":11967},[496],[15,11969,11971],{"className":11970},[346],[15,11972,11974],{"className":11973,"style":4094},[350],[15,11975],{},[15,11977,497],{"className":11978},[496],[15,11980,11982],{"className":11981},[346],[15,11983,11986],{"className":11984,"style":11985},[350],"height:1.1709em;",[15,11987],{},[15,11989],{"className":11990},[178,3837],[15,11992],{"className":11993,"style":128},[127],[15,11995,244],{"className":11996},[132],[15,11998],{"className":11999,"style":128},[127],[15,12001,12003,12006,12012,12015,12018,12021,12024,12064,12067,12070,12073,12076,12079,12082],{"className":12002},[52],[15,12004],{"className":12005,"style":142},[56],[15,12007,12009],{"className":12008},[531],[15,12010,9552],{"className":12011},[61,1620],[15,12013,741],{"className":12014},[146],[15,12016,5858],{"className":12017},[61,122],[15,12019,11294],{"className":12020},[153],[15,12022],{"className":12023,"style":157},[127],[15,12025,12027,12030],{"className":12026},[61],[15,12028,3102],{"className":12029,"style":3187},[61,122],[15,12031,12033],{"className":12032},[338],[15,12034,12036,12056],{"className":12035},[342,472],[15,12037,12039,12053],{"className":12038},[346],[15,12040,12042],{"className":12041,"style":1212},[350],[15,12043,12044,12047],{"style":3202},[15,12045],{"className":12046,"style":358},[357],[15,12048,12050],{"className":12049},[362,363,364,365],[15,12051,278],{"className":12052},[61,365],[15,12054,497],{"className":12055},[496],[15,12057,12059],{"className":12058},[346],[15,12060,12062],{"className":12061,"style":504},[350],[15,12063],{},[15,12065,94],{"className":12066},[153],[15,12068],{"className":12069,"style":157},[127],[15,12071,91],{"className":12072},[61],[15,12074,750],{"className":12075},[178],[15,12077],{"className":12078,"style":772},[127],[15,12080,11205],{"className":12081},[375],[15,12083],{"className":12084,"style":772},[127],[15,12086,12088,12091,12097,12100,12103,12106,12109,12118],{"className":12087},[52],[15,12089],{"className":12090,"style":142},[56],[15,12092,12094],{"className":12093},[531],[15,12095,4182],{"className":12096},[61,1620],[15,12098,741],{"className":12099},[146],[15,12101,1027],{"className":12102},[61,411],[15,12104,11294],{"className":12105},[153],[15,12107],{"className":12108,"style":157},[127],[15,12110,12112],{"className":12111},[61],[15,12113,12115],{"className":12114},[61],[15,12116,3102],{"className":12117},[61,3162],[15,12119,750],{"className":12120},[178],[11,12122,12123,12124,12152,12153,12181,12182,9342,12210,12238],{},"右侧分解为仅含 ",[15,12125,12127,12140],{"className":12126,"translate":19},[18],[15,12128,12130],{"className":12129},[23],[25,12131,12132],{"xmlns":27},[29,12133,12134,12138],{},[32,12135,12136],{},[35,12137,5858],{},[40,12139,5858],{"encoding":42},[15,12141,12143],{"className":12142,"ariaHidden":48},[47],[15,12144,12146,12149],{"className":12145},[52],[15,12147],{"className":12148,"style":6026},[56],[15,12150,5858],{"className":12151},[61,122]," 与仅含 ",[15,12154,12156,12169],{"className":12155,"translate":19},[18],[15,12157,12159],{"className":12158},[23],[25,12160,12161],{"xmlns":27},[29,12162,12163,12167],{},[32,12164,12165],{},[35,12166,1027],{"mathvariant":251},[40,12168,2696],{"encoding":42},[15,12170,12172],{"className":12171,"ariaHidden":48},[47],[15,12173,12175,12178],{"className":12174},[52],[15,12176],{"className":12177,"style":2706},[56],[15,12179,1027],{"className":12180},[61,411]," 的两个因子的乘积，故 ",[15,12183,12185,12198],{"className":12184,"translate":19},[18],[15,12186,12188],{"className":12187},[23],[25,12189,12190],{"xmlns":27},[29,12191,12192,12196],{},[32,12193,12194],{},[35,12195,5858],{},[40,12197,5858],{"encoding":42},[15,12199,12201],{"className":12200,"ariaHidden":48},[47],[15,12202,12204,12207],{"className":12203},[52],[15,12205],{"className":12206,"style":6026},[56],[15,12208,5858],{"className":12209},[61,122],[15,12211,12213,12226],{"className":12212,"translate":19},[18],[15,12214,12216],{"className":12215},[23],[25,12217,12218],{"xmlns":27},[29,12219,12220,12224],{},[32,12221,12222],{},[35,12223,1027],{"mathvariant":251},[40,12225,2696],{"encoding":42},[15,12227,12229],{"className":12228,"ariaHidden":48},[47],[15,12230,12232,12235],{"className":12231},[52],[15,12233],{"className":12234,"style":2706},[56],[15,12236,1027],{"className":12237},[61,411]," 独立，且各自的边缘分布如命题所述。",[11,12240,12241,12242,12288,12289,12317,12318,12500,12501,12504],{},"命题 1.4 给出了三条即时的推论。其一，它提供了一个 ",[15,12243,12245,12266],{"className":12244,"translate":19},[18],[15,12246,12248],{"className":12247},[23],[25,12249,12250],{"xmlns":27},[29,12251,12252,12263],{},[32,12253,12254,12257,12259,12261],{},[35,12255,12256],{},"O",[81,12258,741],{"stretchy":86},[35,12260,102],{},[81,12262,750],{"stretchy":86},[40,12264,12265],{"encoding":42},"O(K)",[15,12267,12269],{"className":12268,"ariaHidden":48},[47],[15,12270,12272,12275,12279,12282,12285],{"className":12271},[52],[15,12273],{"className":12274,"style":142},[56],[15,12276,12256],{"className":12277,"style":12278},[61,122],"margin-right:0.0278em;",[15,12280,741],{"className":12281},[146],[15,12283,102],{"className":12284,"style":174},[61,122],[15,12286,750],{"className":12287},[178]," 的精确采样算法：独立抽取 ",[15,12290,12292,12305],{"className":12291,"translate":19},[18],[15,12293,12295],{"className":12294},[23],[25,12296,12297],{"xmlns":27},[29,12298,12299,12303],{},[32,12300,12301],{},[35,12302,102],{},[40,12304,102],{"encoding":42},[15,12306,12308],{"className":12307,"ariaHidden":48},[47],[15,12309,12311,12314],{"className":12310},[52],[15,12312],{"className":12313,"style":57},[56],[15,12315,102],{"className":12316,"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 ",[15,12319,12321,12353],{"className":12320,"translate":19},[18],[15,12322,12324],{"className":12323},[23],[25,12325,12326],{"xmlns":27},[29,12327,12328,12350],{},[32,12329,12330,12336,12338,12344],{},[266,12331,12332,12334],{},[35,12333,3102],{},[89,12335,278],{},[81,12337,244],{},[266,12339,12340,12342],{},[81,12341,293],{},[35,12343,1068],{},[266,12345,12346,12348],{},[35,12347,3102],{},[35,12349,1068],{},[40,12351,12352],{"encoding":42},"\\alpha_0 = \\sum_i \\alpha_i",[15,12354,12356,12411],{"className":12355,"ariaHidden":48},[47],[15,12357,12359,12362,12402,12405,12408],{"className":12358},[52],[15,12360],{"className":12361,"style":2509},[56],[15,12363,12365,12368],{"className":12364},[61],[15,12366,3102],{"className":12367,"style":3187},[61,122],[15,12369,12371],{"className":12370},[338],[15,12372,12374,12394],{"className":12373},[342,472],[15,12375,12377,12391],{"className":12376},[346],[15,12378,12380],{"className":12379,"style":1212},[350],[15,12381,12382,12385],{"style":3202},[15,12383],{"className":12384,"style":358},[357],[15,12386,12388],{"className":12387},[362,363,364,365],[15,12389,278],{"className":12390},[61,365],[15,12392,497],{"className":12393},[496],[15,12395,12397],{"className":12396},[346],[15,12398,12400],{"className":12399,"style":504},[350],[15,12401],{},[15,12403],{"className":12404,"style":128},[127],[15,12406,244],{"className":12407},[132],[15,12409],{"className":12410,"style":128},[127],[15,12412,12414,12417,12457,12460],{"className":12413},[52],[15,12415],{"className":12416,"style":6045},[56],[15,12418,12420,12423],{"className":12419},[531],[15,12421,293],{"className":12422,"style":537},[531,535,536],[15,12424,12426],{"className":12425},[338],[15,12427,12429,12449],{"className":12428},[342,472],[15,12430,12432,12446],{"className":12431},[346],[15,12433,12435],{"className":12434,"style":6064},[350],[15,12436,12437,12440],{"style":553},[15,12438],{"className":12439,"style":358},[357],[15,12441,12443],{"className":12442},[362,363,364,365],[15,12444,1068],{"className":12445},[61,122,365],[15,12447,497],{"className":12448},[496],[15,12450,12452],{"className":12451},[346],[15,12453,12455],{"className":12454,"style":593},[350],[15,12456],{},[15,12458],{"className":12459,"style":157},[127],[15,12461,12463,12466],{"className":12462},[61],[15,12464,3102],{"className":12465,"style":3187},[61,122],[15,12467,12469],{"className":12468},[338],[15,12470,12472,12492],{"className":12471},[342,472],[15,12473,12475,12489],{"className":12474},[346],[15,12476,12478],{"className":12477,"style":1362},[350],[15,12479,12480,12483],{"style":3202},[15,12481],{"className":12482,"style":358},[357],[15,12484,12486],{"className":12485},[362,363,364,365],[15,12487,1068],{"className":12488},[61,122,365],[15,12490,497],{"className":12491},[496],[15,12493,12495],{"className":12494},[346],[15,12496,12498],{"className":12497,"style":504},[350],[15,12499],{}," 在概率意义上是",[661,12502,12503],{},"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。",[877,12506,12508],{"id":12507},"_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",[11,12510,12511,12514,12515,12565,12566,12722,12723,12793],{},[661,12512,12513],{},"退化到 Beta 分布。"," 当 ",[15,12516,12518,12535],{"className":12517,"translate":19},[18],[15,12519,12521],{"className":12520},[23],[25,12522,12523],{"xmlns":27},[29,12524,12525,12533],{},[32,12526,12527,12529,12531],{},[35,12528,102],{},[81,12530,244],{},[89,12532,906],{},[40,12534,2126],{"encoding":42},[15,12536,12538,12556],{"className":12537,"ariaHidden":48},[47],[15,12539,12541,12544,12547,12550,12553],{"className":12540},[52],[15,12542],{"className":12543,"style":57},[56],[15,12545,102],{"className":12546,"style":174},[61,122],[15,12548],{"className":12549,"style":128},[127],[15,12551,244],{"className":12552},[132],[15,12554],{"className":12555,"style":128},[127],[15,12557,12559,12562],{"className":12558},[52],[15,12560],{"className":12561,"style":859},[56],[15,12563,906],{"className":12564},[61]," 时，约束 ",[15,12567,12569,12599],{"className":12568,"translate":19},[18],[15,12570,12572],{"className":12571},[23],[25,12573,12574],{"xmlns":27},[29,12575,12576,12596],{},[32,12577,12578,12584,12586,12592,12594],{},[266,12579,12580,12582],{},[35,12581,1027],{},[89,12583,91],{},[81,12585,7756],{},[266,12587,12588,12590],{},[35,12589,1027],{},[89,12591,906],{},[81,12593,244],{},[89,12595,91],{},[40,12597,12598],{"encoding":42},"x_1 + x_2 = 1",[15,12600,12602,12658,12713],{"className":12601,"ariaHidden":48},[47],[15,12603,12605,12609,12649,12652,12655],{"className":12604},[52],[15,12606],{"className":12607,"style":12608},[56],"height:0.7333em;vertical-align:-0.15em;",[15,12610,12612,12615],{"className":12611},[61],[15,12613,1027],{"className":12614},[61,122],[15,12616,12618],{"className":12617},[338],[15,12619,12621,12641],{"className":12620},[342,472],[15,12622,12624,12638],{"className":12623},[346],[15,12625,12627],{"className":12626,"style":1212},[350],[15,12628,12629,12632],{"style":482},[15,12630],{"className":12631,"style":358},[357],[15,12633,12635],{"className":12634},[362,363,364,365],[15,12636,91],{"className":12637},[61,365],[15,12639,497],{"className":12640},[496],[15,12642,12644],{"className":12643},[346],[15,12645,12647],{"className":12646,"style":504},[350],[15,12648],{},[15,12650],{"className":12651,"style":772},[127],[15,12653,7756],{"className":12654},[375],[15,12656],{"className":12657,"style":772},[127],[15,12659,12661,12664,12704,12707,12710],{"className":12660},[52],[15,12662],{"className":12663,"style":2509},[56],[15,12665,12667,12670],{"className":12666},[61],[15,12668,1027],{"className":12669},[61,122],[15,12671,12673],{"className":12672},[338],[15,12674,12676,12696],{"className":12675},[342,472],[15,12677,12679,12693],{"className":12678},[346],[15,12680,12682],{"className":12681,"style":1212},[350],[15,12683,12684,12687],{"style":482},[15,12685],{"className":12686,"style":358},[357],[15,12688,12690],{"className":12689},[362,363,364,365],[15,12691,906],{"className":12692},[61,365],[15,12694,497],{"className":12695},[496],[15,12697,12699],{"className":12698},[346],[15,12700,12702],{"className":12701,"style":504},[350],[15,12703],{},[15,12705],{"className":12706,"style":128},[127],[15,12708,244],{"className":12709},[132],[15,12711],{"className":12712,"style":128},[127],[15,12714,12716,12719],{"className":12715},[52],[15,12717],{"className":12718,"style":859},[56],[15,12720,91],{"className":12721},[61]," 使分布由单一变量 ",[15,12724,12726,12744],{"className":12725,"translate":19},[18],[15,12727,12729],{"className":12728},[23],[25,12730,12731],{"xmlns":27},[29,12732,12733,12741],{},[32,12734,12735],{},[266,12736,12737,12739],{},[35,12738,1027],{},[89,12740,91],{},[40,12742,12743],{"encoding":42},"x_1",[15,12745,12747],{"className":12746,"ariaHidden":48},[47],[15,12748,12750,12753],{"className":12749},[52],[15,12751],{"className":12752,"style":2509},[56],[15,12754,12756,12759],{"className":12755},[61],[15,12757,1027],{"className":12758},[61,122],[15,12760,12762],{"className":12761},[338],[15,12763,12765,12785],{"className":12764},[342,472],[15,12766,12768,12782],{"className":12767},[346],[15,12769,12771],{"className":12770,"style":1212},[350],[15,12772,12773,12776],{"style":482},[15,12774],{"className":12775,"style":358},[357],[15,12777,12779],{"className":12778},[362,363,364,365],[15,12780,91],{"className":12781},[61,365],[15,12783,497],{"className":12784},[496],[15,12786,12788],{"className":12787},[346],[15,12789,12791],{"className":12790,"style":504},[350],[15,12792],{}," 完全决定，密度退化为",[15,12795,12797],{"className":12796,"translate":19},[212],[15,12798,12800,12971],{"className":12799,"translate":19},[18],[15,12801,12803],{"className":12802},[23],[25,12804,12805],{"xmlns":27,"display":221},[29,12806,12807,12968],{},[32,12808,12809,12811,12813,12819,12821,12827,12829,12835,12837,12839,12911,12913,12915,12918,12920,12922,12924,12926,12932,12934,12940,12942,12945,12948,12950,12952,12958,12960,12966],{},[35,12810,3703],{},[81,12812,741],{"stretchy":86},[266,12814,12815,12817],{},[35,12816,1027],{},[89,12818,91],{},[81,12820,3710],{},[266,12822,12823,12825],{},[35,12824,3102],{},[89,12826,91],{},[81,12828,94],{"separator":48},[266,12830,12831,12833],{},[35,12832,3102],{},[89,12834,906],{},[81,12836,750],{"stretchy":86},[81,12838,244],{},[3718,12840,12841,12889],{},[32,12842,12843,12861,12863,12865,12867,12873],{},[289,12844,12845,12847,12849],{},[35,12846,1027],{},[89,12848,91],{},[32,12850,12851,12857,12859],{},[266,12852,12853,12855],{},[35,12854,3102],{},[89,12856,91],{},[81,12858,239],{},[89,12860,91],{},[81,12862,741],{"stretchy":86},[89,12864,91],{},[81,12866,239],{},[266,12868,12869,12871],{},[35,12870,1027],{},[89,12872,91],{},[227,12874,12875,12877],{},[81,12876,750],{"stretchy":86},[32,12878,12879,12885,12887],{},[266,12880,12881,12883],{},[35,12882,3102],{},[89,12884,906],{},[81,12886,239],{},[89,12888,91],{},[32,12890,12891,12893,12895,12901,12903,12909],{},[35,12892,3726],{},[81,12894,741],{"stretchy":86},[266,12896,12897,12899],{},[35,12898,3102],{},[89,12900,91],{},[81,12902,94],{"separator":48},[266,12904,12905,12907],{},[35,12906,3102],{},[89,12908,906],{},[81,12910,750],{"stretchy":86},[81,12912,94],{"separator":48},[127,12914],{"width":9838},[282,12916,12917],{},"即",[127,12919],{"width":9838},[35,12921,4182],{"mathvariant":231},[81,12923,1545],{},[81,12925,741],{"stretchy":86},[266,12927,12928,12930],{},[35,12929,3102],{},[89,12931,91],{},[81,12933,94],{"separator":48},[266,12935,12936,12938],{},[35,12937,3102],{},[89,12939,906],{},[81,12941,750],{"stretchy":86},[81,12943,12944],{},"≡",[35,12946,12947],{"mathvariant":231},"Beta",[81,12949,1545],{},[81,12951,741],{"stretchy":86},[266,12953,12954,12956],{},[35,12955,3102],{},[89,12957,91],{},[81,12959,94],{"separator":48},[266,12961,12962,12964],{},[35,12963,3102],{},[89,12965,906],{},[81,12967,750],{"stretchy":86},[40,12969,12970],{"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)",[15,12972,12974,13035,13139,13657],{"className":12973,"ariaHidden":48},[47],[15,12975,12977,12980,12983,12986,13026,13029,13032],{"className":12976},[52],[15,12978],{"className":12979,"style":142},[56],[15,12981,3703],{"className":12982,"style":3781},[61,122],[15,12984,741],{"className":12985},[146],[15,12987,12989,12992],{"className":12988},[61],[15,12990,1027],{"className":12991},[61,122],[15,12993,12995],{"className":12994},[338],[15,12996,12998,13018],{"className":12997},[342,472],[15,12999,13001,13015],{"className":13000},[346],[15,13002,13004],{"className":13003,"style":1212},[350],[15,13005,13006,13009],{"style":482},[15,13007],{"className":13008,"style":358},[357],[15,13010,13012],{"className":13011},[362,363,364,365],[15,13013,91],{"className":13014},[61,365],[15,13016,497],{"className":13017},[496],[15,13019,13021],{"className":13020},[346],[15,13022,13024],{"className":13023,"style":504},[350],[15,13025],{},[15,13027],{"className":13028,"style":128},[127],[15,13030,3710],{"className":13031},[132],[15,13033],{"className":13034,"style":128},[127],[15,13036,13038,13041,13081,13084,13087,13127,13130,13133,13136],{"className":13037},[52],[15,13039],{"className":13040,"style":142},[56],[15,13042,13044,13047],{"className":13043},[61],[15,13045,3102],{"className":13046,"style":3187},[61,122],[15,13048,13050],{"className":13049},[338],[15,13051,13053,13073],{"className":13052},[342,472],[15,13054,13056,13070],{"className":13055},[346],[15,13057,13059],{"className":13058,"style":1212},[350],[15,13060,13061,13064],{"style":3202},[15,13062],{"className":13063,"style":358},[357],[15,13065,13067],{"className":13066},[362,363,364,365],[15,13068,91],{"className":13069},[61,365],[15,13071,497],{"className":13072},[496],[15,13074,13076],{"className":13075},[346],[15,13077,13079],{"className":13078,"style":504},[350],[15,13080],{},[15,13082,94],{"className":13083},[153],[15,13085],{"className":13086,"style":157},[127],[15,13088,13090,13093],{"className":13089},[61],[15,13091,3102],{"className":13092,"style":3187},[61,122],[15,13094,13096],{"className":13095},[338],[15,13097,13099,13119],{"className":13098},[342,472],[15,13100,13102,13116],{"className":13101},[346],[15,13103,13105],{"className":13104,"style":1212},[350],[15,13106,13107,13110],{"style":3202},[15,13108],{"className":13109,"style":358},[357],[15,13111,13113],{"className":13112},[362,363,364,365],[15,13114,906],{"className":13115},[61,365],[15,13117,497],{"className":13118},[496],[15,13120,13122],{"className":13121},[346],[15,13123,13125],{"className":13124,"style":504},[350],[15,13126],{},[15,13128,750],{"className":13129},[178],[15,13131],{"className":13132,"style":128},[127],[15,13134,244],{"className":13135},[132],[15,13137],{"className":13138,"style":128},[127],[15,13140,13142,13146,13527,13530,13533,13536,13544,13547,13550,13556,13559,13599,13602,13605,13645,13648,13651,13654],{"className":13141},[52],[15,13143],{"className":13144,"style":13145},[56],"height:2.509em;vertical-align:-0.936em;",[15,13147,13149,13152,13524],{"className":13148},[61],[15,13150],{"className":13151},[146,3837],[15,13153,13155],{"className":13154},[3718],[15,13156,13158,13516],{"className":13157},[342,472],[15,13159,13161,13513],{"className":13160},[346],[15,13162,13165,13268,13276],{"className":13163,"style":13164},[350],"height:1.573em;",[15,13166,13167,13170],{"style":11393},[15,13168],{"className":13169,"style":3857},[357],[15,13171,13173,13176,13179,13219,13222,13225,13265],{"className":13172},[61],[15,13174,3726],{"className":13175,"style":3867},[61,122],[15,13177,741],{"className":13178},[146],[15,13180,13182,13185],{"className":13181},[61],[15,13183,3102],{"className":13184,"style":3187},[61,122],[15,13186,13188],{"className":13187},[338],[15,13189,13191,13211],{"className":13190},[342,472],[15,13192,13194,13208],{"className":13193},[346],[15,13195,13197],{"className":13196,"style":1212},[350],[15,13198,13199,13202],{"style":3202},[15,13200],{"className":13201,"style":358},[357],[15,13203,13205],{"className":13204},[362,363,364,365],[15,13206,91],{"className":13207},[61,365],[15,13209,497],{"className":13210},[496],[15,13212,13214],{"className":13213},[346],[15,13215,13217],{"className":13216,"style":504},[350],[15,13218],{},[15,13220,94],{"className":13221},[153],[15,13223],{"className":13224,"style":157},[127],[15,13226,13228,13231],{"className":13227},[61],[15,13229,3102],{"className":13230,"style":3187},[61,122],[15,13232,13234],{"className":13233},[338],[15,13235,13237,13257],{"className":13236},[342,472],[15,13238,13240,13254],{"className":13239},[346],[15,13241,13243],{"className":13242,"style":1212},[350],[15,13244,13245,13248],{"style":3202},[15,13246],{"className":13247,"style":358},[357],[15,13249,13251],{"className":13250},[362,363,364,365],[15,13252,906],{"className":13253},[61,365],[15,13255,497],{"className":13256},[496],[15,13258,13260],{"className":13259},[346],[15,13261,13263],{"className":13262,"style":504},[350],[15,13264],{},[15,13266,750],{"className":13267},[178],[15,13269,13270,13273],{"style":3885},[15,13271],{"className":13272,"style":3857},[357],[15,13274],{"className":13275,"style":3893},[3892],[15,13277,13278,13281],{"style":11459},[15,13279],{"className":13280,"style":3857},[357],[15,13282,13284,13383,13386,13389,13392,13395,13398,13438],{"className":13283},[61],[15,13285,13287,13290],{"className":13286},[61],[15,13288,1027],{"className":13289},[61,122],[15,13291,13293],{"className":13292},[338],[15,13294,13296,13374],{"className":13295},[342,472],[15,13297,13299,13371],{"className":13298},[346],[15,13300,13302,13314],{"className":13301,"style":4008},[350],[15,13303,13305,13308],{"style":13304},"top:-2.4337em;margin-left:0em;margin-right:0.05em;",[15,13306],{"className":13307,"style":358},[357],[15,13309,13311],{"className":13310},[362,363,364,365],[15,13312,91],{"className":13313},[61,365],[15,13315,13316,13319],{"style":4023},[15,13317],{"className":13318,"style":358},[357],[15,13320,13322],{"className":13321},[362,363,364,365],[15,13323,13325,13365,13368],{"className":13324},[61,365],[15,13326,13328,13331],{"className":13327},[61,365],[15,13329,3102],{"className":13330,"style":3187},[61,122,365],[15,13332,13334],{"className":13333},[338],[15,13335,13337,13357],{"className":13336},[342,472],[15,13338,13340,13354],{"className":13339},[346],[15,13341,13343],{"className":13342,"style":4421},[350],[15,13344,13345,13348],{"style":4054},[15,13346],{"className":13347,"style":4058},[357],[15,13349,13351],{"className":13350},[362,4062,4063,365],[15,13352,91],{"className":13353},[61,365],[15,13355,497],{"className":13356},[496],[15,13358,13360],{"className":13359},[346],[15,13361,13363],{"className":13362,"style":4076},[350],[15,13364],{},[15,13366,239],{"className":13367},[375,365],[15,13369,91],{"className":13370},[61,365],[15,13372,497],{"className":13373},[496],[15,13375,13377],{"className":13376},[346],[15,13378,13381],{"className":13379,"style":13380},[350],"height:0.2663em;",[15,13382],{},[15,13384,741],{"className":13385},[146],[15,13387,91],{"className":13388},[61],[15,13390],{"className":13391,"style":772},[127],[15,13393,239],{"className":13394},[375],[15,13396],{"className":13397,"style":772},[127],[15,13399,13401,13404],{"className":13400},[61],[15,13402,1027],{"className":13403},[61,122],[15,13405,13407],{"className":13406},[338],[15,13408,13410,13430],{"className":13409},[342,472],[15,13411,13413,13427],{"className":13412},[346],[15,13414,13416],{"className":13415,"style":1212},[350],[15,13417,13418,13421],{"style":482},[15,13419],{"className":13420,"style":358},[357],[15,13422,13424],{"className":13423},[362,363,364,365],[15,13425,91],{"className":13426},[61,365],[15,13428,497],{"className":13429},[496],[15,13431,13433],{"className":13432},[346],[15,13434,13436],{"className":13435,"style":504},[350],[15,13437],{},[15,13439,13441,13444],{"className":13440},[178],[15,13442,750],{"className":13443},[178],[15,13445,13447],{"className":13446},[338],[15,13448,13450],{"className":13449},[342],[15,13451,13453],{"className":13452},[346],[15,13454,13456],{"className":13455,"style":7899},[350],[15,13457,13458,13461],{"style":716},[15,13459],{"className":13460,"style":358},[357],[15,13462,13464],{"className":13463},[362,363,364,365],[15,13465,13467,13507,13510],{"className":13466},[61,365],[15,13468,13470,13473],{"className":13469},[61,365],[15,13471,3102],{"className":13472,"style":3187},[61,122,365],[15,13474,13476],{"className":13475},[338],[15,13477,13479,13499],{"className":13478},[342,472],[15,13480,13482,13496],{"className":13481},[346],[15,13483,13485],{"className":13484,"style":4421},[350],[15,13486,13487,13490],{"style":4054},[15,13488],{"className":13489,"style":4058},[357],[15,13491,13493],{"className":13492},[362,4062,4063,365],[15,13494,906],{"className":13495},[61,365],[15,13497,497],{"className":13498},[496],[15,13500,13502],{"className":13501},[346],[15,13503,13505],{"className":13504,"style":4076},[350],[15,13506],{},[15,13508,239],{"className":13509},[375,365],[15,13511,91],{"className":13512},[61,365],[15,13514,497],{"className":13515},[496],[15,13517,13519],{"className":13518},[346],[15,13520,13522],{"className":13521,"style":11585},[350],[15,13523],{},[15,13525],{"className":13526},[178,3837],[15,13528,94],{"className":13529},[153],[15,13531],{"className":13532,"style":10026},[127],[15,13534],{"className":13535,"style":157},[127],[15,13537,13540],{"className":13538},[61,13539],"text",[15,13541,12917],{"className":13542},[61,13543],"cjk_fallback",[15,13545],{"className":13546,"style":10026},[127],[15,13548],{"className":13549,"style":157},[127],[15,13551,13553],{"className":13552},[531],[15,13554,4182],{"className":13555},[61,1620],[15,13557,741],{"className":13558},[146],[15,13560,13562,13565],{"className":13561},[61],[15,13563,3102],{"className":13564,"style":3187},[61,122],[15,13566,13568],{"className":13567},[338],[15,13569,13571,13591],{"className":13570},[342,472],[15,13572,13574,13588],{"className":13573},[346],[15,13575,13577],{"className":13576,"style":1212},[350],[15,13578,13579,13582],{"style":3202},[15,13580],{"className":13581,"style":358},[357],[15,13583,13585],{"className":13584},[362,363,364,365],[15,13586,91],{"className":13587},[61,365],[15,13589,497],{"className":13590},[496],[15,13592,13594],{"className":13593},[346],[15,13595,13597],{"className":13596,"style":504},[350],[15,13598],{},[15,13600,94],{"className":13601},[153],[15,13603],{"className":13604,"style":157},[127],[15,13606,13608,13611],{"className":13607},[61],[15,13609,3102],{"className":13610,"style":3187},[61,122],[15,13612,13614],{"className":13613},[338],[15,13615,13617,13637],{"className":13616},[342,472],[15,13618,13620,13634],{"className":13619},[346],[15,13621,13623],{"className":13622,"style":1212},[350],[15,13624,13625,13628],{"style":3202},[15,13626],{"className":13627,"style":358},[357],[15,13629,13631],{"className":13630},[362,363,364,365],[15,13632,906],{"className":13633},[61,365],[15,13635,497],{"className":13636},[496],[15,13638,13640],{"className":13639},[346],[15,13641,13643],{"className":13642,"style":504},[350],[15,13644],{},[15,13646,750],{"className":13647},[178],[15,13649],{"className":13650,"style":128},[127],[15,13652,12944],{"className":13653},[132],[15,13655],{"className":13656,"style":128},[127],[15,13658,13660,13663,13669,13672,13712,13715,13718,13758],{"className":13659},[52],[15,13661],{"className":13662,"style":142},[56],[15,13664,13666],{"className":13665},[531],[15,13667,12947],{"className":13668},[61,1620],[15,13670,741],{"className":13671},[146],[15,13673,13675,13678],{"className":13674},[61],[15,13676,3102],{"className":13677,"style":3187},[61,122],[15,13679,13681],{"className":13680},[338],[15,13682,13684,13704],{"className":13683},[342,472],[15,13685,13687,13701],{"className":13686},[346],[15,13688,13690],{"className":13689,"style":1212},[350],[15,13691,13692,13695],{"style":3202},[15,13693],{"className":13694,"style":358},[357],[15,13696,13698],{"className":13697},[362,363,364,365],[15,13699,91],{"className":13700},[61,365],[15,13702,497],{"className":13703},[496],[15,13705,13707],{"className":13706},[346],[15,13708,13710],{"className":13709,"style":504},[350],[15,13711],{},[15,13713,94],{"className":13714},[153],[15,13716],{"className":13717,"style":157},[127],[15,13719,13721,13724],{"className":13720},[61],[15,13722,3102],{"className":13723,"style":3187},[61,122],[15,13725,13727],{"className":13726},[338],[15,13728,13730,13750],{"className":13729},[342,472],[15,13731,13733,13747],{"className":13732},[346],[15,13734,13736],{"className":13735,"style":1212},[350],[15,13737,13738,13741],{"style":3202},[15,13739],{"className":13740,"style":358},[357],[15,13742,13744],{"className":13743},[362,363,364,365],[15,13745,906],{"className":13746},[61,365],[15,13748,497],{"className":13749},[496],[15,13751,13753],{"className":13752},[346],[15,13754,13756],{"className":13755,"style":504},[350],[15,13757],{},[15,13759,750],{"className":13760},[178],[11,13762,13763,13764,13815],{},"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 ",[15,13765,13767,13785],{"className":13766,"translate":19},[18],[15,13768,13770],{"className":13769},[23],[25,13771,13772],{"xmlns":27},[29,13773,13774,13782],{},[32,13775,13776,13778,13780],{},[35,13777,102],{},[81,13779,244],{},[89,13781,906],{},[40,13783,13784],{"encoding":42},"K=2",[15,13786,13788,13806],{"className":13787,"ariaHidden":48},[47],[15,13789,13791,13794,13797,13800,13803],{"className":13790},[52],[15,13792],{"className":13793,"style":57},[56],[15,13795,102],{"className":13796,"style":174},[61,122],[15,13798],{"className":13799,"style":128},[127],[15,13801,244],{"className":13802},[132],[15,13804],{"className":13805,"style":128},[127],[15,13807,13809,13812],{"className":13808},[52],[15,13810],{"className":13811,"style":859},[56],[15,13813,906],{"className":13814},[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。",[11,13817,13818,13821,13822,13892,13893,14100,14101,14254,14255,14421,14422,14491],{},[661,13819,13820],{},"边缘分布仍为 Beta。"," 更一般地，任意单个分量 ",[15,13823,13825,13843],{"className":13824,"translate":19},[18],[15,13826,13828],{"className":13827},[23],[25,13829,13830],{"xmlns":27},[29,13831,13832,13840],{},[32,13833,13834],{},[266,13835,13836,13838],{},[35,13837,1027],{},[35,13839,1068],{},[40,13841,13842],{"encoding":42},"x_i",[15,13844,13846],{"className":13845,"ariaHidden":48},[47],[15,13847,13849,13852],{"className":13848},[52],[15,13850],{"className":13851,"style":2509},[56],[15,13853,13855,13858],{"className":13854},[61],[15,13856,1027],{"className":13857},[61,122],[15,13859,13861],{"className":13860},[338],[15,13862,13864,13884],{"className":13863},[342,472],[15,13865,13867,13881],{"className":13866},[346],[15,13868,13870],{"className":13869,"style":1362},[350],[15,13871,13872,13875],{"style":482},[15,13873],{"className":13874,"style":358},[357],[15,13876,13878],{"className":13877},[362,363,364,365],[15,13879,1068],{"className":13880},[61,122,365],[15,13882,497],{"className":13883},[496],[15,13885,13887],{"className":13886},[346],[15,13888,13890],{"className":13889,"style":504},[350],[15,13891],{}," 的边缘分布是 ",[15,13894,13896,13938],{"className":13895,"translate":19},[18],[15,13897,13899],{"className":13898},[23],[25,13900,13901],{"xmlns":27},[29,13902,13903,13935],{},[32,13904,13905,13907,13909,13911,13917,13919,13925,13927,13933],{},[35,13906,12947],{"mathvariant":231},[81,13908,1545],{},[81,13910,741],{"stretchy":86},[266,13912,13913,13915],{},[35,13914,3102],{},[35,13916,1068],{},[81,13918,94],{"separator":48},[266,13920,13921,13923],{},[35,13922,3102],{},[89,13924,278],{},[81,13926,239],{},[266,13928,13929,13931],{},[35,13930,3102],{},[35,13932,1068],{},[81,13934,750],{"stretchy":86},[40,13936,13937],{"encoding":42},"\\operatorname{Beta}(\\alpha_i, \\alpha_0 - \\alpha_i)",[15,13939,13941,14051],{"className":13940,"ariaHidden":48},[47],[15,13942,13944,13947,13953,13956,13996,13999,14002,14042,14045,14048],{"className":13943},[52],[15,13945],{"className":13946,"style":142},[56],[15,13948,13950],{"className":13949},[531],[15,13951,12947],{"className":13952},[61,1620],[15,13954,741],{"className":13955},[146],[15,13957,13959,13962],{"className":13958},[61],[15,13960,3102],{"className":13961,"style":3187},[61,122],[15,13963,13965],{"className":13964},[338],[15,13966,13968,13988],{"className":13967},[342,472],[15,13969,13971,13985],{"className":13970},[346],[15,13972,13974],{"className":13973,"style":1362},[350],[15,13975,13976,13979],{"style":3202},[15,13977],{"className":13978,"style":358},[357],[15,13980,13982],{"className":13981},[362,363,364,365],[15,13983,1068],{"className":13984},[61,122,365],[15,13986,497],{"className":13987},[496],[15,13989,13991],{"className":13990},[346],[15,13992,13994],{"className":13993,"style":504},[350],[15,13995],{},[15,13997,94],{"className":13998},[153],[15,14000],{"className":14001,"style":157},[127],[15,14003,14005,14008],{"className":14004},[61],[15,14006,3102],{"className":14007,"style":3187},[61,122],[15,14009,14011],{"className":14010},[338],[15,14012,14014,14034],{"className":14013},[342,472],[15,14015,14017,14031],{"className":14016},[346],[15,14018,14020],{"className":14019,"style":1212},[350],[15,14021,14022,14025],{"style":3202},[15,14023],{"className":14024,"style":358},[357],[15,14026,14028],{"className":14027},[362,363,364,365],[15,14029,278],{"className":14030},[61,365],[15,14032,497],{"className":14033},[496],[15,14035,14037],{"className":14036},[346],[15,14038,14040],{"className":14039,"style":504},[350],[15,14041],{},[15,14043],{"className":14044,"style":772},[127],[15,14046,239],{"className":14047},[375],[15,14049],{"className":14050,"style":772},[127],[15,14052,14054,14057,14097],{"className":14053},[52],[15,14055],{"className":14056,"style":142},[56],[15,14058,14060,14063],{"className":14059},[61],[15,14061,3102],{"className":14062,"style":3187},[61,122],[15,14064,14066],{"className":14065},[338],[15,14067,14069,14089],{"className":14068},[342,472],[15,14070,14072,14086],{"className":14071},[346],[15,14073,14075],{"className":14074,"style":1362},[350],[15,14076,14077,14080],{"style":3202},[15,14078],{"className":14079,"style":358},[357],[15,14081,14083],{"className":14082},[362,363,364,365],[15,14084,1068],{"className":14085},[61,122,365],[15,14087,497],{"className":14088},[496],[15,14090,14092],{"className":14091},[346],[15,14093,14095],{"className":14094,"style":504},[350],[15,14096],{},[15,14098,750],{"className":14099},[178],"。这一点可由命题 1.4 直接看出：",[15,14102,14104,14138],{"className":14103,"translate":19},[18],[15,14105,14107],{"className":14106},[23],[25,14108,14109],{"xmlns":27},[29,14110,14111,14135],{},[32,14112,14113,14119,14121,14127,14129,14131,14133],{},[266,14114,14115,14117],{},[35,14116,1027],{},[35,14118,1068],{},[81,14120,244],{},[266,14122,14123,14125],{},[35,14124,9401],{},[35,14126,1068],{},[35,14128,3018],{"mathvariant":231},[81,14130,741],{"stretchy":86},[35,14132,5858],{},[81,14134,750],{"stretchy":86},[40,14136,14137],{"encoding":42},"x_i = \\gamma_i \u002F (s)",[15,14139,14141,14196],{"className":14140,"ariaHidden":48},[47],[15,14142,14144,14147,14187,14190,14193],{"className":14143},[52],[15,14145],{"className":14146,"style":2509},[56],[15,14148,14150,14153],{"className":14149},[61],[15,14151,1027],{"className":14152},[61,122],[15,14154,14156],{"className":14155},[338],[15,14157,14159,14179],{"className":14158},[342,472],[15,14160,14162,14176],{"className":14161},[346],[15,14163,14165],{"className":14164,"style":1362},[350],[15,14166,14167,14170],{"style":482},[15,14168],{"className":14169,"style":358},[357],[15,14171,14173],{"className":14172},[362,363,364,365],[15,14174,1068],{"className":14175},[61,122,365],[15,14177,497],{"className":14178},[496],[15,14180,14182],{"className":14181},[346],[15,14183,14185],{"className":14184,"style":504},[350],[15,14186],{},[15,14188],{"className":14189,"style":128},[127],[15,14191,244],{"className":14192},[132],[15,14194],{"className":14195,"style":128},[127],[15,14197,14199,14202,14242,14245,14248,14251],{"className":14198},[52],[15,14200],{"className":14201,"style":142},[56],[15,14203,14205,14208],{"className":14204},[61],[15,14206,9401],{"className":14207,"style":9435},[61,122],[15,14209,14211],{"className":14210},[338],[15,14212,14214,14234],{"className":14213},[342,472],[15,14215,14217,14231],{"className":14216},[346],[15,14218,14220],{"className":14219,"style":1362},[350],[15,14221,14222,14225],{"style":9450},[15,14223],{"className":14224,"style":358},[357],[15,14226,14228],{"className":14227},[362,363,364,365],[15,14229,1068],{"className":14230},[61,122,365],[15,14232,497],{"className":14233},[496],[15,14235,14237],{"className":14236},[346],[15,14238,14240],{"className":14239,"style":504},[350],[15,14241],{},[15,14243,3018],{"className":14244},[61],[15,14246,741],{"className":14247},[146],[15,14249,5858],{"className":14250},[61,122],[15,14252,750],{"className":14253},[178],"，而 ",[15,14256,14258,14292],{"className":14257,"translate":19},[18],[15,14259,14261],{"className":14260},[23],[25,14262,14263],{"xmlns":27},[29,14264,14265,14289],{},[32,14266,14267,14273,14275,14277,14279,14281,14287],{},[266,14268,14269,14271],{},[35,14270,9401],{},[35,14272,1068],{},[81,14274,10420],{},[81,14276,741],{"stretchy":86},[35,14278,5858],{},[81,14280,239],{},[266,14282,14283,14285],{},[35,14284,9401],{},[35,14286,1068],{},[81,14288,750],{"stretchy":86},[40,14290,14291],{"encoding":42},"\\gamma_i \\perp (s - \\gamma_i)",[15,14293,14295,14351,14372],{"className":14294,"ariaHidden":48},[47],[15,14296,14298,14302,14342,14345,14348],{"className":14297},[52],[15,14299],{"className":14300,"style":14301},[56],"height:0.8889em;vertical-align:-0.1944em;",[15,14303,14305,14308],{"className":14304},[61],[15,14306,9401],{"className":14307,"style":9435},[61,122],[15,14309,14311],{"className":14310},[338],[15,14312,14314,14334],{"className":14313},[342,472],[15,14315,14317,14331],{"className":14316},[346],[15,14318,14320],{"className":14319,"style":1362},[350],[15,14321,14322,14325],{"style":9450},[15,14323],{"className":14324,"style":358},[357],[15,14326,14328],{"className":14327},[362,363,364,365],[15,14329,1068],{"className":14330},[61,122,365],[15,14332,497],{"className":14333},[496],[15,14335,14337],{"className":14336},[346],[15,14338,14340],{"className":14339,"style":504},[350],[15,14341],{},[15,14343],{"className":14344,"style":128},[127],[15,14346,10420],{"className":14347},[132],[15,14349],{"className":14350,"style":128},[127],[15,14352,14354,14357,14360,14363,14366,14369],{"className":14353},[52],[15,14355],{"className":14356,"style":142},[56],[15,14358,741],{"className":14359},[146],[15,14361,5858],{"className":14362},[61,122],[15,14364],{"className":14365,"style":772},[127],[15,14367,239],{"className":14368},[375],[15,14370],{"className":14371,"style":772},[127],[15,14373,14375,14378,14418],{"className":14374},[52],[15,14376],{"className":14377,"style":142},[56],[15,14379,14381,14384],{"className":14380},[61],[15,14382,9401],{"className":14383,"style":9435},[61,122],[15,14385,14387],{"className":14386},[338],[15,14388,14390,14410],{"className":14389},[342,472],[15,14391,14393,14407],{"className":14392},[346],[15,14394,14396],{"className":14395,"style":1362},[350],[15,14397,14398,14401],{"style":9450},[15,14399],{"className":14400,"style":358},[357],[15,14402,14404],{"className":14403},[362,363,364,365],[15,14405,1068],{"className":14406},[61,122,365],[15,14408,497],{"className":14409},[496],[15,14411,14413],{"className":14412},[346],[15,14414,14416],{"className":14415,"style":504},[350],[15,14417],{},[15,14419,750],{"className":14420},[178],"，故 ",[15,14423,14425,14442],{"className":14424,"translate":19},[18],[15,14426,14428],{"className":14427},[23],[25,14429,14430],{"xmlns":27},[29,14431,14432,14440],{},[32,14433,14434],{},[266,14435,14436,14438],{},[35,14437,1027],{},[35,14439,1068],{},[40,14441,13842],{"encoding":42},[15,14443,14445],{"className":14444,"ariaHidden":48},[47],[15,14446,14448,14451],{"className":14447},[52],[15,14449],{"className":14450,"style":2509},[56],[15,14452,14454,14457],{"className":14453},[61],[15,14455,1027],{"className":14456},[61,122],[15,14458,14460],{"className":14459},[338],[15,14461,14463,14483],{"className":14462},[342,472],[15,14464,14466,14480],{"className":14465},[346],[15,14467,14469],{"className":14468,"style":1362},[350],[15,14470,14471,14474],{"style":482},[15,14472],{"className":14473,"style":358},[357],[15,14475,14477],{"className":14476},[362,363,364,365],[15,14478,1068],{"className":14479},[61,122,365],[15,14481,497],{"className":14482},[496],[15,14484,14486],{"className":14485},[346],[15,14487,14489],{"className":14488,"style":504},[350],[15,14490],{}," 是相互独立的两个 Gamma 变量之比，其分布为 Beta。",[11,14493,14494,14497,14498,14501,14502,14536,14537,14567,14568,867,14655],{},[661,14495,14496],{},"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",[661,14499,14500],{},"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」",[15,14503,14505,14518],{"className":14504,"translate":19},[18],[15,14506,14508],{"className":14507},[23],[25,14509,14510],{"xmlns":27},[29,14511,14512,14516],{},[32,14513,14514],{},[35,14515,3102],{"mathvariant":3101},[40,14517,4143],{"encoding":42},[15,14519,14521],{"className":14520,"ariaHidden":48},[47],[15,14522,14524,14527],{"className":14523},[52],[15,14525],{"className":14526,"style":2706},[56],[15,14528,14530],{"className":14529},[61],[15,14531,14533],{"className":14532},[61],[15,14534,3102],{"className":14535},[61,3162]," 描述对各类别出现次数的初始信念，则观测到计数 ",[15,14538,14540,14555],{"className":14539,"translate":19},[18],[15,14541,14543],{"className":14542},[23],[25,14544,14545],{"xmlns":27},[29,14546,14547,14552],{},[32,14548,14549],{},[35,14550,14551],{"mathvariant":251},"n",[40,14553,14554],{"encoding":42},"\\mathbf{n}",[15,14556,14558],{"className":14557,"ariaHidden":48},[47],[15,14559,14561,14564],{"className":14560},[52],[15,14562],{"className":14563,"style":2706},[56],[15,14565,14551],{"className":14566},[61,411]," 后，信念的更新只是 ",[15,14569,14571,14594],{"className":14570,"translate":19},[18],[15,14572,14574],{"className":14573},[23],[25,14575,14576],{"xmlns":27},[29,14577,14578,14591],{},[32,14579,14580,14582,14585,14587,14589],{},[35,14581,3102],{"mathvariant":3101},[81,14583,14584],{},"↦",[35,14586,3102],{"mathvariant":3101},[81,14588,7756],{},[35,14590,14551],{"mathvariant":251},[40,14592,14593],{"encoding":42},"\\boldsymbol{\\alpha} \\mapsto \\boldsymbol{\\alpha} + \\mathbf{n}",[15,14595,14597,14622,14646],{"className":14596,"ariaHidden":48},[47],[15,14598,14600,14604,14613,14616,14619],{"className":14599},[52],[15,14601],{"className":14602,"style":14603},[56],"height:0.522em;vertical-align:-0.011em;",[15,14605,14607],{"className":14606},[61],[15,14608,14610],{"className":14609},[61],[15,14611,3102],{"className":14612},[61,3162],[15,14614],{"className":14615,"style":128},[127],[15,14617,14584],{"className":14618},[132],[15,14620],{"className":14621,"style":128},[127],[15,14623,14625,14628,14637,14640,14643],{"className":14624},[52],[15,14626],{"className":14627,"style":6731},[56],[15,14629,14631],{"className":14630},[61],[15,14632,14634],{"className":14633},[61],[15,14635,3102],{"className":14636},[61,3162],[15,14638],{"className":14639,"style":772},[127],[15,14641,7756],{"className":14642},[375],[15,14644],{"className":14645,"style":772},[127],[15,14647,14649,14652],{"className":14648},[52],[15,14650],{"className":14651,"style":2706},[56],[15,14653,14551],{"className":14654},[61,411],[661,14656,14657],{},"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。",[872,14659,14661],{"id":14660},"二解析性质","二、解析性质",[877,14663,14665],{"id":14664},"_21-矩结构","2.1 矩结构",[882,14667,14669],{"title":14668},"命题 2.1（期望、方差与协方差）",[11,14670,13,14671,14747,14748,14793],{},[15,14672,14674,14699],{"className":14673,"translate":19},[18],[15,14675,14677],{"className":14676},[23],[25,14678,14679],{"xmlns":27},[29,14680,14681,14697],{},[32,14682,14683,14685,14687,14689,14691,14693,14695],{},[35,14684,1027],{"mathvariant":251},[81,14686,4179],{},[35,14688,4182],{"mathvariant":231},[81,14690,1545],{},[81,14692,741],{"stretchy":86},[35,14694,3102],{"mathvariant":3101},[81,14696,750],{"stretchy":86},[40,14698,4193],{"encoding":42},[15,14700,14702,14720],{"className":14701,"ariaHidden":48},[47],[15,14703,14705,14708,14711,14714,14717],{"className":14704},[52],[15,14706],{"className":14707,"style":2706},[56],[15,14709,1027],{"className":14710},[61,411],[15,14712],{"className":14713,"style":128},[127],[15,14715,4179],{"className":14716},[132],[15,14718],{"className":14719,"style":128},[127],[15,14721,14723,14726,14732,14735,14744],{"className":14722},[52],[15,14724],{"className":14725,"style":142},[56],[15,14727,14729],{"className":14728},[531],[15,14730,4182],{"className":14731},[61,1620],[15,14733,741],{"className":14734},[146],[15,14736,14738],{"className":14737},[61],[15,14739,14741],{"className":14740},[61],[15,14742,3102],{"className":14743},[61,3162],[15,14745,750],{"className":14746},[178],"。则对任意 ",[15,14749,14751,14770],{"className":14750,"translate":19},[18],[15,14752,14754],{"className":14753},[23],[25,14755,14756],{"xmlns":27},[29,14757,14758,14767],{},[32,14759,14760,14762,14764],{},[35,14761,1068],{},[81,14763,94],{"separator":48},[35,14765,14766],{},"j",[40,14768,14769],{"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",[15,16079,16081,16136],{"className":16080,"ariaHidden":48},[47],[15,16082,16084,16087,16127,16130,16133],{"className":16083},[52],[15,16085],{"className":16086,"style":2509},[56],[15,16088,16090,16093],{"className":16089},[61],[15,16091,1027],{"className":16092},[61,122],[15,16094,16096],{"className":16095},[338],[15,16097,16099,16119],{"className":16098},[342,472],[15,16100,16102,16116],{"className":16101},[346],[15,16103,16105],{"className":16104,"style":1362},[350],[15,16106,16107,16110],{"style":482},[15,16108],{"className":16109,"style":358},[357],[15,16111,16113],{"className":16112},[362,363,364,365],[15,16114,1068],{"className":16115},[61,122,365],[15,16117,497],{"className":16118},[496],[15,16120,16122],{"className":16121},[346],[15,16123,16125],{"className":16124,"style":504},[350],[15,16126],{},[15,16128],{"className":16129,"style":128},[127],[15,16131,244],{"className":16132},[132],[15,16134],{"className":16135,"style":128},[127],[15,16137,16139,16142,16182,16185],{"className":16138},[52],[15,16140],{"className":16141,"style":142},[56],[15,16143,16145,16148],{"className":16144},[61],[15,16146,9401],{"className":16147,"style":9435},[61,122],[15,16149,16151],{"className":16150},[338],[15,16152,16154,16174],{"className":16153},[342,472],[15,16155,16157,16171],{"className":16156},[346],[15,16158,16160],{"className":16159,"style":1362},[350],[15,16161,16162,16165],{"style":9450},[15,16163],{"className":16164,"style":358},[357],[15,16166,16168],{"className":16167},[362,363,364,365],[15,16169,1068],{"className":16170},[61,122,365],[15,16172,497],{"className":16173},[496],[15,16175,16177],{"className":16176},[346],[15,16178,16180],{"className":16179,"style":504},[350],[15,16181],{},[15,16183,3018],{"className":16184},[61],[15,16186,5858],{"className":16187},[61,122]," 且 ",[15,16190,16192,16214],{"className":16191,"translate":19},[18],[15,16193,16195],{"className":16194},[23],[25,16196,16197],{"xmlns":27},[29,16198,16199,16211],{},[32,16200,16201,16207,16209],{},[266,16202,16203,16205],{},[35,16204,9401],{},[35,16206,1068],{},[81,16208,10420],{},[35,16210,5858],{},[40,16212,16213],{"encoding":42},"\\gamma_i \\perp s",[15,16215,16217,16272],{"className":16216,"ariaHidden":48},[47],[15,16218,16220,16223,16263,16266,16269],{"className":16219},[52],[15,16221],{"className":16222,"style":14301},[56],[15,16224,16226,16229],{"className":16225},[61],[15,16227,9401],{"className":16228,"style":9435},[61,122],[15,16230,16232],{"className":16231},[338],[15,16233,16235,16255],{"className":16234},[342,472],[15,16236,16238,16252],{"className":16237},[346],[15,16239,16241],{"className":16240,"style":1362},[350],[15,16242,16243,16246],{"style":9450},[15,16244],{"className":16245,"style":358},[357],[15,16247,16249],{"className":16248},[362,363,364,365],[15,16250,1068],{"className":16251},[61,122,365],[15,16253,497],{"className":16254},[496],[15,16256,16258],{"className":16257},[346],[15,16259,16261],{"className":16260,"style":504},[350],[15,16262],{},[15,16264],{"className":16265,"style":128},[127],[15,16267,10420],{"className":16268},[132],[15,16270],{"className":16271,"style":128},[127],[15,16273,16275,16278],{"className":16274},[52],[15,16276],{"className":16277,"style":6026},[56],[15,16279,5858],{"className":16280},[61,122],[15,16282,16284,16356],{"className":16283,"translate":19},[18],[15,16285,16287],{"className":16286},[23],[25,16288,16289],{"xmlns":27},[29,16290,16291,16353],{},[32,16292,16293,16295,16297,16303,16305,16307,16309,16311,16317,16319,16321,16323,16325,16327,16329,16331,16333,16335,16341,16343],{},[35,16294,14807],{"mathvariant":258},[81,16296,14810],{"stretchy":86},[266,16298,16299,16301],{},[35,16300,1027],{},[35,16302,1068],{},[81,16304,14819],{"stretchy":86},[81,16306,244],{},[35,16308,14807],{"mathvariant":258},[81,16310,14810],{"stretchy":86},[266,16312,16313,16315],{},[35,16314,9401],{},[35,16316,1068],{},[81,16318,14819],{"stretchy":86},[282,16320,4796],{},[35,16322,14807],{"mathvariant":258},[81,16324,14810],{"stretchy":86},[89,16326,91],{},[35,16328,3018],{"mathvariant":231},[35,16330,5858],{},[81,16332,14819],{"stretchy":86},[81,16334,244],{},[266,16336,16337,16339],{},[35,16338,3102],{},[35,16340,1068],{},[81,16342,11205],{},[3718,16344,16345,16347],{},[89,16346,91],{},[266,16348,16349,16351],{},[35,16350,3102],{},[89,16352,278],{},[40,16354,16355],{"encoding":42},"\\mathbb{E}[x_i] = \\mathbb{E}[\\gamma_i]\\,\\mathbb{E}[1\u002Fs] = \\alpha_i \\cdot \\frac{1}{\\alpha_0}",[15,16357,16359,16423,16505,16561],{"className":16358,"ariaHidden":48},[47],[15,16360,16362,16365,16368,16371,16411,16414,16417,16420],{"className":16361},[52],[15,16363],{"className":16364,"style":142},[56],[15,16366,14807],{"className":16367},[61,427],[15,16369,14810],{"className":16370},[146],[15,16372,16374,16377],{"className":16373},[61],[15,16375,1027],{"className":16376},[61,122],[15,16378,16380],{"className":16379},[338],[15,16381,16383,16403],{"className":16382},[342,472],[15,16384,16386,16400],{"className":16385},[346],[15,16387,16389],{"className":16388,"style":1362},[350],[15,16390,16391,16394],{"style":482},[15,16392],{"className":16393,"style":358},[357],[15,16395,16397],{"className":16396},[362,363,364,365],[15,16398,1068],{"className":16399},[61,122,365],[15,16401,497],{"className":16402},[496],[15,16404,16406],{"className":16405},[346],[15,16407,16409],{"className":16408,"style":504},[350],[15,16410],{},[15,16412,14819],{"className":16413},[178],[15,16415],{"className":16416,"style":128},[127],[15,16418,244],{"className":16419},[132],[15,16421],{"className":16422,"style":128},[127],[15,16424,16426,16429,16432,16435,16475,16478,16481,16484,16487,16490,16493,16496,16499,16502],{"className":16425},[52],[15,16427],{"className":16428,"style":142},[56],[15,16430,14807],{"className":16431},[61,427],[15,16433,14810],{"className":16434},[146],[15,16436,16438,16441],{"className":16437},[61],[15,16439,9401],{"className":16440,"style":9435},[61,122],[15,16442,16444],{"className":16443},[338],[15,16445,16447,16467],{"className":16446},[342,472],[15,16448,16450,16464],{"className":16449},[346],[15,16451,16453],{"className":16452,"style":1362},[350],[15,16454,16455,16458],{"style":9450},[15,16456],{"className":16457,"style":358},[357],[15,16459,16461],{"className":16460},[362,363,364,365],[15,16462,1068],{"className":16463},[61,122,365],[15,16465,497],{"className":16466},[496],[15,16468,16470],{"className":16469},[346],[15,16471,16473],{"className":16472,"style":504},[350],[15,16474],{},[15,16476,14819],{"className":16477},[178],[15,16479],{"className":16480,"style":157},[127],[15,16482,14807],{"className":16483},[61,427],[15,16485,14810],{"className":16486},[146],[15,16488,3047],{"className":16489},[61],[15,16491,5858],{"className":16492},[61,122],[15,16494,14819],{"className":16495},[178],[15,16497],{"className":16498,"style":128},[127],[15,16500,244],{"className":16501},[132],[15,16503],{"className":16504,"style":128},[127],[15,16506,16508,16512,16552,16555,16558],{"className":16507},[52],[15,16509],{"className":16510,"style":16511},[56],"height:0.5945em;vertical-align:-0.15em;",[15,16513,16515,16518],{"className":16514},[61],[15,16516,3102],{"className":16517,"style":3187},[61,122],[15,16519,16521],{"className":16520},[338],[15,16522,16524,16544],{"className":16523},[342,472],[15,16525,16527,16541],{"className":16526},[346],[15,16528,16530],{"className":16529,"style":1362},[350],[15,16531,16532,16535],{"style":3202},[15,16533],{"className":16534,"style":358},[357],[15,16536,16538],{"className":16537},[362,363,364,365],[15,16539,1068],{"className":16540},[61,122,365],[15,16542,497],{"className":16543},[496],[15,16545,16547],{"className":16546},[346],[15,16548,16550],{"className":16549,"style":504},[350],[15,16551],{},[15,16553],{"className":16554,"style":772},[127],[15,16556,11205],{"className":16557},[375],[15,16559],{"className":16560,"style":772},[127],[15,16562,16564,16568],{"className":16563},[52],[15,16565],{"className":16566,"style":16567},[56],"height:1.2902em;vertical-align:-0.4451em;",[15,16569,16571,16574,16670],{"className":16570},[61],[15,16572],{"className":16573},[146,3837],[15,16575,16577],{"className":16576},[3718],[15,16578,16580,16662],{"className":16579},[342,472],[15,16581,16583,16659],{"className":16582},[346],[15,16584,16586,16637,16645],{"className":16585,"style":3850},[350],[15,16587,16588,16591],{"style":3853},[15,16589],{"className":16590,"style":3857},[357],[15,16592,16594],{"className":16593},[362,363,364,365],[15,16595,16597],{"className":16596},[61,365],[15,16598,16600,16603],{"className":16599},[61,365],[15,16601,3102],{"className":16602,"style":3187},[61,122,365],[15,16604,16606],{"className":16605},[338],[15,16607,16609,16629],{"className":16608},[342,472],[15,16610,16612,16626],{"className":16611},[346],[15,16613,16615],{"className":16614,"style":4421},[350],[15,16616,16617,16620],{"style":4054},[15,16618],{"className":16619,"style":4058},[357],[15,16621,16623],{"className":16622},[362,4062,4063,365],[15,16624,278],{"className":16625},[61,365],[15,16627,497],{"className":16628},[496],[15,16630,16632],{"className":16631},[346],[15,16633,16635],{"className":16634,"style":4076},[350],[15,16636],{},[15,16638,16639,16642],{"style":3885},[15,16640],{"className":16641,"style":3857},[357],[15,16643],{"className":16644,"style":3893},[3892],[15,16646,16647,16650],{"style":3896},[15,16648],{"className":16649,"style":3857},[357],[15,16651,16653],{"className"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",[15,16675,16677,16707],{"className":16676,"translate":19},[18],[15,16678,16680],{"className":16679},[23],[25,16681,16682],{"xmlns":27},[29,16683,16684,16704],{},[32,16685,16686,16688,16690,16692,16698,16700,16702],{},[35,16687,9552],{"mathvariant":231},[81,16689,1545],{},[81,16691,741],{"stretchy":86},[266,16693,16694,16696],{},[35,16695,3102],{},[89,16697,278],{},[81,16699,94],{"separator":48},[89,16701,91],{},[81,16703,750],{"stretchy":86},[40,16705,16706],{"encoding":42},"\\operatorname{Gamma}(\\alpha_0,1)",[15,16708,16710],{"className":16709,"ariaHidden":48},[47],[15,16711,16713,16716,16722,16725,16765,16768,16771,16774],{"className":16712},[52],[15,16714],{"className":16715,"style":142},[56],[15,16717,16719],{"className":16718},[531],[15,16720,9552],{"className":16721},[61,1620],[15,16723,741],{"className":16724},[146],[15,16726,16728,16731],{"className":16727},[61],[15,16729,3102],{"className":16730,"style":3187},[61,122],[15,16732,16734],{"className":16733},[338],[15,16735,16737,16757],{"className":16736},[342,472],[15,16738,16740,16754],{"className":16739},[346],[15,16741,16743],{"className":16742,"style":1212},[350],[15,16744,16745,16748],{"style":3202},[15,16746],{"className":16747,"style":358},[357],[15,16749,16751],{"className":16750},[362,363,364,365],[15,16752,278],{"className":16753},[61,365],[15,16755,497],{"className":16756},[496],[15,16758,16760],{"className":16759},[346],[15,16761,16763],{"className":16762,"style":504},[350],[15,16764],{},[15,16766,94],{"className":16767},[153],[15,16769],{"className":16770,"style":157},[127],[15,16772,91],{"className":16773},[61],[15,16775,750],{"className":16776},[178]," 的倒数的期望 ",[15,16779,16781,16825],{"className":16780,"translate":19},[18],[15,16782,16784],{"className":16783},[23],[25,16785,16786],{"xmlns":27},[29,16787,16788,16822],{},[32,16789,16790,16792,16794,16796,16798,16800,16802,16804,16806,16808,16810,16816,16818,16820],{},[35,16791,14807],{"mathvariant":258},[81,16793,14810],{"stretchy":86},[89,16795,91],{},[35,16797,3018],{"mathvariant":231},[35,16799,5858],{},[81,16801,14819],{"stretchy":86},[81,16803,244],{},[89,16805,91],{},[35,16807,3018],{"mathvariant":231},[81,16809,741],{"stretchy":86},[266,16811,16812,16814],{},[35,16813,3102],{},[89,16815,278],{},[81,16817,239],{},[89,16819,91],{},[81,16821,750],{"stretchy":86},[40,16823,16824],{"encoding":42},"\\mathbb{E}[1\u002Fs] = 1\u002F(\\alpha_0 - 1)",[15,16826,16828,16858,16919],{"className":16827,"ariaHidden":48},[47],[15,16829,16831,16834,16837,16840,16843,16846,16849,16852,16855],{"className":16830},[52],[15,16832],{"className":16833,"style":142},[56],[15,16835,14807],{"className":16836},[61,427],[15,16838,14810],{"className":16839},[146],[15,16841,3047],{"className":16842},[61],[15,16844,5858],{"className":16845},[61,122],[15,16847,14819],{"className":16848},[178],[15,16850],{"className":16851,"style":128},[127],[15,16853,244],{"className":16854},[132],[15,16856],{"className":16857,"style":128},[127],[15,16859,16861,16864,16867,16870,16910,16913,16916],{"className":16860},[52],[15,16862],{"className":16863,"style":142},[56],[15,16865,3047],{"className":16866},[61],[15,16868,741],{"className":16869},[146],[15,16871,16873,16876],{"className":16872},[61],[15,16874,3102],{"className":16875,"style":3187},[61,122],[15,16877,16879],{"className":16878},[338],[15,16880,16882,16902],{"className":16881},[342,472],[15,16883,16885,16899],{"className":16884},[346],[15,16886,16888],{"className":16887,"style":1212},[350],[15,16889,16890,16893],{"style":3202},[15,16891],{"className":16892,"style":358},[357],[15,16894,16896],{"className":16895},[362,363,364,365],[15,16897,278],{"className":16898},[61,365],[15,16900,497],{"className":16901},[496],[15,16903,16905],{"className":16904},[346],[15,16906,16908],{"className":16907,"style":504},[350],[15,16909],{},[15,16911],{"className":16912,"style":772},[127],[15,16914,239],{"className":16915},[375],[15,16917],{"className":16918,"style":772},[127],[15,16920,16922,16925,16928],{"className":16921},[52],[15,16923],{"className":16924,"style":142},[56],[15,16926,91],{"className":16927},[61],[15,16929,750],{"className":16930},[178],[15,16932,16934,16964],{"className":16933,"translate":19},[18],[15,16935,16937],{"className":16936},[23],[25,16938,16939],{"xmlns":27},[29,16940,16941,16961],{},[32,16942,16943,16949,16951,16957,16959],{},[266,16944,16945,16947],{},[35,16946,3102],{},[89,16948,278],{},[81,16950,14584],{},[266,16952,16953,16955],{},[35,16954,3102],{},[89,16956,278],{},[81,16958,239],{},[89,16960,91],{},[40,16962,16963],{"encoding":42},"\\alpha_0 \\mapsto \\alpha_0 - 1",[15,16965,16967,17023,17078],{"className":16966,"ariaHidden":48},[47],[15,16968,16970,16974,17014,17017,17020],{"className":16969},[52],[15,16971],{"className":16972,"style":16973},[56],"height:0.661em;vertical-align:-0.15em;",[15,16975,16977,16980],{"className":16976},[61],[15,16978,3102],{"className":16979,"style":3187},[61,122],[15,16981,16983],{"className":16982},[338],[15,16984,16986,17006],{"className":16985},[342,472],[15,16987,16989,17003],{"className":16988},[346],[15,16990,16992],{"className":16991,"style":1212},[350],[15,16993,16994,16997],{"style":3202},[15,16995],{"className":16996,"style":358},[357],[15,16998,17000],{"className":16999},[362,363,364,365],[15,17001,278],{"className":17002},[61,365],[15,17004,497],{"className":17005},[496],[15,17007,17009],{"className":17008},[346],[15,17010,17012],{"className":17011,"style":504},[350],[15,17013],{},[15,17015],{"className":17016,"style":128},[127],[15,17018,14584],{"className":17019},[132],[15,17021],{"className":17022,"style":128},[127],[15,17024,17026,17029,17069,17072,17075],{"className":17025},[52],[15,17027],{"className":17028,"style":12608},[56],[15,17030,17032,17035],{"className":17031},[61],[15,17033,3102],{"className":17034,"style":3187},[61,122],[15,17036,17038],{"className":17037},[338],[15,17039,17041,17061],{"className":17040},[342,472],[15,17042,17044,17058],{"className":17043},[346],[15,17045,17047],{"className":17046,"style":1212},[350],[15,17048,17049,17052],{"style":3202},[15,17050],{"className":17051,"style":358},[357],[15,17053,17055],{"className":17054},[362,363,364,365],[15,17056,278],{"className":17057},[61,365],[15,17059,497],{"className":17060},[496],[15,17062,17064],{"className":17063},[346],[15,17065,17067],{"className":17066,"style":504},[350],[15,17068],{},[15,17070],{"className":17071,"style":772},[127],[15,17073,239],{"className":17074},[375],[15,17076],{"className":17077,"style":772},[127],[15,17079,17081,17084],{"className":17080},[52],[15,17082],{"className":17083,"style":859},[56],[15,17085,91],{"className":17086},[61]," 的平移技巧，或直接由命题 1.3 的矩恒等式 ",[15,17089,17091,17152],{"className":17090,"translate":19},[18],[15,17092,17094],{"className":17093},[23],[25,17095,17096],{"xmlns":27},[29,17097,17098,17149],{},[32,17099,17100,17102,17104,17110,17123,17125,17127,17129,17131,17133,17135,17137,17139,17141,17143,17145,17147],{},[35,17101,14807],{"mathvariant":258},[81,17103,14810],{"stretchy":86},[266,17105,17106,17108],{},[81,17107,3737],{},[35,17109,1068],{},[289,17111,17112,17114,17116],{},[35,17113,1027],{},[35,17115,1068],{},[266,17117,17118,17121],{},[35,17119,17120],{},"β",[35,17122,1068],{},[81,17124,14819],{"stretchy":86},[81,17126,244],{},[35,17128,3726],{},[81,17130,741],{"stretchy":86},[35,17132,3102],{"mathvariant":3101},[81,17134,7756],{},[35,17136,17120],{"mathvariant":3101},[81,17138,750],{"stretchy":86},[35,17140,3018],{"mathvariant":231},[35,17142,3726],{},[81,17144,741],{"stretchy":86},[35,17146,3102],{"mathvariant":3101},[81,17148,750],{"stretchy":86},[40,17150,17151],{"encoding":42},"\\mathbb{E}[\\prod_i x_i^{\\beta_i}] = B(\\boldsymbol{\\alpha} + \\boldsymbol{\\beta})\u002FB(\\boldsymbol{\\alpha})",[15,17153,17155,17318,17348],{"className":17154,"ariaHidden":48},[47],[15,17156,17158,17162,17165,17168,17208,17211,17306,17309,17312,17315],{"className":17157},[52],[15,17159],{"className":17160,"style":17161},[56],"height:1.2667em;vertical-align:-0.2997em;",[15,17163,14807],{"className":17164},[61,427],[15,17166,14810],{"className":17167},[146],[15,17169,17171,17174],{"className":17170},[531],[15,17172,3737],{"className":17173,"style":537},[531,535,536],[15,17175,17177],{"className":17176},[338],[15,17178,17180,17200],{"className":17179},[342,472],[15,17181,17183,17197],{"className":17182},[346],[15,17184,17186],{"className":17185,"style":6064},[350],[15,17187,17188,17191],{"style":553},[15,17189],{"className":17190,"style":358},[357],[15,17192,17194],{"className":17193},[362,363,364,365],[15,17195,1068],{"className":17196},[61,122,365],[15,17198,497],{"className":17199},[496],[15,17201,17203],{"className":17202},[346],[15,17204,17206],{"className":17205,"style":593},[350],[15,17207],{},[15,17209],{"className":17210,"style":157},[127],[15,17212,17214,17217],{"className":17213},[61],[15,17215,1027],{"className":17216},[61,122],[15,17218,17220],{"className":17219},[338],[15,17221,17223,17298],{"className":17222},[342,472],[15,17224,17226,17295],{"className":17225},[346],[15,17227,17230,17241],{"className":17228,"style":17229},[350],"height:0.967em;",[15,17231,17232,17235],{"style":4011},[15,17233],{"className":17234,"style":358},[357],[15,17236,17238],{"className":17237},[362,363,364,365],[15,17239,1068],{"className":17240},[61,122,365],[15,17242,17244,17247],{"style":17243},"top:-3.1809em;margin-right:0.05em;",[15,17245],{"className":17246,"style":358},[357],[15,17248,17250],{"className":17249},[362,363,364,365],[15,17251,17253],{"className":17252},[61,365],[15,17254,17256,17260],{"className":17255},[61,365],[15,17257,17120],{"className":17258,"style":17259},[61,122,365],"margin-right:0.0528em;",[15,17261,17263],{"className":17262},[338],[15,17264,17266,17287],{"className":17265},[342,472],[15,17267,17269,17284],{"className":17268},[346],[15,17270,17272],{"className":17271,"style":4051},[350],[15,17273,17275,17278],{"style":17274},"top:-2.357em;margin-left:-0.0528em;margin-right:0.0714em;",[15,17276],{"className":17277,"style":4058},[357],[15,17279,17281],{"className":17280},[362,4062,4063,365],[15,17282,1068],{"className":17283},[61,122,365],[15,17285,497],{"className":17286},[496],[15,17288,17290],{"className":17289},[346],[15,17291,17293],{"className":17292,"style":4076},[350],[15,17294],{},[15,17296,497],{"className":17297},[496],[15,17299,17301],{"className":17300},[346],[15,17302,17304],{"className":17303,"style":4094},[350],[15,17305],{},[15,17307,14819],{"className":17308},[178],[15,17310],{"className":17311,"style":128},[127],[15,17313,244],{"className":17314},[132],[15,17316],{"className":17317,"style":128},[127],[15,17319,17321,17324,17327,17330,17339,17342,17345],{"className":17320},[52],[15,17322],{"className":17323,"style":142},[56],[15,17325,3726],{"className":17326,"style":3867},[61,122],[15,17328,741],{"className":17329},[146],[15,17331,17333],{"className":17332},[61],[15,17334,17336],{"className":17335},[61],[15,17337,3102],{"className":17338},[61,3162],[15,17340],{"className":17341,"style":772},[127],[15,17343,7756],{"className":17344},[375],[15,17346],{"className":17347,"style":772},[127],[15,17349,17351,17354,17364,17367,17370,17373,17376,17385],{"className":17350},[52],[15,17352],{"className":17353,"style":142},[56],[15,17355,17357],{"className":17356},[61],[15,17358,17360],{"className":17359},[61],[15,17361,17120],{"className":17362,"style":17363},[61,3162],"margin-right:0.034em;",[15,17365,750],{"className":17366},[178],[15,17368,3018],{"className":17369},[61],[15,17371,3726],{"className":17372,"style":3867},[61,122],[15,17374,741],{"className":17375},[146],[15,17377,17379],{"className":17378},[61],[15,17380,17382],{"className":17381},[61],[15,17383,3102],{"className":17384},[61,3162],[15,17386,750],{"className":17387},[178]," 取 ",[15,17390,17392,17414],{"className":17391,"translate":19},[18],[15,17393,17395],{"className":17394},[23],[25,17396,17397],{"xmlns":27},[29,17398,17399,17411],{},[32,17400,17401,17407,17409],{},[266,17402,17403,17405],{},[35,17404,17120],{},[35,17406,1068],{},[81,17408,244],{},[89,17410,91],{},[40,17412,17413],{"encoding":42},"\\beta_i = 1",[15,17415,17417,17473],{"className":17416,"ariaHidden":48},[47],[15,17418,17420,17423,17464,17467,17470],{"className":17419},[52],[15,17421],{"className":17422,"style":14301},[56],[15,17424,17426,17429],{"className":17425},[61],[15,17427,17120],{"className":17428,"style":17259},[61,122],[15,17430,17432],{"className":17431},[338],[15,17433,17435,17456],{"className":17434},[342,472],[15,17436,17438,17453],{"className":17437},[346],[15,17439,17441],{"className":17440,"style":1362},[350],[15,17442,17444,17447],{"style":17443},"top:-2.55em;margin-left:-0.0528em;margin-right:0.05em;",[15,17445],{"className":17446,"style":358},[357],[15,17448,17450],{"className":17449},[362,363,364,365],[15,17451,1068],{"className":17452},[61,122,365],[15,17454,497],{"className":17455},[496],[15,17457,17459],{"className":17458},[346],[15,17460,17462],{"className":17461,"style":504},[350],[15,17463],{},[15,17465],{"className":17466,"style":128},[127],[15,17468,244],{"className":17469},[132],[15,17471],{"className":17472,"style":128},[127],[15,17474,17476,17479],{"className":17475},[52],[15,17477],{"className":17478,"style":859},[56],[15,17480,91],{"className":17481},[61]," 得到。二阶矩由 ",[15,17484,17486,17560],{"className":17485,"translate":19},[18],[15,17487,17489],{"className":17488},[23],[25,17490,17491],{"xmlns":27},[29,17492,17493,17557],{},[32,17494,17495,17497,17499,17507,17509,17511],{},[35,17496,14807],{"mathvariant":258},[81,17498,14810],{"stretchy":86},[289,17500,17501,17503,17505],{},[35,17502,1027],{},[35,17504,1068],{},[89,17506,906],{},[81,17508,14819],{"stretchy":86},[81,17510,244],{},[3718,17512,17513,17535],{},[32,17514,17515,17521,17523,17529,17531,17533],{},[266,17516,17517,17519],{},[35,17518,3102],{},[35,17520,1068],{},[81,17522,741],{"stretchy":86},[266,17524,17525,17527],{},[35,17526,3102],{},[35,17528,1068],{},[81,17530,7756],{},[89,17532,91],{},[81,17534,750],{"stretchy":86},[32,17536,17537,17543,17545,17551,17553,17555],{},[266,17538,17539,17541],{},[35,17540,3102],{},[89,17542,278],{},[81,17544,741],{"stretchy":86},[266,17546,17547,17549],{},[35,17548,3102],{},[89,17550,278],{},[81,17552,7756],{},[89,17554,91],{},[81,17556,750],{"stretchy":86},[40,17558,17559],{"encoding":42},"\\mathbb{E}[x_i^2] = \\frac{\\alpha_i(\\alpha_i+1)}{\\alpha_0(\\alpha_0+1)}",[15,17561,17563,17641],{"className":17562,"ariaHidden":48},[47],[15,17564,17566,17570,17573,17576,17629,17632,17635,17638],{"className":17565},[52],[15,17567],{"className":17568,"style":17569},[56],"height:1.0728em;vertical-align:-0.2587em;",[15,17571,14807],{"className":17572},[61,427],[15,17574,14810],{"className":17575},[146],[15,17577,17579,17582],{"className":17578},[61],[15,17580,1027],{"className":17581},[61,122],[15,17583,17585],{"className":17584},[338],[15,17586,17588,17620],{"className":17587},[342,472],[15,17589,17591,17617],{"className":17590},[346],[15,17592,17594,17606],{"className":17593,"style":7899},[350],[15,17595,17597,17600],{"style":17596},"top:-2.4413em;margin-left:0em;margin-right:0.05em;",[15,17598],{"className":17599,"style":358},[357],[15,17601,17603],{"className":17602},[362,363,364,365],[15,17604,1068],{"className":17605},[61,122,365],[15,17607,17608,17611],{"style":716},[15,17609],{"className":17610,"style":358},[357],[15,17612,17614],{"className":17613},[362,363,364,365],[15,17615,906],{"className":17616},[61,365],[15,17618,497],{"className":17619},[496],[15,17621,17623],{"className":17622},[346],[15,17624,17627],{"className":17625,"style":17626},[350],"height:0.2587em;",[15,17628],{},[15,17630,14819],{"className":17631},[178],[15,17633],{"className":17634,"style":128},[127],[15,17636,244],{"className":17637},[132],[15,17639],{"className":17640,"style":128},[127],[15,17642,17644,17648],{"className":17643},[52],[15,17645],{"className":17646,"style":17647},[56],"height:1.53em;vertical-align:-0.52em;",[15,17649,17651,17654,17891],{"className":17650},[61],[15,17652],{"className":17653},[146,3837],[15,17655,17657],{"className":17656},[3718],[15,17658,17660,17883],{"className":17659},[342,472],[15,17661,17663,17880],{"className":17662},[346],[15,17664,17666,17769,17777],{"className":17665,"style":15310},[350],[15,17667,17668,17671],{"style":3853},[15,17669],{"className":17670,"style":3857},[357],[15,17672,17674],{"className":17673},[362,363,364,365],[15,17675,17677,17717,17720,17760,17763,17766],{"className":17676},[61,365],[15,17678,17680,17683],{"className":17679},[61,365],[15,17681,3102],{"className":17682,"style":3187},[61,122,365],[15,17684,17686],{"className":17685},[338],[15,17687,17689,17709],{"className":17688},[342,472],[15,17690,17692,17706],{"className":17691},[346],[15,17693,17695],{"className":17694,"style":4421},[350],[15,17696,17697,17700],{"style":4054},[15,17698],{"className":17699,"style":4058},[357],[15,17701,17703],{"className":17702},[362,4062,4063,365],[15,17704,278],{"className":17705},[61,365],[15,17707,497],{"className":17708},[496],[15,17710,17712],{"className":17711},[346],[15,17713,17715],{"className":17714,"style":4076},[350],[15,17716],{},[15,17718,741],{"className":17719},[146,365],[15,17721,17723,17726],{"className":17722},[61,365],[15,17724,3102],{"className":17725,"style":3187},[61,122,365],[15,17727,17729],{"className":17728},[338],[15,17730,17732,17752],{"className":17731},[342,472],[15,17733,17735,17749],{"className":17734},[346],[15,17736,17738],{"className":17737,"style":4421},[350],[15,17739,17740,17743],{"style":4054},[15,17741],{"className":17742,"style":4058},[357],[15,17744,17746],{"className":17745},[362,4062,4063,365],[15,17747,278],{"className":17748},[61,365],[15,17750,497],{"className":17751},[496],[15,17753,17755],{"className":17754},[346],[15,17756,17758],{"className":17757,"style":4076},[350],[15,17759],{},[15,17761,7756],{"className":17762},[375,365],[15,17764,91],{"className":17765},[61,365],[15,17767,750],{"className":17768},[178,365],[15,17770,17771,17774],{"style":3885},[15,17772],{"className":17773,"style":3857},[357],[15,17775],{"className":17776,"style":3893},[3892],[15,17778,17779,17782],{"style":15439},[15,17780],{"className":17781,"style":3857},[357],[15,17783,17785],{"className":17784},[362,363,364,365],[15,17786,17788,17828,17831,17871,17874,17877],{"className":17787},[61,365],[15,17789,17791,17794],{"className":17790},[61,365],[15,17792,3102],{"className":17793,"style":3187},[61,122,365],[15,17795,17797],{"className":17796},[338],[15,17798,17800,17820],{"className":17799},[342,472],[15,17801,17803,17817],{"className":17802},[346],[15,17804,17806],{"className":17805,"style":4051},[350],[15,17807,17808,17811],{"style":4054},[15,17809],{"className":17810,"style":4058},[357],[15,17812,17814],{"className":17813},[362,4062,4063,365],[15,17815,1068],{"className":17816},[61,122,365],[15,17818,497],{"className":17819},[496],[15,17821,17823],{"className":17822},[346],[15,17824,17826],{"className":17825,"style":4076},[350],[15,17827],{},[15,17829,741],{"className":17830},[146,365],[15,17832,17834,17837],{"className":17833},[61,365],[15,17835,3102],{"className":17836,"style":3187},[61,122,365],[15,17838,17840],{"className":17839},[338],[15,17841,17843,17863],{"className":17842},[342,472],[15,17844,17846,17860],{"className":17845},[346],[15,17847,17849],{"className":17848,"s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x_j] = \\frac{\\alpha_i \\alpha_j}{\\alpha_0(\\alpha_0+1)}",[15,17968,17970,18075],{"className":17969,"ariaHidden":48},[47],[15,17971,17973,17977,17980,17983,18023,18063,18066,18069,18072],{"className":17972},[52],[15,17974],{"className":17975,"style":17976},[56],"height:1.0361em;vertical-align:-0.2861em;",[15,17978,14807],{"className":17979},[61,427],[15,17981,14810],{"className":17982},[146],[15,17984,17986,17989],{"className":17985},[61],[15,17987,1027],{"className":17988},[61,122],[15,17990,17992],{"className":17991},[338],[15,17993,17995,18015],{"className":17994},[342,472],[15,17996,17998,18012],{"className":17997},[346],[15,17999,18001],{"className":18000,"style":1362},[350],[15,18002,18003,18006],{"style":482},[15,18004],{"className":18005,"style":358},[357],[15,18007,18009],{"className":18008},[362,363,364,365],[15,18010,1068],{"className":18011},[61,122,365],[15,18013,497],{"className":18014},[496],[15,18016,18018],{"className":18017},[346],[15,18019,18021],{"className":18020,"style":504},[350],[15,18022],{},[15,18024,18026,18029],{"className":18025},[61],[15,18027,1027],{"className":18028},[61,122],[15,18030,18032],{"className":18031},[338],[15,18033,18035,18055],{"className":18034},[342,472],[15,18036,18038,18052],{"className":18037},[346],[15,18039,18041],{"className":18040,"style":1362},[350],[15,18042,18043,18046],{"style":482},[15,18044],{"className":18045,"style":358},[357],[15,18047,18049],{"className":18048},[362,363,364,365],[15,18050,14766],{"className":18051,"style":14792},[61,122,365],[15,18053,497],{"className":18054},[496],[15,18056,18058],{"className":18057},[346],[15,18059,18061],{"className":18060,"style":15696},[350],[15,18062],{},[15,18064,14819],{"className":18065},[178],[15,18067],{"className":18068,"style":128},[127],[15,18070,244],{"className":18071},[132],[15,18073],{"className":18074,"style":128},[127],[15,18076,18078,18082],{"className":18077},[52],[15,18079],{"className":18080,"style":18081},[56],"height:1.3287em;vertical-align:-0.52em;",[15,18083,18085,18088,18314],{"className":18084},[61],[15,18086],{"className":18087},[146,3837],[15,18089,18091],{"className":18090},[3718],[15,18092,18094,18306],{"className":18093},[342,472],[15,18095,18097,18303],{"className":18096},[346],[15,18098,18101,18204,18212],{"className":18099,"style":18100},[350],"height:0.8087em;",[15,18102,18103,18106],{"style":3853},[15,18104],{"className":18105,"style":3857},[357],[15,18107,18109],{"className":18108},[362,363,364,365],[15,18110,18112,18152,18155,18195,18198,18201],{"className":18111},[61,365],[15,18113,18115,18118],{"className":18114},[61,365],[15,18116,3102],{"className":18117,"style":3187},[61,122,365],[15,18119,18121],{"className":18120},[338],[15,18122,18124,18144],{"className":18123},[342,472],[15,18125,18127,18141],{"className":18126},[346],[15,18128,18130],{"className":18129,"style":4421},[350],[15,18131,18132,18135],{"style":4054},[15,18133],{"className":18134,"style":4058},[357],[15,18136,18138],{"className":18137},[362,4062,4063,365],[15,18139,278],{"className":18140},[61,365],[15,18142,497],{"className":18143},[496],[15,18145,18147],{"className":18146},[346],[15,18148,18150],{"className":18149,"style":4076},[350],[15,18151],{},[15,18153,741],{"className":18154},[146,365],[15,18156,18158,18161],{"className":18157},[61,365],[15,18159,3102],{"className":18160,"style":3187},[61,122,365],[15,18162,18164],{"className":18163},[338],[15,18165,18167,18187],{"className":18166},[342,472],[15,18168,18170,18184],{"className":18169},[346],[15,18171,18173],{"className":18172,"style":4421},[350],[15,18174,18175,18178],{"style":4054},[15,18176],{"className":18177,"style":4058},[357],[15,18179,18181],{"className":18180},[362,4062,4063,365],[15,18182,278],{"className":18183},[61,365],[15,18185,497],{"className":18186},[496],[15,18188,18190],{"className":18189},[346],[15,18191,18193],{"className":18192,"style":4076},[350],[15,18194],{},[15,18196,7756],{"className":18197},[375,365],[15,18199,91],{"className":18200},[61,365],[15,18202,750],{"className":18203},[178,365],[15,18205,18206,18209],{"style":3885},[15,18207],{"className":18208,"style":3857},[357],[15,18210],{"className":18211,"style":3893},[3892],[15,18213,18214,18217],{"style":15861},[15,18215],{"className":18216,"style":3857},[357],[15,18218,18220],{"className":18219},[362,363,364,365],[15,18221,18223,18263],{"className":18222},[61,365],[15,18224,18226,18229],{"className":18225},[61,365],[15,18227,3102],{"className":18228,"style":3187},[61,122,365],[15,18230,18232],{"className":18231},[338],[15,18233,18235,18255],{"className":18234},[342,472],[15,18236,18238,18252],{"className":18237},[346],[15,18239,18241],{"className":18240,"style":4051},[350],[15,18242,18243,18246],{"style":4054},[15,18244],{"className":18245,"style":4058},[357],[15,18247,18249],{"className":18248},[362,4062,4063,365],[15,18250,1068],{"className":18251},[61,122,365],[15,18253,497],{"className":18254},[496],[15,18256,18258],{"className":18257},[346],[15,18259,18261],{"className":18260,"style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相减即得。",[11,18319,18320,18321,18516],{},"上述结果可写成更紧凑的矩阵形式。记 ",[15,18322,18324,18369],{"className":18323,"translate":19},[18],[15,18325,18327],{"className":18326},[23],[25,18328,18329],{"xmlns":27},[29,18330,18331,18366],{},[32,18332,18333,18336,18338,18340,18342,18348,18350,18352,18354],{},[35,18334,18335],{"mathvariant":3101},"μ",[81,18337,244],{},[35,18339,3102],{"mathvariant":3101},[35,18341,3018],{"mathvariant":231},[266,18343,18344,18346],{},[35,18345,3102],{},[89,18347,278],{},[81,18349,83],{},[35,18351,1542],{"mathvariant":231},[81,18353,1545],{},[227,18355,18356,18358],{},[35,18357,232],{"mathvariant":231},[32,18359,18360,18362,18364],{},[35,18361,102],{},[81,18363,239],{},[89,18365,91],{},[40,18367,18368],{"encoding":42},"\\boldsymbol{\\mu} = \\boldsymbol{\\alpha}\u002F\\alpha_0 \\in \\operatorname{int}\\Delta^{K-1}",[15,18370,18372,18396,18463],{"className":18371,"ariaHidden":48},[47],[15,18373,18375,18378,18387,18390,18393],{"className":18374},[52],[15,18376],{"className":18377,"style":829},[56],[15,18379,18381],{"className":18380},[61],[15,18382,18384],{"className":18383},[61],[15,18385,18335],{"className":18386},[61,3162],[15,18388],{"className":18389,"style":128},[127],[15,18391,244],{"className":18392},[132],[15,18394],{"className":18395,"style":128},[127],[15,18397,18399,18402,18411,18414,18454,18457,18460],{"className":18398},[52],[15,18400],{"className":18401,"style":142},[56],[15,18403,18405],{"className":18404},[61],[15,18406,18408],{"className":18407},[61],[15,18409,3102],{"className":18410},[61,3162],[15,18412,3018],{"className":18413},[61],[15,18415,18417,18420],{"className":18416},[61],[15,18418,3102],{"className":18419,"style":3187},[61,122],[15,18421,18423],{"className":18422},[338],[15,18424,18426,18446],{"className":18425},[342,472],[15,18427,18429,18443],{"className":18428},[346],[15,18430,18432],{"className":18431,"style":1212},[350],[15,18433,18434,18437],{"style":3202},[15,18435],{"className":18436,"style":358},[357],[15,18438,18440],{"className":18439},[362,363,364,365],[15,18441,278],{"className":18442},[61,365],[15,18444,497],{"className":18445},[496],[15,18447,18449],{"className":18448},[346],[15,18450,18452],{"className":18451,"style":504},[350],[15,18453],{},[15,18455],{"className":18456,"style":128},[127],[15,18458,83],{"className":18459},[132],[15,18461],{"className":18462,"style":128},[127],[15,18464,18466,18469,18475,18478],{"className":18465},[52],[15,18467],{"className":18468,"style":695},[56],[15,18470,18472],{"className":18471},[531],[15,18473,1542],{"className":18474},[61,1620],[15,18476],{"className":18477,"style":157},[127],[15,18479,18481,18484],{"className":18480},[61],[15,18482,232],{"className":18483},[61],[15,18485,18487],{"className":18486},[338],[15,18488,18490],{"className":18489},[342],[15,18491,18493],{"className":18492},[346],[15,18494,18496],{"className":18495,"style":695},[350],[15,18497,18498,18501],{"style":716},[15,18499],{"className":18500,"style":358},[357],[15,18502,18504],{"className":18503},[362,363,364,365],[15,18505,18507,18510,18513],{"className":18506},[61,365],[15,18508,102],{"className":18509,"style":174},[61,122,365],[15,18511,239],{"className":18512},[375,365],[15,18514,91],{"className":18515},[61,365]," 为均值向量，则",[15,18518,18520],{"className":18519,"translate":19},[212],[15,18521,18523,18596],{"className":18522,"translate":19},[18],[15,18524,18526],{"className":18525},[23],[25,18527,18528],{"xmlns":27,"display":221},[29,18529,18530,18593],{},[32,18531,18532,18535,18537,18539,18541,18543,18545,18547,18549,18565],{},[35,18533,18534],{"mathvariant":251},"Σ",[81,18536,244],{},[35,18538,14915],{"mathvariant":231},[81,18540,1545],{},[81,18542,741],{"stretchy":86},[35,18544,1027],{"mathvariant":251},[81,18546,750],{"stretchy":86},[81,18548,244],{},[3718,18550,18551,18553],{},[89,18552,91],{},[32,18554,18555,18561,18563],{},[266,18556,18557,18559],{},[35,18558,3102],{},[89,18560,278],{},[81,18562,7756],{},[89,18564,91],{},[32,18566,18567,18569,18572,18574,18576,18578,18580,18582,18584,18591],{},[81,18568,741],{"fence":48},[35,18570,18571],{"mathvariant":231},"diag",[81,18573,1545],{},[81,18575,741],{"stretchy":86},[35,18577,18335],{"mathvariant":3101},[81,18579,750],{"stretchy":86},[81,18581,239],{},[35,18583,18335],{"mathvariant":3101},[227,18585,18586,18588],{},[35,18587,18335],{"mathvariant":3101},[35,18589,18590],{"mathvariant":231},"⊤",[81,18592,750],{"fence":48},[40,18594,18595],{"encoding":42},"\\boldsymbol{\\Sigma} = \\operatorname{Cov}(\\mathbf{x}) = \\frac{1}{\\alpha_0 + 1}\\left(\\operatorname{diag}(\\boldsymbol{\\mu}) - \\boldsymbol{\\mu}\\boldsymbol{\\mu}^\\top\\right)",[15,18597,18599,18624,18654],{"className":18598,"ariaHidden":48},[47],[15,18600,18602,18606,18615,18618,18621],{"className":18601},[52],[15,18603],{"className":18604,"style":18605},[56],"height:0.6861em;",[15,18607,18609],{"className":18608},[61],[15,18610,18612],{"className":18611},[61],[15,18613,18534],{"className":18614},[61,411],[15,18616],{"className":18617,"style":128},[127],[15,18619,244],{"className":18620},[132],[15,18622],{"className":18623,"style":128},[127],[15,18625,18627,18630,18636,18639,18642,18645,18648,18651],{"className":18626},[52],[15,18628],{"className":18629,"style":142},[56],[15,18631,18633],{"className":18632},[531],[15,18634,14915],{"className":18635,"style":15608},[61,1620],[15,18637,741],{"className":18638},[146],[15,18640,1027],{"className":18641},[61,411],[15,18643,750],{"className":18644},[178],[15,18646],{"className":18647,"style":128},[127],[15,18649,244],{"className":18650},[132],[15,18652],{"className":18653,"style":128},[127],[15,18655,18657,18661,18774,18777],{"className":18656},[52],[15,18658],{"className":18659,"style":18660},[56],"height:2.1574em;vertical-align:-0.836em;",[15,18662,18664,18667,18771],{"className":18663},[61],[15,18665],{"className":18666},[146,3837],[15,18668,18670],{"className":18669},[3718],[15,18671,18673,18762],{"className":18672},[342,472],[15,18674,18676,18759],{"className":18675},[346],[15,18677,18680,18740,18748],{"className":18678,"style":18679},[350],"height:1.3214em;",[15,18681,18682,18685],{"style":11393},[15,18683],{"className":18684,"style":3857},[357],[15,18686,18688,18728,18731,18734,18737],{"className":18687},[61],[15,18689,18691,18694],{"className":18690},[61],[15,18692,3102],{"className":18693,"style":3187},[61,122],[15,18695,18697],{"className":18696},[338],[15,18698,18700,18720],{"className":18699},[342,472],[15,18701,18703,18717],{"className":18702},[346],[15,18704,18706],{"className":18705,"style":1212},[350],[15,18707,18708,18711],{"style":3202},[15,18709],{"className":18710,"style":358},[357],[15,18712,18714],{"className":18713},[362,363,364,365],[15,18715,278],{"className":18716},[61,365],[15,18718,497],{"className":18719},[496],[15,18721,18723],{"className":18722},[346],[15,18724,18726],{"className":18725,"style":504},[350],[15,18727],{},[15,18729],{"className":18730,"style":772},[127],[15,18732,7756],{"className":18733},[375],[15,18735],{"className":18736,"style":772},[127],[15,18738,91],{"className":18739},[61],[15,18741,18742,18745],{"style":3885},[15,18743],{"className":18744,"style":3857},[357],[15,18746],{"className":18747,"style":3893},[3892],[15,18749,18750,18753],{"style":11459},[15,18751],{"className":18752,"style":3857},[357],[15,18754,18756],{"className":18755},[61],[15,18757,91],{"className":18758},[61],[15,18760,497],{"className":18761},[496],[15,18763,18765],{"className":18764},[346],[15,18766,18769],{"className":18767,"style":18768},[350],"height:0.836em;",[15,18770],{},[15,18772],{"className":18773},[178,3837],[15,18775],{"className":18776,"style":157},[127],[15,18778,18780,18786,18792,18795,18804,18807,18810,18813,18816,18825,18861],{"className":18779},[161],[15,18781,18783],{"className":18782,"style":402},[146,401],[15,18784,741],{"className":18785},[406,4063],[15,18787,18789],{"className":18788},[531],[15,18790,18571],{"className":18791,"style":15608},[61,1620],[15,18793,741],{"className":18794},[146],[15,18796,18798],{"className":18797},[61],[15,18799,18801],{"className":18800},[61],[15,18802,18335],{"className":18803},[61,3162],[15,18805,750],{"className":18806},[178],[15,18808],{"className":18809,"style":772},[127],[15,18811,239],{"className":18812},[375],[15,18814],{"className":18815,"style":772},[127],[15,18817,18819],{"className":18818},[61],[15,18820,18822],{"className":18821},[61],[15,18823,18335],{"className":18824},[61,3162],[15,18826,18828,18837],{"className":18827},[61],[15,18829,18831],{"className":18830},[61],[15,18832,18834],{"className":18833},[61],[15,18835,18335],{"className":18836},[61,3162],[15,18838,18840],{"className":18839},[338],[15,18841,18843],{"className":18842},[342],[15,18844,18846],{"className":18845},[346],[15,18847,18850],{"className":18848,"style":18849},[350],"height:0.8991em;",[15,18851,18852,18855],{"style":353},[15,18853],{"className":18854,"style":358},[357],[15,18856,18858],{"className":18857},[362,363,364,365],[15,18859,18590],{"className":18860},[61,365],[15,18862,18864],{"className":18863,"style":402},[178,401],[15,18865,750],{"className":18866},[406,4063],[11,18868,18869,18870,18920,18921,941,19019,19054,19055,19084,19085,19145,19146,19387,19388,19416],{},"这一形式揭示了两个结构性事实。第一，协方差矩阵的秩为 ",[15,18871,18873,18890],{"className":18872,"translate":19},[18],[15,18874,18876],{"className":18875},[23],[25,18877,18878],{"xmlns":27},[29,18879,18880,18888],{},[32,18881,18882,18884,18886],{},[35,18883,102],{},[81,18885,239],{},[89,18887,91],{},[40,18889,962],{"encoding":42},[15,18891,18893,18911],{"className":18892,"ariaHidden":48},[47],[15,18894,18896,18899,18902,18905,18908],{"className":18895},[52],[15,18897],{"className":18898,"style":972},[56],[15,18900,102],{"className":18901,"style":174},[61,122],[15,18903],{"className":18904,"style":772},[127],[15,18906,239],{"className":18907},[375],[15,18909],{"className":18910,"style":772},[127],[15,18912,18914,18917],{"className":18913},[52],[15,18915],{"className":18916,"style":859},[56],[15,18918,91],{"className":18919},[61],"：由于 ",[15,18922,18924,18950],{"className":18923,"translate":19},[18],[15,18925,18927],{"className":18926},[23],[25,18928,18929],{"xmlns":27},[29,18930,18931,18947],{},[32,18932,18933,18939,18941,18943,18945],{},[227,18934,18935,18937],{},[89,18936,91],{"mathvariant":251},[35,18938,18590],{"mathvariant":231},[35,18940,18534],{"mathvariant":251},[89,18942,91],{"mathvariant":251},[81,18944,244],{},[89,18946,278],{},[40,18948,18949],{"encoding":42},"\\mathbf{1}^\\top \\boldsymbol{\\Sigma} \\mathbf{1} = 0",[15,18951,18953,19010],{"className":18952,"ariaHidden":48},[47],[15,18954,18956,18960,18989,18998,19001,19004,19007],{"className":18955},[52],[15,18957],{"className":18958,"style":18959},[56],"height:0.8491em;",[15,18961,18963,18966],{"className":18962},[61],[15,18964,91],{"className":18965},[61,411],[15,18967,18969],{"className":18968},[338],[15,18970,18972],{"className":18971},[342],[15,18973,18975],{"className":18974},[346],[15,18976,18978],{"className":18977,"style":18959},[350],[15,18979,18980,18983],{"style":716},[15,18981],{"className":18982,"style":358},[357],[15,18984,18986],{"className":18985},[362,363,364,365],[15,18987,18590],{"className":18988},[61,365],[15,18990,18992],{"className":18991},[61],[15,18993,18995],{"className":18994},[61],[15,18996,18534],{"className":18997},[61,411],[15,18999,91],{"className":19000},[61,411],[15,19002],{"className":19003,"style":128},[127],[15,19005,244],{"className":19006},[132],[15,19008],{"className":19009,"style":128},[127],[15,19011,19013,19016],{"className":19012},[52],[15,19014],{"className":19015,"style":859},[56],[15,19017,278],{"className":19018},[61],[15,19020,19022,19036],{"className":19021,"translate":19},[18],[15,19023,19025],{"className":19024},[23],[25,19026,19027],{"xmlns":27},[29,19028,19029,19033],{},[32,19030,19031],{},[35,19032,18534],{"mathvariant":251},[40,19034,19035],{"encoding":42},"\\boldsymbol{\\Sigma}",[15,19037,19039],{"className":19038,"ariaHidden":48},[47],[15,19040,19042,19045],{"className":19041},[52],[15,19043],{"className":19044,"style":18605},[56],[15,19046,19048],{"className":19047},[61],[15,19049,19051],{"className":19050},[61],[15,19052,18534],{"className":19053},[61,411]," 必然奇异，其零空间由 ",[15,19056,19058,19072],{"className":19057,"translate":19},[18],[15,19059,19061],{"className":19060},[23],[25,19062,19063],{"xmlns":27},[29,19064,19065,19069],{},[32,19066,19067],{},[89,19068,91],{"mathvariant":251},[40,19070,19071],{"encoding":42},"\\mathbf{1}",[15,19073,19075],{"className":19074,"ariaHidden":48},[47],[15,19076,19078,19081],{"className":19077},[52],[15,19079],{"className":19080,"style":859},[56],[15,19082,91],{"className":19083},[61,411]," 张成——这正是「分布在 ",[15,19086,19088,19109],{"className":19087,"translate":19},[18],[15,19089,19091],{"className":19090},[23],[25,19092,19093],{"xmlns":27},[29,19094,19095,19107],{},[32,19096,19097,19099,19101,19103,19105],{},[81,19098,741],{"stretchy":86},[35,19100,102],{},[81,19102,239],{},[89,19104,91],{},[81,19106,750],{"stretchy":86},[40,19108,753],{"encoding":42},[15,19110,19112,19133],{"className":19111,"ariaHidden":48},[47],[15,19113,19115,19118,19121,19124,19127,19130],{"className":19114},[52],[15,19116],{"className":19117,"style":142},[56],[15,19119,741],{"className":19120},[146],[15,19122,102],{"className":19123,"style":174},[61,122],[15,19125],{"className":19126,"style":772},[127],[15,19128,239],{"className":19129},[375],[15,19131],{"className":19132,"style":772},[127],[15,19134,19136,19139,19142],{"className":19135},[52],[15,19137],{"className":19138,"style":142},[56],[15,19140,91],{"className":19141},[61],[15,19143,750],{"className":19144},[178]," 维对象上」的代数痕迹。第二，全部协方差为负：",[15,19147,19149,19199],{"className":19148,"translate":19},[18],[15,19150,19152],{"className":19151},[23],[25,19153,19154],{"xmlns":27},[29,19155,19156,19196],{},[32,19157,19158,19160,19162,19164,19170,19172,19178,19180,19182,19184,19186,19188,19190,19192,19194],{},[35,19159,14915],{"mathvariant":231},[81,19161,1545],{},[81,19163,741],{"stretchy":86},[266,19165,19166,19168],{},[35,19167,1027],{},[35,19169,1068],{},[81,19171,94],{"separator":48},[266,19173,19174,19176],{},[35,19175,1027],{},[35,19177,14766],{},[81,19179,750],{"stretchy":86},[81,19181,2488],{},[89,19183,278],{},[282,19185,284],{},[81,19187,741],{"stretchy":86},[35,19189,1068],{},[81,19191,14988],{"mathvariant":231},[35,19193,14766],{},[81,19195,750],{"stretchy":86},[40,19197,19198],{"encoding":42},"\\operatorname{Cov}(x_i, x_j) \u003C 0\\ (i \\ne j)",[15,19200,19202,19315,19375],{"className":19201,"ariaHidden":48},[47],[15,19203,19205,19208,19214,19217,19257,19260,19263,19303,19306,19309,19312],{"className":19204},[52],[15,19206],{"className":19207,"style":17976},[56],[15,19209,19211],{"className":19210},[531],[15,19212,14915],{"className":19213,"style":15608},[61,1620],[15,19215,741],{"className":19216},[146],[15,19218,19220,19223],{"className":19219},[61],[15,19221,1027],{"className":19222},[61,122],[15,19224,19226],{"className":19225},[338],[15,19227,19229,19249],{"className":19228},[342,472],[15,19230,19232,19246],{"className":19231},[346],[15,19233,19235],{"className":19234,"style":1362},[350],[15,19236,19237,19240],{"style":482},[15,19238],{"className":19239,"style":358},[357],[15,19241,19243],{"className":19242},[362,363,364,365],[15,19244,1068],{"className":19245},[61,122,365],[15,19247,497],{"className":19248},[496],[15,19250,19252],{"className":19251},[346],[15,19253,19255],{"className":19254,"style":504},[350],[15,19256],{},[15,19258,94],{"className":19259},[153],[15,19261],{"className":19262,"style":157},[127],[15,19264,19266,19269],{"className":19265},[61],[15,19267,1027],{"className":19268},[61,122],[15,19270,19272],{"className":19271},[338],[15,19273,19275,19295],{"className":19274},[342,472],[15,19276,19278,19292],{"className":19277},[346],[15,19279,19281],{"className":19280,"style":1362},[350],[15,19282,19283,19286],{"style":482},[15,19284],{"className":19285,"style":358},[357],[15,19287,19289],{"className":19288},[362,363,364,365],[15,19290,14766],{"className":19291,"style":14792},[61,122,365],[15,19293,497],{"className":19294},[496],[15,19296,19298],{"className":19297},[346],[15,19299,19301],{"className":19300,"style":15696},[350],[15,19302],{},[15,19304,750],{"className":19305},[178],[15,19307],{"className":19308,"style":128},[127],[15,19310,2488],{"className":19311},[132],[15,19313],{"className":19314,"style":128},[127],[15,19316,19318,19321,19324,19327,19330,19333,19336,19372],{"className":19317},[52],[15,19319],{"className":19320,"style":142},[56],[15,19322,278],{"className":19323},[61],[15,19325,284],{"className":19326},[127],[15,19328,741],{"className":19329},[146],[15,19331,1068],{"className":19332},[61,122],[15,19334],{"className":19335,"style":128},[127],[15,19337,19339,19366,19369],{"className":19338},[132],[15,19340,19342],{"className":19341},[132],[15,19343,19345],{"className":19344},[61,15991],[15,19346,19348],{"className":19347},[15995],[15,19349,19351,19354,19363],{"className":19350},[15999],[15,19352],{"className":19353,"style":14301},[56],[15,19355,19357],{"className":19356},[16006],[15,19358,19360],{"className":19359},[61],[15,19361,16013],{"className":19362},[132],[15,19364],{"className":19365},[16017],[15,19367],{"className":19368},[127,16021],[15,19370,244],{"className":19371},[132],[15,19373],{"className":19374,"style":128},[127],[15,19376,19378,19381,19384],{"className":19377},[52],[15,19379],{"className":19380,"style":142},[56],[15,19382,14766],{"className":19383,"style":14792},[61,122],[15,19385,750],{"className":19386},[178],"，因为分量之间必须通过和为 ",[15,19389,19391,19404],{"className":19390,"translate":19},[18],[15,19392,19394],{"className":19393},[23],[25,19395,19396],{"xmlns":27},[29,19397,19398,19402],{},[32,19399,19400],{},[89,19401,91],{},[40,19403,91],{"encoding":42},[15,19405,19407],{"className":19406,"ariaHidden":48},[47],[15,19408,19410,19413],{"className":19409},[52],[15,19411],{"className":19412,"style":859},[56],[15,19414,91],{"className":19415},[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。",[877,19418,19420],{"id":19419},"_22-均值精度重参数化","2.2 均值—精度重参数化",[11,19422,19423,19424,9342,19459,19529],{},"命题 2.1 中的 ",[15,19425,19427,19441],{"className":19426,"translate":19},[18],[15,19428,19430],{"className":19429},[23],[25,19431,19432],{"xmlns":27},[29,19433,19434,19438],{},[32,19435,19436],{},[35,19437,18335],{"mathvariant":3101},[40,19439,19440],{"encoding":42},"\\boldsymbol{\\mu}",[15,19442,19444],{"className":19443,"ariaHidden":48},[47],[15,19445,19447,19450],{"className":19446},[52],[15,19448],{"className":19449,"style":829},[56],[15,19451,19453],{"className":19452},[61],[15,19454,19456],{"className":19455},[61],[15,19457,18335],{"className":19458},[61,3162],[15,19460,19462,19480],{"className":19461,"translate":19},[18],[15,19463,19465],{"className":19464},[23],[25,19466,19467],{"xmlns":27},[29,19468,19469,19477],{},[32,19470,19471],{},[266,19472,19473,19475],{},[35,19474,3102],{},[89,19476,278],{},[40,19478,19479],{"encoding":42},"\\alpha_0",[15,19481,19483],{"className":19482,"ariaHidden":48},[47],[15,19484,19486,19489],{"className":19485},[52],[15,19487],{"className":19488,"style":2509},[56],[15,19490,19492,19495],{"className":19491},[61],[15,19493,3102],{"className":19494,"style":3187},[61,122],[15,19496,19498],{"className":19497},[338],[15,19499,19501,19521],{"className":19500},[342,472],[15,19502,19504,19518],{"className":19503},[346],[15,19505,19507],{"className":19506,"style":1212},[350],[15,19508,19509,19512],{"style":3202},[15,19510],{"className":19511,"style":358},[357],[15,19513,19515],{"className":19514},[362,363,364,365],[15,19516,278],{"className":19517},[61,365],[15,19519,497],{"className":19520},[496],[15,19522,19524],{"className":19523},[346],[15,19525,19527],{"className":19526,"style":504},[350],[15,19528],{}," 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",[15,19531,19533],{"className":19532,"translate":19},[212],[15,19534,19536,19598],{"className":19535,"translate":19},[18],[15,19537,19539],{"className":19538},[23],[25,19540,19541],{"xmlns":27,"display":221},[29,19542,19543,19595],{},[32,19544,19545,19547,19549,19555,19557,19559,19561,19567,19569,19571,19573,19575,19577,19579,19581,19583],{},[35,19546,3102],{"mathvariant":3101},[81,19548,244],{},[266,19550,19551,19553],{},[35,19552,3102],{},[89,19554,278],{},[35,19556,18335],{"mathvariant":3101},[81,19558,94],{"separator":48},[127,19560],{"width":9838},[266,19562,19563,19565],{},[35,19564,3102],{},[89,19566,278],{},[81,19568,1588],{},[89,19570,278],{},[81,19572,94],{"separator":48},[282,19574,284],{},[35,19576,18335],{"mathvariant":3101},[81,19578,83],{},[35,19580,1542],{"mathvariant":231},[81,19582,1545],{},[227,19584,19585,19587],{},[35,19586,232],{"mathvariant":231},[32,19588,19589,19591,19593],{},[35,19590,102],{},[81,19592,239],{},[89,19594,91],{},[40,19596,19597],{"encoding":42},"\\boldsymbol{\\alpha} = \\alpha_0 \\boldsymbol{\\mu}, \\qquad \\alpha_0 > 0,\\ \\boldsymbol{\\mu} \\in \\operatorname{int}\\Delta^{K-1}",[15,19599,19601,19625,19738,19775],{"className":19600,"ariaHidden":48},[47],[15,19602,19604,19607,19616,19619,19622],{"className":19603},[52],[15,19605],{"className":19606,"style":2706},[56],[15,19608,19610],{"className":19609},[61],[15,19611,19613],{"className":19612},[61],[15,19614,3102],{"className":19615},[61,3162],[15,19617],{"className":19618,"style":128},[127],[15,19620,244],{"className":19621},[132],[15,19623],{"className":19624,"style":128},[127],[15,19626,19628,19631,19671,19680,19683,19686,19689,19729,19732,19735],{"className":19627},[52],[15,19629],{"className":19630,"style":118},[56],[15,19632,19634,19637],{"className":19633},[61],[15,19635,3102],{"className":19636,"style":3187},[61,122],[15,19638,19640],{"className":19639},[338],[15,19641,19643,19663],{"className":19642},[342,472],[15,19644,19646,19660],{"className":19645},[346],[15,19647,19649],{"className":19648,"style":1212},[350],[15,19650,19651,19654],{"style":3202},[15,19652],{"className":19653,"style":358},[357],[15,19655,19657],{"className":19656},[362,363,364,365],[15,19658,278],{"className":19659},[61,365],[15,19661,497],{"className":19662},[496],[15,19664,19666],{"className":19665},[346],[15,19667,19669],{"className":19668,"style":504},[350],[15,19670],{},[15,19672,19674],{"className":19673},[61],[15,19675,19677],{"className":19676},[61],[15,19678,18335],{"className":19679},[61,3162],[15,19681,94],{"className":19682},[153],[15,19684],{"className":19685,"style":10026},[127],[15,19687],{"className":19688,"style":157},[127],[15,19690,19692,19695],{"className":19691},[61],[15,19693,3102],{"className":19694,"style":3187},[61,122],[15,19696,19698],{"className":19697},[338],[15,19699,19701,19721],{"className":19700},[342,472],[15,19702,19704,19718],{"className":19703},[346],[15,19705,19707],{"className":19706,"style":1212},[350],[15,19708,19709,19712],{"style":3202},[15,19710],{"className":19711,"style":358},[357],[15,19713,19715],{"className":19714},[362,363,364,365],[15,19716,278],{"className":19717},[61,365],[15,19719,497],{"className":19720},[496],[15,19722,19724],{"className":19723},[346],[15,19725,19727],{"className":19726,"style":504},[350],[15,19728],{},[15,19730],{"className":19731,"style":128},[127],[15,19733,1588],{"className":19734},[132],[15,19736],{"className":19737,"style":128},[127],[15,19739,19741,19745,19748,19751,19754,19757,19766,19769,19772],{"className":19740},[52],[15,19742],{"className":19743,"style":19744},[56],"height:0.8389em;vertical-align:-0.1944em;",[15,19746,278],{"className":19747},[61],[15,19749,94],{"className":19750},[153],[15,19752,284],{"className":19753},[127],[15,19755],{"className":19756,"style":157},[127],[15,19758,19760],{"className":19759},[61],[15,19761,19763],{"className":19762},[61],[15,19764,18335],{"className":19765},[61,3162],[15,19767],{"className":19768,"style":128},[127],[15,19770,83],{"className":19771},[132],[15,19773],{"className":19774,"style":128},[127],[15,19776,19778,19781,19787,19790],{"className":19777},[52],[15,19779],{"className":19780,"style":328},[56],[15,19782,19784],{"className":19783},[531],[15,19785,1542],{"className":19786},[61,1620],[15,19788],{"className":19789,"style":157},[127],[15,19791,19793,19796],{"className":19792},[61],[15,19794,232],{"className":19795},[61],[15,19797,19799],{"className":19798},[338],[15,19800,19802],{"className":19801},[342],[15,19803,19805],{"className":19804},[346],[15,19806,19808],{"className":19807,"style":328},[350],[15,19809,19810,19813],{"style":353},[15,19811],{"className":19812,"style":358},[357],[15,19814,19816],{"className":19815},[362,363,364,365],[15,19817,19819,19822,19825],{"className":19818},[61,365],[15,19820,102],{"className":19821,"style":174},[61,122,365],[15,19823,239],{"className":19824},[375,365],[15,19826,91],{"className":19827},[61,365],[11,19829,19830,19831,19834,19835,19885,19886,19889,19890,941,20173,20281,20282,20351],{},"它把参数空间分解为一个",[661,19832,19833],{},"方向","（",[15,19836,19838,19855],{"className":19837,"translate":19},[18],[15,19839,19841],{"className":19840},[23],[25,19842,19843],{"xmlns":27},[29,19844,19845,19853],{},[32,19846,19847,19849,19851],{},[35,19848,102],{},[81,19850,239],{},[89,19852,91],{},[40,19854,962],{"encoding":42},[15,19856,19858,19876],{"className":19857,"ariaHidden":48},[47],[15,19859,19861,19864,19867,19870,19873],{"className":19860},[52],[15,19862],{"className":19863,"style":972},[56],[15,19865,102],{"className":19866,"style":174},[61,122],[15,19868],{"className":19869,"style":772},[127],[15,19871,239],{"className":19872},[375],[15,19874],{"className":19875,"style":772},[127],[15,19877,19879,19882],{"className":19878},[52],[15,19880],{"className":19881,"style":859},[56],[15,19883,91],{"className":19884},[61]," 维单纯形）与一个",[661,19887,19888],{},"尺度","（正实数）。在此参数化下，协方差为 ",[15,19891,19893,19945],{"className":19892,"translate":19},[18],[15,19894,19896],{"className":19895},[23],[25,19897,19898],{"xmlns":27},[29,19899,19900,19942],{},[32,19901,19902,19904,19906,19922,19924,19926,19928,19930,19932,19934,19940],{},[35,19903,18534],{"mathvariant":251},[81,19905,244],{},[3718,19907,19908,19910],{},[89,19909,91],{},[32,19911,19912,19918,19920],{},[266,19913,19914,19916],{},[35,19915,3102],{},[89,19917,278],{},[81,19919,7756],{},[89,19921,91],{},[81,19923,741],{"stretchy":86},[35,19925,18571],{"mathvariant":231},[81,19927,1545],{},[35,19929,18335],{"mathvariant":3101},[81,19931,239],{},[35,19933,18335],{"mathvariant":3101},[227,19935,19936,19938],{},[35,19937,18335],{"mathvariant":3101},[35,19939,18590],{"mathvariant":231},[81,19941,750],{"stretchy":86},[40,19943,19944],{"encoding":42},"\\boldsymbol{\\Sigma} = \\frac{1}{\\alpha_0+1}(\\operatorname{diag}\\boldsymbol{\\mu} - \\boldsymbol{\\mu}\\boldsymbol{\\mu}^\\top)",[15,19946,19948,19972,20119],{"className":19947,"ariaHidden":48},[47],[15,19949,19951,19954,19963,19966,19969],{"className":19950},[52],[15,19952],{"className":19953,"style":18605},[56],[15,19955,19957],{"className":19956},[61],[15,19958,19960],{"className":19959},[61],[15,19961,18534],{"className":19962},[61,411],[15,19964],{"className":19965,"style":128},[127],[15,19967,244],{"className":19968},[132],[15,19970],{"className":19971,"style":128},[127],[15,19973,19975,19978,20089,20092,20098,20101,20110,20113,20116],{"className":19974},[52],[15,19976],{"className":19977,"style":16567},[56],[15,19979,19981,19984,20086],{"className":19980},[61],[15,19982],{"className":19983},[146,3837],[15,19985,19987],{"className":19986},[3718],[15,19988,19990,20078],{"className":19989},[342,472],[15,19991,19993,20075],{"className":19992},[346],[15,19994,19996,20053,20061],{"className":19995,"style":3850},[350],[15,19997,19998,20001],{"style":3853},[15,19999],{"className":20000,"style":3857},[357],[15,20002,20004],{"className":20003},[362,363,364,365],[15,20005,20007,20047,20050],{"className":20006},[61,365],[15,20008,20010,20013],{"className":20009},[61,365],[15,20011,3102],{"className":20012,"style":3187},[61,122,365],[15,20014,20016],{"className":20015},[338],[15,20017,20019,20039],{"className":20018},[342,472],[15,20020,20022,20036],{"className":20021},[346],[15,20023,20025],{"className":20024,"style":4421},[350],[15,20026,20027,20030],{"style":4054},[15,20028],{"className":20029,"style":4058},[357],[15,20031,20033],{"className":20032},[362,4062,4063,365],[15,20034,278],{"className":20035},[61,365],[15,20037,497],{"className":20038},[496],[15,20040,20042],{"className":20041},[346],[15,20043,20045],{"className":20044,"style":4076},[350],[15,20046],{},[15,20048,7756],{"className":20049},[375,365],[15,20051,91],{"className":20052},[61,365],[15,20054,20055,20058],{"style":3885},[15,20056],{"className":20057,"style":3857},[357],[15,20059],{"className":20060,"style":3893},[3892],[15,20062,20063,20066],{"style":3896},[15,20064],{"className":20065,"style":3857},[357],[15,20067,20069],{"className":20068},[362,363,364,365],[15,20070,20072],{"className":20071},[61,365],[15,20073,91],{"className":20074},[61,365],[15,20076,497],{"className":20077},[496],[15,20079,20081],{"className":20080},[346],[15,20082,20084],{"className":20083,"style":15209},[350],[15,20085],{},[15,20087],{"className":20088},[178,3837],[15,20090,741],{"className":20091},[146],[15,20093,20095],{"className":20094},[531],[15,20096,18571],{"className":20097,"style":15608},[61,1620],[15,20099],{"className":20100,"style":157},[127],[15,20102,20104],{"className":20103},[61],[15,20105,20107],{"className":20106},[61],[15,20108,18335],{"className":20109},[61,3162],[15,20111],{"className":20112,"style":772},[127],[15,20114,239],{"className":20115},[375],[15,20117],{"className":20118,"style":772},[127],[15,20120,20122,20126,20135,20170],{"className":20121},[52],[15,20123],{"className":20124,"style":20125},[56],"height:1.0991em;vertical-align:-0.25em;",[15,20127,20129],{"className":20128},[61],[15,20130,20132],{"className":20131},[61],[15,20133,18335],{"className":20134},[61,3162],[15,20136,20138,20147],{"className":20137},[61],[15,20139,20141],{"className":20140},[61],[15,20142,20144],{"className":20143},[61],[15,20145,18335],{"className":20146},[61,3162],[15,20148,20150],{"className":20149},[338],[15,20151,20153],{"className":20152},[342],[15,20154,20156],{"className":20155},[346],[15,20157,20159],{"className":20158,"style":18959},[350],[15,20160,20161,20164],{"style":716},[15,20162],{"className":20163,"style":358},[357],[15,20165,20167],{"className":20166},[362,363,364,365],[15,20168,18590],{"className":20169},[61,365],[15,20171,750],{"className":20172},[178],[661,20174,20175,20176,20210,20211,20280],{},"与 ",[15,20177,20179,20192],{"className":20178,"translate":19},[18],[15,20180,20182],{"className":20181},[23],[25,20183,20184],{"xmlns":27},[29,20185,20186,20190],{},[32,20187,20188],{},[35,20189,18335],{"mathvariant":3101},[40,20191,19440],{"encoding":42},[15,20193,20195],{"className":20194,"ariaHidden":48},[47],[15,20196,20198,20201],{"className":20197},[52],[15,20199],{"className":20200,"style":829},[56],[15,20202,20204],{"className":20203},[61],[15,20205,20207],{"className":20206},[61],[15,20208,18335],{"className":20209},[61,3162]," 的取值无关、只随 ",[15,20212,20214,20231],{"className":20213,"translate":19},[18],[15,20215,20217],{"className":20216},[23],[25,20218,20219],{"xmlns":27},[29,20220,20221,20229],{},[32,20222,20223],{},[266,20224,20225,20227],{},[35,20226,3102],{},[89,20228,278],{},[40,20230,19479],{"encoding":42},[15,20232,20234],{"className":20233,"ariaHidden":48},[47],[15,20235,20237,20240],{"className":20236},[52],[15,20238],{"className":20239,"style":2509},[56],[15,20241,20243,20246],{"className":20242},[61],[15,20244,3102],{"className":20245,"style":3187},[61,122],[15,20247,20249],{"className":20248},[338],[15,20250,20252,20272],{"className":20251},[342,472],[15,20253,20255,20269],{"className":20254},[346],[15,20256,20258],{"className":20257,"style":1212},[350],[15,20259,20260,20263],{"style":3202},[15,20261],{"className":20262,"style":358},[357],[15,20264,20266],{"className":20265},[362,363,364,365],[15,20267,278],{"className":20268},[61,365],[15,20270,497],{"className":20271},[496],[15,20273,20275],{"className":20274},[346],[15,20276,20278],{"className":20277,"style":504},[350],[15,20279],{}," 单调收缩","。因此 ",[15,20283,20285,20302],{"className":20284,"translate":19},[18],[15,20286,20288],{"className":20287},[23],[25,20289,20290],{"xmlns":27},[29,20291,20292,20300],{},[32,20293,20294],{},[266,20295,20296,20298],{},[35,20297,3102],{},[89,20299,278],{},[40,20301,19479],{"encoding":42},[15,20303,20305],{"className":20304,"ariaHidden":48},[47],[15,20306,20308,20311],{"className":20307},[52],[15,20309],{"className":20310,"style":2509},[56],[15,20312,20314,20317],{"className":20313},[61],[15,20315,3102],{"className":20316,"style":3187},[61,122],[15,20318,20320],{"className":20319},[338],[15,20321,20323,20343],{"className":20322},[342,472],[15,20324,20326,20340],{"className":20325},[346],[15,20327,20329],{"className":20328,"style":1212},[350],[15,20330,20331,20334],{"style":3202},[15,20332],{"className":20333,"style":358},[357],[15,20335,20337],{"className":20336},[362,363,364,365],[15,20338,278],{"className":20339},[61,365],[15,20341,497],{"className":20342},[496],[15,20344,20346],{"className":20345},[346],[15,20347,20349],{"className":20348,"style":504},[350],[15,20350],{}," 是一个纯粹的集中度（或称精度、强度）参数：",[882,20353,20355],{"title":20354},"命题 2.2（集中性极限）",[11,20356,20357,20358,20473,20474,20670,20671,20764,20765,20866,20867,20901,20902,867],{},"固定 ",[15,20359,20361,20393],{"className":20360,"translate":19},[18],[15,20362,20364],{"className":20363},[23],[25,20365,20366],{"xmlns":27},[29,20367,20368,20390],{},[32,20369,20370,20372,20374,20376,20378],{},[35,20371,18335],{"mathvariant":3101},[81,20373,83],{},[35,20375,1542],{"mathvariant":231},[81,20377,1545],{},[227,20379,20380,20382],{},[35,20381,232],{"mathvariant":231},[32,20383,20384,20386,20388],{},[35,20385,102],{},[81,20387,239],{},[89,20389,91],{},[40,20391,20392],{"encoding":42},"\\boldsymbol{\\mu} \\in \\operatorname{int}\\Delta^{K-1}",[15,20394,20396,20420],{"className":20395,"ariaHidden":48},[47],[15,20397,20399,20402,20411,20414,20417],{"className":20398},[52],[15,20400],{"className":20401,"style":118},[56],[15,20403,20405],{"className":20404},[61],[15,20406,20408],{"className":20407},[61],[15,20409,18335],{"className":20410},[61,3162],[15,20412],{"className":20413,"style":128},[127],[15,20415,83],{"className":20416},[132],[15,20418],{"className":20419,"style":128},[127],[15,20421,20423,20426,20432,20435],{"className":20422},[52],[15,20424],{"className":20425,"style":695},[56],[15,20427,20429],{"className":20428},[531],[15,20430,1542],{"className":20431},[61,1620],[15,20433],{"className":20434,"style":157},[127],[15,20436,20438,20441],{"className":20437},[61],[15,20439,232],{"className":20440},[61],[15,20442,20444],{"className":20443},[338],[15,20445,20447],{"className":20446},[342],[15,20448,20450],{"className":20449},[346],[15,20451,20453],{"className":20452,"style":695},[350],[15,20454,20455,20458],{"style":716},[15,20456],{"className":20457,"style":358},[357],[15,20459,20461],{"className":20460},[362,363,364,365],[15,20462,20464,20467,20470],{"className":20463},[61,365],[15,20465,102],{"className":20466,"style":174},[61,122,365],[15,20468,239],{"className":20469},[375,365],[15,20471,91],{"className":20472},[61,365],"，令 ",[15,20475,20477,20515],{"className":20476,"translate":19},[18],[15,20478,20480],{"className":20479},[23],[25,20481,20482],{"xmlns":27},[29,20483,20484,20512],{},[32,20485,20486,20502,20504,20510],{},[227,20487,20488,20490],{},[35,20489,3102],{"mathvariant":3101},[32,20491,20492,20494,20500],{},[81,20493,741],{"stretchy":86},[266,20495,20496,20498],{},[35,20497,3102],{},[89,20499,278],{},[81,20501,750],{"stretchy":86},[81,20503,244],{},[266,20505,20506,20508],{},[35,20507,3102],{},[89,20509,278],{},[35,20511,18335],{"mathvariant":3101},[40,20513,20514],{"encoding":42},"\\boldsymbol{\\alpha}^{(\\alpha_0)} = \\alpha_0 \\boldsymbol{\\mu}",[15,20516,20518,20615],{"className":20517,"ariaHidden":48},[47],[15,20519,20521,20525,20606,20609,20612],{"className":20520},[52],[15,20522],{"className":20523,"style":20524},[56],"height:0.888em;",[15,20526,20528,20537],{"className":20527},[61],[15,20529,20531],{"className":20530},[61],[15,20532,20534],{"className":20533},[61],[15,20535,3102],{"className":20536},[61,3162],[15,20538,20540],{"className":20539},[338],[15,20541,20543],{"className":20542},[342],[15,20544,20546],{"className":20545},[346],[15,20547,20549],{"className":20548,"style":20524},[350],[15,20550,20551,20554],{"style":716},[15,20552],{"className":20553,"style":358},[357],[15,20555,20557],{"className":20556},[362,363,364,365],[15,20558,20560,20563,20603],{"className":20559},[61,365],[15,20561,741],{"className":20562},[146,365],[15,20564,20566,20569],{"className":20565},[61,365],[15,20567,3102],{"className":20568,"style":3187},[61,122,365],[15,20570,20572],{"className":20571},[338],[15,20573,20575,20595],{"className":20574},[342,472],[15,20576,20578,20592],{"className":20577},[346],[15,20579,20581],{"className":20580,"style":4421},[350],[15,20582,20583,20586],{"style":4054},[15,20584],{"className":20585,"style":4058},[357],[15,20587,20589],{"className":20588},[362,4062,4063,365],[15,20590,278],{"className":20591},[61,365],[15,20593,497],{"className":20594},[496],[15,20596,20598],{"className":20597},[346],[15,20599,20601],{"className":20600,"style":4076},[350],[15,20602],{},[15,20604,750],{"className":20605},[178,365],[15,20607],{"className":20608,"style":128},[127],[15,20610,244],{"className":20611},[132],[15,20613],{"className":20614,"style":128},[127],[15,20616,20618,20621,20661],{"className":20617},[52],[15,20619],{"className":20620,"style":829},[56],[15,20622,20624,20627],{"className":20623},[61],[15,20625,3102],{"className":20626,"style":3187},[61,122],[15,20628,20630],{"className":20629},[338],[15,20631,20633,20653],{"className":20632},[342,472],[15,20634,20636,20650],{"className":20635},[346],[15,20637,20639],{"className":20638,"style":1212},[350],[15,20640,20641,20644],{"style":3202},[15,20642],{"className":20643,"style":358},[357],[15,20645,20647],{"className":20646},[362,363,364,365],[15,20648,278],{"className":20649},[61,365],[15,20651,497],{"className":20652},[496],[15,20654,20656],{"className":20655},[346],[15,20657,20659],{"className":20658,"style":504},[350],[15,20660],{},[15,20662,20664],{"className":20663},[61],[15,20665,20667],{"className":20666},[61],[15,20668,18335],{"className":20669},[61,3162],"。则当 ",[15,20672,20674,20697],{"className":20673,"translate":19},[18],[15,20675,20677],{"className":20676},[23],[25,20678,20679],{"xmlns":27},[29,20680,20681,20694],{},[32,20682,20683,20689,20692],{},[266,20684,20685,20687],{},[35,20686,3102],{},[89,20688,278],{},[81,20690,20691],{},"→",[35,20693,6790],{"mathvariant":231},[40,20695,20696],{"encoding":42},"\\alpha_0 \\to \\infty",[15,20698,20700,20755],{"className":20699,"ariaHidden":48},[47],[15,20701,20703,20706,20746,20749,20752],{"className":20702},[52],[15,20704],{"className":20705,"style":2509},[56],[15,20707,20709,20712],{"className":20708},[61],[15,20710,3102],{"className":20711,"style":3187},[61,122],[15,20713,20715],{"className":20714},[338],[15,20716,20718,20738],{"className":20717},[342,472],[15,20719,20721,20735],{"className":20720},[346],[15,20722,20724],{"className":20723,"style":1212},[350],[15,20725,20726,20729],{"style":3202},[15,20727],{"className":20728,"style":358},[357],[15,20730,20732],{"className":20731},[362,363,364,365],[15,20733,278],{"className":20734},[61,365],[15,20736,497],{"className":20737},[496],[15,20739,20741],{"className":20740},[346],[15,20742,20744],{"className":20743,"style":504},[350],[15,20745],{},[15,20747],{"className":20748,"style":128},[127],[15,20750,20691],{"className":20751},[132],[15,20753],{"className":20754,"style":128},[127],[15,20756,20758,20761],{"className":20757},[52],[15,20759],{"className":20760,"style":6026},[56],[15,20762,6790],{"className":20763},[61]," 时，",[15,20766,20768,20796],{"className":20767,"translate":19},[18],[15,20769,20771],{"className":20770},[23],[25,20772,20773],{"xmlns":27},[29,20774,20775,20793],{},[32,20776,20777,20779,20781,20783,20789,20791],{},[35,20778,4182],{"mathvariant":231},[81,20780,1545],{},[81,20782,741],{"stretchy":86},[266,20784,20785,20787],{},[35,20786,3102],{},[89,20788,278],{},[35,20790,18335],{"mathvariant":3101},[81,20792,750],{"stretchy":86},[40,20794,20795],{"encoding":42},"\\operatorname{Dir}(\\alpha_0 \\boldsymbol{\\mu})",[15,20797,20799],{"className":20798,"ariaHidden":48},[47],[15,20800,20802,20805,20811,20814,20854,20863],{"className":20801},[52],[15,20803],{"className":20804,"style":142},[56],[15,20806,20808],{"className":20807},[531],[15,20809,4182],{"className":20810},[61,1620],[15,20812,741],{"className":20813},[146],[15,20815,20817,20820],{"className":20816},[61],[15,20818,3102],{"className":20819,"style":3187},[61,122],[15,20821,20823],{"className":20822},[338],[15,20824,20826,20846],{"className":20825},[342,472],[15,20827,20829,20843],{"className":20828},[346],[15,20830,20832],{"className":20831,"style":1212},[350],[15,20833,20834,20837],{"style":3202},[15,20835],{"className":20836,"style":358},[357],[15,20838,20840],{"className":20839},[362,363,364,365],[15,20841,278],{"className":20842},[61,365],[15,20844,497],{"className":20845},[496],[15,20847,20849],{"className":20848},[346],[15,20850,20852],{"className":20851,"style":504},[350],[15,20853],{},[15,20855,20857],{"className":20856},[61],[15,20858,20860],{"className":20859},[61],[15,20861,18335],{"className":20862},[61,3162],[15,20864,750],{"className":20865},[178]," 弱收敛于位于 ",[15,20868,20870,20883],{"className":20869,"translate":19},[18],[15,20871,20873],{"className":20872},[23],[25,20874,20875],{"xmlns":27},[29,20876,20877,20881],{},[32,20878,20879],{},[35,20880,18335],{"mathvariant":3101},[40,20882,19440],{"encoding":42},[15,20884,20886],{"className":20885,"ariaHidden":48},[47],[15,20887,20889,20892],{"className":20888},[52],[15,20890],{"className":20891,"style":829},[56],[15,20893,20895],{"className":20894},[61],[15,20896,20898],{"className":20897},[61],[15,20899,18335],{"className":20900},[61,3162]," 的退化测度 ",[15,20903,20905,20924],{"className":20904,"translate":19},[18],[15,20906,20908],{"className":20907},[23],[25,20909,20910],{"xmlns":27},[29,20911,20912,20921],{},[32,20913,20914],{},[266,20915,20916,20919],{},[35,20917,20918],{},"δ",[35,20920,18335],{"mathvariant":3101},[40,20922,20923],{"encoding":42},"\\delta_{\\boldsymbol{\\mu}}",[15,20925,20927],{"className":20926,"ariaHidden":48},[47],[15,20928,20930,20934],{"className":20929},[52],[15,20931],{"className":20932,"style":20933},[56],"height:0.9805em;vertical-align:-0.2861em;",[15,20935,20937,20941],{"className":20936},[61],[15,20938,20918],{"className":20939,"style":20940},[61,122],"margin-right:0.0379em;",[15,20942,20944],{"className":20943},[338],[15,20945,20947,20978],{"className":20946},[342,472],[15,20948,20950,20975],{"className":20949},[346],[15,20951,20954],{"className":20952,"style":20953},[350],"height:0.1611em;",[15,20955,20957,20960],{"style":20956},"top:-2.55em;margin-left:-0.0379em;margin-right:0.05em;",[15,20958],{"className":20959,"style":358},[357],[15,20961,20963],{"className":20962},[362,363,364,365],[15,20964,20966],{"className":20965},[61,365],[15,20967,20969],{"className":20968},[61,365],[15,20970,20972],{"className":20971},[61,365],[15,20973,18335],{"className":20974},[61,3162,365],[15,20976,497],{"className":20977},[496],[15,20979,20981],{"className":20980},[346],[15,20982,20984],{"className":20983,"style":15696},[350],[15,20985],{},[11,20987,20988,20990,20991,16188,21063,21205,21206,21258,21259,21471,21472,867],{},[661,20989,5117],{}," 由命题 2.1，",[15,20992,20994,21018],{"className":20993,"translate":19},[18],[15,20995,20997],{"className":20996},[23],[25,20998,20999],{"xmlns":27},[29,21000,21001,21015],{},[32,21002,21003,21005,21007,21009,21011,21013],{},[35,21004,14807],{"mathvariant":258},[81,21006,14810],{"stretchy":86},[35,21008,1027],{"mathvariant":251},[81,21010,14819],{"stretchy":86},[81,21012,244],{},[35,21014,18335],{"mathvariant":3101},[40,21016,21017],{"encoding":42},"\\mathbb{E}[\\mathbf{x}] = \\boldsymbol{\\mu}",[15,21019,21021,21048],{"className":21020,"ariaHidden":48},[47],[15,21022,21024,21027,21030,21033,21036,21039,21042,21045],{"className":21023},[52],[15,21025],{"className":21026,"style":142},[56],[15,21028,14807],{"className":21029},[61,427],[15,21031,14810],{"className":21032},[146],[15,21034,1027],{"className":21035},[61,411],[15,21037,14819],{"className":21038},[178],[15,21040],{"className":21041,"style":128},[127],[15,21043,244],{"className":21044},[132],[15,21046],{"className":21047,"style":128},[127],[15,21049,21051,21054],{"className":21050},[52],[15,21052],{"className":21053,"style":829},[56],[15,21055,21057],{"className":21056},[61],[15,21058,21060],{"className":21059},[61],[15,21061,18335],{"className":21062},[61,3162],[15,21064,21066,21102],{"className":21065,"translate":19},[18],[15,21067,21069],{"className":21068},[23],[25,21070,21071],{"xmlns":27},[29,21072,21073,21099],{},[32,21074,21075,21077,21079,21081,21083,21085,21087,21093,21095,21097],{},[35,21076,18534],{"mathvariant":251},[81,21078,244],{},[35,21080,12256],{},[81,21082,741],{"stretchy":86},[89,21084,91],{},[35,21086,3018],{"mathvariant":231},[266,21088,21089,21091],{},[35,21090,3102],{},[89,21092,278],{},[81,21094,750],{"stretchy":86},[81,21096,20691],{},[89,21098,278],{"mathvariant":251},[40,21100,21101],{"encoding":42},"\\boldsymbol{\\Sigma} = O(1\u002F\\alpha_0) \\to \\mathbf{0}",[15,21103,21105,21129,21196],{"className":21104,"ariaHidden":48},[47],[15,21106,21108,21111,21120,21123,21126],{"className":21107},[52],[15,21109],{"className":21110,"style":18605},[56],[15,21112,21114],{"className":21113},[61],[15,21115,21117],{"className":21116},[61],[15,21118,18534],{"className":21119},[61,411],[15,21121],{"className":21122,"style":128},[127],[15,21124,244],{"className":21125},[132],[15,21127],{"className":21128,"style":128},[127],[15,21130,21132,21135,21138,21141,21144,21184,21187,21190,21193],{"className":21131},[52],[15,21133],{"className":21134,"style":142},[56],[15,21136,12256],{"className":21137,"style":12278},[61,122],[15,21139,741],{"className":21140},[146],[15,21142,3047],{"className":21143},[61],[15,21145,21147,21150],{"className":21146},[61],[15,21148,3102],{"className":21149,"style":3187},[61,122],[15,21151,21153],{"className":21152},[338],[15,21154,21156,21176],{"className":21155},[342,472],[15,21157,21159,21173],{"className":21158},[346],[15,21160,21162],{"className":21161,"style":1212},[350],[15,21163,21164,21167],{"style":3202},[15,21165],{"className":21166,"style":358},[357],[15,21168,21170],{"className":21169},[362,363,364,365],[15,21171,278],{"className":21172},[61,365],[15,21174,497],{"className":21175},[496],[15,21177,21179],{"className":21178},[346],[15,21180,21182],{"className":21181,"style":504},[350],[15,21183],{},[15,21185,750],{"className":21186},[178],[15,21188],{"className":21189,"style":128},[127],[15,21191,20691],{"className":21192},[132],[15,21194],{"className":21195,"style":128},[127],[15,21197,21199,21202],{"className":21198},[52],[15,21200],{"className":21201,"style":859},[56],[15,21203,278],{"className":21204},[61,411],"。对任意 ",[15,21207,21209,21228],{"className":21208,"translate":19},[18],[15,21210,21212],{"className":21211},[23],[25,21213,21214],{"xmlns":27},[29,21215,21216,21225],{},[32,21217,21218,21221,21223],{},[35,21219,21220],{},"ε",[81,21222,1588],{},[89,21224,278],{},[40,21226,21227],{"encoding":42},"\\varepsilon > 0",[15,21229,21231,21249],{"className":21230,"ariaHidden":48},[47],[15,21232,21234,21237,21240,21243,21246],{"className":21233},[52],[15,21235],{"className":21236,"style":4650},[56],[15,21238,21220],{"className":21239},[61,122],[15,21241],{"className":21242,"style":128},[127],[15,21244,1588],{"className":21245},[132],[15,21247],{"className":21248,"style":128},[127],[15,21250,21252,21255],{"className":21251},[52],[15,21253],{"className":21254,"style":859},[56],[15,21256,278],{"className":21257},[61],"，由 Chebyshev 不等式 ",[15,21260,21262,21318],{"className":21261,"translate":19},[18],[15,21263,21265],{"className":21264},[23],[25,21266,21267],{"xmlns":27},[29,21268,21269,21315],{},[32,21270,21271,21274,21276,21279,21281,21283,21285,21287,21289,21291,21293,21296,21299,21301,21303,21305,21311,21313],{},[35,21272,21273],{"mathvariant":258},"P",[81,21275,741],{"stretchy":86},[35,21277,21278],{"mathvariant":231},"∥",[35,21280,1027],{"mathvariant":251},[81,21282,239],{},[35,21284,18335],{"mathvariant":3101},[35,21286,21278],{"mathvariant":231},[81,21288,1588],{},[35,21290,21220],{},[81,21292,750],{"stretchy":86},[81,21294,21295],{},"≤",[35,21297,21298],{"mathvariant":231},"tr",[81,21300,1545],{},[35,21302,18534],{"mathvariant":251},[35,21304,3018],{"mathvariant":231},[227,21306,21307,21309],{},[35,21308,21220],{},[89,21310,906],{},[81,21312,20691],{},[89,21314,278],{},[40,21316,21317],{"encoding":42},"\\mathbb{P}(\\|\\mathbf{x} - \\boldsymbol{\\mu}\\| > \\varepsilon) \\le \\operatorname{tr}\\boldsymbol{\\Sigma}\u002F\\varepsilon^2 \\to 0",[15,21319,21321,21348,21375,21396,21462],{"className":21320,"ariaHidden":48},[47],[15,21322,21324,21327,21330,21333,21336,21339,21342,21345],{"className":21323},[52],[15,21325],{"className":21326,"style":142},[56],[15,21328,21273],{"className":21329},[61,427],[15,21331,741],{"className":21332},[146],[15,21334,21278],{"className":21335},[61],[15,21337,1027],{"className":21338},[61,411],[15,21340],{"className":21341,"style":772},[127],[15,21343,239],{"className":21344},[375],[15,21346],{"className":21347,"style":772},[127],[15,21349,21351,21354,21363,21366,21369,21372],{"className":21350},[52],[15,21352],{"className":21353,"style":142},[56],[15,21355,21357],{"className":21356},[61],[15,21358,21360],{"className":21359},[61],[15,21361,18335],{"className":21362},[61,3162],[15,21364,21278],{"className":21365},[61],[15,21367],{"className":21368,"style":128},[127],[15,21370,1588],{"className":21371},[132],[15,21373],{"className":21374,"style":128},[127],[15,21376,21378,21381,21384,21387,21390,21393],{"className":21377},[52],[15,21379],{"className":21380,"style":142},[56],[15,21382,21220],{"className":21383},[61,122],[15,21385,750],{"className":21386},[178],[15,21388],{"className":21389,"style":128},[127],[15,21391,21295],{"className":21392},[132],[15,21394],{"className":21395,"style":128},[127],[15,21397,21399,21403,21409,21412,21421,21424,21453,21456,21459],{"className":21398},[52],[15,21400],{"className":21401,"style":21402},[56],"height:1.0641em;vertical-align:-0.25em;",[15,21404,21406],{"className":21405},[531],[15,21407,21298],{"className":21408},[61,1620],[15,21410],{"className":21411,"style":157},[127],[15,21413,21415],{"className":21414},[61],[15,21416,21418],{"className":21417},[61],[15,21419,18534],{"className":21420},[61,411],[15,21422,3018],{"className":21423},[61],[15,21425,21427,21430],{"className":21426},[61],[15,21428,21220],{"className":21429},[61,122],[15,21431,21433],{"className":21432},[338],[15,21434,21436],{"className":21435},[342],[15,21437,21439],{"className":21438},[346],[15,21440,21442],{"className":21441,"style":7899},[350],[15,21443,21444,21447],{"style":716},[15,21445],{"className":21446,"style":358},[357],[15,21448,21450],{"className":21449},[362,363,364,365],[15,21451,906],{"className":21452},[61,365],[15,21454],{"className":21455,"style":128},[127],[15,21457,20691],{"className":21458},[132],[15,21460],{"className":21461,"style":128},[127],[15,21463,21465,21468],{"className":21464},[52],[15,21466],{"className":21467,"style":859},[56],[15,21469,278],{"className":21470},[61],"，故依概率收敛，进而弱收敛于 ",[15,21473,21475,21493],{"className":21474,"translate":19},[18],[15,21476,21478],{"className":21477},[23],[25,21479,21480],{"xmlns":27},[29,21481,21482,21490],{},[32,21483,21484],{},[266,21485,21486,21488],{},[35,21487,20918],{},[35,21489,18335],{"mathvariant":3101},[40,21491,21492],{"encoding":42},"\\delta_{\\boldsymbol\\mu}",[15,21494,21496],{"className":21495,"ariaHidden":48},[47],[15,21497,21499,21502],{"className":21498},[52],[15,21500],{"className":21501,"style":20933},[56],[15,21503,21505,21508],{"className":21504},[61],[15,21506,20918],{"className":21507,"style":20940},[61,122],[15,21509,21511],{"className":21510},[338],[15,21512,21514,21543],{"className":21513},[342,472],[15,21515,21517,21540],{"className":21516},[346],[15,21518,21520],{"className":21519,"style":20953},[350],[15,21521,21522,21525],{"style":20956},[15,21523],{"className":21524,"style":358},[357],[15,21526,21528],{"className":21527},[362,363,364,365],[15,21529,21531],{"className":21530},[61,365],[15,21532,21534],{"className":21533},[61,365],[15,21535,21537],{"className":21536},[61,365],[15,21538,18335],{"className":21539},[61,3162,365],[15,21541,497],{"className":21542},[496],[15,21544,21546],{"className":21545},[346],[15,21547,21549],{"className":21548,"style":15696},[350],[15,21550],{},[11,21552,21553,21554,21670],{},"这一命题给出了本文后续所有「不确定性」语汇的几何基础：",[661,21555,21556,21557,21634,21635,21669],{},"迪利克雷分布的「模糊程度」等于 ",[15,21558,21560,21582],{"className":21559,"translate":19},[18],[15,21561,21563],{"className":21562},[23],[25,21564,21565],{"xmlns":27},[29,21566,21567,21579],{},[32,21568,21569,21571,21573],{},[89,21570,91],{},[35,21572,3018],{"mathvariant":231},[266,21574,21575,21577],{},[35,21576,3102],{},[89,21578,278],{},[40,21580,21581],{"encoding":42},"1\u002F\\alpha_0",[15,21583,21585],{"className":21584,"ariaHidden":48},[47],[15,21586,21588,21591,21594],{"className":21587},[52],[15,21589],{"className":21590,"style":142},[56],[15,21592,3047],{"className":21593},[61],[15,21595,21597,21600],{"className":21596},[61],[15,21598,3102],{"className":21599,"style":3187},[61,122],[15,21601,21603],{"className":21602},[338],[15,21604,21606,21626],{"className":21605},[342,472],[15,21607,21609,21623],{"className":21608},[346],[15,21610,21612],{"className":21611,"style":1212},[350],[15,21613,21614,21617],{"style":3202},[15,21615],{"className":21616,"style":358},[357],[15,21618,21620],{"className":21619},[362,363,364,365],[15,21621,278],{"className":21622},[61,365],[15,21624,497],{"className":21625},[496],[15,21627,21629],{"className":21628},[346],[15,21630,21632],{"className":21631,"style":504},[350],[15,21633],{}," 的量级，而它的「中心」由 ",[15,21636,21638,21651],{"className":21637,"translate":19},[18],[15,21639,21641],{"className":21640},[23],[25,21642,21643],{"xmlns":27},[29,21644,21645,21649],{},[32,21646,21647],{},[35,21648,18335],{"mathvariant":3101},[40,21650,19440],{"encoding":42},[15,21652,21654],{"className":21653,"ariaHidden":48},[47],[15,21655,21657,21660],{"className":21656},[52],[15,21658],{"className":21659,"style":829},[56],[15,21661,21663],{"className":21662},[61],[15,21664,21666],{"className":21665},[61],[15,21667,18335],{"className":21668},[61,3162]," 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。",[877,21672,21674],{"id":21673},"_23-众数与熵","2.3 众数与熵",[882,21676,21678],{"title":21677},"命题 2.3（众数）",[11,21679,21680,21681,21773,21774,21803,21804,21876,21877,21950,21951],{},"若 ",[15,21682,21684,21706],{"className":21683,"translate":19},[18],[15,21685,21687],{"className":21686},[23],[25,21688,21689],{"xmlns":27},[29,21690,21691,21703],{},[32,21692,21693,21699,21701],{},[266,21694,21695,21697],{},[35,21696,3102],{},[35,21698,1068],{},[81,21700,1588],{},[89,21702,91],{},[40,21704,21705],{"encoding":42},"\\alpha_i > 1",[15,21707,21709,21764],{"className":21708,"ariaHidden":48},[47],[15,21710,21712,21715,21755,21758,21761],{"className":21711},[52],[15,21713],{"className":21714,"style":1751},[56],[15,21716,21718,21721],{"className":21717},[61],[15,21719,3102],{"className":21720,"style":3187},[61,122],[15,21722,21724],{"className":21723},[338],[15,21725,21727,21747],{"className":21726},[342,472],[15,21728,21730,21744],{"className":21729},[346],[15,21731,21733],{"className":21732,"style":1362},[350],[15,21734,21735,21738],{"style":3202},[15,21736],{"className":21737,"style":358},[357],[15,21739,21741],{"className":21740},[362,363,364,365],[15,21742,1068],{"className":21743},[61,122,365],[15,21745,497],{"className":21746},[496],[15,21748,21750],{"className":21749},[346],[15,21751,21753],{"className":21752,"style":504},[350],[15,21754],{},[15,21756],{"className":21757,"style":128},[127],[15,21759,1588],{"className":21760},[132],[15,21762],{"className":21763,"style":128},[127],[15,21765,21767,21770],{"className":21766},[52],[15,21768],{"className":21769,"style":859},[56],[15,21771,91],{"className":21772},[61]," 对一切 ",[15,21775,21777,21790],{"className":21776,"translate":19},[18],[15,21778,21780],{"className":21779},[23],[25,21781,21782],{"xmlns":27},[29,21783,21784,21788],{},[32,21785,21786],{},[35,21787,1068],{},[40,21789,1068],{"encoding":42},[15,21791,21793],{"className":21792,"ariaHidden":48},[47],[15,21794,21796,21800],{"className":21795},[52],[15,21797],{"className":21798,"style":21799},[56],"height:0.6595em;",[15,21801,1068],{"className":21802},[61,122]," 成立，则 ",[15,21805,21807,21831],{"className":21806,"translate":19},[18],[15,21808,21810],{"className":21809},[23],[25,21811,21812],{"xmlns":27},[29,21813,21814,21828],{},[32,21815,21816,21818,21820,21822,21824,21826],{},[35,21817,3703],{},[81,21819,741],{"stretchy":86},[35,21821,1027],{"mathvariant":251},[81,21823,3710],{},[35,21825,3102],{"mathvariant":3101},[81,21827,750],{"stretchy":86},[40,21829,21830],{"encoding":42},"f(\\mathbf{x} \\mid 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",[15,21878,21880,21903],{"className":21879,"translate":19},[18],[15,21881,21883],{"className":21882},[23],[25,21884,21885],{"xmlns":27},[29,21886,21887,21901],{},[32,21888,21889],{},[227,21890,21891,21893],{},[35,21892,232],{"mathvariant":231},[32,21894,21895,21897,21899],{},[35,21896,102],{},[81,21898,239],{},[89,21900,91],{},[40,21902,1856],{"encoding":42},[15,21904,21906],{"className":21905,"ariaHidden":48},[47],[15,21907,21909,21912],{"className":21908},[52],[15,21910],{"className":21911,"style":695},[56],[15,21913,21915,21918],{"className":21914},[61],[15,21916,232],{"className":21917},[61],[15,21919,21921],{"className":21920},[338],[15,21922,21924],{"className":21923},[342],[15,21925,21927],{"className":21926},[346],[15,21928,21930],{"className":21929,"style":695},[350],[15,21931,21932,21935],{"style":716},[15,21933],{"className":21934,"style":358},[357],[15,21936,21938],{"className":21937},[362,363,364,365],[15,21939,21941,21944,21947],{"className":21940},[61,365],[15,21942,102],{"className":21943,"style":174},[61,122,365],[15,21945,239],{"className":21946},[375,365],[15,21948,91],{"className":21949},[61,365]," 的内部有唯一众数\n",[15,21952,21954,22039],{"className":21953,"translate":19},[18],[15,21955,21957],{"className":21956},[23],[25,21958,21959],{"xmlns":27},[29,21960,21961,22036],{},[32,21962,21963,21970,21972],{},[227,21964,21965,21967],{},[35,21966,1027],{"mathvariant":251},[81,21968,21969],{},"⋆",[81,21971,244],{},[32,21973,21974,21976,22002,22004,22006,22008,22034],{},[81,21975,741],{"fence":48},[3718,21977,21978,21990],{},[32,21979,21980,21986,21988],{},[266,21981,21982,21984],{},[35,21983,3102],{},[89,21985,91],{},[81,21987,239],{},[89,21989,91],{},[32,21991,21992,21998,22000],{},[266,21993,21994,21996],{},[35,21995,3102],{},[89,21997,278],{},[81,21999,239],{},[35,22001,102],{},[81,22003,94],{"separator":48},[81,22005,97],{},[81,22007,94],{"separator":48},[3718,22009,22010,22022],{},[32,22011,22012,22018,22020],{},[266,22013,22014,22016],{},[35,22015,3102],{},[35,22017,102],{},[81,22019,239],{},[89,22021,91],{},[32,22023,22024,22030,22032],{},[266,22025,22026,22028],{},[35,22027,3102],{},[89,22029,278],{},[81,22031,239],{},[35,22033,102],{},[81,22035,750],{"fence":48},[40,22037,22038],{"encoding":42},"\\mathbf{x}^\\star = \\left( \\frac{\\alpha_1 - 1}{\\alpha_0 - K}, \\dots, \\frac{\\alpha_K - 1}{\\alpha_0 - K} \\right)",[15,22040,22042,22087],{"className":22041,"ariaHidden":48},[47],[15,22043,22045,22049,22078,22081,22084],{"className":22044},[52],[15,22046],{"className":22047,"style":22048},[56],"height:0.6887em;",[15,22050,22052,22055],{"className":22051},[61],[15,22053,1027],{"className":22054},[61,411],[15,22056,22058],{"className":22057},[338],[15,22059,22061],{"className":22060},[342],[15,22062,22064],{"className":22063},[346],[15,22065,22067],{"className":22066,"style":22048},[350],[15,22068,22069,22072],{"style":716},[15,22070],{"className":22071,"style":358},[357],[15,22073,22075],{"className":22074},[362,363,364,365],[15,22076,21969],{"className":22077},[375,365],[15,22079],{"className":22080,"style":128},[127],[15,22082,244],{"className":22083},[132],[15,22085],{"className":22086,"style":128},[127],[15,22088,22090,22093],{"className":22089},[52],[15,22091],{"className":22092,"style":394},[56],[15,22094,22096,22102,22257,22260,22263,22266,22269,22272,22275,22434],{"className":22095},[161],[15,22097,22099],{"className":22098,"style":402},[146,401],[15,22100,741],{"className":22101},[406,407],[15,22103,22105,22108,22254],{"className":22104},[61],[15,22106],{"className":22107},[146,3837],[15,22109,22111],{"className":22110},[3718],[15,22112,22114,22246],{"className":22113},[342,472],[15,22115,22117,22243],{"className":22116},[346],[15,22118,22121,22178,22186],{"className":22119,"style":22120},[350],"height:0.8612em;",[15,22122,22123,22126],{"style":3853},[15,22124],{"className":22125,"style":3857},[357],[15,22127,22129],{"className":22128},[362,363,364,365],[15,22130,22132,22172,22175],{"className":22131},[61,365],[15,22133,22135,22138],{"className":22134},[61,365],[15,22136,3102],{"className":22137,"style":3187},[61,122,365],[15,22139,22141],{"className":22140},[338],[15,22142,22144,22164],{"className":22143},[342,472],[15,22145,22147,22161],{"className":22146},[346],[15,22148,22150],{"className":22149,"style":4421},[350],[15,22151,22152,22155],{"style":4054},[15,22153],{"className":22154,"style":4058},[357],[15,22156,22158],{"className":22157},[362,4062,4063,365],[15,22159,278],{"className":22160},[61,365],[15,22162,497],{"className":22163},[496],[15,22165,22167],{"className":22166},[346],[15,22168,22170],{"className":22169,"style":4076},[350],[15,22171],{},[15,22173,239],{"className":22174},[375,365],[15,22176,102],{"className":22177,"style":174},[61,122,365],[15,22179,22180,22183],{"style":3885},[15,22181],{"className":22182,"style":3857},[357],[15,22184],{"className":22185,"style":3893},[3892],[15,22187,22188,22191],{"style":15150},[15,22189],{"className":22190,"style":3857},[357],[15,22192,22194],{"className":22193},[362,363,364,365],[15,22195,22197,22237,22240],{"className":22196},[61,365],[15,22198,22200,22203],{"className":22199},[61,365],[15,22201,3102],{"className":22202,"style":3187},[61,122,365],[15,22204,22206],{"className":22205},[338],[15,22207,22209,22229],{"className":22208},[342,472],[15,22210,22212,22226],{"className":22211},[346],[15,22213,22215],{"className":22214,"style":4421},[350],[15,22216,22217,22220],{"style":4054},[15,22218],{"className":22219,"style":4058},[357],[15,22221,22223],{"className":22222},[362,4062,4063,365],[15,22224,91],{"className":22225},[61,365],[15,22227,497],{"className":22228},[496],[15,22230,22232],{"className":22231},[346],[15,22233,22235],{"className":22234,"style":4076},[350],[15,22236],{},[15,22238,239],{"className":22239},[375,365],[15,22241,91],{"className":22242},[61,365],[15,22244,497],{"className":22245},[496],[15,22247,22249],{"className":22248},[346],[15,22250,22252],{"className":22251,"style":15209},[350],[15,22253],{},[15,22255],{"className":22256},[178,3837],[15,22258,94],{"className":22259},[153],[15,22261],{"className":22262,"style":157},[127],[15,22264,97],{"className":22265},[161],[15,22267],{"className":22268,"style":157},[127],[15,22270,94],{"className":22271},[153],[15,22273],{"className":22274,"style":157},[127],[15,22276,22278,22281,22431],{"className":22277},[61],[15,22279],{"className":22280},[146,3837],[15,22282,22284],{"className":22283},[3718],[15,22285,22287,22423],{"className":22286},[342,472],[15,22288,22290,22420],{"className":22289},[346],[15,22291,22294,22351,22359],{"className":22292,"style":22293},[350],"height:0.8614em;",[15,22295,22296,22299],{"style":3853},[15,22297],{"className":22298,"style":3857},[357],[15,22300,22302],{"className":22301},[362,363,364,365],[15,22303,22305,22345,22348],{"className":22304},[61,365],[15,22306,22308,22311],{"className":22307},[61,365],[15,22309,3102],{"className":22310,"style":3187},[61,122,365],[15,22312,22314],{"className":22313},[338],[15,22315,22317,22337],{"className":22316},[342,472],[15,22318,22320,22334],{"className":22319},[346],[15,22321,22323],{"className":22322,"style":4421},[350],[15,22324,22325,22328],{"style":4054},[15,22326],{"className":22327,"style":4058},[357],[15,22329,22331],{"className":22330},[362,4062,4063,365],[15,22332,278],{"className":22333},[61,365],[15,22335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",[15,22444,22446,22468],{"className":22445,"translate":19},[18],[15,22447,22449],{"className":22448},[23],[25,22450,22451],{"xmlns":27},[29,22452,22453,22465],{},[32,22454,22455,22461,22463],{},[266,22456,22457,22459],{},[35,22458,3102],{},[35,22460,1068],{},[81,22462,21295],{},[89,22464,91],{},[40,22466,22467],{"encoding":42},"\\alpha_i \\le 1",[15,22469,22471,22527],{"className":22470,"ariaHidden":48},[47],[15,22472,22474,22478,22518,22521,22524],{"className":22473},[52],[15,22475],{"className":22476,"style":22477},[56],"height:0.786em;vertical-align:-0.15em;",[15,22479,22481,22484],{"className":22480},[61],[15,22482,3102],{"className":22483,"style":3187},[61,122],[15,22485,22487],{"className":22486},[338],[15,22488,22490,22510],{"className":22489},[342,472],[15,22491,22493,22507],{"className":22492},[346],[15,22494,22496],{"className":22495,"style":1362},[350],[15,22497,22498,22501],{"style":3202},[15,22499],{"className":22500,"style":358},[357],[15,22502,22504],{"className":22503},[362,363,364,365],[15,22505,1068],{"className":22506},[61,122,365],[15,22508,497],{"className":22509},[496],[15,22511,22513],{"className":22512},[346],[15,22514,22516],{"className":22515,"style":504},[350],[15,22517],{},[15,22519],{"className":22520,"style":128},[127],[15,22522,21295],{"className":22523},[132],[15,22525],{"className":22526,"style":128},[127],[15,22528,22530,22533],{"className":22529},[52],[15,22531],{"className":22532,"style":859},[56],[15,22534,91],{"className":22535},[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。",[882,22538,22540],{"title":22539},"命题 2.4（微分熵）",[11,22541,22542,22543,22619,22620,23121,23122,23217],{},"在 1.1 节的测度约定下，",[15,22544,22546,22571],{"className":22545,"translate":19},[18],[15,22547,22549],{"className":22548},[23],[25,22550,22551],{"xmlns":27},[29,22552,22553,22569],{},[32,22554,22555,22557,22559,22561,22563,22565,22567],{},[35,22556,1027],{"mathvariant":251},[81,22558,4179],{},[35,22560,4182],{"mathvariant":231},[81,22562,1545],{},[81,22564,741],{"stretchy":86},[35,22566,3102],{"mathvariant":3101},[81,22568,750],{"stretchy":86},[40,22570,4193],{"encoding":42},[15,22572,22574,22592],{"className":22573,"ariaHidden":48},[47],[15,22575,22577,22580,22583,22586,22589],{"className":22576},[52],[15,22578],{"className":22579,"style":2706},[56],[15,22581,1027],{"className":22582},[61,411],[15,22584],{"className":22585,"style":128},[127],[15,22587,4179],{"className":22588},[132],[15,22590],{"className":22591,"style":128},[127],[15,22593,22595,22598,22604,22607,22616],{"className":22594},[52],[15,22596],{"className":22597,"style":142},[56],[15,22599,22601],{"className":22600},[531],[15,22602,4182],{"className":22603},[61,1620],[15,22605,741],{"className":22606},[146],[15,22608,22610],{"className":22609},[61],[15,22611,22613],{"className":22612},[61],[15,22614,3102],{"className":22615},[61,3162],[15,22617,750],{"className":22618},[178]," 的微分熵为\n",[15,22621,22623,22734],{"className":22622,"translate":19},[18],[15,22624,22626],{"className":22625},[23],[25,22627,22628],{"xmlns":27},[29,22629,22630,22731],{},[32,22631,22632,22635,22637,22639,22641,22643,22646,22648,22650,22652,22654,22656,22658,22660,22666,22668,22670,22672,22674,22677,22679,22685,22687,22689,22703,22705,22711,22713,22715,22717,22719,22721,22723,22729],{},[35,22633,22634],{},"H",[81,22636,741],{"stretchy":86},[35,22638,1027],{"mathvariant":251},[81,22640,750],{"stretchy":86},[81,22642,244],{},[35,22644,22645],{},"log",[81,22647,1545],{},[35,22649,3726],{},[81,22651,741],{"stretchy":86},[35,22653,3102],{"mathvariant":3101},[81,22655,750],{"stretchy":86},[81,22657,7756],{},[81,22659,741],{"stretchy":86},[266,22661,22662,22664],{},[35,22663,3102],{},[89,22665,278],{},[81,22667,239],{},[35,22669,102],{},[81,22671,750],{"stretchy":86},[282,22673,4796],{},[35,22675,22676],{},"ψ",[81,22678,741],{"stretchy":86},[266,22680,22681,22683],{},[35,22682,3102],{},[89,22684,278],{},[81,22686,750],{"stretchy":86},[81,22688,239],{},[289,22690,22691,22693,22701],{},[81,22692,293],{},[32,22694,22695,22697,22699],{},[35,22696,1068],{},[81,22698,244],{},[89,22700,91],{},[35,22702,102],{},[81,22704,741],{"stretchy":86},[266,22706,22707,22709],{},[35,22708,3102],{},[35,22710,1068],{},[81,22712,239],{},[89,22714,91],{},[81,22716,750],{"stretchy":86},[282,22718,4796],{},[35,22720,22676],{},[81,22722,741],{"stretchy":86},[266,22724,22725,22727],{},[35,22726,3102],{},[35,22728,1068],{},[81,22730,750],{"stretchy":86},[40,22732,22733],{"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)",[15,22735,22737,22765,22808,22866,22939,23057],{"className":22736,"ariaHidden":48},[47],[15,22738,22740,22743,22747,22750,22753,22756,22759,22762],{"className":22739},[52],[15,22741],{"className":22742,"style":142},[56],[15,22744,22634],{"className":22745,"style":22746},[61,122],"margin-right:0.0813em;",[15,22748,741],{"className":22749},[146],[15,22751,1027],{"className":22752},[61,411],[15,22754,750],{"className":22755},[178],[15,22757],{"className":22758,"style":128},[127],[15,22760,244],{"className":22761},[132],[15,22763],{"className":22764,"style":128},[127],[15,22766,22768,22771,22778,22781,22784,22787,22796,22799,22802,22805],{"className":22767},[52],[15,22769],{"className":22770,"style":142},[56],[15,22772,22774,22775],{"className":22773},[531],"lo",[15,22776,22777],{"style":15608},"g",[15,22779],{"className":22780,"style":157},[127],[15,22782,3726],{"className":22783,"style":3867},[61,122],[15,22785,741],{"className":22786},[146],[15,22788,22790],{"className":22789},[61],[15,22791,22793],{"className":22792},[61],[15,22794,3102],{"className":22795},[61,3162],[15,22797,750],{"className":22798},[178],[15,22800],{"className":22801,"style":772},[127],[15,22803,7756],{"className":22804},[375],[15,22806],{"className":22807,"style":772},[127],[15,22809,22811,22814,22817,22857,22860,22863],{"className":22810},[52],[15,22812],{"className":22813,"style":142},[56],[15,22815,741],{"className":22816},[146],[15,22818,22820,22823],{"className":22819},[61],[15,22821,3102],{"className":22822,"style":3187},[61,122],[15,22824,22826],{"className":22825},[338],[15,22827,22829,22849],{"className":22828},[342,472],[15,22830,22832,22846],{"className":22831},[346],[15,22833,22835],{"className":22834,"style":1212},[350],[15,22836,22837,22840],{"style":3202},[15,22838],{"className":22839,"style":358},[357],[15,22841,22843],{"className":22842},[362,363,364,365],[15,22844,278],{"className":22845},[61,365],[15,22847,497],{"className":22848},[496],[15,22850,22852],{"className":22851},[346],[15,22853,22855],{"className":22854,"style":504},[350],[15,22856],{},[15,22858],{"className":22859,"style":772},[127],[15,22861,239],{"className":22862},[375],[15,22864],{"className":22865,"style":772},[127],[15,22867,22869,22872,22875,22878,22881,22884,22887,22927,22930,22933,22936],{"className":22868},[52],[15,22870],{"className":22871,"style":142},[56],[15,22873,102],{"className":22874,"style":174},[61,122],[15,22876,750],{"className":22877},[178],[15,22879],{"className":22880,"style":157},[127],[15,22882,22676],{"className":22883,"style":123},[61,122],[15,22885,741],{"className":22886},[146],[15,22888,22890,22893],{"className":22889},[61],[15,22891,3102],{"className":22892,"style":3187},[61,122],[15,22894,22896],{"className":22895},[338],[15,22897,22899,22919],{"className":22898},[342,472],[15,22900,22902,22916],{"className":22901},[346],[15,22903,22905],{"className":22904,"style":1212},[350],[15,22906,22907,22910],{"style":3202},[15,22908],{"className":22909,"style":358},[357],[15,22911,22913],{"className":22912},[362,363,364,365],[15,22914,278],{"className":22915},[61,365],[15,22917,497],{"className":22918},[496],[15,22920,22922],{"className":22921},[346],[15,22923,22925],{"className":22924,"style":504},[350],[15,22926],{},[15,22928,750],{"className":22929},[178],[15,22931],{"className":22932,"style":772},[127],[15,22934,239],{"className":22935},[375],[15,22937],{"className":22938,"style":772},[127],[15,22940,22942,22945,23005,23008,23048,23051,23054],{"className":22941},[52],[15,22943],{"className":22944,"style":3467},[56],[15,22946,22948,22951],{"className":22947},[531],[15,22949,293],{"className":22950,"style":537},[531,535,536],[15,22952,22954],{"className":22953},[338],[15,22955,22957,22997],{"className":22956},[342,472],[15,22958,22960,22994],{"className":22959},[346],[15,22961,22963,22983],{"className":22962,"style":550},[350],[15,22964,22965,22968],{"style":553},[15,22966],{"className":22967,"style":358},[357],[15,22969,22971],{"className":22970},[362,363,364,365],[15,22972,22974,22977,22980],{"className":22973},[61,365],[15,22975,1068],{"className":22976},[61,122,365],[15,22978,244],{"className":22979},[132,365],[15,22981,91],{"className":22982},[61,365],[15,22984,22985,22988],{"style":574},[15,22986],{"className":22987,"style":358},[357],[15,22989,22991],{"className":22990},[362,363,364,365],[15,22992,102],{"className":22993,"style":174},[61,122,365],[15,22995,497],{"className":22996},[496],[15,22998,23000],{"className":22999},[346],[15,23001,23003],{"className":23002,"style":593},[350],[15,23004],{},[15,23006,741],{"className":23007},[146],[15,23009,23011,23014],{"className":23010},[61],[15,23012,3102],{"className":23013,"style":3187},[61,122],[15,23015,23017],{"className":23016},[338],[15,23018,23020,23040],{"className":23019},[342,472],[15,23021,23023,23037],{"className":23022},[346],[15,23024,23026],{"className":23025,"style":1362},[350],[15,23027,23028,23031],{"style":3202},[15,23029],{"className":23030,"style":358},[357],[15,23032,23034],{"className":23033},[362,363,364,365],[15,23035,1068],{"className":23036},[61,122,365],[15,23038,497],{"className":23039},[496],[15,23041,23043],{"className":23042},[346],[15,23044,23046],{"className":23045,"style":504},[350],[15,23047],{},[15,23049],{"className":23050,"style":772},[127],[15,23052,239],{"className":23053},[375],[15,23055],{"className":23056,"style":772},[127],[15,23058,23060,23063,23066,23069,23072,23075,23078,23118],{"className":23059},[52],[15,23061],{"className":23062,"style":142},[56],[15,23064,91],{"className":23065},[61],[15,23067,750],{"className":23068},[178],[15,23070],{"className":23071,"style":157},[127],[15,23073,22676],{"className":23074,"style":123},[61,122],[15,23076,741],{"className":23077},[146],[15,23079,23081,23084],{"className":23080},[61],[15,23082,3102],{"className":23083,"style":3187},[61,122],[15,23085,23087],{"className":23086},[338],[15,23088,23090,23110],{"className":23089},[342,472],[15,23091,23093,23107],{"className":23092},[346],[15,23094,23096],{"className":23095,"style":1362},[350],[15,23097,23098,23101],{"style":3202},[15,23099],{"className":23100,"style":358},[357],[15,23102,23104],{"className":23103},[362,363,364,365],[15,23105,1068],{"className":23106},[61,122,365],[15,23108,497],{"className":23109},[496],[15,23111,23113],{"className":23112},[346],[15,23114,23116],{"className":23115,"style":504},[350],[15,23117],{},[15,23119,750],{"className":23120},[178],"\n其中 ",[15,23123,23125,23153],{"className":23124,"translate":19},[18],[15,23126,23128],{"className":23127},[23],[25,23129,23130],{"xmlns":27},[29,23131,23132,23150],{},[32,23133,23134,23136,23138,23146,23148],{},[35,23135,22676],{},[81,23137,244],{},[227,23139,23140,23142],{},[35,23141,4288],{"mathvariant":231},[81,23143,23145],{"mathvariant":231,"lspace":23144,"rspace":23144},"0em","′",[35,23147,3018],{"mathvariant":231},[35,23149,4288],{"mathvariant":231},[40,23151,23152],{"encoding":42},"\\psi = \\Gamma'\u002F\\Gamma",[15,23154,23156,23174],{"className":23155,"ariaHidden":48},[47],[15,23157,23159,23162,23165,23168,23171],{"className":23158},[52],[15,23160],{"className":23161,"style":14301},[56],[15,23163,22676],{"className":23164,"style":123},[61,122],[15,23166],{"className":23167,"style":128},[127],[15,23169,244],{"className":23170},[132],[15,23172],{"className":23173,"style":128},[127],[15,23175,23177,23181,23214],{"className":23176},[52],[15,23178],{"className":23179,"style":23180},[56],"height:1.0019em;vertical-align:-0.25em;",[15,23182,23184,23187],{"className":23183},[61],[15,23185,4288],{"className":23186},[61],[15,23188,23190],{"className":23189},[338],[15,23191,23193],{"className":23192},[342],[15,23194,23196],{"className":23195},[346],[15,23197,23200],{"className":23198,"style":23199},[350],"height:0.7519em;",[15,23201,23202,23205],{"style":716},[15,23203],{"className":23204,"style":358},[357],[15,23206,23208],{"className":23207},[362,363,364,365],[15,23209,23211],{"className":23210},[61,365],[15,23212,23145],{"className":23213},[61,365],[15,23215,10943],{"className":23216},[61]," 为 digamma 函数。",[11,23219,23220,23222,23223,23311,23312,9342,23618,23871,23872,23942],{},[661,23221,5117],{}," 直接计算 ",[15,23224,23226,23254],{"className":23225,"translate":19},[18],[15,23227,23229],{"className":23228},[23],[25,23230,23231],{"xmlns":27},[29,23232,23233,23251],{},[32,23234,23235,23237,23239,23241,23243,23245,23247,23249],{},[35,23236,22634],{},[81,23238,244],{},[81,23240,239],{},[81,23242,4749],{},[35,23244,3703],{},[35,23246,22645],{},[81,23248,1545],{},[35,23250,3703],{},[40,23252,23253],{"encoding":42},"H = -\\int f \\log f",[15,23255,23257,23275],{"className":23256,"ariaHidden":48},[47],[15,23258,23260,23263,23266,23269,23272],{"className":23259},[52],[15,23261],{"className":23262,"style":57},[56],[15,23264,22634],{"className":23265,"style":22746},[61,122],[15,23267],{"className":23268,"style":128},[127],[15,23270,244],{"className":23271},[132],[15,23273],{"className":23274,"style":128},[127],[15,23276,23278,23282,23285,23288,23291,23294,23297,23300,23305,23308],{"className":23277},[52],[15,23279],{"className":23280,"style":23281},[56],"height:1.1111em;vertical-align:-0.3061em;",[15,23283,239],{"className":23284},[61],[15,23286],{"className":23287,"style":157},[127],[15,23289,4749],{"className":23290,"style":4830},[531,535,536],[15,23292],{"className":23293,"style":157},[127],[15,23295,3703],{"className":23296,"style":3781},[61,122],[15,23298],{"className":23299,"style":157},[127],[15,23301,22774,23303],{"className":23302},[531],[15,23304,22777],{"style":15608},[15,23306],{"className":23307,"style":157},[127],[15,23309,3703],{"className":23310,"style":3781},[61,122],"，代入 ",[15,23313,23315,23381],{"className":23314,"translate":19},[18],[15,23316,23318],{"className":23317},[23],[25,23319,23320],{"xmlns":27},[29,23321,23322,23378],{},[32,23323,23324,23326,23328,23330,23332,23334,23336,23338,23340,23342,23344,23346,23348,23354,23356,23362,23364,23366,23368,23370,23372],{},[81,23325,239],{},[35,23327,22645],{},[81,23329,1545],{},[35,23331,3703],{},[81,23333,244],{},[35,23335,22645],{},[81,23337,1545],{},[35,23339,3726],{},[81,23341,741],{"stretchy":86},[35,23343,3102],{"mathvariant":3101},[81,23345,750],{"stretchy":86},[81,23347,239],{},[266,23349,23350,23352],{},[81,23351,293],{},[35,23353,1068],{},[81,23355,741],{"stretchy":86},[266,23357,23358,23360],{},[35,23359,3102],{},[35,23361,1068],{},[81,23363,239],{},[89,23365,91],{},[81,23367,750],{"stretchy":86},[35,23369,22645],{},[81,23371,1545],{},[266,23373,23374,23376],{},[35,23375,1027],{},[35,23377,1068],{},[40,23379,23380],{"encoding":42},"-\\log f = \\log B(\\boldsymbol{\\alpha}) - \\sum_i (\\alpha_i - 1)\\log x_i",[15,23382,23384,23416,23457,23555],{"className":23383,"ariaHidden":48},[47],[15,23385,23387,23390,23393,23396,23401,23404,23407,23410,23413],{"className":23386},[52],[15,23388],{"className":23389,"style":14301},[56],[15,23391,239],{"className":23392},[61],[15,23394],{"className":23395,"style":157},[127],[15,23397,22774,23399],{"className":23398},[531],[15,23400,22777],{"style":15608},[15,23402],{"className":23403,"style":157},[127],[15,23405,3703],{"className":23406,"style":3781},[61,122],[15,23408],{"className":23409,"style":128},[127],[15,23411,244],{"className":23412},[132],[15,23414],{"className":23415,"style":128},[127],[15,23417,23419,23422,23427,23430,23433,23436,23445,23448,23451,23454],{"className":23418},[52],[15,23420],{"className":23421,"style":142},[56],[15,23423,22774,23425],{"className":23424},[531],[15,23426,22777],{"style":15608},[15,23428],{"className":23429,"style":157},[127],[15,23431,3726],{"className":23432,"style":3867},[61,122],[15,23434,741],{"className":23435},[146],[15,23437,23439],{"className":23438},[61],[15,23440,23442],{"className":23441},[61],[15,23443,3102],{"className":23444},[61,3162],[15,23446,750],{"className":23447},[178],[15,23449],{"className":23450,"style":772},[127],[15,23452,239],{"className":23453},[375],[15,23455],{"className":23456,"style":772},[127],[15,23458,23460,23463,23503,23506,23546,23549,23552],{"className":23459},[52],[15,23461],{"className":23462,"style":6045},[56],[15,23464,23466,23469],{"className":23465},[531],[15,23467,293],{"className":23468,"style":537},[531,535,536],[15,23470,23472],{"className":23471},[338],[15,23473,23475,23495],{"className":23474},[342,472],[15,23476,23478,23492],{"className":23477},[346],[15,23479,23481],{"className":23480,"style":6064},[350],[15,23482,23483,23486],{"style":553},[15,23484],{"className":23485,"style":358},[357],[15,23487,23489],{"className":23488},[362,363,364,365],[15,23490,1068],{"className":23491},[61,122,365],[15,23493,497],{"className":23494},[496],[15,23496,23498],{"className":23497},[346],[15,23499,23501],{"className":23500,"style":593},[350],[15,23502],{},[15,23504,741],{"className":23505},[146],[15,23507,23509,23512],{"className":23508},[61],[15,23510,3102],{"className":23511,"style":3187},[61,122],[15,23513,23515],{"className":23514},[338],[15,23516,23518,23538],{"className":23517},[342,472],[15,23519,23521,23535],{"className":23520},[346],[15,23522,23524],{"className":23523,"style":1362},[350],[15,23525,23526,23529],{"style":3202},[15,23527],{"className":23528,"style":358},[357],[15,23530,23532],{"className":23531},[362,363,364,365],[15,23533,1068],{"className":23534},[61,122,365],[15,23536,497],{"className":23537},[496],[15,23539,23541],{"className":23540},[346],[15,23542,23544],{"className":23543,"style":504},[350],[15,23545],{},[15,23547],{"className":23548,"style":772},[127],[15,23550,239],{"className":23551},[375],[15,23553],{"className":23554,"style":772},[127],[15,23556,23558,23561,23564,23567,23570,23575,23578],{"className":23557},[52],[15,23559],{"className":23560,"style":142},[56],[15,23562,91],{"className":23563},[61],[15,23565,750],{"className":23566},[178],[15,23568],{"className":23569,"style":157},[127],[15,23571,22774,23573],{"className":23572},[531],[15,23574,22777],{"style":15608},[15,23576],{"className":23577,"style":157},[127],[15,23579,23581,23584],{"className":23580},[61],[15,23582,1027],{"className":23583},[61,122],[15,23585,23587],{"className":23586},[338],[15,23588,23590,23610],{"className":23589},[342,472],[15,23591,23593,23607],{"className":23592},[346],[15,23594,23596],{"className":23595,"style":1362},[350],[15,23597,23598,23601],{"style":482},[15,23599],{"className":23600,"style":358},[357],[15,23602,23604],{"className":23603},[362,363,364,365],[15,23605,1068],{"className":23606},[61,122,365],[15,23608,497],{"className":23609},[496],[15,23611,23613],{"className":23612},[346],[15,23614,23616],{"className":23615,"style":504},[350],[15,23617],{},[15,23619,23621,23677],{"className":23620,"translate":19},[18],[15,23622,23624],{"className":23623},[23],[25,23625,23626],{"xmlns":27},[29,23627,23628,23674],{},[32,23629,23630,23632,23634,23636,23638,23644,23646,23648,23650,23652,23658,23660,23662,23664,23666,23672],{},[35,23631,14807],{"mathvariant":258},[81,23633,14810],{"stretchy":86},[35,23635,22645],{},[81,23637,1545],{},[266,23639,23640,23642],{},[35,23641,1027],{},[35,23643,1068],{},[81,23645,14819],{"stretchy":86},[81,23647,244],{},[35,23649,22676],{},[81,23651,741],{"stretchy":86},[266,23653,23654,23656],{},[35,23655,3102],{},[35,23657,1068],{},[81,23659,750],{"stretchy":86},[81,23661,239],{},[35,23663,22676],{},[81,23665,741],{"stretchy":86},[266,23667,23668,23670],{},[35,23669,3102],{},[89,23671,278],{},[81,23673,750],{"stretchy":86},[40,23675,23676],{"encoding":42},"\\mathbb{E}[\\log x_i] = \\psi(\\alpha_i) - \\psi(\\alpha_0)",[15,23678,23680,23752,23816],{"className":23679,"ariaHidden":48},[47],[15,23681,23683,23686,23689,23692,23697,23700,23740,23743,23746,23749],{"className":23682},[52],[15,23684],{"className":23685,"style":142},[56],[15,23687,14807],{"className":23688},[61,427],[15,23690,14810],{"className":23691},[146],[15,23693,22774,23695],{"className":23694},[531],[15,23696,22777],{"style":15608},[15,23698],{"className":23699,"style":157},[127],[15,23701,23703,23706],{"className":23702},[61],[15,23704,1027],{"className":23705},[61,122],[15,23707,23709],{"className":23708},[338],[15,23710,23712,23732],{"className":23711},[342,472],[15,23713,23715,23729],{"className":23714},[346],[15,23716,23718],{"className":23717,"style":1362},[350],[15,23719,23720,23723],{"style":482},[15,23721],{"className":23722,"style":358},[357],[15,23724,23726],{"className":23725},[362,363,364,365],[15,23727,1068],{"className":23728},[61,122,365],[15,23730,497],{"className":23731},[496],[15,23733,23735],{"className":23734},[346],[15,23736,23738],{"className":23737,"style":504},[350],[15,23739],{},[15,23741,14819],{"className":23742},[178],[15,23744],{"className":23745,"style":128},[127],[15,23747,244],{"className":23748},[132],[15,23750],{"className":23751,"style":128},[127],[15,23753,23755,23758,23761,23764,23804,23807,23810,23813],{"className":23754},[52],[15,23756],{"className":23757,"style":142},[56],[15,23759,22676],{"className":23760,"style":123},[61,122],[15,23762,741],{"className":23763},[146],[15,23765,23767,23770],{"className":23766},[61],[15,23768,3102],{"className":23769,"style":3187},[61,122],[15,23771,23773],{"className":23772},[338],[15,23774,23776,23796],{"className":23775},[342,472],[15,23777,23779,23793],{"className":23778},[346],[15,23780,23782],{"className":23781,"style":1362},[350],[15,23783,23784,23787],{"style":3202},[15,23785],{"className":23786,"style":358},[357],[15,23788,23790],{"className":23789},[362,363,364,365],[15,23791,1068],{"className":23792},[61,122,365],[15,23794,497],{"className":23795},[496],[15,23797,23799],{"className":23798},[346],[15,23800,23802],{"className":23801,"style":504},[350],[15,23803],{},[15,23805,750],{"className":23806},[178],[15,23808],{"className":23809,"style":772},[127],[15,23811,239],{"className":23812},[375],[15,23814],{"className":23815,"style":772},[127],[15,23817,23819,23822,23825,23828,23868],{"className":23818},[52],[15,23820],{"className":23821,"style":142},[56],[15,23823,22676],{"className":23824,"style":123},[61,122],[15,23826,741],{"className":23827},[146],[15,23829,23831,23834],{"className":23830},[61],[15,23832,3102],{"className":23833,"style":3187},[61,122],[15,23835,23837],{"className":23836},[338],[15,23838,23840,23860],{"className":23839},[342,472],[15,23841,23843,23857],{"className":23842},[346],[15,23844,23846],{"className":23845,"style":1212},[350],[15,23847,23848,23851],{"style":3202},[15,23849],{"className":23850,"style":358},[357],[15,23852,23854],{"className":23853},[362,363,364,365],[15,23855,278],{"className":23856},[61,365],[15,23858,497],{"className":23859},[496],[15,23861,23863],{"className":23862},[346],[15,23864,23866],{"className":23865,"style":504},[350],[15,23867],{},[15,23869,750],{"className":23870},[178],"（由命题 1.3 的矩恒等式对 ",[15,23873,23875,23893],{"className":23874,"translate":19},[18],[15,23876,23878],{"className":23877},[23],[25,23879,23880],{"xmlns":27},[29,23881,23882,23890],{},[32,23883,23884],{},[266,23885,23886,23888],{},[35,23887,17120],{},[35,23889,1068],{},[40,23891,23892],{"encoding":42},"\\beta_i",[15,23894,23896],{"className":23895,"ariaHidden":48},[47],[15,23897,23899,23902],{"className":23898},[52],[15,23900],{"className":23901,"style":14301},[56],[15,23903,23905,23908],{"className":23904},[61],[15,23906,17120],{"className":23907,"style":17259},[61,122],[15,23909,23911],{"className":23910},[338],[15,23912,23914,23934],{"className":23913},[342,472],[15,23915,23917,23931],{"className":23916},[346],[15,23918,23920],{"className":23919,"style":1362},[350],[15,23921,23922,23925],{"style":17443},[15,23923],{"className":23924,"style":358},[357],[15,23926,23928],{"className":23927},[362,363,364,365],[15,23929,1068],{"className":23930},[61,122,365],[15,23932,497],{"className":23933},[496],[15,23935,23937],{"className":23936},[346],[15,23938,23940],{"className":23939,"style":504},[350],[15,23941],{}," 求导得到），整理即得。",[11,23944,23945,23946,24008,24009,21876,24037,24088,24089,24226,24227,24319,24320,24370,24371,24374,24375,24448],{},"在对称情形 ",[15,23947,23949,23969],{"className":23948,"translate":19},[18],[15,23950,23952],{"className":23951},[23],[25,23953,23954],{"xmlns":27},[29,23955,23956,23966],{},[32,23957,23958,23960,23962,23964],{},[35,23959,3102],{"mathvariant":3101},[81,23961,244],{},[35,23963,3102],{},[89,23965,91],{"mathvariant":251},[40,23967,23968],{"encoding":42},"\\boldsymbol{\\alpha} = \\alpha \\mathbf{1}",[15,23970,23972,23996],{"className":23971,"ariaHidden":48},[47],[15,23973,23975,23978,23987,23990,23993],{"className":23974},[52],[15,23976],{"className":23977,"style":2706},[56],[15,23979,23981],{"className":23980},[61],[15,23982,23984],{"className":23983},[61],[15,23985,3102],{"className":23986},[61,3162],[15,23988],{"className":23989,"style":128},[127],[15,23991,244],{"className":23992},[132],[15,23994],{"className":23995,"style":128},[127],[15,23997,23999,24002,24005],{"className":23998},[52],[15,24000],{"className":24001,"style":859},[56],[15,24003,3102],{"className":24004,"style":3187},[61,122],[15,24006,91],{"className":24007},[61,411]," 下，",[15,24010,24012,24025],{"className":24011,"translate":19},[18],[15,24013,24015],{"className":24014},[23],[25,24016,24017],{"xmlns":27},[29,24018,24019,24023],{},[32,24020,24021],{},[35,24022,22634],{},[40,24024,22634],{"encoding":42},[15,24026,24028],{"className":24027,"ariaHidden":48},[47],[15,24029,24031,24034],{"className":24030},[52],[15,24032],{"className":24033,"style":57},[56],[15,24035,22634],{"className":24036,"style":22746},[61,122],[15,24038,24040,24058],{"className":24039,"translate":19},[18],[15,24041,24043],{"className":24042},[23],[25,24044,24045],{"xmlns":27},[29,24046,24047,24055],{},[32,24048,24049,24051,24053],{},[35,24050,3102],{},[81,24052,244],{},[89,24054,91],{},[40,24056,24057],{"encoding":42},"\\alpha = 1",[15,24059,24061,24079],{"className":24060,"ariaHidden":48},[47],[15,24062,24064,24067,24070,24073,24076],{"className":24063},[52],[15,24065],{"className":24066,"style":6026},[56],[15,24068,3102],{"className":24069,"style":3187},[61,122],[15,24071],{"className":24072,"style":128},[127],[15,24074,244],{"className":24075},[132],[15,24077],{"className":24078,"style":128},[127],[15,24080,24082,24085],{"className":24081},[52],[15,24083],{"className":24084,"style":859},[56],[15,24086,91],{"className":24087},[61]," 处取最大值 ",[15,24090,24092,24138],{"className":24091,"translate":19},[18],[15,24093,24095],{"className":24094},[23],[25,24096,24097],{"xmlns":27},[29,24098,24099,24135],{},[32,24100,24101,24103,24105,24107,24109,24111,24113,24115,24117,24119,24121,24123,24125,24127,24129,24131,24133],{},[81,24102,239],{},[35,24104,22645],{},[81,24106,1545],{},[35,24108,4288],{"mathvariant":231},[81,24110,741],{"stretchy":86},[35,24112,102],{},[81,24114,750],{"stretchy":86},[81,24116,244],{},[81,24118,239],{},[35,24120,22645],{},[81,24122,1545],{},[81,24124,741],{"stretchy":86},[35,24126,102],{},[81,24128,239],{},[89,24130,91],{},[81,24132,750],{"stretchy":86},[81,24134,3031],{"stretchy":86},[40,24136,24137],{"encoding":42},"-\\log \\Gamma(K) = -\\log (K-1)!",[15,24139,24141,24182,24214],{"className":24140,"ariaHidden":48},[47],[15,24142,24144,24147,24150,24153,24158,24161,24164,24167,24170,24173,24176,24179],{"className":24143},[52],[15,24145],{"className":24146,"style":142},[56],[15,24148,239],{"className":24149},[61],[15,24151],{"className":24152,"style":157},[127],[15,24154,22774,24156],{"className":24155},[531],[15,24157,22777],{"style":15608},[15,24159],{"className":24160,"style":157},[127],[15,24162,4288],{"className":24163},[61],[15,24165,741],{"className":24166},[146],[15,24168,102],{"className":24169,"style":174},[61,122],[15,24171,750],{"className":24172},[178],[15,24174],{"className":24175,"style":128},[127],[15,24177,244],{"className":24178},[132],[15,24180],{"className":24181,"style":128},[127],[15,24183,24185,24188,24191,24194,24199,24202,24205,24208,24211],{"className":24184},[52],[15,24186],{"className":24187,"style":142},[56],[15,24189,239],{"className":24190},[61],[15,24192],{"className":24193,"style":157},[127],[15,24195,22774,24197],{"className":24196},[531],[15,24198,22777],{"style":15608},[15,24200,741],{"className":24201},[146],[15,24203,102],{"className":24204,"style":174},[61,122],[15,24206],{"className":24207,"style":772},[127],[15,24209,239],{"className":24210},[375],[15,24212],{"className":24213,"style":772},[127],[15,24215,24217,24220,24223],{"className":24216},[52],[15,24218],{"className":24219,"style":142},[56],[15,24221,91],{"className":24222},[61],[15,24224,3075],{"className":24225},[178],"。该值为负，并不矛盾：它是相对于体积为 ",[15,24228,24230,24262],{"className":24229,"translate":19},[18],[15,24231,24233],{"className":24232},[23],[25,24234,24235],{"xmlns":27},[29,24236,24237,24259],{},[32,24238,24239,24241,24243,24245,24247,24249,24251,24253,24255,24257],{},[89,24240,91],{},[35,24242,3018],{"mathvariant":231},[81,24244,741],{"stretchy":86},[35,24246,102],{},[81,24248,239],{},[89,24250,91],{},[81,24252,750],{"stretchy":86},[81,24254,3031],{"stretchy":86},[81,24256,2488],{},[89,24258,91],{},[40,24260,24261],{"encoding":42},"1\u002F(K-1)! \u003C 1",[15,24263,24265,24289,24310],{"className":24264,"ariaHidden":48},[47],[15,24266,24268,24271,24274,24277,24280,24283,24286],{"className":24267},[52],[15,24269],{"className":24270,"style":142},[56],[15,24272,3047],{"className":24273},[61],[15,24275,741],{"className":24276},[146],[15,24278,102],{"className":24279,"style":174},[61,122],[15,24281],{"className":24282,"style":772},[127],[15,24284,239],{"className":24285},[375],[15,24287],{"className":24288,"style":772},[127],[15,24290,24292,24295,24298,24301,24304,24307],{"className":24291},[52],[15,24293],{"className":24294,"style":142},[56],[15,24296,91],{"className":24297},[61],[15,24299,3075],{"className":24300},[178],[15,24302],{"className":24303,"style":128},[127],[15,24305,2488],{"className":24306},[132],[15,24308],{"className":24309,"style":128},[127],[15,24311,24313,24316],{"className":24312},[52],[15,24314],{"className":24315,"style":859},[56],[15,24317,91],{"className":24318},[61]," 的支撑集而言的。",[15,24321,24323,24340],{"className":24322,"translate":19},[18],[15,24324,24326],{"className":24325},[23],[25,24327,24328],{"xmlns":27},[29,24329,24330,24338],{},[32,24331,24332,24334,24336],{},[35,24333,3102],{},[81,24335,244],{},[89,24337,91],{},[40,24339,24057],{"encoding":42},[15,24341,24343,24361],{"className":24342,"ariaHidden":48},[47],[15,24344,24346,24349,24352,24355,24358],{"className":24345},[52],[15,24347],{"className":24348,"style":6026},[56],[15,24350,3102],{"className":24351,"style":3187},[61,122],[15,24353],{"className":24354,"style":128},[127],[15,24356,244],{"className":24357},[132],[15,24359],{"className":24360,"style":128},[127],[15,24362,24364,24367],{"className":24363},[52],[15,24365],{"className":24366,"style":859},[56],[15,24368,91],{"className":24369},[61]," 对应单纯形上的",[661,24372,24373],{},"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 ",[15,24376,24378,24401],{"className":24377,"translate":19},[18],[15,24379,24381],{"className":24380},[23],[25,24382,24383],{"xmlns":27},[29,24384,24385,24399],{},[32,24386,24387],{},[227,24388,24389,24391],{},[35,24390,232],{"mathvariant":231},[32,24392,24393,24395,24397],{},[35,24394,102],{},[81,24396,239],{},[89,24398,91],{},[40,24400,1856],{"encoding":42},[15,24402,24404],{"className":24403,"ariaHidden":48},[47],[15,24405,24407,24410],{"className":24406},[52],[15,24408],{"className":24409,"style":695},[56],[15,24411,24413,24416],{"className":24412},[61],[15,24414,232],{"className":24415},[61],[15,24417,24419],{"className":24418},[338],[15,24420,24422],{"className":24421},[342],[15,24423,24425],{"className":24424},[346],[15,24426,24428],{"className":24427,"style":695},[350],[15,24429,24430,24433],{"style":716},[15,24431],{"className":24432,"style":358},[357],[15,24434,24436],{"className":24435},[362,363,364,365],[15,24437,24439,24442,24445],{"className":24438},[61,365],[15,24440,102],{"className":24441,"style":174},[61,122,365],[15,24443,239],{"className":24444},[375,365],[15,24446,91],{"className":24447},[61,365]," 紧且测度有限，均匀分布是最大熵分布，故",[15,24450,24452],{"className":24451,"translate":19},[212],[15,24453,24455,24512],{"className":24454,"translate":19},[18],[15,24456,24458],{"className":24457},[23],[25,24459,24460],{"xmlns":27,"display":221},[29,24461,24462,24509],{},[32,24463,24464,24466,24468,24470,24472,24474,24476,24478,24480,24482,24484,24486,24488,24490,24492,24495,24498,24501,24503,24505,24507],{},[35,24465,22634],{},[81,24467,741],{"stretchy":86},[35,24469,1027],{"mathvariant":251},[81,24471,750],{"stretchy":86},[81,24473,21295],{},[81,24475,239],{},[35,24477,22645],{},[81,24479,1545],{},[35,24481,4288],{"mathvariant":231},[81,24483,741],{"stretchy":86},[35,24485,102],{},[81,24487,750],{"stretchy":86},[81,24489,94],{"separator":48},[127,24491],{"width":9838},[282,24493,24494],{},"等号成立",[282,24496,24497],{},"  ",[81,24499,24500],{},"⟺",[282,24502,24497],{},[35,24504,3102],{"mathvariant":3101},[81,24506,244],{},[89,24508,91],{"mathvariant":251},[40,24510,24511],{"encoding":42},"H(\\mathbf{x}) \\le -\\log \\Gamma(K), \\qquad \\text{等号成立} \\iff \\boldsymbol{\\alpha} = \\mathbf{1}",[15,24513,24515,24542,24604,24628],{"className":24514,"ariaHidden":48},[47],[15,24516,24518,24521,24524,24527,24530,24533,24536,24539],{"className":24517},[52],[15,24519],{"className":24520,"style":142},[56],[15,24522,22634],{"className":24523,"style":22746},[61,122],[15,24525,741],{"className":24526},[146],[15,24528,1027],{"className":24529},[61,411],[15,24531,750],{"className":24532},[178],[15,24534],{"className":24535,"style":128},[127],[15,24537,21295],{"className":24538},[132],[15,24540],{"className":24541,"style":128},[127],[15,24543,24545,24548,24551,24554,24559,24562,24565,24568,24571,24574,24577,24580,24583,24589,24592,24595,24598,24601],{"className":24544},[52],[15,24546],{"className":24547,"style":142},[56],[15,24549,239],{"className":24550},[61],[15,24552],{"className":24553,"style":157},[127],[15,24555,22774,24557],{"className":24556},[531],[15,24558,22777],{"style":15608},[15,24560],{"className":24561,"style":157},[127],[15,24563,4288],{"className":24564},[61],[15,24566,741],{"className":24567},[146],[15,24569,102],{"className":24570,"style":174},[61,122],[15,24572,750],{"className":24573},[178],[15,24575,94],{"className":24576},[153],[15,24578],{"className":24579,"style":10026},[127],[15,24581],{"className":24582,"style":157},[127],[15,24584,24586],{"className":24585},[61,13539],[15,24587,24494],{"className":24588},[61,13543],[15,24590],{"className":24591,"style":128},[127],[15,24593],{"className":24594,"style":128},[127],[15,24596,24500],{"className":24597},[132],[15,24599],{"className":24600,"style":128},[127],[15,24602],{"className":24603,"style":128},[127],[15,24605,24607,24610,24619,24622,24625],{"className":24606},[52],[15,24608],{"className":24609,"style":2706},[56],[15,24611,24613],{"className":24612},[61],[15,24614,24616],{"className":24615},[61],[15,24617,3102],{"className":24618},[61,3162],[15,24620],{"className":24621,"style":128},[127],[15,24623,244],{"className":24624},[132],[15,24626],{"className":24627,"style":128},[127],[15,24629,24631,24634],{"className":24630},[52],[15,24632],{"className":24633,"style":859},[56],[15,24635,91],{"className":24636},[61,411],[877,24638,24640],{"id":24639},"_24-共轭性与贝叶斯更新","2.4 共轭性与贝叶斯更新",[882,24642,24644],{"title":24643},"命题 2.5（Dirichlet–Multinomial 共轭）",[11,24645,24646,24647,24732,24733,24768,24769,19834,24941,941,25035,25177,25178],{},"设先验 ",[15,24648,24650,24677],{"className":24649,"translate":19},[18],[15,24651,24653],{"className":24652},[23],[25,24654,24655],{"xmlns":27},[29,24656,24657,24674],{},[32,24658,24659,24662,24664,24666,24668,24670,24672],{},[35,24660,24661],{"mathvariant":3101},"π",[81,24663,4179],{},[35,24665,4182],{"mathvariant":231},[81,24667,1545],{},[81,24669,741],{"stretchy":86},[35,24671,3102],{"mathvariant":3101},[81,24673,750],{"stretchy":86},[40,24675,24676],{"encoding":42},"\\boldsymbol{\\pi} \\sim \\operatorname{Dir}(\\boldsymbol{\\alpha})",[15,24678,24680,24705],{"className":24679,"ariaHidden":48},[47],[15,24681,24683,24686,24696,24699,24702],{"className":24682},[52],[15,24684],{"className":24685,"style":2706},[56],[15,24687,24689],{"className":24688},[61],[15,24690,24692],{"className":24691},[61],[15,24693,24661],{"className":24694,"style":24695},[61,3162],"margin-right:0.037em;",[15,24697],{"className":24698,"style":128},[127],[15,24700,4179],{"className":24701},[132],[15,24703],{"className":24704,"style":128},[127],[15,24706,24708,24711,24717,24720,24729],{"className":24707},[52],[15,24709],{"className":24710,"style":142},[56],[15,24712,24714],{"className":24713},[531],[15,24715,4182],{"className":24716},[61,1620],[15,24718,741],{"className":24719},[146],[15,24721,24723],{"className":24722},[61],[15,24724,24726],{"className":24725},[61],[15,24727,3102],{"className":24728},[61,3162],[15,24730,750],{"className":24731},[178],"，给定 ",[15,24734,24736,24750],{"className":24735,"translate":19},[18],[15,24737,24739],{"className":24738},[23],[25,24740,24741],{"xmlns":27},[29,24742,24743,24747],{},[32,24744,24745],{},[35,24746,24661],{"mathvariant":3101},[40,24748,24749],{"encoding":42},"\\boldsymbol{\\pi}",[15,24751,24753],{"className":24752,"ariaHidden":48},[47],[15,24754,24756,24759],{"className":24755},[52],[15,24757],{"className":24758,"style":2706},[56],[15,24760,24762],{"className":24761},[61],[15,24763,24765],{"className":24764},[61],[15,24766,24661],{"className":24767,"style":24695},[61,3162]," 观测到计数向量 ",[15,24770,24772,24810],{"className":24771,"translate":19},[18],[15,24773,24775],{"className":24774},[23],[25,24776,24777],{"xmlns":27},[29,24778,24779,24807],{},[32,24780,24781,24783,24785,24787,24793,24795,24797,24799,24805],{},[35,24782,14551],{"mathvariant":251},[81,24784,244],{},[81,24786,741],{"stretchy":86},[266,24788,24789,24791],{},[35,24790,14551],{},[89,24792,91],{},[81,24794,94],{"separator":48},[81,24796,97],{},[81,24798,94],{"separator":48},[266,24800,24801,24803],{},[35,24802,14551],{},[35,24804,102],{},[81,24806,750],{"stretchy":86},[40,24808,24809],{"encoding":42},"\\mathbf{n} = (n_1, \\dots, n_K)",[15,24811,24813,24831],{"className":24812,"ariaHidden":48},[47],[15,24814,24816,24819,24822,24825,24828],{"className":24815},[52],[15,24817],{"className":24818,"style":2706},[56],[15,24820,14551],{"className":24821},[61,411],[15,24823],{"className":24824,"style":128},[127],[15,24826,244],{"className":24827},[132],[15,24829],{"className":24830,"style":128},[127],[15,24832,24834,24837,24840,24880,24883,24886,24889,24892,24895,24898,24938],{"className":24833},[52],[15,24835],{"className":24836,"style":142},[56],[15,24838,741],{"className":24839},[146],[15,24841,24843,24846],{"className":24842},[61],[15,24844,14551],{"className":24845},[61,122],[15,24847,24849],{"className":24848},[338],[15,24850,24852,24872],{"className":24851},[342,472],[15,24853,24855,24869],{"className":24854},[346],[15,24856,24858],{"className":24857,"style":1212},[350],[15,24859,24860,24863],{"style":482},[15,24861],{"className":24862,"style":358},[357],[15,24864,24866],{"className":24865},[362,363,364,365],[15,24867,91],{"className":24868},[61,365],[15,24870,497],{"className":24871},[496],[15,24873,24875],{"className":24874},[346],[15,24876,24878],{"className":24877,"style":504},[350],[15,24879],{},[15,24881,94],{"className":24882},[153],[15,24884],{"className":24885,"style":157},[127],[15,24887,97],{"className":24888},[161],[15,24890],{"className":24891,"style":157},[127],[15,24893,94],{"className":24894},[153],[15,24896],{"className":24897,"style":157},[127],[15,24899,24901,24904],{"className":24900},[61],[15,24902,14551],{"className":24903},[61,122],[15,24905,24907],{"className":24906},[338],[15,24908,24910,24930],{"className":24909},[342,472],[15,24911,24913,24927],{"className":24912},[346],[15,24914,24916],{"className":24915,"style":1271},[350],[15,24917,24918,24921],{"style":482},[15,24919],{"className":24920,"style":358},[357],[15,24922,24924],{"className":24923},[362,363,364,365],[15,24925,102],{"className":24926,"style":174},[61,122,365],[15,24928,497],{"className":24929},[496],[15,24931,24933],{"className":24932},[346],[15,24934,24936],{"className":24935,"style":504},[350],[15,24937],{},[15,24939,750],{"className":24940},[178],[15,24942,24944,24967],{"className":24943,"translate":19},[18],[15,24945,24947],{"className":24946},[23],[25,24948,24949],{"xmlns":27},[29,24950,24951,24964],{},[32,24952,24953,24959,24961],{},[266,24954,24955,24957],{},[35,24956,14551],{},[35,24958,1068],{},[81,24960,83],{},[35,24962,24963],{"mathvariant":258},"N",[40,24965,24966],{"encoding":42},"n_i \\in \\mathbb{N}",[15,24968,24970,25025],{"className":24969,"ariaHidden":48},[47],[15,24971,24973,24976,25016,25019,25022],{"className":24972},[52],[15,24974],{"className":24975,"style":1751},[56],[15,24977,24979,24982],{"className":24978},[61],[15,24980,14551],{"className":24981},[61,122],[15,24983,24985],{"className":24984},[338],[15,24986,24988,25008],{"className":24987},[342,472],[15,24989,24991,25005],{"className":24990},[346],[15,24992,24994],{"className":24993,"style":1362},[350],[15,24995,24996,24999],{"style":482},[15,24997],{"className":24998,"style":358},[357],[15,25000,25002],{"className":25001},[362,363,364,365],[15,25003,1068],{"className":25004},[61,122,365],[15,25006,497],{"className":25007},[496],[15,25009,25011],{"className":25010},[346],[15,25012,25014],{"className":25013,"style":504},[350],[15,25015],{},[15,25017],{"className":25018,"style":128},[127],[15,25020,83],{"className":25021},[132],[15,25023],{"className":25024,"style":128},[127],[15,25026,25028,25032],{"className":25027},[52],[15,25029],{"className":25030,"style":25031},[56],"height:0.6889em;",[15,25033,24963],{"className":25034},[61,427],[15,25036,25038,25066],{"className":25037,"translate":19},[18],[15,25039,25041],{"className":25040},[23],[25,25042,25043],{"xmlns":27},[29,25044,25045,25063],{},[32,25046,25047,25049,25051,25057],{},[35,25048,24963],{},[81,25050,244],{},[266,25052,25053,25055],{},[81,25054,293],{},[35,25056,1068],{},[266,25058,25059,25061],{},[35,25060,14551],{},[35,25062,1068],{},[40,25064,25065],{"encoding":42},"N = \\sum_i n_i",[15,25067,25069,25088],{"className":25068,"ariaHidden":48},[47],[15,25070,25072,25075,25079,25082,25085],{"className":25071},[52],[15,25073],{"className":25074,"style":57},[56],[15,25076,24963],{"className":25077,"style":25078},[61,122],"margin-right:0.109em;",[15,25080],{"className":25081,"style":128},[127],[15,25083,244],{"className":25084},[132],[15,25086],{"className":25087,"style":128},[127],[15,25089,25091,25094,25134,25137],{"className":25090},[52],[15,25092],{"className":25093,"style":6045},[56],[15,25095,25097,25100],{"className":25096},[531],[15,25098,293],{"className":25099,"style":537},[531,535,536],[15,25101,25103],{"className":25102},[338],[15,25104,25106,25126],{"className":25105},[342,472],[15,25107,25109,25123],{"className":25108},[346],[15,25110,25112],{"className":25111,"style":6064},[350],[15,25113,25114,25117],{"style":553},[15,25115],{"className":25116,"style":358},[357],[15,25118,25120],{"className":25119},[362,363,364,365],[15,25121,1068],{"className":25122},[61,122,365],[15,25124,497],{"className":25125},[496],[15,25127,25129],{"className":25128},[346],[15,25130,25132],{"className":25131,"style":593},[350],[15,25133],{},[15,25135],{"className":25136,"style":157},[127],[15,25138,25140,25143],{"className":25139},[61],[15,25141,14551],{"className":25142},[61,122],[15,25144,25146],{"className":25145},[338],[15,25147,25149,25169],{"className":25148},[342,472],[15,25150,25152,25166],{"className":25151},[346],[15,25153,25155],{"className":25154,"style":1362},[350],[15,25156,25157,25160],{"style":482},[15,25158],{"className":25159,"style":358},[357],[15,25161,25163],{"className":25162},[362,363,364,365],[15,25164,1068],{"className":25165},[61,122,365],[15,25167,497],{"className":25168},[496],[15,25170,25172],{"className":25171},[346],[15,25173,25175],{"className":25174,"style":504},[350],[15,25176],{},"），则后验为\n",[15,25179,25181,25215],{"className":25180,"translate":19},[18],[15,25182,25184],{"className":25183},[23],[25,25185,25186],{"xmlns":27},[29,25187,25188,25212],{},[32,25189,25190,25192,25194,25196,25198,25200,25202,25204,25206,25208,25210],{},[35,25191,24661],{"mathvariant":3101},[81,25193,3710],{},[35,25195,14551],{"mathvariant":251},[81,25197,4179],{},[35,25199,4182],{"mathvariant":231},[81,25201,1545],{},[81,25203,741],{"stretchy":86},[35,25205,3102],{"mathvariant":3101},[81,25207,7756],{},[35,25209,14551],{"mathvariant":251},[81,25211,750],{"stretchy":86},[40,25213,25214],{"encoding":42},"\\boldsymbol{\\pi} \\mid \\mathbf{n} \\sim \\operatorname{Dir}(\\boldsymbol{\\alpha} + \\mathbf{n})",[15,25216,25218,25242,25260,25293],{"className":25217,"ariaHidden":48},[47],[15,25219,25221,25224,25233,25236,25239],{"className":25220},[52],[15,25222],{"className":25223,"style":142},[56],[15,25225,25227],{"className":25226},[61],[15,25228,25230],{"className":25229},[61],[15,25231,24661],{"className":25232,"style":24695},[61,3162],[15,25234],{"className":25235,"style":128},[127],[15,25237,3710],{"className":25238},[132],[15,25240],{"className":25241,"style":128},[127],[15,25243,25245,25248,25251,25254,25257],{"className":25244},[52],[15,25246],{"className":25247,"style":2706},[56],[15,25249,14551],{"className":25250},[61,411],[15,25252],{"className":25253,"style":128},[127],[15,25255,4179],{"className":25256},[132],[15,25258],{"className":25259,"style":128},[127],[15,25261,25263,25266,25272,25275,25284,25287,25290],{"className":25262},[52],[15,25264],{"className":25265,"style":142},[56],[15,25267,25269],{"className":25268},[531],[15,25270,4182],{"className":25271},[61,1620],[15,25273,741],{"className":25274},[146],[15,25276,25278],{"className":25277},[61],[15,25279,25281],{"className":25280},[61],[15,25282,3102],{"className":25283},[61,3162],[15,25285],{"className":25286,"style":772},[127],[15,25288,7756],{"className":25289},[375],[15,25291],{"className":25292,"style":772},[127],[15,25294,25296,25299,25302],{"className":25295},[52],[15,25297],{"className":25298,"style":142},[56],[15,25300,14551],{"className":25301},[61,411],[15,25303,750],{"className":25304},[178],[11,25306,25307,25309,25310,25488],{},[661,25308,5117],{}," 多项似然正比于 ",[15,25311,25313,25343],{"className":25312,"translate":19},[18],[15,25314,25316],{"className":25315},[23],[25,25317,25318],{"xmlns":27},[29,25319,25320,25340],{},[32,25321,25322,25328],{},[266,25323,25324,25326],{},[81,25325,3737],{},[35,25327,1068],{},[289,25329,25330,25332,25334],{},[35,25331,24661],{},[35,25333,1068],{},[266,25335,25336,25338],{},[35,25337,14551],{},[35,25339,1068],{},[40,25341,25342],{"encoding":42},"\\prod_i \\pi_i^{n_i}",[15,25344,25346],{"className":25345,"ariaHidden":48},[47],[15,25347,25349,25352,25392,25395],{"className":25348},[52],[15,25350],{"className":25351,"style":6045},[56],[15,25353,25355,25358],{"className":25354},[531],[15,25356,3737],{"className":25357,"style":537},[531,535,536],[15,25359,25361],{"className":25360},[338],[15,25362,25364,25384],{"className":25363},[342,472],[15,25365,25367,25381],{"className":25366},[346],[15,25368,25370],{"className":25369,"style":6064},[350],[15,25371,25372,25375],{"style":553},[15,25373],{"className":25374,"style":358},[357],[15,25376,25378],{"className":25377},[362,363,364,365],[15,25379,1068],{"className":25380},[61,122,365],[15,25382,497],{"className":25383},[496],[15,25385,25387],{"className":25386},[346],[15,25388,25390],{"className":25389,"style":593},[350],[15,25391],{},[15,25393],{"className":25394,"style":157},[127],[15,25396,25398,25401],{"className":25397},[61],[15,25399,24661],{"className":25400,"style":123},[61,122],[15,25402,25404],{"className":25403},[338],[15,25405,25407,25480],{"className":25406},[342,472],[15,25408,25410,25477],{"className":25409},[346],[15,25411,25414,25426],{"className":25412,"style":25413},[350],"height:0.7463em;",[15,25415,25417,25420],{"style":25416},"top:-2.4231em;margin-left:-0.0359em;margin-right:0.05em;",[15,25418],{"className":25419,"style":358},[357],[15,25421,25423],{"className":25422},[362,363,364,365],[15,25424,1068],{"className":25425},[61,122,365],[15,25427,25428,25431],{"style":4023},[15,25429],{"className":25430,"style":358},[357],[15,25432,25434],{"className":25433},[362,363,364,365],[15,25435,25437],{"className":25436},[61,365],[15,25438,25440,25443],{"className":25439},[61,365],[15,25441,14551],{"className":25442},[61,122,365],[15,25444,25446],{"className":25445},[338],[15,25447,25449,25469],{"className":25448},[342,472],[15,25450,25452,25466],{"className":25451},[346],[15,25453,25455],{"className":25454,"style":4051},[350],[15,25456,25457,25460],{"style":5803},[15,25458],{"className":25459,"style":4058},[357],[15,25461,25463],{"className":25462},[362,4062,4063,365],[15,25464,1068],{"className":25465},[61,122,365],[15,25467,497],{"className":25468},[496],[15,25470,25472],{"className":25471},[346],[15,25473,25475],{"className":25474,"style":4076},[350],[15,25476],{},[15,25478,497],{"className":25479},[496],[15,25481,25483],{"className":25482},[346],[15,25484,25486],{"className":25485,"style":4094},[350],[15,25487],{},"，与先验密度相乘得",[15,25490,25492],{"className":25491,"translate":19},[212],[15,25493,25495,25562],{"className":25494,"translate":19},[18],[15,25496,25498],{"className":25497},[23],[25,25499,25500],{"xmlns":27,"display":221},[29,25501,25502,25559],{},[32,25503,25504,25506,25508,25510,25512,25514,25516,25519,25533],{},[35,25505,11],{},[81,25507,741],{"stretchy":86},[35,25509,24661],{"mathvariant":3101},[81,25511,3710],{},[35,25513,14551],{"mathvariant":251},[81,25515,750],{"stretchy":86},[81,25517,25518],{},"∝",[5229,25520,25521,25523,25531],{},[81,25522,3737],{},[32,25524,25525,25527,25529],{},[35,25526,1068],{},[81,25528,244],{},[89,25530,91],{},[35,25532,102],{},[289,25534,25535,25537,25539],{},[35,25536,24661],{},[35,25538,1068],{},[32,25540,25541,25547,25549,25555,25557],{},[266,25542,25543,25545],{},[35,25544,3102],{},[35,25546,1068],{},[81,25548,7756],{},[266,25550,25551,25553],{},[35,25552,14551],{},[35,25554,1068],{},[81,25556,239],{},[89,25558,91],{},[40,25560,25561],{"encoding":42},"p(\\boldsymbol{\\pi} \\mid \\mathbf{n}) \\propto \\prod_{i=1}^K \\pi_i^{\\alpha_i + n_i - 1}",[15,25563,25565,25595,25616],{"className":25564,"ariaHidden":48},[47],[15,25566,25568,25571,25574,25577,25586,25589,25592],{"className":25567},[52],[15,25569],{"className":25570,"style":142},[56],[15,25572,11],{"className":25573},[61,122],[15,25575,741],{"className":25576},[146],[15,25578,25580],{"className":25579},[61],[15,25581,25583],{"className":25582},[61],[15,25584,24661],{"className":25585,"style":24695},[61,3162],[15,25587],{"className":25588,"style":128},[127],[15,25590,3710],{"className":25591},[132],[15,25593],{"className":25594,"style":128},[127],[15,25596,25598,25601,25604,25607,25610,25613],{"className":25597},[52],[15,25599],{"className":25600,"style":142},[56],[15,25602,14551],{"className":25603},[61,411],[15,25605,750],{"className":25606},[178],[15,25608],{"className":25609,"style":128},[127],[15,25611,25518],{"className":25612},[132],[15,25614],{"className":25615,"style":128},[127],[15,25617,25619,25622,25686,25689],{"className":25618},[52],[15,25620],{"className":25621,"style":5338},[56],[15,25623,25625],{"className":25624},[531,5342],[15,25626,25628,25678],{"className":25627},[342,472],[15,25629,25631,25675],{"className":25630},[346],[15,25632,25634,25654,25664],{"className":25633,"style":5352},[350],[15,25635,25636,25639],{"style":5355},[15,25637],{"className":25638,"style":5359},[357],[15,25640,25642],{"className":25641},[362,363,364,365],[15,25643,25645,25648,25651],{"className":25644},[61,365],[15,25646,1068],{"className":25647},[61,122,365],[15,25649,244],{"className":25650},[132,365],[15,25652,91],{"className":25653},[61,365],[15,25655,25656,25659],{"style":5377},[15,25657],{"className":25658,"style":5359},[357],[15,25660,25661],{},[15,25662,3737],{"className":25663},[531,535,5386],[15,25665,25666,25669],{"style":5389},[15,25667],{"className":25668,"style":5359},[357],[15,25670,25672],{"className":25671},[362,363,364,365],[15,25673,102],{"className":25674,"style":174},[61,122,365],[15,25676,497],{"className":25677},[496],[15,25679,25681],{"className":25680},[346],[15,25682,25684],{"className":25683,"style":5408},[350],[15,25685],{},[15,25687],{"className":25688,"style":157},[127],[15,25690,25692,25695],{"className":25691},[61],[15,25693,24661],{"className":25694,"style":123},[61,122],[15,25696,25698],{"className":25697},[338],[15,25699,25701,25821],{"className":25700},[342,472],[15,25702,25704,25818],{"className":25703},[346],[15,25705,25707,25718],{"className":25706,"style":4008},[350],[15,25708,25709,25712],{"style":25416},[15,25710],{"className":25711,"style":358},[357],[15,25713,25715],{"className":25714},[362,363,364,365],[15,25716,1068],{"className":25717},[61,122,365],[15,25719,25720,25723],{"style":4023},[15,25721],{"className":25722,"style":358},[357],[15,25724,25726],{"className":25725},[362,363,364,365],[15,25727,25729,25769,25772,25812,25815],{"className":25728},[61,365],[15,25730,25732,25735],{"className":25731},[61,365],[15,25733,3102],{"className":25734,"style":3187},[61,122,365],[15,25736,25738],{"className":25737},[338],[15,25739,25741,25761],{"className":25740},[342,472],[15,25742,25744,25758],{"className":25743},[346],[15,25745,25747],{"className":25746,"style":4051},[350],[15,25748,25749,25752],{"style":4054},[15,25750],{"className":25751,"style":4058},[357],[15,25753,25755],{"className":25754},[362,4062,4063,365],[15,25756,1068],{"className":25757},[61,122,365],[15,25759,497],{"className":25760},[496],[15,25762,25764],{"className":25763},[346],[15,25765,25767],{"className":25766,"style":4076},[350],[15,25768],{},[15,25770,7756],{"className":25771},[375,365],[15,25773,25775,25778],{"className":25774},[61,365],[15,25776,14551],{"className":25777},[61,122,365],[15,25779,25781],{"className":25780},[338],[15,25782,25784,25804],{"className":25783},[342,472],[15,25785,25787,25801],{"className":25786},[346],[15,25788,25790],{"className":25789,"style":4051},[350],[15,25791,25792,25795],{"style":5803},[15,25793],{"className":25794,"style":4058},[357],[15,25796,25798],{"className":25797},[362,4062,4063,365],[15,25799,1068],{"className":25800},[61,122,365],[15,25802,497],{"className":25803},[496],[15,25805,25807],{"className":25806},[346],[15,25808,25810],{"className":25809,"style":4076},[350],[15,25811],{},[15,25813,239],{"className":25814},[375,365],[15,25816,91],{"className":25817},[61,365],[15,25819,497],{"className":25820},[496],[15,25822,25824],{"className":25823},[346],[15,25825,25827],{"className":25826,"style":4094},[350],[15,25828],{},[11,25830,25831,25832,25909],{},"该核与 ",[15,25833,25835,25861],{"className":25834,"translate":19},[18],[15,25836,25838],{"className":25837},[23],[25,25839,25840],{"xmlns":27},[29,25841,25842,25858],{},[32,25843,25844,25846,25848,25850,25852,25854,25856],{},[35,25845,4182],{"mathvariant":231},[81,25847,1545],{},[81,25849,741],{"stretchy":86},[35,25851,3102],{"mathvariant":3101},[81,25853,7756],{},[35,25855,14551],{"mathvariant":251},[81,25857,750],{"stretchy":86},[40,25859,25860],{"encoding":42},"\\operatorname{Dir}(\\boldsymbol{\\alpha} + \\mathbf{n})",[15,25862,25864,25897],{"className":25863,"ariaHidden":48},[47],[15,25865,25867,25870,25876,25879,25888,25891,25894],{"className":25866},[52],[15,25868],{"className":25869,"style":142},[56],[15,25871,25873],{"className":25872},[531],[15,25874,4182],{"className":25875},[61,1620],[15,25877,741],{"className":25878},[146],[15,25880,25882],{"className":25881},[61],[15,25883,25885],{"className":25884},[61],[15,25886,3102],{"className":25887},[61,3162],[15,25889],{"className":25890,"style":772},[127],[15,25892,7756],{"className":25893},[375],[15,25895],{"className":25896,"style":772},[127],[15,25898,25900,25903,25906],{"className":25899},[52],[15,25901],{"className":25902,"style":142},[56],[15,25904,14551],{"className":25905},[61,411],[15,25907,750],{"className":25908},[178]," 的核成比例，归一化即得。",[11,25911,25912],{},"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。",[882,25914,25916],{"title":25915},"推论 2.6（Dirichlet–Multinomial 分布）",[11,25917,25918,25919,25947,25948],{},"计数向量 ",[15,25920,25922,25935],{"className":25921,"translate":19},[18],[15,25923,25925],{"className":25924},[23],[25,25926,25927],{"xmlns":27},[29,25928,25929,25933],{},[32,25930,25931],{},[35,25932,14551],{"mathvariant":251},[40,25934,14554],{"encoding":42},[15,25936,25938],{"className":25937,"ariaHidden":48},[47],[15,25939,25941,25944],{"className":25940},[52],[15,25942],{"className":25943,"style":2706},[56],[15,25945,14551],{"className":25946},[61,411]," 的边缘似然为\n",[15,25949,25951,26117],{"className":25950,"translate":19},[18],[15,25952,25954],{"className":25953},[23],[25,25955,25956],{"xmlns":27},[29,25957,25958,26114],{},[32,25959,25960,25962,25964,25966,25968,25970,25972,25974,25998,26000,26026,26028,26062,26076],{},[35,25961,11],{},[81,25963,741],{"stretchy":86},[35,25965,14551],{"mathvariant":251},[81,25967,3710],{},[35,25969,3102],{"mathvariant":3101},[81,25971,750],{"stretchy":86},[81,25973,244],{},[3718,25975,25976,25982],{},[32,25977,25978,25980],{},[35,25979,24963],{},[81,25981,3031],{"stretchy":86},[32,25983,25984,25990,25996],{},[266,25985,25986,25988],{},[81,25987,3737],{},[35,25989,1068],{},[266,25991,25992,25994],{},[35,25993,14551],{},[35,25995,1068],{},[81,25997,3031],{"stretchy":86},[81,25999,11205],{},[3718,26001,26002,26016],{},[32,26003,26004,26006,26008,26010,26012,26014],{},[35,26005,3726],{},[81,26007,741],{"stretchy":86},[35,26009,3102],{"mathvariant":3101},[81,26011,7756],{},[35,26013,14551],{"mathvariant":251},[81,26015,750],{"stretchy":86},[32,26017,26018,26020,26022,26024],{},[35,26019,3726],{},[81,26021,741],{"stretchy":86},[35,26023,3102],{"mathvariant":3101},[81,26025,750],{"stretchy":86},[81,26027,244],{},[3718,26029,26030,26044],{},[32,26031,26032,26034,26036,26042],{},[35,26033,4288],{"mathvariant":231},[81,26035,741],{"stretchy":86},[266,26037,26038,26040],{},[35,26039,3102],{},[89,26041,278],{},[81,26043,750],{"stretchy":86},[32,26045,26046,26048,26050,26056,26058,26060],{},[35,26047,4288],{"mathvariant":231},[81,26049,741],{"stretchy":86},[266,26051,26052,26054],{},[35,26053,3102],{},[89,26055,278],{},[81,26057,7756],{},[35,26059,24963],{},[81,26061,750],{"stretchy":86},[289,26063,26064,26066,26074],{},[81,26065,3737],{},[32,26067,26068,26070,26072],{},[35,26069,1068],{},[81,26071,244],{},[89,26073,91],{},[35,26075,102],{},[3718,26077,26078,26100],{},[32,26079,26080,26082,26084,26090,26092,26098],{},[35,26081,4288],{"mathvariant":231},[81,26083,741],{"stretchy":86},[266,26085,26086,26088],{},[35,26087,3102],{},[35,26089,1068],{},[81,26091,7756],{},[266,26093,26094,26096],{},[35,26095,14551],{},[35,26097,1068],{},[81,26099,750],{"stretchy":86},[32,26101,26102,26104,26106,26112],{},[35,26103,4288],{"mathvariant":231},[81,26105,741],{"stretchy":86},[266,26107,26108,26110],{},[35,26109,3102],{},[35,26111,1068],{},[81,26113,750],{"stretchy":86},[40,26115,26116],{"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)}",[15,26118,26120,26144,26171,26343,26462],{"className":26119,"ariaHidden":48},[47],[15,26121,26123,26126,26129,26132,26135,26138,26141],{"className":26122},[52],[15,26124],{"className":26125,"style":142},[56],[15,26127,11],{"className":26128},[61,122],[15,26130,741],{"className":26131},[146],[15,26133,14551],{"className":26134},[61,411],[15,26136],{"className":26137,"style":128},[127],[15,26139,3710],{"className":26140},[132],[15,26142],{"className":26143,"style":128},[127],[15,26145,26147,26150,26159,26162,26165,26168],{"className":26146},[52],[15,26148],{"className":26149,"style":142},[56],[15,26151,26153],{"className":26152},[61],[15,26154,26156],{"className":26155},[61],[15,26157,3102],{"className":26158},[61,3162],[15,26160,750],{"className":26161},[178],[15,26163],{"className":26164,"style":128},[127],[15,26166,244],{"className":26167},[132],[15,26169],{"className":26170,"style":128},[127],[15,26172,26174,26178,26334,26337,26340],{"className":26173},[52],[15,26175],{"className":26176,"style":26177},[56],"height:1.4501em;vertical-align:-0.57em;",[15,26179,26181,26184,26331],{"className":26180},[61],[15,26182],{"className":26183},[146,3837],[15,26185,26187],{"className":26186},[3718],[15,26188,26190,26322],{"className":26189},[342,472],[15,26191,26193,26319],{"className":26192},[346],[15,26194,26197,26294,26302],{"className":26195,"style":26196},[350],"height:0.8801em;",[15,26198,26199,26202],{"style":3853},[15,26200],{"className":26201,"style":3857},[357],[15,26203,26205],{"className":26204},[362,363,364,365],[15,26206,26208,26248,26251,26291],{"className":26207},[61,365],[15,26209,26211,26214],{"className":26210},[531,365],[15,26212,3737],{"className":26213,"style":537},[531,535,536,365],[15,26215,26217],{"className":26216},[338],[15,26218,26220,26240],{"className":26219},[342,472],[15,26221,26223,26237],{"className":26222},[346],[15,26224,26226],{"className":26225,"style":7472},[350],[15,26227,26228,26231],{"style":4491},[15,26229],{"className":26230,"style":4058},[357],[15,26232,26234],{"className":26233},[362,4062,4063,365],[15,26235,1068],{"className":26236},[61,122,365],[15,26238,497],{"className":26239},[496],[15,26241,26243],{"className":26242},[346],[15,26244,26246],{"className":26245,"style":4531},[350],[15,26247],{},[15,26249],{"className":26250,"style":4537},[127,365],[15,26252,26254,26257],{"className":26253},[61,365],[15,26255,14551],{"className":26256},[61,122,365],[15,26258,26260],{"className":26259},[338],[15,26261,26263,26283],{"className":26262},[342,472],[15,26264,26266,26280],{"className":26265},[346],[15,26267,26269],{"className":26268,"style":4051},[350],[15,26270,26271,26274],{"style":5803},[15,26272],{"className":26273,"style":4058},[357],[15,26275,26277],{"className":26276},[362,4062,4063,365],[15,26278,1068],{"className":26279},[61,122,365],[15,26281,497],{"className":26282},[496],[15,26284,26286],{"className":26285},[346],[15,26287,26289],{"className":26288,"style":4076},[350],[15,26290],{},[15,26292,3031],{"className":26293},[178,365],[15,26295,26296,26299],{"style":3885},[15,26297],{"className":26298,"style":3857},[357],[15,26300],{"className":26301,"style":3893},[3892],[15,26303,26304,26307],{"style":3896},[15,26305],{"className":26306,"style":3857},[357],[15,26308,26310],{"className":26309},[362,363,364,365],[15,26311,26313,26316],{"className":26312},[61,365],[15,26314,24963],{"className":26315,"style":25078},[61,122,365],[15,26317,3031],{"className":26318},[178,365],[15,26320,497],{"className":26321},[496],[15,26323,26325],{"className":26324},[346],[15,26326,26329],{"className":26327,"style":26328},[350],"height:0.57em;",[15,26330],{},[15,26332],{"className":26333},[178,3837],[15,26335],{"className":26336,"style":772},[127],[15,26338,11205],{"className":26339},[375],[15,26341],{"className":26342,"style":772},[127],[15,26344,26346,26349,26453,26456,26459],{"className":26345},[52],[15,26347],{"className":26348,"style":17647},[56],[15,26350,26352,26355,26450],{"className":26351},[61],[15,26353],{"className":26354},[146,3837],[15,26356,26358],{"className":26357},[3718],[15,26359,26361,26442],{"className":26360},[342,472],[15,26362,26364,26439],{"className":26363},[346],[15,26365,26367,26396,26404],{"className":26366,"style":15310},[350],[15,26368,26369,26372],{"style":3853},[15,26370],{"className":26371,"style":3857},[357],[15,26373,26375],{"className":26374},[362,363,364,365],[15,26376,26378,26381,26384,26393],{"className":26377},[61,365],[15,26379,3726],{"className":26380,"style":3867},[61,122,365],[15,26382,741],{"className":26383},[146,365],[15,26385,26387],{"className":26386},[61,365],[15,26388,26390],{"className":26389},[61,365],[15,26391,3102],{"className":26392},[61,3162,365],[15,26394,750],{"className":26395},[178,365],[15,26397,26398,26401],{"style":3885},[15,26399],{"className":26400,"style":3857},[357],[15,26402],{"className":26403,"style":3893},[3892],[15,26405,26406,26409],{"style":15439},[15,26407],{"className":26408,"style":3857},[3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2.7（后验预测与收缩）",[11,26907,26908,26909,27291,27292,27399,27400,27403,27404],{},"对未见样本，\n",[15,26910,26912,26977],{"className":26911,"translate":19},[18],[15,26913,26915],{"className":26914},[23],[25,26916,26917],{"xmlns":27},[29,26918,26919,26974],{},[32,26920,26921,26923,26925,26932,26934,26936,26938,26940,26942,26944],{},[35,26922,21273],{"mathvariant":258},[81,26924,741],{"stretchy":86},[266,26926,26927,26929],{},[35,26928,79],{},[282,26930,26931],{},"new",[81,26933,244],{},[35,26935,272],{},[81,26937,3710],{},[35,26939,14551],{"mathvariant":251},[81,26941,750],{"stretchy":86},[81,26943,244],{},[3718,26945,26946,26962],{},[32,26947,26948,26954,26956],{},[266,26949,26950,26952],{},[35,26951,3102],{},[35,26953,272],{},[81,26955,7756],{},[266,26957,26958,26960],{},[35,26959,14551],{},[35,26961,272],{},[32,26963,26964,26970,26972],{},[266,26965,26966,26968],{},[35,26967,3102],{},[89,26969,278],{},[81,26971,7756],{},[35,26973,24963],{},[40,26975,26976],{"encoding":42},"\\mathbb{P}(y_{\\text{new}} = k \\mid \\mathbf{n}) = \\frac{\\alpha_k + n_k}{\\alpha_0 + N}",[15,26978,26980,27049,27067,27088],{"className":26979,"ariaHidden":48},[47],[15,26981,26983,26986,26989,26992,27040,27043,27046],{"className":26982},[52],[15,26984],{"className":26985,"style":142},[56],[15,26987,21273],{"className":26988},[61,427],[15,26990,741],{"className":26991},[146],[15,26993,26995,26998],{"className":26994},[61],[15,26996,79],{"className":26997,"style":123},[61,122],[15,26999,27001],{"className":27000},[338],[15,27002,27004,27032],{"className":27003},[342,472],[15,27005,27007,27029],{"className":27006},[346],[15,27008,27011],{"className":27009,"style":27010},[350],"height:0.1514em;",[15,27012,27014,27017],{"style":27013},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[15,27015],{"className":27016,"style":358},[357],[15,27018,27020],{"className":27019},[362,363,364,365],[15,27021,27023],{"className":27022},[61,365],[15,27024,27026],{"className":27025},[61,13539,365],[15,27027,26931],{"className":27028},[61,365],[15,27030,497],{"className":27031},[496],[15,27033,27035],{"className":27034},[346],[15,27036,27038],{"className":27037,"style":504},[350],[15,27039],{},[15,27041],{"className":27042,"style":128},[127],[15,27044,244],{"className":27045},[132],[15,27047],{"className":27048,"style":128},[127],[15,27050,27052,27055,27058,27061,27064],{"className":27051},[52],[15,27053],{"className":27054,"style":142},[56],[15,27056,272],{"className":27057,"style":492},[61,122],[15,27059],{"className":27060,"style":128},[127],[15,27062,3710],{"className":27063},[132],[15,27065],{"className":27066,"style":128},[127],[15,27068,27070,27073,27076,27079,27082,27085],{"className":27069},[52],[15,27071],{"className":27072,"style":142},[56],[15,27074,14551],{"className":27075},[61,411],[15,27077,750],{"className":27078},[178],[15,27080],{"className":27081,"style":128},[127],[15,27083,244],{"className":27084},[132],[15,27086],{"className":27087,"style":128},[127],[15,27089,27091,27095],{"className":27090},[52],[15,27092],{"className":27093,"style":27094},[56],"height:1.2693em;vertical-align:-0.4451em;",[15,27096,27098,27101,27288],{"className":27097},[61],[15,27099],{"className":27100},[146,3837],[15,27102,27104],{"className":27103},[3718],[15,27105,27107,27280],{"className":27106},[342,472],[15,27108,27110,27277],{"className":27109},[346],[15,27111,27114,27171,27179],{"className":27112,"style":27113},[350],"height:0.8242em;",[15,27115,27116,27119],{"style":3853},[15,27117],{"className":27118,"style":3857},[357],[15,27120,27122],{"className":27121},[362,363,364,365],[15,27123,27125,27165,27168],{"className":27124},[61,365],[15,27126,27128,27131],{"className":27127},[61,365],[15,27129,3102],{"className":27130,"style":3187},[61,122,365],[15,27132,27134],{"className":27133},[338],[15,27135,27137,27157],{"className":27136},[342,472],[15,27138,27140,27154],{"className":27139},[346],[15,27141,27143],{"className":27142,"style":4421},[350],[15,27144,27145,27148],{"style":4054},[15,27146],{"className":27147,"style":4058},[357],[15,27149,27151],{"className":27150},[362,4062,4063,365],[15,27152,278],{"className":27153},[61,365],[15,27155,497],{"className":27156},[496],[15,27158,27160],{"className":27159},[346],[15,27161,27163],{"className":27162,"style":4076},[350],[15,27164],{},[15,27166,7756],{"className":27167},[375,365],[15,27169,24963],{"className":27170,"style":25078},[61,122,365],[15,27172,27173,27176],{"style":3885},[15,27174],{"className":27175,"style":3857},[357],[15,27177],{"className":27178,"style":3893},[3892],[15,27180,27182,27185],{"style":27181},"top:-3.4159em;",[15,27183],{"className":27184,"style":3857},[357],[15,27186,27188],{"className":27187},[362,363,364,365],[15,27189,27191,27233,27236],{"className":27190},[61,365],[15,27192,27194,27197],{"className":27193},[61,365],[15,27195,3102],{"className":27196,"style":3187},[61,122,365],[15,27198,27200],{"className":27199},[338],[15,27201,27203,27224],{"className":27202},[342,472],[15,27204,27206,27221],{"className":27205},[346],[15,27207,27209],{"className":27208,"style":22389},[350],[15,27210,27212,27215],{"style":27211},"top:-2.3488em;margin-left:-0.0037em;margin-right:0.0714em;",[15,27213],{"className":27214,"style":4058},[357],[15,27216,27218],{"className":27217},[362,4062,4063,365],[15,27219,272],{"className":27220,"style":492},[61,122,365],[15,27222,497],{"className":27223},[496],[15,27225,27227],{"className":27226},[346],[15,27228,27231],{"className":27229,"style":27230},[350],"height:0.1512em;",[15,27232],{},[15,27234,7756],{"className":27235},[375,365],[15,27237,27239,27242],{"className":27238},[61,365],[15,27240,14551],{"className":27241},[61,122,365],[15,27243,27245],{"className":27244},[338],[15,27246,27248,27269],{"className":27247},[342,472],[15,27249,27251,27266],{"className":27250},[346],[15,27252,27254],{"className":27253,"style":22389},[350],[15,27255,27257,27260],{"style":27256},"top:-2.3488em;margin-left:0em;margin-right:0.0714em;",[15,27258],{"className":27259,"style":4058},[357],[15,27261,27263],{"className":27262},[362,4062,4063,365],[15,27264,272],{"className":27265,"style":492},[61,122,365],[15,27267,497],{"className":27268},[496],[15,27270,27272],{"className":27271},[346],[15,27273,27275],{"className":27274,"style":27230},[350],[15,27276],{},[15,27278,497],{"className":27279},[496],[15,27281,27283],{"className":27282},[346],[15,27284,27286],{"className":27285,"style":15209},[350],[15,27287],{},[15,27289],{"className":27290},[178,3837],"\n而 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\\mathbf{n}]",[15,27324,27326,27387],{"className":27325,"ariaHidden":48},[47],[15,27327,27329,27332,27335,27338,27378,27381,27384],{"className":27328},[52],[15,27330],{"className":27331,"style":142},[56],[15,27333,14807],{"className":27334},[61,427],[15,27336,14810],{"className":27337},[146],[15,27339,27341,27344],{"className":27340},[61],[15,27342,24661],{"className":27343,"style":123},[61,122],[15,27345,27347],{"className":27346},[338],[15,27348,27350,27370],{"className":27349},[342,472],[15,27351,27353,27367],{"className":27352},[346],[15,27354,27356],{"className":27355,"style":479},[350],[15,27357,27358,27361],{"style":27013},[15,27359],{"className":27360,"style":358},[357],[15,27362,27364],{"className":27363},[362,363,364,365],[15,27365,272],{"className":27366,"style":492},[61,122,365],[15,27368,497],{"className":27369},[496],[15,27371,27373],{"className":27372},[346],[15,27374,27376],{"className":27375,"style":504},[350],[15,27377],{},[15,27379],{"className":27380,"style":128},[127],[15,27382,3710],{"className":27383},[132],[15,27385],{"className":27386,"style":128},[127],[15,27388,27390,27393,27396],{"className":27389},[52],[15,27391],{"className":27392,"style":142},[56],[15,27394,14551],{"className":27395},[61,411],[15,27397,14819],{"className":27398},[178]," 是",[661,27401,27402],{},"先验均值与经验频率的凸组合","：\n",[15,27405,27407,27522],{"className":27406,"translate":19},[18],[15,27408,27410],{"className":27409},[23],[25,27411,27412],{"xmlns":27},[29,27413,27414,27519],{},[32,27415,27416,27418,27420,27426,27428,27430,27432,27434,27464,27466,27480,27482,27507,27509],{},[35,27417,14807],{"mathvariant":258},[81,27419,14810],{"stretchy":86},[266,27421,27422,27424],{},[35,27423,24661],{},[35,27425,272],{},[81,27427,3710],{},[35,27429,14551],{"mathvariant":251},[81,27431,14819],{"stretchy":86},[81,27433,244],{},[6875,27435,27436,27461],{},[6875,27437,27438,27458],{},[3718,27439,27440,27446],{},[266,27441,27442,27444],{},[35,27443,3102],{},[89,27445,278],{},[32,27447,27448,27454,27456],{},[266,27449,27450,27452],{},[35,27451,3102],{},[89,27453,278],{},[81,27455,7756],{},[35,27457,24963],{},[81,27459,27460],{"stretchy":48},"⏟",[282,27462,27463],{},"收缩权重",[81,27465,11205],{},[3718,27467,27468,27474],{},[266,27469,27470,27472],{},[35,27471,3102],{},[35,27473,272],{},[266,27475,27476,27478],{},[35,27477,3102],{},[89,27479,278],{},[81,27481,7756],{},[6875,27483,27484,27504],{},[6875,27485,27486,27502],{},[3718,27487,27488,27490],{},[35,27489,24963],{},[32,27491,27492,27498,27500],{},[266,27493,27494,27496],{},[35,27495,3102],{},[89,27497,278],{},[81,27499,7756],{},[35,27501,24963],{},[81,27503,27460],{"stretchy":48},[282,27505,27506],{},"数据权重",[81,27508,11205],{},[3718,27510,27511,27517],{},[266,27512,27513,27515],{},[35,27514,14551],{},[35,27516,272],{},[35,27518,24963],{},[40,27520,27521],{"encoding":42},"\\mathbb{E}[\\pi_k \\mid \\mathbf{n}] = \\underbrace{\\frac{\\alpha_0}{\\alpha_0 + N}}_{\\text{收缩权重}} \\cdot \\frac{\\alpha_k}{\\alpha_0} + \\underbrace{\\frac{N}{\\alpha_0 + N}}_{\\text{数据权重}} \\cdot 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lassName":28016,"style":22389},[350],[15,28018,28019,28022],{"style":27211},[15,28020],{"className":28021,"style":4058},[357],[15,28023,28025],{"className":28024},[362,4062,4063,365],[15,28026,272],{"className":28027,"style":492},[61,122,365],[15,28029,497],{"className":28030},[496],[15,28032,28034],{"className":28033},[346],[15,28035,28037],{"className":28036,"style":27230},[350],[15,28038],{},[15,28040,497],{"className":28041},[496],[15,28043,28045],{"className":28044},[346],[15,28046,28048],{"className":28047,"style":15209},[350],[15,28049],{},[15,28051],{"className":28052},[178,3837],[15,28054],{"className":28055,"style":772},[127],[15,28057,7756],{"className":28058},[375],[15,28060],{"className":28061,"style":772},[127],[15,28063,28065,28069,28291,28294,28297],{"className":28064},[52],[15,28066],{"className":28067,"style":28068},[56],"height:3.5227em;vertical-align:-2.1623em;",[15,28070,28072],{"className":28071},[161,6875],[15,28073,28075,28283],{"className":28074},[342,472],[15,28076,28078,28280],{"className":28077},[346],[15,28079,28082,28101],{"className":28080,"style":28081},[350],"height:1.3603em;",[15,28083,28085,28089],{"style":28084},"top:-1.198em;",[15,28086],{"className":28087,"style":28088},[357],"height:3.3603em;",[15,28090,28092],{"className":28091},[362,363,364,365],[15,28093,28095],{"className":28094},[61,365],[15,28096,28098],{"className":28097},[61,13539,365],[15,28099,27506],{"className":28100},[61,13543,365],[15,28102,28104,28107],{"style":28103},"top:-3.3603em;",[15,28105],{"className":28106,"style":28088},[357],[15,28108,28110],{"className":28109},[161,6875],[15,28111,28113,28272],{"className":28112},[342,472],[15,28114,28116,28269],{"className":28115},[346],[15,28117,28119,28150],{"className":28118,"style":28081},[350],[15,28120,28123,28126],{"className":28121,"style":28122},[27667],"top:-1.8763em;",[15,28124],{"className":28125,"style":28088},[357],[15,28127,28129,28136,28143],{"className":28128,"style":27676},[27675],[15,28130,28132],{"className":28131,"style":27681},[27680],[27683,28133,28134],{"xmlns":27685,"width":27686,"height":27687,"viewBox":27688,"preserveAspectRatio":27689},[27691,28135],{"d":27693},[15,28137,28139],{"className":28138,"style":27681},[27697],[27683,28140,28141],{"xmlns":27685,"width":27686,"height":27687,"viewBox":27688,"preserveAspectRatio":27700},[27691,28142],{"d":27703},[15,28144,28146],{"className":28145,"style":27681},[27707],[27683,28147,28148],{"xmlns":27685,"width":27686,"height":27687,"viewBox":27688,"preserveAspectRatio":27710},[27691,28149],{"d":27713},[15,28151,28152,28155],{"style":28103},[15,28153],{"className":28154,"style":28088},[357],[15,28156,28158],{"className":28157},[61],[15,28159,28161,28164,28266],{"className":28160},[61],[15,28162],{"className":28163},[146,3837],[15,28165,28167],{"className":28166},[3718],[15,28168,28170,28258],{"className":28169},[342,472],[15,28171,28173,28255],{"className":28172},[346],[15,28174,28176,28236,28244],{"className":28175,"style":28081},[350],[15,28177,28178,28181],{"style":11393},[15,28179],{"className":28180,"style":3857},[357],[15,28182,28184,28224,28227,28230,28233],{"className":28183},[61],[15,28185,28187,28190],{"className":28186},[61],[15,28188,3102],{"className":28189,"style":3187},[61,122],[15,28191,28193],{"className":28192},[338],[15,28194,28196,28216],{"className":28195},[342,472],[15,28197,28199,28213],{"className":28198},[346],[15,28200,28202],{"className":28201,"style":1212},[350],[15,28203,28204,28207],{"style":3202},[15,28205],{"className":28206,"style":358},[357],[15,28208,28210],{"className":28209},[362,363,364,365],[15,28211,278],{"className":28212},[61,365],[15,28214,497],{"className":28215},[496],[15,28217,28219],{"className":28218},[346],[15,28220,28222],{"className":28221,"style":504},[350],[15,28223],{},[15,28225],{"className":28226,"style":772},[127],[15,28228,7756],{"className":28229},[375],[15,28231],{"className":28232,"style":772},[127],[15,28234,24963],{"className":28235,"style":25078},[61,122],[15,28237,28238,28241],{"style":3885},[15,28239],{"className":28240,"style":3857},[357],[15,28242],{"className":28243,"style":3893},[3892],[15,28245,28246,28249],{"style":11459},[15,28247],{"className":28248,"style":3857},[357],[15,28250,28252],{"className":28251},[61],[15,28253,24963],{"className":28254,"style":25078},[61,122],[15,28256,497],{"className":28257},[496],[15,28259,28261],{"className":28260},[346],[15,28262,28264],{"className":28263,"style":18768},[350],[15,28265],{},[15,28267],{"className":28268},[178,3837],[15,28270,497],{"className":28271},[496],[15,28273,28275],{"className":28274},[346],[15,28276,28278],{"className":28277,"style":27879},[350],[15,28279],{},[15,28281,497],{"className":28282},[496],[15,28284,28286],{"className":28285},[346],[15,28287,28289],{"className":28288,"style":27891},[350],[15,28290],{},[15,28292],{"className":28293,"style":772},[127],[15,28295,11205],{"className":28296},[375],[15,28298],{"className":28299,"style":772},[127],[15,28301,28303,28307],{"className":28302},[52],[15,28304],{"className":28305,"style":28306},[56],"height:1.0623em;vertical-align:-0.345em;",[15,28308,28310,28313,28409],{"className":28309},[61],[15,28311],{"className":28312},[146,3837],[15,28314,28316],{"className":28315},[3718],[15,28317,28319,28401],{"className":28318},[342,472],[15,28320,28322,28398],{"className":28321},[346],[15,28323,28325,28339,28347],{"className":28324,"style":27928},[350],[15,28326,28327,28330],{"style":3853},[15,28328],{"className":28329,"style":3857},[357],[15,28331,28333],{"className":28332},[362,363,364,365],[15,28334,28336],{"className":28335},[61,365],[15,28337,24963],{"className":28338,"style":25078},[61,122,365],[15,28340,28341,28344],{"style":3885},[15,28342],{"className":28343,"style":3857},[357],[15,28345],{"className":28346,"style":3893},[3892],[15,28348,28349,28352],{"style":27181},[15,28350],{"className":28351,"style":3857},[357],[15,28353,28355],{"className":28354},[362,363,364,365],[15,28356,28358],{"className":28357},[61,365],[15,28359,28361,28364],{"className":28360},[61,365],[15,28362,14551],{"className":28363},[61,122,365],[15,28365,28367],{"className":28366},[338],[15,28368,28370,28390],{"className":28369},[342,472],[15,28371,28373,28387],{"className":28372},[346],[15,28374,28376],{"className":28375,"style":22389},[350],[15,28377,28378,28381],{"style":27256},[15,28379],{"className":28380,"style":4058},[357],[15,28382,28384],{"className":28383},[362,4062,4063,365],[15,28385,272],{"className":28386,"style":492},[61,122,365],[15,28388,497],{"className":28389},[496],[15,28391,28393],{"className":28392},[346],[15,28394,28396],{"className":28395,"style":27230},[350],[15,28397],{},[15,28399,497],{"className":28400},[496],[15,28402,28404],{"className":28403},[346],[15,28405,28407],{"className":28406,"style":10189},[350],[15,28408],{},[15,28410],{"className":28411},[178,3837],[11,28413,28414,28415,17388,28517,28609,28610,28701,28702,28794],{},"推论 2.7 用一句话概括了加法平滑的全部含义：",[661,28416,28417,28418,28487,28488,28516],{},"先验强度的倒数 ",[15,28419,28421,28438],{"className":28420,"translate":19},[18],[15,28422,28424],{"className":28423},[23],[25,28425,28426],{"xmlns":27},[29,28427,28428,28436],{},[32,28429,28430],{},[266,28431,28432,28434],{},[35,28433,3102],{},[89,28435,278],{},[40,28437,19479],{"encoding":42},[15,28439,28441],{"className":28440,"ariaHidden":48},[47],[15,28442,28444,28447],{"className":28443},[52],[15,28445],{"className":28446,"style":2509},[56],[15,28448,28450,28453],{"className":28449},[61],[15,28451,3102],{"className":28452,"style":3187},[61,122],[15,28454,28456],{"className":28455},[338],[15,28457,28459,28479],{"className":28458},[342,472],[15,28460,28462,28476],{"className":28461},[346],[15,28463,28465],{"className":28464,"style":1212},[350],[15,28466,28467,28470],{"style":3202},[15,28468],{"className":28469,"style":358},[357],[15,28471,28473],{"className":28472},[362,363,364,365],[15,28474,278],{"className":28475},[61,365],[15,28477,497],{"className":28478},[496],[15,28480,28482],{"className":28481},[346],[15,28483,28485],{"className":28484,"style":504},[350],[15,28486],{}," 相对于样本量 ",[15,28489,28491,28504],{"className":28490,"translate":19},[18],[15,28492,28494],{"className":28493},[23],[25,28495,28496],{"xmlns":27},[29,28497,28498,28502],{},[32,28499,28500],{},[35,28501,24963],{},[40,28503,24963],{"encoding":42},[15,28505,28507],{"className":28506,"ariaHidden":48},[47],[15,28508,28510,28513],{"className":28509},[52],[15,28511],{"className":28512,"style":57},[56],[15,28514,24963],{"className":28515,"style":25078},[61,122]," 的大小，就是数据与先验的信任分配比例。",[15,28518,28520,28542],{"className":28519,"translate":19},[18],[15,28521,28523],{"className":28522},[23],[25,28524,28525],{"xmlns":27},[29,28526,28527,28539],{},[32,28528,28529,28535,28537],{},[266,28530,28531,28533],{},[35,28532,3102],{},[35,28534,1068],{},[81,28536,244],{},[89,28538,91],{},[40,28540,28541],{"encoding":42},"\\alpha_i = 1",[15,28543,28545,28600],{"className":28544,"ariaHidden":48},[47],[15,28546,28548,28551,28591,28594,28597],{"className":28547},[52],[15,28549],{"className":28550,"style":2509},[56],[15,28552,28554,28557],{"className":28553},[61],[15,28555,3102],{"className":28556,"style":3187},[61,122],[15,28558,28560],{"className":28559},[338],[15,28561,28563,28583],{"className":28562},[342,472],[15,28564,28566,28580],{"className":28565},[346],[15,28567,28569],{"className":28568,"style":1362},[350],[15,28570,28571,28574],{"style":3202},[15,28572],{"className":28573,"style":358},[357],[15,28575,28577],{"className":28576},[362,363,364,365],[15,28578,1068],{"className":28579},[61,122,365],[15,28581,497],{"className":28582},[496],[15,28584,28586],{"className":28585},[346],[15,28587,28589],{"className":28588,"style":504},[350],[15,28590],{},[15,28592],{"className":28593,"style":128},[127],[15,28595,244],{"className":28596},[132],[15,28598],{"className":28599,"style":128},[127],[15,28601,28603,28606],{"className":28602},[52],[15,28604],{"className":28605,"style":859},[56],[15,28607,91],{"className":28608},[61]," 得 Laplace 平滑，取 ",[15,28611,28613,28634],{"className":28612,"translate":19},[18],[15,28614,28616],{"className":28615},[23],[25,28617,28618],{"xmlns":27},[29,28619,28620,28632],{},[32,28621,28622,28628,28630],{},[266,28623,28624,28626],{},[35,28625,3102],{},[35,28627,1068],{},[81,28629,1588],{},[89,28631,91],{},[40,28633,21705],{"encoding":42},[15,28635,28637,28692],{"className":28636,"ariaHidden":48},[47],[15,28638,28640,28643,28683,28686,28689],{"className":28639},[52],[15,28641],{"className":28642,"style":1751},[56],[15,28644,28646,28649],{"className":28645},[61],[15,28647,3102],{"className":28648,"style":3187},[61,122],[15,28650,28652],{"className":28651},[338],[15,28653,28655,28675],{"className":28654},[342,472],[15,28656,28658,28672],{"className":28657},[346],[15,28659,28661],{"className":28660,"style":1362},[350],[15,28662,28663,28666],{"style":3202},[15,28664],{"className":28665,"style":358},[357],[15,28667,28669],{"className":28668},[362,363,364,365],[15,28670,1068],{"className":28671},[61,122,365],[15,28673,497],{"className":28674},[496],[15,28676,28678],{"className":28677},[346],[15,28679,28681],{"className":28680,"style":504},[350],[15,28682],{},[15,28684],{"className":28685,"style":128},[127],[15,28687,1588],{"className":28688},[132],[15,28690],{"className":28691,"style":128},[127],[15,28693,28695,28698],{"className":28694},[52],[15,28696],{"className":28697,"style":859},[56],[15,28699,91],{"className":28700},[61]," 得 Lidstone 平滑，取 ",[15,28703,28705,28727],{"className":28704,"translate":19},[18],[15,28706,28708],{"className":28707},[23],[25,28709,28710],{"xmlns":27},[29,28711,28712,28724],{},[32,28713,28714,28720,28722],{},[266,28715,28716,28718],{},[35,28717,3102],{},[35,28719,1068],{},[81,28721,20691],{},[89,28723,278],{},[40,28725,28726],{"encoding":42},"\\alpha_i \\to 0",[15,28728,28730,28785],{"className":28729,"ariaHidden":48},[47],[15,28731,28733,28736,28776,28779,28782],{"className":28732},[52],[15,28734],{"className":28735,"style":2509},[56],[15,28737,28739,28742],{"className":28738},[61],[15,28740,3102],{"className":28741,"style":3187},[61,122],[15,28743,28745],{"className":28744},[338],[15,28746,28748,28768],{"className":28747},[342,472],[15,28749,28751,28765],{"className":28750},[346],[15,28752,28754],{"className":28753,"style":1362},[350],[15,28755,28756,28759],{"style":3202},[15,28757],{"className":28758,"style":358},[357],[15,28760,28762],{"className":28761},[362,363,364,365],[15,28763,1068],{"className":28764},[61,122,365],[15,28766,497],{"className":28767},[496],[15,28769,28771],{"className":28770},[346],[15,28772,28774],{"className":28773,"style":504},[350],[15,28775],{},[15,28777],{"className":28778,"style":128},[127],[15,28780,20691],{"className":28781},[132],[15,28783],{"className":28784,"style":128},[127],[15,28786,28788,28791],{"className":28787},[52],[15,28789],{"className":28790,"style":859},[56],[15,28792,278],{"className":28793},[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。",[877,28796,28798],{"id":28797},"_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",[882,28800,28802],{"title":28801},"命题 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相互独立，且后者服从 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而 ",[15,30141,30143,30193],{"className":30142,"translate":19},[18],[15,30144,30146],{"className":30145},[23],[25,30147,30148],{"xmlns":27},[29,30149,30150,30190],{},[32,30151,30152,30158,30160,30162,30164,30166,30172,30174,30180,30182,30188],{},[266,30153,30154,30156],{},[35,30155,1027],{},[35,30157,272],{},[81,30159,4179],{},[35,30161,12947],{"mathvariant":231},[81,30163,1545],{},[81,30165,741],{"stretchy":86},[266,30167,30168,30170],{},[35,30169,3102],{},[35,30171,272],{},[81,30173,94],{"separator":48},[266,30175,30176,30178],{},[35,30177,3102],{},[89,30179,278],{},[81,30181,239],{},[266,30183,30184,30186],{},[35,30185,3102],{},[35,30187,272],{},[81,30189,750],{"stretchy":86},[40,30191,30192],{"encoding":42},"x_k \\sim \\operatorname{Beta}(\\alpha_k, \\alpha_0 - \\alpha_k)",[15,30194,30196,30251,30361],{"className":30195,"ariaHidden":48},[47],[15,30197,30199,30202,30242,30245,30248],{"className":30198},[52],[15,30200],{"className":30201,"style":2509},[56],[15,30203,30205,30208],{"className":30204},[61],[15,30206,1027],{"className":30207},[61,122],[15,30209,30211],{"className":30210},[338],[15,30212,30214,30234],{"className":30213},[342,472],[15,30215,30217,30231],{"className":30216},[346],[15,30218,30220],{"className":30219,"style":479},[350],[15,30221,30222,30225],{"style":482},[15,30223],{"className":30224,"style":358},[357],[15,30226,30228],{"className":30227},[362,363,364,365],[15,30229,272],{"className":30230,"style":492},[61,122,365],[15,30232,497],{"className":30233},[496],[15,30235,30237],{"className":30236},[346],[15,30238,30240],{"className":30239,"style":504},[350],[15,30241],{},[15,30243],{"className":30244,"style":128},[127],[15,30246,4179],{"className":30247},[132],[15,30249],{"className":30250,"style":128},[127],[15,30252,30254,30257,30263,30266,30306,30309,30312,30352,30355,30358],{"className":30253},[52],[15,30255],{"className":30256,"style":142},[56],[15,30258,30260],{"className":30259},[531],[15,30261,12947],{"className":30262},[61,1620],[15,30264,741],{"className":30265},[146],[15,30267,30269,30272],{"className":30268},[61],[15,30270,3102],{"className":30271,"style":3187},[61,122],[15,30273,30275],{"className":30274},[338],[15,30276,30278,30298],{"className":30277},[342,472],[15,30279,30281,30295],{"className":30280},[346],[15,30282,30284],{"className":30283,"style":479},[350],[15,30285,30286,30289],{"style":3202},[15,30287],{"className":30288,"style":358},[357],[15,30290,30292],{"className":30291},[362,363,364,365],[15,30293,272],{"className":30294,"style":492},[61,122,365],[15,30296,497],{"className":30297},[496],[15,30299,30301],{"className":30300},[346],[15,30302,30304],{"className":30303,"style":504},[350],[15,30305],{},[15,30307,94],{"className":30308},[153],[15,30310],{"className":30311,"style":157},[127],[15,30313,30315,30318],{"className":30314},[61],[15,30316,3102],{"className":30317,"style":3187},[61,122],[15,30319,30321],{"className":30320},[338],[15,30322,30324,30344],{"className":30323},[342,472],[15,30325,30327,30341],{"className":30326},[346],[15,30328,30330],{"className":30329,"style":1212},[350],[15,30331,30332,30335],{"style":3202},[15,30333],{"className":30334,"style":358},[357],[15,30336,30338],{"className":30337},[362,363,364,365],[15,30339,278],{"className":30340},[61,365],[15,30342,497],{"className":30343},[496],[15,30345,30347],{"className":30346},[346],[15,30348,30350],{"className":30349,"style":504},[350],[15,30351],{},[15,30353],{"className":30354,"style":772},[127],[15,30356,239],{"className":30357},[375],[15,30359],{"className":30360,"style":772},[127],[15,30362,30364,30367,30407],{"className":30363},[52],[15,30365],{"className":30366,"style":142},[56],[15,30368,30370,30373],{"className":30369},[61],[15,30371,3102],{"className":30372,"style":3187},[61,122],[15,30374,30376],{"className":30375},[338],[15,30377,30379,30399],{"className":30378},[342,472],[15,30380,30382,30396],{"className":30381},[346],[15,30383,30385],{"className":30384,"style":479},[350],[15,30386,30387,30390],{"style":3202},[15,30388],{"className":30389,"style":358},[357],[15,30391,30393],{"className":30392},[362,363,364,365],[15,30394,272],{"className":30395,"style":492},[61,122,365],[15,30397,497],{"className":30398},[496],[15,30400,30402],{"className":30401},[346],[15,30403,30405],{"className":30404,"style":504},[350],[15,30406],{},[15,30408,750],{"className":30409},[178],"。反复施用中立性，得到如下构造性表示：",[882,30412,30414],{"title":30413},"命题 2.9（stick-breaking 表示）",[11,30415,13,30416,30707,30708,10326,31176,867],{},[15,30417,30419,30478],{"className":30418,"translate":19},[18],[15,30420,30422],{"className":30421},[23],[25,30423,30424],{"xmlns":27},[29,30425,30426,30475],{},[32,30427,30428,30435,30437,30439,30441,30443],{},[266,30429,30430,30433],{},[35,30431,30432],{},"v",[35,30434,272],{},[81,30436,4179],{},[35,30438,12947],{"mathvariant":231},[81,30440,1545],{},[282,30442,6793],{},[32,30444,30445,30447,30453,30455,30467,30473],{},[81,30446,741],{"fence":48},[266,30448,30449,30451],{},[35,30450,3102],{},[35,30452,272],{},[81,30454,94],{"separator":48},[266,30456,30457,30459],{},[81,30458,293],{},[32,30460,30461,30463,30465],{},[35,30462,14766],{},[81,30464,1588],{},[35,30466,272],{},[266,30468,30469,30471],{},[35,30470,3102],{},[35,30472,14766],{},[81,30474,750],{"fence":48},[40,30476,30477],{"encoding":42},"v_k \\sim \\operatorname{Beta}\\!\\left(\\alpha_k, \\sum_{j>k} \\alpha_j\\right)",[15,30479,30481,30536],{"className":30480,"ariaHidden":48},[47],[15,30482,30484,30487,30527,30530,30533],{"className":30483},[52],[15,30485],{"className":30486,"style":2509},[56],[15,30488,30490,30493],{"className":30489},[61],[15,30491,30432],{"className":30492,"style":123},[61,122],[15,30494,30496],{"className":30495},[338],[15,30497,30499,30519],{"className":30498},[342,472],[15,30500,30502,30516],{"className":30501},[346],[15,30503,30505],{"className":30504,"style":479},[350],[15,30506,30507,30510],{"style":27013},[15,30508],{"className":30509,"style":358},[357],[15,30511,30513],{"className":30512},[362,363,364,365],[15,30514,272],{"className":30515,"style":492},[61,122,365],[15,30517,497],{"className":30518},[496],[15,30520,30522],{"className":30521},[346],[15,30523,30525],{"className":30524,"style":504},[350],[15,30526],{},[15,30528],{"className":30529,"style":128},[127],[15,30531,4179],{"className":30532},[132],[15,30534],{"className":30535,"style":128},[127],[15,30537,30539,30542,30548,30551,30554],{"className":30538},[52],[15,30540],{"className":30541,"style":394},[56],[15,30543,30545],{"className":30544},[531],[15,30546,12947],{"className":30547},[61,1620],[15,30549],{"className":30550,"style":7099},[127],[15,30552],{"className":30553,"style":157},[127],[15,30555,30557,30563,30603,30606,30609,30658,30661,30701],{"className":30556},[161],[15,30558,30560],{"className":30559,"style":402},[146,401],[15,30561,741],{"className":30562},[406,407],[15,30564,30566,30569],{"className":30565},[61],[15,30567,3102],{"className":30568,"style":3187},[61,122],[15,30570,30572],{"className":30571},[338],[15,30573,30575,30595],{"className":30574},[342,472],[15,30576,30578,30592],{"className":30577},[346],[15,30579,30581],{"className":30580,"style":479},[350],[15,30582,30583,30586],{"style":3202},[15,30584],{"className":30585,"style":358},[357],[15,30587,30589],{"className":30588},[362,363,364,365],[15,30590,272],{"className":30591,"style":492},[61,122,365],[15,30593,497],{"className":30594},[496],[15,30596,30598],{"className":30597},[346],[15,30599,30601],{"className":30600,"style":504},[350],[15,30602],{},[15,30604,94],{"className":30605},[153],[15,30607],{"className":30608,"style":157},[127],[15,30610,30612,30615],{"className":30611},[531],[15,30613,293],{"className":30614,"style":537},[531,535,536],[15,30616,30618],{"className":30617},[338],[15,30619,30621,30650],{"className":30620},[342,472],[15,30622,30624,30647],{"className":30623},[346],[15,30625,30627],{"className":30626,"style":29964},[350],[15,30628,30629,30632],{"style":553},[15,30630],{"className":30631,"style":358},[357],[15,30633,30635],{"className":30634},[362,363,364,365],[15,30636,30638,30641,30644],{"className":30637},[61,365],[15,30639,14766],{"className":30640,"style":14792},[61,122,365],[15,30642,1588],{"className":30643},[132,365],[15,30645,272],{"className":30646,"style":492},[61,122,365],[15,30648,497],{"className":30649},[496],[15,30651,30653],{"className":30652},[346],[15,30654,30656],{"className":30655,"style":30028},[350],[15,30657],{},[15,30659],{"className":30660,"style":157},[127],[15,30662,30664,30667],{"className":30663},[61],[15,30665,3102],{"className":30666,"style":3187},[61,122],[15,30668,30670],{"className":30669},[338],[15,30671,30673,30693],{"className":30672},[342,472],[15,30674,30676,30690],{"className":30675},[346],[15,30677,30679],{"className":30678,"style":1362},[350],[15,30680,30681,30684],{"style":3202},[15,30682],{"className":30683,"style":358},[357],[15,30685,30687],{"className":30686},[362,363,364,365],[15,30688,14766],{"className":30689,"style":14792},[61,122,365],[15,30691,497],{"className":30692},[496],[15,30694,30696],{"className":30695},[346],[15,30697,30699],{"className":30698,"style":15696},[350],[15,30700],{},[15,30702,30704],{"className":30703,"style":402},[178,401],[15,30705,750],{"className":30706},[406,407]," 相互独立。令\n",[15,30709,30711,30801],{"className":30710,"translate":19},[18],[15,30712,30714],{"className":30713},[23],[25,30715,30716],{"xmlns":27},[29,30717,30718,30798],{},[32,30719,30720,30726,30728,30734,30736,30738,30744,30746,30752,30764,30766,30768,30770,30776,30778,30780,30782,30784,30786,30788,30790,30792,30794,30796],{},[266,30721,30722,30724],{},[35,30723,1027],{},[89,30725,91],{},[81,30727,244],{},[266,30729,30730,30732],{},[35,30731,30432],{},[89,30733,91],{},[81,30735,94],{"separator":48},[127,30737],{"width":9838},[266,30739,30740,30742],{},[35,30741,1027],{},[35,30743,272],{},[81,30745,244],{},[266,30747,30748,30750],{},[35,30749,30432],{},[35,30751,272],{},[266,30753,30754,30756],{},[81,30755,3737],{},[32,30757,30758,30760,30762],{},[35,30759,14766],{},[81,30761,2488],{},[35,30763,272],{},[81,30765,741],{"stretchy":86},[89,30767,91],{},[81,30769,239],{},[266,30771,30772,30774],{},[35,30773,30432],{},[35,30775,14766],{},[81,30777,750],{"stretchy":86},[127,30779],{"width":6445},[81,30781,741],{"stretchy":86},[35,30783,272],{},[81,30785,244],{},[89,30787,906],{},[81,30789,94],{"separator":48},[81,30791,97],{},[81,30793,94],{"separator":48},[35,30795,102],{},[81,30797,750],{"stretchy":86},[40,30799,30800],{"encoding":42},"x_1 = v_1, \\qquad x_k = v_k \\prod_{j \u003C k}(1 - v_j) \\quad (k = 2, \\dots, K)",[15,30802,30804,30859,30963,31076,31143],{"className":30803,"ariaHidden":48},[47],[15,30805,30807,30810,30850,30853,30856],{"className":30806},[52],[15,30808],{"className":30809,"style":2509},[56],[15,30811,30813,30816],{"className":30812},[61],[15,30814,1027],{"className":30815},[61,122],[15,30817,30819],{"className":30818},[338],[15,30820,30822,30842],{"className":30821},[342,472],[15,30823,30825,30839],{"className":30824},[346],[15,30826,30828],{"className":30827,"style":1212},[350],[15,30829,30830,30833],{"style":482},[15,30831],{"className":30832,"style":358},[357],[15,30834,30836],{"className":30835},[362,363,364,365],[15,30837,91],{"className":30838},[61,365],[15,30840,497],{"className":30841},[496],[15,30843,30845],{"className":30844},[346],[15,30846,30848],{"className":30847,"style":504},[350],[15,30849],{},[15,30851],{"className":30852,"style":128},[127],[15,30854,244],{"className":30855},[132],[15,30857],{"className":30858,"style":128},[127],[15,30860,30862,30865,30905,30908,30911,30914,30954,30957,30960],{"className":30861},[52],[15,30863],{"className":30864,"style":9428},[56],[15,30866,30868,30871],{"className":30867},[61],[15,30869,30432],{"className":30870,"style":123},[61,122],[15,30872,30874],{"className":30873},[338],[15,30875,30877,30897],{"className":30876},[342,472],[15,30878,30880,30894],{"className":30879},[346],[15,30881,30883],{"className":30882,"style":1212},[350],[15,30884,30885,30888],{"style":27013},[15,30886],{"className":30887,"style":358},[357],[15,30889,30891],{"className":30890},[362,363,364,365],[15,30892,91],{"className":30893},[61,365],[15,30895,497],{"className":30896},[496],[15,30898,30900],{"className":30899},[346],[15,30901,30903],{"className":30902,"style":504},[350],[15,30904],{},[15,30906,94],{"className":30907},[153],[15,30909],{"className":30910,"style":10026},[127],[15,30912],{"className":30913,"style":157},[127],[15,30915,30917,30920],{"className":30916},[61],[15,30918,1027],{"className":30919},[61,122],[15,30921,30923],{"className":30922},[338],[15,30924,30926,30946],{"className":30925},[342,472],[15,30927,30929,30943],{"className":30928},[346],[15,30930,30932],{"className":30931,"style":479},[350],[15,30933,30934,30937],{"style":482},[15,30935],{"className":30936,"style":358},[357],[15,30938,30940],{"className":30939},[362,363,364,365],[15,30941,272],{"className":30942,"style":492},[61,122,365],[15,30944,497],{"className":30945},[496],[15,30947,30949],{"className":30948},[346],[15,30950,30952],{"className":30951,"style":504},[350],[15,30953],{},[15,30955],{"className":30956,"style":128},[127],[15,30958,244],{"className":30959},[132],[15,30961],{"className":30962,"style":128},[127],[15,30964,30966,30969,31009,31012,31061,31064,31067,31070,31073],{"className":30965},[52],[15,30967],{"className":30968,"style":29838},[56],[15,30970,30972,30975],{"className":30971},[61],[15,30973,30432],{"className":30974,"style":123},[61,122],[15,30976,30978],{"className":30977},[338],[15,30979,30981,31001],{"className":30980},[342,472],[15,30982,30984,30998],{"className":30983},[346],[15,30985,30987],{"className":30986,"style":479},[350],[15,30988,30989,30992],{"style":27013},[15,30990],{"className":30991,"style":358},[357],[15,30993,30995],{"className":30994},[362,363,364,365],[15,30996,272],{"className":30997,"style":492},[61,122,365],[15,30999,497],{"className":31000},[496],[15,31002,31004],{"className":31003},[346],[15,31005,31007],{"className":31006,"style":504},[350],[15,31008],{},[15,31010],{"className":31011,"style":157},[127],[15,31013,31015,31018],{"className":31014},[531],[15,31016,3737],{"className":31017,"style":537},[531,535,536],[15,31019,31021],{"className":31020},[338],[15,31022,31024,31053],{"className":31023},[342,472],[15,31025,31027,31050],{"className":31026},[346],[15,31028,31030],{"className":31029,"style":29964},[350],[15,31031,31032,31035],{"style":553},[15,31033],{"className":31034,"style":358},[357],[15,31036,31038],{"className":31037},[362,363,364,365],[15,31039,31041,31044,31047],{"className":31040},[61,365],[15,31042,14766],{"className":31043,"style":14792},[61,122,365],[15,31045,2488],{"className":31046},[132,365],[15,31048,272],{"className":31049,"style":492},[61,122,365],[15,31051,497],{"className":31052},[496],[15,31054,31056],{"className":31055},[346],[15,31057,31059],{"className":31058,"style":30028},[350],[15,31060],{},[15,31062,741],{"className":31063},[146],[15,31065,91],{"className":31066},[61],[15,31068],{"className":31069,"style":772},[127],[15,31071,239],{"className":31072},[375],[15,31074],{"className":31075,"style":772},[127],[15,31077,31079,31082,31122,31125,31128,31131,31134,31137,31140],{"className":31078},[52],[15,31080],{"className":31081,"style":17976},[56],[15,31083,31085,31088],{"className":31084},[61],[15,31086,30432],{"className":31087,"style":123},[61,122],[15,31089,31091],{"className":31090},[338],[15,31092,31094,31114],{"className":31093},[342,472],[15,31095,31097,31111],{"className":31096},[346],[15,31098,31100],{"c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\\dots, x_K) \\sim \\operatorname{Dir}(\\boldsymbol{\\alpha})",[15,31226,31228,31347],{"className":31227,"ariaHidden":48},[47],[15,31229,31231,31234,31237,31277,31280,31283,31286,31289,31292,31295,31335,31338,31341,31344],{"className":31230},[52],[15,31232],{"className":31233,"style":142},[56],[15,31235,741],{"className":31236},[146],[15,31238,31240,31243],{"className":31239},[61],[15,31241,1027],{"className":31242},[61,122],[15,31244,31246],{"className":31245},[338],[15,31247,31249,31269],{"className":31248},[342,472],[15,31250,31252,31266],{"className":31251},[346],[15,31253,31255],{"className":31254,"style":1212},[350],[15,31256,31257,31260],{"style":482},[15,31258],{"className":31259,"style":358},[357],[15,31261,31263],{"className":31262},[362,363,364,365],[15,31264,91],{"className":31265},[61,365],[15,31267,497],{"className":31268},[496],[15,31270,31272],{"className":31271},[346],[15,31273,31275],{"className":31274,"style":504},[350],[15,31276],{},[15,31278,94],{"className":31279},[153],[15,31281],{"className":31282,"style":157},[127],[15,31284,97],{"className":31285},[161],[15,31287],{"className":31288,"style":157},[127],[15,31290,94],{"className":31291},[153],[15,31293],{"className":31294,"style":157},[127],[15,31296,31298,31301],{"className":31297},[61],[15,31299,1027],{"className":31300},[61,122],[15,31302,31304],{"className":31303},[338],[15,31305,31307,31327],{"className":31306},[342,472],[15,31308,31310,31324],{"className":31309},[346],[15,31311,31313],{"className":31312,"style":1271},[350],[15,31314,31315,31318],{"style":482},[15,31316],{"className":31317,"style":358},[357],[15,31319,31321],{"className":31320},[362,363,364,365],[15,31322,102],{"className":31323,"style":174},[61,122,365],[15,31325,497],{"className":31326},[496],[15,31328,31330],{"className":31329},[346],[15,31331,31333],{"className":31332,"style":504},[350],[15,31334],{},[15,31336,750],{"className":31337},[178],[15,31339],{"className":31340,"style":128},[127],[15,31342,4179],{"className":31343},[132],[15,31345],{"className":31346,"style":128},[127],[15,31348,31350,31353,31359,31362,31371],{"className":31349},[52],[15,31351],{"className":31352,"style":142},[56],[15,31354,31356],{"className":31355},[531],[15,31357,4182],{"className":31358},[61,1620],[15,31360,741],{"className":31361},[146],[15,31363,31365],{"className":31364},[61],[15,31366,31368],{"className":31367},[61],[15,31369,3102],{"className":31370},[61,3162],[15,31372,750],{"className":31373},[178],[11,31375,31376,31377,31405,31406,31476],{},"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 ",[15,31378,31380,31393],{"className":31379,"translate":19},[18],[15,31381,31383],{"className":31382},[23],[25,31384,31385],{"xmlns":27},[29,31386,31387,31391],{},[32,31388,31389],{},[89,31390,91],{},[40,31392,91],{"encoding":42},[15,31394,31396],{"className":31395,"ariaHidden":48},[47],[15,31397,31399,31402],{"className":31398},[52],[15,31400],{"className":31401,"style":859},[56],[15,31403,91],{"className":31404},[61]," 的木棍中依次折断出比例 ",[15,31407,31409,31427],{"className":31408,"translate":19},[18],[15,31410,31412],{"className":31411},[23],[25,31413,31414],{"xmlns":27},[29,31415,31416,31424],{},[32,31417,31418],{},[266,31419,31420,31422],{},[35,31421,30432],{},[35,31423,272],{},[40,31425,31426],{"encoding":42},"v_k",[15,31428,31430],{"className":31429,"ariaHidden":48},[47],[15,31431,31433,31436],{"className":31432},[52],[15,31434],{"className":31435,"style":2509},[56],[15,31437,31439,31442],{"className":31438},[61],[15,31440,30432],{"className":31441,"style":123},[61,122],[15,31443,31445],{"className":31444},[338],[15,31446,31448,31468],{"className":31447},[342,472],[15,31449,31451,31465],{"className":31450},[346],[15,31452,31454],{"className":31453,"style":479},[350],[15,31455,31456,31459],{"style":27013},[15,31457],{"className":31458,"style":358},[357],[15,31460,31462],{"className":31461},[362,363,364,365],[15,31463,272],{"className":31464,"style":492},[61,122,365],[15,31466,497],{"className":31467},[496],[15,31469,31471],{"className":31470},[346],[15,31472,31474],{"className":31473,"style":504},[350],[15,31475],{}," 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。",[877,31478,31480],{"id":31479},"_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",[11,31482,31483,31484,31518],{},"综合 2.1 至 2.5 节，可以把形状参数 ",[15,31485,31487,31500],{"className":31486,"translate":19},[18],[15,31488,31490],{"className":31489},[23],[25,31491,31492],{"xmlns":27},[29,31493,31494,31498],{},[32,31495,31496],{},[35,31497,3102],{"mathvariant":3101},[40,31499,4143],{"encoding":42},[15,31501,31503],{"className":31502,"ariaHidden":48},[47],[15,31504,31506,31509],{"className":31505},[52],[15,31507],{"className":31508,"style":2706},[56],[15,31510,31512],{"className":31511},[61],[15,31513,31515],{"className":31514},[61],[15,31516,3102],{"className":31517},[61,3162]," 的语义概括为三点：",[31520,31521,31522,31634,31810],"ol",{},[31523,31524,31525,31563,31564,31633],"li",{},[661,31526,31527,31528,31562],{},"在共轭更新中，",[15,31529,31531,31544],{"className":31530,"translate":19},[18],[15,31532,31534],{"className":31533},[23],[25,31535,31536],{"xmlns":27},[29,31537,31538,31542],{},[32,31539,31540],{},[35,31541,3102],{"mathvariant":3101},[40,31543,4143],{"encoding":42},[15,31545,31547],{"className":31546,"ariaHidden":48},[47],[15,31548,31550,31553],{"className":31549},[52],[15,31551],{"className":31552,"style":2706},[56],[15,31554,31556],{"className":31555},[61],[15,31557,31559],{"className":31558},[61],[15,31560,3102],{"className":31561},[61,3162]," 起先验样本计数的角色","（命题 2.5），其总强度 ",[15,31565,31567,31584],{"className":31566,"translate":19},[18],[15,31568,31570],{"className":31569},[23],[25,31571,31572],{"xmlns":27},[29,31573,31574,31582],{},[32,31575,31576],{},[266,31577,31578,31580],{},[35,31579,3102],{},[89,31581,278],{},[40,31583,19479],{"encoding":42},[15,31585,31587],{"className":31586,"ariaHidden":48},[47],[15,31588,31590,31593],{"className":31589},[52],[15,31591],{"className":31592,"style":2509},[56],[15,31594,31596,31599],{"className":31595},[61],[15,31597,3102],{"className":31598,"style":3187},[61,122],[15,31600,31602],{"className":31601},[338],[15,31603,31605,31625],{"className":31604},[342,472],[15,31606,31608,31622],{"className":31607},[346],[15,31609,31611],{"className":31610,"style":1212},[350],[15,31612,31613,31616],{"style":3202},[15,31614],{"className":31615,"style":358},[357],[15,31617,31619],{"className":31618},[362,363,364,365],[15,31620,278],{"className":31621},[61,365],[15,31623,497],{"className":31624},[496],[15,31626,31628],{"className":31627},[346],[15,31629,31631],{"className":31630,"style":504},[350],[15,31632],{}," 是「先验等价样本量」（推论 2.7）。",[31523,31635,31636,31774,31775,31809],{},[661,31637,31638,31639,31668,31669,31703,31704,31773],{},"在几何上，",[15,31640,31642,31656],{"className":31641,"translate":19},[18],[15,31643,31645],{"className":31644},[23],[25,31646,31647],{"xmlns":27},[29,31648,31649,31653],{},[32,31650,31651],{},[35,31652,3102],{},[40,31654,31655],{"encoding":42},"\\alpha",[15,31657,31659],{"className":31658,"ariaHidden":48},[47],[15,31660,31662,31665],{"className":31661},[52],[15,31663],{"className":31664,"style":6026},[56],[15,31666,3102],{"className":31667,"style":3187},[61,122]," 由方向 ",[15,31670,31672,31685],{"className":31671,"translate":19},[18],[15,31673,31675],{"className":31674},[23],[25,31676,31677],{"xmlns":27},[29,31678,31679,31683],{},[32,31680,31681],{},[35,31682,18335],{"mathvariant":3101},[40,31684,19440],{"encoding":42},[15,31686,31688],{"className":31687,"ariaHidden":48},[47],[15,31689,31691,31694],{"className":31690},[52],[15,31692],{"className":31693,"style":829},[56],[15,31695,31697],{"className":31696},[61],[15,31698,31700],{"className":31699},[61],[15,31701,18335],{"className":31702},[61,3162]," 与精度 ",[15,31705,31707,31724],{"className":31706,"translate":19},[18],[15,31708,31710],{"className":31709},[23],[25,31711,31712],{"xmlns":27},[29,31713,31714,31722],{},[32,31715,31716],{},[266,31717,31718,31720],{},[35,31719,3102],{},[89,31721,278],{},[40,31723,19479],{"encoding":42},[15,31725,31727],{"className":31726,"ariaHidden":48},[47],[15,31728,31730,31733],{"className":31729},[52],[15,31731],{"className":31732,"style":2509},[56],[15,31734,31736,31739],{"className":31735},[61],[15,31737,3102],{"className":31738,"style":3187},[61,122],[15,31740,31742],{"className":31741},[338],[15,31743,31745,31765],{"className":31744},[342,472],[15,31746,31748,31762],{"className":31747},[346],[15,31749,31751],{"className":31750,"style":1212},[350],[15,31752,31753,31756],{"style":3202},[15,31754],{"className":31755,"style":358},[357],[15,31757,31759],{"className":31758},[362,363,364,365],[15,31760,278],{"className":31761},[61,365],[15,31763,497],{"className":31764},[496],[15,31766,31768],{"className":31767},[346],[15,31769,31771],{"className":31770,"style":504},[350],[15,31772],{}," 两个正交分量构成","（2.2 节），精度决定围绕 ",[15,31776,31778,31791],{"className":31777,"translate":19},[18],[15,31779,31781],{"className":31780},[23],[25,31782,31783],{"xmlns":27},[29,31784,31785,31789],{},[32,31786,31787],{},[35,31788,18335],{"mathvariant":3101},[40,31790,19440],{"encoding":42},[15,31792,31794],{"className":31793,"ariaHidden":48},[47],[15,31795,31797,31800],{"className":31796},[52],[15,31798],{"className":31799,"style":829},[56],[15,31801,31803],{"className":31802},[61],[15,31804,31806],{"className":31805},[61],[15,31807,18335],{"className":31808},[61,3162]," 的收缩程度。",[31523,31811,31812,14421,31978,32070,32071,32163],{},[661,31813,31814,31815,31907,31908,31977],{},"在似然意义上，",[15,31816,31818,31840],{"className":31817,"translate":19},[18],[15,31819,31821],{"className":31820},[23],[25,31822,31823],{"xmlns":27},[29,31824,31825,31837],{},[32,31826,31827,31833,31835],{},[266,31828,31829,31831],{},[35,31830,3102],{},[35,31832,1068],{},[81,31834,239],{},[89,31836,91],{},[40,31838,31839],{"encoding":42},"\\alpha_i - 1",[15,31841,31843,31898],{"className":31842,"ariaHidden":48},[47],[15,31844,31846,31849,31889,31892,31895],{"className":31845},[52],[15,31847],{"className":31848,"style":12608},[56],[15,31850,31852,31855],{"className":31851},[61],[15,31853,3102],{"className":31854,"style":3187},[61,122],[15,31856,31858],{"className":31857},[338],[15,31859,31861,31881],{"className":31860},[342,472],[15,31862,31864,31878],{"className":31863},[346],[15,31865,31867],{"className":31866,"style":1362},[350],[15,31868,31869,31872],{"style":3202},[15,31870],{"className":31871,"style":358},[357],[15,31873,31875],{"className":31874},[362,363,364,365],[15,31876,1068],{"className":31877},[61,122,365],[15,31879,497],{"className":31880},[496],[15,31882,31884],{"className":31883},[346],[15,31885,31887],{"className":31886,"style":504},[350],[15,31888],{},[15,31890],{"className":31891,"style":772},[127],[15,31893,239],{"className":31894},[375],[15,31896],{"className":31897,"style":772},[127],[15,31899,31901,31904],{"className":31900},[52],[15,31902],{"className":31903,"style":859},[56],[15,31905,91],{"className":31906},[61]," 是对数密度中 ",[15,31909,31911,31928],{"className":31910,"translate":19},[18],[15,31912,31914],{"className":31913},[23],[25,31915,31916],{"xmlns":27},[29,31917,31918,31926],{},[32,31919,31920],{},[266,31921,31922,31924],{},[35,31923,1027],{},[35,31925,1068],{},[40,31927,13842],{"encoding":42},[15,31929,31931],{"className":31930,"ariaHidden":48},[47],[15,31932,31934,31937],{"className":31933},[52],[15,31935],{"className":31936,"style":2509},[56],[15,31938,31940,31943],{"className":31939},[61],[15,31941,1027],{"className":31942},[61,122],[15,31944,31946],{"className":31945},[338],[15,31947,31949,31969],{"className":31948},[342,472],[15,31950,31952,31966],{"className":31951},[346],[15,31953,31955],{"className":31954,"style":1362},[350],[15,31956,31957,31960],{"style":482},[15,31958],{"className":31959,"style":358},[357],[15,31961,31963],{"className":31962},[362,363,364,365],[15,31964,1068],{"className":31965},[61,122,365],[15,31967,497],{"className":31968},[496],[15,31970,31972],{"className":31971},[346],[15,31973,31975],{"className":31974,"style":504},[350],[15,31976],{}," 的指数",[15,31979,31981,32003],{"className":31980,"translate":19},[18],[15,31982,31984],{"className":31983},[23],[25,31985,31986],{"xmlns":27},[29,31987,31988,32000],{},[32,31989,31990,31996,31998],{},[266,31991,31992,31994],{},[35,31993,3102],{},[35,31995,1068],{},[81,31997,2488],{},[89,31999,91],{},[40,32001,32002],{"encoding":42},"\\alpha_i \u003C 1",[15,32004,32006,32061],{"className":32005,"ariaHidden":48},[47],[15,32007,32009,32012,32052,32055,32058],{"className":32008},[52],[15,32010],{"className":32011,"style":1751},[56],[15,32013,32015,32018],{"className":32014},[61],[15,32016,3102],{"className":32017,"style":3187},[61,122],[15,32019,32021],{"className":32020},[338],[15,32022,32024,32044],{"className":32023},[342,472],[15,32025,32027,32041],{"className":32026},[346],[15,32028,32030],{"className":32029,"style":1362},[350],[15,32031,32032,32035],{"style":3202},[15,32033],{"className":32034,"style":358},[357],[15,32036,32038],{"className":32037},[362,363,364,365],[15,32039,1068],{"className":32040},[61,122,365],[15,32042,497],{"className":32043},[496],[15,32045,32047],{"className":32046},[346],[15,32048,32050],{"className":32049,"style":504},[350],[15,32051],{},[15,32053],{"className":32054,"style":128},[127],[15,32056,2488],{"className":32057},[132],[15,32059],{"className":32060,"style":128},[127],[15,32062,32064,32067],{"className":32063},[52],[15,32065],{"className":32066,"style":859},[56],[15,32068,91],{"className":32069},[61]," 意味着密度在 ",[15,32072,32074,32096],{"className":32073,"translate":19},[18],[15,32075,32077],{"className":32076},[23],[25,32078,32079],{"xmlns":27},[29,32080,32081,32093],{},[32,32082,32083,32089,32091],{},[266,32084,32085,32087],{},[35,32086,1027],{},[35,32088,1068],{},[81,32090,20691],{},[89,32092,278],{},[40,32094,32095],{"encoding":42},"x_i \\to 0",[15,32097,32099,32154],{"className":32098,"ariaHidden":48},[47],[15,32100,32102,32105,32145,32148,32151],{"className":32101},[52],[15,32103],{"className":32104,"style":2509},[56],[15,32106,32108,32111],{"className":32107},[61],[15,32109,1027],{"className":32110},[61,122],[15,32112,32114],{"className":32113},[338],[15,32115,32117,32137],{"className":32116},[342,472],[15,32118,32120,32134],{"className":32119},[346],[15,32121,32123],{"className":32122,"style":1362},[350],[15,32124,32125,32128],{"style":482},[15,32126],{"className":32127,"style":358},[357],[15,32129,32131],{"className":32130},[362,363,364,365],[15,32132,1068],{"className":32133},[61,122,365],[15,32135,497],{"className":32136},[496],[15,32138,32140],{"className":32139},[346],[15,32141,32143],{"className":32142,"style":504},[350],[15,32144],{},[15,32146],{"className":32147,"style":128},[127],[15,32149,20691],{"className":32150},[132],[15,32152],{"className":32153,"style":128},[127],[15,32155,32157,32160],{"className":32156},[52],[15,32158],{"className":32159,"style":859},[56],[15,32161,278],{"className":32162},[61]," 处发散。",[11,32165,32166,32167,32170,32171,32232],{},"对于",[661,32168,32169],{},"对称","情形 ",[15,32172,32174,32193],{"className":32173,"translate":19},[18],[15,32175,32177],{"className":32176},[23],[25,32178,32179],{"xmlns":27},[29,32180,32181,32191],{},[32,32182,32183,32185,32187,32189],{},[35,32184,3102],{"mathvariant":3101},[81,32186,244],{},[35,32188,3102],{},[89,32190,91],{"mathvariant":251},[40,32192,23968],{"encoding":42},[15,32194,32196,32220],{"className":32195,"ariaHidden":48},[47],[15,32197,32199,32202,32211,32214,32217],{"className":32198},[52],[15,32200],{"className":32201,"style":2706},[56],[15,32203,32205],{"className":32204},[61],[15,32206,32208],{"className":32207},[61],[15,32209,3102],{"className":32210},[61,3162],[15,32212],{"className":32213,"style":128},[127],[15,32215,244],{"className":32216},[132],[15,32218],{"className":32219,"style":128},[127],[15,32221,32223,32226,32229],{"className":32222},[52],[15,32224],{"className":32225,"style":859},[56],[15,32227,3102],{"className":32228,"style":3187},[61,122],[15,32230,91],{"className":32231},[61,411],"，这三重语义给出一条完整的形态谱：",[32234,32235,32236,32255],"table",{},[32237,32238,32239],"thead",{},[21298,32240,32241,32246,32249,32252],{},[32242,32243,32245],"th",{"align":32244},"left","参数区间",[32242,32247,32248],{"align":32244},"分布的形态",[32242,32250,32251],{"align":32244},"采样的典型特征",[32242,32253,32254],{"align":32244},"典型用途",[32256,32257,32258,32432,32495],"tbody",{},[21298,32259,32260,32314,32398,32429],{},[32261,32262,32263],"td",{"align":32244},[15,32264,32266,32284],{"className":32265,"translate":19},[18],[15,32267,32269],{"className":32268},[23],[25,32270,32271],{"xmlns":27},[29,32272,32273,32281],{},[32,32274,32275,32277,32279],{},[35,32276,3102],{},[81,32278,20691],{},[89,32280,278],{},[40,32282,32283],{"encoding":42},"\\alpha \\to 0",[15,32285,32287,32305],{"className":32286,"ariaHidden":48},[47],[15,32288,32290,32293,32296,32299,32302],{"className":32289},[52],[15,32291],{"className":32292,"style":6026},[56],[15,32294,3102],{"className":32295,"style":3187},[61,122],[15,32297],{"className":32298,"style":128},[127],[15,32300,20691],{"className":32301},[132],[15,32303],{"className":32304,"style":128},[127],[15,32306,32308,32311],{"className":32307},[52],[15,32309],{"className":32310,"style":859},[56],[15,32312,278],{"className":32313},[61],[32261,32315,32316,32317,32397],{"align":32244},"质量集中于顶点 ",[15,32318,32320,32342],{"className":32319,"translate":19},[18],[15,32321,32323],{"className":32322},[23],[25,32324,32325],{"xmlns":27},[29,32326,32327,32339],{},[32,32328,32329,32331,32337],{},[81,32330,87],{"stretchy":86},[266,32332,32333,32335],{},[35,32334,1980],{"mathvariant":251},[35,32336,1068],{},[81,32338,105],{"stretchy":86},[40,32340,32341],{"encoding":42},"\\{\\mathbf{e}_i\\}",[15,32343,32345],{"className":32344,"ariaHidden":48},[47],[15,32346,32348,32351,32354,32394],{"className":32347},[52],[15,32349],{"className":32350,"style":142},[56],[15,32352,87],{"className":32353},[146],[15,32355,32357,32360],{"className":32356},[61],[15,32358,1980],{"className":32359},[61,411],[15,32361,32363],{"className":32362},[338],[15,32364,32366,32386],{"className":32365},[342,472],[15,32367,32369,32383],{"className":32368},[346],[15,32370,32372],{"className":32371,"style":1362},[350],[15,32373,32374,32377],{"style":482},[15,32375],{"className":32376,"style":358},[357],[15,32378,32380],{"className":32379},[362,363,364,365],[15,32381,1068],{"className":32382},[61,122,365],[15,32384,497],{"className":32385},[496],[15,32387,32389],{"className":32388},[346],[15,32390,32392],{"className":32391,"style":504},[350],[15,32393],{},[15,32395,105],{"className":32396},[178]," 邻域",[32261,32399,32400,32401],{"align":32244},"极端稀疏，单个分量接近 ",[15,32402,32404,32417],{"className":32403,"translate":19},[18],[15,32405,32407],{"className":32406},[23],[25,32408,32409],{"xmlns":27},[29,32410,32411,32415],{},[32,32412,32413],{},[89,32414,91],{},[40,32416,91],{"encoding":42},[15,32418,32420],{"className":32419,"ariaHidden":48},[47],[15,32421,32423,32426],{"className":32422},[52],[15,32424],{"className":32425,"style":859},[56],[15,32427,91],{"className":32428},[61],[32261,32430,32431],{"align":32244},"LDA 文档—主题混合（文档只涉少数主题）",[21298,32433,32434,32486,32489,32492],{},[32261,32435,32436],{"align":32244},[15,32437,32439,32456],{"className":32438,"translate":19},[18],[15,32440,32442],{"className":32441},[23],[25,32443,32444],{"xmlns":27},[29,32445,32446,32454],{},[32,32447,32448,32450,32452],{},[35,32449,3102],{},[81,32451,244],{},[89,32453,91],{},[40,32455,24057],{"encoding":42},[15,32457,32459,32477],{"className":32458,"ariaHidden":48},[47],[15,32460,32462,32465,32468,32471,32474],{"className":32461},[52],[15,32463],{"className":32464,"style":6026},[56],[15,32466,3102],{"className":32467,"style":3187},[61,122],[15,32469],{"className":32470,"style":128},[127],[15,32472,244],{"className":32473},[132],[15,32475],{"className":32476,"style":128},[127],[15,32478,32480,32483],{"className":32479},[52],[15,32481],{"className":32482,"style":859},[56],[15,32484,91],{"className":32485},[61],[32261,32487,32488],{"align":32244},"单纯形上的均匀分布",[32261,32490,32491],{"align":32244},"各分量量级相当，无偏好",[32261,32493,32494],{"align":32244},"无信息先验",[21298,32496,32497,32550,32622,32662],{},[32261,32498,32499],{"align":32244},[15,32500,32502,32520],{"className":32501,"translate":19},[18],[15,32503,32505],{"className":32504},[23],[25,32506,32507],{"xmlns":27},[29,32508,32509,32517],{},[32,32510,32511,32513,32515],{},[35,32512,3102],{},[81,32514,1588],{},[89,32516,91],{},[40,32518,32519],{"encoding":42},"\\alpha > 1",[15,32521,32523,32541],{"className":32522,"ariaHidden":48},[47],[15,32524,32526,32529,32532,32535,32538],{"className":32525},[52],[15,32527],{"className":32528,"style":4650},[56],[15,32530,3102],{"className":32531,"style":3187},[61,122],[15,32533],{"className":32534,"style":128},[127],[15,32536,1588],{"className":32537},[132],[15,32539],{"className":32540,"style":128},[127],[15,32542,32544,32547],{"className":32543},[52],[15,32545],{"className":32546,"style":859},[56],[15,32548,91],{"className":32549},[61],[32261,32551,32552,32553,32592,32593,32621],{"align":32244},"集中于中心 ",[15,32554,32556,32574],{"className":32555,"translate":19},[18],[15,32557,32559],{"className":32558},[23],[25,32560,32561],{"xmlns":27},[29,32562,32563,32571],{},[32,32564,32565,32567,32569],{},[89,32566,91],{"mathvariant":251},[35,32568,3018],{"mathvariant":231},[35,32570,102],{},[40,32572,32573],{"encoding":42},"\\mathbf{1}\u002FK",[15,32575,32577],{"className":32576,"ariaHidden":48},[47],[15,32578,32580,32583,32586,32589],{"className":32579},[52],[15,32581],{"className":32582,"style":142},[56],[15,32584,91],{"className":32585},[61,411],[15,32587,3018],{"className":32588},[61],[15,32590,102],{"className":32591,"style":174},[61,122],"，随 ",[15,32594,32596,32609],{"className":32595,"translate":19},[18],[15,32597,32599],{"className":32598},[23],[25,32600,32601],{"xmlns":27},[29,32602,32603,32607],{},[32,32604,32605],{},[35,32606,3102],{},[40,32608,31655],{"encoding":42},[15,32610,32612],{"className":32611,"ariaHidden":48},[47],[15,32613,32615,32618],{"className":32614},[52],[15,32616],{"className":32617,"style":6026},[56],[15,32619,3102],{"className":32620,"style":3187},[61,122]," 增大而收缩",[32261,32623,32624,32625,32661],{"align":32244},"各分量接近 ",[15,32626,32628,32646],{"className":32627,"translate":19},[18],[15,32629,32631],{"className":32630},[23],[25,32632,32633],{"xmlns":27},[29,32634,32635,32643],{},[32,32636,32637,32639,32641],{},[89,32638,91],{},[35,32640,3018],{"mathvariant":231},[35,32642,102],{},[40,32644,32645],{"encoding":42},"1\u002FK",[15,32647,32649],{"className":32648,"ariaHidden":48},[47],[15,32650,32652,32655,32658],{"className":32651},[52],[15,32653],{"className":32654,"style":142},[56],[15,32656,3047],{"className":32657},[61],[15,32659,102],{"className":32660,"style":174},[61,122]," 的平滑向量",[32261,32663,32664],{"align":32244},"强正则化、收缩估计",[11,32666,32667,32668,32737,32738,32830,32831,20764,32922,33037,33038,33259],{},"此处必须澄清一个常见但过强的说法：把 ",[15,32669,32671,32688],{"className":32670,"translate":19},[18],[15,32672,32674],{"className":32673},[23],[25,32675,32676],{"xmlns":27},[29,32677,32678,32686],{},[32,32679,32680],{},[266,32681,32682,32684],{},[35,32683,3102],{},[35,32685,1068],{},[40,32687,9718],{"encoding":42},[15,32689,32691],{"className":32690,"ariaHidden":48},[47],[15,32692,32694,32697],{"className":32693},[52],[15,32695],{"className":32696,"style":2509},[56],[15,32698,32700,32703],{"className":32699},[61],[15,32701,3102],{"className":32702,"style":3187},[61,122],[15,32704,32706],{"className":32705},[338],[15,32707,32709,32729],{"className":32708},[342,472],[15,32710,32712,32726],{"className":32711},[346],[15,32713,32715],{"className":32714,"style":1362},[350],[15,32716,32717,32720],{"style":3202},[15,32718],{"className":32719,"style":358},[357],[15,32721,32723],{"className":32722},[362,363,364,365],[15,32724,1068],{"className":32725},[61,122,365],[15,32727,497],{"className":32728},[496],[15,32730,32732],{"className":32731},[346],[15,32733,32735],{"className":32734,"style":504},[350],[15,32736],{}," 直接称为「观测到的次数」只在 ",[15,32739,32741,32763],{"className":32740,"translate":19},[18],[15,32742,32744],{"className":32743},[23],[25,32745,32746],{"xmlns":27},[29,32747,32748,32760],{},[32,32749,32750,32756,32758],{},[266,32751,32752,32754],{},[35,32753,3102],{},[35,32755,1068],{},[81,32757,275],{},[89,32759,91],{},[40,32761,32762],{"encoding":42},"\\alpha_i \\ge 1",[15,32764,32766,32821],{"className":32765,"ariaHidden":48},[47],[15,32767,32769,32772,32812,32815,32818],{"className":32768},[52],[15,32770],{"className":32771,"style":22477},[56],[15,32773,32775,32778],{"className":32774},[61],[15,32776,3102],{"className":32777,"style":3187},[61,122],[15,32779,32781],{"className":32780},[338],[15,32782,32784,32804],{"className":32783},[342,472],[15,32785,32787,32801],{"className":32786},[346],[15,32788,32790],{"className":32789,"style":1362},[350],[15,32791,32792,32795],{"style":3202},[15,32793],{"className":32794,"style":358},[357],[15,32796,32798],{"className":32797},[362,363,364,365],[15,32799,1068],{"className":32800},[61,122,365],[15,32802,497],{"className":32803},[496],[15,32805,32807],{"className":32806},[346],[15,32808,32810],{"className":32809,"style":504},[350],[15,32811],{},[15,32813],{"className":32814,"style":128},[127],[15,32816,275],{"className":32817},[132],[15,32819],{"className":32820,"style":128},[127],[15,32822,32824,32827],{"className":32823},[52],[15,32825],{"className":32826,"style":859},[56],[15,32828,91],{"className":32829},[61]," 时在字面上成立。当 ",[15,32832,32834,32855],{"className":32833,"translate":19},[18],[15,32835,32837],{"className":32836},[23],[25,32838,32839],{"xmlns":27},[29,32840,32841,32853],{},[32,32842,32843,32849,32851],{},[266,32844,32845,32847],{},[35,32846,3102],{},[35,32848,1068],{},[81,32850,2488],{},[89,32852,91],{},[40,32854,32002],{"encoding":42},[15,32856,32858,32913],{"className":32857,"ariaHidden":48},[47],[15,32859,32861,32864,32904,32907,32910],{"className":32860},[52],[15,32862],{"className":32863,"style":1751},[56],[15,32865,32867,32870],{"className":32866},[61],[15,32868,3102],{"className":32869,"style":3187},[61,122],[15,32871,32873],{"className":32872},[338],[15,32874,32876,32896],{"className":32875},[342,472],[15,32877,32879,32893],{"className":32878},[346],[15,32880,32882],{"className":32881,"style":1362},[350],[15,32883,32884,32887],{"style":3202},[15,32885],{"className":32886,"style":358},[357],[15,32888,32890],{"className":32889},[362,363,364,365],[15,32891,1068],{"className":32892},[61,122,365],[15,32894,497],{"className":32895},[496],[15,32897,32899],{"className":32898},[346],[15,32900,32902],{"className":32901,"style":504},[350],[15,32903],{},[15,32905],{"className":32906,"style":128},[127],[15,32908,2488],{"className":32909},[132],[15,32911],{"className":32912,"style":128},[127],[15,32914,32916,32919],{"className":32915},[52],[15,32917],{"className":32918,"style":859},[56],[15,32920,91],{"className":32921},[61],[15,32923,32925,32951],{"className":32924,"translate":19},[18],[15,32926,32928],{"className":32927},[23],[25,32929,32930],{"xmlns":27},[29,32931,32932,32948],{},[32,32933,32934,32940,32942,32944,32946],{},[266,32935,32936,32938],{},[35,32937,3102],{},[35,32939,1068],{},[81,32941,239],{},[89,32943,91],{},[81,32945,2488],{},[89,32947,278],{},[40,32949,32950],{"encoding":42},"\\alpha_i - 1 \u003C 0",[15,32952,32954,33009,33028],{"className":32953,"ariaHidden":48},[47],[15,32955,32957,32960,33000,33003,33006],{"className":32956},[52],[15,32958],{"className":32959,"style":12608},[56],[15,32961,32963,32966],{"className":32962},[61],[15,32964,3102],{"className":32965,"style":3187},[61,122],[15,32967,32969],{"className":32968},[338],[15,32970,32972,32992],{"className":32971},[342,472],[15,32973,32975,32989],{"className":32974},[346],[15,32976,32978],{"className":32977,"style":1362},[350],[15,32979,32980,32983],{"style":3202},[15,32981],{"className":32982,"style":358},[357],[15,32984,32986],{"className":32985},[362,363,364,365],[15,32987,1068],{"className":32988},[61,122,365],[15,32990,497],{"className":32991},[496],[15,32993,32995],{"className":32994},[346],[15,32996,32998],{"className":32997,"style":504},[350],[15,32999],{},[15,33001],{"className":33002,"style":772},[127],[15,33004,239],{"className":33005},[375],[15,33007],{"className":33008,"style":772},[127],[15,33010,33012,33016,33019,33022,33025],{"className":33011},[52],[15,33013],{"className":33014,"style":33015},[56],"height:0.6835em;vertical-align:-0.0391em;",[15,33017,91],{"className":33018},[61],[15,33020],{"className":33021,"style":128},[127],[15,33023,2488],{"className":33024},[132],[15,33026],{"className":33027,"style":128},[127],[15,33029,33031,33034],{"className":33030},[52],[15,33032],{"className":33033,"style":859},[56],[15,33035,278],{"className":33036},[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：",[661,33039,33040,33074,33075,33166,33167,33258],{},[15,33041,33043,33056],{"className":33042,"translate":19},[18],[15,33044,33046],{"className":33045},[23],[25,33047,33048],{"xmlns":27},[29,33049,33050,33054],{},[32,33051,33052],{},[35,33053,3102],{"mathvariant":3101},[40,33055,4143],{"encoding":42},[15,33057,33059],{"className":33058,"ariaHidden":48},[47],[15,33060,33062,33065],{"className":33061},[52],[15,33063],{"className":33064,"style":2706},[56],[15,33066,33068],{"className":33067},[61],[15,33069,33071],{"className":33070},[61],[15,33072,3102],{"className":33073},[61,3162]," 是伪计数概念的连续化，其在 ",[15,33076,33078,33099],{"className":33077,"translate":19},[18],[15,33079,33081],{"className":33080},[23],[25,33082,33083],{"xmlns":27},[29,33084,33085,33097],{},[32,33086,33087,33093,33095],{},[266,33088,33089,33091],{},[35,33090,3102],{},[35,33092,1068],{},[81,33094,275],{},[89,33096,91],{},[40,33098,32762],{"encoding":42},[15,33100,33102,33157],{"className":33101,"ariaHidden":48},[47],[15,33103,33105,33108,33148,33151,33154],{"className":33104},[52],[15,33106],{"className":33107,"style":22477},[56],[15,33109,33111,33114],{"className":33110},[61],[15,33112,3102],{"className":33113,"style":3187},[61,122],[15,33115,33117],{"className":33116},[338],[15,33118,33120,33140],{"className":33119},[342,472],[15,33121,33123,33137],{"className":33122},[346],[15,33124,33126],{"className":33125,"style":1362},[350],[15,33127,33128,33131],{"style":3202},[15,33129],{"className":33130,"style":358},[357],[15,33132,33134],{"className":33133},[362,363,364,365],[15,33135,1068],{"className":33136},[61,122,365],[15,33138,497],{"className":33139},[496],[15,33141,33143],{"className":33142},[346],[15,33144,33146],{"className":33145,"style":504},[350],[15,33147],{},[15,33149],{"className":33150,"style":128},[127],[15,33152,275],{"className":33153},[132],[15,33155],{"className":33156,"style":128},[127],[15,33158,33160,33163],{"className":33159},[52],[15,33161],{"className":33162,"style":859},[56],[15,33164,91],{"className":33165},[61]," 区域退化回字面意义上的计数解释，而在 ",[15,33168,33170,33191],{"className":33169,"translate":19},[18],[15,33171,33173],{"className":33172},[23],[25,33174,33175],{"xmlns":27},[29,33176,33177,33189],{},[32,33178,33179,33185,33187],{},[266,33180,33181,33183],{},[35,33182,3102],{},[35,33184,1068],{},[81,33186,2488],{},[89,33188,91],{},[40,33190,32002],{"encoding":42},[15,33192,33194,33249],{"className":33193,"ariaHidden":48},[47],[15,33195,33197,33200,33240,33243,33246],{"className":33196},[52],[15,33198],{"className":33199,"style":1751},[56],[15,33201,33203,33206],{"className":33202},[61],[15,33204,3102],{"className":33205,"style":3187},[61,122],[15,33207,33209],{"className":33208},[338],[15,33210,33212,33232],{"className":33211},[342,472],[15,33213,33215,33229],{"className":33214},[346],[15,33216,33218],{"className":33217,"style":1362},[350],[15,33219,33220,33223],{"style":3202},[15,33221],{"className":33222,"style":358},[357],[15,33224,33226],{"className":33225},[362,363,364,365],[15,33227,1068],{"className":33228},[61,122,365],[15,33230,497],{"className":33231},[496],[15,33233,33235],{"className":33234},[346],[15,33236,33238],{"className":33237,"style":504},[350],[15,33239],{},[15,33241],{"className":33242,"style":128},[127],[15,33244,2488],{"className":33245},[132],[15,33247],{"className":33248,"style":128},[127],[15,33250,33252,33255],{"className":33251},[52],[15,33253],{"className":33254,"style":859},[56],[15,33256,91],{"className":33257},[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。",[872,33261,33263],{"id":33262},"三主观逻辑到证据深度学习","三、主观逻辑到证据深度学习",[877,33265,33267],{"id":33266},"_31-主观逻辑的-opinion-表示","3.1 主观逻辑的 opinion 表示",[11,33269,33270,33271,33274],{},"证据深度学习的直接理论前身是 Josang 的",[661,33272,33273],{},"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。",[882,33276,33278,33467,34054],{"title":33277},"定义 3.1（opinion）",[11,33279,33280,33281,33374,33375,33466],{},"在有限域 ",[15,33282,33284,33315],{"className":33283,"translate":19},[18],[15,33285,33287],{"className":33286},[23],[25,33288,33289],{"xmlns":27},[29,33290,33291,33312],{},[32,33292,33293,33296,33298,33300,33302,33304,33306,33308,33310],{},[35,33294,33295],{"mathvariant":37},"Y",[81,33297,244],{},[81,33299,87],{"stretchy":86},[89,33301,91],{},[81,33303,94],{"separator":48},[81,33305,97],{},[81,33307,94],{"separator":48},[35,33309,102],{},[81,33311,105],{"stretchy":86},[40,33313,33314],{"encoding":42},"\\mathcal{Y} = \\{1, \\dots, K\\}",[15,33316,33318,33338],{"className":33317,"ariaHidden":48},[47],[15,33319,33321,33325,33329,33332,33335],{"className":33320},[52],[15,33322],{"className":33323,"style":33324},[56],"height:0.7805em;vertical-align:-0.0972em;",[15,33326,33295],{"className":33327,"style":33328},[61,62],"margin-right:0.0822em;",[15,33330],{"className":33331,"style":128},[127],[15,33333,244],{"className":33334},[132],[15,33336],{"className":33337,"style":128},[127],[15,33339,33341,33344,33347,33350,33353,33356,33359,33362,33365,33368,33371],{"className":33340},[52],[15,33342],{"className":33343,"style":142},[56],[15,33345,87],{"className":33346},[146],[15,33348,91],{"className":33349},[61],[15,33351,94],{"className":33352},[153],[15,33354],{"className":33355,"style":157},[127],[15,33357,97],{"className":33358},[161],[15,33360],{"className":33361,"style":157},[127],[15,33363,94],{"className":33364},[153],[15,33366],{"className":33367,"style":157},[127],[15,33369,102],{"className":33370,"style":174},[61,122],[15,33372,105],{"className":33373},[178]," 上的一个 opinion 是三元组 ",[15,33376,33378,33412],{"className":33377,"translate":19},[18],[15,33379,33381],{"className":33380},[23],[25,33382,33383],{"xmlns":27},[29,33384,33385,33409],{},[32,33386,33387,33390,33392,33394,33397,33399,33402,33404,33407],{},[35,33388,33389],{},"ω",[81,33391,244],{},[81,33393,741],{"stretchy":86},[35,33395,33396],{"mathvariant":251},"b",[81,33398,94],{"separator":48},[35,33400,33401],{},"u",[81,33403,94],{"separator":48},[35,33405,33406],{"mathvariant":251},"a",[81,33408,750],{"stretchy":86},[40,33410,33411],{"encoding":42},"\\omega = (\\mathbf{b}, u, \\mathbf{a})",[15,33413,33415,33433],{"className":33414,"ariaHidden":48},[47],[15,33416,33418,33421,33424,33427,33430],{"className":33417},[52],[15,33419],{"className":33420,"style":6026},[56],[15,33422,33389],{"className":33423,"style":123},[61,122],[15,33425],{"className":33426,"style":128},[127],[15,33428,244],{"className":33429},[132],[15,33431],{"className":33432,"style":128},[127],[15,33434,33436,33439,33442,33445,33448,33451,33454,33457,33460,33463],{"className":33435},[52],[15,33437],{"className":33438,"style":142},[56],[15,33440,741],{"className":33441},[146],[15,33443,33396],{"className":33444},[61,411],[15,33446,94],{"className":33447},[153],[15,33449],{"className":33450,"style":157},[127],[15,33452,33401],{"className":33453},[61,122],[15,33455,94],{"className":33456},[153],[15,33458],{"className":33459,"style":157},[127],[15,33461,33406],{"className":33462},[61,411],[15,33464,750],{"className":33465},[178],"，其中",[33468,33469,33470,33724,33803],"ul",{},[31523,33471,33472,33720,33721,867],{},[15,33473,33475,33529],{"className":33474,"translate":19},[18],[15,33476,33478],{"className":33477},[23],[25,33479,33480],{"xmlns":27},[29,33481,33482,33526],{},[32,33483,33484,33486,33488,33490,33496,33498,33500,33502,33508,33510,33512,33514,33516,33518,33520],{},[35,33485,33396],{"mathvariant":251},[81,33487,244],{},[81,33489,741],{"stretchy":86},[266,33491,33492,33494],{},[35,33493,33396],{},[89,33495,91],{},[81,33497,94],{"separator":48},[81,33499,97],{},[81,33501,94],{"separator":48},[266,33503,33504,33506],{},[35,33505,33396],{},[35,33507,102],{},[81,33509,750],{"stretchy":86},[81,33511,83],{},[81,33513,14810],{"stretchy":86},[89,33515,278],{},[81,33517,94],{"separator":48},[89,33519,91],{},[227,33521,33522,33524],{},[81,33523,14819],{"stretchy":86},[35,33525,102],{},[40,33527,33528],{"encoding":42},"\\mathbf{b} = (b_1, \\dots, b_K) \\in [0,1]^K",[15,33530,33532,33550,33669],{"className":33531,"ariaHidden":48},[47],[15,33533,33535,33538,33541,33544,33547],{"className":33534},[52],[15,33536],{"className":33537,"style":10443},[56],[15,33539,33396],{"className":33540},[61,411],[15,33542],{"className":33543,"style":128},[127],[15,33545,244],{"className":33546},[132],[15,33548],{"className":33549,"style":128},[127],[15,33551,33553,33556,33559,33599,33602,33605,33608,33611,33614,33617,33657,33660,33663,33666],{"className":33552},[52],[15,33554],{"className":33555,"style":142},[56],[15,33557,741],{"className":33558},[146],[15,33560,33562,33565],{"className":33561},[61],[15,33563,33396],{"className":33564},[61,122],[15,33566,33568],{"className":33567},[338],[15,33569,33571,33591],{"className":33570},[342,472],[15,33572,33574,33588],{"className":33573},[346],[15,33575,33577],{"className":33576,"style":1212},[350],[15,33578,33579,33582],{"style":482},[15,33580],{"className":33581,"style":358},[357],[15,33583,33585],{"className":33584},[362,363,364,365],[15,33586,91],{"className":33587},[61,365],[15,33589,497],{"className":33590},[496],[15,33592,33594],{"className":33593},[346],[15,33595,33597],{"className":33596,"style":504},[350],[15,33598],{},[15,33600,94],{"className":33601},[153],[15,33603],{"className":33604,"style":157},[127],[15,33606,97],{"className":33607},[161],[15,33609],{"className":33610,"style":157},[127],[15,33612,94],{"className":33613},[153],[15,33615],{"className":33616,"style":157},[127],[15,33618,33620,33623],{"className":33619},[61],[15,33621,33396],{"className":33622},[61,122],[15,33624,33626],{"className":33625},[338],[15,33627,33629,33649],{"className":33628},[342,472],[15,33630,33632,33646],{"className":33631},[346],[15,33633,33635],{"className":33634,"style":1271},[350],[15,33636,33637,33640],{"style":482},[15,33638],{"className":33639,"style":358},[357],[15,33641,33643],{"className":33642},[362,363,364,365],[15,33644,102],{"className":33645,"style":174},[61,122,365],[15,33647,497],{"className":33648},[496],[15,33650,33652],{"className":33651},[346],[15,33653,33655],{"className":33654,"style":504},[350],[15,33656],{},[15,33658,750],{"className":33659},[178],[15,33661],{"className":33662,"style":128},[127],[15,33664,83],{"className":33665},[132],[15,33667],{"className":33668,"style":128},[127],[15,33670,33672,33676,33679,33682,33685,33688,33691],{"className":33671},[52],[15,33673],{"className":33674,"style":33675},[56],"height:1.0913em;vertical-align:-0.25em;",[15,33677,14810],{"className":33678},[146],[15,33680,278],{"className":33681},[61],[15,33683,94],{"className":33684},[153],[15,33686],{"className":33687,"style":157},[127],[15,33689,91],{"className":33690},[61],[15,33692,33694,33697],{"className":33693},[178],[15,33695,14819],{"className":33696},[178],[15,33698,33700],{"className":33699},[338],[15,33701,33703],{"className":33702},[342],[15,33704,33706],{"className":33705},[346],[15,33707,33709],{"className":33708,"style":695},[350],[15,33710,33711,33714],{"style":716},[15,33712],{"className":33713,"style":358},[357],[15,33715,33717],{"className":33716},[362,363,364,365],[15,33718,102],{"className":33719,"style":174},[61,122,365]," 为",[661,33722,33723],{},"信念质量（belief mass）",[31523,33725,33726,33720,33800,867],{},[15,33727,33729,33755],{"className":33728,"translate":19},[18],[15,33730,33732],{"className":33731},[23],[25,33733,33734],{"xmlns":27},[29,33735,33736,33752],{},[32,33737,33738,33740,33742,33744,33746,33748,33750],{},[35,33739,33401],{},[81,33741,83],{},[81,33743,14810],{"stretchy":86},[89,33745,278],{},[81,33747,94],{"separator":48},[89,33749,91],{},[81,33751,14819],{"stretchy":86},[40,33753,33754],{"encoding":42},"u \\in [0,1]",[15,33756,33758,33776],{"className":33757,"ariaHidden":48},[47],[15,33759,33761,33764,33767,33770,33773],{"className":33760},[52],[15,33762],{"className":33763,"style":4650},[56],[15,33765,33401],{"className":33766},[61,122],[15,33768],{"className":33769,"style":128},[127],[15,33771,83],{"className":33772},[132],[15,33774],{"className":33775,"style":128},[127],[15,33777,33779,33782,33785,33788,33791,33794,33797],{"className":33778},[52],[15,33780],{"className":33781,"style":142},[56],[15,33783,14810],{"className":33784},[146],[15,33786,278],{"className":33787},[61],[15,33789,94],{"className":33790},[153],[15,33792],{"className":33793,"style":157},[127],[15,33795,91],{"className":33796},[61],[15,33798,14819],{"className":33799},[178],[661,33801,33802],{},"不确定性质量（uncertainty mass）",[31523,33804,33805,941,33909,34050,34051,867],{},[15,33806,33808,33838],{"className":33807,"translate":19},[18],[15,33809,33811],{"className":33810},[23],[25,33812,33813],{"xmlns":27},[29,33814,33815,33835],{},[32,33816,33817,33819,33821,33823,33825,33827,33829],{},[35,33818,33406],{"mathvariant":251},[81,33820,83],{},[81,33822,14810],{"stretchy":86},[89,33824,278],{},[81,33826,94],{"separator":48},[89,33828,91],{},[227,33830,33831,33833],{},[81,33832,14819],{"stretchy":86},[35,33834,102],{},[40,33836,33837],{"encoding":42},"\\mathbf{a} \\in [0,1]^K",[15,33839,33841,33859],{"className":33840,"ariaHidden":48},[47],[15,33842,33844,33847,33850,33853,33856],{"className":33843},[52],[15,33845],{"className":33846,"style":4650},[56],[15,33848,33406],{"className":33849},[61,411],[15,33851],{"className":33852,"style":128},[127],[15,33854,83],{"className":33855},[132],[15,33857],{"className":33858,"style":128},[127],[15,33860,33862,33865,33868,33871,33874,33877,33880],{"className":33861},[52],[15,33863],{"className":33864,"style":33675},[56],[15,33866,14810],{"className":33867},[146],[15,33869,278],{"className":33870},[61],[15,33872,94],{"className":33873},[153],[15,33875],{"className":33876,"style":157},[127],[15,33878,91],{"className":33879},[61],[15,33881,33883,33886],{"className":33882},[178],[15,33884,14819],{"className":33885},[178],[15,33887,33889],{"className":33888},[338],[15,33890,33892],{"className":33891},[342],[15,33893,33895],{"className":33894},[346],[15,33896,33898],{"className":33897,"style":695},[350],[15,33899,33900,33903],{"style":716},[15,33901],{"className":33902,"style":358},[357],[15,33904,33906],{"className":33905},[362,363,364,365],[15,33907,102],{"className":33908,"style":174},[61,122,365],[15,33910,33912,33940],{"className":33911,"translate":19},[18],[15,33913,33915],{"className":33914},[23],[25,33916,33917],{"xmlns":27},[29,33918,33919,33937],{},[32,33920,33921,33927,33933,33935],{},[266,33922,33923,33925],{},[81,33924,293],{},[35,33926,272],{},[266,33928,33929,33931],{},[35,33930,33406],{},[35,33932,272],{},[81,33934,244],{},[89,33936,91],{},[40,33938,33939],{"encoding":42},"\\sum_k a_k = 1",[15,33941,33943,34041],{"className":33942,"ariaHidden":48},[47],[15,33944,33946,33949,33989,33992,34032,34035,34038],{"className":33945},[52],[15,33947],{"className":33948,"style":6045},[56],[15,33950,33952,33955],{"className":33951},[531],[15,33953,293],{"className":33954,"style":537},[531,535,536],[15,33956,33958],{"className":33957},[338],[15,33959,33961,33981],{"className":33960},[342,472],[15,33962,33964,33978],{"className":33963},[346],[15,33965,33967],{"className":33966,"style":29964},[350],[15,33968,33969,33972],{"style":553},[15,33970],{"className":33971,"style":358},[357],[15,33973,33975],{"className":33974},[362,363,364,365],[15,33976,272],{"className":33977,"style":492},[61,122,365],[15,33979,497],{"className":33980},[496],[15,33982,33984],{"className":33983},[346],[15,33985,33987],{"className":33986,"style":593},[350],[15,33988],{},[15,33990],{"className":33991,"style":157},[127],[15,33993,33995,33998],{"className":33994},[61],[15,33996,33406],{"className":33997},[61,122],[15,33999,34001],{"className":34000},[338],[15,34002,34004,34024],{"className":34003},[342,472],[15,34005,34007,34021],{"className":34006},[346],[15,34008,34010],{"className":34009,"style":479},[350],[15,34011,34012,34015],{"style":482},[15,34013],{"className":34014,"style":358},[357],[15,34016,34018],{"className":34017},[362,363,364,365],[15,34019,272],{"className":34020,"style":492},[61,122,365],[15,34022,497],{"className":34023},[496],[15,34025,34027],{"className":34026},[346],[15,34028,34030],{"className":34029,"style":504},[350],[15,34031],{},[15,34033],{"className":34034,"style":128},[127],[15,34036,244],{"className":34037},[132],[15,34039],{"className":34040,"style":128},[127],[15,34042,34044,34047],{"className":34043},[52],[15,34045],{"className":34046,"style":859},[56],[15,34048,91],{"className":34049},[61],"，为",[661,34052,34053],{},"基准率（base rate）",[11,34055,34056,34057,867],{},"并满足单纯形约束 ",[15,34058,34060,34100],{"className":34059,"translate":19},[18],[15,34061,34063],{"className":34062},[23],[25,34064,34065],{"xmlns":27},[29,34066,34067,34097],{},[32,34068,34069,34083,34089,34091,34093,34095],{},[289,34070,34071,34073,34081],{},[81,34072,293],{},[32,34074,34075,34077,34079],{},[35,34076,272],{},[81,34078,244],{},[89,34080,91],{},[35,34082,102],{},[266,34084,34085,34087],{},[35,34086,33396],{},[35,34088,272],{},[81,34090,7756],{},[35,34092,33401],{},[81,34094,244],{},[89,34096,91],{},[40,34098,34099],{"encoding":42},"\\sum_{k=1}^K b_k + u = 1",[15,34101,34103,34221,34239],{"className":34102,"ariaHidden":48},[47],[15,34104,34106,34109,34169,34172,34212,34215,34218],{"className":34105},[52],[15,34107],{"className":34108,"style":3467},[56],[15,34110,34112,34115],{"className":34111},[531],[15,34113,293],{"className":34114,"style":537},[531,535,536],[15,34116,34118],{"className":34117},[338],[15,34119,34121,34161],{"className":34120},[342,472],[15,34122,34124,34158],{"className":34123},[346],[15,34125,34127,34147],{"className":34126,"style":550},[350],[15,34128,34129,34132],{"style":553},[15,34130],{"className":34131,"style":358},[357],[15,34133,34135],{"className":34134},[362,363,364,365],[15,34136,34138,34141,34144],{"className":34137},[61,365],[15,34139,272],{"className":34140,"style":492},[61,122,365],[15,34142,244],{"className":34143},[132,365],[15,34145,91],{"className":34146},[61,365],[15,34148,34149,34152],{"style":574},[15,34150],{"className":34151,"style":358},[357],[15,34153,34155],{"className":34154},[362,363,364,365],[15,34156,102],{"className":34157,"style":174},[61,122,365],[15,34159,497],{"className":34160},[496],[15,34162,34164],{"className":34163},[346],[15,34165,34167],{"className":34166,"style":593},[350],[15,34168],{},[15,34170],{"className":34171,"style":157},[127],[15,34173,34175,34178],{"className":34174},[61],[15,34176,33396],{"className":34177},[61,122],[15,34179,34181],{"className":34180},[338],[15,34182,34184,34204],{"className":34183},[342,472],[15,34185,34187,34201],{"className":34186},[346],[15,34188,34190],{"className":34189,"style":479},[350],[15,34191,34192,34195],{"style":482},[15,34193],{"className":34194,"style":358},[357],[15,34196,34198],{"className":34197},[362,363,364,365],[15,34199,272],{"className":34200,"style":492},[61,122,365],[15,34202,497],{"className":34203},[496],[15,34205,34207],{"className":34206},[346],[15,34208,34210],{"className":34209,"style":504},[350],[15,34211],{},[15,34213],{"className":34214,"style":772},[127],[15,34216,7756],{"className":34217},[375],[15,34219],{"className":34220,"style":772},[127],[15,34222,34224,34227,34230,34233,34236],{"className":34223},[52],[15,34225],{"className":34226,"style":6026},[56],[15,34228,33401],{"className":34229},[61,122],[15,34231],{"className":34232,"style":128},[127],[15,34234,244],{"className":34235},[132],[15,34237],{"className":34238,"style":128},[127],[15,34240,34242,34245],{"className":34241},[52],[15,34243],{"className":34244,"style":859},[56],[15,34246,91],{"className":34247},[61],[11,34249,34250,9371,34279,34282,34283,34488,34489,20764,34588,867],{},[15,34251,34253,34267],{"className":34252,"translate":19},[18],[15,34254,34256],{"className":34255},[23],[25,34257,34258],{"xmlns":27},[29,34259,34260,34264],{},[32,34261,34262],{},[35,34263,33389],{},[40,34265,34266],{"encoding":42},"\\omega",[15,34268,34270],{"className":34269,"ariaHidden":48},[47],[15,34271,34273,34276],{"className":34272},[52],[15,34274],{"className":34275,"style":6026},[56],[15,34277,33389],{"className":34278,"style":123},[61,122],[661,34280,34281],{},"概率投影（probability projection）"," 定义为 ",[15,34284,34286,34322],{"className":34285,"translate":19},[18],[15,34287,34289],{"className":34288},[23],[25,34290,34291],{"xmlns":27},[29,34292,34293,34319],{},[32,34294,34295,34301,34303,34309,34311,34317],{},[266,34296,34297,34299],{},[35,34298,21273],{},[35,34300,272],{},[81,34302,244],{},[266,34304,34305,34307],{},[35,34306,33396],{},[35,34308,272],{},[81,34310,7756],{},[266,34312,34313,34315],{},[35,34314,33406],{},[35,34316,272],{},[35,34318,33401],{},[40,34320,34321],{"encoding":42},"P_k = b_k + a_k u",[15,34323,34325,34383,34439],{"className":34324,"ariaHidden":48},[47],[15,34326,34328,34332,34374,34377,34380],{"className":34327},[52],[15,34329],{"className":34330,"style":34331},[56],"height:0.8333em;vertical-align:-0.15em;",[15,34333,34335,34339],{"className":34334},[61],[15,34336,21273],{"className":34337,"style":34338},[61,122],"margin-right:0.1389em;",[15,34340,34342],{"className":34341},[338],[15,34343,34345,34366],{"className":34344},[342,472],[15,34346,34348,34363],{"className":34347},[346],[15,34349,34351],{"className":34350,"style":479},[350],[15,34352,34354,34357],{"style":34353},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[15,34355],{"className":34356,"style":358},[357],[15,34358,34360],{"className":34359},[362,363,364,365],[15,34361,272],{"className":34362,"style":492},[61,122,365],[15,34364,497],{"className":34365},[496],[15,34367,34369],{"className":34368},[346],[15,34370,34372],{"className":34371,"style":504},[350],[15,34373],{},[15,34375],{"className":34376,"style":128},[127],[15,34378,244],{"className":34379},[132],[15,34381],{"className":34382,"style":128},[127],[15,34384,34386,34390,34430,34433,34436],{"className":34385},[52],[15,34387],{"className":34388,"style":34389},[56],"height:0.8444em;vertical-align:-0.15em;",[15,34391,34393,34396],{"className":34392},[61],[15,34394,33396],{"className":34395},[61,122],[15,34397,34399],{"className":34398},[338],[15,34400,34402,34422],{"className":34401},[342,472],[15,34403,34405,34419],{"className":34404},[346],[15,34406,34408],{"className":34407,"style":479},[350],[15,34409,34410,34413],{"style":482},[15,34411],{"className":34412,"style":358},[357],[15,34414,34416],{"className":34415},[362,363,364,365],[15,34417,272],{"className":34418,"style":492},[61,122,365],[15,34420,497],{"className":34421},[496],[15,34423,34425],{"className":34424},[346],[15,34426,34428],{"className":34427,"style":504},[350],[15,34429],{},[15,34431],{"className":34432,"style":772},[127],[15,34434,7756],{"className":34435},[375],[15,34437],{"className":34438,"style":772},[127],[15,34440,34442,34445,34485],{"className":34441},[52],[15,34443],{"className":34444,"style":2509},[56],[15,34446,34448,34451],{"className":34447},[61],[15,34449,33406],{"className":34450},[61,122],[15,34452,34454],{"className":34453},[338],[15,34455,34457,34477],{"className":34456},[342,472],[15,34458,34460,34474],{"className":34459},[346],[15,34461,34463],{"className":34462,"style":479},[350],[15,34464,34465,34468],{"style":482},[15,34466],{"className":34467,"style":358},[357],[15,34469,34471],{"className":34470},[362,363,364,365],[15,34472,272],{"className":34473,"style":492},[61,122,365],[15,34475,497],{"className":34476},[496],[15,34478,34480],{"className":34479},[346],[15,34481,34483],{"className":34482,"style":504},[350],[15,34484],{},[15,34486,33401],{"className":34487},[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 ",[15,34490,34492,34518],{"className":34491,"translate":19},[18],[15,34493,34495],{"className":34494},[23],[25,34496,34497],{"xmlns":27},[29,34498,34499,34515],{},[32,34500,34501,34507,34509,34511,34513],{},[266,34502,34503,34505],{},[35,34504,33406],{},[35,34506,272],{},[81,34508,244],{},[89,34510,91],{},[35,34512,3018],{"mathvariant":231},[35,34514,102],{},[40,34516,34517],{"encoding":42},"a_k = 1\u002FK",[15,34519,34521,34576],{"className":34520,"ariaHidden":48},[47],[15,34522,34524,34527,34567,34570,34573],{"className":34523},[52],[15,34525],{"className":34526,"style":2509},[56],[15,34528,34530,34533],{"className":34529},[61],[15,34531,33406],{"className":34532},[61,122],[15,34534,34536],{"className":34535},[338],[15,34537,34539,34559],{"className":34538},[342,472],[15,34540,34542,34556],{"className":34541},[346],[15,34543,34545],{"className":34544,"style":479},[350],[15,34546,34547,34550],{"style":482},[15,34548],{"className":34549,"style":358},[357],[15,34551,34553],{"className":34552},[362,363,364,365],[15,34554,272],{"className":34555,"style":492},[61,122,365],[15,34557,497],{"className":34558},[496],[15,34560,34562],{"className":34561},[346],[15,34563,34565],{"className":34564,"style":504},[350],[15,34566],{},[15,34568],{"className":34569,"style":128},[127],[15,34571,244],{"className":34572},[132],[15,34574],{"className":34575,"style":128},[127],[15,34577,34579,34582,34585],{"className":34578},[52],[15,34580],{"className":34581,"style":142},[56],[15,34583,3047],{"className":34584},[61],[15,34586,102],{"className":34587,"style":174},[61,122],[15,34589,34591,34625],{"className":34590,"translate":19},[18],[15,34592,34594],{"className":34593},[23],[25,34595,34596],{"xmlns":27},[29,34597,34598,34622],{},[32,34599,34600,34606,34608,34614,34616,34618,34620],{},[266,34601,34602,34604],{},[35,34603,21273],{},[35,34605,272],{},[81,34607,244],{},[266,34609,34610,34612],{},[35,34611,33396],{},[35,34613,272],{},[81,34615,7756],{},[35,34617,33401],{},[35,34619,3018],{"mathvariant":231},[35,34621,102],{},[40,34623,34624],{"encoding":42},"P_k = b_k + u\u002FK",[15,34626,34628,34683,34738],{"className":34627,"ariaHidden":48},[47],[15,34629,34631,34634,34674,34677,34680],{"className":34630},[52],[15,34632],{"className":34633,"style":34331},[56],[15,34635,34637,34640],{"className":34636},[61],[15,34638,21273],{"className":34639,"style":34338},[61,122],[15,34641,34643],{"className":34642},[338],[15,34644,34646,34666],{"className":34645},[342,472],[15,34647,34649,34663],{"className":34648},[346],[15,34650,34652],{"className":34651,"style":479},[350],[15,34653,34654,34657],{"style":34353},[15,34655],{"className":34656,"style":358},[357],[15,34658,34660],{"className":34659},[362,363,364,365],[15,34661,272],{"className":34662,"style":492},[61,122,365],[15,34664,497],{"className":34665},[496],[15,34667,34669],{"className":34668},[346],[15,34670,34672],{"className":34671,"style":504},[350],[15,34673],{},[15,34675],{"className":34676,"style":128},[127],[15,34678,244],{"className":34679},[132],[15,34681],{"className":34682,"style":128},[127],[15,34684,34686,34689,34729,34732,34735],{"className":34685},[52],[15,34687],{"className":34688,"style":34389},[56],[15,34690,34692,34695],{"className":34691},[61],[15,34693,33396],{"className":34694},[61,122],[15,34696,34698],{"className":34697},[338],[15,34699,34701,34721],{"className":34700},[342,472],[15,34702,34704,34718],{"className":34703},[346],[15,34705,34707],{"className":34706,"style":479},[350],[15,34708,34709,34712],{"style":482},[15,34710],{"className":34711,"style":358},[357],[15,34713,34715],{"className":34714},[362,363,364,365],[15,34716,272],{"className":34717,"style":492},[61,122,365],[15,34719,497],{"className":34720},[496],[15,34722,34724],{"className":34723},[346],[15,34725,34727],{"className":34726,"style":504},[350],[15,34728],{},[15,34730],{"className":34731,"style":772},[127],[15,34733,7756],{"className":34734},[375],[15,34736],{"className":34737,"style":772},[127],[15,34739,34741,34744,34747,34750],{"className":34740},[52],[15,34742],{"className":34743,"style":142},[56],[15,34745,33401],{"className":34746},[61,122],[15,34748,3018],{"className":34749},[61],[15,34751,102],{"className":34752,"style":174},[61,122],[11,34754,34755,34756,34818,34819,34847,34848,34921],{},"定义 3.1 的关键在于：opinion 所栖居的空间是 ",[15,34757,34759,34777],{"className":34758,"translate":19},[18],[15,34760,34762],{"className":34761},[23],[25,34763,34764],{"xmlns":27},[29,34765,34766,34774],{},[32,34767,34768],{},[227,34769,34770,34772],{},[35,34771,232],{"mathvariant":231},[35,34773,102],{},[40,34775,34776],{"encoding":42},"\\Delta^{K}",[15,34778,34780],{"className":34779,"ariaHidden":48},[47],[15,34781,34783,34786],{"className":34782},[52],[15,34784],{"className":34785,"style":695},[56],[15,34787,34789,34792],{"className":34788},[61],[15,34790,232],{"className":34791},[61],[15,34793,34795],{"className":34794},[338],[15,34796,34798],{"className":34797},[342],[15,34799,34801],{"className":34800},[346],[15,34802,34804],{"className":34803,"style":695},[350],[15,34805,34806,34809],{"style":716},[15,34807],{"className":34808,"style":358},[357],[15,34810,34812],{"className":34811},[362,363,364,365],[15,34813,34815],{"className":34814},[61,365],[15,34816,102],{"className":34817,"style":174},[61,122,365],"——一个 ",[15,34820,34822,34835],{"className":34821,"translate":19},[18],[15,34823,34825],{"className":34824},[23],[25,34826,34827],{"xmlns":27},[29,34828,34829,34833],{},[32,34830,34831],{},[35,34832,102],{},[40,34834,102],{"encoding":42},[15,34836,34838],{"className":34837,"ariaHidden":48},[47],[15,34839,34841,34844],{"className":34840},[52],[15,34842],{"className":34843,"style":57},[56],[15,34845,102],{"className":34846,"style":174},[61,122]," 维单纯形，比 ",[15,34849,34851,34874],{"className":34850,"translate":19},[18],[15,34852,34854],{"className":34853},[23],[25,34855,34856],{"xmlns":27},[29,34857,34858,34872],{},[32,34859,34860],{},[227,34861,34862,34864],{},[35,34863,232],{"mathvariant":231},[32,34865,34866,34868,34870],{},[35,34867,102],{},[81,34869,239],{},[89,34871,91],{},[40,34873,1856],{"encoding":42},[15,34875,34877],{"className":34876,"ariaHidden":48},[47],[15,34878,34880,34883],{"className":34879},[52],[15,34881],{"className":34882,"style":695},[56],[15,34884,34886,34889],{"className":34885},[61],[15,34887,232],{"className":34888},[61],[15,34890,34892],{"className":34891},[338],[15,34893,34895],{"className":34894},[342],[15,34896,34898],{"className":34897},[346],[15,34899,34901],{"className":34900,"style":695},[350],[15,34902,34903,34906],{"style":716},[15,34904],{"className":34905,"style":358},[357],[15,34907,34909],{"className":34908},[362,363,364,365],[15,34910,34912,34915,34918],{"className":34911},[61,365],[15,34913,102],{"className":34914,"style":174},[61,122,365],[15,34916,239],{"className":34917},[375,365],[15,34919,91],{"className":34920},[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。",[877,34923,34925],{"id":34924},"_32-dirichlet-与-opinion-的双射","3.2 Dirichlet 与 opinion 的双射",[11,34927,34928,34929,867],{},"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",[661,34930,34931],{},"双射",[882,34933,34935],{"title":34934},"命题 3.2（双射）",[11,34936,34937,34938,34990,34991,35089,35090],{},"设先验权重 ",[15,34939,34941,34960],{"className":34940,"translate":19},[18],[15,34942,34944],{"className":34943},[23],[25,34945,34946],{"xmlns":27},[29,34947,34948,34957],{},[32,34949,34950,34953,34955],{},[35,34951,34952],{},"W",[81,34954,244],{},[35,34956,102],{},[40,34958,34959],{"encoding":42},"W = K",[15,34961,34963,34981],{"className":34962,"ariaHidden":48},[47],[15,34964,34966,34969,34972,34975,34978],{"className":34965},[52],[15,34967],{"className":34968,"style":57},[56],[15,34970,34952],{"className":34971,"style":34338},[61,122],[15,34973],{"className":34974,"style":128},[127],[15,34976,244],{"className":34977},[132],[15,34979],{"className":34980,"style":128},[127],[15,34982,34984,34987],{"className":34983},[52],[15,34985],{"className":34986,"style":57},[56],[15,34988,102],{"className":34989,"style":174},[61,122],"、基准率 ",[15,34992,34994,35019],{"className":34993,"translate":19},[18],[15,34995,34997],{"className":34996},[23],[25,34998,34999],{"xmlns":27},[29,35000,35001,35017],{},[32,35002,35003,35009,35011,35013,35015],{},[266,35004,35005,35007],{},[35,35006,33406],{},[35,35008,272],{},[81,35010,244],{},[89,35012,91],{},[35,35014,3018],{"mathvariant":231},[35,35016,102],{},[40,35018,34517],{"encoding":42},[15,35020,35022,35077],{"className":35021,"ariaHidden":48},[47],[15,35023,35025,35028,35068,35071,35074],{"className":35024},[52],[15,35026],{"className":35027,"style":2509},[56],[15,35029,35031,35034],{"className":35030},[61],[15,35032,33406],{"className":35033},[61,122],[15,35035,35037],{"className":35036},[338],[15,35038,35040,35060],{"className":35039},[342,472],[15,35041,35043,35057],{"className":35042},[346],[15,35044,35046],{"className":35045,"style":479},[350],[15,35047,35048,35051],{"style":482},[15,35049],{"className":35050,"style":358},[357],[15,35052,35054],{"className":35053},[362,363,364,365],[15,35055,272],{"className":35056,"style":492},[61,122,365],[15,35058,497],{"className":35059},[496],[15,35061,35063],{"className":35062},[346],[15,35064,35066],{"className":35065,"style":504},[350],[15,35067],{},[15,35069],{"className":35070,"style":128},[127],[15,35072,244],{"className":35073},[132],[15,35075],{"className":35076,"style":128},[127],[15,35078,35080,35083,35086],{"className":35079},[52],[15,35081],{"className":35082,"style":142},[56],[15,35084,3047],{"className":35085},[61],[15,35087,102],{"className":35088,"style":174},[61,122],"。则如下两个映射互逆：\n",[15,35091,35093,35195],{"className":35092,"translate":19},[18],[15,35094,35096],{"className":35095},[23],[25,35097,35098],{"xmlns":27},[29,35099,35100,35192],{},[32,35101,35102,35104,35106,35122,35124,35126,35128,35130,35140,35142,35144,35146,35152,35154,35170,35172,35174,35176,35178,35184,35186],{},[35,35103,33396],{"mathvariant":251},[81,35105,244],{},[3718,35107,35108,35116],{},[32,35109,35110,35112,35114],{},[35,35111,3102],{"mathvariant":3101},[81,35113,239],{},[89,35115,91],{"mathvariant":251},[266,35117,35118,35120],{},[35,35119,3102],{},[89,35121,278],{},[81,35123,94],{"separator":48},[127,35125],{"width":9838},[35,35127,33401],{},[81,35129,244],{},[3718,35131,35132,35134],{},[35,35133,102],{},[266,35135,35136,35138],{},[35,35137,3102],{},[89,35139,278],{},[127,35141],{"width":9838},[81,35143,24500],{},[127,35145],{"width":9838},[266,35147,35148,35150],{},[35,35149,3102],{},[35,35151,272],{},[81,35153,244],{},[3718,35155,35156,35168],{},[32,35157,35158,35160,35162],{},[35,35159,102],{},[282,35161,4796],{},[266,35163,35164,35166],{},[35,35165,33396],{},[35,35167,272],{},[35,35169,33401],{},[81,35171,7756],{},[89,35173,91],{},[81,35175,94],{"separator":48},[127,35177],{"width":9838},[266,35179,35180,35182],{},[35,35181,3102],{},[89,35183,278],{},[81,35185,244],{},[3718,35187,35188,35190],{},[35,35189,102],{},[35,35191,33401],{},[40,35193,35194],{"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}",[15,35196,35198,35216,35360,35488,35543,35671,35738],{"className":35197,"ariaHidden":48},[47],[15,35199,35201,35204,35207,35210,35213],{"className":35200},[52],[15,35202],{"className":35203,"style":10443},[56],[15,35205,33396],{"className":35206},[61,411],[15,35208],{"className":35209,"style":128},[127],[15,35211,244],{"className":35212},[132],[15,35214],{"className":35215,"style":128},[127],[15,35217,35219,35222,35339,35342,35345,35348,35351,35354,35357],{"className":35218},[52],[15,35220],{"className":35221,"style":16567},[56],[15,35223,35225,35228,35336],{"className":35224},[61],[15,35226],{"className":35227},[146,3837],[15,35229,35231],{"className":35230},[3718],[15,35232,35234,35328],{"className":35233},[342,472],[15,35235,35237,35325],{"className":35236},[346],[15,35238,35240,35291,35299],{"className":35239,"style":3850},[350],[15,35241,35242,35245],{"style":3853},[15,35243],{"className":35244,"style":3857},[357],[15,35246,35248],{"className":35247},[362,363,364,365],[15,35249,35251],{"className":35250},[61,365],[15,35252,35254,35257],{"className":35253},[61,365],[15,35255,3102],{"className":35256,"style":3187},[61,122,365],[15,35258,35260],{"className":35259},[338],[15,35261,35263,35283],{"className":35262},[342,472],[15,35264,35266,35280],{"className":35265},[346],[15,35267,35269],{"className":35268,"style":4421},[350],[15,35270,35271,35274],{"style":4054},[15,35272],{"className":35273,"style":4058},[357],[15,35275,35277],{"className":35276},[362,4062,4063,365],[15,35278,278],{"className":35279},[61,365],[15,35281,497],{"className":35282},[496],[15,35284,35286],{"className":35285},[346],[15,35287,35289],{"className":35288,"style":4076},[350],[15,35290],{},[15,35292,35293,35296],{"style":3885},[15,35294],{"className":35295,"style":3857},[357],[15,35297],{"className":35298,"style":3893},[3892],[15,35300,35301,35304],{"style":3896},[15,35302],{"className":35303,"style":3857},[357],[15,35305,35307],{"className":35306},[362,363,364,365],[15,35308,35310,35319,35322],{"className":35309},[61,365],[15,35311,35313],{"className":35312},[61,365],[15,35314,35316],{"className":35315},[61,365],[15,35317,3102],{"className":35318},[61,3162,365],[15,35320,239],{"className":35321},[375,365],[15,35323,91],{"className":35324},[61,411,365],[15,35326,497],{"className":35327},[496],[15,35329,35331],{"className":35330},[346],[15,35332,35334],{"className":35333,"style":15209},[350],[15,35335],{},[15,35337],{"className":35338},[178,3837],[15,35340,94],{"className":35341},[153],[15,35343],{"className":35344,"style":10026},[127],[15,35346],{"className":35347,"style":157},[127],[15,35349,33401],{"className":35350},[61,122],[15,35352],{"className":35353,"style":128},[127],[15,35355,244],{"className":35356},[132],[15,35358],{"className":35359,"style":128},[127],[15,35361,35363,35367,35473,35476,35479,35482,35485],{"className":35362},[52],[15,35364],{"className":35365,"style":35366},[56],"height:1.3174em;vertical-align:-0.4451em;",[15,35368,35370,35373,35470],{"className":35369},[61],[15,35371],{"className":35372},[146,3837],[15,35374,35376],{"className":35375},[3718],[15,35377,35379,35462],{"className":35378},[342,472],[15,35380,35382,35459],{"className":35381},[346],[15,35383,35386,35437,35445],{"className":35384,"style":35385},[350],"height:0.8723em;",[15,35387,35388,35391],{"style":3853},[15,35389],{"className":35390,"style":3857},[357],[15,35392,35394],{"className":35393},[362,363,364,365],[15,35395,35397],{"className":35396},[61,365],[15,35398,35400,35403],{"className":35399},[61,365],[15,35401,3102],{"className":35402,"style":3187},[61,122,365],[15,35404,35406],{"className":35405},[338],[15,35407,35409,35429],{"className":35408},[342,472],[15,35410,35412,35426],{"className":35411},[346],[15,35413,35415],{"className":35414,"style":4421},[350],[15,35416,35417,35420],{"style":4054},[15,35418],{"className":35419,"style":4058},[357],[15,35421,35423],{"className":35422},[362,4062,4063,365],[15,35424,278],{"className":35425},[61,365],[15,35427,497],{"className":35428},[496],[15,35430,35432],{"className":35431},[346],[15,35433,35435],{"className":35434,"style":4076},[350],[15,35436],{},[15,35438,35439,35442],{"style":3885},[15,35440],{"className":35441,"style":3857},[357],[15,35443],{"className":35444,"style":3893},[3892],[15,35446,35447,35450],{"style":3896},[15,35448],{"className":35449,"style":3857},[357],[15,35451,35453],{"className":35452},[362,363,364,365],[15,35454,35456],{"className":35455},[61,365],[15,35457,102],{"className":35458,"style":174},[61,122,365],[15,35460,497],{"className":35461},[496],[15,35463,35465],{"className":35464},[346],[15,35466,35468],{"className":35467,"style":15209},[350],[15,35469],{},[15,35471],{"className":35472},[178,3837],[15,35474],{"className":35475,"style":10026},[127],[15,35477],{"className":35478,"style":128},[127],[15,35480,24500],{"className":35481},[132],[15,35483],{"className":35484,"style":10026},[127],[15,35486],{"className":35487,"style":128},[127],[15,35489,35491,35494,35534,35537,35540],{"className":35490},[52],[15,35492],{"className":35493,"style":2509},[56],[15,35495,35497,35500],{"className":35496},[61],[15,35498,3102],{"className":35499,"style":3187},[61,122],[15,35501,35503],{"className":35502},[338],[15,35504,35506,35526],{"className":35505},[342,472],[15,35507,35509,35523],{"className":35508},[346],[15,35510,35512],{"className":35511,"style":479},[350],[15,35513,35514,35517],{"style":3202},[15,35515],{"className":35516,"style":358},[357],[15,35518,35520],{"className":35519},[362,363,364,365],[15,35521,272],{"className":35522,"style":492},[61,122,365],[15,35524,497],{"className":35525},[496],[15,35527,35529],{"className":35528},[346],[15,35530,35532],{"className":35531,"style":504},[350],[15,35533],{},[15,35535],{"className":35536,"style":128},[127],[15,35538,244],{"className":35539},[132],[15,35541],{"className":35542,"style":128},[127],[15,35544,35546,35550,35662,35665,35668],{"className":35545},[52],[15,35547],{"className":35548,"style":35549},[56],"height:1.247em;vertical-align:-0.345em;",[15,35551,35553,35556,35659],{"className":35552},[61],[15,35554],{"className":35555},[146,3837],[15,35557,35559],{"className":35558},[3718],[15,35560,35562,35651],{"className":35561},[342,472],[15,35563,35565,35648],{"className":35564},[346],[15,35566,35569,35583,35591],{"className":35567,"style":35568},[350],"height:0.902em;",[15,35570,35571,35574],{"style":3853},[15,35572],{"className":35573,"style":3857},[357],[15,35575,35577],{"className":35576},[362,363,364,365],[15,35578,35580],{"className":35579},[61,365],[15,35581,33401],{"className":35582},[61,122,365],[15,35584,35585,35588],{"style":3885},[15,35586],{"className":35587,"style":3857},[357],[15,35589],{"className":35590,"style":3893},[3892],[15,35592,35593,35596],{"style":27181},[15,35594],{"className":35595,"style":3857},[357],[15,35597,35599],{"className":35598},[362,363,364,365],[15,35600,35602,35605,35608],{"className":35601},[61,365],[15,35603,102],{"className":35604,"style":174},[61,122,365],[15,35606],{"className":35607,"style":4537},[127,365],[15,35609,35611,35614],{"className":35610},[61,365],[15,35612,33396],{"className":35613},[61,122,365],[15,35615,35617],{"className":35616},[338],[15,35618,35620,35640],{"className":35619},[342,472],[15,35621,35623,35637],{"className":35622},[346],[15,35624,35626],{"className":35625,"style":22389},[350],[15,35627,35628,35631],{"style":27256},[15,35629],{"className":35630,"style":4058},[357],[15,35632,35634],{"className":35633},[362,4062,4063,365],[15,35635,272],{"className":35636,"style":492},[61,122,365],[15,35638,497],{"className":35639},[496],[15,35641,35643],{"className":35642},[346],[15,35644,35646],{"className":35645,"style":27230},[350],[15,35647],{},[15,35649,497],{"className":35650},[496],[15,35652,35654],{"className":35653},[346],[15,35655,35657],{"className":35656,"style":10189},[350],[15,35658],{},[15,35660],{"className":35661},[178,3837],[15,35663],{"className":35664,"style":772},[127],[15,35666,7756],{"className":35667},[375],[15,35669],{"className":35670,"style":772},[127],[15,35672,35674,35677,35680,35683,35686,35689,35729,35732,35735],{"className":35673},[52],[15,35675],{"className":35676,"style":19744},[56],[15,35678,91],{"className":35679},[61],[15,35681,94],{"className":35682},[153],[15,35684],{"className":35685,"style":10026},[127],[15,35687],{"className":35688,"style":157},[127],[15,35690,35692,35695],{"className":35691},[61],[15,35693,3102],{"className":35694,"style":3187},[61,122],[15,35696,35698],{"className":35697},[338],[15,35699,35701,35721],{"className":35700},[342,472],[15,35702,35704,35718],{"className":35703},[346],[15,35705,35707],{"className":35706,"style":1212},[350],[15,35708,35709,35712],{"style":3202},[15,35710],{"className":35711,"style":358},[357],[15,35713,35715],{"className":35714},[362,363,364,365],[15,35716,278],{"className":35717},[61,365],[15,35719,497],{"className":35720},[496],[15,35722,35724],{"className":35723},[346],[15,35725,35727],{"className":35726,"style":504},[350],[15,35728],{},[15,35730],{"className":35731,"style":128},[127],[15,35733,244],{"className":35734},[132],[15,35736],{"className":35737,"style":128},[127],[15,35739,35741,35745],{"className":35740},[52],[15,35742],{"className":35743,"style":35744},[56],"height:1.2173em;vertical-align:-0.345em;",[15,35746,35748,35751,35810],{"className":35747},[61],[15,35749],{"className":35750},[146,3837],[15,35752,35754],{"className":35753},[3718],[15,35755,35757,35802],{"className":35756},[342,472],[15,35758,35760,35799],{"className":35759},[346],[15,35761,35763,35777,35785],{"className":35762,"style":35385},[350],[15,35764,35765,35768],{"style":3853},[15,35766],{"className":35767,"style":3857},[357],[15,35769,35771],{"className":35770},[362,363,364,365],[15,35772,35774],{"className":35773},[61,365],[15,35775,33401],{"className":35776},[61,122,365],[15,35778,35779,35782],{"style":3885},[15,35780],{"className":35781,"style":3857},[357],[15,35783],{"className":35784,"style":3893},[3892],[15,35786,35787,35790],{"style":3896},[15,35788],{"className":35789,"style":3857},[357],[15,35791,35793],{"className":35792},[362,363,364,365],[15,35794,35796],{"className":35795},[61,365],[15,35797,102],{"className":35798,"style":174},[61,122,365],[15,35800,497],{"className":35801},[496],[15,35803,35805],{"className":35804},[346],[15,35806,35808],{"className":35807,"style":10189},[350],[15,35809],{},[15,35811],{"className":35812},[178,3837],[11,35814,35815,35818,35819,36301],{},[661,35816,35817],{},"验证"," 正向：",[15,35820,35822,35888],{"className":35821,"translate":19},[18],[15,35823,35825],{"className":35824},[23],[25,35826,35827],{"xmlns":27},[29,35828,35829,35885],{},[32,35830,35831,35837,35843,35845,35847,35849,35869,35871,35881,35883],{},[266,35832,35833,35835],{},[81,35834,293],{},[35,35836,272],{},[266,35838,35839,35841],{},[35,35840,33396],{},[35,35842,272],{},[81,35844,7756],{},[35,35846,33401],{},[81,35848,244],{},[3718,35850,35851,35863],{},[32,35852,35853,35859,35861],{},[266,35854,35855,35857],{},[35,35856,3102],{},[89,35858,278],{},[81,35860,239],{},[35,35862,102],{},[266,35864,35865,35867],{},[35,35866,3102],{},[89,35868,278],{},[81,35870,7756],{},[3718,35872,35873,35875],{},[35,35874,102],{},[266,35876,35877,35879],{},[35,35878,3102],{},[89,35880,278],{},[81,35882,244],{},[89,35884,91],{},[40,35886,35887],{"encoding":42},"\\sum_k b_k + u = \\frac{\\alpha_0 - K}{\\alpha_0} + \\frac{K}{\\alpha_0} = 1",[15,35889,35891,35989,36007,36172,36292],{"className":35890,"ariaHidden":48},[47],[15,35892,35894,35897,35937,35940,35980,35983,35986],{"className":35893},[52],[15,35895],{"className":35896,"style":6045},[56],[15,35898,35900,35903],{"className":35899},[531],[15,35901,293],{"className":35902,"style":537},[531,535,536],[15,35904,35906],{"className":35905},[338],[15,35907,35909,35929],{"className":35908},[342,472],[15,35910,35912,35926],{"className":35911},[346],[15,35913,35915],{"className":35914,"style":29964},[350],[15,35916,35917,35920],{"style":553},[15,35918],{"className":35919,"style":358},[357],[15,35921,35923],{"className":35922},[362,363,364,365],[15,35924,272],{"className":35925,"style":492},[61,122,365],[15,35927,497],{"className":35928},[496],[15,35930,35932],{"className":35931},[346],[15,35933,35935],{"className":35934,"style":593},[350],[15,35936],{},[15,35938],{"className":35939,"style":157},[127],[15,35941,35943,35946],{"className":35942},[61],[15,35944,33396],{"className":35945},[61,122],[15,35947,35949],{"className":35948},[338],[15,35950,35952,35972],{"className":35951},[342,472],[15,35953,35955,35969],{"className":35954},[346],[15,35956,35958],{"className":35957,"style":479},[350],[15,35959,35960,35963],{"style":482},[15,35961],{"className":35962,"style":358},[357],[15,35964,35966],{"className":35965},[362,363,364,365],[15,35967,272],{"className":35968,"style":492},[61,122,365],[15,35970,497],{"className":35971},[496],[15,35973,35975],{"className":35974},[346],[15,35976,35978],{"className":35977,"style":504},[350],[15,35979],{},[15,35981],{"className":35982,"style":772},[127],[15,35984,7756],{"className":35985},[375],[15,35987],{"className":35988,"style":772},[127],[15,35990,35992,35995,35998,36001,36004],{"className":35991},[52],[15,35993],{"className":35994,"style":6026},[56],[15,35996,33401],{"className":35997},[61,122],[15,35999],{"className":36000,"style":128},[127],[15,36002,244],{"className":36003},[132],[15,36005],{"className":36006,"style":128},[127],[15,36008,36010,36014,36163,36166,36169],{"className":36009},[52],[15,36011],{"className":36012,"style":36013},[56],"height:1.3335em;vertical-align:-0.4451em;",[15,36015,36017,36020,36160],{"className":36016},[61],[15,36018],{"className":36019},[146,3837],[15,36021,36023],{"className":36022},[3718],[15,36024,36026,36152],{"className":36025},[342,472],[15,36027,36029,36149],{"className":36028},[346],[15,36030,36033,36084,36092],{"className":36031,"style":36032},[350],"height:0.8884em;",[15,36034,36035,36038],{"style":3853},[15,36036],{"className":36037,"style":3857},[357],[15,36039,36041],{"className":36040},[362,363,364,365],[15,36042,36044],{"className":36043},[61,365],[15,36045,36047,36050],{"className":36046},[61,365],[15,36048,3102],{"className":36049,"style":3187},[61,122,365],[15,36051,36053],{"className":36052},[338],[15,36054,36056,36076],{"className":36055},[342,472],[15,36057,36059,36073],{"className":36058},[346],[15,36060,36062],{"className":36061,"style":4421},[350],[15,36063,36064,36067],{"style":4054},[15,36065],{"className":36066,"style":4058},[357],[15,36068,36070],{"className":36069},[362,4062,4063,365],[15,36071,278],{"className":36072},[61,365],[15,36074,497],{"className":36075},[496],[15,36077,36079],{"className":36078},[346],[15,36080,36082],{"className":36081,"style":4076},[350],[15,36083],{},[15,36085,36086,36089],{"style":3885},[15,36087],{"className":36088,"style":3857},[357],[15,36090],{"className":36091,"style":3893},[3892],[15,36093,36094,36097],{"style":15150},[15,36095],{"className":36096,"style":3857},[357],[15,36098,36100],{"className":36099},[362,363,364,365],[15,36101,36103,36143,36146],{"className":36102},[61,365],[15,36104,36106,36109],{"className":36105},[61,365],[15,36107,3102],{"className":36108,"style":3187},[61,122,365],[15,36110,36112],{"className":36111},[338],[15,36113,36115,36135],{"className":36114},[342,472],[15,36116,36118,36132],{"className":36117},[346],[15,36119,36121],{"className":36120,"style":4421},[350],[15,36122,36123,36126],{"style":4054},[15,36124],{"className":36125,"style":4058},[357],[15,36127,36129],{"className":36128},[362,4062,4063,365],[15,36130,278],{"className":36131},[61,365],[15,36133,497],{"className":36134},[496],[15,36136,36138],{"className":36137},[346],[15,36139,36141],{"className":36140,"style":4076},[350],[15,36142],{},[15,36144,239],{"className":36145},[375,365],[15,36147,102],{"className":36148,"style":174},[61,122,365],[15,36150,497],{"className":36151},[496],[15,36153,36155],{"className":36154},[346],[15,36156,36158],{"className":36157,"style":15209},[350],[15,36159],{},[15,36161],{"className":36162},[178,3837],[15,36164],{"className":36165,"style":772},[127],[15,36167,7756],{"className":36168},[375],[15,36170],{"className":36171,"style":772},[127],[15,36173,36175,36178,36283,36286,36289],{"className":36174},[52],[15,36176],{"className":36177,"style":35366},[56],[15,36179,36181,36184,36280],{"className":36180},[61],[15,36182],{"className":36183},[146,3837],[15,36185,36187],{"className":36186},[3718],[15,36188,36190,36272],{"className":36189},[342,472],[15,36191,36193,36269],{"className":36192},[346],[15,36194,36196,36247,36255],{"className":36195,"style":35385},[350],[15,36197,36198,36201],{"style":3853},[15,36199],{"className":36200,"style":3857},[357],[15,36202,36204],{"className":36203},[362,363,364,365],[15,36205,36207],{"className":36206},[61,365],[15,36208,36210,36213],{"className":36209},[61,365],[15,36211,3102],{"className":36212,"style":3187},[61,122,365],[15,36214,36216],{"className":36215},[338],[15,36217,36219,36239],{"className":36218},[342,472],[15,36220,36222,36236],{"className":36221},[346],[15,36223,36225],{"className":36224,"style":4421},[350],[15,36226,36227,36230],{"style":4054},[15,36228],{"className":36229,"style":4058},[357],[15,36231,36233],{"className":36232},[362,4062,4063,365],[15,36234,278],{"className":36235},[61,365],[15,36237,497],{"className":36238},[496],[15,36240,36242],{"className":36241},[346],[15,36243,36245],{"className":36244,"style":4076},[350],[15,36246],{},[15,36248,36249,36252],{"style":3885},[15,36250],{"className":36251,"style":3857},[357],[15,36253],{"className":36254,"style":3893},[3892],[15,36256,36257,36260],{"style":3896},[15,36258],{"className":36259,"style":3857},[357],[15,36261,36263],{"className":36262},[362,363,364,365],[15,36264,36266],{"className":36265},[61,365],[15,36267,102],{"className":36268,"style":174},[61,122,365],[15,36270,497],{"className":36271},[496],[15,36273,36275],{"className":36274},[346],[15,36276,36278],{"className":36277,"style":15209},[350],[15,36279],{},[15,36281],{"className":36282},[178,3837],[15,36284],{"className":36285,"style":128},[127],[15,36287,244],{"className":36288},[132],[15,36290],{"className":36291,"style":128},[127],[15,36293,36295,36298],{"className":36294},[52],[15,36296],{"className":36297,"style":859},[56],[15,36299,91],{"className":36300},[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：",[15,36303,36305],{"className":36304,"translate":19},[212],[15,36306,36308,36406],{"className":36307,"translate":19},[18],[15,36309,36311],{"className":36310},[23],[25,36312,36313],{"xmlns":27,"display":221},[29,36314,36315,36403],{},[32,36316,36317,36323,36325,36331,36333,36339,36341,36361,36363,36373,36375,36389,36391,36393,36395,36401],{},[266,36318,36319,36321],{},[35,36320,21273],{},[35,36322,272],{},[81,36324,244],{},[266,36326,36327,36329],{},[35,36328,33396],{},[35,36330,272],{},[81,36332,7756],{},[3718,36334,36335,36337],{},[35,36336,33401],{},[35,36338,102],{},[81,36340,244],{},[3718,36342,36343,36355],{},[32,36344,36345,36351,36353],{},[266,36346,36347,36349],{},[35,36348,3102],{},[35,36350,272],{},[81,36352,239],{},[89,36354,91],{},[266,36356,36357,36359],{},[35,36358,3102],{},[89,36360,278],{},[81,36362,7756],{},[3718,36364,36365,36367],{},[89,36366,91],{},[266,36368,36369,36371],{},[35,36370,3102],{},[89,36372,278],{},[81,36374,244],{},[3718,36376,36377,36383],{},[266,36378,36379,36381],{},[35,36380,3102],{},[35,36382,272],{},[266,36384,36385,36387],{},[35,36386,3102],{},[89,36388,278],{},[81,36390,244],{},[35,36392,14807],{"mathvariant":258},[81,36394,14810],{"stretchy":86},[266,36396,36397,36399],{},[35,36398,1027],{},[35,36400,272],{},[81,36402,14819],{"stretchy":86},[40,36404,36405],{"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]",[15,36407,36409,36464,36519,36598,36761,36875,37027],{"className":36408,"ariaHidden":48},[47],[15,36410,36412,36415,36455,36458,36461],{"className":36411},[52],[15,36413],{"className":36414,"style":34331},[56],[15,36416,36418,36421],{"className":36417},[61],[15,36419,21273],{"className":36420,"style":34338},[61,122],[15,36422,36424],{"className":36423},[338],[15,36425,36427,36447],{"className":36426},[342,472],[15,36428,36430,36444],{"className":36429},[346],[15,36431,36433],{"className":36432,"style":479},[350],[15,36434,36435,36438],{"style":34353},[15,36436],{"className":36437,"style":358},[357],[15,36439,36441],{"className":36440},[362,363,364,365],[15,36442,272],{"className":36443,"style":492},[61,122,365],[15,36445,497],{"className":36446},[496],[15,36448,36450],{"className":36449},[346],[15,36451,36453],{"className":36452,"style":504},[350],[15,36454],{},[15,36456],{"className":36457,"style":128},[127],[15,36459,244],{"className":36460},[132],[15,36462],{"className":36463,"style":128},[127],[15,36465,36467,36470,36510,36513,36516],{"className":36466},[52],[15,36468],{"className":36469,"style":34389},[56],[15,36471,36473,36476],{"className":36472},[61],[15,36474,33396],{"className":36475},[61,122],[15,36477,36479],{"className":36478},[338],[15,36480,36482,36502],{"className":36481},[342,472],[15,36483,36485,36499],{"className":36484},[346],[15,36486,36488],{"className":36487,"style":479},[350],[15,36489,36490,36493],{"style":482},[15,36491],{"className":36492,"style":358},[357],[15,36494,36496],{"className":36495},[362,363,364,365],[15,36497,272],{"className":36498,"style":492},[61,122,365],[15,36500,497],{"className":36501},[496],[15,36503,36505],{"className":36504},[346],[15,36506,36508],{"className":36507,"style":504},[350],[15,36509],{},[15,36511],{"className":36512,"style":772},[127],[15,36514,7756],{"className":36515},[375],[15,36517],{"className":36518,"style":772},[127],[15,36520,36522,36526,36589,36592,36595],{"className":36521},[52],[15,36523],{"className":36524,"style":36525},[56],"height:1.7936em;vertical-align:-0.686em;",[15,36527,36529,36532,36586],{"className":36528},[61],[15,36530],{"className":36531},[146,3837],[15,36533,36535],{"className":36534},[3718],[15,36536,36538,36577],{"className":36537},[342,472],[15,36539,36541,36574],{"className":36540},[346],[15,36542,36544,36555,36563],{"className":36543,"style":27626},[350],[15,36545,36546,36549],{"style":11393},[15,36547],{"className":36548,"style":3857},[357],[15,36550,36552],{"className":36551},[61],[15,36553,102],{"className":36554,"style":174},[61,122],[15,36556,36557,36560],{"style":3885},[15,36558],{"className":36559,"style":3857},[357],[15,36561],{"className":36562,"style":3893},[3892],[15,36564,36565,36568],{"style":11459},[15,36566],{"className":36567,"style":3857},[357],[15,36569,36571],{"className":36570},[61],[15,36572,33401],{"className":36573},[61,122],[15,36575,497],{"className":36576},[496],[15,36578,36580],{"className":36579},[346],[15,36581,36584],{"className":36582,"style":36583},[350],"height:0.686em;",[15,36585],{},[15,36587],{"className":36588},[178,3837],[15,36590],{"className":36591,"style":128},[127],[15,36593,244],{"className":36594},[132],[15,36596],{"className":36597,"style":128},[127],[15,36599,36601,36604,36752,36755,36758],{"className":36600},[52],[15,36602],{"className":36603,"style":18660},[56],[15,36605,36607,36610,36749],{"className":36606},[61],[15,36608],{"className":36609},[146,3837],[15,36611,36613],{"className":36612},[3718],[15,36614,36616,36741],{"className":36615},[342,472],[15,36617,36619,36738],{"className":36618},[346],[15,36620,36622,36670,36678],{"className":36621,"style":18679},[350],[15,36623,36624,36627],{"style":11393},[15,36625],{"className":36626,"style":3857},[357],[15,36628,36630],{"className":36629},[61],[15,36631,36633,36636],{"className":36632},[61],[15,36634,3102],{"className":36635,"style":3187},[61,122],[15,36637,36639],{"className":36638},[338],[15,36640,36642,36662],{"className":36641},[342,472],[15,36643,36645,36659],{"className":36644},[346],[15,36646,36648],{"className":36647,"style":1212},[350],[15,36649,36650,36653],{"style":3202},[15,36651],{"className":36652,"style":358},[357],[15,36654,36656],{"className":36655},[362,363,364,365],[15,36657,278],{"className":36658},[61,365],[15,36660,497],{"className":36661},[496],[15,36663,36665],{"className":36664},[346],[15,36666,36668],{"className":36667,"style":504},[350],[15,36669],{},[15,36671,36672,36675],{"style":3885},[15,36673],{"className":36674,"style":3857},[357],[15,36676],{"className":36677,"style":3893},[3892],[15,36679,36680,36683],{"style":11459},[15,36681],{"className":36682,"style":3857},[357],[15,36684,36686,36726,36729,36732,36735],{"className":36685},[61],[15,36687,36689,36692],{"className":36688},[61],[15,36690,3102],{"className":36691,"style":3187},[61,122],[15,36693,36695],{"className":36694},[338],[15,36696,36698,36718],{"className":36697},[342,472],[15,36699,36701,36715],{"className":36700},[346],[15,36702,36704],{"className":36703,"style":479},[350],[15,36705,36706,36709],{"style":3202},[15,36707],{"className":36708,"style":358},[357],[15,36710,36712],{"className":36711},[362,363,364,365],[15,36713,272],{"className":36714,"style":492},[61,122,365],[15,36716,497],{"className":36717},[496],[15,36719,36721],{"className":36720},[346],[15,36722,36724],{"className":36723,"style":504},[350],[15,36725],{},[15,36727],{"className":36728,"style":772},[127],[15,36730,239],{"className":36731},[375],[15,36733],{"className":36734,"style":772},[127],[15,36736,91],{"className":36737},[61],[15,36739,497],{"className":36740},[496],[15,36742,36744],{"className":36743},[346],[15,36745,36747],{"className":36746,"style":18768},[350],[15,36748],{},[15,36750],{"className":36751},[178,3837],[15,36753],{"className":36754,"style":772},[127],[15,36756,7756],{"className":36757},[375],[15,36759],{"className":36760,"style":772},[127],[15,36762,36764,36767,36866,36869,36872],{"className":36763},[52],[15,36765],{"className":36766,"style":18660},[56],[15,36768,36770,36773,36863],{"className":36769},[61],[15,36771],{"className":36772},[146,3837],[15,36774,36776],{"className":36775},[3718],[15,36777,36779,36855],{"className":36778},[342,472],[15,36780,36782,36852],{"className":36781},[346],[15,36783,36785,36833,36841],{"className":36784,"style":18679},[350],[15,36786,36787,36790],{"style":11393},[15,36788],{"className":36789,"style":3857},[357],[15,36791,36793],{"className":36792},[61],[15,36794,36796,36799],{"className":36795},[61],[15,36797,3102],{"className":36798,"style":3187},[61,122],[15,36800,36802],{"className":36801},[338],[15,36803,36805,36825],{"className":36804},[342,472],[15,36806,36808,36822],{"className":36807},[346],[15,36809,36811],{"className":36810,"style":1212},[350],[15,36812,36813,36816],{"style":3202},[15,36814],{"className":36815,"style":358},[357],[15,36817,36819],{"className":36818},[362,363,364,365],[15,36820,278],{"className":36821},[61,365],[15,36823,497],{"className":36824},[496],[15,36826,36828],{"className":36827},[346],[15,36829,36831],{"className":36830,"style":504},[350],[15,36832],{},[15,36834,36835,36838],{"style":3885},[15,36836],{"className":36837,"style":3857},[357],[15,36839],{"className":36840,"style":3893},[3892],[15,36842,36843,36846],{"style":11459},[15,36844],{"className":36845,"style":3857},[357],[15,36847,36849],{"className":36848},[61],[15,36850,91],{"className":36851},[61],[15,36853,497],{"className":36854},[496],[15,36856,36858],{"className":36857},[346],[15,36859,36861],{"className":36860,"style":18768},[350],[15,36862],{},[15,36864],{"className":36865},[178,3837],[15,36867],{"className":36868,"style":128},[127],[15,36870,244],{"className":36871},[132],[15,36873],{"className":36874,"style":128},[127],[15,36876,36878,36882,37018,37021,37024],{"className":36877},[52],[15,36879],{"className":36880,"style":36881},[56],"height:1.9436em;vertical-align:-0.836em;",[15,36883,36885,36888,37015],{"className":36884},[61],[15,36886],{"className":36887},[146,3837],[15,36889,36891],{"className":36890},[3718],[15,36892,36894,37007],{"className":36893},[342,472],[15,36895,36897,37004],{"className":36896},[346],[15,36898,36900,36948,36956],{"className":36899,"style":27626},[350],[15,36901,36902,36905],{"style":11393},[15,36903],{"className":36904,"style":3857},[357],[15,36906,36908],{"className":36907},[61],[15,36909,36911,36914],{"className":36910},[61],[15,36912,3102],{"className":36913,"style":3187},[61,122],[15,36915,36917],{"className":36916},[338],[15,36918,36920,36940],{"className":36919},[342,472],[15,36921,36923,36937],{"className":36922},[346],[15,36924,36926],{"className":36925,"style":1212},[350],[15,36927,36928,36931],{"style":3202},[15,36929],{"className":36930,"style":358},[357],[15,36932,36934],{"className":36933},[362,363,364,365],[15,36935,278],{"className":36936},[61,365],[15,36938,497],{"className":36939},[496],[15,36941,36943],{"className":36942},[346],[15,36944,36946],{"className":36945,"style":504},[350],[15,36947],{},[15,36949,36950,36953],{"style":3885},[15,36951],{"className":36952,"style":3857},[357],[15,36954],{"className":36955,"style":3893},[3892],[15,36957,36958,36961],{"style":11459},[15,36959],{"className":36960,"style":3857},[357],[15,36962,36964],{"className":36963},[61],[15,36965,36967,36970],{"className":36966},[61],[15,36968,3102],{"className":36969,"style":3187},[61,122],[15,36971,36973],{"className":36972},[338],[15,36974,36976,36996],{"className":36975},[342,472],[15,36977,36979,36993],{"className":36978},[346],[15,36980,36982],{"className":36981,"style":479},[350],[15,36983,36984,36987],{"style":3202},[15,36985],{"className":36986,"style":358},[357],[15,36988,36990],{"className":36989},[362,363,364,365],[15,36991,272],{"className":36992,"style":492},[61,122,365],[15,36994,497],{"className":36995},[496],[15,36997,36999],{"className":36998},[346],[15,37000,37002],{"className":37001,"style":504},[350],[15,37003],{},[15,37005,497],{"className":37006},[496],[15,37008,37010],{"className":37009},[346],[15,37011,37013],{"className":37012,"style":18768},[350],[15,37014],{},[15,37016],{"className":37017},[178,3837],[15,37019],{"className":37020,"style":128},[127],[15,37022,244],{"className":37023},[132],[15,37025],{"className":37026,"style":128},[127],[15,37028,37030,37033,37036,37039,37079],{"className":37029},[52],[15,37031],{"className":37032,"style":142},[56],[15,37034,14807],{"className":37035},[61,427],[15,37037,14810],{"className":37038},[146],[15,37040,37042,37045],{"className":37041},[61],[15,37043,1027],{"className":37044},[61,122],[15,37046,37048],{"className":37047},[338],[15,37049,37051,37071],{"className":37050},[342,472],[15,37052,37054,37068],{"className":37053},[346],[15,37055,37057],{"className":37056,"style":479},[350],[15,37058,37059,37062],{"style":482},[15,37060],{"className":37061,"style":358},[357],[15,37063,37065],{"className":37064},[362,363,364,365],[15,37066,272],{"className":37067,"style":492},[61,122,365],[15,37069,497],{"className":37070},[496],[15,37072,37074],{"className":37073},[346],[15,37075,37077],{"className":37076,"style":504},[350],[15,37078],{},[15,37080,14819],{"className":37081},[178],[11,37083,37084],{},"其中最后一个等号由命题 2.1 给出。",[11,37086,37087,37088,37091],{},"命题 3.2 是整篇文章的枢纽：它说明",[661,37089,37090],{},"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。",[877,37093,37095],{"id":37094},"_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",[11,37097,37098,37099,37255,37256,37373,37374,37446,37447,37753],{},"设神经网络 ",[15,37100,37102,37133],{"className":37101,"translate":19},[18],[15,37103,37105],{"className":37104},[23],[25,37106,37107],{"xmlns":27},[29,37108,37109,37130],{},[32,37110,37111,37118,37120,37122,37124],{},[266,37112,37113,37115],{},[35,37114,3703],{},[35,37116,37117],{"mathvariant":3101},"θ",[81,37119,264],{},[35,37121,38],{"mathvariant":37},[81,37123,20691],{},[227,37125,37126,37128],{},[35,37127,259],{"mathvariant":258},[35,37129,102],{},[40,37131,37132],{"encoding":42},"f_{\\boldsymbol{\\theta}}: \\mathcal{X} \\to \\mathbb{R}^K",[15,37134,37136,37202,37220],{"className":37135,"ariaHidden":48},[47],[15,37137,37139,37142,37193,37196,37199],{"className":37138},[52],[15,37140],{"className":37141,"style":14301},[56],[15,37143,37145,37148],{"className":37144},[61],[15,37146,3703],{"className":37147,"style":3781},[61,122],[15,37149,37151],{"className":37150},[338],[15,37152,37154,37185],{"className":37153},[342,472],[15,37155,37157,37182],{"className":37156},[346],[15,37158,37160],{"className":37159,"style":479},[350],[15,37161,37163,37166],{"style":37162},"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;",[15,37164],{"className":37165,"style":358},[357],[15,37167,37169],{"className":37168},[362,363,364,365],[15,37170,37172],{"className":37171},[61,365],[15,37173,37175],{"className":37174},[61,365],[15,37176,37178],{"className":37177},[61,365],[15,37179,37117],{"className":37180,"style":37181},[61,3162,365],"margin-right:0.0319em;",[15,37183,497],{"className":37184},[496],[15,37186,37188],{"className":37187},[346],[15,37189,37191],{"className":37190,"style":504},[350],[15,37192],{},[15,37194],{"className":37195,"style":128},[127],[15,37197,264],{"className":37198},[132],[15,37200],{"className":37201,"style":128},[127],[15,37203,37205,37208,37211,37214,37217],{"className":37204},[52],[15,37206],{"className":37207,"style":57},[56],[15,37209,38],{"className":37210,"style":63},[61,62],[15,37212],{"className":37213,"style":128},[127],[15,37215,20691],{"className":37216},[132],[15,37218],{"className":37219,"style":128},[127],[15,37221,37223,37226],{"className":37222},[52],[15,37224],{"className":37225,"style":695},[56],[15,37227,37229,37232],{"className":37228},[61],[15,37230,259],{"className":37231},[61,427],[15,37233,37235],{"className":37234},[338],[15,37236,37238],{"className":37237},[342],[15,37239,37241],{"className":37240},[346],[15,37242,37244],{"className":37243,"style":695},[350],[15,37245,37246,37249],{"style":716},[15,37247],{"className":37248,"style":358},[357],[15,37250,37252],{"className":37251},[362,363,364,365],[15,37253,102],{"className":37254,"style":174},[61,122,365]," 输出 logits ",[15,37257,37259,37288],{"className":37258,"translate":19},[18],[15,37260,37262],{"className":37261},[23],[25,37263,37264],{"xmlns":27},[29,37265,37266,37285],{},[32,37267,37268,37271,37273,37279,37281,37283],{},[35,37269,37270],{"mathvariant":251},"z",[81,37272,244],{},[266,37274,37275,37277],{},[35,37276,3703],{},[35,37278,37117],{"mathvariant":3101},[81,37280,741],{"stretchy":86},[35,37282,1027],{"mathvariant":251},[81,37284,750],{"stretchy":86},[40,37286,37287],{"encoding":42},"\\mathbf{z} = f_{\\boldsymbol{\\theta}}(\\mathbf{x})",[15,37289,37291,37309],{"className":37290,"ariaHidden":48},[47],[15,37292,37294,37297,37300,37303,37306],{"className":37293},[52],[15,37295],{"className":37296,"style":2706},[56],[15,37298,37270],{"className":37299},[61,411],[15,37301],{"className":37302,"style":128},[127],[15,37304,244],{"className":37305},[132],[15,37307],{"className":37308,"style":128},[127],[15,37310,37312,37315,37364,37367,37370],{"className":37311},[52],[15,37313],{"className":37314,"style":142},[56],[15,37316,37318,37321],{"className":37317},[61],[15,37319,3703],{"className":37320,"style":3781},[61,122],[15,37322,37324],{"className":37323},[338],[15,37325,37327,37356],{"className":37326},[342,472],[15,37328,37330,37353],{"className":37329},[346],[15,37331,37333],{"className":37332,"style":479},[350],[15,37334,37335,37338],{"style":37162},[15,37336],{"className":37337,"style":358},[357],[15,37339,37341],{"className":37340},[362,363,364,365],[15,37342,37344],{"className":37343},[61,365],[15,37345,37347],{"className":37346},[61,365],[15,37348,37350],{"className":37349},[61,365],[15,37351,37117],{"className":37352,"style":37181},[61,3162,365],[15,37354,497],{"className":37355},[496],[15,37357,37359],{"className":37358},[346],[15,37360,37362],{"className":37361,"style":504},[350],[15,37363],{},[15,37365,741],{"className":37366},[146],[15,37368,1027],{"className":37369},[61,411],[15,37371,750],{"className":37372},[178],"，softmax 给出 ",[15,37375,37377,37404],{"className":37376,"translate":19},[18],[15,37378,37380],{"className":37379},[23],[25,37381,37382],{"xmlns":27},[29,37383,37384,37401],{},[32,37385,37386,37388,37390,37393,37395,37397,37399],{},[35,37387,11],{"mathvariant":251},[81,37389,244],{},[35,37391,37392],{"mathvariant":231},"softmax",[81,37394,1545],{},[81,37396,741],{"stretchy":86},[35,37398,37270],{"mathvariant":251},[81,37400,750],{"stretchy":86},[40,37402,37403],{"encoding":42},"\\mathbf{p} = \\operatorname{softmax}(\\mathbf{z})",[15,37405,37407,37425],{"className":37406,"ariaHidden":48},[47],[15,37408,37410,37413,37416,37419,37422],{"className":37409},[52],[15,37411],{"className":37412,"style":829},[56],[15,37414,11],{"className":37415},[61,411],[15,37417],{"className":37418,"style":128},[127],[15,37420,244],{"className":37421},[132],[15,37423],{"className":37424,"style":128},[127],[15,37426,37428,37431,37437,37440,37443],{"className":37427},[52],[15,37429],{"className":37430,"style":142},[56],[15,37432,37434],{"className":37433},[531],[15,37435,37392],{"className":37436},[61,1620],[15,37438,741],{"className":37439},[146],[15,37441,37270],{"className":37442},[61,411],[15,37444,750],{"className":37445},[178],"，即 ",[15,37448,37450,37498],{"className":37449,"translate":19},[18],[15,37451,37453],{"className":37452},[23],[25,37454,37455],{"xmlns":27},[29,37456,37457,37495],{},[32,37458,37459,37465,37467,37477,37479,37485],{},[266,37460,37461,37463],{},[35,37462,11],{},[35,37464,272],{},[81,37466,244],{},[227,37468,37469,37471],{},[35,37470,1980],{},[266,37472,37473,37475],{},[35,37474,37270],{},[35,37476,272],{},[35,37478,3018],{"mathvariant":231},[266,37480,37481,37483],{},[81,37482,293],{},[35,37484,14766],{},[227,37486,37487,37489],{},[35,37488,1980],{},[266,37490,37491,37493],{},[35,37492,37270],{},[35,37494,14766],{},[40,37496,37497],{"encoding":42},"p_k = e^{z_k}\u002F\\sum_j e^{z_j}",[15,37499,37501,37556],{"className":37500,"ariaHidden":48},[47],[15,37502,37504,37507,37547,37550,37553],{"className":37503},[52],[15,37505],{"className":37506,"style":9428},[56],[15,37508,37510,37513],{"className":37509},[61],[15,37511,11],{"className":37512},[61,122],[15,37514,37516],{"className":37515},[338],[15,37517,37519,37539],{"className":37518},[342,472],[15,37520,37522,37536],{"className":37521},[346],[15,37523,37525],{"className":37524,"style":479},[350],[15,37526,37527,37530],{"style":482},[15,37528],{"className":37529,"style":358},[357],[15,37531,37533],{"className":37532},[362,363,364,365],[15,37534,272],{"className":37535,"style":492},[61,122,365],[15,37537,497],{"className":37538},[496],[15,37540,37542],{"className":37541},[346],[15,37543,37545],{"className":37544,"style":504},[350],[15,37546],{},[15,37548],{"className":37549,"style":128},[127],[15,37551,244],{"className":37552},[132],[15,37554],{"className":37555,"style":128},[127],[15,37557,37559,37562,37634,37637,37640,37680,37683],{"className":37558},[52],[15,37560],{"className":37561,"style":29838},[56],[15,37563,37565,37568],{"className":37564},[61],[15,37566,1980],{"className":37567},[61,122],[15,37569,37571],{"className":37570},[338],[15,37572,37574],{"className":37573},[342],[15,37575,37577],{"className":37576},[346],[15,37578,37581],{"className":37579,"style":37580},[350],"height:0.6644em;",[15,37582,37583,37586],{"style":716},[15,37584],{"className":37585,"style":358},[357],[15,37587,37589],{"className":37588},[362,363,364,365],[15,37590,37592],{"className":37591},[61,365],[15,37593,37595,37599],{"className":37594},[61,365],[15,37596,37270],{"className":37597,"style":37598},[61,122,365],"margin-right:0.044em;",[15,37600,37602],{"className":37601},[338],[15,37603,37605,37626],{"className":37604},[342,472],[15,37606,37608,37623],{"className":37607},[346],[15,37609,37611],{"className":37610,"style":22389},[350],[15,37612,37614,37617],{"style":37613},"top:-2.3488em;margin-left:-0.044em;margin-right:0.0714em;",[15,37615],{"className":37616,"style":4058},[357],[15,37618,37620],{"className":37619},[362,4062,4063,365],[15,37621,272],{"className":37622,"style":492},[61,122,365],[15,37624,497],{"className":37625},[496],[15,37627,37629],{"className":37628},[346],[15,37630,37632],{"className":37631,"style":27230},[350],[15,37633],{},[15,37635,3018],{"className":37636},[61],[15,37638],{"className":37639,"style":157},[127],[15,37641,37643,37646],{"className":37642},[531],[15,37644,293],{"className":37645,"style":537},[531,535,536],[15,37647,37649],{"className":37648},[338],[15,37650,37652,37672],{"className":37651},[342,472],[15,37653,37655,37669],{"className":37654},[346],[15,37656,37658],{"className":37657,"style":6064},[350],[15,37659,37660,37663],{"style":553},[15,37661],{"className":37662,"style":358},[357],[15,37664,37666],{"className":37665},[362,363,364,365],[15,37667,14766],{"className":37668,"style":14792},[61,122,365],[15,37670,497],{"className":37671},[496],[15,37673,37675],{"className":37674},[346],[15,37676,37678],{"className":37677,"style":30028},[350],[15,37679],{},[15,37681],{"className":37682,"style":157},[127],[15,37684,37686,37689],{"className":37685},[61],[15,37687,1980],{"className":37688},[61,122],[15,37690,37692],{"className":37691},[338],[15,37693,37695],{"className":37694},[342],[15,37696,37698],{"className":37697},[346],[15,37699,37701],{"className":37700,"style":37580},[350],[15,37702,37703,37706],{"style":716},[15,37704],{"className":37705,"style":358},[357],[15,37707,37709],{"className":37708},[362,363,364,365],[15,37710,37712],{"className":37711},[61,365],[15,37713,37715,37718],{"className":37714},[61,365],[15,37716,37270],{"className":37717,"style":37598},[61,122,365],[15,37719,37721],{"className":37720},[338],[15,37722,37724,37745],{"className":37723},[342,472],[15,37725,37727,37742],{"className":37726},[346],[15,37728,37730],{"className":37729,"style":4051},[350],[15,37731,37733,37736],{"style":37732},"top:-2.357em;margin-left:-0.044em;margin-right:0.0714em;",[15,37734],{"className":37735,"style":4058},[357],[15,37737,37739],{"className":37738},[362,4062,4063,365],[15,37740,14766],{"className":37741,"style":14792},[61,122,365],[15,37743,497],{"className":37744},[496],[15,37746,37748],{"className":37747},[346],[15,37749,37751],{"className":37750,"style":15952},[350],[15,37752],{},"。这一映射存在两个结构性缺陷。",[11,37755,37756,37759,37760,941,37789,37907,37908,9371,37989,37992,37993,38044,38045,38119,38120,38158],{},[661,37757,37758],{},"其一，平移不变性抹去了「无信息」这一状态。"," 对任意常数 ",[15,37761,37763,37777],{"className":37762,"translate":19},[18],[15,37764,37766],{"className":37765},[23],[25,37767,37768],{"xmlns":27},[29,37769,37770,37775],{},[32,37771,37772],{},[35,37773,37774],{},"c",[40,37776,37774],{"encoding":42},[15,37778,37780],{"className":37779,"ariaHidden":48},[47],[15,37781,37783,37786],{"className":37782},[52],[15,37784],{"className":37785,"style":6026},[56],[15,37787,37774],{"className":37788},[61,122],[15,37790,37792,37832],{"className":37791,"translate":19},[18],[15,37793,37795],{"className":37794},[23],[25,37796,37797],{"xmlns":27},[29,37798,37799,37829],{},[32,37800,37801,37803,37805,37807,37809,37811,37813,37815,37817,37819,37821,37823,37825,37827],{},[35,37802,37392],{"mathvariant":231},[81,37804,1545],{},[81,37806,741],{"stretchy":86},[35,37808,37270],{"mathvariant":251},[81,37810,7756],{},[35,37812,37774],{},[89,37814,91],{"mathvariant":251},[81,37816,750],{"stretchy":86},[81,37818,244],{},[35,37820,37392],{"mathvariant":231},[81,37822,1545],{},[81,37824,741],{"stretchy":86},[35,37826,37270],{"mathvariant":251},[81,37828,750],{"stretchy":86},[40,37830,37831],{"encoding":42},"\\operatorname{softmax}(\\mathbf{z} + c\\mathbf{1}) = \\operatorname{softmax}(\\mathbf{z})",[15,37833,37835,37862,37886],{"className":37834,"ariaHidden":48},[47],[15,37836,37838,37841,37847,37850,37853,37856,37859],{"className":37837},[52],[15,37839],{"className":37840,"style":142},[56],[15,37842,37844],{"className":37843},[531],[15,37845,37392],{"className":37846},[61,1620],[15,37848,741],{"className":37849},[146],[15,37851,37270],{"className":37852},[61,411],[15,37854],{"className":37855,"style":772},[127],[15,37857,7756],{"className":37858},[375],[15,37860],{"className":37861,"style":772},[127],[15,37863,37865,37868,37871,37874,37877,37880,37883],{"className":37864},[52],[15,37866],{"className":37867,"style":142},[56],[15,37869,37774],{"className":37870},[61,122],[15,37872,91],{"className":37873},[61,411],[15,37875,750],{"className":37876},[178],[15,37878],{"className":37879,"style":128},[127],[15,37881,244],{"className":37882},[132],[15,37884],{"className":37885,"style":128},[127],[15,37887,37889,37892,37898,37901,37904],{"className":37888},[52],[15,37890],{"className":37891,"style":142},[56],[15,37893,37895],{"className":37894},[531],[15,37896,37392],{"className":37897},[61,1620],[15,37899,741],{"className":37900},[146],[15,37902,37270],{"className":37903},[61,411],[15,37905,750],{"className":37906},[178],"。因此 logits 的绝对水平不含任何信息，网络的输出只依赖于 ",[15,37909,37911,37933],{"className":37910,"translate":19},[18],[15,37912,37914],{"className":37913},[23],[25,37915,37916],{"xmlns":27},[29,37917,37918,37930],{},[32,37919,37920,37922,37928],{},[81,37921,87],{"stretchy":86},[266,37923,37924,37926],{},[35,37925,37270],{},[35,37927,272],{},[81,37929,105],{"stretchy":86},[40,37931,37932],{"encoding":42},"\\{z_k\\}",[15,37934,37936],{"className":37935,"ariaHidden":48},[47],[15,37937,37939,37942,37945,37986],{"className":37938},[52],[15,37940],{"className":37941,"style":142},[56],[15,37943,87],{"className":37944},[146],[15,37946,37948,37951],{"className":37947},[61],[15,37949,37270],{"className":37950,"style":37598},[61,122],[15,37952,37954],{"className":37953},[338],[15,37955,37957,37978],{"className":37956},[342,472],[15,37958,37960,37975],{"className":37959},[346],[15,37961,37963],{"className":37962,"style":479},[350],[15,37964,37966,37969],{"style":37965},"top:-2.55em;margin-left:-0.044em;margin-right:0.05em;",[15,37967],{"className":37968,"style":358},[357],[15,37970,37972],{"className":37971},[362,363,364,365],[15,37973,272],{"className":37974,"style":492},[61,122,365],[15,37976,497],{"className":37977},[496],[15,37979,37981],{"className":37980},[346],[15,37982,37984],{"className":37983,"style":504},[350],[15,37985],{},[15,37987,105],{"className":37988},[178],[661,37990,37991],{},"相对差","。后果是：",[15,37994,37996,38014],{"className":37995,"translate":19},[18],[15,37997,37999],{"className":37998},[23],[25,38000,38001],{"xmlns":27},[29,38002,38003,38011],{},[32,38004,38005,38007,38009],{},[35,38006,37270],{"mathvariant":251},[81,38008,244],{},[89,38010,278],{"mathvariant":251},[40,38012,38013],{"encoding":42},"\\mathbf{z} = \\mathbf{0}",[15,38015,38017,38035],{"className":38016,"ariaHidden":48},[47],[15,38018,38020,38023,38026,38029,38032],{"className":38019},[52],[15,38021],{"className":38022,"style":2706},[56],[15,38024,37270],{"className":38025},[61,411],[15,38027],{"className":38028,"style":128},[127],[15,38030,244],{"className":38031},[132],[15,38033],{"className":38034,"style":128},[127],[15,38036,38038,38041],{"className":38037},[52],[15,38039],{"className":38040,"style":859},[56],[15,38042,278],{"className":38043},[61,411],"（「全零 logits」）与 ",[15,38046,38048,38071],{"className":38047,"translate":19},[18],[15,38049,38051],{"className":38050},[23],[25,38052,38053],{"xmlns":27},[29,38054,38055,38068],{},[32,38056,38057,38059,38061,38064,38066],{},[35,38058,37270],{"mathvariant":251},[81,38060,244],{},[89,38062,38063],{},"100",[81,38065,11205],{},[89,38067,91],{"mathvariant":251},[40,38069,38070],{"encoding":42},"\\mathbf{z} = 100 \\cdot \\mathbf{1}",[15,38072,38074,38092,38110],{"className":38073,"ariaHidden":48},[47],[15,38075,38077,38080,38083,38086,38089],{"className":38076},[52],[15,38078],{"className":38079,"style":2706},[56],[15,38081,37270],{"className":38082},[61,411],[15,38084],{"className":38085,"style":128},[127],[15,38087,244],{"className":38088},[132],[15,38090],{"className":38091,"style":128},[127],[15,38093,38095,38098,38101,38104,38107],{"className":38094},[52],[15,38096],{"className":38097,"style":859},[56],[15,38099,38063],{"className":38100},[61],[15,38102],{"className":38103,"style":772},[127],[15,38105,11205],{"className":38106},[375],[15,38108],{"className":38109,"style":772},[127],[15,38111,38113,38116],{"className":38112},[52],[15,38114],{"className":38115,"style":859},[56],[15,38117,91],{"className":38118},[61,411]," 不可区分，二者都对应均匀分布 ",[15,38121,38123,38140],{"className":38122,"translate":19},[18],[15,38124,38126],{"className":38125},[23],[25,38127,38128],{"xmlns":27},[29,38129,38130,38138],{},[32,38131,38132,38134,38136],{},[89,38133,91],{"mathvariant":251},[35,38135,3018],{"mathvariant":231},[35,38137,102],{},[40,38139,32573],{"encoding":42},[15,38141,38143],{"className":38142,"ariaHidden":48},[47],[15,38144,38146,38149,38152,38155],{"className":38145},[52],[15,38147],{"className":38148,"style":142},[56],[15,38150,91],{"className":38151},[61,411],[15,38153,3018],{"className":38154},[61],[15,38156,102],{"className":38157,"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。",[11,38160,38161,38164,38165,38297,38298,38301,38302,867],{},[661,38162,38163],{},"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 ",[15,38166,38168,38204],{"className":38167,"translate":19},[18],[15,38169,38171],{"className":38170},[23],[25,38172,38173],{"xmlns":27},[29,38174,38175,38201],{},[32,38176,38177,38179,38181,38183,38185,38191,38193,38195,38197,38199],{},[35,38178,37270],{"mathvariant":251},[81,38180,20691],{},[35,38182,37774],{},[282,38184,4796],{},[227,38186,38187,38189],{},[35,38188,37270],{"mathvariant":251},[81,38190,21969],{},[81,38192,94],{"separator":48},[282,38194,284],{},[35,38196,37774],{},[81,38198,20691],{},[35,38200,6790],{"mathvariant":231},[40,38202,38203],{"encoding":42},"\\mathbf{z} \\to c\\,\\mathbf{z}^\\star,\\ c \\to \\infty",[15,38205,38207,38225,38288],{"className":38206,"ariaHidden":48},[47],[15,38208,38210,38213,38216,38219,38222],{"className":38209},[52],[15,38211],{"className":38212,"style":2706},[56],[15,38214,37270],{"className":38215},[61,411],[15,38217],{"className":38218,"style":128},[127],[15,38220,20691],{"className":38221},[132],[15,38223],{"className":38224,"style":128},[127],[15,38226,38228,38232,38235,38238,38267,38270,38273,38276,38279,38282,38285],{"className":38227},[52],[15,38229],{"className":38230,"style":38231},[56],"height:0.8831em;vertical-align:-0.1944em;",[15,38233,37774],{"className":38234},[61,122],[15,38236],{"className":38237,"style":157},[127],[15,38239,38241,38244],{"className":38240},[61],[15,38242,37270],{"className":38243},[61,411],[15,38245,38247],{"className":38246},[338],[15,38248,38250],{"className":38249},[342],[15,38251,38253],{"className":38252},[346],[15,38254,38256],{"className":38255,"style":22048},[350],[15,38257,38258,38261],{"style":716},[15,38259],{"className":38260,"style":358},[357],[15,38262,38264],{"className":38263},[362,363,364,365],[15,38265,21969],{"className":38266},[375,365],[15,38268,94],{"className":38269},[153],[15,38271,284],{"className":38272},[127],[15,38274],{"className":38275,"style":157},[127],[15,38277,37774],{"className":38278},[61,122],[15,38280],{"className":38281,"style":128},[127],[15,38283,20691],{"className":38284},[132],[15,38286],{"className":38287,"style":128},[127],[15,38289,38291,38294],{"className":38290},[52],[15,38292],{"className":38293,"style":6026},[56],[15,38295,6790],{"className":38296},[61]," 可将损失单调压向零。换言之，",[661,38299,38300],{},"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",[661,38303,38304],{},"完全不受输入是否为分布内样本的约束",[11,38306,38307],{},"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」。",[877,38309,38311],{"id":38310},"_34-证据参数化","3.4 证据参数化",[11,38313,38314,38315],{},"证据深度学习对上述缺陷的回应是：",[661,38316,38317],{},"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。",[882,38319,38321],{"title":38320},"定义 3.3（证据网络）",[11,38322,13,38323,38512,38513,38516,38517,38776,38777,39014,39015,867],{},[15,38324,38326,38362],{"className":38325,"translate":19},[18],[15,38327,38329],{"className":38328},[23],[25,38330,38331],{"xmlns":27},[29,38332,38333,38359],{},[32,38334,38335,38341,38343,38345,38347],{},[266,38336,38337,38339],{},[35,38338,22777],{},[35,38340,37117],{"mathvariant":3101},[81,38342,264],{},[35,38344,38],{"mathvariant":37},[81,38346,20691],{},[289,38348,38349,38351,38357],{},[35,38350,259],{"mathvariant":258},[32,38352,38353,38355],{},[81,38354,275],{},[89,38356,278],{},[35,38358,102],{},[40,38360,38361],{"encoding":42},"g_{\\boldsymbol{\\theta}}: \\mathcal{X} \\to \\mathbb{R}_{\\ge 0}^K",[15,38363,38365,38429,38447],{"className":38364,"ariaHidden":48},[47],[15,38366,38368,38371,38420,38423,38426],{"className":38367},[52],[15,38369],{"className":38370,"style":9428},[56],[15,38372,38374,38377],{"className":38373},[61],[15,38375,22777],{"className":38376,"style":123},[61,122],[15,38378,38380],{"className":38379},[338],[15,38381,38383,38412],{"className":38382},[342,472],[15,38384,38386,38409],{"className":38385},[346],[15,38387,38389],{"className":38388,"style":479},[350],[15,38390,38391,38394],{"style":27013},[15,38392],{"className":38393,"style":358},[357],[15,38395,38397],{"className":38396},[362,363,364,365],[15,38398,38400],{"className":38399},[61,365],[15,38401,38403],{"className":38402},[61,365],[15,38404,38406],{"className":38405},[61,365],[15,38407,37117],{"className":38408,"style":37181},[61,3162,365],[15,38410,497],{"className":38411},[496],[15,38413,38415],{"className":38414},[346],[15,38416,38418],{"className":38417,"style":504},[350],[15,38419],{},[15,38421],{"className":38422,"style":128},[127],[15,38424,264],{"className":38425},[132],[15,38427],{"className":38428,"style":128},[127],[15,38430,38432,38435,38438,38441,38444],{"className":38431},[52],[15,38433],{"className":38434,"style":57},[56],[15,38436,38],{"className":38437,"style":63},[61,62],[15,38439],{"className":38440,"style":128},[127],[15,38442,20691],{"className":38443},[132],[15,38445],{"className":38446,"style":128},[127],[15,38448,38450,38454],{"className":38449},[52],[15,38451],{"className":38452,"style":38453},[56],"height:1.1846em;vertical-align:-0.3433em;",[15,38455,38457,38460],{"className":38456},[61],[15,38458,259],{"className":38459},[61,427],[15,38461,38463],{"className":38462},[338],[15,38464,38466,38503],{"className":38465},[342,472],[15,38467,38469,38500],{"className":38468},[346],[15,38470,38472,38489],{"className":38471,"style":695},[350],[15,38473,38474,38477],{"style":3320},[15,38475],{"className":38476,"style":358},[357],[15,38478,38480],{"className":38479},[362,363,364,365],[15,38481,38483,38486],{"className":38482},[61,365],[15,38484,275],{"className":38485},[132,365],[15,38487,278],{"className":38488},[61,365],[15,38490,38491,38494],{"style":716},[15,38492],{"className":38493,"style":358},[357],[15,38495,38497],{"className":38496},[362,363,364,365],[15,38498,102],{"className":38499,"style":174},[61,122,365],[15,38501,497],{"className":38502},[496],[15,38504,38506],{"className":38505},[346],[15,38507,38510],{"className":38508,"style":38509},[350],"height:0.3433em;",[15,38511],{}," 是由主干网络与",[661,38514,38515],{},"非负激活函数","复合而成的证据函数，输出 ",[15,38518,38520,38572],{"className":38519,"translate":19},[18],[15,38521,38523],{"className":38522},[23],[25,38524,38525],{"xmlns":27},[29,38526,38527,38569],{},[32,38528,38529,38531,38533,38539,38541,38543,38545,38547,38549,38555,38557,38559,38561,38567],{},[35,38530,1980],{"mathvariant":251},[81,38532,244],{},[266,38534,38535,38537],{},[35,38536,22777],{},[35,38538,37117],{"mathvariant":3101},[81,38540,741],{"stretchy":86},[35,38542,1027],{"mathvariant":251},[81,38544,750],{"stretchy":86},[81,38546,244],{},[81,38548,741],{"stretchy":86},[266,38550,38551,38553],{},[35,38552,1980],{},[89,38554,91],{},[81,38556,94],{"separator":48},[81,38558,97],{},[81,38560,94],{"separator":48},[266,38562,38563,38565],{},[35,38564,1980],{},[35,38566,102],{},[81,38568,750],{"stretchy":86},[40,38570,38571],{"encoding":42},"\\mathbf{e} = g_{\\boldsymbol{\\theta}}(\\mathbf{x}) = (e_1, \\dots, e_K)",[15,38573,38575,38593,38666],{"className":38574,"ariaHidden":48},[47],[15,38576,38578,38581,38584,38587,38590],{"className":38577},[52],[15,38579],{"className":38580,"style":2706},[56],[15,38582,1980],{"className":38583},[61,411],[15,38585],{"className":38586,"style":128},[127],[15,38588,244],{"className":38589},[132],[15,38591],{"className":38592,"style":128},[127],[15,38594,38596,38599,38648,38651,38654,38657,38660,38663],{"className":38595},[52],[15,38597],{"className":38598,"style":142},[56],[15,38600,38602,38605],{"className":38601},[61],[15,38603,22777],{"className":38604,"style":123},[61,122],[15,38606,38608],{"className":38607},[338],[15,38609,38611,38640],{"className":38610},[342,472],[15,38612,38614,38637],{"className":38613},[346],[15,38615,38617],{"className":38616,"style":479},[350],[15,38618,38619,38622],{"style":27013},[15,38620],{"className":38621,"style":358},[357],[15,38623,38625],{"className":38624},[362,363,364,365],[15,38626,38628],{"className":38627},[61,365],[15,38629,38631],{"className":38630},[61,365],[15,38632,38634],{"className":38633},[61,365],[15,38635,37117],{"className":38636,"style":37181},[61,3162,365],[15,38638,497],{"className":38639},[496],[15,38641,38643],{"className":38642},[346],[15,38644,38646],{"className":38645,"style":504},[350],[15,38647],{},[15,38649,741],{"className":38650},[146],[15,38652,1027],{"className":38653},[61,411],[15,38655,750],{"className":38656},[178],[15,38658],{"className":38659,"style":128},[127],[15,38661,244],{"className":38662},[132],[15,38664],{"className":38665,"style":128},[127],[15,38667,38669,38672,38675,38715,38718,38721,38724,38727,38730,38733,38773],{"className":38668},[52],[15,38670],{"className":38671,"style":142},[56],[15,38673,741],{"className":38674},[146],[15,38676,38678,38681],{"className":38677},[61],[15,38679,1980],{"className":38680},[61,122],[15,38682,38684],{"className":38683},[338],[15,38685,38687,38707],{"className":38686},[342,472],[15,38688,38690,38704],{"className":38689},[346],[15,38691,38693],{"className":38692,"style":1212},[350],[15,38694,38695,38698],{"style":482},[15,38696],{"className":38697,"style":358},[357],[15,38699,38701],{"className":38700},[362,363,364,365],[15,38702,91],{"className":38703},[61,365],[15,38705,497],{"className":38706},[496],[15,38708,38710],{"className":38709},[346],[15,38711,38713],{"className":38712,"style":504},[350],[15,38714],{},[15,38716,94],{"className":38717},[153],[15,38719],{"className":38720,"style":157},[127],[15,38722,97],{"className":38723},[161],[15,38725],{"className":38726,"style":157},[127],[15,38728,94],{"className":38729},[153],[15,38731],{"className":38732,"style":157},[127],[15,38734,38736,38739],{"className":38735},[61],[15,38737,1980],{"className":38738},[61,122],[15,38740,38742],{"className":38741},[338],[15,38743,38745,38765],{"className":38744},[342,472],[15,38746,38748,38762],{"className":38747},[346],[15,38749,38751],{"className":38750,"style":1271},[350],[15,38752,38753,38756],{"style":482},[15,38754],{"className":38755,"style":358},[357],[15,38757,38759],{"className":38758},[362,363,364,365],[15,38760,102],{"className":38761,"style":174},[61,122,365],[15,38763,497],{"className":38764},[496],[15,38766,38768],{"className":38767},[346],[15,38769,38771],{"className":38770,"style":504},[350],[15,38772],{},[15,38774,750],{"className":38775},[178],"。迪利克雷参数取为\n",[15,38778,38780,38828],{"className":38779,"translate":19},[18],[15,38781,38783],{"className":38782},[23],[25,38784,38785],{"xmlns":27},[29,38786,38787,38825],{},[32,38788,38789,38795,38797,38803,38805,38807,38809,38811,38813,38815,38817,38819,38821,38823],{},[266,38790,38791,38793],{},[35,38792,3102],{},[35,38794,272],{},[81,38796,244],{},[266,38798,38799,38801],{},[35,38800,1980],{},[35,38802,272],{},[81,38804,7756],{},[89,38806,91],{},[81,38808,94],{"separator":48},[127,38810],{"width":9838},[282,38812,12917],{},[127,38814],{"width":9838},[35,38816,3102],{"mathvariant":3101},[81,38818,244],{},[35,38820,1980],{"mathvariant":251},[81,38822,7756],{},[89,38824,91],{"mathvariant":251},[40,38826,38827],{"encoding":42},"\\alpha_k = e_k + 1, \\qquad \\text{即} \\qquad \\boldsymbol{\\alpha} = \\mathbf{e} + \\mathbf{1}",[15,38829,38831,38886,38941,38987,39005],{"className":38830,"ariaHidden":48},[47],[15,38832,38834,38837,38877,38880,38883],{"className":38833},[52],[15,38835],{"className":38836,"style":2509},[56],[15,38838,38840,38843],{"className":38839},[61],[15,38841,3102],{"className":38842,"style":3187},[61,122],[15,38844,38846],{"className":38845},[338],[15,38847,38849,38869],{"className":38848},[342,472],[15,38850,38852,38866],{"className":38851},[346],[15,38853,38855],{"className":38854,"style":479},[350],[15,38856,38857,38860],{"style":3202},[15,38858],{"className":38859,"style":358},[357],[15,38861,38863],{"className":38862},[362,363,364,365],[15,38864,272],{"className":38865,"style":492},[61,122,365],[15,38867,497],{"className":38868},[496],[15,38870,38872],{"className":38871},[346],[15,38873,38875],{"className":38874,"style":504},[350],[15,38876],{},[15,38878],{"className":38879,"style":128},[127],[15,38881,244],{"className":38882},[132],[15,38884],{"className":38885,"style":128},[127],[15,38887,38889,38892,38932,38935,38938],{"className":38888},[52],[15,38890],{"className":38891,"style":12608},[56],[15,38893,38895,38898],{"className":38894},[61],[15,38896,1980],{"className":38897},[61,122],[15,38899,38901],{"className":38900},[338],[15,38902,38904,38924],{"className":38903},[342,472],[15,38905,38907,38921],{"className":38906},[346],[15,38908,38910],{"className":38909,"style":479},[350],[15,38911,38912,38915],{"style":482},[15,38913],{"className":38914,"style":358},[357],[15,38916,38918],{"className":38917},[362,363,364,365],[15,38919,272],{"className":38920,"style":492},[61,122,365],[15,38922,497],{"className":38923},[496],[15,38925,38927],{"className":38926},[346],[15,38928,38930],{"className":38929,"style":504},[350],[15,38931],{},[15,38933],{"className":38934,"style":772},[127],[15,38936,7756],{"className":38937},[375],[15,38939],{"className":38940,"style":772},[127],[15,38942,38944,38948,38951,38954,38957,38960,38966,38969,38978,38981,38984],{"className":38943},[52],[15,38945],{"className":38946,"style":38947},[56],"height:0.8778em;vertical-align:-0.1944em;",[15,38949,91],{"className":38950},[61],[15,38952,94],{"className":38953},[153],[15,38955],{"className":38956,"style":10026},[127],[15,38958],{"className":38959,"style":157},[127],[15,38961,38963],{"className":38962},[61,13539],[15,38964,12917],{"className":38965},[61,13543],[15,38967],{"className":38968,"style":10026},[127],[15,38970,38972],{"className":38971},[61],[15,38973,38975],{"className":38974},[61],[15,38976,3102],{"className":38977},[61,3162],[15,38979],{"className":38980,"style":128},[127],[15,38982,244],{"className":38983},[132],[15,38985],{"className":38986,"style":128},[127],[15,38988,38990,38993,38996,38999,39002],{"className":38989},[52],[15,38991],{"className":38992,"style":6731},[56],[15,38994,1980],{"className":38995},[61,411],[15,38997],{"className":38998,"style":772},[127],[15,39000,7756],{"className":39001},[375],[15,39003],{"className":39004,"style":772},[127],[15,39006,39008,39011],{"className":39007},[52],[15,39009],{"className":39010,"style":859},[56],[15,39012,91],{"className":39013},[61,411],"\n网络对输入的预测由此被定义为 ",[15,39016,39018,39040],{"className":39017,"translate":19},[18],[15,39019,39021],{"className":39020},[23],[25,39022,39023],{"xmlns":27},[29,39024,39025,39037],{},[32,39026,39027,39029,39031,39033,39035],{},[35,39028,4182],{"mathvariant":231},[81,39030,1545],{},[81,39032,741],{"stretchy":86},[35,39034,3102],{"mathvariant":3101},[81,39036,750],{"stretchy":86},[40,39038,39039],{"encoding":42},"\\operatorname{Dir}(\\boldsymbol{\\alpha})",[15,39041,39043],{"className":39042,"ariaHidden":48},[47],[15,39044,39046,39049,39055,39058,39067],{"className":39045},[52],[15,39047],{"className":39048,"style":142},[56],[15,39050,39052],{"className":39051},[531],[15,39053,4182],{"className":39054},[61,1620],[15,39056,741],{"className":39057},[146],[15,39059,39061],{"className":39060},[61],[15,39062,39064],{"className":39063},[61],[15,39065,3102],{"className":39066},[61,3162],[15,39068,750],{"className":39069},[178],[11,39071,39072,39073,39152,39153,39254,39255,39325,39326,39354],{},"参数化的语义可直接由命题 3.2 读出：",[15,39074,39076,39098],{"className":39075,"translate":19},[18],[15,39077,39079],{"className":39078},[23],[25,39080,39081],{"xmlns":27},[29,39082,39083,39095],{},[32,39084,39085,39087,39089,39091,39093],{},[35,39086,1980],{"mathvariant":251},[81,39088,244],{},[35,39090,3102],{"mathvariant":3101},[81,39092,239],{},[89,39094,91],{"mathvariant":251},[40,39096,39097],{"encoding":42},"\\mathbf{e} = \\boldsymbol{\\alpha} - \\mathbf{1}",[15,39099,39101,39119,39143],{"className":39100,"ariaHidden":48},[47],[15,39102,39104,39107,39110,39113,39116],{"className":39103},[52],[15,39105],{"className":39106,"style":2706},[56],[15,39108,1980],{"className":39109},[61,411],[15,39111],{"className":39112,"style":128},[127],[15,39114,244],{"className":39115},[132],[15,39117],{"className":39118,"style":128},[127],[15,39120,39122,39125,39134,39137,39140],{"className":39121},[52],[15,39123],{"className":39124,"style":6731},[56],[15,39126,39128],{"className":39127},[61],[15,39129,39131],{"className":39130},[61],[15,39132,3102],{"className":39133},[61,3162],[15,39135],{"className":39136,"style":772},[127],[15,39138,239],{"className":39139},[375],[15,39141],{"className":39142,"style":772},[127],[15,39144,39146,39149],{"className":39145},[52],[15,39147],{"className":39148,"style":859},[56],[15,39150,91],{"className":39151},[61,411]," 正是",[661,39154,39155,39156,39253],{},"超出均匀先验 ",[15,39157,39159,39183],{"className":39158,"translate":19},[18],[15,39160,39162],{"className":39161},[23],[25,39163,39164],{"xmlns":27},[29,39165,39166,39180],{},[32,39167,39168,39174,39176,39178],{},[266,39169,39170,39172],{},[35,39171,33406],{},[35,39173,272],{},[35,39175,34952],{},[81,39177,244],{},[89,39179,91],{},[40,39181,39182],{"encoding":42},"a_k W = 1",[15,39184,39186,39244],{"className":39185,"ariaHidden":48},[47],[15,39187,39189,39192,39232,39235,39238,39241],{"className":39188},[52],[15,39190],{"className":39191,"style":34331},[56],[15,39193,39195,39198],{"className":39194},[61],[15,39196,33406],{"className":39197},[61,122],[15,39199,39201],{"className":39200},[338],[15,39202,39204,39224],{"className":39203},[342,472],[15,39205,39207,39221],{"className":39206},[346],[15,39208,39210],{"className":39209,"style":479},[350],[15,39211,39212,39215],{"style":482},[15,39213],{"className":39214,"style":358},[357],[15,39216,39218],{"className":39217},[362,363,364,365],[15,39219,272],{"className":39220,"style":492},[61,122,365],[15,39222,497],{"className":39223},[496],[15,39225,39227],{"className":39226},[346],[15,39228,39230],{"className":39229,"style":504},[350],[15,39231],{},[15,39233,34952],{"className":39234,"style":34338},[61,122],[15,39236],{"className":39237,"style":128},[127],[15,39239,244],{"className":39240},[132],[15,39242],{"className":39243,"style":128},[127],[15,39245,39247,39250],{"className":39246},[52],[15,39248],{"className":39249,"style":859},[56],[15,39251,91],{"className":39252},[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：",[15,39256,39258,39276],{"className":39257,"translate":19},[18],[15,39259,39261],{"className":39260},[23],[25,39262,39263],{"xmlns":27},[29,39264,39265,39273],{},[32,39266,39267],{},[266,39268,39269,39271],{},[35,39270,1980],{},[35,39272,272],{},[40,39274,39275],{"encoding":42},"e_k",[15,39277,39279],{"className":39278,"ariaHidden":48},[47],[15,39280,39282,39285],{"className":39281},[52],[15,39283],{"className":39284,"style":2509},[56],[15,39286,39288,39291],{"className":39287},[61],[15,39289,1980],{"className":39290},[61,122],[15,39292,39294],{"className":39293},[338],[15,39295,39297,39317],{"className":39296},[342,472],[15,39298,39300,39314],{"className":39299},[346],[15,39301,39303],{"className":39302,"style":479},[350],[15,39304,39305,39308],{"style":482},[15,39306],{"className":39307,"style":358},[357],[15,39309,39311],{"className":39310},[362,363,364,365],[15,39312,272],{"className":39313,"style":492},[61,122,365],[15,39315,497],{"className":39316},[496],[15,39318,39320],{"className":39319},[346],[15,39321,39323],{"className":39322,"style":504},[350],[15,39324],{}," 是与类别 ",[15,39327,39329,39342],{"className":39328,"translate":19},[18],[15,39330,39332],{"className":39331},[23],[25,39333,39334],{"xmlns":27},[29,39335,39336,39340],{},[32,39337,39338],{},[35,39339,272],{},[40,39341,272],{"encoding":42},[15,39343,39345],{"className":39344,"ariaHidden":48},[47],[15,39346,39348,39351],{"className":39347},[52],[15,39349],{"className":39350,"style":10443},[56],[15,39352,272],{"className":39353,"style":492},[61,122]," 相关的、由网络从输入中提取出的支持性观测。",[11,39356,39357],{},"非负激活函数的选择影响显著，值得逐一说明：",[33468,39359,39360,39680,39800,40102],{},[31523,39361,39362,39505,39506,39562,39563,39622,39623,39679],{},[15,39363,39365,39408],{"className":39364,"translate":19},[18],[15,39366,39368],{"className":39367},[23],[25,39369,39370],{"xmlns":27},[29,39371,39372,39405],{},[32,39373,39374,39377,39379,39381,39383,39385,39387,39389,39391,39393,39395,39397,39403],{},[35,39375,39376],{"mathvariant":231},"softplus",[81,39378,1545],{},[81,39380,741],{"stretchy":86},[35,39382,37270],{},[81,39384,750],{"stretchy":86},[81,39386,244],{},[35,39388,22645],{},[81,39390,1545],{},[81,39392,741],{"stretchy":86},[89,39394,91],{},[81,39396,7756],{},[227,39398,39399,39401],{},[35,39400,1980],{},[35,39402,37270],{},[81,39404,750],{"stretchy":86},[40,39406,39407],{"encoding":42},"\\operatorname{softplus}(z) = \\log(1 + e^z)",[15,39409,39411,39441,39467],{"className":39410,"ariaHidden":48},[47],[15,39412,39414,39417,39423,39426,39429,39432,39435,39438],{"className":39413},[52],[15,39415],{"className":39416,"style":142},[56],[15,39418,39420],{"className":39419},[531],[15,39421,39376],{"className":39422},[61,1620],[15,39424,741],{"className":39425},[146],[15,39427,37270],{"className":39428,"style":37598},[61,122],[15,39430,750],{"className":39431},[178],[15,39433],{"className":39434,"style":128},[127],[15,39436,244],{"className":39437},[132],[15,39439],{"className":39440,"style":128},[127],[15,39442,39444,39447,39452,39455,39458,39461,39464],{"className":39443},[52],[15,39445],{"className":39446,"style":142},[56],[15,39448,22774,39450],{"className":39449},[531],[15,39451,22777],{"style":15608},[15,39453,741],{"className":39454},[146],[15,39456,91],{"className":39457},[61],[15,39459],{"className":39460,"style":772},[127],[15,39462,7756],{"className":39463},[375],[15,39465],{"className":39466,"style":772},[127],[15,39468,39470,39473,39502],{"className":39469},[52],[15,39471],{"className":39472,"style":142},[56],[15,39474,39476,39479],{"className":39475},[61],[15,39477,1980],{"className":39478},[61,122],[15,39480,39482],{"className":39481},[338],[15,39483,39485],{"className":39484},[342],[15,39486,39488],{"className":39487},[346],[15,39489,39491],{"className":39490,"style":37580},[350],[15,39492,39493,39496],{"style":716},[15,39494],{"className":39495,"style":358},[357],[15,39497,39499],{"className":39498},[362,363,364,365],[15,39500,37270],{"className":39501,"style":37598},[61,122,365],[15,39503,750],{"className":39504},[178],"：恒正、处处可微、在 ",[15,39507,39509,39529],{"className":39508,"translate":19},[18],[15,39510,39512],{"className":39511},[23],[25,39513,39514],{"xmlns":27},[29,39515,39516,39526],{},[32,39517,39518,39520,39522,39524],{},[35,39519,37270],{},[81,39521,20691],{},[81,39523,239],{},[35,39525,6790],{"mathvariant":231},[40,39527,39528],{"encoding":42},"z \\to -\\infty",[15,39530,39532,39550],{"className":39531,"ariaHidden":48},[47],[15,39533,39535,39538,39541,39544,39547],{"className":39534},[52],[15,39536],{"className":39537,"style":6026},[56],[15,39539,37270],{"className":39540,"style":37598},[61,122],[15,39542],{"className":39543,"style":128},[127],[15,39545,20691],{"className":39546},[132],[15,39548],{"className":39549,"style":128},[127],[15,39551,39553,39556,39559],{"className":39552},[52],[15,39554],{"className":39555,"style":6731},[56],[15,39557,239],{"className":39558},[61],[15,39560,6790],{"className":39561},[61]," 时以 ",[15,39564,39566,39584],{"className":39565,"translate":19},[18],[15,39567,39569],{"className":39568},[23],[25,39570,39571],{"xmlns":27},[29,39572,39573,39581],{},[32,39574,39575],{},[227,39576,39577,39579],{},[35,39578,1980],{},[35,39580,37270],{},[40,39582,39583],{"encoding":42},"e^z",[15,39585,39587],{"className":39586,"ariaHidden":48},[47],[15,39588,39590,39593],{"className":39589},[52],[15,39591],{"className":39592,"style":37580},[56],[15,39594,39596,39599],{"className":39595},[61],[15,39597,1980],{"className":39598},[61,122],[15,39600,39602],{"className":39601},[338],[15,39603,39605],{"className":39604},[342],[15,39606,39608],{"className":39607},[346],[15,39609,39611],{"className":39610,"style":37580},[350],[15,39612,39613,39616],{"style":716},[15,39614],{"className":39615,"style":358},[357],[15,39617,39619],{"className":39618},[362,363,364,365],[15,39620,37270],{"className":39621,"style":37598},[61,122,365]," 的速度趋零、在 ",[15,39624,39626,39646],{"className":39625,"translate":19},[18],[15,39627,39629],{"className":39628},[23],[25,39630,39631],{"xmlns":27},[29,39632,39633,39643],{},[32,39634,39635,39637,39639,39641],{},[35,39636,37270],{},[81,39638,20691],{},[81,39640,7756],{},[35,39642,6790],{"mathvariant":231},[40,39644,39645],{"encoding":42},"z \\to +\\infty",[15,39647,39649,39667],{"className":39648,"ariaHidden":48},[47],[15,39650,39652,39655,39658,39661,39664],{"className":39651},[52],[15,39653],{"className":39654,"style":6026},[56],[15,39656,37270],{"className":39657,"style":37598},[61,122],[15,39659],{"className":39660,"style":128},[127],[15,39662,20691],{"className":39663},[132],[15,39665],{"className":39666,"style":128},[127],[15,39668,39670,39673,39676],{"className":39669},[52],[15,39671],{"className":39672,"style":6731},[56],[15,39674,7756],{"className":39675},[61],[15,39677,6790],{"className":39678},[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。",[31523,39681,39682,39729,39730,39799],{},[15,39683,39685,39708],{"className":39684,"translate":19},[18],[15,39686,39688],{"className":39687},[23],[25,39689,39690],{"xmlns":27},[29,39691,39692,39705],{},[32,39693,39694,39697,39699,39701,39703],{},[35,39695,39696],{},"exp",[81,39698,1545],{},[81,39700,741],{"stretchy":86},[35,39702,37270],{},[81,39704,750],{"stretchy":86},[40,39706,39707],{"encoding":42},"\\exp(z)",[15,39709,39711],{"className":39710,"ariaHidden":48},[47],[15,39712,39714,39717,39720,39723,39726],{"className":39713},[52],[15,39715],{"className":39716,"style":142},[56],[15,39718,39696],{"className":39719},[531],[15,39721,741],{"className":39722},[146],[15,39724,37270],{"className":39725,"style":37598},[61,122],[15,39727,750],{"className":39728},[178],"：严格正、光滑，但无上界，使 ",[15,39731,39733,39750],{"className":39732,"translate":19},[18],[15,39734,39736],{"className":39735},[23],[25,39737,39738],{"xmlns":27},[29,39739,39740,39748],{},[32,39741,39742],{},[266,39743,39744,39746],{},[35,39745,3102],{},[89,39747,278],{},[40,39749,19479],{"encoding":42},[15,39751,39753],{"className":39752,"ariaHidden":48},[47],[15,39754,39756,39759],{"className":39755},[52],[15,39757],{"className":39758,"style":2509},[56],[15,39760,39762,39765],{"className":39761},[61],[15,39763,3102],{"className":39764,"style":3187},[61,122],[15,39766,39768],{"className":39767},[338],[15,39769,39771,39791],{"className":39770},[342,472],[15,39772,39774,39788],{"className":39773},[346],[15,39775,39777],{"className":39776,"style":1212},[350],[15,39778,39779,39782],{"style":3202},[15,39780],{"className":39781,"style":358},[357],[15,39783,39785],{"className":39784},[362,363,364,365],[15,39786,278],{"className":39787},[61,365],[15,39789,497],{"className":39790},[496],[15,39792,39794],{"className":39793},[346],[15,39795,39797],{"className":39796,"style":504},[350],[15,39798],{}," 可任意大，易造成数值溢出与损失对证据的不稳定响应。",[31523,39801,39802,39852,39853,39904,39905,39997,39998,40001,40002,40072,40073,40101],{},[15,39803,39805,39828],{"className":39804,"translate":19},[18],[15,39806,39808],{"className":39807},[23],[25,39809,39810],{"xmlns":27},[29,39811,39812,39825],{},[32,39813,39814,39817,39819,39821,39823],{},[35,39815,39816],{"mathvariant":231},"ReLU",[81,39818,1545],{},[81,39820,741],{"stretchy":86},[35,39822,37270],{},[81,39824,750],{"stretchy":86},[40,39826,39827],{"encoding":42},"\\operatorname{ReLU}(z)",[15,39829,39831],{"className":39830,"ariaHidden":48},[47],[15,39832,39834,39837,39843,39846,39849],{"className":39833},[52],[15,39835],{"className":39836,"style":142},[56],[15,39838,39840],{"className":39839},[531],[15,39841,39816],{"className":39842},[61,1620],[15,39844,741],{"className":39845},[146],[15,39847,37270],{"className":39848,"style":37598},[61,122],[15,39850,750],{"className":39851},[178],"：计算廉价，但在 ",[15,39854,39856,39874],{"className":39855,"translate":19},[18],[15,39857,39859],{"className":39858},[23],[25,39860,39861],{"xmlns":27},[29,39862,39863,39871],{},[32,39864,39865,39867,39869],{},[35,39866,37270],{},[81,39868,2488],{},[89,39870,278],{},[40,39872,39873],{"encoding":42},"z \u003C 0",[15,39875,39877,39895],{"className":39876,"ariaHidden":48},[47],[15,39878,39880,39883,39886,39889,39892],{"className":39879},[52],[15,39881],{"className":39882,"style":4650},[56],[15,39884,37270],{"className":39885,"style":37598},[61,122],[15,39887],{"className":39888,"style":128},[127],[15,39890,2488],{"className":39891},[132],[15,39893],{"className":39894,"style":128},[127],[15,39896,39898,39901],{"className":39897},[52],[15,39899],{"className":39900,"style":859},[56],[15,39902,278],{"className":39903},[61]," 处梯度恒为零。处于该区域的单元输出证据 ",[15,39906,39908,39930],{"className":39907,"translate":19},[18],[15,39909,39911],{"className":39910},[23],[25,39912,39913],{"xmlns":27},[29,39914,39915,39927],{},[32,39916,39917,39923,39925],{},[266,39918,39919,39921],{},[35,39920,1980],{},[35,39922,272],{},[81,39924,244],{},[89,39926,278],{},[40,39928,39929],{"encoding":42},"e_k = 0",[15,39931,39933,39988],{"className":39932,"ariaHidden":48},[47],[15,39934,39936,39939,39979,39982,39985],{"className":39935},[52],[15,39937],{"className":39938,"style":2509},[56],[15,39940,39942,39945],{"className":39941},[61],[15,39943,1980],{"className":39944},[61,122],[15,39946,39948],{"className":39947},[338],[15,39949,39951,39971],{"className":39950},[342,472],[15,39952,39954,39968],{"className":39953},[346],[15,39955,39957],{"className":39956,"style":479},[350],[15,39958,39959,39962],{"style":482},[15,39960],{"className":39961,"style":358},[357],[15,39963,39965],{"className":39964},[362,363,364,365],[15,39966,272],{"className":39967,"style":492},[61,122,365],[15,39969,497],{"className":39970},[496],[15,39972,39974],{"className":39973},[346],[15,39975,39977],{"className":39976,"style":504},[350],[15,39978],{},[15,39980],{"className":39981,"style":128},[127],[15,39983,244],{"className":39984},[132],[15,39986],{"className":39987,"style":128},[127],[15,39989,39991,39994],{"className":39990},[52],[15,39992],{"className":39993,"style":859},[56],[15,39995,278],{"className":39996},[61]," 且",[661,39999,40000],{},"永远无法通过梯度逃离","——",[15,40003,40005,40023],{"className":40004,"translate":19},[18],[15,40006,40008],{"className":40007},[23],[25,40009,40010],{"xmlns":27},[29,40011,40012,40020],{},[32,40013,40014],{},[266,40015,40016,40018],{},[35,40017,3102],{},[35,40019,272],{},[40,40021,40022],{"encoding":42},"\\alpha_k",[15,40024,40026],{"className":40025,"ariaHidden":48},[47],[15,40027,40029,40032],{"className":40028},[52],[15,40030],{"className":40031,"style":2509},[56],[15,40033,40035,40038],{"className":40034},[61],[15,40036,3102],{"className":40037,"style":3187},[61,122],[15,40039,40041],{"className":40040},[338],[15,40042,40044,40064],{"className":40043},[342,472],[15,40045,40047,40061],{"className":40046},[346],[15,40048,40050],{"className":40049,"style":479},[350],[15,40051,40052,40055],{"style":3202},[15,40053],{"className":40054,"style":358},[357],[15,40056,40058],{"className":40057},[362,363,364,365],[15,40059,272],{"className":40060,"style":492},[61,122,365],[15,40062,497],{"className":40063},[496],[15,40065,40067],{"className":40066},[346],[15,40068,40070],{"className":40069,"style":504},[350],[15,40071],{}," 被锁在 ",[15,40074,40076,40089],{"className":40075,"translate":19},[18],[15,40077,40079],{"className":40078},[23],[25,40080,40081],{"xmlns":27},[29,40082,40083,40087],{},[32,40084,40085],{},[89,40086,91],{},[40,40088,91],{"encoding":42},[15,40090,40092],{"className":40091,"ariaHidden":48},[47],[15,40093,40095,40098],{"className":40094},[52],[15,40096],{"className":40097,"style":859},[56],[15,40099,91],{"className":40100},[61],"，形成死区。这使 ReLU 在证据函数中通常劣于 softplus。",[31523,40103,40104,40176,40177,40205],{},[15,40105,40107,40134],{"className":40106,"translate":19},[18],[15,40108,40110],{"className":40109},[23],[25,40111,40112],{"xmlns":27},[29,40113,40114,40131],{},[32,40115,40116,40119,40121,40123,40125,40127,40129],{},[35,40117,40118],{"mathvariant":231},"ELU",[81,40120,1545],{},[81,40122,741],{"stretchy":86},[35,40124,37270],{},[81,40126,750],{"stretchy":86},[81,40128,7756],{},[89,40130,91],{},[40,40132,40133],{"encoding":42},"\\operatorname{ELU}(z) + 1",[15,40135,40137,40167],{"className":40136,"ariaHidden":48},[47],[15,40138,40140,40143,40149,40152,40155,40158,40161,40164],{"className":40139},[52],[15,40141],{"className":40142,"style":142},[56],[15,40144,40146],{"className":40145},[531],[15,40147,40118],{"className":40148},[61,1620],[15,40150,741],{"className":40151},[146],[15,40153,37270],{"className":40154,"style":37598},[61,122],[15,40156,750],{"className":40157},[178],[15,40159],{"className":40160,"style":772},[127],[15,40162,7756],{"className":40163},[375],[15,40165],{"className":40166,"style":772},[127],[15,40168,40170,40173],{"className":40169},[52],[15,40171],{"className":40172,"style":859},[56],[15,40174,91],{"className":40175},[61],"：具有负饱和下界 ",[15,40178,40180,40193],{"className":40179,"translate":19},[18],[15,40181,40183],{"className":40182},[23],[25,40184,40185],{"xmlns":27},[29,40186,40187,40191],{},[32,40188,40189],{},[89,40190,278],{},[40,40192,278],{"encoding":42},[15,40194,40196],{"className":40195,"ariaHidden":48},[47],[15,40197,40199,40202],{"className":40198},[52],[15,40200],{"className":40201,"style":859},[56],[15,40203,278],{"className":40204},[61],"，梯度在负半轴非零，是介于二者之间的折中。",[877,40207,40209],{"id":40208},"_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",[882,40211,40213],{"title":40212},"命题 3.4（EDL 的预测与不确定性）",[11,40214,40215,40216,40612,40613],{},"在定义 3.3 的参数化下，记总强度 ",[15,40217,40219,40290],{"className":40218,"translate":19},[18],[15,40220,40222],{"className":40221},[23],[25,40223,40224],{"xmlns":27},[29,40225,40226,40287],{},[32,40227,40228,40231,40233,40239,40241,40255,40261,40263,40265,40267,40281],{},[35,40229,40230],{},"S",[81,40232,244],{},[266,40234,40235,40237],{},[35,40236,3102],{},[89,40238,278],{},[81,40240,244],{},[289,40242,40243,40245,40253],{},[81,40244,293],{},[32,40246,40247,40249,40251],{},[35,40248,272],{},[81,40250,244],{},[89,40252,91],{},[35,40254,102],{},[266,40256,40257,40259],{},[35,40258,3102],{},[35,40260,272],{},[81,40262,244],{},[35,40264,102],{},[81,40266,7756],{},[289,40268,40269,40271,40279],{},[81,40270,293],{},[32,40272,40273,40275,40277],{},[35,40274,272],{},[81,40276,244],{},[89,40278,91],{},[35,40280,102],{},[266,40282,40283,40285],{},[35,40284,1980],{},[35,40286,272],{},[40,40288,40289],{"encoding":42},"S = \\alpha_0 = \\sum_{k=1}^K \\alpha_k = K + \\sum_{k=1}^K e_k",[15,40291,40293,40312,40367,40485,40503],{"className":40292,"ariaHidden":48},[47],[15,40294,40296,40299,40303,40306,40309],{"className":40295},[52],[15,40297],{"className":40298,"style":57},[56],[15,40300,40230],{"className":40301,"style":40302},[61,122],"margin-right:0.0576em;",[15,40304],{"className":40305,"style":128},[127],[15,40307,244],{"className":40308},[132],[15,40310],{"className":40311,"style":128},[127],[15,40313,40315,40318,40358,40361,40364],{"className":40314},[52],[15,40316],{"className":40317,"style":2509},[56],[15,40319,40321,40324],{"className":40320},[61],[15,40322,3102],{"className":40323,"style":3187},[61,122],[15,40325,40327],{"className":40326},[338],[15,40328,40330,40350],{"className":40329},[342,472],[15,40331,40333,40347],{"className":40332},[346],[15,40334,40336],{"className":40335,"style":1212},[350],[15,40337,40338,40341],{"style":3202},[15,40339],{"className":40340,"style":358},[357],[15,40342,40344],{"className":40343},[362,363,364,365],[15,40345,278],{"className":40346},[61,365],[15,40348,497],{"className":40349},[496],[15,40351,40353],{"className":40352},[346],[15,40354,40356],{"className":40355,"style":504},[350],[15,40357],{},[15,40359],{"className":40360,"style":128},[127],[15,40362,244],{"className":40363},[132],[15,40365],{"className":40366,"style":128},[127],[15,40368,40370,40373,40433,40436,40476,40479,40482],{"className":40369},[52],[15,40371],{"className":40372,"style":3467},[56],[15,40374,40376,40379],{"className":40375},[531],[15,40377,293],{"className":40378,"style":537},[531,535,536],[15,40380,40382],{"className":40381},[338],[15,40383,40385,40425],{"className":40384},[342,472],[15,40386,40388,40422],{"className":40387},[346],[15,40389,40391,40411],{"className":40390,"style":550},[350],[15,40392,40393,40396],{"style":553},[15,40394],{"className":40395,"style":358},[357],[15,40397,40399],{"className":40398},[362,363,364,365],[15,40400,40402,40405,40408],{"className":40401},[61,365],[15,40403,272],{"className":40404,"style":492},[61,122,365],[15,40406,244],{"className":40407},[132,365],[15,40409,91],{"className":40410},[61,365],[15,40412,40413,40416],{"style":574},[15,40414],{"className":40415,"style":358},[357],[15,40417,40419],{"className":40418},[362,363,364,365],[15,40420,102],{"className":40421,"style":174},[61,122,365],[15,40423,497],{"className":40424},[496],[15,40426,40428],{"className":40427},[346],[15,40429,40431],{"className":40430,"style":593},[350],[15,40432],{},[15,40434],{"className":40435,"style":157},[127],[15,40437,40439,40442],{"className":40438},[61],[15,40440,3102],{"className":40441,"style":3187},[61,122],[15,40443,40445],{"className":40444},[338],[15,40446,40448,40468],{"className":40447},[342,472],[15,40449,40451,40465],{"className":40450},[346],[15,40452,40454],{"className":40453,"style":479},[350],[15,40455,40456,40459],{"style":3202},[15,40457],{"className":40458,"style":358},[357],[15,40460,40462],{"className":40461},[362,363,364,365],[15,40463,272],{"className":40464,"style":492},[61,122,365],[15,40466,497],{"className":40467},[496],[15,40469,40471],{"className":40470},[346],[15,40472,40474],{"className":40473,"style":504},[350],[15,40475],{},[15,40477],{"className":40478,"style":128},[127],[15,40480,244],{"className":40481},[132],[15,40483],{"className":40484,"style":128},[127],[15,40486,40488,40491,40494,40497,40500],{"className":40487},[52],[15,40489],{"className":40490,"style":972},[56],[15,40492,102],{"className":40493,"style":174},[61,122],[15,40495],{"className":40496,"style":772},[127],[15,40498,7756],{"className":40499},[375],[15,40501],{"className":40502,"style":772},[127],[15,40504,40506,40509,40569,40572],{"className":40505},[52],[15,40507],{"className":40508,"style":3467},[56],[15,40510,40512,40515],{"className":40511},[531],[15,40513,293],{"className":40514,"style":537},[531,535,536],[15,40516,40518],{"className":40517},[338],[15,40519,40521,40561],{"className":40520},[342,472],[15,40522,40524,40558],{"className":40523},[346],[15,40525,40527,40547],{"className":40526,"style":550},[350],[15,40528,40529,40532],{"style":553},[15,40530],{"className":40531,"style":358},[357],[15,40533,40535],{"className":40534},[362,363,364,365],[15,40536,40538,40541,40544],{"className":40537},[61,365],[15,40539,272],{"className":40540,"style":492},[61,122,365],[15,40542,244],{"className":40543},[132,365],[15,40545,91],{"className":40546},[61,365],[15,40548,40549,40552],{"style":574},[15,40550],{"className":40551,"style":358},[357],[15,40553,40555],{"className":40554},[362,363,364,365],[15,40556,102],{"className":40557,"style":174},[61,122,365],[15,40559,497],{"className":40560},[496],[15,40562,40564],{"className":40563},[346],[15,40565,40567],{"className":40566,"style":593},[350],[15,40568],{},[15,40570],{"className":40571,"style":157},[127],[15,40573,40575,40578],{"className":40574},[61],[15,40576,1980],{"className":40577},[61,122],[15,40579,40581],{"className":40580},[338],[15,40582,40584,40604],{"className":40583},[342,472],[15,40585,40587,40601],{"className":40586},[346],[15,40588,40590],{"className":40589,"style":479},[350],[15,40591,40592,40595],{"style":482},[15,40593],{"className":40594,"style":358},[357],[15,40596,40598],{"className":40597},[362,363,364,365],[15,40599,272],{"className":40600,"style":492},[61,122,365],[15,40602,497],{"className":40603},[496],[15,40605,40607],{"className":40606},[346],[15,40608,40610],{"className":40609,"style":504},[350],[15,40611],{},"。则\n",[15,40614,40616,40674],{"className":40615,"translate":19},[18],[15,40617,40619],{"className":40618},[23],[25,40620,40621],{"xmlns":27},[29,40622,40623,40671],{},[32,40624,40625,40631,40633,40635,40637,40643,40645,40647,40657,40659,40661,40663,40665],{},[266,40626,40627,40629],{},[35,40628,11],{},[35,40630,272],{},[81,40632,244],{},[35,40634,14807],{"mathvariant":258},[81,40636,14810],{"stretchy":86},[266,40638,40639,40641],{},[35,40640,1027],{},[35,40642,272],{},[81,40644,14819],{"stretchy":86},[81,40646,244],{},[3718,40648,40649,40655],{},[266,40650,40651,40653],{},[35,40652,3102],{},[35,40654,272],{},[35,40656,40230],{},[81,40658,94],{"separator":48},[127,40660],{"width":9838},[35,40662,33401],{},[81,40664,244],{},[3718,40666,40667,40669],{},[35,40668,102],{},[35,40670,40230],{},[40,40672,40673],{"encoding":42},"p_k = \\mathbb{E}[x_k] = \\frac{\\alpha_k}{S}, \\qquad u = \\frac{K}{S}",[15,40675,40677,40732,40796,40928],{"className":40676,"ariaHidden":48},[47],[15,40678,40680,40683,40723,40726,40729],{"className":40679},[52],[15,40681],{"className":40682,"style":9428},[56],[15,40684,40686,40689],{"className":40685},[61],[15,40687,11],{"className":40688},[61,122],[15,40690,40692],{"className":40691},[338],[15,40693,40695,40715],{"className":40694},[342,472],[15,40696,40698,40712],{"className":40697},[346],[15,40699,40701],{"className":40700,"style":479},[350],[15,40702,40703,40706],{"style":482},[15,40704],{"className":40705,"style":358},[357],[15,40707,40709],{"className":40708},[362,363,364,365],[15,40710,272],{"className":40711,"style":492},[61,122,365],[15,40713,497],{"className":40714},[496],[15,40716,40718],{"className":40717},[346],[15,40719,40721],{"className":40720,"style":504},[350],[15,40722],{},[15,40724],{"className":40725,"style":128},[127],[15,40727,244],{"className":40728},[132],[15,40730],{"className":40731,"style":128},[127],[15,40733,40735,40738,40741,40744,40784,40787,40790,40793],{"className":40734},[52],[15,40736],{"className":40737,"style":142},[56],[15,40739,14807],{"className":40740},[61,427],[15,40742,14810],{"className":40743},[146],[15,40745,40747,40750],{"className":40746},[61],[15,40748,1027],{"className":40749},[61,122],[15,40751,40753],{"className":40752},[338],[15,40754,40756,40776],{"className":40755},[342,472],[15,40757,40759,40773],{"className":40758},[346],[15,40760,40762],{"className":40761,"style":479},[350],[15,40763,40764,40767],{"style":482},[15,40765],{"className":40766,"style":358},[357],[15,40768,40770],{"className":40769},[362,363,364,365],[15,40771,272],{"className":40772,"style":492},[61,122,365],[15,40774,497],{"className":40775},[496],[15,40777,40779],{"className":40778},[346],[15,40780,40782],{"className":40781,"style":504},[350],[15,40783],{},[15,40785,14819],{"className":40786},[178],[15,40788],{"className":40789,"style":128},[127],[15,40791,244],{"className":40792},[132],[15,40794],{"className":40795,"style":128},[127],[15,40797,40799,40802,40907,40910,40913,40916,40919,40922,40925],{"className":40798},[52],[15,40800],{"className":40801,"style":28306},[56],[15,40803,40805,40808,40904],{"className":40804},[61],[15,40806],{"className":40807},[146,3837],[15,40809,40811],{"className":40810},[3718],[15,40812,40814,40896],{"className":40813},[342,472],[15,40815,40817,40893],{"className":40816},[346],[15,40818,40820,40834,40842],{"className":40819,"style":27928},[350],[15,40821,40822,40825],{"style":3853},[15,40823],{"className":40824,"style":3857},[357],[15,40826,40828],{"className":40827},[362,363,364,365],[15,40829,40831],{"className":40830},[61,365],[15,40832,40230],{"className":40833,"style":40302},[61,122,365],[15,40835,40836,40839],{"style":3885},[15,40837],{"className":40838,"style":3857},[357],[15,40840],{"className":40841,"style":3893},[3892],[15,40843,40844,40847],{"style":27181},[15,40845],{"className":40846,"style":3857},[357],[15,40848,40850],{"className":40849},[362,363,364,365],[15,40851,40853],{"className":40852},[61,365],[15,40854,40856,40859],{"className":40855},[61,365],[15,40857,3102],{"className":40858,"style":3187},[61,122,365],[15,40860,40862],{"className":40861},[338],[15,40863,40865,40885],{"className":40864},[342,472],[15,40866,40868,40882],{"className":40867},[346],[15,40869,40871],{"className":40870,"style":22389},[350],[15,40872,40873,40876],{"style":27211},[15,40874],{"className":40875,"style":4058},[357],[15,40877,40879],{"className":40878},[362,4062,4063,365],[15,40880,272],{"className":40881,"style":492},[61,122,365],[15,40883,497],{"className":40884},[496],[15,40886,40888],{"className":40887},[346],[15,40889,40891],{"className":40890,"style":27230},[350],[15,40892],{},[15,40894,497],{"className":40895},[496],[15,40897,40899],{"className":40898},[346],[15,40900,40902],{"className":40901,"style":10189},[350],[15,40903],{},[15,40905],{"className":40906},[178,3837],[15,40908,94],{"className":40909},[153],[15,40911],{"className":40912,"style":10026},[127],[15,40914],{"className":40915,"style":157},[127],[15,40917,33401],{"className":40918},[61,122],[15,40920],{"className":40921,"style":128},[127],[15,40923,244],{"className":40924},[132],[15,40926],{"className":40927,"style":128},[127],[15,40929,40931,40934],{"className":40930},[52],[15,40932],{"className":40933,"style":35744},[56],[15,40935,40937,40940,40999],{"className":40936},[61],[15,40938],{"className":40939},[146,3837],[15,40941,40943],{"className":40942},[3718],[15,40944,40946,40991],{"className":40945},[342,472],[15,40947,40949,40988],{"className":40948},[346],[15,40950,40952,40966,40974],{"className":40951,"style":35385},[350],[15,40953,40954,40957],{"style":3853},[15,40955],{"className":40956,"style":3857},[357],[15,40958,40960],{"className":40959},[362,363,364,365],[15,40961,40963],{"className":40962},[61,365],[15,40964,40230],{"className":40965,"style":40302},[61,122,365],[15,40967,40968,40971],{"style":3885},[15,40969],{"className":40970,"style":3857},[357],[15,40972],{"className":40973,"style":3893},[3892],[15,40975,40976,40979],{"style":3896},[15,40977],{"className":40978,"style":3857},[357],[15,40980,40982],{"className":40981},[362,363,364,365],[15,40983,40985],{"className":40984},[61,365],[15,40986,102],{"className":40987,"style":174},[61,122,365],[15,40989,497],{"className":40990},[496],[15,40992,40994],{"className":40993},[346],[15,40995,40997],{"className":40996,"style":10189},[350],[15,40998],{},[15,41000],{"className":41001},[178,3837],[11,41003,41004,41006,41007,41035,41036,41105,41106,867],{},[661,41005,5117],{}," 第一式即命题 2.1。第二式即命题 3.2。对 ",[15,41008,41010,41023],{"className":41009,"translate":19},[18],[15,41011,41013],{"className":41012},[23],[25,41014,41015],{"xmlns":27},[29,41016,41017,41021],{},[32,41018,41019],{},[35,41020,33401],{},[40,41022,33401],{"encoding":42},[15,41024,41026],{"className":41025,"ariaHidden":48},[47],[15,41027,41029,41032],{"className":41028},[52],[15,41030],{"className":41031,"style":6026},[56],[15,41033,33401],{"className":41034},[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 ",[15,41037,41039,41056],{"className":41038,"translate":19},[18],[15,41040,41042],{"className":41041},[23],[25,41043,41044],{"xmlns":27},[29,41045,41046,41054],{},[32,41047,41048],{},[266,41049,41050,41052],{},[35,41051,3102],{},[89,41053,278],{},[40,41055,19479],{"encoding":42},[15,41057,41059],{"className":41058,"ariaHidden":48},[47],[15,41060,41062,41065],{"className":41061},[52],[15,41063],{"className":41064,"style":2509},[56],[15,41066,41068,41071],{"className":41067},[61],[15,41069,3102],{"className":41070,"style":3187},[61,122],[15,41072,41074],{"className":41073},[338],[15,41075,41077,41097],{"className":41076},[342,472],[15,41078,41080,41094],{"className":41079},[346],[15,41081,41083],{"className":41082,"style":1212},[350],[15,41084,41085,41088],{"style":3202},[15,41086],{"className":41087,"style":358},[357],[15,41089,41091],{"className":41090},[362,363,364,365],[15,41092,278],{"className":41093},[61,365],[15,41095,497],{"className":41096},[496],[15,41098,41100],{"className":41099},[346],[15,41101,41103],{"className":41102,"style":504},[350],[15,41104],{},"，不确定性质量即为 ",[15,41107,41109,41131],{"className":41108,"translate":19},[18],[15,41110,41112],{"className":41111},[23],[25,41113,41114],{"xmlns":27},[29,41115,41116,41128],{},[32,41117,41118,41120,41122],{},[35,41119,102],{},[35,41121,3018],{"mathvariant":231},[266,41123,41124,41126],{},[35,41125,3102],{},[89,41127,278],{},[40,41129,41130],{"encoding":42},"K\u002F\\alpha_0",[15,41132,41134],{"className":41133,"ariaHidden":48},[47],[15,41135,41137,41140,41143,41146],{"className":41136},[52],[15,41138],{"className":41139,"style":142},[56],[15,41141,102],{"className":41142,"style":174},[61,122],[15,41144,3018],{"className":41145},[61],[15,41147,41149,41152],{"className":41148},[61],[15,41150,3102],{"className":41151,"style":3187},[61,122],[15,41153,41155],{"className":41154},[338],[15,41156,41158,41178],{"className":41157},[342,472],[15,41159,41161,41175],{"className":41160},[346],[15,41162,41164],{"className":41163,"style":1212},[350],[15,41165,41166,41169],{"style":3202},[15,41167],{"className":41168,"style":358},[357],[15,41170,41172],{"className":41171},[362,363,364,365],[15,41173,278],{"className":41174},[61,365],[15,41176,497],{"className":41177},[496],[15,41179,41181],{"className":41180},[346],[15,41182,41184],{"className":41183,"style":504},[350],[15,41185],{},[11,41187,41188,41189,41217],{},"在 ",[15,41190,41192,41205],{"className":41191,"translate":19},[18],[15,41193,41195],{"className":41194},[23],[25,41196,41197],{"xmlns":27},[29,41198,41199,41203],{},[32,41200,41201],{},[35,41202,33401],{},[40,41204,33401],{"encoding":42},[15,41206,41208],{"className":41207,"ariaHidden":48},[47],[15,41209,41211,41214],{"className":41210},[52],[15,41212],{"className":41213,"style":6026},[56],[15,41215,33401],{"className":41216},[61,122]," 的定义下，预测概率具有一个在解释上极有用的分解：",[15,41219,41221],{"className":41220,"translate":19},[212],[15,41222,41224,41338],{"className":41223,"translate":19},[18],[15,41225,41227],{"className":41226},[23],[25,41228,41229],{"xmlns":27,"display":221},[29,41230,41231,41335],{},[32,41232,41233,41239,41241,41266,41268,41283,41285,41287,41289,41291,41293,41295,41297,41323,41325,41327,41329],{},[266,41234,41235,41237],{},[35,41236,11],{},[35,41238,272],{},[81,41240,244],{},[6875,41242,41243,41263],{},[6875,41244,41245,41261],{},[3718,41246,41247,41259],{},[32,41248,41249,41255,41257],{},[266,41250,41251,41253],{},[35,41252,3102],{},[35,41254,272],{},[81,41256,239],{},[89,41258,91],{},[35,41260,40230],{},[81,41262,27460],{"stretchy":48},[282,41264,41265],{},"证据部分",[81,41267,7756],{},[6875,41269,41270,41280],{},[6875,41271,41272,41278],{},[3718,41273,41274,41276],{},[89,41275,91],{},[35,41277,40230],{},[81,41279,27460],{"stretchy":48},[282,41281,41282],{},"先验部分",[81,41284,244],{},[81,41286,741],{"stretchy":86},[89,41288,91],{},[81,41290,239],{},[35,41292,33401],{},[81,41294,750],{"stretchy":86},[282,41296,4796],{},[3718,41298,41299,41311],{},[32,41300,41301,41307,41309],{},[266,41302,41303,41305],{},[35,41304,3102],{},[35,41306,272],{},[81,41308,239],{},[89,41310,91],{},[32,41312,41313,41319,41321],{},[266,41314,41315,41317],{},[35,41316,3102],{},[89,41318,278],{},[81,41320,239],{},[35,41322,102],{},[81,41324,7756],{},[35,41326,33401],{},[81,41328,11205],{},[3718,41330,41331,41333],{},[89,41332,91],{},[35,41334,102],{},[40,41336,41337],{"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}",[15,41339,41341,41396,41635,41818,41839,42023,42042],{"className":41340,"ariaHidden":48},[47],[15,41342,41344,41347,41387,41390,41393],{"className":41343},[52],[15,41345],{"className":41346,"style":9428},[56],[15,41348,41350,41353],{"className":41349},[61],[15,41351,11],{"className":41352},[61,122],[15,41354,41356],{"className":41355},[338],[15,41357,41359,41379],{"className":41358},[342,472],[15,41360,41362,41376],{"className":41361},[346],[15,41363,41365],{"className":41364,"style":479},[350],[15,41366,41367,41370],{"style":482},[15,41368],{"className":41369,"style":358},[357],[15,41371,41373],{"className":41372},[362,363,364,365],[15,41374,272],{"className":41375,"style":492},[61,122,365],[15,41377,497],{"className":41378},[496],[15,41380,41382],{"className":41381},[346],[15,41383,41385],{"className":41384,"style":504},[350],[15,41386],{},[15,41388],{"className":41389,"style":128},[127],[15,41391,244],{"className":41392},[132],[15,41394],{"className":41395,"style":128},[127],[15,41397,41399,41403,41626,41629,41632],{"className":41398},[52],[15,41400],{"className":41401,"style":41402},[56],"height:3.3338em;vertical-align:-2.0123em;",[15,41404,41406],{"className":41405},[161,6875],[15,41407,41409,41617],{"className":41408},[342,472],[15,41410,41412,41614],{"className":41411},[346],[15,41413,41415,41434],{"className":41414,"style":18679},[350],[15,41416,41418,41422],{"style":41417},"top:-1.3091em;",[15,41419],{"className":41420,"style":41421},[357],"height:3.3214em;",[15,41423,41425],{"className":41424},[362,363,364,365],[15,41426,41428],{"className":41427},[61,365],[15,41429,41431],{"className":41430},[61,13539,365],[15,41432,41265],{"className":41433},[61,13543,365],[15,41435,41437,41440],{"style":41436},"top:-3.3214em;",[15,41438],{"className":41439,"style":41421},[357],[15,41441,41443],{"className":41442},[161,6875],[15,41444,41446,41605],{"className":41445},[342,472],[15,41447,41449,41602],{"className":41448},[346],[15,41450,41452,41483],{"className":41451,"style":18679},[350],[15,41453,41456,41459],{"className":41454,"style":41455},[27667],"top:-1.9874em;",[15,41457],{"className":41458,"style":41421},[357],[15,41460,41462,41469,41476],{"className":41461,"style":27676},[27675],[15,41463,41465],{"className":41464,"style":27681},[27680],[27683,41466,41467],{"xmlns":27685,"width":27686,"height":27687,"viewBox":27688,"preserveAspectRatio":27689},[27691,41468],{"d":27693},[15,41470,41472],{"className":41471,"style":27681},[27697],[27683,41473,41474],{"xmlns":27685,"width":27686,"height":27687,"viewBox":27688,"preserveAspectRatio":27700},[27691,41475],{"d":27703},[15,41477,41479],{"className":41478,"style":27681},[27707],[27683,41480,41481],{"xmlns":27685,"width":27686,"height":27687,"viewBox":27688,"preserveAspectRatio":27710},[27691,41482],{"d":27713},[15,41484,41485,41488],{"style":41436},[15,41486],{"className":41487,"style":41421},[357],[15,41489,41491],{"className":41490},[61],[15,41492,41494,41497,41599],{"className":41493},[61],[15,41495],{"className":41496},[146,3837],[15,41498,41500],{"className":41499},[3718],[15,41501,41503,41591],{"className":41502},[342,472],[15,41504,41506,41588],{"className":41505},[346],[15,41507,41509,41520,41528],{"className":41508,"style":18679},[350],[15,41510,41511,41514],{"style":11393},[15,41512],{"className":41513,"style":3857},[357],[15,41515,41517],{"className":41516},[61],[15,41518,40230],{"className":41519,"style":40302},[61,122],[15,41521,41522,41525],{"style":3885},[15,41523],{"className":41524,"style":3857},[357],[15,41526],{"className":41527,"style":3893},[3892],[15,41529,41530,41533],{"style":11459},[15,41531],{"className":41532,"style":3857},[357],[15,41534,41536,41576,41579,41582,41585],{"className":41535},[61],[15,41537,41539,41542],{"className":41538},[61],[15,41540,3102],{"className":41541,"style":3187},[61,122],[15,41543,41545],{"className":41544},[338],[15,41546,41548,41568],{"className":41547},[342,472],[15,41549,41551,41565],{"className":41550},[346],[15,41552,41554],{"className":41553,"style":479},[350],[15,41555,41556,41559],{"style":3202},[15,41557],{"className":41558,"style":358},[357],[15,41560,41562],{"className":41561},[362,363,364,365],[15,41563,272],{"className":41564,"style":492},[61,122,365],[15,41566,497],{"className":41567},[496],[15,41569,41571],{"className":41570},[346],[15,41572,41574],{"className":41573,"style":504},[350],[15,41575],{},[15,41577],{"className":41578,"style":772},[127],[15,41580,239],{"className":41581},[375],[15,41583],{"className":41584,"style":772},[127],[15,41586,91],{"className":41587},[61],[15,41589,497],{"className":41590},[496],[15,41592,41594],{"className":41593},[346],[15,41595,41597],{"className":41596,"style":36583},[350],[15,41598],{},[15,41600],{"className":41601},[178,3837],[15,41603,497],{"className":41604},[496],[15,41606,41608],{"className":41607},[346],[15,41609,41612],{"className":41610,"style":41611},[350],"height:1.334em;",[15,41613],{},[15,41615,497],{"className":41616},[496],[15,41618,41620],{"className":41619},[346],[15,41621,41624],{"className":41622,"style":41623},[350],"height:2.0123em;",[15,41625],{},[15,41627],{"className":41628,"style":772},[127],[15,41630,7756],{"className":41631},[375],[15,41633],{"className":41634,"style":772},[127],[15,41636,41638,41641,41809,41812,41815],{"className":41637},[52],[15,41639],{"className":41640,"style":41402},[56],[15,41642,41644],{"className":41643},[161,6875],[15,41645,41647,41801],{"className":41646},[342,472],[15,41648,41650,41798],{"className":41649},[346],[15,41651,41653,41670],{"className":41652,"style":18679},[350],[15,41654,41655,41658],{"style":41417},[15,41656],{"className":41657,"style":41421},[357],[15,41659,41661],{"className":41660},[362,363,364,365],[15,41662,41664],{"className":41663},[61,365],[15,41665,41667],{"className":41666},[61,13539,365],[15,41668,41282],{"className":41669},[61,13543,365],[15,41671,41672,41675],{"style":41436},[15,41673],{"className":41674,"style":41421},[357],[15,41676,41678],{"className":41677},[161,6875],[15,41679,41681,41790],{"className":41680},[342,472],[15,41682,41684,41787],{"className":41683},[346],[15,41685,41687,41717],{"className":41686,"style":18679},[350],[15,41688,41690,41693],{"className":41689,"style":41455},[27667],[15,41691],{"className":41692,"style":41421},[357],[15,41694,41696,41703,41710],{"className":41695,"style":27676},[27675],[15,41697,41699],{"className":41698,"style":27681},[27680],[27683,41700,41701],{"xmlns":27685,"width":27686,"height":27687,"viewBox":27688,"preserveAspectRatio":27689},[27691,41702],{"d":27693},[15,41704,41706],{"className":41705,"style":27681},[27697],[27683,41707,41708],{"xmlns":27685,"width":27686,"height":27687,"viewBox":27688,"preserveAspectRatio":27700},[27691,41709],{"d":27703},[15,41711,41713],{"className":41712,"style":27681},[27707],[27683,41714,41715],{"xmlns":27685,"width":27686,"height":27687,"viewBox":27688,"preserveAspectRatio":27710},[27691,41716],{"d":27713},[15,41718,41719,41722],{"style":41436},[15,41720],{"className":41721,"style":41421},[357],[15,41723,41725],{"className":41724},[61],[15,41726,41728,41731,41784],{"className":41727},[61],[15,41729],{"className":41730},[146,3837],[15,41732,41734],{"className":41733},[3718],[15,41735,41737,41776],{"className":41736},[342,472],[15,41738,41740,41773],{"className":41739},[346],[15,41741,41743,41754,41762],{"className":41742,"style":18679},[350],[15,41744,41745,41748],{"style":11393},[15,41746],{"className":41747,"style":3857},[357],[15,41749,41751],{"className":41750},[61],[15,41752,40230],{"className":41753,"style":40302},[61,122],[15,41755,41756,41759],{"style":3885},[15,41757],{"className":41758,"style":3857},[357],[15,41760],{"className":41761,"style":3893},[3892],[15,41763,41764,41767],{"style":11459},[15,41765],{"className":41766,"style":3857},[357],[15,41768,41770],{"className":41769},[61],[15,41771,91],{"className":41772},[61],[15,41774,497],{"className":41775},[496],[15,41777,41779],{"className":41778},[346],[15,41780,41782],{"className":41781,"style":36583},[350],[15,41783],{},[15,41785],{"className":41786},[178,3837],[15,41788,497],{"className":41789},[496],[15,41791,41793],{"className":41792},[346],[15,41794,41796],{"className":41795,"style":41611},[350],[15,41797],{},[15,41799,497],{"className":41800},[496],[15,41802,41804],{"className":41803},[346],[15,41805,41807],{"className":41806,"style":41623},[350],[15,41808],{},[15,41810],{"className":41811,"style":128},[127],[15,41813,244],{"className":41814},[132],[15,41816],{"className":41817,"style":128},[127],[15,41819,41821,41824,41827,41830,41833,41836],{"className":41820},[52],[15,41822],{"className":41823,"style":142},[56],[15,41825,741],{"className":41826},[146],[15,41828,91],{"className":41829},[61],[15,41831],{"className":41832,"style":772},[127],[15,41834,239],{"className":41835},[375],[15,41837],{"className":41838,"style":772},[127],[15,41840,41842,41845,41848,41851,41854,42014,42017,42020],{"className":41841},[52],[15,41843],{"className":41844,"style":18660},[56],[15,41846,33401],{"className":41847},[61,122],[15,41849,750],{"className":41850},[178],[15,41852],{"className":41853,"style":157},[127],[15,41855,41857,41860,42011],{"className":41856},[61],[15,41858],{"className":41859},[146,3837],[15,41861,41863],{"className":41862},[3718],[15,41864,41866,42003],{"className":41865},[342,472],[15,41867,41869,42000],{"className":41868},[346],[15,41870,41872,41932,41940],{"className":41871,"style":18679},[350],[15,41873,41874,41877],{"style":11393},[15,41875],{"className":41876,"style":3857},[357],[15,41878,41880,41920,41923,41926,41929],{"className":41879},[61],[15,41881,41883,41886],{"className":41882},[61],[15,41884,3102],{"className":41885,"style":3187},[61,122],[15,41887,41889],{"className":41888},[338],[15,41890,41892,41912],{"className":41891},[342,472],[15,41893,41895,41909],{"className":41894},[346],[15,41896,41898],{"className":41897,"style":1212},[350],[15,41899,41900,41903],{"style":3202},[15,41901],{"className":41902,"style":358},[357],[15,41904,41906],{"className":41905},[362,363,364,365],[15,41907,278],{"className":41908},[61,365],[15,41910,497],{"className":41911},[496],[15,41913,41915],{"className":41914},[346],[15,41916,41918],{"className":41917,"style":504},[350],[15,41919],{},[15,41921],{"className":41922,"style":772},[127],[15,41924,239],{"className":41925},[375],[15,41927],{"className":41928,"style":772},[127],[15,41930,102],{"className":41931,"style":174},[61,122],[15,41933,41934,41937],{"style":3885},[15,41935],{"className":41936,"style":3857},[357],[15,41938],{"className":41939,"style":3893},[3892],[15,41941,41942,41945],{"style":11459},[15,41943],{"className":41944,"style":3857},[357],[15,41946,41948,41988,41991,41994,41997],{"className":41947},[61],[15,41949,41951,41954],{"className":41950},[61],[15,41952,3102],{"className":41953,"style":3187},[61,122],[15,41955,41957],{"className":41956},[338],[15,41958,41960,41980],{"className":41959},[342,472],[15,41961,41963,41977],{"className":41962},[346],[15,41964,41966],{"className":41965,"style":479},[350],[15,41967,41968,41971],{"style":3202},[15,41969],{"className":41970,"style":358},[357],[15,41972,41974],{"className":41973},[362,363,364,365],[15,41975,272],{"className":41976,"style":492},[61,122,365],[15,41978,497],{"className":41979},[496],[15,41981,41983],{"className":41982},[346],[15,41984,41986],{"className":41985,"style":504},[350],[15,41987],{},[15,41989],{"className":41990,"style":772},[127],[15,41992,239],{"className":41993},[375],[15,41995],{"className":41996,"style":772},[127],[15,41998,91],{"className":41999},[61],[15,42001,497],{"className":42002},[496],[15,42004,42006],{"className":42005},[346],[15,42007,42009],{"className":42008,"style":18768},[350],[15,42010],{},[15,42012],{"className":42013},[178,3837],[15,42015],{"className":42016,"style":772},[127],[15,42018,7756],{"className":42019},[375],[15,42021],{"className":42022,"style":772},[127],[15,42024,42026,42030,42033,42036,42039],{"className":42025},[52],[15,42027],{"className":42028,"style":42029},[56],"height:0.4445em;",[15,42031,33401],{"className":42032},[61,122],[15,42034],{"className":42035,"style":772},[127],[15,42037,11205],{"className":42038},[375],[15,42040],{"className":42041,"style":772},[127],[15,42043,42045,42049],{"className":42044},[52],[15,42046],{"className":42047,"style":42048},[56],"height:2.0074em;vertical-align:-0.686em;",[15,42050,42052,42055,42108],{"className":42051},[61],[15,42053],{"className":42054},[146,3837],[15,42056,42058],{"className":42057},[3718],[15,42059,42061,42100],{"className":42060},[342,472],[15,42062,42064,42097],{"className":42063},[346],[15,42065,42067,42078,42086],{"className":42066,"style":18679},[350],[15,42068,42069,42072],{"style":11393},[15,42070],{"className":42071,"style":3857},[357],[15,42073,42075],{"className":42074},[61],[15,42076,102],{"className":42077,"style":174},[61,122],[15,42079,42080,42083],{"style":3885},[15,42081],{"className":42082,"style":3857},[357],[15,42084],{"className":42085,"style":3893},[3892],[15,42087,42088,42091],{"style":11459},[15,42089],{"className":42090,"style":3857},[357],[15,42092,42094],{"className":42093},[61],[15,42095,91],{"className":42096},[61],[15,42098,497],{"className":42099},[496],[15,42101,42103],{"className":42102},[346],[15,42104,42106],{"className":42105,"style":36583},[350],[15,42107],{},[15,42109],{"className":42110},[178,3837],[11,42112,12917,42113,42191,42192,42243,42244,42295],{},[661,42114,42115,42116,42190],{},"预测概率是「证据归一化的类比例」与「均匀分布」按 ",[15,42117,42119,42145],{"className":42118,"translate":19},[18],[15,42120,42122],{"className":42121},[23],[25,42123,42124],{"xmlns":27},[29,42125,42126,42142],{},[32,42127,42128,42130,42132,42134,42136,42138,42140],{},[81,42129,741],{"stretchy":86},[89,42131,91],{},[81,42133,239],{},[35,42135,33401],{},[81,42137,94],{"separator":48},[35,42139,33401],{},[81,42141,750],{"stretchy":86},[40,42143,42144],{"encoding":42},"(1-u, u)",[15,42146,42148,42169],{"className":42147,"ariaHidden":48},[47],[15,42149,42151,42154,42157,42160,42163,42166],{"className":42150},[52],[15,42152],{"className":42153,"style":142},[56],[15,42155,741],{"className":42156},[146],[15,42158,91],{"className":42159},[61],[15,42161],{"className":42162,"style":772},[127],[15,42164,239],{"className":42165},[375],[15,42167],{"className":42168,"style":772},[127],[15,42170,42172,42175,42178,42181,42184,42187],{"className":42171},[52],[15,42173],{"className":42174,"style":142},[56],[15,42176,33401],{"className":42177},[61,122],[15,42179,94],{"className":42180},[153],[15,42182],{"className":42183,"style":157},[127],[15,42185,33401],{"className":42186},[61,122],[15,42188,750],{"className":42189},[178]," 的凸组合","。当证据充分时（",[15,42193,42195,42213],{"className":42194,"translate":19},[18],[15,42196,42198],{"className":42197},[23],[25,42199,42200],{"xmlns":27},[29,42201,42202,42210],{},[32,42203,42204,42206,42208],{},[35,42205,33401],{},[81,42207,20691],{},[89,42209,278],{},[40,42211,42212],{"encoding":42},"u \\to 0",[15,42214,42216,42234],{"className":42215,"ariaHidden":48},[47],[15,42217,42219,42222,42225,42228,42231],{"className":42218},[52],[15,42220],{"className":42221,"style":6026},[56],[15,42223,33401],{"className":42224},[61,122],[15,42226],{"className":42227,"style":128},[127],[15,42229,20691],{"className":42230},[132],[15,42232],{"className":42233,"style":128},[127],[15,42235,42237,42240],{"className":42236},[52],[15,42238],{"className":42239,"style":859},[56],[15,42241,278],{"className":42242},[61],"），预测趋于证据的方向。当证据消失时（",[15,42245,42247,42265],{"className":42246,"translate":19},[18],[15,42248,42250],{"className":42249},[23],[25,42251,42252],{"xmlns":27},[29,42253,42254,42262],{},[32,42255,42256,42258,42260],{},[35,42257,33401],{},[81,42259,20691],{},[89,42261,91],{},[40,42263,42264],{"encoding":42},"u \\to 1",[15,42266,42268,42286],{"className":42267,"ariaHidden":48},[47],[15,42269,42271,42274,42277,42280,42283],{"className":42270},[52],[15,42272],{"className":42273,"style":6026},[56],[15,42275,33401],{"className":42276},[61,122],[15,42278],{"className":42279,"style":128},[127],[15,42281,20691],{"className":42282},[132],[15,42284],{"className":42285,"style":128},[127],[15,42287,42289,42292],{"className":42288},[52],[15,42290],{"className":42291,"style":859},[56],[15,42293,91],{"className":42294},[61],"），预测退回均匀分布。极限行为因此完全符合直觉：",[15,42297,42299],{"className":42298,"translate":19},[212],[15,42300,42302,42407],{"className":42301,"translate":19},[18],[15,42303,42305],{"className":42304},[23],[25,42306,42307],{"xmlns":27,"display":221},[29,42308,42309,42404],{},[32,42310,42311,42313,42315,42317,42319,42321,42323,42325,42327,42329,42331,42333,42335,42337,42339,42341,42344,42346,42348,42356,42358,42360,42362,42364,42366,42368,42370,42372,42374,42376,42378,42380,42382,42384,42386],{},[35,42312,40230],{},[81,42314,20691],{},[35,42316,102],{},[282,42318,284],{},[81,42320,741],{"stretchy":86},[35,42322,1980],{"mathvariant":251},[81,42324,20691],{},[89,42326,278],{"mathvariant":251},[81,42328,750],{"stretchy":86},[81,42330,264],{},[127,42332],{"width":6445},[35,42334,33401],{},[81,42336,20691],{},[89,42338,91],{},[81,42340,94],{"separator":48},[282,42342,42343],{},"  ",[35,42345,11],{"mathvariant":251},[81,42347,20691],{},[286,42349,42350],{"scriptlevel":278,"displaystyle":86},[3718,42351,42352,42354],{},[89,42353,91],{},[35,42355,102],{},[89,42357,91],{"mathvariant":251},[127,42359],{"width":9838},[127,42361],{"width":9838},[35,42363,40230],{},[81,42365,20691],{},[35,42367,6790],{"mathvariant":231},[81,42369,264],{},[127,42371],{"width":6445},[35,42373,33401],{},[81,42375,20691],{},[89,42377,278],{},[81,42379,94],{"separator":48},[282,42381,42343],{},[35,42383,11],{"mathvariant":251},[81,42385,20691],{},[3718,42387,42388,42390],{},[35,42389,1980],{"mathvariant":251},[32,42391,42392,42398],{},[6875,42393,42394,42396],{},[81,42395,293],{},[35,42397,14766],{},[266,42399,42400,42402],{},[35,42401,1980],{},[35,42403,14766],{},[40,42405,42406],{"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}",[15,42408,42410,42428,42455,42479,42497,42530,42626,42647,42665,42698],{"className":42409,"ariaHidden":48},[47],[15,42411,42413,42416,42419,42422,42425],{"className":42412},[52],[15,42414],{"className":42415,"style":57},[56],[15,42417,40230],{"className":42418,"style":40302},[61,122],[15,42420],{"className":42421,"style":128},[127],[15,42423,20691],{"className":42424},[132],[15,42426],{"className":42427,"style":128},[127],[15,42429,42431,42434,42437,42440,42443,42446,42449,42452],{"className":42430},[52],[15,42432],{"className":42433,"style":142},[56],[15,42435,102],{"className":42436,"style":174},[61,122],[15,42438,284],{"className":42439},[127],[15,42441,741],{"className":42442},[146],[15,42444,1980],{"className":42445},[61,411],[15,42447],{"className":42448,"style":128},[127],[15,42450,20691],{"className":42451},[132],[15,42453],{"className":42454,"style":128},[127],[15,42456,42458,42461,42464,42467,42470,42473,42476],{"className":42457},[52],[15,42459],{"className":42460,"style":142},[56],[15,42462,278],{"className":42463},[61,411],[15,42465,750],{"className":42466},[178],[15,42468],{"className":42469,"style":128},[127],[15,42471,264],{"className":42472},[132],[15,42474],{"className":42475,"style":15972},[127],[15,42477],{"className":42478,"style":128},[127],[15,42480,42482,42485,42488,42491,42494],{"className":42481},[52],[15,42483],{"className":42484,"style":6026},[56],[15,42486,33401],{"className":42487},[61,122],[15,42489],{"className":42490,"style":128},[127],[15,42492,20691],{"className":42493},[132],[15,42495],{"className":42496,"style":128},[127],[15,42498,42500,42503,42506,42509,42512,42515,42518,42521,42524,42527],{"className":42499},[52],[15,42501],{"className":42502,"style":19744},[56],[15,42504,91],{"className":42505},[61],[15,42507,94],{"className":42508},[153],[15,42510,284],{"className":42511},[127],[15,42513,284],{"className":42514},[127],[15,42516],{"className":42517,"style":157},[127],[15,42519,11],{"className":42520},[61,411],[15,42522],{"className":42523,"style":128},[127],[15,42525,20691],{"className":42526},[132],[15,42528],{"className":42529,"style":128},[127],[15,42531,42533,42537,42605,42608,42611,42614,42617,42620,42623],{"className":42532},[52],[15,42534],{"className":42535,"style":42536},[56],"height:1.1901em;vertical-align:-0.345em;",[15,42538,42540,42543,42602],{"className":42539},[61],[15,42541],{"className":42542},[146,3837],[15,42544,42546],{"className":42545},[3718],[15,42547,42549,42594],{"className":42548},[342,472],[15,42550,42552,42591],{"className":42551},[346],[15,42553,42555,42569,42577],{"className":42554,"style":3850},[350],[15,42556,42557,42560],{"style":3853},[15,42558],{"className":42559,"style":3857},[357],[15,42561,42563],{"className":42562},[362,363,364,365],[15,42564,42566],{"className":42565},[61,365],[15,42567,102],{"className":42568,"style":174},[61,122,365],[15,42570,42571,42574],{"style":3885},[15,42572],{"className":42573,"style":3857},[357],[15,42575],{"className":42576,"style":3893},[3892],[15,42578,42579,42582],{"style":3896},[15,42580],{"className":42581,"style":3857},[357],[15,42583,42585],{"className":42584},[362,363,364,365],[15,42586,42588],{"className":42587},[61,365],[15,42589,91],{"className":42590},[61,365],[15,42592,497],{"className":42593},[496],[15,42595,42597],{"className":42596},[346],[15,42598,42600],{"className":42599,"style":10189},[350],[15,42601],{},[15,42603],{"className":42604},[178,3837],[15,42606,91],{"className":42607},[61,411],[15,42609],{"className":42610,"style":10026},[127],[15,42612],{"className":42613,"style":10026},[127],[15,42615,40230],{"className":42616,"style":40302},[61,122],[15,42618],{"className":42619,"style":128},[127],[15,42621,20691],{"className":42622},[132],[15,42624],{"className":42625,"style":128},[127],[15,42627,42629,42632,42635,42638,42641,42644],{"className":42628},[52],[15,42630],{"className":42631,"style":6026},[56],[15,42633,6790],{"className":42634},[61],[15,42636],{"className":42637,"style":128},[127],[15,42639,264],{"className":42640},[132],[15,42642],{"className":42643,"style":15972},[127],[15,42645],{"className":42646,"style":128},[127],[15,42648,42650,42653,42656,42659,42662],{"className":42649},[52],[15,42651],{"className":42652,"style":6026},[56],[15,42654,33401],{"className":42655},[61,122],[15,42657],{"className":42658,"style":128},[127],[15,42660,20691],{"className":42661},[132],[15,42663],{"className":42664,"style":128},[127],[15,42666,42668,42671,42674,42677,42680,42683,42686,42689,42692,42695],{"className":42667},[52],[15,42669],{"className":42670,"style":19744},[56],[15,42672,278],{"className":42673},[61],[15,42675,94],{"className":42676},[153],[15,42678,284],{"className":42679},[127],[15,42681,284],{"className":42682},[127],[15,42684],{"className":42685,"style":157},[127],[15,42687,11],{"className":42688},[61,411],[15,42690],{"className":42691,"style":128},[127],[15,42693,20691],{"className":42694},[132],[15,42696],{"className":42697,"style":128},[127],[15,42699,42701,42705],{"className":42700},[52],[15,42702],{"className":42703,"style":42704},[56],"height:2.2433em;vertical-align:-1.1218em;",[15,42706,42708,42711,42846],{"className":42707},[61],[15,42709],{"className":42710},[146,3837],[15,42712,42714],{"className":42713},[3718],[15,42715,42717,42837],{"className":42716},[342,472],[15,42718,42720,42834],{"className":42719},[346],[15,42721,42724,42815,42823],{"className":42722,"style":42723},[350],"height:1.1214em;",[15,42725,42726,42729],{"style":11393},[15,42727],{"className":42728,"style":3857},[357],[15,42730,42732,42772,42775],{"className":42731},[61],[15,42733,42735,42738],{"className":42734},[531],[15,42736,293],{"className":42737,"style":537},[531,535,536],[15,42739,42741],{"className":42740},[338],[15,42742,42744,42764],{"className":42743},[342,472],[15,42745,42747,42761],{"className":42746},[346],[15,42748,42750],{"className":42749,"style":6064},[350],[15,42751,42752,42755],{"style":553},[15,42753],{"className":42754,"style":358},[357],[15,42756,42758],{"className":42757},[362,363,364,365],[15,42759,14766],{"className":42760,"style":14792},[61,122,365],[15,42762,497],{"className":42763},[496],[15,42765,42767],{"className":42766},[346],[15,42768,42770],{"className":42769,"style":30028},[350],[15,42771],{},[15,42773],{"className":42774,"style":157},[127],[15,42776,42778,42781],{"className":42777},[61],[15,42779,1980],{"className":42780},[61,122],[15,42782,42784],{"className":42783},[338],[15,42785,42787,42807],{"className":42786},[342,472],[15,42788,42790,42804],{"className":42789},[346],[15,42791,42793],{"className":42792,"style":1362},[350],[15,42794,42795,42798],{"style":482},[15,42796],{"className":42797,"style":358},[357],[15,42799,42801],{"className":42800},[362,363,364,365],[15,42802,14766],{"className":42803,"style":14792},[61,122,365],[15,42805,497],{"className":42806},[496],[15,42808,42810],{"className":42809},[346],[15,42811,42813],{"className":42812,"style":15696},[350],[15,42814],{},[15,42816,42817,42820],{"style":3885},[15,42818],{"className":42819,"style":3857},[357],[15,42821],{"className":42822,"style":3893},[3892],[15,42824,42825,42828],{"style":11459},[15,42826],{"className":42827,"style":3857},[357],[15,42829,42831],{"className":42830},[61],[15,42832,1980],{"className":42833},[61,411],[15,42835,497],{"className":42836},[496],[15,42838,42840],{"className":42839},[346],[15,42841,42844],{"className":42842,"style":42843},[350],"height:1.1218em;",[15,42845],{},[15,42847],{"className":42848},[178,3837],[11,42850,42851,42852,42855,42856,42884,42885,42913,42914,42917],{},"命题 3.4 同时也暴露了该参数化的一个",[661,42853,42854],{},"结构性局限","：",[15,42857,42859,42872],{"className":42858,"translate":19},[18],[15,42860,42862],{"className":42861},[23],[25,42863,42864],{"xmlns":27},[29,42865,42866,42870],{},[32,42867,42868],{},[35,42869,33401],{},[40,42871,33401],{"encoding":42},[15,42873,42875],{"className":42874,"ariaHidden":48},[47],[15,42876,42878,42881],{"className":42877},[52],[15,42879],{"className":42880,"style":6026},[56],[15,42882,33401],{"className":42883},[61,122]," 只是总强度 ",[15,42886,42888,42901],{"className":42887,"translate":19},[18],[15,42889,42891],{"className":42890},[23],[25,42892,42893],{"xmlns":27},[29,42894,42895,42899],{},[32,42896,42897],{},[35,42898,40230],{},[40,42900,40230],{"encoding":42},[15,42902,42904],{"className":42903,"ariaHidden":48},[47],[15,42905,42907,42910],{"className":42906},[52],[15,42908],{"className":42909,"style":57},[56],[15,42911,40230],{"className":42912,"style":40302},[61,122]," 的函数，与证据在各类别之间的",[661,42915,42916],{},"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。",[877,42919,42921],{"id":42920},"_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",[11,42923,42924,42925,42986,42987,42990],{},"命题 3.4 给出的 ",[15,42926,42928,42950],{"className":42927,"translate":19},[18],[15,42929,42931],{"className":42930},[23],[25,42932,42933],{"xmlns":27},[29,42934,42935,42947],{},[32,42936,42937,42939,42941,42943,42945],{},[35,42938,33401],{},[81,42940,244],{},[35,42942,102],{},[35,42944,3018],{"mathvariant":231},[35,42946,40230],{},[40,42948,42949],{"encoding":42},"u = K\u002FS",[15,42951,42953,42971],{"className":42952,"ariaHidden":48},[47],[15,42954,42956,42959,42962,42965,42968],{"className":42955},[52],[15,42957],{"className":42958,"style":6026},[56],[15,42960,33401],{"className":42961},[61,122],[15,42963],{"className":42964,"style":128},[127],[15,42966,244],{"className":42967},[132],[15,42969],{"className":42970,"style":128},[127],[15,42972,42974,42977,42980,42983],{"className":42973},[52],[15,42975],{"className":42976,"style":142},[56],[15,42978,102],{"className":42979,"style":174},[61,122],[15,42981,3018],{"className":42982},[61],[15,42984,40230],{"className":42985,"style":40302},[61,122]," 是一个",[661,42988,42989],{},"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。",[11,42992,13,42993,16188,43075,43181],{},[15,42994,42996,43021],{"className":42995,"translate":19},[18],[15,42997,42999],{"className":42998},[23],[25,43000,43001],{"xmlns":27},[29,43002,43003,43019],{},[32,43004,43005,43007,43009,43011,43013,43015,43017],{},[35,43006,24661],{"mathvariant":3101},[81,43008,4179],{},[35,43010,4182],{"mathvariant":231},[81,43012,1545],{},[81,43014,741],{"stretchy":86},[35,43016,3102],{"mathvariant":3101},[81,43018,750],{"stretchy":86},[40,43020,24676],{"encoding":42},[15,43022,43024,43048],{"className":43023,"ariaHidden":48},[47],[15,43025,43027,43030,43039,43042,43045],{"className":43026},[52],[15,43028],{"className":43029,"style":2706},[56],[15,43031,43033],{"className":43032},[61],[15,43034,43036],{"className":43035},[61],[15,43037,24661],{"className":43038,"style":24695},[61,3162],[15,43040],{"className":43041,"style":128},[127],[15,43043,4179],{"className":43044},[132],[15,43046],{"className":43047,"style":128},[127],[15,43049,43051,43054,43060,43063,43072],{"className":43050},[52],[15,43052],{"className":43053,"style":142},[56],[15,43055,43057],{"className":43056},[531],[15,43058,4182],{"className":43059},[61,1620],[15,43061,741],{"className":43062},[146],[15,43064,43066],{"className":43065},[61],[15,43067,43069],{"className":43068},[61],[15,43070,3102],{"className":43071},[61,3162],[15,43073,750],{"className":43074},[178],[15,43076,43078,43109],{"className":43077,"translate":19},[18],[15,43079,43081],{"className":43080},[23],[25,43082,43083],{"xmlns":27},[29,43084,43085,43106],{},[32,43086,43087,43089,43091,43093,43095,43098,43100,43102,43104],{},[35,43088,79],{},[81,43090,3710],{},[35,43092,24661],{"mathvariant":3101},[81,43094,4179],{},[35,43096,43097],{"mathvariant":231},"Cat",[81,43099,1545],{},[81,43101,741],{"stretchy":86},[35,43103,24661],{"mathvariant":3101},[81,43105,750],{"stretchy":86},[40,43107,43108],{"encoding":42},"y \\mid \\boldsymbol{\\pi} \\sim \\operatorname{Cat}(\\boldsymbol{\\pi})",[15,43110,43112,43130,43154],{"className":43111,"ariaHidden":48},[47],[15,43113,43115,43118,43121,43124,43127],{"className":43114},[52],[15,43116],{"className":43117,"style":142},[56],[15,43119,79],{"className":43120,"style":123},[61,122],[15,43122],{"className":43123,"style":128},[127],[15,43125,3710],{"className":43126},[132],[15,43128],{"className":43129,"style":128},[127],[15,43131,43133,43136,43145,43148,43151],{"className":43132},[52],[15,43134],{"className":43135,"style":2706},[56],[15,43137,43139],{"className":43138},[61],[15,43140,43142],{"className":43141},[61],[15,43143,24661],{"className":43144,"style":24695},[61,3162],[15,43146],{"className":43147,"style":128},[127],[15,43149,4179],{"className":43150},[132],[15,43152],{"className":43153,"style":128},[127],[15,43155,43157,43160,43166,43169,43178],{"className":43156},[52],[15,43158],{"className":43159,"style":142},[56],[15,43161,43163],{"className":43162},[531],[15,43164,43097],{"className":43165},[61,1620],[15,43167,741],{"className":43168},[146],[15,43170,43172],{"className":43171},[61],[15,43173,43175],{"className":43174},[61],[15,43176,24661],{"className":43177,"style":24695},[61,3162],[15,43179,750],{"className":43180},[178],"。考虑三个量：",[31520,43183,43184,43534,43687],{},[31523,43185,43186,43189,43190,5981,43269,867],{},[661,43187,43188],{},"总不确定性（total uncertainty）","：预测分布的熵 ",[15,43191,43193,43223],{"className":43192,"translate":19},[18],[15,43194,43196],{"className":43195},[23],[25,43197,43198],{"xmlns":27},[29,43199,43200,43220],{},[32,43201,43202,43204,43206,43208,43210,43212,43214,43216,43218],{},[35,43203,22634],{},[81,43205,14810],{"stretchy":86},[35,43207,21273],{"mathvariant":258},[81,43209,741],{"stretchy":86},[35,43211,79],{},[81,43213,3710],{},[35,43215,1027],{"mathvariant":251},[81,43217,750],{"stretchy":86},[81,43219,14819],{"stretchy":86},[40,43221,43222],{"encoding":42},"H[\\mathbb{P}(y \\mid \\mathbf{x})]",[15,43224,43226,43256],{"className":43225,"ariaHidden":48},[47],[15,43227,43229,43232,43235,43238,43241,43244,43247,43250,43253],{"className":43228},[52],[15,43230],{"className":43231,"style":142},[56],[15,43233,22634],{"className":43234,"style":22746},[61,122],[15,43236,14810],{"className":43237},[146],[15,43239,21273],{"className":43240},[61,427],[15,43242,741],{"className":43243},[146],[15,43245,79],{"className":43246,"style":123},[61,122],[15,43248],{"className":43249,"style":128},[127],[15,43251,3710],{"className":43252},[132],[15,43254],{"className":43255,"style":128},[127],[15,43257,43259,43262,43265],{"className":43258},[52],[15,43260],{"className":43261,"style":142},[56],[15,43263,1027],{"className":43264},[61,411],[15,43266,43268],{"className":43267},[178],")]",[15,43270,43272,43324],{"className":43271,"translate":19},[18],[15,43273,43275],{"className":43274},[23],[25,43276,43277],{"xmlns":27},[29,43278,43279,43321],{},[32,43280,43281,43283,43285,43287,43289,43291,43293,43295,43297,43299,43305,43307,43313,43315],{},[35,43282,21273],{"mathvariant":258},[81,43284,741],{"stretchy":86},[35,43286,79],{},[81,43288,244],{},[35,43290,272],{},[81,43292,3710],{},[35,43294,1027],{"mathvariant":251},[81,43296,750],{"stretchy":86},[81,43298,244],{},[266,43300,43301,43303],{},[35,43302,11],{},[35,43304,272],{},[81,43306,244],{},[266,43308,43309,43311],{},[35,43310,3102],{},[35,43312,272],{},[35,43314,3018],{"mathvariant":231},[266,43316,43317,43319],{},[35,43318,3102],{},[89,43320,278],{},[40,43322,43323],{"encoding":42},"\\mathbb{P}(y = k \\mid \\mathbf{x}) = p_k = \\alpha_k\u002F\\alpha_0",[15,43325,43327,43351,43369,43390,43445],{"className":43326,"ariaHidden":48},[47],[15,43328,43330,43333,43336,43339,43342,43345,43348],{"className":43329},[52],[15,43331],{"className":43332,"style":142},[56],[15,43334,21273],{"className":43335},[61,427],[15,43337,741],{"className":43338},[146],[15,43340,79],{"className":43341,"style":123},[61,122],[15,43343],{"className":43344,"style":128},[127],[15,43346,244],{"className":43347},[132],[15,43349],{"className":43350,"style":128},[127],[15,43352,43354,43357,43360,43363,43366],{"className":43353},[52],[15,43355],{"className":43356,"style":142},[56],[15,43358,272],{"className":43359,"style":492},[61,122],[15,43361],{"className":43362,"style":128},[127],[15,43364,3710],{"className":43365},[132],[15,43367],{"className":43368,"style":128},[127],[15,43370,43372,43375,43378,43381,43384,43387],{"className":43371},[52],[15,43373],{"className":43374,"style":142},[56],[15,43376,1027],{"className":43377},[61,411],[15,43379,750],{"className":43380},[178],[15,43382],{"className":43383,"style":128},[127],[15,43385,244],{"className":43386},[132],[15,43388],{"className":43389,"style":128},[127],[15,43391,43393,43396,43436,43439,43442],{"className":43392},[52],[15,43394],{"className":43395,"style":9428},[56],[15,43397,43399,43402],{"className":43398},[61],[15,43400,11],{"className":43401},[61,122],[15,43403,43405],{"className":43404},[338],[15,43406,43408,43428],{"className":43407},[342,472],[15,43409,43411,43425],{"className":43410},[346],[15,43412,43414],{"className":43413,"style":479},[350],[15,43415,43416,43419],{"style":482},[15,43417],{"className":43418,"style":358},[357],[15,43420,43422],{"className":43421},[362,363,364,365],[15,43423,272],{"className":43424,"style":492},[61,122,365],[15,43426,497],{"className":43427},[496],[15,43429,43431],{"className":43430},[346],[15,43432,43434],{"className":43433,"style":504},[350],[15,43435],{},[15,43437],{"className":43438,"style":128},[127],[15,43440,244],{"className":43441},[132],[15,43443],{"className":43444,"style":128},[127],[15,43446,43448,43451,43491,43494],{"className":43447},[52],[15,43449],{"className":43450,"style":142},[56],[15,43452,43454,43457],{"className":43453},[61],[15,43455,3102],{"className":43456,"style":3187},[61,122],[15,43458,43460],{"className":43459},[338],[15,43461,43463,43483],{"className":43462},[342,472],[15,43464,43466,43480],{"className":43465},[346],[15,43467,43469],{"className":43468,"style":479},[350],[15,43470,43471,43474],{"style":3202},[15,43472],{"className":43473,"style":358},[357],[15,43475,43477],{"className":43476},[362,363,364,365],[15,43478,272],{"className":43479,"style":492},[61,122,365],[15,43481,497],{"className":43482},[496],[15,43484,43486],{"className":43485},[346],[15,43487,43489],{"className":43488,"style":504},[350],[15,43490],{},[15,43492,3018],{"className":43493},[61],[15,43495,43497,43500],{"className":43496},[61],[15,43498,3102],{"className":43499,"style":3187},[61,122],[15,43501,43503],{"className":43502},[338],[15,43504,43506,43526],{"className":43505},[342,472],[15,43507,43509,43523],{"className":43508},[346],[15,43510,43512],{"className":43511,"style":1212},[350],[15,43513,43514,43517],{"style":3202},[15,43515],{"className":43516,"style":358},[357],[15,43518,43520],{"className":43519},[362,363,364,365],[15,43521,278],{"className":43522},[61,365],[15,43524,497],{"className":43525},[496],[15,43527,43529],{"className":43528},[346],[15,43530,43532],{"className":43531,"style":504},[350],[15,43533],{},[31523,43535,43536,43539,43540,43574,43575,867],{},[661,43537,43538],{},"偶然不确定性（aleatoric uncertainty）","：给定 ",[15,43541,43543,43556],{"className":43542,"translate":19},[18],[15,43544,43546],{"className":43545},[23],[25,43547,43548],{"xmlns":27},[29,43549,43550,43554],{},[32,43551,43552],{},[35,43553,24661],{"mathvariant":3101},[40,43555,24749],{"encoding":42},[15,43557,43559],{"className":43558,"ariaHidden":48},[47],[15,43560,43562,43565],{"className":43561},[52],[15,43563],{"className":43564,"style":2706},[56],[15,43566,43568],{"className":43567},[61],[15,43569,43571],{"className":43570},[61],[15,43572,24661],{"className":43573,"style":24695},[61,3162]," 后仍存在的熵的期望 ",[15,43576,43578,43608],{"className":43577,"translate":19},[18],[15,43579,43581],{"className":43580},[23],[25,43582,43583],{"xmlns":27},[29,43584,43585,43605],{},[32,43586,43587,43593,43595,43597,43599,43601,43603],{},[266,43588,43589,43591],{},[35,43590,14807],{"mathvariant":258},[35,43592,24661],{"mathvariant":3101},[81,43594,14810],{"stretchy":86},[35,43596,22634],{},[81,43598,741],{"stretchy":86},[35,43600,24661],{"mathvariant":3101},[81,43602,750],{"stretchy":86},[81,43604,14819],{"stretchy":86},[40,43606,43607],{"encoding":42},"\\mathbb{E}_{\\boldsymbol{\\pi}}[H(\\boldsymbol{\\pi})]",[15,43609,43611],{"className":43610,"ariaHidden":48},[47],[15,43612,43614,43617,43666,43669,43672,43675,43684],{"className":43613},[52],[15,43615],{"className":43616,"style":142},[56],[15,43618,43620,43623],{"className":43619},[61],[15,43621,14807],{"className":43622},[61,427],[15,43624,43626],{"className":43625},[338],[15,43627,43629,43658],{"className":43628},[342,472],[15,43630,43632,43655],{"className":43631},[346],[15,43633,43635],{"className":43634,"style":20953},[350],[15,43636,43637,43640],{"style":482},[15,43638],{"className":43639,"style":358},[357],[15,43641,43643],{"className":43642},[362,363,364,365],[15,43644,43646],{"className":43645},[61,365],[15,43647,43649],{"className":43648},[61,365],[15,43650,43652],{"className":43651},[61,365],[15,43653,24661],{"className":43654,"style":24695},[61,3162,365],[15,43656,497],{"className":43657},[496],[15,43659,43661],{"className":43660},[346],[15,43662,43664],{"className":43663,"style":504},[350],[15,43665],{},[15,43667,14810],{"className":43668},[146],[15,43670,22634],{"className":43671,"style":22746},[61,122],[15,43673,741],{"className":43674},[146],[15,43676,43678],{"className":43677},[61],[15,43679,43681],{"className":43680},[61],[15,43682,24661],{"className":43683,"style":24695},[61,3162],[15,43685,43268],{"className":43686},[178],[31523,43688,43689,43692,43693,9342,43721,43755],{},[661,43690,43691],{},"认知不确定性（epistemic uncertainty）","：二者的差，即 ",[15,43694,43696,43709],{"className":43695,"translate":19},[18],[15,43697,43699],{"className":43698},[23],[25,43700,43701],{"xmlns":27},[29,43702,43703,43707],{},[32,43704,43705],{},[35,43706,79],{},[40,43708,79],{"encoding":42},[15,43710,43712],{"className":43711,"ariaHidden":48},[47],[15,43713,43715,43718],{"className":43714},[52],[15,43716],{"className":43717,"style":9428},[56],[15,43719,79],{"className":43720,"style":123},[61,122],[15,43722,43724,43737],{"className":43723,"translate":19},[18],[15,43725,43727],{"className":43726},[23],[25,43728,43729],{"xmlns":27},[29,43730,43731,43735],{},[32,43732,43733],{},[35,43734,24661],{"mathvariant":3101},[40,43736,24749],{"encoding":42},[15,43738,43740],{"className":43739,"ariaHidden":48},[47],[15,43741,43743,43746],{"className":43742},[52],[15,43744],{"className":43745,"style":2706},[56],[15,43747,43749],{"className":43748},[61],[15,43750,43752],{"className":43751},[61],[15,43753,24661],{"className":43754,"style":24695},[61,3162]," 之间的互信息。",[882,43757,43759],{"title":43758},"命题 3.5（Dirichlet 族内的不确定性分解）",[11,43760,43761,43762,941,43946,40612,44038,44325,44724],{},"记 ",[15,43763,43765,43799],{"className":43764,"translate":19},[18],[15,43766,43768],{"className":43767},[23],[25,43769,43770],{"xmlns":27},[29,43771,43772,43796],{},[32,43773,43774,43780,43782,43788,43790],{},[266,43775,43776,43778],{},[35,43777,11],{},[35,43779,272],{},[81,43781,244],{},[266,43783,43784,43786],{},[35,43785,3102],{},[35,43787,272],{},[35,43789,3018],{"mathvariant":231},[266,43791,43792,43794],{},[35,43793,3102],{},[89,43795,278],{},[40,43797,43798],{"encoding":42},"p_k = \\alpha_k\u002F\\alpha_0",[15,43800,43802,43857],{"className":43801,"ariaHidden":48},[47],[15,43803,43805,43808,43848,43851,43854],{"className":43804},[52],[15,43806],{"className":43807,"style":9428},[56],[15,43809,43811,43814],{"className":43810},[61],[15,43812,11],{"className":43813},[61,122],[15,43815,43817],{"className":43816},[338],[15,43818,43820,43840],{"className":43819},[342,472],[15,43821,43823,43837],{"className":43822},[346],[15,43824,43826],{"className":43825,"style":479},[350],[15,43827,43828,43831],{"style":482},[15,43829],{"className":43830,"style":358},[357],[15,43832,43834],{"className":43833},[362,363,364,365],[15,43835,272],{"className":43836,"style":492},[61,122,365],[15,43838,497],{"className":43839},[496],[15,43841,43843],{"className":43842},[346],[15,43844,43846],{"className":43845,"style":504},[350],[15,43847],{},[15,43849],{"className":43850,"style":128},[127],[15,43852,244],{"className":43853},[132],[15,43855],{"className":43856,"style":128},[127],[15,43858,43860,43863,43903,43906],{"className":43859},[52],[15,43861],{"className":43862,"style":142},[56],[15,43864,43866,43869],{"className":43865},[61],[15,43867,3102],{"className":43868,"style":3187},[61,122],[15,43870,43872],{"className":43871},[338],[15,43873,43875,43895],{"className":43874},[342,472],[15,43876,43878,43892],{"className":43877},[346],[15,43879,43881],{"className":43880,"style":479},[350],[15,43882,43883,43886],{"style":3202},[15,43884],{"className":43885,"style":358},[357],[15,43887,43889],{"className":43888},[362,363,364,365],[15,43890,272],{"className":43891,"style":492},[61,122,365],[15,43893,497],{"className":43894},[496],[15,43896,43898],{"className":43897},[346],[15,43899,43901],{"className":43900,"style":504},[350],[15,43902],{},[15,43904,3018],{"className":43905},[61],[15,43907,43909,43912],{"className":43908},[61],[15,43910,3102],{"className":43911,"style":3187},[61,122],[15,43913,43915],{"className":43914},[338],[15,43916,43918,43938],{"className":43917},[342,472],[15,43919,43921,43935],{"className":43920},[346],[15,43922,43924],{"className":43923,"style":1212},[350],[15,43925,43926,43929],{"style":3202},[15,43927],{"className":43928,"style":358},[357],[15,43930,43932],{"className":43931},[362,363,364,365],[15,43933,278],{"className":43934},[61,365],[15,43936,497],{"className":43937},[496],[15,43939,43941],{"className":43940},[346],[15,43942,43944],{"className":43943,"style":504},[350],[15,43945],{},[15,43947,43949,43971],{"className":43948,"translate":19},[18],[15,43950,43952],{"className":43951},[23],[25,43953,43954],{"xmlns":27},[29,43955,43956,43968],{},[32,43957,43958,43960,43962],{},[35,43959,40230],{},[81,43961,244],{},[266,43963,43964,43966],{},[35,43965,3102],{},[89,43967,278],{},[40,43969,43970],{"encoding":42},"S = \\alpha_0",[15,43972,43974,43992],{"className":43973,"ariaHidden":48},[47],[15,43975,43977,43980,43983,43986,43989],{"className":43976},[52],[15,43978],{"className":43979,"style":57},[56],[15,43981,40230],{"className":43982,"style":40302},[61,122],[15,43984],{"className":43985,"style":128},[127],[15,43987,244],{"className":43988},[132],[15,43990],{"className":43991,"style":128},[127],[15,43993,43995,43998],{"className":43994},[52],[15,43996],{"className":43997,"style":2509},[56],[15,43999,44001,44004],{"className":44000},[61],[15,44002,3102],{"className":44003,"style":3187},[61,122],[15,44005,44007],{"className":44006},[338],[15,44008,44010,44030],{"className":44009},[342,472],[15,44011,44013,44027],{"className":44012},[346],[15,44014,44016],{"className":44015,"style":1212},[350],[15,44017,44018,44021],{"style":3202},[15,44019],{"className":44020,"style":358},[357],[15,44022,44024],{"className":44023},[362,363,364,365],[15,44025,278],{"className":44026},[61,365],[15,44028,497],{"className":44029},[496],[15,44031,44033],{"className":44032},[346],[15,44034,44036],{"className":44035,"style":504},[350],[15,44037],{},[15,44039,44041,44105],{"className":44040,"translate":19},[18],[15,44042,44044],{"className":44043},[23],[25,44045,44046],{"xmlns":27},[29,44047,44048,44102],{},[32,44049,44050,44052,44054,44056,44058,44060,44062,44064,44066,44068,44070,44072,44086,44092,44094,44096],{},[35,44051,22634],{},[81,44053,14810],{"stretchy":86},[35,44055,21273],{"mathvariant":258},[81,44057,741],{"stretchy":86},[35,44059,79],{},[81,44061,3710],{},[35,44063,1027],{"mathvariant":251},[81,44065,750],{"stretchy":86},[81,44067,14819],{"stretchy":86},[81,44069,244],{},[81,44071,239],{},[289,44073,44074,44076,44084],{},[81,44075,293],{},[32,44077,44078,44080,44082],{},[35,44079,272],{},[81,44081,244],{},[89,44083,91],{},[35,44085,102],{},[266,44087,44088,44090],{},[35,44089,11],{},[35,44091,272],{},[35,44093,22645],{},[81,44095,1545],{},[266,44097,44098,44100],{},[35,44099,11],{},[35,44101,272],{},[40,44103,44104],{"encoding":42},"H[\\mathbb{P}(y \\mid \\mathbf{x})] = -\\sum_{k=1}^K p_k \\log p_k",[15,44106,44108,44138,44159],{"className":44107,"ariaHidden":48},[47],[15,44109,44111,44114,44117,44120,44123,44126,44129,44132,44135],{"className":44110},[52],[15,44112],{"className":44113,"style":142},[56],[15,44115,22634],{"className":44116,"style":22746},[61,122],[15,44118,14810],{"className":44119},[146],[15,44121,21273],{"className":44122},[61,427],[15,44124,741],{"className":44125},[146],[15,44127,79],{"className":44128,"style":123},[61,122],[15,44130],{"className":44131,"style":128},[127],[15,44133,3710],{"className":44134},[132],[15,44136],{"className":44137,"style":128},[127],[15,44139,44141,44144,44147,44150,44153,44156],{"className":44140},[52],[15,44142],{"className":44143,"style":142},[56],[15,44145,1027],{"className":44146},[61,411],[15,44148,43268],{"className":44149},[178],[15,44151],{"className":44152,"style":128},[127],[15,44154,244],{"className":44155},[132],[15,44157],{"className":44158,"style":128},[127],[15,44160,44162,44165,44168,44171,44231,44234,44274,44277,44282,44285],{"className":44161},[52],[15,44163],{"className":44164,"style":3467},[56],[15,44166,239],{"className":44167},[61],[15,44169],{"className":44170,"style":157},[127],[15,44172,44174,44177],{"className":44173},[531],[15,44175,293],{"className":44176,"style":537},[531,535,536],[15,44178,44180],{"className":44179},[338],[15,44181,44183,44223],{"className":44182},[342,472],[15,44184,44186,44220],{"className":44185},[346],[15,44187,44189,44209],{"className":44188,"style":550},[350],[15,44190,44191,44194],{"style":553},[15,44192],{"className":44193,"style":358},[357],[15,44195,44197],{"className":44196},[362,363,364,365],[15,44198,44200,44203,44206],{"className":44199},[61,365],[15,44201,272],{"className":44202,"style":492},[61,122,365],[15,44204,244],{"className":44205},[132,365],[15,44207,91],{"className":44208},[61,365],[15,44210,44211,44214],{"style":574},[15,44212],{"className":44213,"style":358},[357],[15,44215,44217],{"className":44216},[362,363,364,365],[15,44218,102],{"className":44219,"style":174},[61,122,365],[15,44221,497],{"className":44222},[496],[15,44224,44226],{"className":44225},[346],[15,44227,44229],{"className":44228,"style":593},[350],[15,44230],{},[15,44232],{"className":44233,"style":157},[127],[15,44235,44237,44240],{"className":44236},[61],[15,44238,11],{"className":44239},[61,122],[15,44241,44243],{"className":44242},[338],[15,44244,44246,44266],{"className":44245},[342,472],[15,44247,44249,44263],{"className":44248},[346],[15,44250,44252],{"className":44251,"style":479},[350],[15,44253,44254,44257],{"style":482},[15,44255],{"className":44256,"style":358},[357],[15,44258,44260],{"className":44259},[362,363,364,365],[15,44261,272],{"className":44262,"style":492},[61,122,365],[15,44264,497],{"className":44265},[496],[15,44267,44269],{"className":44268},[346],[15,44270,44272],{"className":44271,"style":504},[350],[15,44273],{},[15,44275],{"className":44276,"style":157},[127],[15,44278,22774,44280],{"className":44279},[531],[15,44281,22777],{"style":15608},[15,44283],{"className":44284,"style":157},[127],[15,44286,44288,44291],{"className":44287},[61],[15,44289,11],{"className":44290},[61,122],[15,44292,44294],{"className":44293},[338],[15,44295,44297,44317],{"className":44296},[342,472],[15,44298,44300,44314],{"className":44299},[346],[15,44301,44303],{"className":44302,"style":479},[350],[15,44304,44305,44308],{"style":482},[15,44306],{"className":44307,"style":358},[357],[15,44309,44311],{"className":44310},[362,363,364,365],[15,44312,272],{"className":44313,"style":492},[61,122,365],[15,44315,497],{"className":44316},[496],[15,44318,44320],{"className":44319},[346],[15,44321,44323],{"className":44322,"style":504},[350],[15,44324],{},[15,44326,44328,44412],{"className":44327,"translate":19},[18],[15,44329,44331],{"className":44330},[23],[25,44332,44333],{"xmlns":27},[29,44334,44335,44409],{},[32,44336,44337,44343,44345,44347,44349,44351,44353,44355,44357,44359,44361,44363,44365,44367,44369,44371,44385,44391,44393,44395,44397,44403,44405,44407],{},[266,44338,44339,44341],{},[35,44340,14807],{"mathvariant":258},[35,44342,24661],{"mathvariant":3101},[81,44344,14810],{"stretchy":86},[35,44346,22634],{},[81,44348,741],{"stretchy":86},[35,44350,24661],{"mathvariant":3101},[81,44352,750],{"stretchy":86},[81,44354,14819],{"stretchy":86},[81,44356,244],{},[35,44358,22676],{},[81,44360,741],{"stretchy":86},[35,44362,40230],{},[81,44364,7756],{},[89,44366,91],{},[81,44368,750],{"stretchy":86},[81,44370,239],{},[289,44372,44373,44375,44383],{},[81,44374,293],{},[32,44376,44377,44379,44381],{},[35,44378,272],{},[81,44380,244],{},[89,44382,91],{},[35,44384,102],{},[266,44386,44387,44389],{},[35,44388,11],{},[35,44390,272],{},[282,44392,4796],{},[35,44394,22676],{},[81,44396,741],{"stretchy":86},[266,44398,44399,44401],{},[35,44400,3102],{},[35,44402,272],{},[81,44404,7756],{},[89,44406,91],{},[81,44408,750],{"stretchy":86},[40,44410,44411],{"encoding":42},"\\mathbb{E}_{\\boldsymbol{\\pi}}[H(\\boldsymbol{\\pi})] = \\psi(S + 1) - \\sum_{k=1}^K p_k \\,\\psi(\\alpha_k + 1)",[15,44413,44415,44500,44524,44545,44712],{"className":44414,"ariaHidden":48},[47],[15,44416,44418,44421,44470,44473,44476,44479,44488,44491,44494,44497],{"className":44417},[52],[15,44419],{"className":44420,"style":142},[56],[15,44422,44424,44427],{"className":44423},[61],[15,44425,14807],{"className":44426},[61,427],[15,44428,44430],{"className":44429},[338],[15,44431,44433,44462],{"className":44432},[342,472],[15,44434,44436,44459],{"className":44435},[346],[15,44437,44439],{"className":44438,"style":20953},[350],[15,44440,44441,44444],{"style":482},[15,44442],{"className":44443,"style":358},[357],[15,44445,44447],{"className":44446},[362,363,364,365],[15,44448,44450],{"className":44449},[61,365],[15,44451,44453],{"className":44452},[61,365],[15,44454,44456],{"className":44455},[61,365],[15,44457,24661],{"className":44458,"style":24695},[61,3162,365],[15,44460,497],{"className":44461},[496],[15,44463,44465],{"className":44464},[346],[15,44466,44468],{"className":44467,"style":504},[350],[15,44469],{},[15,44471,14810],{"className":44472},[146],[15,44474,22634],{"className":44475,"style":22746},[61,122],[15,44477,741],{"className":44478},[146],[15,44480,44482],{"className":44481},[61],[15,44483,44485],{"className":44484},[61],[15,44486,24661],{"className":44487,"style":24695},[61,3162],[15,44489,43268],{"className":44490},[178],[15,44492],{"className":44493,"style":128},[127],[15,44495,244],{"className":44496},[132],[15,44498],{"className":44499,"style":128},[127],[15,44501,44503,44506,44509,44512,44515,44518,44521],{"className":44502},[52],[15,44504],{"className":44505,"style":142},[56],[15,44507,22676],{"className":44508,"style":123},[61,122],[15,44510,741],{"className":44511},[146],[15,44513,40230],{"className":44514,"style":40302},[61,122],[15,44516],{"className":44517,"style":772},[127],[15,44519,7756],{"className":44520},[375],[15,44522],{"className":44523,"style":772},[127],[15,44525,44527,44530,44533,44536,44539,44542],{"className":44526},[52],[15,44528],{"className":44529,"style":142},[56],[15,44531,91],{"className":44532},[61],[15,44534,750],{"className":44535},[178],[15,44537],{"className":44538,"style":772},[127],[15,44540,239],{"className":44541},[375],[15,44543],{"className":44544,"style":772},[127],[15,44546,44548,44551,44611,44614,44654,44657,44660,44663,44703,44706,44709],{"className":44547},[52],[15,44549],{"className":44550,"style":3467},[56],[15,44552,44554,44557],{"className":44553},[531],[15,44555,293],{"className":44556,"style":537},[531,535,536],[15,44558,44560],{"className":44559},[338],[15,44561,44563,44603],{"className":44562},[342,472],[15,44564,44566,44600],{"className":44565},[346],[15,44567,44569,44589],{"className":44568,"style":550},[350],[15,44570,44571,44574],{"style":553},[15,44572],{"className":44573,"style":358},[357],[15,44575,44577],{"className":44576},[362,363,364,365],[15,44578,44580,44583,44586],{"className":44579},[61,365],[15,44581,272],{"className":44582,"style":492},[61,122,365],[15,44584,244],{"className":44585},[132,365],[15,44587,91],{"className":44588},[61,365],[15,44590,44591,44594],{"style":574},[15,44592],{"className":44593,"style":358},[357],[15,44595,44597],{"className":44596},[362,363,364,365],[15,44598,102],{"className":44599,"style":174},[61,122,365],[15,44601,497],{"className":44602},[496],[15,44604,44606],{"className":44605},[346],[15,44607,44609],{"className":44608,"style":593},[350],[15,44610],{},[15,44612],{"className":44613,"style":157},[127],[15,44615,44617,44620],{"className":44616},[61],[15,44618,11],{"className":44619},[61,122],[15,44621,44623],{"className":44622},[338],[15,44624,44626,44646],{"className":44625},[342,472],[15,44627,44629,44643],{"className":44628},[346],[15,44630,44632],{"className":44631,"style":479},[350],[15,44633,44634,44637],{"style":482},[15,44635],{"className":44636,"style":358},[357],[15,44638,44640],{"className":44639},[362,363,364,365],[15,44641,272],{"className":44642,"style":492},[61,122,365],[15,44644,497],{"className":44645},[496],[15,44647,44649],{"className":44648},[346],[15,44650,44652],{"className":44651,"style":504},[350],[15,44653],{},[15,44655],{"className":44656,"style":157},[127],[15,44658,22676],{"className":44659,"style":123},[61,122],[15,44661,741],{"className":44662},[146],[15,44664,44666,44669],{"className":44665},[61],[15,44667,3102],{"className":44668,"style":3187},[61,122],[15,44670,44672],{"className":44671},[338],[15,44673,44675,44695],{"className":44674},[342,472],[15,44676,44678,44692],{"className":44677},[346],[15,44679,44681],{"className":44680,"style":479},[350],[15,44682,44683,44686],{"style":3202},[15,44684],{"className":44685,"style":358},[357],[15,44687,44689],{"className":44688},[362,363,364,365],[15,44690,272],{"className":44691,"style":492},[61,122,365],[15,44693,497],{"className":44694},[496],[15,44696,44698],{"className":44697},[346],[15,44699,44701],{"className":44700,"style":504},[350],[15,44702],{},[15,44704],{"className":44705,"style":772},[127],[15,44707,7756],{"clas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\\boldsymbol{\\pi}) = H[\\mathbb{P}(y \\mid \\mathbf{x})] - \\mathbb{E}_{\\boldsymbol{\\pi}}[H(\\boldsymbol{\\pi})] = \\sum_{k=1}^K p_k\\big(\\psi(\\alpha_k + 1) - \\log p_k\\big) - \\psi(S + 1)",[15,44861,44863,44905,44935,44956,45041,45212,45233,45303,45327],{"className":44862,"ariaHidden":48},[47],[15,44864,44866,44869,44872,44875,44878,44881,44884,44893,44896,44899,44902],{"className":44865},[52],[15,44867],{"className":44868,"style":142},[56],[15,44870,28898],{"className":44871,"style":28937},[61,122],[15,44873,741],{"className":44874},[146],[15,44876,79],{"className":44877,"style":123},[61,122],[15,44879,11294],{"className":44880},[153],[15,44882],{"className":44883,"style":157},[127],[15,44885,44887],{"className":44886},[61],[15,44888,44890],{"className":44889},[61],[15,44891,24661],{"className":44892,"style":24695},[61,3162],[15,44894,750],{"className":44895},[178],[15,44897],{"className":44898,"style":128},[127],[15,44900,244],{"className":44901},[132],[15,44903],{"className":44904,"style":128},[127],[15,44906,44908,44911,44914,44917,44920,44923,44926,44929,44932],{"className":44907},[52],[15,44909],{"className":44910,"style":142},[56],[15,44912,22634],{"className":44913,"style":22746},[61,122],[15,44915,14810],{"className":44916},[146],[15,44918,21273],{"className":44919},[61,427],[15,44921,741],{"className":44922},[146],[15,44924,79],{"className":44925,"style":123},[61,122],[15,44927],{"className":44928,"style":128},[127],[15,44930,3710],{"className":44931},[132],[15,44933],{"className":44934,"style":128},[127],[15,44936,44938,44941,44944,44947,44950,44953],{"className":44937},[52],[15,44939],{"className":44940,"style":142},[56],[15,44942,1027],{"className":44943},[61,411],[15,44945,43268],{"className":44946},[178],[15,44948],{"className":44949,"style":772},[127],[15,44951,239],{"className":44952},[375],[15,44954],{"className":44955,"style":772},[127],[15,44957,44959,44962,45011,45014,45017,45020,45029,45032,45035,45038],{"className":44958},[52],[15,44960],{"className":44961,"style":142},[56],[15,44963,44965,44968],{"className":44964},[61],[15,44966,14807],{"className":44967},[61,427],[15,44969,44971],{"className":44970},[338],[15,44972,44974,45003],{"className":44973},[342,472],[15,44975,44977,45000],{"className":44976},[346],[15,44978,44980],{"className":44979,"style":20953},[350],[15,44981,44982,44985],{"style":482},[15,44983],{"className":44984,"style":358},[357],[15,44986,44988],{"className":44987},[362,363,364,365],[15,44989,44991],{"className":44990},[61,365],[15,44992,44994],{"className":44993},[61,365],[15,44995,44997],{"className":44996},[61,365],[15,44998,24661],{"className":44999,"style":24695},[61,3162,365],[15,45001,497],{"className":45002},[496],[15,45004,45006],{"className":45005},[346],[15,45007,45009],{"className":45008,"style":504},[350],[15,45010],{},[15,45012,14810],{"className":45013},[146],[15,45015,22634],{"className":45016,"style":22746},[61,122],[15,45018,741],{"className":45019},[146],[15,45021,45023],{"className":45022},[61],[15,45024,45026],{"className":45025},[61],[15,45027,24661],{"className":45028,"style":24695},[61,3162],[15,45030,43268],{"className":45031},[178],[15,45033],{"className":45034,"style":128},[127],[15,45036,244],{"className":45037},[132],[15,45039],{"className":45040,"style":128},[127],[15,45042,45044,45048,45108,45111,45151,45157,45160,45163,45203,45206,45209],{"className":45043},[52],[15,45045],{"className":45046,"style":45047},[56],"height:1.3312em;vertical-align:-0.35em;",[15,45049,45051,45054],{"className":45050},[531],[15,45052,293],{"className":45053,"style":537},[531,535,536],[15,45055,45057],{"className":45056},[338],[15,45058,45060,45100],{"className":45059},[342,472],[15,45061,45063,45097],{"className":45062},[346],[15,45064,45066,45086],{"className":45065,"style":550},[350],[15,45067,45068,45071],{"style":553},[15,45069],{"className":45070,"style":358},[357],[15,45072,45074],{"className":45073},[362,363,364,365],[15,45075,45077,45080,45083],{"className":45076},[61,365],[15,45078,272],{"className":45079,"style":492},[61,122,365],[15,45081,244],{"className":45082},[132,365],[15,45084,91],{"className":45085},[61,365],[15,45087,45088,45091],{"style":574},[15,45089],{"className":45090,"style":358},[357],[15,45092,45094],{"className":45093},[362,363,364,365],[15,45095,102],{"className":45096,"style":174},[61,122,365],[15,45098,497],{"className":45099},[496],[15,45101,45103],{"className":45102},[346],[15,45104,45106],{"className":45105,"style":593},[350],[15,45107],{},[15,45109],{"className":45110,"style":157},[127],[15,45112,45114,45117],{"className":45113},[61],[15,45115,11],{"className":45116},[61,122],[15,45118,45120],{"className":45119},[338],[15,45121,45123,45143],{"className":45122},[342,472],[15,45124,45126,45140],{"className":45125},[346],[15,45127,45129],{"className":45128,"style":479},[350],[15,45130,45131,45134],{"style":482},[15,45132],{"className":45133,"style":358},[357],[15,45135,45137],{"className":45136},[362,363,364,365],[15,45138,272],{"className":45139,"style":492},[61,122,365],[15,45141,497],{"className":45142},[496],[15,45144,45146],{"className":45145},[346],[15,45147,45149],{"className":45148,"style":504},[350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关键在于 ",[15,45345,45347,45393],{"className":45346,"translate":19},[18],[15,45348,45350],{"className":45349},[23],[25,45351,45352],{"xmlns":27},[29,45353,45354,45390],{},[32,45355,45356,45358,45360,45362,45364,45366,45368,45374,45380,45382,45384],{},[35,45357,22634],{},[81,45359,741],{"stretchy":86},[35,45361,24661],{"mathvariant":3101},[81,45363,750],{"stretchy":86},[81,45365,244],{},[81,45367,239],{},[266,45369,45370,45372],{},[81,45371,293],{},[35,45373,272],{},[266,45375,45376,45378],{},[35,45377,24661],{},[35,45379,272],{},[35,45381,22645],{},[81,45383,1545],{},[266,45385,45386,45388],{},[35,45387,24661],{},[35,45389,272],{},[40,45391,45392],{"encoding":42},"H(\\boldsymbol{\\pi}) = -\\sum_k \\pi_k \\log 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= -\\sum_k \\mathbb{E}[\\pi_k \\log \\pi_k]",[15,45637,45639,45678],{"className":45638,"ariaHidden":48},[47],[15,45640,45642,45645,45648,45651,45654,45657,45666,45669,45672,45675],{"className":45641},[52],[15,45643],{"className":45644,"style":142},[56],[15,45646,14807],{"className":45647},[61,427],[15,45649,14810],{"className":45650},[146],[15,45652,22634],{"className":45653,"style":22746},[61,122],[15,45655,741],{"className":45656},[146],[15,45658,45660],{"className":45659},[61],[15,45661,45663],{"className":45662},[61],[15,45664,24661],{"className":45665,"style":24695},[61,3162],[15,45667,43268],{"className":45668},[178],[15,45670],{"className":45671,"style":128},[127],[15,45673,244],{"className":45674},[132],[15,45676],{"className":45677,"style":128},[127],[15,45679,45681,45684,45687,45690,45730,45733,45736,45739,45779,45782,45787,45790,45830],{"className":45680},[52],[15,45682],{"className":45683,"style":6045},[56],[15,45685,239],{"className":45686},[61],[15,45688],{"className":45689,"style":157},[127],[15,45691,45693,45696],{"className":45692},[531],[15,45694,293],{"className":45695,"style":537},[531,535,536],[15,45697,45699],{"className":45698},[338],[15,45700,45702,45722],{"className":45701},[342,472],[15,45703,45705,45719],{"className":45704},[346],[15,45706,45708],{"className":45707,"style":29964},[350],[15,45709,45710,45713],{"style":553},[15,45711],{"className":45712,"style":358},[357],[15,45714,45716],{"className":45715},[362,363,364,365],[15,45717,272],{"className":45718,"style":492},[61,122,365],[15,45720,497],{"className":45721},[496],[15,45723,45725],{"className":45724},[346],[15,45726,45728],{"className":45727,"style":593},[350],[15,45729],{},[15,45731],{"className":45732,"style":157},[127],[15,45734,14807],{"className":45735},[61,427],[15,45737,14810],{"className":45738},[146],[15,45740,45742,45745],{"className":45741},[61],[15,45743,24661],{"className":45744,"style":123},[61,122],[15,45746,45748],{"className":45747},[338],[15,45749,45751,45771],{"className":45750},[342,472],[15,45752,45754,45768],{"className":45753},[346],[15,45755,45757],{"className":45756,"style":479},[350],[15,45758,45759,45762],{"style":27013},[15,45760],{"className":45761,"style":358},[357],[15,45763,45765],{"className":45764},[362,363,364,365],[15,45766,272],{"className":45767,"style":492},[61,122,365],[15,45769,497],{"className":45770},[496],[15,45772,45774],{"className":45773},[346],[15,45775,45777],{"className":45776,"style":504},[350],[15,45778],{},[15,45780],{"className":45781,"style":157},[127],[15,45783,22774,45785],{"className":45784},[531],[15,45786,22777],{"style":15608},[15,45788],{"className":45789,"style":157},[127],[15,45791,45793,45796],{"className":45792},[61],[15,45794,24661],{"className":45795,"style":123},[61,122],[15,45797,45799],{"className":45798},[338],[15,45800,45802,45822],{"className":45801},[342,472],[15,45803,45805,45819],{"className":45804},[346],[15,45806,45808],{"className":45807,"style":479},[350],[15,45809,45810,45813],{"style":27013},[15,45811],{"className":45812,"style":358},[357],[15,45814,45816],{"className":45815},[362,363,364,365],[15,45817,272],{"className":45818,"style":492},[61,122,365],[15,45820,497],{"className":45821},[496],[15,45823,45825],{"className":45824},[346],[15,45826,45828],{"className":45827,"style":504},[350],[15,45829],{},[15,45831,14819],{"className":45832},[178],"。由 1.4 节的边缘 Beta 性，",[15,45835,45837,45883],{"className":45836,"translate":19},[18],[15,45838,45840],{"className":45839},[23],[25,45841,45842],{"xmlns":27},[29,45843,45844,45880],{},[32,45845,45846,45852,45854,45856,45858,45860,45866,45868,45870,45872,45878],{},[266,45847,45848,45850],{},[35,45849,24661],{},[35,45851,272],{},[81,45853,4179],{},[35,45855,12947],{"mathvariant":231},[81,45857,1545],{},[81,45859,741],{"stretchy":86},[266,45861,45862,45864],{},[35,45863,3102],{},[35,45865,272],{},[81,45867,94],{"separator":48},[35,45869,40230],{},[81,45871,239],{},[266,45873,45874,45876],{},[35,45875,3102],{},[35,45877,272],{},[81,45879,750],{"stretchy":86},[40,45881,45882],{"encoding":42},"\\pi_k \\sim \\operatorname{Beta}(\\alpha_k, S - \\alpha_k)",[15,45884,45886,45941,46014],{"className":45885,"ariaHidden":48},[47],[15,45887,45889,45892,45932,45935,45938],{"className":45888},[52],[15,45890],{"className":45891,"style":2509},[56],[15,45893,45895,45898],{"className":45894},[61],[15,45896,24661],{"className":45897,"style":123},[61,122],[15,45899,45901],{"className":45900},[338],[15,45902,45904,45924],{"className":45903},[342,472],[15,45905,45907,45921],{"className":45906},[346],[15,45908,45910],{"className":45909,"style":479},[350],[15,45911,45912,45915],{"style":27013},[15,45913],{"className":45914,"style":358},[357],[15,45916,45918],{"className":45917},[362,363,364,365],[15,45919,272],{"className":45920,"style":492},[61,122,365],[15,45922,497],{"className":45923},[496],[15,45925,45927],{"className":45926},[346],[15,45928,45930],{"className":45929,"style":504},[350],[15,45931],{},[15,45933],{"className":45934,"style":128},[127],[15,45936,4179],{"className":45937},[132],[15,45939],{"className":45940,"style":128},[127],[15,45942,45944,45947,45953,45956,45996,45999,46002,46005,46008,46011],{"className":45943},[52],[15,45945],{"className":45946,"style":142},[56],[15,45948,45950],{"className":45949},[531],[15,45951,12947],{"className":45952},[61,1620],[15,45954,741],{"className":45955},[146],[15,45957,45959,45962],{"className":45958},[61],[15,45960,3102],{"className":45961,"style":3187},[61,122],[15,45963,45965],{"className":45964},[338],[15,45966,45968,45988],{"className":45967},[342,472],[15,45969,45971,45985],{"className":45970},[346],[15,45972,45974],{"className":45973,"style":479},[350],[15,45975,45976,45979],{"style":3202},[15,45977],{"className":45978,"style":358},[357],[15,45980,45982],{"className":45981},[362,363,364,365],[15,45983,272],{"className":45984,"style":492},[61,122,365],[15,45986,497],{"className":45987},[496],[15,45989,45991],{"className":45990},[346],[15,45992,45994],{"className":45993,"style":504},[350],[15,45995],{},[15,45997,94],{"className":45998},[153],[15,46000],{"className":46001,"style":157},[127],[15,46003,40230],{"className":46004,"style":40302},[61,122],[15,46006],{"className":46007,"style":772},[127],[15,46009,239],{"className":46010},[375],[15,46012],{"className":46013,"style":772},[127],[15,46015,46017,46020,46060],{"className":46016},[52],[15,46018],{"className":46019,"style":142},[56],[15,46021,46023,46026],{"className":46022},[61],[15,46024,3102],{"className":46025,"style":3187},[61,122],[15,46027,46029],{"className":46028},[338],[15,46030,46032,46052],{"className":46031},[342,472],[15,46033,46035,46049],{"className":46034},[346],[15,46036,46038],{"className":46037,"style":479},[350],[15,46039,46040,46043],{"style":3202},[15,46041],{"className":46042,"style":358},[357],[15,46044,46046],{"className":46045},[362,363,364,365],[15,46047,272],{"className":46048,"style":492},[61,122,365],[15,46050,497],{"className":46051},[496],[15,46053,46055],{"className":46054},[346],[15,46056,46058],{"className":46057,"style":504},[350],[15,46059],{},[15,46061,750],{"className":46062},[178],"，而对该 Beta 分布有恒等式 ",[15,46065,46067,46141],{"className":46066,"translate":19},[18],[15,46068,46070],{"className":46069},[23],[25,46071,46072],{"xmlns":27},[29,46073,46074,46138],{},[32,46075,46076,46078,46080,46082,46084,46086,46088,46090,46092,46104,46106,46108,46110,46112,46114,46116,46118,46120,46122,46124,46126,46128,46130,46132,46134,46136],{},[35,46077,14807],{"mathvariant":258},[81,46079,14810],{"stretchy":86},[35,46081,38],{},[35,46083,22645],{},[81,46085,1545],{},[35,46087,38],{},[81,46089,14819],{"stretchy":86},[81,46091,244],{},[3718,46093,46094,46096],{},[35,46095,33406],{},[32,46097,46098,46100,46102],{},[35,46099,33406],{},[81,46101,7756],{},[35,46103,33396],{},[81,46105,741],{"fence":86,"stretchy":48,"minsize":44812,"maxsize":44812},[35,46107,22676],{},[81,46109,741],{"stretchy":86},[35,46111,33406],{},[81,46113,7756],{},[89,46115,91],{},[81,46117,750],{"stretchy":86},[81,46119,239],{},[35,46121,22676],{},[81,46123,741],{"stretchy":86},[35,46125,33406],{},[81,46127,7756],{},[35,46129,33396],{},[81,46131,7756],{},[89,46133,91],{},[81,46135,750],{"stretchy":86},[81,46137,750],{"fence":86,"stretchy":48,"minsize":44812,"maxsize":44812},[40,46139,46140],{"encoding":42},"\\mathbb{E}[X \\log X] = \\frac{a}{a+b}\\big(\\psi(a+1) - \\psi(a+b+1)\\big)",[15,46142,46144,46185,46292,46313,46337,46356],{"className":46143,"ariaHidden":48},[47],[15,46145,46147,46150,46153,46156,46159,46162,46167,46170,46173,46176,46179,46182],{"className":46146},[52],[15,46148],{"className":46149,"style":142},[56],[15,46151,14807],{"className":46152},[61,427],[15,46154,14810],{"className":46155},[146],[15,46157,38],{"className":46158,"style":28937},[61,122],[15,46160],{"className":46161,"style":157},[127],[15,46163,22774,46165],{"className":46164},[531],[15,46166,22777],{"style":15608},[15,46168],{"className":46169,"style":157},[127],[15,46171,38],{"className":46172,"style":28937},[61,122],[15,46174,14819],{"className":46175},[178],[15,46177],{"className":46178,"style":128},[127],[15,46180,244],{"className":46181},[132],[15,46183],{"className":46184,"style":128},[127],[15,46186,46188,46192,46268,46274,46277,46280,46283,46286,46289],{"className":46187},[52],[15,46189],{"className":46190,"style":46191},[56],"height:1.2533em;vertical-align:-0.4033em;",[15,46193,46195,46198,46265],{"className":46194},[61],[15,46196],{"className":46197},[146,3837],[15,46199,46201],{"className":46200},[3718],[15,46202,46204,46256],{"className":46203},[342,472],[15,46205,46207,46253],{"className":46206},[346],[15,46208,46211,46231,46239],{"className":46209,"style":46210},[350],"height:0.6954em;",[15,46212,46213,46216],{"style":3853},[15,46214],{"className":46215,"style":3857},[357],[15,46217,46219],{"className":46218},[362,363,364,365],[15,46220,46222,46225,46228],{"className":46221},[61,365],[15,46223,33406],{"className":46224},[61,122,365],[15,46226,7756],{"className":46227},[375,365],[15,46229,33396],{"className":46230},[61,122,365],[15,46232,46233,46236],{"style":3885},[15,46234],{"className":46235,"style":3857},[357],[15,46237],{"className":46238,"style":3893},[3892],[15,46240,46241,46244],{"style":3896},[15,46242],{"className":46243,"style":3857},[357],[15,46245,46247],{"className":46246},[362,363,364,365],[15,46248,46250],{"className":46249},[61,365],[15,46251,33406],{"className":46252},[61,122,365],[15,46254,497],{"className":46255},[496],[15,46257,46259],{"className":46258},[346],[15,46260,46263],{"className":46261,"style":46262},[350],"height:0.4033em;",[15,46264],{},[15,46266],{"className":46267},[178,3837],[15,46269,46271],{"className":46270},[61],[15,46272,741],{"className":46273},[406,4063],[15,46275,22676],{"className":46276,"style":123},[61,122],[15,46278,741],{"className":46279},[146],[15,46281,33406],{"className":46282},[61,122],[15,46284],{"className":46285,"style":772},[127],[15,46287,7756],{"className":46288},[375],[15,46290],{"className":46291,"style":772},[127],[15,46293,46295,46298,46301,46304,46307,46310],{"className":46294},[52],[15,46296],{"className":46297,"style":142},[56],[15,46299,91],{"className":46300},[61],[15,46302,750],{"className":46303},[178],[15,46305],{"className":46306,"style":772},[127],[15,46308,239],{"className":46309},[375],[15,46311],{"className":46312,"style":772},[127],[15,46314,46316,46319,46322,46325,46328,46331,46334],{"className":46315},[52],[15,46317],{"className":46318,"style":142},[56],[15,46320,22676],{"className":46321,"style":123},[61,122],[15,46323,741],{"className":46324},[146],[15,46326,33406],{"className":46327},[61,122],[15,46329],{"className":46330,"style":772},[127],[15,46332,7756],{"className":46333},[375],[15,46335],{"className":46336,"style":772},[127],[15,46338,46340,46344,46347,46350,46353],{"className":46339},[52],[15,46341],{"className":46342,"style":46343},[56],"height:0.7778em;vertical-align:-0.0833em;",[15,46345,33396],{"className":46346},[61,122],[15,46348],{"className":46349,"style":772},[127],[15,46351,7756],{"className":46352},[375],[15,46354],{"className":46355,"style":772},[127],[15,46357,46359,46362,46365,46368],{"className":46358},[52],[15,46360],{"className":46361,"style":45239},[56],[15,46363,91],{"className":46364},[61],[15,46366,750],{"className":46367},[178],[15,46369,46371],{"className":46370},[61],[15,46372,750],{"className":46373},[406,4063],"（此处 ",[15,46376,46378,46400],{"className":46377,"translate":19},[18],[15,46379,46381],{"className":46380},[23],[25,46382,46383],{"xmlns":27},[29,46384,46385,46397],{},[32,46386,46387,46389,46391],{},[35,46388,33406],{},[81,46390,244],{},[266,46392,46393,46395],{},[35,46394,3102],{},[35,46396,272],{},[40,46398,46399],{"encoding":42},"a = \\alpha_k",[15,46401,46403,46421],{"className":46402,"ariaHidden":48},[47],[15,46404,46406,46409,46412,46415,46418],{"className":46405},[52],[15,46407],{"className":46408,"style":6026},[56],[15,46410,33406],{"className":46411},[61,122],[15,46413],{"className":46414,"style":128},[127],[15,46416,244],{"className":46417},[132],[15,46419],{"className":46420,"style":128},[127],[15,46422,46424,46427],{"className":46423},[52],[15,46425],{"className":46426,"style":2509},[56],[15,46428,46430,46433],{"className":46429},[61],[15,46431,3102],{"className":46432,"style":3187},[61,122],[15,46434,46436],{"className":46435},[338],[15,46437,46439,46459],{"className":46438},[342,472],[15,46440,46442,46456],{"className":46441},[346],[15,46443,46445],{"className":46444,"style":479},[350],[15,46446,46447,46450],{"style":3202},[15,46448],{"className":46449,"style":358},[357],[15,46451,46453],{"className":46452},[362,363,364,365],[15,46454,272],{"className":46455,"style":492},[61,122,365],[15,46457,497],{"className":46458},[496],[15,46460,46462],{"className":46461},[346],[15,46463,46465],{"className":46464,"style":504},[350],[15,46466],{},[15,46468,46470,46492],{"className":46469,"translate":19},[18],[15,46471,46473],{"className":46472},[23],[25,46474,46475],{"xmlns":27},[29,46476,46477,46489],{},[32,46478,46479,46481,46483,46485,46487],{},[35,46480,33406],{},[81,46482,7756],{},[35,46484,33396],{},[81,46486,244],{},[35,46488,40230],{},[40,46490,46491],{"encoding":42},"a + b = S",[15,46493,46495,46513,46531],{"className":46494,"ariaHidden":48},[47],[15,46496,46498,46501,46504,46507,46510],{"className":46497},[52],[15,46499],{"className":46500,"style":6731},[56],[15,46502,33406],{"className":46503},[61,122],[15,46505],{"className":46506,"style":772},[127],[15,46508,7756],{"className":46509},[375],[15,46511],{"className":46512,"style":772},[127],[15,46514,46516,46519,46522,46525,46528],{"className":46515},[52],[15,46517],{"className":46518,"style":10443},[56],[15,46520,33396],{"className":46521},[61,122],[15,46523],{"className":46524,"style":128},[127],[15,46526,244],{"className":46527},[132],[15,46529],{"className":46530,"style":128},[127],[15,46532,46534,46537],{"className":46533},[52],[15,46535],{"className":46536,"style":57},[56],[15,46538,40230],{"className":46539,"style":40302},[61,122],"）。代入 ",[15,46542,46544,46578],{"className":46543,"translate":19},[18],[15,46545,46547],{"className":46546},[23],[25,46548,46549],{"xmlns":27},[29,46550,46551,46575],{},[32,46552,46553,46555,46557,46559,46561,46563,46565,46567,46569],{},[35,46554,33406],{},[35,46556,3018],{"mathvariant":231},[81,46558,741],{"stretchy":86},[35,46560,33406],{},[81,46562,7756],{},[35,46564,33396],{},[81,46566,750],{"stretchy":86},[81,46568,244],{},[266,46570,46571,46573],{},[35,46572,11],{},[35,46574,272],{},[40,46576,46577],{"encoding":42},"a\u002F(a+b) = p_k",[15,46579,46581,46608,46629],{"className":46580,"ariaHidden":48},[47],[15,46582,46584,46587,46590,46593,46596,46599,46602,46605],{"className":46583},[52],[15,46585],{"className":46586,"style":142},[56],[15,46588,33406],{"className":46589},[61,122],[15,46591,3018],{"className":46592},[61],[15,46594,741],{"className":46595},[146],[15,46597,33406],{"className":46598},[61,122],[15,46600],{"className":46601,"style":772},[127],[15,46603,7756],{"className":46604},[375],[15,46606],{"className":46607,"style":772},[127],[15,46609,46611,46614,46617,46620,46623,46626],{"className":46610},[52],[15,46612],{"className":46613,"style":142},[56],[15,46615,33396],{"className":46616},[61,122],[15,46618,750],{"className":46619},[178],[15,46621],{"className":46622,"style":128},[127],[15,46624,244],{"className":46625},[132],[15,46627],{"className":46628,"style":128},[127],[15,46630,46632,46635],{"className":46631},[52],[15,46633],{"className":46634,"style":9428},[56],[15,46636,46638,46641],{"className":46637},[61],[15,46639,11],{"className":46640},[61,122],[15,46642,46644],{"className":46643},[338],[15,46645,46647,46667],{"className":46646},[342,472],[15,46648,46650,46664],{"className":46649},[346],[15,46651,46653],{"className":46652,"style":479},[350],[15,46654,46655,46658],{"style":482},[15,46656],{"className":46657,"style":358},[357],[15,46659,46661],{"className":46660},[362,363,364,365],[15,46662,272],{"className":46663,"style":492},[61,122,365],[15,46665,497],{"className":46666},[496],[15,46668,46670],{"className":46669},[346],[15,46671,46673],{"className":46672,"style":504},[350],[15,46674],{}," 即得第二式。互信息由 ",[15,46677,46679,46723],{"className":46678,"translate":19},[18],[15,46680,46682],{"className":46681},[23],[25,46683,46684],{"xmlns":27},[29,46685,46686,46720],{},[32,46687,46688,46690,46692,46694,46696,46698,46700,46702,46704,46706,46708,46710,46712,46714,46716,46718],{},[35,46689,28898],{},[81,46691,244],{},[35,46693,22634],{},[81,46695,14810],{"stretchy":86},[35,46697,14807],{"mathvariant":258},[282,46699,4796],{},[35,46701,24661],{"mathvariant":3101},[81,46703,14819],{"stretchy":86},[81,46705,239],{},[35,46707,14807],{"mathvariant":258},[81,46709,14810],{"stretchy":86},[35,46711,22634],{},[81,46713,741],{"stretchy":86},[35,46715,24661],{"mathvariant":3101},[81,46717,750],{"stretchy":86},[81,46719,14819],{"stretchy":86},[40,46721,46722],{"encoding":42},"I = H[\\mathbb{E}\\,\\boldsymbol{\\pi}] - \\mathbb{E}[H(\\boldsymbol{\\pi})]",[15,46724,46726,46744,46783],{"className":46725,"ariaHidden":48},[47],[15,46727,46729,46732,46735,46738,46741],{"className":46728},[52],[15,46730],{"className":46731,"style":57},[56],[15,46733,28898],{"className":46734,"style":28937},[61,122],[15,46736],{"className":46737,"style":128},[127],[15,46739,244],{"className":46740},[132],[15,46742],{"className":46743,"style":128},[127],[15,46745,46747,46750,46753,46756,46759,46762,46771,46774,46777,46780],{"className":46746},[52],[15,46748],{"className":46749,"style":142},[56],[15,46751,22634],{"className":46752,"style":22746},[61,122],[15,46754,14810],{"className":46755},[146],[15,46757,14807],{"className":46758},[61,427],[15,46760],{"className":46761,"style":157},[127],[15,46763,46765],{"className":46764},[61],[15,46766,46768],{"className":46767},[61],[15,46769,24661],{"className":46770,"style":24695},[61,3162],[15,46772,14819],{"className":46773},[178],[15,46775],{"className":46776,"style":772},[127],[15,46778,239],{"className":46779},[375],[15,46781],{"className":46782,"style":772},[127],[15,46784,46786,46789,46792,46795,46798,46801,46810],{"className":46785},[52],[15,46787],{"className":46788,"style":142},[56],[15,46790,14807],{"className":46791},[61,427],[15,46793,14810],{"className":46794},[146],[15,46796,22634],{"className":46797,"style":22746},[61,122],[15,46799,741],{"className":46800},[146],[15,46802,46804],{"className":46803},[61],[15,46805,46807],{"className":46806},[61],[15,46808,24661],{"className":46809,"style":24695},[61,3162],[15,46811,43268],{"className":46812},[178]," 得到。",[11,46815,46816,46817,46820,46821,21876,46884,46935,46936,46996,46997,47026,47027,47055,47056,47059,47060,47088],{},"这一分解的意义在于：",[661,46818,46819],{},"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 ",[15,46822,46824,46848],{"className":46823,"translate":19},[18],[15,46825,46827],{"className":46826},[23],[25,46828,46829],{"xmlns":27},[29,46830,46831,46845],{},[32,46832,46833,46835,46837,46839,46841,46843],{},[35,46834,28898],{},[81,46836,741],{"stretchy":86},[35,46838,79],{},[81,46840,11294],{"separator":48},[35,46842,24661],{"mathvariant":3101},[81,46844,750],{"stretchy":86},[40,46846,46847],{"encoding":42},"I(y;\\boldsymbol{\\pi})",[15,46849,46851],{"className":46850,"ariaHidden":48},[47],[15,46852,46854,46857,46860,46863,46866,46869,46872,46881],{"className":46853},[52],[15,46855],{"className":46856,"style":142},[56],[15,46858,28898],{"className":46859,"style":28937},[61,122],[15,46861,741],{"className":46862},[146],[15,46864,79],{"className":46865,"style":123},[61,122],[15,46867,11294],{"className":46868},[153],[15,46870],{"className":46871,"style":157},[127],[15,46873,46875],{"className":46874},[61],[15,46876,46878],{"className":46877},[61],[15,46879,24661],{"className":46880,"style":24695},[61,3162],[15,46882,750],{"className":46883},[178],[15,46885,46887,46905],{"className":46886,"translate":19},[18],[15,46888,46890],{"className":46889},[23],[25,46891,46892],{"xmlns":27},[29,46893,46894,46902],{},[32,46895,46896,46898,46900],{},[35,46897,40230],{},[81,46899,20691],{},[35,46901,6790],{"mathvariant":231},[40,46903,46904],{"encoding":42},"S \\to \\infty",[15,46906,46908,46926],{"className":46907,"ariaHidden":48},[47],[15,46909,46911,46914,46917,46920,46923],{"className":46910},[52],[15,46912],{"className":46913,"style":57},[56],[15,46915,40230],{"className":46916,"style":40302},[61,122],[15,46918],{"className":46919,"style":128},[127],[15,46921,20691],{"className":46922},[132],[15,46924],{"className":46925,"style":128},[127],[15,46927,46929,46932],{"className":46928},[52],[15,46930],{"className":46931,"style":6026},[56],[15,46933,6790],{"className":46934},[61],"（证据充分）时趋于零，在证据缺失时增大，与 ",[15,46937,46939,46960],{"className":46938,"translate":19},[18],[15,46940,46942],{"className":46941},[23],[25,46943,46944],{"xmlns":27},[29,46945,46946,46958],{},[32,46947,46948,46950,46952,46954,46956],{},[35,46949,33401],{},[81,46951,244],{},[35,46953,102],{},[35,46955,3018],{"mathvariant":231},[35,46957,40230],{},[40,46959,42949],{"encoding":42},[15,46961,46963,46981],{"className":46962,"ariaHidden":48},[47],[15,46964,46966,46969,46972,46975,46978],{"className":46965},[52],[15,46967],{"className":46968,"style":6026},[56],[15,46970,33401],{"className":46971},[61,122],[15,46973],{"className":46974,"style":128},[127],[15,46976,244],{"className":46977},[132],[15,46979],{"className":46980,"style":128},[127],[15,46982,46984,46987,46990,46993],{"className":46983},[52],[15,46985],{"className":46986,"style":142},[56],[15,46988,102],{"className":46989,"style":174},[61,122],[15,46991,3018],{"className":46992},[61],[15,46994,40230],{"className":46995,"style":40302},[61,122]," 的定性行为一致。但它是通过 ",[15,46998,47000,47014],{"className":46999,"translate":19},[18],[15,47001,47003],{"className":47002},[23],[25,47004,47005],{"xmlns":27},[29,47006,47007,47011],{},[32,47008,47009],{},[35,47010,22676],{},[40,47012,47013],{"encoding":42},"\\psi",[15,47015,47017],{"className":47016,"ariaHidden":48},[47],[15,47018,47020,47023],{"className":47019},[52],[15,47021],{"className":47022,"style":14301},[56],[15,47024,22676],{"className":47025,"style":123},[61,122]," 函数的差构造的，捕捉了 ",[15,47028,47030,47043],{"className":47029,"translate":19},[18],[15,47031,47033],{"className":47032},[23],[25,47034,47035],{"xmlns":27},[29,47036,47037,47041],{},[32,47038,47039],{},[35,47040,33401],{},[40,47042,33401],{"encoding":42},[15,47044,47046],{"className":47045,"ariaHidden":48},[47],[15,47047,47049,47052],{"className":47048},[52],[15,47050],{"className":47051,"style":6026},[56],[15,47053,33401],{"className":47054},[61,122]," 所忽略的",[661,47057,47058],{},"证据分布","信息，因而是比 ",[15,47061,47063,47076],{"className":47062,"translate":19},[18],[15,47064,47066],{"className":47065},[23],[25,47067,47068],{"xmlns":27},[29,47069,47070,47074],{},[32,47071,47072],{},[35,47073,33401],{},[40,47075,33401],{"encoding":42},[15,47077,47079],{"className":47078,"ariaHidden":48},[47],[15,47080,47082,47085],{"className":47081},[52],[15,47083],{"className":47084,"style":6026},[56],[15,47086,33401],{"className":47087},[61,122]," 更精细的分数。",[11,47090,47091,47092,47126,47127,47161,47162,47256,47257,20281,47260,47322,47323,47327,47328,47331],{},"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 ",[15,47093,47095,47108],{"className":47094,"translate":19},[18],[15,47096,47098],{"className":47097},[23],[25,47099,47100],{"xmlns":27},[29,47101,47102,47106],{},[32,47103,47104],{},[35,47105,24661],{"mathvariant":3101},[40,47107,24749],{"encoding":42},[15,47109,47111],{"className":47110,"ariaHidden":48},[47],[15,47112,47114,47117],{"className":47113},[52],[15,47115],{"className":47116,"style":2706},[56],[15,47118,47120],{"className":47119},[61],[15,47121,47123],{"className":47122},[61],[15,47124,24661],{"className":47125,"style":24695},[61,3162]," 的不确定性」，而 ",[15,47128,47130,47143],{"className":47129,"translate":19},[18],[15,47131,47133],{"className":47132},[23],[25,47134,47135],{"xmlns":27},[29,47136,47137,47141],{},[32,47138,47139],{},[35,47140,24661],{"mathvariant":3101},[40,47142,24749],{"encoding":42},[15,47144,47146],{"className":47145,"ariaHidden":48},[47],[15,47147,47149,47152],{"className":47148},[52],[15,47150],{"className":47151,"style":2706},[56],[15,47153,47155],{"className":47154},[61],[15,47156,47158],{"className":47157},[61],[15,47159,24661],{"className":47160,"style":24695},[61,3162]," 在 EDL 中是 ",[15,47163,47165,47189],{"className":47164,"translate":19},[18],[15,47166,47168],{"className":47167},[23],[25,47169,47170],{"xmlns":27},[29,47171,47172,47186],{},[32,47173,47174,47180,47182,47184],{},[266,47175,47176,47178],{},[35,47177,22777],{},[35,47179,37117],{"mathvariant":3101},[81,47181,741],{"stretchy":86},[35,47183,1027],{"mathvariant":251},[81,47185,750],{"stretchy":86},[40,47187,47188],{"encoding":42},"g_{\\boldsymbol{\\theta}}(\\mathbf{x})",[15,47190,47192],{"className":47191,"ariaHidden":48},[47],[15,47193,47195,47198,47247,47250,47253],{"className":47194},[52],[15,47196],{"className":47197,"style":142},[56],[15,47199,47201,47204],{"className":47200},[61],[15,47202,22777],{"className":47203,"style":123},[61,122],[15,47205,47207],{"className":47206},[338],[15,47208,47210,47239],{"className":47209},[342,472],[15,47211,47213,47236],{"className":47212},[346],[15,47214,47216],{"className":47215,"style":479},[350],[15,47217,47218,47221],{"style":27013},[15,47219],{"className":47220,"style":358},[357],[15,47222,47224],{"className":47223},[362,363,364,365],[15,47225,47227],{"className":47226},[61,365],[15,47228,47230],{"className":47229},[61,365],[15,47231,47233],{"className":47232},[61,365],[15,47234,37117],{"className":47235,"style":37181},[61,3162,365],[15,47237,497],{"className":47238},[496],[15,47240,47242],{"className":47241},[346],[15,47243,47245],{"className":47244,"style":504},[350],[15,47246],{},[15,47248,741],{"className":47249},[146],[15,47251,1027],{"className":47252},[61,411],[15,47254,750],{"className":47255},[178]," 的输出，是",[661,47258,47259],{},"网络参数的确定性函数",[15,47261,47263,47286],{"className":47262,"translate":19},[18],[15,47264,47266],{"className":47265},[23],[25,47267,47268],{"xmlns":27},[29,47269,47270,47284],{},[32,47271,47272,47274,47276,47278,47280,47282],{},[35,47273,28898],{},[81,47275,741],{"stretchy":86},[35,47277,79],{},[81,47279,11294],{"separator":48},[35,47281,24661],{"mathvariant":3101},[81,47283,750],{"stretchy":86},[40,47285,46847],{"encoding":42},[15,47287,47289],{"className":47288,"ariaHidden":48},[47],[15,47290,47292,47295,47298,47301,47304,47307,47310,47319],{"className":47291},[52],[15,47293],{"className":47294,"style":142},[56],[15,47296,28898],{"className":47297,"style":28937},[61,122],[15,47299,741],{"className":47300},[146],[15,47302,79],{"className":47303,"style":123},[61,122],[15,47305,11294],{"className":47306},[153],[15,47308],{"className":47309,"style":157},[127],[15,47311,47313],{"className":47312},[61],[15,47314,47316],{"className":47315},[61],[15,47317,24661],{"className":47318,"style":24695},[61,3162],[15,47320,750],{"className":47321},[178]," 度量的是",[47324,47325,47326],"em",{},"类概率向量","的不确定，而不是",[47324,47329,47330],{},"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。",[872,47333,47335],{"id":47334},"四证据深度学习的损失函数","四、证据深度学习的损失函数",[11,47337,47338,47339,47367,47368,47422],{},"EDL 的训练目标不再是「使 ",[15,47340,47342,47355],{"className":47341,"translate":19},[18],[15,47343,47345],{"className":47344},[23],[25,47346,47347],{"xmlns":27},[29,47348,47349,47353],{},[32,47350,47351],{},[35,47352,11],{"mathvariant":251},[40,47354,819],{"encoding":42},[15,47356,47358],{"className":47357,"ariaHidden":48},[47],[15,47359,47361,47364],{"className":47360},[52],[15,47362],{"className":47363,"style":829},[56],[15,47365,11],{"className":47366},[61,411]," 逼近 one-hot 标签」，而是「使整族 ",[15,47369,47371,47392],{"className":47370,"translate":19},[18],[15,47372,47374],{"className":47373},[23],[25,47375,47376],{"xmlns":27},[29,47377,47378,47390],{},[32,47379,47380,47382,47384,47386,47388],{},[35,47381,4182],{"mathvariant":231},[81,47383,1545],{},[81,47385,741],{"stretchy":86},[35,47387,3102],{"mathvariant":3101},[81,47389,750],{"stretchy":86},[40,47391,39039],{"encoding":42},[15,47393,47395],{"className":47394,"ariaHidden":48},[47],[15,47396,47398,47401,47407,47410,47419],{"className":47397},[52],[15,47399],{"className":47400,"style":142},[56],[15,47402,47404],{"className":47403},[531],[15,47405,4182],{"className":47406},[61,1620],[15,47408,741],{"className":47409},[146],[15,47411,47413],{"className":47412},[61],[15,47414,47416],{"className":47415},[61],[15,47417,3102],{"className":47418},[61,3162],[15,47420,750],{"className":47421},[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。",[877,47424,47426],{"id":47425},"_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",[11,47428,47429,47430,47433],{},"最自然的构造是极小化对数损失在迪利克雷先验下的",[661,47431,47432],{},"贝叶斯风险","（亦称 Type-II 极大似然）：",[15,47435,47437],{"className":47436,"translate":19},[212],[15,47438,47440,47498],{"className":47439,"translate":19},[18],[15,47441,47443],{"className":47442},[23],[25,47444,47445],{"xmlns":27,"display":221},[29,47446,47447,47495],{},[32,47448,47449,47457,47459,47479,47481,47483,47485,47487,47493],{},[227,47450,47451,47454],{},[35,47452,47453],{"mathvariant":37},"L",[282,47455,47456],{},"NLL",[81,47458,244],{},[266,47460,47461,47463],{},[35,47462,14807],{"mathvariant":258},[32,47464,47465,47467,47469,47471,47473,47475,47477],{},[35,47466,24661],{"mathvariant":3101},[81,47468,4179],{},[35,47470,4182],{"mathvariant":231},[81,47472,1545],{},[81,47474,741],{"stretchy":86},[35,47476,3102],{"mathvariant":3101},[81,47478,750],{"stretchy":86},[81,47480,14810],{"fence":86,"stretchy":48,"minsize":44812,"maxsize":44812},[81,47482,239],{},[35,47484,22645],{},[81,47486,1545],{},[266,47488,47489,47491],{},[35,47490,24661],{},[35,47492,79],{},[81,47494,14819],{"fence":86,"stretchy":48,"minsize":44812,"maxsize":44812},[40,47496,47497],{"encoding":42},"\\mathcal{L}^{\\text{NLL}} = \\mathbb{E}_{\\boldsymbol{\\pi} \\sim \\operatorname{Dir}(\\boldsymbol{\\alpha})}\\big[-\\log \\pi_y\\big]",[15,47499,47501,47551,47648],{"className":47500,"ariaHidden":48},[47],[15,47502,47504,47507,47542,47545,47548],{"className":47503},[52],[15,47505],{"className":47506,"style":328},[56],[15,47508,47510,47513],{"className":47509},[61],[15,47511,47453],{"className":47512},[61,62],[15,47514,47516],{"className":47515},[338],[15,47517,47519],{"className":47518},[342],[15,47520,47522],{"className":47521},[346],[15,47523,47525],{"className":47524,"style":328},[350],[15,47526,47527,47530],{"style":353},[15,47528],{"className":47529,"style":358},[357],[15,47531,47533],{"className":47532},[362,363,364,365],[15,47534,47536],{"className":47535},[61,365],[15,47537,47539],{"className":47538},[61,13539,365],[15,47540,47456],{"className":47541},[61,365],[15,47543],{"className":47544,"style":128},[127],[15,47546,244],{"className":47547},[132],[15,47549],{"className":47550,"style":128},[127],[15,47552,47554,47558,47633,47639,47642,47645],{"className":47553},[52],[15,47555],{"className":47556,"style":47557},[56],"height:1.2052em;vertical-align:-0.3552em;",[15,47559,47561,47564],{"className":47560},[61],[15,47562,14807],{"className":47563},[61,427],[15,47565,47567],{"className":47566},[338],[15,47568,47570,47624],{"className":47569},[342,472],[15,47571,47573,47621],{"className":47572},[346],[15,47574,47576],{"className":47575,"style":22389},[350],[15,47577,47579,47582],{"style":47578},"top:-2.5198em;margin-left:0em;margin-right:0.05em;",[15,47580],{"className":47581,"style":358},[357],[15,47583,47585],{"className":47584},[362,363,364,365],[15,47586,47588,47597,47600,47606,47609,47618],{"className":47587},[61,365],[15,47589,47591],{"className":47590},[61,365],[15,47592,47594],{"className":47593},[61,365],[15,47595,24661],{"className":47596,"style":24695},[61,3162,365],[15,47598,4179],{"className":47599},[132,365],[15,47601,47603],{"className":47602},[531,365],[15,47604,4182],{"className":47605},[61,1620,365],[15,47607,741],{"className":47608},[146,365],[15,47610,47612],{"className":47611},[61,365],[15,47613,47615],{"className":47614},[61,365],[15,47616,3102],{"className":47617},[61,3162,365],[15,47619,750],{"className":47620},[178,365],[15,47622,497],{"className":47623},[496],[15,47625,47627],{"className":47626},[346],[15,47628,47631],{"className":47629,"style":47630},[350],"height:0.3552em;",[15,47632],{},[15,47634,47636],{"className":47635},[61],[15,47637,14810],{"className":47638},[406,4063],[15,47640],{"className":47641,"style":772},[127],[15,47643,239],{"className":47644},[375],[15,47646],{"className":47647,"style":772},[127],[15,47649,47651,47654,47659,47662,47702],{"className":47650},[52],[15,47652],{"className":47653,"style":45239},[56],[15,47655,22774,47657],{"className":47656},[531],[15,47658,22777],{"style":15608},[15,47660],{"className":47661,"style":157},[127],[15,47663,47665,47668],{"className":47664},[61],[15,47666,24661],{"className":47667,"style":123},[61,122],[15,47669,47671],{"className":47670},[338],[15,47672,47674,47694],{"className":47673},[342,472],[15,47675,47677,47691],{"className":47676},[346],[15,47678,47680],{"className":47679,"style":27010},[350],[15,47681,47682,47685],{"style":27013},[15,47683],{"className":47684,"style":358},[357],[15,47686,47688],{"className":47687},[362,363,364,365],[15,47689,79],{"className":47690,"style":123},[61,122,365],[15,47692,497],{"className":47693},[496],[15,47695,47697],{"className":47696},[346],[15,47698,47700],{"className":47699,"style":15696},[350],[15,47701],{},[15,47703,47705],{"className":47704},[61],[15,47706,14819],{"className":47707},[406,4063],[882,47709,47711],{"title":47710},"命题 4.1（对数风险的闭式）",[11,47712,47713,47714,47742,47743],{},"设真实标签为 ",[15,47715,47717,47730],{"className":47716,"translate":19},[18],[15,47718,47720],{"className":47719},[23],[25,47721,47722],{"xmlns":27},[29,47723,47724,47728],{},[32,47725,47726],{},[35,47727,79],{},[40,47729,79],{"encoding":42},[15,47731,47733],{"className":47732,"ariaHidden":48},[47],[15,47734,47736,47739],{"className":47735},[52],[15,47737],{"className":47738,"style":9428},[56],[15,47740,79],{"className":47741,"style":123},[61,122],"，则\n",[15,47744,47746,47788],{"className":47745,"translate":19},[18],[15,47747,47749],{"className":47748},[23],[25,47750,47751],{"xmlns":27},[29,47752,47753,47785],{},[32,47754,47755,47761,47763,47765,47767,47769,47771,47773,47775,47777,47783],{},[227,47756,47757,47759],{},[35,47758,47453],{"mathvariant":37},[282,47760,47456],{},[81,47762,244],{},[35,47764,22676],{},[81,47766,741],{"stretchy":86},[35,47768,40230],{},[81,47770,750],{"stretchy":86},[81,47772,239],{},[35,47774,22676],{},[81,47776,741],{"stretchy":86},[266,47778,47779,47781],{},[35,47780,3102],{},[35,47782,79],{},[81,47784,750],{"stretchy":86},[40,47786,47787],{"encoding":42},"\\mathcal{L}^{\\text{NLL}} = \\psi(S) - \\psi(\\alpha_y)",[15,47789,47791,47841,47868],{"className":47790,"ariaHidden":48},[47],[15,47792,47794,47797,47832,47835,47838],{"className":47793},[52],[15,47795],{"className":47796,"style":695},[56],[15,47798,47800,47803],{"className":47799},[61],[15,47801,47453],{"className":47802},[61,62],[15,47804,47806],{"className":47805},[338],[15,47807,47809],{"className":47808},[342],[15,47810,47812],{"className":47811},[346],[15,47813,47815],{"className":47814,"style":695},[350],[15,47816,47817,47820],{"style":716},[15,47818],{"className":47819,"style":358},[357],[15,47821,47823],{"className":47822},[362,363,364,365],[15,47824,47826],{"className":47825},[61,365],[15,47827,47829],{"className":47828},[61,13539,365],[15,47830,47456],{"className":47831},[61,365],[15,47833],{"className":47834,"style":128},[127],[15,47836,244],{"className":47837},[132],[15,47839],{"className":47840,"style":128},[127],[15,47842,47844,47847,47850,47853,47856,47859,47862,47865],{"className":47843},[52],[15,47845],{"className":47846,"style":142},[56],[15,47848,22676],{"className":47849,"style":123},[61,122],[15,47851,741],{"className":47852},[146],[15,47854,40230],{"className":47855,"style":40302},[61,122],[15,47857,750],{"className":47858},[178],[15,47860],{"className":47861,"style":772},[127],[15,47863,239],{"className":47864},[375],[15,47866],{"className":47867,"style":772},[127],[15,47869,47871,47874,47877,47880,47920],{"className":47870},[52],[15,47872],{"className":47873,"style":17976},[56],[15,47875,22676],{"className":47876,"style":123},[61,122],[15,47878,741],{"className":47879},[146],[15,47881,47883,47886],{"className":47882},[61],[15,47884,3102],{"className":47885,"style":3187},[61,122],[15,47887,47889],{"className":47888},[338],[15,47890,47892,47912],{"className":47891},[342,472],[15,47893,47895,47909],{"className":47894},[346],[15,47896,47898],{"className":47897,"style":27010},[350],[15,47899,47900,47903],{"style":3202},[15,47901],{"className":47902,"style":358},[357],[15,47904,47906],{"className":47905},[362,363,364,365],[15,47907,79],{"className":47908,"style":123},[61,122,365],[15,47910,497],{"className":47911},[496],[15,47913,47915],{"className":47914},[346],[15,47916,47918],{"className":47917,"style":15696},[350],[15,47919],{},[15,47921,750],{"className":47922},[178],[11,47924,47925,47927,47928,48181,48182,48233,48234,48268],{},[661,47926,5117],{}," 由 ",[15,47929,47931,47987],{"className":47930,"translate":19},[18],[15,47932,47934],{"className":47933},[23],[25,47935,47936],{"xmlns":27},[29,47937,47938,47984],{},[32,47939,47940,47942,47944,47946,47948,47954,47956,47958,47960,47962,47968,47970,47972,47974,47976,47982],{},[35,47941,14807],{"mathvariant":258},[81,47943,14810],{"stretchy":86},[35,47945,22645],{},[81,47947,1545],{},[266,47949,47950,47952],{},[35,47951,1027],{},[35,47953,272],{},[81,47955,14819],{"stretchy":86},[81,47957,244],{},[35,47959,22676],{},[81,47961,741],{"stretchy":86},[266,47963,47964,47966],{},[35,47965,3102],{},[35,47967,272],{},[81,47969,750],{"stretchy":86},[81,47971,239],{},[35,47973,22676],{},[81,47975,741],{"stretchy":86},[266,47977,47978,47980],{},[35,47979,3102],{},[89,47981,278],{},[81,47983,750],{"stretchy":86},[40,47985,47986],{"encoding":42},"\\mathbb{E}[\\log x_k] = \\psi(\\alpha_k) - \\psi(\\alpha_0)",[15,47988,47990,48062,48126],{"className":47989,"ariaHidden":48},[47],[15,47991,47993,47996,47999,48002,48007,48010,48050,48053,48056,48059],{"className":47992},[52],[15,47994],{"className":47995,"style":142},[56],[15,47997,14807],{"className":47998},[61,427],[15,48000,14810],{"className":48001},[146],[15,48003,22774,48005],{"className":48004},[531],[15,48006,22777],{"style":15608},[15,48008],{"className":48009,"style":157},[127],[15,48011,48013,48016],{"className":48012},[61],[15,48014,1027],{"className":48015},[61,122],[15,48017,48019],{"className":48018},[338],[15,48020,48022,48042],{"className":48021},[342,472],[15,48023,48025,48039],{"className":48024},[346],[15,48026,48028],{"className":48027,"style":479},[350],[15,48029,48030,48033],{"style":482},[15,48031],{"className":48032,"style":358},[357],[15,48034,48036],{"className":48035},[362,363,364,365],[15,48037,272],{"className":48038,"style":492},[61,122,365],[15,48040,497],{"className":48041},[496],[15,48043,48045],{"className":48044},[346],[15,48046,48048],{"className":48047,"style":504},[350],[15,48049],{},[15,48051,14819],{"className":48052},[178],[15,48054],{"className":48055,"style":128},[127],[15,48057,244],{"className":48058},[132],[15,48060],{"className":48061,"style":128},[127],[15,48063,48065,48068,48071,48074,48114,48117,48120,48123],{"className":48064},[52],[15,48066],{"className":48067,"style":142},[56],[15,48069,22676],{"className":48070,"style":123},[61,122],[15,48072,741],{"className":48073},[146],[15,48075,48077,48080],{"className":48076},[61],[15,48078,3102],{"className":48079,"style":3187},[61,122],[15,48081,48083],{"className":48082},[338],[15,48084,48086,48106],{"className":48085},[342,472],[15,48087,48089,48103],{"className":48088},[346],[15,48090,48092],{"className":48091,"style":479},[350],[15,48093,48094,48097],{"style":3202},[15,48095],{"className":48096,"style":358},[357],[15,48098,48100],{"className":48099},[362,363,364,365],[15,48101,272],{"className":48102,"style":492},[61,122,365],[15,48104,497],{"className":48105},[496],[15,48107,48109],{"className":48108},[346],[15,48110,48112],{"className":48111,"style":504},[350],[15,48113],{},[15,48115,750],{"className":48116},[178],[15,48118],{"className":48119,"style":772},[127],[15,48121,239],{"className":48122},[375],[15,48124],{"className":48125,"style":772},[127],[15,48127,48129,48132,48135,48138,48178],{"className":48128},[52],[15,48130],{"className":48131,"style":142},[56],[15,48133,22676],{"className":48134,"style":123},[61,122],[15,48136,741],{"className":48137},[146],[15,48139,48141,48144],{"className":48140},[61],[15,48142,3102],{"className":48143,"style":3187},[61,122],[15,48145,48147],{"className":48146},[338],[15,48148,48150,48170],{"className":48149},[342,472],[15,48151,48153,48167],{"className":48152},[346],[15,48154,48156],{"className":48155,"style":1212},[350],[15,48157,48158,48161],{"style":3202},[15,48159],{"className":48160,"style":358},[357],[15,48162,48164],{"className":48163},[362,363,364,365],[15,48165,278],{"className":48166},[61,365],[15,48168,497],{"className":48169},[496],[15,48171,48173],{"className":48172},[346],[15,48174,48176],{"className":48175,"style":504},[350],[15,48177],{},[15,48179,750],{"className":48180},[178],"（命题 2.4 的证明中已给出），取 ",[15,48183,48185,48203],{"className":48184,"translate":19},[18],[15,48186,48188],{"className":48187},[23],[25,48189,48190],{"xmlns":27},[29,48191,48192,48200],{},[32,48193,48194,48196,48198],{},[35,48195,272],{},[81,48197,244],{},[35,48199,79],{},[40,48201,48202],{"encoding":42},"k = y",[15,48204,48206,48224],{"className":48205,"ariaHidden":48},[47],[15,48207,48209,48212,48215,48218,48221],{"className":48208},[52],[15,48210],{"className":48211,"style":10443},[56],[15,48213,272],{"className":48214,"style":492},[61,122],[15,48216],{"className":48217,"style":128},[127],[15,48219,244],{"className":48220},[132],[15,48222],{"className":48223,"style":128},[127],[15,48225,48227,48230],{"className":48226},[52],[15,48228],{"className":48229,"style":9428},[56],[15,48231,79],{"className":48232,"style":123},[61,122]," 并乘以 ",[15,48235,48237,48253],{"className":48236,"translate":19},[18],[15,48238,48240],{"className":48239},[23],[25,48241,48242],{"xmlns":27},[29,48243,48244,48250],{},[32,48245,48246,48248],{},[81,48247,239],{},[89,48249,91],{},[40,48251,48252],{"encoding":42},"-1",[15,48254,48256],{"className":48255,"ariaHidden":48},[47],[15,48257,48259,48262,48265],{"className":48258},[52],[15,48260],{"className":48261,"style":2565},[56],[15,48263,239],{"className":48264},[61],[15,48266,91],{"className":48267},[61]," 即得。",[11,48270,48271],{},"该损失的梯度结构值得注意：",[15,48273,48275],{"className":48274,"translate":19},[212],[15,48276,48278,48405],{"className":48277,"translate":19},[18],[15,48279,48281],{"className":48280},[23],[25,48282,48283],{"xmlns":27,"display":221},[29,48284,48285,48402],{},[32,48286,48287,48310,48312,48318,48320,48322,48324,48326,48332,48334,48340,48342,48344,48346,48348,48350,48372,48374,48380,48382,48384,48386,48388,48390,48392,48394,48396,48398,48400],{},[3718,48288,48289,48300],{},[32,48290,48291,48294],{},[35,48292,48293],{"mathvariant":231},"∂",[227,48295,48296,48298],{},[35,48297,47453],{"mathvariant":37},[282,48299,47456],{},[32,48301,48302,48304],{},[35,48303,48293],{"mathvariant":231},[266,48305,48306,48308],{},[35,48307,3102],{},[35,48309,79],{},[81,48311,244],{},[227,48313,48314,48316],{},[35,48315,22676],{},[81,48317,23145],{"mathvariant":231,"lspace":23144,"rspace":23144},[81,48319,741],{"stretchy":86},[35,48321,40230],{},[81,48323,750],{"stretchy":86},[81,48325,239],{},[227,48327,48328,48330],{},[35,48329,22676],{},[81,48331,23145],{"mathvariant":231,"lspace":23144,"rspace":23144},[81,48333,741],{"stretchy":86},[266,48335,48336,48338],{},[35,48337,3102],{},[35,48339,79],{},[81,48341,750],{"stretchy":86},[81,48343,21295],{},[89,48345,278],{},[81,48347,94],{"separator":48},[127,48349],{"width":9838},[3718,48351,48352,48362],{},[32,48353,48354,48356],{},[35,48355,48293],{"mathvariant":231},[227,48357,48358,48360],{},[35,48359,47453],{"mathvariant":37},[282,48361,47456],{},[32,48363,48364,48366],{},[35,48365,48293],{"mathvariant":231},[266,48367,48368,48370],{},[35,48369,3102],{},[35,48371,14766],{},[81,48373,244],{},[227,48375,48376,48378],{},[35,48377,22676],{},[81,48379,23145],{"mathvariant":231,"lspace":23144,"rspace":23144},[81,48381,741],{"stretchy":86},[35,48383,40230],{},[81,48385,750],{"stretchy":86},[81,48387,1588],{},[89,48389,278],{},[127,48391],{"width":6445},[81,48393,741],{"stretchy":86},[35,48395,14766],{},[81,48397,14988],{"mathvariant":231},[35,48399,79],{},[81,48401,750],{"stretchy":86},[40,48403,48404],{"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)",[15,48406,48408,48563,48621,48715,48879,48935,48995],{"className":48407,"ariaHidden":48},[47],[15,48409,48411,48415,48554,48557,48560],{"className":48410},[52],[15,48412],{"className":48413,"style":48414},[56],"height:2.4904em;vertical-align:-0.9721em;",[15,48416,48418,48421,48551],{"className":48417},[61],[15,48419],{"className":48420},[146,3837],[15,48422,48424],{"className":48423},[3718],[15,48425,48427,48542],{"className":48426},[342,472],[15,48428,48430,48539],{"className":48429},[346],[15,48431,48434,48485,48493],{"className":48432,"style":48433},[350],"height:1.5183em;",[15,48435,48436,48439],{"style":11393},[15,48437],{"className":48438,"style":3857},[357],[15,48440,48442,48445],{"className":48441},[61],[15,48443,48293],{"className":48444,"style":9435},[61],[15,48446,48448,48451],{"className":48447},[61],[15,48449,3102],{"className":48450,"style":3187},[61,122],[15,48452,48454],{"className":48453},[338],[15,48455,48457,48477],{"className":48456},[342,472],[15,48458,48460,48474],{"className":48459},[346],[15,48461,48463],{"className":48462,"style":27010},[350],[15,48464,48465,48468],{"style":3202},[15,48466],{"className":48467,"style":358},[357],[15,48469,48471],{"className":48470},[362,363,364,365],[15,48472,79],{"className":48473,"style":123},[61,122,365],[15,48475,497],{"className":48476},[496],[15,48478,48480],{"className":48479},[346],[15,48481,48483],{"className":48482,"style":15696},[350],[15,48484],{},[15,48486,48487,48490],{"style":3885},[15,48488],{"className":48489,"style":3857},[357],[15,48491],{"className":48492,"style":3893},[3892],[15,48494,48495,48498],{"style":11459},[15,48496],{"className":48497,"style":3857},[357],[15,48499,48501,48504],{"className":48500},[61],[15,48502,48293],{"className":48503,"style":9435},[61],[15,48505,48507,48510],{"className":48506},[61],[15,48508,47453],{"className":48509},[61,62],[15,48511,48513],{"className":48512},[338],[15,48514,48516],{"className":48515},[342],[15,48517,48519],{"className":48518},[346],[15,48520,48522],{"className":48521,"style":695},[350],[15,48523,48524,48527],{"style":716},[15,48525],{"className":48526,"style":358},[357],[15,48528,48530],{"className":48529},[362,363,364,365],[15,48531,48533],{"className":48532},[61,365],[15,48534,48536],{"className":48535},[61,13539,365],[15,48537,47456],{"className":48538},[61,365],[15,48540,497],{"className":48541},[496],[15,48543,48545],{"className":48544},[346],[15,48546,48549],{"className":48547,"style":48548},[350],"height:0.9721em;",[15,48550],{},[15,48552],{"className":48553},[178,3837],[15,48555],{"className":48556,"style":128},[127],[15,48558,244],{"className":48559},[132],[15,48561],{"className":48562,"style":128},[127],[15,48564,48566,48570,48603,48606,48609,48612,48615,48618],{"className":48565},[52],[15,48567],{"className":48568,"style":48569},[56],"height:1.0519em;vertical-align:-0.25em;",[15,48571,48573,48576],{"className":48572},[61],[15,48574,22676],{"className":48575,"style":123},[61,122],[15,48577,48579],{"className":48578},[338],[15,48580,48582],{"className":48581},[342],[15,48583,48585],{"className":48584},[346],[15,48586,48589],{"className":48587,"style":48588},[350],"height:0.8019em;",[15,48590,48591,48594],{"style":353},[15,48592],{"className":48593,"style":358},[357],[15,48595,48597],{"className":48596},[362,363,364,365],[15,48598,48600],{"className":48599},[61,365],[15,48601,23145],{"className":48602},[61,365],[15,48604,741],{"className":48605},[146],[15,48607,40230],{"className":48608,"style":40302},[61,122],[15,48610,750],{"className":48611},[178],[15,48613],{"className":48614,"style":772},[127],[15,48616,239],{"className":48617},[375],[15,48619],{"className":48620,"style":772},[127],[15,48622,48624,48628,48660,48663,48703,48706,48709,48712],{"className":48623},[52],[15,48625],{"className":48626,"style":48627},[56],"height:1.088em;vertical-align:-0.2861em;",[15,48629,48631,48634],{"className":48630},[61],[15,48632,22676],{"className":48633,"style":123},[61,122],[15,48635,48637],{"className":48636},[338],[15,48638,48640],{"className":48639},[342],[15,48641,48643],{"className":48642},[346],[15,48644,48646],{"className":48645,"style":48588},[350],[15,48647,48648,48651],{"style":353},[15,48649],{"className":48650,"style":358},[357],[15,48652,48654],{"className":48653},[362,363,364,365],[15,48655,48657],{"className":48656},[61,365],[15,48658,23145],{"className":48659},[61,365],[15,48661,741],{"className":48662},[146],[15,48664,48666,48669],{"className":48665},[61],[15,48667,3102],{"className":48668,"style":3187},[61,122],[15,48670,48672],{"className":48671},[338],[15,48673,48675,48695],{"className":48674},[342,472],[15,48676,48678,48692],{"className":48677},[346],[15,48679,48681],{"className":48680,"style":27010},[350],[15,48682,48683,48686],{"style":3202},[15,48684],{"className":48685,"style":358},[357],[15,48687,48689],{"className":48688},[362,363,364,365],[15,48690,79],{"className":48691,"style":123},[61,122,365],[15,48693,497],{"className":48694},[496],[15,48696,48698],{"className":48697},[346],[15,48699,48701],{"className":48700,"style":15696},[350],[15,48702],{},[15,48704,750],{"className":48705},[178],[15,48707],{"className":48708,"style":128},[127],[15,48710,21295],{"className":48711},[132],[15,48713],{"className":48714,"style":128},[127],[15,48716,48718,48721,48724,48727,48730,48733,48870,48873,48876],{"className":48717},[52],[15,48719],{"className":48720,"style":48414},[56],[15,48722,278],{"className":48723},[61],[15,48725,94],{"className":48726},[153],[15,48728],{"className":48729,"style":10026},[127],[15,48731],{"className":48732,"style":157},[127],[15,48734,48736,48739,48867],{"className":48735},[61],[15,48737],{"className":48738},[146,3837],[15,48740,48742],{"className":48741},[3718],[15,48743,48745,48859],{"className":48744},[342,472],[15,48746,48748,48856],{"className":48747},[346],[15,48749,48751,48802,48810],{"className":48750,"style":48433},[350],[15,48752,48753,48756],{"style":11393},[15,48754],{"className":48755,"style":3857},[357],[15,48757,48759,48762],{"className":48758},[61],[15,48760,48293],{"className":48761,"style":9435},[61],[15,48763,48765,48768],{"className":48764},[61],[15,48766,3102],{"className":48767,"style":3187},[61,122],[15,48769,48771],{"className":48770},[338],[15,48772,48774,48794],{"className":48773},[342,472],[15,48775,48777,48791],{"className":48776},[346],[15,48778,48780],{"className":48779,"style":1362},[350],[15,48781,48782,48785],{"style":3202},[15,48783],{"className":48784,"style":358},[357],[15,48786,48788],{"className":48787},[362,363,364,365],[15,48789,14766],{"className":48790,"style":14792},[61,122,365],[15,48792,497],{"className":48793},[496],[15,48795,48797],{"className":48796},[346],[15,48798,48800],{"className":48799,"style":15696},[350],[15,48801],{},[15,48803,48804,48807],{"style":3885},[15,48805],{"className":48806,"style":3857},[357],[15,48808],{"className":48809,"style":3893},[3892],[15,48811,48812,48815],{"style":11459},[15,48813],{"className":48814,"style":3857},[357],[15,48816,48818,48821],{"className":48817},[61],[15,48819,48293],{"className":48820,"style":9435},[61],[15,48822,48824,48827],{"className":48823},[61],[15,48825,47453],{"className":48826},[61,62],[15,48828,48830],{"className":48829},[338],[15,48831,48833],{"className":48832},[342],[15,48834,48836],{"className":48835},[346],[15,48837,48839],{"className":48838,"style":695},[350],[15,48840,48841,48844],{"style":716},[15,48842],{"className":48843,"style":358},[357],[15,48845,48847],{"className":48846},[362,363,364,365],[15,48848,48850],{"className":48849},[61,365],[15,48851,48853],{"className":48852},[61,13539,365],[15,48854,47456],{"className":48855},[61,365],[15,48857,497],{"className":48858},[496],[15,48860,48862],{"className":48861},[346],[15,48863,48865],{"className":48864,"style":48548},[350],[15,48866],{},[15,48868],{"className":48869},[178,3837],[15,48871],{"className":48872,"style":128},[127],[15,48874,244],{"className":48875},[132],[15,48877],{"className":48878,"style":128},[127],[15,48880,48882,48885,48917,48920,48923,48926,48929,48932],{"className":48881},[52],[15,48883],{"className":48884,"style":48569},[56],[15,48886,48888,48891],{"className":48887},[61],[15,48889,22676],{"className":48890,"style":123},[61,122],[15,48892,48894],{"className":48893},[338],[15,48895,48897],{"className":48896},[342],[15,48898,48900],{"className":48899},[346],[15,48901,48903],{"className":48902,"style":48588},[350],[15,48904,48905,48908],{"style":353},[15,48906],{"className":48907,"style":358},[357],[15,48909,48911],{"className":48910},[362,363,364,365],[15,48912,48914],{"className":48913},[61,365],[15,48915,23145],{"className":48916},[61,365],[15,48918,741],{"className":48919},[146],[15,48921,40230],{"className":48922,"style":40302},[61,122],[15,48924,750],{"className":48925},[178],[15,48927],{"className":48928,"style":128},[127],[15,48930,1588],{"className":48931},[132],[15,48933],{"className":48934,"style":128},[127],[15,48936,48938,48941,48944,48947,48950,48953,48956,48992],{"className":48937},[52],[15,48939],{"className":48940,"style":142},[56],[15,48942,278],{"className":48943},[61],[15,48945],{"className":48946,"style":15972},[127],[15,48948,741],{"className":48949},[146],[15,48951,14766],{"className":48952,"style":14792},[61,122],[15,48954],{"className":48955,"style":128},[127],[15,48957,48959,48986,48989],{"className":48958},[132],[15,48960,48962],{"className":48961},[132],[15,48963,48965],{"className":48964},[61,15991],[15,48966,48968],{"className":48967},[15995],[15,48969,48971,48974,48983],{"className":48970},[15999],[15,48972],{"className":48973,"style":14301},[56],[15,48975,48977],{"className":48976},[16006],[15,48978,48980],{"className":48979},[61],[15,48981,16013],{"className":48982},[132],[15,48984],{"className":48985},[16017],[15,48987],{"className":48988},[127,16021],[15,48990,244],{"className":48991},[132],[15,48993],{"className":48994,"style":128},[127],[15,48996,48998,49001,49004],{"className":48997},[52],[15,48999],{"className":49000,"style":142},[56],[15,49002,79],{"className":49003,"style":123},[61,122],[15,49005,750],{"className":49006},[178],[11,49008,49009,49010,49073,49074,19834,49077,49148,49149,19834,49152,49222,49223,49292,49293,49321,49322,49350],{},"其中 ",[15,49011,49013,49031],{"className":49012,"translate":19},[18],[15,49014,49016],{"className":49015},[23],[25,49017,49018],{"xmlns":27},[29,49019,49020,49028],{},[32,49021,49022],{},[227,49023,49024,49026],{},[35,49025,22676],{},[81,49027,23145],{"mathvariant":231,"lspace":23144,"rspace":23144},[40,49029,49030],{"encoding":42},"\\psi'",[15,49032,49034],{"className":49033,"ariaHidden":48},[47],[15,49035,49037,49041],{"className":49036},[52],[15,49038],{"className":49039,"style":49040},[56],"height:0.9463em;vertical-align:-0.1944em;",[15,49042,49044,49047],{"className":49043},[61],[15,49045,22676],{"className":49046,"style":123},[61,122],[15,49048,49050],{"className":49049},[338],[15,49051,49053],{"className":49052},[342],[15,49054,49056],{"className":49055},[346],[15,49057,49059],{"className":49058,"style":23199},[350],[15,49060,49061,49064],{"style":716},[15,49062],{"className":49063,"style":358},[357],[15,49065,49067],{"className":49066},[362,363,364,365],[15,49068,49070],{"className":49069},[61,365],[15,49071,23145],{"className":49072},[61,365]," 为 trigamma 函数（恒正、单调递减）。因此梯度沿两个方向作用：",[661,49075,49076],{},"提升真实类别的证据",[15,49078,49080,49098],{"className":49079,"translate":19},[18],[15,49081,49083],{"className":49082},[23],[25,49084,49085],{"xmlns":27},[29,49086,49087,49095],{},[32,49088,49089],{},[266,49090,49091,49093],{},[35,49092,3102],{},[35,49094,79],{},[40,49096,49097],{"encoding":42},"\\alpha_y",[15,49099,49101],{"className":49100,"ariaHidden":48},[47],[15,49102,49104,49108],{"className":49103},[52],[15,49105],{"className":49106,"style":49107},[56],"height:0.7167em;vertical-align:-0.2861em;",[15,49109,49111,49114],{"className":49110},[61],[15,49112,3102],{"className":49113,"style":3187},[61,122],[15,49115,49117],{"className":49116},[338],[15,49118,49120,49140],{"className":49119},[342,472],[15,49121,49123,49137],{"className":49122},[346],[15,49124,49126],{"className":49125,"style":27010},[350],[15,49127,49128,49131],{"style":3202},[15,49129],{"className":49130,"style":358},[357],[15,49132,49134],{"className":49133},[362,363,364,365],[15,49135,79],{"className":49136,"style":123},[61,122,365],[15,49138,497],{"className":49139},[496],[15,49141,49143],{"className":49142},[346],[15,49144,49146],{"className":49145,"style":15696},[350],[15,49147],{}," 增大），",[661,49150,49151],{},"压低其余类别的证据",[15,49153,49155,49173],{"className":49154,"translate":19},[18],[15,49156,49158],{"className":49157},[23],[25,49159,49160],{"xmlns":27},[29,49161,49162,49170],{},[32,49163,49164],{},[266,49165,49166,49168],{},[35,49167,3102],{},[35,49169,14766],{},[40,49171,49172],{"encoding":42},"\\alpha_j",[15,49174,49176],{"className":49175,"ariaHidden":48},[47],[15,49177,49179,49182],{"className":49178},[52],[15,49180],{"className":49181,"style":49107},[56],[15,49183,49185,49188],{"className":49184},[61],[15,49186,3102],{"className":49187,"style":3187},[61,122],[15,49189,49191],{"className":49190},[338],[15,49192,49194,49214],{"className":49193},[342,472],[15,49195,49197,49211],{"className":49196},[346],[15,49198,49200],{"className":49199,"style":1362},[350],[15,49201,49202,49205],{"style":3202},[15,49203],{"className":49204,"style":358},[357],[15,49206,49208],{"className":49207},[362,363,364,365],[15,49209,14766],{"className":49210,"style":14792},[61,122,365],[15,49212,497],{"className":49213},[496],[15,49215,49217],{"className":49216},[346],[15,49218,49220],{"className":49219,"style":15696},[350],[15,49221],{}," 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 ",[15,49224,49226,49243],{"className":49225,"translate":19},[18],[15,49227,49229],{"className":49228},[23],[25,49230,49231],{"xmlns":27},[29,49232,49233,49241],{},[32,49234,49235],{},[266,49236,49237,49239],{},[35,49238,3102],{},[35,49240,14766],{},[40,49242,49172],{"encoding":42},[15,49244,49246],{"className":49245,"ariaHidden":48},[47],[15,49247,49249,49252],{"className":49248},[52],[15,49250],{"className":49251,"style":49107},[56],[15,49253,49255,49258],{"className":49254},[61],[15,49256,3102],{"className":49257,"style":3187},[61,122],[15,49259,49261],{"className":49260},[338],[15,49262,49264,49284],{"className":49263},[342,472],[15,49265,49267,49281],{"className":49266},[346],[15,49268,49270],{"className":49269,"style":1362},[350],[15,49271,49272,49275],{"style":3202},[15,49273],{"className":49274,"style":358},[357],[15,49276,49278],{"className":49277},[362,363,364,365],[15,49279,14766],{"className":49280,"style":14792},[61,122,365],[15,49282,497],{"className":49283},[496],[15,49285,49287],{"className":49286},[346],[15,49288,49290],{"className":49289,"style":15696},[350],[15,49291],{}," 全部驱向 ",[15,49294,49296,49309],{"className":49295,"translate":19},[18],[15,49297,49299],{"className":49298},[23],[25,49300,49301],{"xmlns":27},[29,49302,49303,49307],{},[32,49304,49305],{},[89,49306,91],{},[40,49308,91],{"encoding":42},[15,49310,49312],{"className":49311,"ariaHidden":48},[47],[15,49313,49315,49318],{"className":49314},[52],[15,49316],{"className":49317,"style":859},[56],[15,49319,91],{"className":49320},[61],"，从而收缩总证据 ",[15,49323,49325,49338],{"className":49324,"translate":19},[18],[15,49326,49328],{"className":49327},[23],[25,49329,49330],{"xmlns":27},[29,49331,49332,49336],{},[32,49333,49334],{},[35,49335,40230],{},[40,49337,40230],{"encoding":42},[15,49339,49341],{"className":49340,"ariaHidden":48},[47],[15,49342,49344,49347],{"className":49343},[52],[15,49345],{"className":49346,"style":57},[56],[15,49348,40230],{"className":49349,"style":40302},[61,122],"——这正是 4.3 节引入正则项的直接原因。",[882,49352,49354],{"title":49353},"推论 4.2（与插值交叉熵的关系）",[11,49355,49356,49421,49422],{},[15,49357,49359,49377],{"className":49358,"translate":19},[18],[15,49360,49362],{"className":49361},[23],[25,49363,49364],{"xmlns":27},[29,49365,49366,49374],{},[32,49367,49368],{},[227,49369,49370,49372],{},[35,49371,47453],{"mathvariant":37},[282,49373,47456],{},[40,49375,49376],{"encoding":42},"\\mathcal{L}^{\\text{NLL}}",[15,49378,49380],{"className":49379,"ariaHidden":48},[47],[15,49381,49383,49386],{"className":49382},[52],[15,49384],{"className":49385,"style":695},[56],[15,49387,49389,49392],{"className":49388},[61],[15,49390,47453],{"className":49391},[61,62],[15,49393,49395],{"className":49394},[338],[15,49396,49398],{"className":49397},[342],[15,49399,49401],{"className":49400},[346],[15,49402,49404],{"className":49403,"style":695},[350],[15,49405,49406,49409],{"style":716},[15,49407],{"className":49408,"style":358},[357],[15,49410,49412],{"className":49411},[362,363,364,365],[15,49413,49415],{"className":49414},[61,365],[15,49416,49418],{"className":49417},[61,13539,365],[15,49419,47456],{"className":49420},[61,365]," 是 one-hot 标签下插值交叉熵的 Jensen 上界：\n",[15,49423,49425,49529],{"className":49424,"translate":19},[18],[15,49426,49428],{"className":49427},[23],[25,49429,49430],{"xmlns":27},[29,49431,49432,49526],{},[32,49433,49434,49440,49442,49444,49446,49448,49454,49456,49462,49486,49488,49490,49492,49508,49510,49512,49514,49516,49518,49520],{},[227,49435,49436,49438],{},[35,49437,47453],{"mathvariant":37},[282,49439,47456],{},[81,49441,244],{},[81,49443,239],{},[35,49445,22645],{},[81,49447,1545],{},[266,49449,49450,49452],{},[35,49451,11],{},[35,49453,79],{},[81,49455,7756],{},[3718,49457,49458,49460],{},[89,49459,91],{},[89,49461,906],{},[32,49463,49464,49466,49476,49478,49484],{},[81,49465,741],{"fence":48},[3718,49467,49468,49470],{},[89,49469,91],{},[266,49471,49472,49474],{},[35,49473,3102],{},[35,49475,79],{},[81,49477,239],{},[3718,49479,49480,49482],{},[89,49481,91],{},[35,49483,40230],{},[81,49485,750],{"fence":48},[81,49487,7756],{},[35,49489,12256],{},[282,49491,6793],{},[32,49493,49494,49496,49506],{},[81,49495,741],{"fence":48},[227,49497,49498,49500],{},[35,49499,40230],{},[32,49501,49502,49504],{},[81,49503,239],{},[89,49505,906],{},[81,49507,750],{"fence":48},[282,49509,284],{},[81,49511,275],{},[282,49513,284],{},[81,49515,239],{},[35,49517,22645],{},[81,49519,1545],{},[266,49521,49522,49524],{},[35,49523,11],{},[35,49525,79],{},[40,49527,49528],{"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",[15,49530,49532,49582,49651,49935,50015],{"className":49531,"ariaHidden":48},[47],[15,49533,49535,49538,49573,49576,49579],{"className":49534},[52],[15,49536],{"className":49537,"style":695},[56],[15,49539,49541,49544],{"className":49540},[61],[15,49542,47453],{"className":49543},[61,62],[15,49545,49547],{"className":49546},[338],[15,49548,49550],{"className":49549},[342],[15,49551,49553],{"className":49552},[346],[15,49554,49556],{"className":49555,"style":695},[350],[15,49557,49558,49561],{"style":716},[15,49559],{"className":49560,"style":358},[357],[15,49562,49564],{"className":49563},[362,363,364,365],[15,49565,49567],{"className":49566},[61,365],[15,49568,49570],{"className":49569},[61,13539,365],[15,49571,47456],{"className":49572},[61,365],[15,49574],{"className":49575,"style":128},[127],[15,49577,244],{"className":49578},[132],[15,49580],{"className":49581,"style":128},[127],[15,49583,49585,49588,49591,49594,49599,49602,49642,49645,49648],{"className":49584},[52],[15,49586],{"className":49587,"style":20933},[56],[15,49589,239],{"className":49590},[61],[15,49592],{"className":49593,"style":157},[127],[15,49595,22774,49597],{"className":49596},[531],[15,49598,22777],{"style":15608},[15,49600],{"className":49601,"style":157},[127],[15,49603,49605,49608],{"className":49604},[61],[15,49606,11],{"className":49607},[61,122],[15,49609,49611],{"className":49610},[338],[15,49612,49614,49634],{"className":49613},[342,472],[15,49615,49617,49631],{"className":49616},[346],[15,49618,49620],{"className":49619,"style":27010},[350],[15,49621,49622,49625],{"style":482},[15,49623],{"className":49624,"style":358},[357],[15,49626,49628],{"className":49627},[362,363,364,365],[15,49629,79],{"className":49630,"style":123},[61,122,365],[15,49632,497],{"className":49633},[496],[15,49635,49637],{"className":49636},[346],[15,49638,49640],{"className":49639,"style":15696},[350],[15,49641],{},[15,49643],{"className":49644,"style":772},[127],[15,49646,7756],{"className":49647},[375],[15,49649],{"className":49650,"style":772},[127],[15,49652,49654,49657,49725,49728,49926,49929,49932],{"className":49653},[52],[15,49655],{"className":49656,"style":394},[56],[15,49658,49660,49663,49722],{"className":49659},[61],[15,49661],{"className":49662},[146,3837],[15,49664,49666],{"className":49665},[3718],[15,49667,49669,49714],{"className":49668},[342,472],[15,49670,49672,49711],{"className":49671},[346],[15,49673,49675,49689,49697],{"className":49674,"style":3850},[350],[15,49676,49677,49680],{"style":3853},[15,49678],{"className":49679,"style":3857},[357],[15,49681,49683],{"className":49682},[362,363,364,365],[15,49684,49686],{"className":49685},[61,365],[15,49687,906],{"className":49688},[61,365],[15,49690,49691,49694],{"style":3885},[15,49692],{"className":49693,"style":3857},[357],[15,49695],{"className":49696,"style":3893},[3892],[15,49698,49699,49702],{"style":3896},[15,49700],{"className":49701,"style":3857},[357],[15,49703,49705],{"className":49704},[362,363,364,365],[15,49706,49708],{"className":49707},[61,365],[15,49709,91],{"className":49710},[61,365],[15,49712,497],{"className":49713},[496],[15,49715,49717],{"className":49716},[346],[15,49718,49720],{"className":49719,"style":10189},[350],[15,49721],{},[15,49723],{"className":49724},[178,3837],[15,49726],{"className":49727,"style":157},[127],[15,49729,49731,49737,49843,49846,49849,49852,49920],{"className":49730},[161],[15,49732,49734],{"className":49733,"style":402},[146,401],[15,49735,741],{"className":49736},[406,407],[15,49738,49740,49743,49840],{"className":49739},[61],[15,49741],{"className":49742},[146,3837],[15,49744,49746],{"className":49745},[3718],[15,49747,49749,49831],{"className":49748},[342,472],[15,49750,49752,49828],{"className":49751},[346],[15,49753,49755,49806,49814],{"className":49754,"style":3850},[350],[15,49756,49757,49760],{"style":3853},[15,49758],{"className":49759,"style":3857},[357],[15,49761,49763],{"className":49762},[362,363,364,365],[15,49764,49766],{"className":49765},[61,365],[15,49767,49769,49772],{"className":49768},[61,365],[15,49770,3102],{"className":49771,"style":3187},[61,122,365],[15,49773,49775],{"className":49774},[338],[15,49776,49778,49798],{"className":49777},[342,472],[15,49779,49781,49795],{"className":49780},[346],[15,49782,49784],{"className":49783,"style":29267},[350],[15,49785,49786,49789],{"style":4054},[15,49787],{"className":49788,"style":4058},[357],[15,49790,49792],{"className":49791},[362,4062,4063,365],[15,49793,79],{"className":49794,"style":123},[61,122,365],[15,49796,497],{"className":49797},[496],[15,49799,49801],{"className":49800},[346],[15,49802,49804],{"className":49803,"style":15952},[350],[15,49805],{},[15,49807,49808,49811],{"style":3885},[15,49809],{"className":49810,"style":3857},[357],[15,49812],{"className":49813,"style":3893},[3892],[15,49815,49816,49819],{"style":3896},[15,49817],{"className":49818,"style":3857},[357],[15,49820,49822],{"className":49821},[362,363,364,365],[15,49823,49825],{"className":49824},[61,365],[15,49826,91],{"className":49827},[61,365],[15,49829,497],{"className":49830},[496],[15,49832,49834],{"className":49833},[346],[15,49835,49838],{"className":49836,"style":49837},[350],"height:0.5423em;",[15,49839],{},[15,49841],{"className":49842},[178,3837],[15,49844],{"className":49845,"style":772},[127],[15,49847,239],{"className":49848},[375],[15,49850],{"className":49851,"style":772},[127],[15,49853,49855,49858,49917],{"className":49854},[61],[15,49856],{"className":49857},[146,3837],[15,49859,49861],{"className":49860},[3718],[15,49862,49864,49909],{"className":49863},[342,472],[15,49865,49867,49906],{"className":49866},[346],[15,49868,49870,49884,49892],{"className":49869,"style":3850},[350],[15,49871,49872,49875],{"style":3853},[15,49873],{"className":49874,"style":3857},[357],[15,49876,49878],{"className":49877},[362,363,364,365],[15,49879,49881],{"className":49880},[61,365],[15,49882,40230],{"className":49883,"style":40302},[61,122,365],[15,49885,49886,49889],{"style":3885},[15,49887],{"className":49888,"style":3857},[357],[15,49890],{"className":49891,"style":3893},[3892],[15,49893,49894,49897],{"style":3896},[15,49895],{"className":49896,"style":3857},[357],[15,49898,49900],{"className":49899},[362,363,364,365],[15,49901,49903],{"className":49902},[61,365],[15,49904,91],{"className":49905},[61,365],[15,49907,497],{"className":49908},[496],[15,49910,49912],{"className":49911},[346],[15,49913,49915],{"className":49914,"style":10189},[350],[15,49916],{},[15,49918],{"className":49919},[178,3837],[15,49921,49923],{"className":49922,"style":402},[178,401],[15,49924,750],{"className":49925},[406,407],[15,49927],{"className":49928,"style":772},[127],[15,49930,7756],{"className":49931},[375],[15,49933],{"className":49934,"style":772},[127],[15,49936,49938,49941,49944,49947,49950,50000,50003,50006,50009,50012],{"className":49937},[52],[15,49939],{"className":49940,"style":45239},[56],[15,49942,12256],{"className":49943,"style":12278},[61,122],[15,49945],{"className":49946,"style":7099},[127],[15,49948],{"className":49949,"style":157},[127],[15,49951,49953,49959,49994],{"className":49952},[161],[15,49954,49956],{"className":49955,"style":402},[146,401],[15,49957,741],{"className":49958},[406,4063],[15,49960,49962,49965],{"className":49961},[61],[15,49963,40230],{"className":49964,"style":40302},[61,122],[15,49966,49968],{"className":49967},[338],[15,49969,49971],{"className":49970},[342],[15,49972,49974],{"className":49973},[346],[15,49975,49977],{"className":49976,"style":7899},[350],[15,49978,49979,49982],{"style":716},[15,49980],{"className":49981,"style":358},[357],[15,49983,49985],{"className":49984},[362,363,364,365],[15,49986,49988,49991],{"className":49987},[61,365],[15,49989,239],{"className":49990},[61,365],[15,49992,906],{"className":49993},[61,365],[15,49995,49997],{"className":49996,"style":402},[178,401],[15,49998,750],{"className":49999},[406,4063],[15,50001,284],{"className":50002},[127],[15,50004],{"className":50005,"style":128},[127],[15,50007,275],{"className":50008},[132],[15,50010,284],{"className":50011},[127],[15,50013],{"className":50014,"style":128},[127],[15,50016,50018,50021,50024,50027,50032,50035],{"className":50017},[52],[15,50019],{"className":50020,"style":20933},[56],[15,50022,239],{"className":50023},[61],[15,50025],{"className":50026,"style":157},[127],[15,50028,22774,50030],{"className":50029},[531],[15,50031,22777],{"style":15608},[15,50033],{"className":50034,"style":157},[127],[15,50036,50038,50041],{"className":50037},[61],[15,50039,11],{"className":50040},[61,122],[15,50042,50044],{"className":50043},[338],[15,50045,50047,50067],{"className":50046},[342,472],[15,50048,50050,50064],{"className":50049},[346],[15,50051,50053],{"className":50052,"style":27010},[350],[15,50054,50055,50058],{"style":482},[15,50056],{"className":50057,"style":358},[357],[15,50059,50061],{"className":50060},[362,363,364,365],[15,50062,79],{"className":50063,"style":123},[61,122,365],[15,50065,497],{"className":50066},[496],[15,50068,50070],{"className":50069},[346],[15,50071,50073],{"className":50072,"style":15696},[350],[15,50074],{},[11,50076,50077,50079,50080,14421,50342,50491,50492,50745,50746,50839],{},[661,50078,5117],{}," 由 Jensen 不等式 ",[15,50081,50083,50141],{"className":50082,"translate":19},[18],[15,50084,50086],{"className":50085},[23],[25,50087,50088],{"xmlns":27},[29,50089,50090,50138],{},[32,50091,50092,50094,50096,50098,50100,50106,50108,50110,50112,50114,50116,50118,50124,50126,50128,50130,50132],{},[35,50093,14807],{"mathvariant":258},[81,50095,14810],{"stretchy":86},[35,50097,22645],{},[81,50099,1545],{},[266,50101,50102,50104],{},[35,50103,24661],{},[35,50105,79],{},[81,50107,14819],{"stretchy":86},[81,50109,21295],{},[35,50111,22645],{},[81,50113,1545],{},[35,50115,14807],{"mathvariant":258},[81,50117,14810],{"stretchy":86},[266,50119,50120,50122],{},[35,50121,24661],{},[35,50123,79],{},[81,50125,14819],{"stretchy":86},[81,50127,244],{},[35,50129,22645],{},[81,50131,1545],{},[266,50133,50134,50136],{},[35,50135,11],{},[35,50137,79],{},[40,50139,50140],{"encoding":42},"\\mathbb{E}[\\log \\pi_y] \\le \\log \\mathbb{E}[\\pi_y] = \\log p_y",[15,50142,50144,50216,50288],{"className":50143,"ariaHidden":48},[47],[15,50145,50147,50150,50153,50156,50161,50164,50204,50207,50210,50213],{"className":50146},[52],[15,50148],{"className":50149,"style":17976},[56],[15,50151,14807],{"className":50152},[61,427],[15,50154,14810],{"className":50155},[146],[15,50157,22774,50159],{"className":50158},[531],[15,50160,22777],{"style":15608},[15,50162],{"className":50163,"style":157},[127],[15,50165,50167,50170],{"className":50166},[61],[15,50168,24661],{"className":50169,"style":123},[61,122],[15,50171,50173],{"className":50172},[338],[15,50174,50176,50196],{"className":50175},[342,472],[15,50177,50179,50193],{"className":50178},[346],[15,50180,50182],{"className":50181,"style":27010},[350],[15,50183,50184,50187],{"style":27013},[15,50185],{"className":50186,"style":358},[357],[15,50188,50190],{"className":50189},[362,363,364,365],[15,50191,79],{"className":50192,"style":123},[61,122,365],[15,50194,497],{"className":50195},[496],[15,50197,50199],{"className":50198},[346],[15,50200,50202],{"className":50201,"style":15696},[350],[15,50203],{},[15,50205,14819],{"className":50206},[178],[15,50208],{"className":50209,"style":128},[127],[15,50211,21295],{"className":50212},[132],[15,50214],{"className":50215,"style":128},[127],[15,50217,50219,50222,50227,50230,50233,50236,50276,50279,50282,50285],{"className":50218},[52],[15,50220],{"className":50221,"style":17976},[56],[15,50223,22774,50225],{"className":50224},[531],[15,50226,22777],{"style":15608},[15,50228],{"className":50229,"style":157},[127],[15,50231,14807],{"className":50232},[61,427],[15,50234,14810],{"className":50235},[146],[15,50237,50239,50242],{"className":50238},[61],[15,50240,24661],{"className":50241,"style":123},[61,122],[15,50243,50245],{"className":50244},[338],[15,50246,50248,50268],{"className":50247},[342,472],[15,50249,50251,50265],{"className":50250},[346],[15,50252,50254],{"className":50253,"style":27010},[350],[15,50255,50256,50259],{"style":27013},[15,50257],{"className":50258,"style":358},[357],[15,50260,50262],{"className":50261},[362,363,364,365],[15,50263,79],{"className":50264,"style":123},[61,122,365],[15,50266,497],{"className":50267},[496],[15,50269,50271],{"className":50270},[346],[15,50272,50274],{"className":50273,"style":15696},[350],[15,50275],{},[15,50277,14819],{"className":50278},[178],[15,50280],{"className":50281,"style":128},[127],[15,50283,244],{"className":50284},[132],[15,50286],{"className":50287,"style":128},[127],[15,50289,50291,50294,50299,50302],{"className":50290},[52],[15,50292],{"className":50293,"style":20933},[56],[15,50295,22774,50297],{"className":50296},[531],[15,50298,22777],{"style":15608},[15,50300],{"className":50301,"style":157},[127],[15,50303,50305,50308],{"className":50304},[61],[15,50306,11],{"className":50307},[61,122],[15,50309,50311],{"className":50310},[338],[15,50312,50314,50334],{"className":50313},[342,472],[15,50315,50317,50331],{"className":50316},[346],[15,50318,50320],{"className":50319,"style":27010},[350],[15,50321,50322,50325],{"style":482},[15,50323],{"className":50324,"style":358},[357],[15,50326,50328],{"className":50327},[362,363,364,365],[15,50329,79],{"className":50330,"style":123},[61,122,365],[15,50332,497],{"className":50333},[496],[15,50335,50337],{"className":50336},[346],[15,50338,50340],{"className":50339,"style":15696},[350],[15,50341],{},[15,50343,50345,50377],{"className":50344,"translate":19},[18],[15,50346,50348],{"className":50347},[23],[25,50349,50350],{"xmlns":27},[29,50351,50352,50374],{},[32,50353,50354,50360,50362,50364,50366,50368],{},[227,50355,50356,50358],{},[35,50357,47453],{"mathvariant":37},[282,50359,47456],{},[81,50361,275],{},[81,50363,239],{},[35,50365,22645],{},[81,50367,1545],{},[266,50369,50370,50372],{},[35,50371,11],{},[35,50373,79],{},[40,50375,50376],{"encoding":42},"\\mathcal{L}^{\\text{NLL}} \\ge -\\log p_y",[15,50378,50380,50431],{"className":50379,"ariaHidden":48},[47],[15,50381,50383,50387,50422,50425,50428],{"className":50382},[52],[15,50384],{"className":50385,"style":50386},[56],"height:0.9773em;vertical-align:-0.136em;",[15,50388,50390,50393],{"className":50389},[61],[15,50391,47453],{"className":50392},[61,62],[15,50394,50396],{"className":50395},[338],[15,50397,50399],{"className":50398},[342],[15,50400,50402],{"className":50401},[346],[15,50403,50405],{"className":50404,"style":695},[350],[15,50406,50407,50410],{"style":716},[15,50408],{"className":50409,"style":358},[357],[15,50411,50413],{"className":50412},[362,363,364,365],[15,50414,50416],{"className":50415},[61,365],[15,50417,50419],{"className":50418},[61,13539,365],[15,50420,47456],{"className":50421},[61,365],[15,50423],{"className":50424,"style":128},[127],[15,50426,275],{"className":50427},[132],[15,50429],{"className":50430,"style":128},[127],[15,50432,50434,50437,50440,50443,50448,50451],{"className":50433},[52],[15,50435],{"className":50436,"style":20933},[56],[15,50438,239],{"className":50439},[61],[15,50441],{"className":50442,"style":157},[127],[15,50444,22774,50446],{"className":50445},[531],[15,50447,22777],{"style":15608},[15,50449],{"className":50450,"style":157},[127],[15,50452,50454,50457],{"className":50453},[61],[15,50455,11],{"className":50456},[61,122],[15,50458,50460],{"className":50459},[338],[15,50461,50463,50483],{"className":50462},[342,472],[15,50464,50466,50480],{"className":50465},[346],[15,50467,50469],{"className":50468,"style":27010},[350],[15,50470,50471,50474],{"style":482},[15,50472],{"className":50473,"style":358},[357],[15,50475,50477],{"className":50476},[362,363,364,365],[15,50478,79],{"className":50479,"style":123},[61,122,365],[15,50481,497],{"className":50482},[496],[15,50484,50486],{"className":50485},[346],[15,50487,50489],{"className":50488,"style":15696},[350],[15,50490],{},"。渐近展开由 ",[15,50493,50495,50553],{"className":50494,"translate":19},[18],[15,50496,50498],{"className":50497},[23],[25,50499,50500],{"xmlns":27},[29,50501,50502,50550],{},[32,50503,50504,50506,50508,50510,50512,50514,50516,50518,50520,50522,50532,50534,50536,50538,50548],{},[35,50505,22676],{},[81,50507,741],{"stretchy":86},[35,50509,33406],{},[81,50511,750],{"stretchy":86},[81,50513,244],{},[35,50515,22645],{},[81,50517,1545],{},[35,50519,33406],{},[81,50521,239],{},[3718,50523,50524,50526],{},[89,50525,91],{},[32,50527,50528,50530],{},[89,50529,906],{},[35,50531,33406],{},[81,50533,7756],{},[35,50535,12256],{},[81,50537,741],{"stretchy":86},[227,50539,50540,50542],{},[35,50541,33406],{},[32,50543,50544,50546],{},[81,50545,239],{},[89,50547,906],{},[81,50549,750],{"stretchy":86},[40,50551,50552],{"encoding":42},"\\psi(a) = \\log a - \\frac{1}{2a} + O(a^{-2})",[15,50554,50556,50583,50609,50695],{"className":50555,"ariaHidden":48},[47],[15,50557,50559,50562,50565,50568,50571,50574,50577,50580],{"className":50558},[52],[15,50560],{"className":50561,"style":142},[56],[15,50563,22676],{"className":50564,"style":123},[61,122],[15,50566,741],{"className":50567},[146],[15,50569,33406],{"className":50570},[61,122],[15,50572,750],{"className":50573},[178],[15,50575],{"className":50576,"style":128},[127],[15,50578,244],{"className":50579},[132],[15,50581],{"className":50582,"style":128},[127],[15,50584,50586,50589,50594,50597,50600,50603,50606],{"className":50585},[52],[15,50587],{"className":50588,"style":14301},[56],[15,50590,22774,50592],{"className":50591},[531],[15,50593,22777],{"style":15608},[15,50595],{"className":50596,"style":157},[127],[15,50598,33406],{"className":50599},[61,122],[15,50601],{"className":50602,"style":772},[127],[15,50604,239],{"className":50605},[375],[15,50607],{"className":50608,"style":772},[127],[15,50610,50612,50615,50686,50689,50692],{"className":50611},[52],[15,50613],{"className":50614,"style":42536},[56],[15,50616,50618,50621,50683],{"className":50617},[61],[15,50619],{"className":50620},[146,3837],[15,50622,50624],{"className":50623},[3718],[15,50625,50627,50675],{"className":50626},[342,472],[15,50628,50630,50672],{"className":50629},[346],[15,50631,50633,50650,50658],{"className":50632,"style":3850},[350],[15,50634,50635,50638],{"style":3853},[15,50636],{"className":50637,"style":3857},[357],[15,50639,50641],{"className":50640},[362,363,364,365],[15,50642,50644,50647],{"className":50643},[61,365],[15,50645,906],{"className":50646},[61,365],[15,50648,33406],{"className":50649},[61,122,365],[15,50651,50652,50655],{"style":3885},[15,50653],{"className":50654,"style":3857},[357],[15,50656],{"className":50657,"style":3893},[3892],[15,50659,50660,50663],{"style":3896},[15,50661],{"className":50662,"style":3857},[357],[15,50664,50666],{"className":50665},[362,363,364,365],[15,50667,50669],{"className":50668},[61,365],[15,50670,91],{"className":50671},[61,365],[15,50673,497],{"className":50674},[496],[15,50676,50678],{"className":50677},[346],[15,50679,50681],{"className":50680,"style":10189},[350],[15,50682],{},[15,50684],{"className":50685},[178,3837],[15,50687],{"className":50688,"style":772},[127],[15,50690,7756],{"className":50691},[375],[15,50693],{"className":50694,"style":772},[127],[15,50696,50698,50701,50704,50707,50742],{"className":50697},[52],[15,50699],{"className":50700,"style":21402},[56],[15,50702,12256],{"className":50703,"style":12278},[61,122],[15,50705,741],{"className":50706},[146],[15,50708,50710,50713],{"className":50709},[61],[15,50711,33406],{"className":50712},[61,122],[15,50714,50716],{"className":50715},[338],[15,50717,50719],{"className":50718},[342],[15,50720,50722],{"className":50721},[346],[15,50723,50725],{"className":50724,"style":7899},[350],[15,50726,50727,50730],{"style":716},[15,50728],{"className":50729,"style":358},[357],[15,50731,50733],{"className":50732},[362,363,364,365],[15,50734,50736,50739],{"className":50735},[61,365],[15,50737,239],{"className":50738},[61,365],[15,50740,906],{"className":50741},[61,365],[15,50743,750],{"className":50744},[178]," 给出，而 ",[15,50747,50749,50771],{"className":50748,"translate":19},[18],[15,50750,50752],{"className":50751},[23],[25,50753,50754],{"xmlns":27},[29,50755,50756,50768],{},[32,50757,50758,50764,50766],{},[266,50759,50760,50762],{},[35,50761,3102],{},[35,50763,79],{},[81,50765,21295],{},[35,50767,40230],{},[40,50769,50770],{"encoding":42},"\\alpha_y \\le S",[15,50772,50774,50830],{"className":50773,"ariaHidden":48},[47],[15,50775,50777,50781,50821,50824,50827],{"className":50776},[52],[15,50778],{"className":50779,"style":50780},[56],"height:0.9221em;vertical-align:-0.2861em;",[15,50782,50784,50787],{"className":50783},[61],[15,50785,3102],{"className":50786,"style":3187},[61,122],[15,50788,50790],{"className":50789},[338],[15,50791,50793,50813],{"className":50792},[342,472],[15,50794,50796,50810],{"className":50795},[346],[15,50797,50799],{"className":50798,"style":27010},[350],[15,50800,50801,50804],{"style":3202},[15,50802],{"className":50803,"style":358},[357],[15,50805,50807],{"className":50806},[362,363,364,365],[15,50808,79],{"className":50809,"style":123},[61,122,365],[15,50811,497],{"className":50812},[496],[15,50814,50816],{"className":50815},[346],[15,50817,50819],{"className":50818,"style":15696},[350],[15,50820],{},[15,50822],{"className":50823,"style":128},[127],[15,50825,21295],{"className":50826},[132],[15,50828],{"className":50829,"style":128},[127],[15,50831,50833,50836],{"className":50832},[52],[15,50834],{"className":50835,"style":57},[56],[15,50837,40230],{"className":50838,"style":40302},[61,122]," 保证修正项非负。",[11,50841,50842,50843,50846,50847,50875],{},"推论 4.2 揭示了一个在文献中常被忽略的事实：",[661,50844,50845],{},"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 ",[15,50848,50850,50863],{"className":50849,"translate":19},[18],[15,50851,50853],{"className":50852},[23],[25,50854,50855],{"xmlns":27},[29,50856,50857,50861],{},[32,50858,50859],{},[35,50860,40230],{},[40,50862,40230],{"encoding":42},[15,50864,50866],{"className":50865,"ariaHidden":48},[47],[15,50867,50869,50872],{"className":50868},[52],[15,50870],{"className":50871,"style":57},[56],[15,50873,40230],{"className":50874,"style":40302},[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。",[877,50877,50879],{"id":50878},"_42-平方损失的贝叶斯风险","4.2 平方损失的贝叶斯风险",[11,50881,50882],{},"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。",[882,50884,50886],{"title":50885},"命题 4.3（平方风险的闭式）",[11,50887,13,50888,50917,50918],{},[15,50889,50891,50905],{"className":50890,"translate":19},[18],[15,50892,50894],{"className":50893},[23],[25,50895,50896],{"xmlns":27},[29,50897,50898,50902],{},[32,50899,50900],{},[35,50901,79],{"mathvariant":251},[40,50903,50904],{"encoding":42},"\\mathbf{y}",[15,50906,50908],{"className":50907,"ariaHidden":48},[47],[15,50909,50911,50914],{"className":50910},[52],[15,50912],{"className":50913,"style":829},[56],[15,50915,79],{"className":50916,"style":29388},[61,411]," 为 one-hot 标签，则\n",[15,50919,50921,51046],{"className":50920,"translate":19},[18],[15,50922,50924],{"className":50923},[23],[25,50925,50926],{"xmlns":27},[29,50927,50928,51043],{},[32,50929,50930,50937,50939,50945,50965,50967,50981],{},[227,50931,50932,50934],{},[35,50933,47453],{"mathvariant":37},[282,50935,50936],{},"MSE",[81,50938,244],{},[266,50940,50941,50943],{},[35,50942,14807],{"mathvariant":258},[35,50944,24661],{"mathvariant":3101},[32,50946,50947,50949,50951,50953,50955,50957,50963],{},[81,50948,14810],{"fence":48},[35,50950,21278],{"mathvariant":231},[35,50952,79],{"mathvariant":251},[81,50954,239],{},[35,50956,24661],{"mathvariant":3101},[227,50958,50959,50961],{},[35,50960,21278],{"mathvariant":231},[89,50962,906],{},[81,50964,14819],{"fence":48},[81,50966,244],{},[289,50968,50969,50971,50979],{},[81,50970,293],{},[32,50972,50973,50975,50977],{},[35,50974,272],{},[81,50976,244],{},[89,50978,91],{},[35,50980,102],{},[32,50982,50983,50985,50987,50993,50995,51001,51007,51009,51041],{},[81,50984,14810],{"fence":48},[81,50986,741],{"stretchy":86},[266,50988,50989,50991],{},[35,50990,79],{},[35,50992,272],{},[81,50994,239],{},[266,50996,50997,50999],{},[35,50998,11],{},[35,51000,272],{},[227,51002,51003,51005],{},[81,51004,750],{"stretchy":86},[89,51006,906],{},[81,51008,7756],{},[3718,51010,51011,51033],{},[32,51012,51013,51019,51021,51023,51025,51031],{},[266,51014,51015,51017],{},[35,51016,11],{},[35,51018,272],{},[81,51020,741],{"stretchy":86},[89,51022,91],{},[81,51024,239],{},[266,51026,51027,51029],{},[35,51028,11],{},[35,51030,272],{},[81,51032,750],{"stretchy":86},[32,51034,51035,51037,51039],{},[35,51036,40230],{},[81,51038,7756],{},[89,51040,91],{},[81,51042,14819],{"fence":48},[40,51044,51045],{"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]",[15,51047,51049,51099,51234],{"className":51048,"ariaHidden":48},[47],[15,51050,51052,51055,51090,51093,51096],{"className":51051},[52],[15,51053],{"className":51054,"style":695},[56],[15,51056,51058,51061],{"className":51057},[61],[15,51059,47453],{"className":51060},[61,62],[15,51062,51064],{"className":51063},[338],[15,51065,51067],{"className":51066},[342],[15,51068,51070],{"className":51069},[346],[15,51071,51073],{"className":51072,"style":695},[350],[15,51074,51075,51078],{"style":716},[15,51076],{"className":51077,"style":358},[357],[15,51079,51081],{"className":51080},[362,363,364,365],[15,51082,51084],{"className":51083},[61,365],[15,51085,51087],{"className":51086},[61,13539,365],[15,51088,50936],{"className":51089},[61,365],[15,51091],{"className":51092,"style":128},[127],[15,51094,244],{"className":51095},[132],[15,51097],{"className":51098,"style":128},[127],[15,51100,51102,51105,51154,51157,51225,51228,51231],{"className":51101},[52],[15,51103],{"className":51104,"style":45239},[56],[15,51106,51108,51111],{"className":51107},[61],[15,51109,14807],{"className":51110},[61,427],[15,51112,51114],{"className":51113},[338],[15,51115,51117,51146],{"className":51116},[342,472],[15,51118,51120,51143],{"className":51119},[346],[15,51121,51123],{"className":51122,"style":20953},[350],[15,51124,51125,51128],{"style":482},[15,51126],{"className":51127,"style":358},[357],[15,51129,51131],{"className":51130},[362,363,364,365],[15,51132,51134],{"className":51133},[61,365],[15,51135,51137],{"className":51136},[61,365],[15,51138,51140],{"className":51139},[61,365],[15,51141,24661],{"className":51142,"style":24695},[61,3162,365],[15,51144,497],{"className":51145},[496],[15,51147,51149],{"className":51148},[346],[15,51150,51152],{"className":51151,"style":504},[350],[15,51153],{},[15,51155],{"className":51156,"style":157},[127],[15,51158,51160,51166,51169,51172,51175,51178,51181,51190,51219],{"className":51159},[161],[15,51161,51163],{"className":51162,"style":402},[146,401],[15,51164,14810],{"className":51165},[406,4063],[15,51167,21278],{"className":51168},[61],[15,51170,79],{"className":51171,"style":29388},[61,411],[15,51173],{"className":51174,"style":772},[127],[15,51176,239],{"className":51177},[375],[15,51179],{"className":51180,"style":772},[127],[15,51182,51184],{"className":51183},[61],[15,51185,51187],{"className":51186},[61],[15,51188,24661],{"className":51189,"style":24695},[61,3162],[15,51191,51193,51196],{"className":51192},[61],[15,51194,21278],{"className":51195},[61],[15,51197,51199],{"className":51198},[338],[15,51200,51202],{"className":51201},[342],[15,51203,51205],{"className":51204},[346],[15,51206,51208],{"className":51207,"style":7899},[350],[15,51209,51210,51213],{"style":716},[15,51211],{"className":51212,"style":358},[357],[15,51214,51216],{"className":51215},[362,363,364,365],[15,51217,906],{"className":51218},[61,365],[15,51220,51222],{"className":51221,"style":402},[178,401],[15,51223,14819],{"className":51224},[406,4063],[15,51226],{"className":51227,"style":128},[127],[15,51229,244],{"className":51230},[132],[15,51232],{"className":51233,"style":128},[127],[15,51235,51237,51240,51300,51303],{"className":51236},[52],[15,51238],{"className":51239,"style":394},[56],[15,51241,51243,51246],{"className":51242},[531],[15,51244,293],{"className":51245,"style":537},[531,535,536],[15,51247,51249],{"className":51248},[338],[15,51250,51252,51292],{"className":51251},[342,472],[15,51253,51255,51289],{"className":51254},[346],[15,51256,51258,51278],{"className":51257,"style":550},[350],[15,51259,51260,51263],{"style":553},[15,51261],{"className":51262,"style":358},[357],[15,51264,51266],{"className":51265},[362,363,364,365],[15,51267,51269,51272,51275],{"className":51268},[61,365],[15,51270,272],{"className":51271,"style":492},[61,122,365],[15,51273,244],{"className":51274},[132,365],[15,51276,91],{"className":51277},[61,365],[15,51279,51280,51283],{"style":574},[15,51281],{"className":51282,"style":358},[357],[15,51284,51286],{"className":51285},[362,363,364,365],[15,51287,102],{"className":51288,"style":174},[61,122,365],[15,51290,497],{"className":51291},[496],[15,51293,51295],{"className":51294},[346],[15,51296,51298],{"className":51297,"style":593},[350],[15,51299],{},[15,51301],{"className":51302,"style":157},[127],[15,51304,51306,51312,51315,51355,51358,51361,51364,51404,51433,51436,51439,51442,51605],{"className":51305},[161],[15,51307,51309],{"className":51308,"style":402},[146,401],[15,51310,14810],{"className":51311},[406,407],[15,51313,741],{"className":51314},[146],[15,51316,51318,51321],{"className":51317},[61],[15,51319,79],{"className":51320,"style":123},[61,122],[15,51322,51324],{"className":51323},[338],[15,51325,51327,51347],{"className":51326},[342,472],[15,51328,51330,51344],{"className":51329},[346],[15,51331,51333],{"className":51332,"style":479},[350],[15,51334,51335,51338],{"style":27013},[15,51336],{"className":51337,"style":358},[357],[15,51339,51341],{"className":51340},[362,363,364,365],[15,51342,272],{"className":51343,"style":492},[61,122,365],[15,51345,497],{"className":51346},[496],[15,51348,51350],{"className":51349},[346],[15,51351,51353],{"className":51352,"style":504},[350],[15,51354],{},[15,51356],{"className":51357,"style":772},[127],[15,51359,239],{"className":51360},[375],[15,51362],{"className":51363,"style":772},[127],[15,51365,51367,51370],{"className":51366},[61],[15,51368,11],{"className":51369},[61,122],[15,51371,51373],{"className":51372},[338],[15,51374,51376,51396],{"className":51375},[342,472],[15,51377,51379,51393],{"className":51378},[346],[15,51380,51382],{"className":51381,"style":479},[350],[15,51383,51384,51387],{"style":482},[15,51385],{"className":51386,"style":358},[357],[15,51388,51390],{"className":51389},[362,363,364,365],[15,51391,272],{"className":51392,"style":492},[61,122,365],[15,51394,497],{"className":51395},[496],[15,51397,51399],{"className":51398},[346],[15,51400,51402],{"className":51401,"style":504},[350],[15,51403],{},[15,51405,51407,51410],{"className":51406},[178],[15,51408,750],{"className":51409},[178],[15,51411,51413],{"className":51412},[338],[15,51414,51416],{"className":51415},[342],[15,51417,51419],{"className":51418},[346],[15,51420,51422],{"className":51421,"style":7899},[350],[15,51423,51424,51427],{"style":716},[15,51425],{"className":51426,"style":358},[357],[15,51428,51430],{"className":51429},[362,363,364,365],[15,51431,906],{"className":51432},[61,365],[15,51434],{"className":51435,"style":772},[127],[15,51437,7756],{"className":51438},[375],[15,51440],{"className":51441,"style":772},[127],[15,51443,51445,51448,51602],{"className":51444},[61],[15,51446],{"className":51447},[146,3837],[15,51449,51451],{"className":51450},[3718],[15,51452,51454,51594],{"className":51453},[342,472],[15,51455,51457,51591],{"className":51456},[346],[15,51458,51460,51480,51488],{"className":51459,"style":15310},[350],[15,51461,51462,51465],{"style":3853},[15,51463],{"className":51464,"style":3857},[357],[15,51466,51468],{"className":51467},[362,363,364,365],[15,51469,51471,51474,51477],{"className":51470},[61,365],[15,51472,40230],{"className":51473,"style":40302},[61,122,365],[15,51475,7756],{"className":51476},[375,365],[15,51478,91],{"className":51479},[61,365],[15,51481,51482,51485],{"style":3885},[15,51483],{"className":51484,"style":3857},[357],[15,51486],{"className":51487,"style":3893},[3892],[15,51489,51490,51493],{"style":15439},[15,51491],{"className":51492,"style":3857},[357],[15,51494,51496],{"className":51495},[362,363,364,365],[15,51497,51499,51539,51542,51545,51548,51588],{"className":51498},[61,365],[15,51500,51502,51505],{"className":51501},[61,365],[15,51503,11],{"className":51504},[61,122,365],[15,51506,51508],{"className":51507},[338],[15,51509,51511,51531],{"className":51510},[342,472],[15,51512,51514,51528],{"className":51513},[346],[15,51515,51517],{"className":51516,"style":22389},[350],[15,51518,51519,51522],{"style":27256},[15,51520],{"className":51521,"style":4058},[357],[15,51523,51525],{"className":51524},[362,4062,4063,365],[15,51526,272],{"className":51527,"style":492},[61,122,365],[15,51529,497],{"className":51530},[496],[15,51532,51534],{"className":51533},[346],[15,51535,51537],{"className":51536,"style":27230},[350],[15,51538],{},[15,51540,741],{"className":51541},[146,365],[15,51543,91],{"className":51544},[61,365],[15,51546,239],{"className":51547},[375,365],[15,51549,51551,51554],{"className":51550},[61,365],[15,51552,11],{"className":51553},[61,122,365],[15,51555,51557],{"className":51556},[338],[15,51558,51560,51580],{"className":51559},[342,472],[15,51561,51563,51577],{"className":51562},[346],[15,51564,51566],{"className":51565,"style":22389},[350],[15,51567,51568,51571],{"style":27256},[15,51569],{"className":51570,"style":4058},[357],[15,51572,51574],{"className":51573},[362,4062,4063,365],[15,51575,272],{"className":51576,"style":492},[61,122,365],[15,51578,497],{"className":51579},[496],[15,51581,51583],{"className":51582},[346],[15,51584,51586],{"className":51585,"style":27230},[350],[15,51587],{},[15,51589,750],{"className":51590},[178,365],[15,51592,497],{"className":51593},[496],[15,51595,51597],{"className":51596},[346],[15,51598,51600],{"className":51599,"style":46262},[350],[15,51601],{},[15,51603],{"className":51604},[178,3837],[15,51606,51608],{"className":51607,"style":402},[178,401],[15,51609,14819],{"className":51610},[406,407],[11,51612,51613,51615,51616,52051,52052,52612],{},[661,51614,5117],{}," 逐项展开：",[15,51617,51619,51699],{"className":51618,"translate":19},[18],[15,51620,51622],{"className":51621},[23],[25,51623,51624],{"xmlns":27},[29,51625,51626,51696],{},[32,51627,51628,51630,51632,51634,51640,51642,51648,51654,51656,51658,51660,51666,51668,51674,51680,51682,51684,51686,51688,51694],{},[35,51629,14807],{"mathvariant":258},[81,51631,14810],{"stretchy":86},[81,51633,741],{"stretchy":86},[266,51635,51636,51638],{},[35,51637,79],{},[35,51639,272],{},[81,51641,239],{},[266,51643,51644,51646],{},[35,51645,24661],{},[35,51647,272],{},[227,51649,51650,51652],{},[81,51651,750],{"stretchy":86},[89,51653,906],{},[81,51655,14819],{"stretchy":86},[81,51657,244],{},[81,51659,741],{"stretchy":86},[266,51661,51662,51664],{},[35,51663,79],{},[35,51665,272],{},[81,51667,239],{},[266,51669,51670,51672],{},[35,51671,11],{},[35,51673,272],{},[227,51675,51676,51678],{},[81,51677,750],{"stretchy":86},[89,51679,906],{},[81,51681,7756],{},[35,51683,14842],{"mathvariant":231},[81,51685,1545],{},[81,51687,741],{"stretchy":86},[266,51689,51690,51692],{},[35,51691,24661],{},[35,51693,272],{},[81,51695,750],{"stretchy":86},[40,51697,51698],{"encoding":42},"\\mathbb{E}[(y_k - \\pi_k)^2] = (y_k - p_k)^2 + \\operatorname{Var}(\\pi_k)",[15,51700,51702,51764,51851,51909,51993],{"className":51701,"ariaHidden":48},[47],[15,51703,51705,51708,51711,51715,51755,51758,51761],{"className":51704},[52],[15,51706],{"className":51707,"style":142},[56],[15,51709,14807],{"className":51710},[61,427],[15,51712,51714],{"className":51713},[146],"[(",[15,51716,51718,51721],{"className":51717},[61],[15,51719,79],{"className":51720,"style":123},[61,122],[15,51722,51724],{"className":51723},[338],[15,51725,51727,51747],{"className":51726},[342,472],[15,51728,51730,51744],{"className":51729},[346],[15,51731,51733],{"className":51732,"style":479},[350],[15,51734,51735,51738],{"style":27013},[15,51736],{"className":51737,"style":358},[357],[15,51739,51741],{"className":51740},[362,363,364,365],[15,51742,272],{"className":51743,"style":492},[61,122,365],[15,51745,497],{"className":51746},[496],[15,51748,51750],{"className":51749},[346],[15,51751,51753],{"className":51752,"style":504},[350],[15,51754],{},[15,51756],{"className":51757,"style":772},[127],[15,51759,239],{"className":51760},[375],[15,51762],{"className":51763,"style":772},[127],[15,51765,51767,51770,51810,51839,51842,51845,51848],{"className":51766},[52],[15,51768],{"className":51769,"style":21402},[56],[15,51771,51773,51776],{"className":51772},[61],[15,51774,24661],{"className":51775,"style":123},[61,122],[15,51777,51779],{"className":51778},[338],[15,51780,51782,51802],{"className":51781},[342,472],[15,51783,51785,51799],{"className":51784},[346],[15,51786,51788],{"className":51787,"style":479},[350],[15,51789,51790,51793],{"style":27013},[15,51791],{"className":51792,"style":358},[357],[15,51794,51796],{"className":51795},[362,363,364,365],[15,51797,272],{"className":51798,"style":492},[61,122,365],[15,51800,497],{"className":51801},[496],[15,51803,51805],{"className":51804},[346],[15,51806,51808],{"className":51807,"style":504},[350],[15,51809],{},[15,51811,51813,51816],{"className":51812},[178],[15,51814,750],{"className":51815},[178],[15,51817,51819],{"className":51818},[338],[15,51820,51822],{"className":51821},[342],[15,51823,51825],{"className":51824},[346],[15,51826,51828],{"className":51827,"style":7899},[350],[15,51829,51830,51833],{"style":716},[15,51831],{"className":51832,"style":358},[357],[15,51834,51836],{"className":51835},[362,363,364,365],[15,51837,906],{"className":51838},[61,365],[15,51840,14819],{"className":51841},[178],[15,51843],{"className":51844,"style":128},[127],[15,51846,244],{"className":51847},[132],[15,51849],{"className":51850,"style":128},[127],[15,51852,51854,51857,51860,51900,51903,51906],{"className":51853},[52],[15,51855],{"className":51856,"style":142},[56],[15,51858,741],{"className":51859},[146],[15,51861,51863,51866],{"className":51862},[61],[15,51864,79],{"className":51865,"style":123},[61,122],[15,51867,51869],{"className":51868},[338],[15,51870,51872,51892],{"className":51871},[342,472],[15,51873,51875,51889],{"className":51874},[346],[15,51876,51878],{"className":51877,"style":479},[350],[15,51879,51880,51883],{"style":27013},[15,51881],{"className":51882,"style":358},[357],[15,51884,51886],{"className":51885},[362,363,364,365],[15,51887,272],{"className":51888,"style":492},[61,122,365],[15,51890,497],{"className":51891},[496],[15,51893,51895],{"className":51894},[346],[15,51896,51898],{"className":51897,"style":504},[350],[15,51899],{},[15,51901],{"className":51902,"style":772},[127],[15,51904,239],{"className":51905},[375],[15,51907],{"className":51908,"style":772},[127],[15,51910,51912,51915,51955,51984,51987,51990],{"className":51911},[52],[15,51913],{"className":51914,"style":21402},[56],[15,51916,51918,51921],{"className":51917},[61],[15,51919,11],{"className":51920},[61,122],[15,51922,51924],{"className":51923},[338],[15,51925,51927,51947],{"className":51926},[342,472],[15,51928,51930,51944],{"className":51929},[346],[15,51931,51933],{"className":51932,"style":479},[350],[15,51934,51935,51938],{"style":482},[15,51936],{"className":51937,"style":358},[357],[15,51939,51941],{"className":51940},[362,363,364,365],[15,51942,272],{"className":51943,"style":492},[61,122,365],[15,51945,497],{"className":51946},[496],[15,51948,51950],{"className":51949},[346],[15,51951,51953],{"className":51952,"style":504},[350],[15,51954],{},[15,51956,51958,51961],{"className":51957},[178],[15,51959,750],{"className":51960},[178],[15,51962,51964],{"className":51963},[338],[15,51965,51967],{"className":51966},[342],[15,51968,51970],{"className":51969},[346],[15,51971,51973],{"className":51972,"style":7899},[350],[15,51974,51975,51978],{"style":716},[15,51976],{"className":51977,"style":358},[357],[15,51979,51981],{"className":51980},[362,363,364,365],[15,51982,906],{"className":51983},[61,365],[15,51985],{"className":51986,"style":772},[127],[15,51988,7756],{"className":51989},[375],[15,51991],{"className":51992,"style":772},[127],[15,51994,51996,51999,52005,52008,52048],{"className":51995},[52],[15,51997],{"className":51998,"style":142},[56],[15,52000,52002],{"className":52001},[531],[15,52003,14842],{"className":52004},[61,1620],[15,52006,741],{"className":52007},[146],[15,52009,52011,52014],{"className":52010},[61],[15,52012,24661],{"className":52013,"style":123},[61,122],[15,52015,52017],{"className":52016},[338],[15,52018,52020,52040],{"className":52019},[342,472],[15,52021,52023,52037],{"className":52022},[346],[15,52024,52026],{"className":52025,"style":479},[350],[15,52027,52028,52031],{"style":27013},[15,52029],{"className":52030,"style":358},[357],[15,52032,52034],{"className":52033},[362,363,364,365],[15,52035,272],{"className":52036,"style":492},[61,122,365],[15,52038,497],{"className":52039},[496],[15,52041,52043],{"className":52042},[346],[15,52044,52046],{"className":52045,"style":504},[350],[15,52047],{},[15,52049,750],{"className":52050},[178],"。由命题 2.1，",[15,52053,52055,52159],{"className":52054,"translate":19},[18],[15,52056,52058],{"className":52057},[23],[25,52059,52060],{"xmlns":27},[29,52061,52062,52156],{},[32,52063,52064,52066,52068,52070,52076,52078,52080,52122,52124],{},[35,52065,14842],{"mathvariant":231},[81,52067,1545],{},[81,52069,741],{"stretchy":86},[266,52071,52072,52074],{},[35,52073,24661],{},[35,52075,272],{},[81,52077,750],{"stretchy":86},[81,52079,244],{},[3718,52081,52082,52104],{},[32,52083,52084,52090,52092,52094,52096,52102],{},[266,52085,52086,52088],{},[35,52087,3102],{},[35,52089,272],{},[81,52091,741],{"stretchy":86},[35,52093,40230],{},[81,52095,239],{},[266,52097,52098,52100],{},[35,52099,3102],{},[35,52101,272],{},[81,52103,750],{"stretchy":86},[32,52105,52106,52112,52114,52116,52118,52120],{},[227,52107,52108,52110],{},[35,52109,40230],{},[89,52111,906],{},[81,52113,741],{"stretchy":86},[35,52115,40230],{},[81,52117,7756],{},[89,52119,91],{},[81,52121,750],{"stretchy":86},[81,52123,244],{},[3718,52125,52126,52148],{},[32,52127,52128,52134,52136,52138,52140,52146],{},[266,52129,52130,52132],{},[35,52131,11],{},[35,52133,272],{},[81,52135,741],{"stretchy":86},[89,52137,91],{},[81,52139,239],{},[266,52141,52142,52144],{},[35,52143,11],{},[35,52145,272],{},[81,52147,750],{"stretchy":86},[32,52149,52150,52152,52154],{},[35,52151,40230],{},[81,52153,7756],{},[89,52155,91],{},[40,52157,52158],{"encoding":42},"\\operatorname{Var}(\\pi_k) = \\frac{\\alpha_k(S - \\alpha_k)}{S^2(S+1)} = \\frac{p_k(1-p_k)}{S+1}",[15,52160,52162,52229,52442],{"className":52161,"ariaHidden":48},[47],[15,52163,52165,52168,52174,52177,52217,52220,52223,52226],{"className":52164},[52],[15,52166],{"className":52167,"style":142},[56],[15,52169,52171],{"className":52170},[531],[15,52172,14842],{"className":52173},[61,1620],[15,52175,741],{"className":52176},[146],[15,52178,52180,52183],{"className":52179},[61],[15,52181,24661],{"className":52182,"style":123},[61,122],[15,52184,52186],{"className":52185},[338],[15,52187,52189,52209],{"className":52188},[342,472],[15,52190,52192,52206],{"className":52191},[346],[15,52193,52195],{"className":52194,"style":479},[350],[15,52196,52197,52200],{"style":27013},[15,52198],{"className":52199,"style":358},[357],[15,52201,52203],{"className":52202},[362,363,364,365],[15,52204,272],{"className":52205,"style":492},[61,122,365],[15,52207,497],{"className":52208},[496],[15,52210,52212],{"className":52211},[346],[15,52213,52215],{"className":52214,"style":504},[350],[15,52216],{},[15,52218,750],{"className":52219},[178],[15,52221],{"className":52222,"style":128},[127],[15,52224,244],{"className":52225},[132],[15,52227],{"className":52228,"style":128},[127],[15,52230,52232,52235,52433,52436,52439],{"className":52231},[52],[15,52233],{"className":52234,"style":17647},[56],[15,52236,52238,52241,52430],{"className":52237},[61],[15,52239],{"className":52240},[146,3837],[15,52242,52244],{"className":52243},[3718],[15,52245,52247,52422],{"className":52246},[342,472],[15,52248,52250,52419],{"className":52249},[346],[15,52251,52253,52308,52316],{"className":52252,"style":15310},[350],[15,52254,52255,52258],{"style":3853},[15,52256],{"className":52257,"style":3857},[357],[15,52259,52261],{"className":52260},[362,363,364,365],[15,52262,52264,52293,52296,52299,52302,52305],{"className":52263},[61,365],[15,52265,52267,52270],{"className":52266},[61,365],[15,52268,40230],{"className":52269,"style":40302},[61,122,365],[15,52271,52273],{"className":52272},[338],[15,52274,52276],{"className":52275},[342],[15,52277,52279],{"className":52278},[346],[15,52280,52282],{"className":52281,"style":25413},[350],[15,52283,52284,52287],{"style":4877},[15,52285],{"className":52286,"style":4058},[357],[15,52288,52290],{"className":52289},[362,4062,4063,365],[15,52291,906],{"className":52292},[61,365],[15,52294,741],{"className":52295},[146,365],[15,52297,40230],{"className":52298,"style":40302},[61,122,365],[15,52300,7756],{"className":52301},[375,365],[15,52303,91],{"className":52304},[61,365],[15,52306,750],{"className":52307},[178,365],[15,52309,52310,52313],{"style":3885},[15,52311],{"className":52312,"style":3857},[357],[15,52314],{"className":52315,"style":3893},[3892],[15,52317,52318,52321],{"style":15439},[15,52319],{"className":52320,"style":3857},[357],[15,52322,52324],{"className":52323},[362,363,364,365],[15,52325,52327,52367,52370,52373,52376,52416],{"className":52326},[61,365],[15,52328,52330,52333],{"className":52329},[61,365],[15,52331,3102],{"className":52332,"style":3187},[61,122,365],[15,52334,52336],{"className":52335},[338],[15,52337,52339,52359],{"className":52338},[342,472],[15,52340,52342,52356],{"className":52341},[346],[15,52343,52345],{"className":52344,"style":22389},[350],[15,52346,52347,52350],{"style":27211},[15,52348],{"className":52349,"style":4058},[357],[15,52351,52353],{"className":52352},[362,4062,4063,365],[15,52354,272],{"className":52355,"style":492},[61,122,365],[15,52357,497],{"className":52358},[496],[15,52360,52362],{"className":52361},[346],[15,52363,52365],{"className":52364,"style":27230},[350],[15,52366],{},[15,52368,741],{"className":52369},[146,365],[15,52371,40230],{"className":52372,"style":40302},[61,122,365],[15,52374,239],{"className":52375},[375,365],[15,52377,52379,52382],{"className":52378},[61,365],[15,52380,3102],{"className":52381,"style":3187},[61,122,365],[15,52383,52385],{"className":52384},[338],[15,52386,52388,52408],{"className":52387},[342,472],[15,52389,52391,52405],{"className":52390},[346],[15,52392,52394],{"className":52393,"style":22389},[350],[15,52395,52396,52399],{"style":27211},[15,52397],{"className":52398,"style":4058},[357],[15,52400,52402],{"className":52401},[362,4062,4063,365],[15,52403,272],{"className":52404,"style":492},[61,122,365],[15,52406,497],{"className":52407},[496],[15,52409,52411],{"className":52410},[346],[15,52412,52414],{"className":52413,"style":27230},[350],[15,52415],{},[15,52417,750],{"className":52418},[178,365],[15,52420,497],{"className":52421},[496],[15,52423,52425],{"className":52424},[346],[15,52426,52428],{"className":52427,"style":3918},[350],[15,52429],{},[15,52431],{"className":52432},[178,3837],[15,52434],{"className":52435,"style":128},[127],[15,52437,244],{"className":52438},[132],[15,52440],{"className":52441,"style":128},[127],[15,52443,52445,52449],{"className":52444},[52],[15,52446],{"className":52447,"style":52448},[56],"height:1.4133em;vertical-align:-0.4033em;",[15,52450,52452,52455,52609],{"className":52451},[61],[15,52453],{"className":52454},[146,3837],[15,52456,52458],{"className":52457},[3718],[15,52459,52461,52601],{"className":52460},[342,472],[15,52462,52464,52598],{"className":52463},[346],[15,52465,52467,52487,52495],{"className":52466,"style":15310},[350],[15,52468,52469,52472],{"style":3853},[15,52470],{"className":52471,"style":3857},[357],[15,52473,52475],{"className":52474},[362,363,364,365],[15,52476,52478,52481,52484],{"className":52477},[61,365],[15,52479,40230],{"className":52480,"style":40302},[61,122,365],[15,52482,7756],{"className":52483},[375,365],[15,52485,91],{"className":52486},[61,365],[15,52488,52489,52492],{"style":3885},[15,52490],{"className":52491,"style":3857},[357],[15,52493],{"className":52494,"style":3893},[3892],[15,52496,52497,52500],{"style":15439},[15,52498],{"className":52499,"style":3857},[357],[15,52501,52503],{"className":52502},[362,363,364,365],[15,52504,52506,52546,52549,52552,52555,52595],{"className":52505},[61,365],[15,52507,52509,52512],{"className":52508},[61,365],[15,52510,11],{"className":52511},[61,122,365],[15,52513,52515],{"className":52514},[338],[15,52516,52518,52538],{"className":52517},[342,472],[15,52519,52521,52535],{"className":52520},[346],[15,52522,52524],{"className":52523,"style":22389},[350],[15,52525,52526,52529],{"style":27256},[15,52527],{"className":52528,"style":4058},[357],[15,52530,52532],{"className":52531},[362,4062,4063,365],[15,52533,272],{"className":52534,"style":492},[61,122,365],[15,52536,497],{"className":52537},[496],[15,52539,52541],{"className":52540},[346],[15,52542,52544],{"className":52543,"style":27230},[350],[15,52545],{},[15,52547,741],{"className":52548},[146,365],[15,52550,91],{"className":52551},[61,365],[15,52553,239],{"className":52554},[375,365],[15,52556,52558,52561],{"className":52557},[61,365],[15,52559,11],{"className":52560},[61,122,365],[15,52562,52564],{"className":52563},[338],[15,52565,52567,52587],{"className":52566},[342,472],[15,52568,52570,52584],{"className":52569},[346],[15,52571,52573],{"className":52572,"style":22389},[350],[15,52574,52575,52578],{"style":27256},[15,52576],{"className":52577,"style":4058},[357],[15,52579,52581],{"className":52580},[362,4062,4063,365],[15,52582,272],{"className":52583,"style":492},[61,122,365],[15,52585,497],{"className":52586},[496],[15,52588,52590],{"className":52589},[346],[15,52591,52593],{"className":52592,"style":27230},[350],[15,52594],{},[15,52596,750],{"className":52597},[178,365],[15,52599,497],{"className":52600},[496],[15,52602,52604],{"className":52603},[346],[15,52605,52607],{"className":52606,"style":46262},[350],[15,52608],{},[15,52610],{"className":52611},[178,3837],"。求和即得。",[11,52614,52615,52616,52619,52620,52690,52691,52694,52695,52843,52844,52895],{},"该损失的两项具有不同的角色。",[661,52617,52618],{},"第一项是拟合项","：在 ",[15,52621,52623,52641],{"className":52622,"translate":19},[18],[15,52624,52626],{"className":52625},[23],[25,52627,52628],{"xmlns":27},[29,52629,52630,52638],{},[32,52631,52632],{},[266,52633,52634,52636],{},[35,52635,11],{},[35,52637,272],{},[40,52639,52640],{"encoding":42},"p_k",[15,52642,52644],{"className":52643,"ariaHidden":48},[47],[15,52645,52647,52650],{"className":52646},[52],[15,52648],{"className":52649,"style":9428},[56],[15,52651,52653,52656],{"className":52652},[61],[15,52654,11],{"className":52655},[61,122],[15,52657,52659],{"className":52658},[338],[15,52660,52662,52682],{"className":52661},[342,472],[15,52663,52665,52679],{"className":52664},[346],[15,52666,52668],{"className":52667,"style":479},[350],[15,52669,52670,52673],{"style":482},[15,52671],{"className":52672,"style":358},[357],[15,52674,52676],{"className":52675},[362,363,364,365],[15,52677,272],{"className":52678,"style":492},[61,122,365],[15,52680,497],{"className":52681},[496],[15,52683,52685],{"className":52684},[346],[15,52686,52688],{"className":52687,"style":504},[350],[15,52689],{}," 退化为确定性预测时，它退化为 one-hot 编码下的均方误差。",[661,52692,52693],{},"第二项是方差惩罚项","：它正比于每个分量的伯努利方差 ",[15,52696,52698,52730],{"className":52697,"translate":19},[18],[15,52699,52701],{"className":52700},[23],[25,52702,52703],{"xmlns":27},[29,52704,52705,52727],{},[32,52706,52707,52713,52715,52717,52719,52725],{},[266,52708,52709,52711],{},[35,52710,11],{},[35,52712,272],{},[81,52714,741],{"stretchy":86},[89,52716,91],{},[81,52718,239],{},[266,52720,52721,52723],{},[35,52722,11],{},[35,52724,272],{},[81,52726,750],{"stretchy":86},[40,52728,52729],{"encoding":42},"p_k(1-p_k)",[15,52731,52733,52794],{"className":52732,"ariaHidden":48},[47],[15,52734,52736,52739,52779,52782,52785,52788,52791],{"className":52735},[52],[15,52737],{"className":52738,"style":142},[56],[15,52740,52742,52745],{"className":52741},[61],[15,52743,11],{"className":52744},[61,122],[15,52746,52748],{"className":52747},[338],[15,52749,52751,52771],{"className":52750},[342,472],[15,52752,52754,52768],{"className":52753},[346],[15,52755,52757],{"className":52756,"style":479},[350],[15,52758,52759,52762],{"style":482},[15,52760],{"className":52761,"style":358},[357],[15,52763,52765],{"className":52764},[362,363,364,365],[15,52766,272],{"className":52767,"style":492},[61,122,365],[15,52769,497],{"className":52770},[496],[15,52772,52774],{"className":52773},[346],[15,52775,52777],{"className":52776,"style":504},[350],[15,52778],{},[15,52780,741],{"className":52781},[146],[15,52783,91],{"className":52784},[61],[15,52786],{"className":52787,"style":772},[127],[15,52789,239],{"className":52790},[375],[15,52792],{"className":52793,"style":772},[127],[15,52795,52797,52800,52840],{"className":52796},[52],[15,52798],{"className":52799,"style":142},[56],[15,52801,52803,52806],{"className":52802},[61],[15,52804,11],{"className":52805},[61,122],[15,52807,52809],{"className":52808},[338],[15,52810,52812,52832],{"className":52811},[342,472],[15,52813,52815,52829],{"className":52814},[346],[15,52816,52818],{"className":52817,"style":479},[350],[15,52819,52820,52823],{"style":482},[15,52821],{"className":52822,"style":358},[357],[15,52824,52826],{"className":52825},[362,363,364,365],[15,52827,272],{"className":52828,"style":492},[61,122,365],[15,52830,497],{"className":52831},[496],[15,52833,52835],{"className":52834},[346],[15,52836,52838],{"className":52837,"style":504},[350],[15,52839],{},[15,52841,750],{"className":52842},[178],"，反比于总证据 ",[15,52845,52847,52865],{"className":52846,"translate":19},[18],[15,52848,52850],{"className":52849},[23],[25,52851,52852],{"xmlns":27},[29,52853,52854,52862],{},[32,52855,52856,52858,52860],{},[35,52857,40230],{},[81,52859,7756],{},[89,52861,91],{},[40,52863,52864],{"encoding":42},"S + 1",[15,52866,52868,52886],{"className":52867,"ariaHidden":48},[47],[15,52869,52871,52874,52877,52880,52883],{"className":52870},[52],[15,52872],{"className":52873,"style":972},[56],[15,52875,40230],{"className":52876,"style":40302},[61,122],[15,52878],{"className":52879,"style":772},[127],[15,52881,7756],{"className":52882},[375],[15,52884],{"className":52885,"style":772},[127],[15,52887,52889,52892],{"className":52888},[52],[15,52890],{"className":52891,"style":859},[56],[15,52893,91],{"className":52894},[61],"。其作用是双重的：",[31520,52897,52898,52934],{},[31523,52899,52900,52901,52929,52930,52933],{},"当证据稀缺（",[15,52902,52904,52917],{"className":52903,"translate":19},[18],[15,52905,52907],{"className":52906},[23],[25,52908,52909],{"xmlns":27},[29,52910,52911,52915],{},[32,52912,52913],{},[35,52914,40230],{},[40,52916,40230],{"encoding":42},[15,52918,52920],{"className":52919,"ariaHidden":48},[47],[15,52921,52923,52926],{"className":52922},[52],[15,52924],{"className":52925,"style":57},[56],[15,52927,40230],{"className":52928,"style":40302},[61,122]," 小）时，方差项较大，迫使模型",[661,52931,52932],{},"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。",[31523,52935,52936,52937,9342,52965,53034,53035,53063],{},"当证据变化时，",[15,52938,52940,52953],{"className":52939,"translate":19},[18],[15,52941,52943],{"className":52942},[23],[25,52944,52945],{"xmlns":27},[29,52946,52947,52951],{},[32,52948,52949],{},[35,52950,40230],{},[40,52952,40230],{"encoding":42},[15,52954,52956],{"className":52955,"ariaHidden":48},[47],[15,52957,52959,52962],{"className":52958},[52],[15,52960],{"className":52961,"style":57},[56],[15,52963,40230],{"className":52964,"style":40302},[61,122],[15,52966,52968,52985],{"className":52967,"translate":19},[18],[15,52969,52971],{"className":52970},[23],[25,52972,52973],{"xmlns":27},[29,52974,52975,52983],{},[32,52976,52977],{},[266,52978,52979,52981],{},[35,52980,11],{},[35,52982,272],{},[40,52984,52640],{"encoding":42},[15,52986,52988],{"className":52987,"ariaHidden":48},[47],[15,52989,52991,52994],{"className":52990},[52],[15,52992],{"className":52993,"style":9428},[56],[15,52995,52997,53000],{"className":52996},[61],[15,52998,11],{"className":52999},[61,122],[15,53001,53003],{"className":53002},[338],[15,53004,53006,53026],{"className":53005},[342,472],[15,53007,53009,53023],{"className":53008},[346],[15,53010,53012],{"className":53011,"style":479},[350],[15,53013,53014,53017],{"style":482},[15,53015],{"className":53016,"style":358},[357],[15,53018,53020],{"className":53019},[362,363,364,365],[15,53021,272],{"className":53022,"style":492},[61,122,365],[15,53024,497],{"className":53025},[496],[15,53027,53029],{"className":53028},[346],[15,53030,53032],{"className":53031,"style":504},[350],[15,53033],{}," 同时进入两项，形成张力：增大 ",[15,53036,53038,53051],{"className":53037,"translate":19},[18],[15,53039,53041],{"className":53040},[23],[25,53042,53043],{"xmlns":27},[29,53044,53045,53049],{},[32,53046,53047],{},[35,53048,40230],{},[40,53050,40230],{"encoding":42},[15,53052,53054],{"className":53053,"ariaHidden":48},[47],[15,53055,53057,53060],{"className":53056},[52],[15,53058],{"className":53059,"style":57},[56],[15,53061,40230],{"className":53062,"style":40302},[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。",[877,53065,53067],{"id":53066},"_43-证据正则化与退火-kl","4.3 证据正则化与退火 KL",[11,53069,53070,53071,53074,53075,53152,53153,53156],{},"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",[661,53072,53073],{},"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 ",[15,53076,53078,53102],{"className":53077,"translate":19},[18],[15,53079,53081],{"className":53080},[23],[25,53082,53083],{"xmlns":27},[29,53084,53085,53099],{},[32,53086,53087,53093,53095,53097],{},[227,53088,53089,53091],{},[35,53090,22676],{},[81,53092,23145],{"mathvariant":231,"lspace":23144,"rspace":23144},[81,53094,741],{"stretchy":86},[35,53096,40230],{},[81,53098,750],{"stretchy":86},[40,53100,53101],{"encoding":42},"\\psi'(S)",[15,53103,53105],{"className":53104,"ariaHidden":48},[47],[15,53106,53108,53111,53143,53146,53149],{"className":53107},[52],[15,53109],{"className":53110,"style":23180},[56],[15,53112,53114,53117],{"className":53113},[61],[15,53115,22676],{"className":53116,"style":123},[61,122],[15,53118,53120],{"className":53119},[338],[15,53121,53123],{"className":53122},[342],[15,53124,53126],{"className":53125},[346],[15,53127,53129],{"className":53128,"style":23199},[350],[15,53130,53131,53134],{"style":716},[15,53132],{"className":53133,"style":358},[357],[15,53135,53137],{"className":53136},[362,363,364,365],[15,53138,53140],{"className":53139},[61,365],[15,53141,23145],{"className":53142},[61,365],[15,53144,741],{"className":53145},[146],[15,53147,40230],{"className":53148,"style":40302},[61,122],[15,53150,750],{"className":53151},[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：",[661,53154,53155],{},"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。",[11,53158,53159],{},"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。",[882,53161,53163],{"title":53162},"定义 4.4（反事实参数与证据正则）",[11,53164,53165,53166,53347,53348,53376,53377],{},"令 ",[15,53167,53169,53210],{"className":53168,"translate":19},[18],[15,53170,53172],{"className":53171},[23],[25,53173,53174],{"xmlns":27},[29,53175,53176,53207],{},[32,53177,53178,53186,53188,53190,53192,53194,53196,53198,53200,53202,53205],{},[53179,53180,53181,53183],"mover",{"accent":48},[35,53182,3102],{"mathvariant":3101},[81,53184,53185],{},"~",[81,53187,244],{},[35,53189,79],{"mathvariant":251},[81,53191,7756],{},[81,53193,741],{"stretchy":86},[89,53195,91],{"mathvariant":251},[81,53197,239],{},[35,53199,79],{"mathvariant":251},[81,53201,750],{"stretchy":86},[81,53203,53204],{},"⊙",[35,53206,3102],{"mathvariant":3101},[40,53208,53209],{"encoding":42},"\\tilde{\\boldsymbol{\\alpha}} = \\mathbf{y} + (\\mathbf{1} - \\mathbf{y}) \\odot \\boldsymbol{\\alpha}",[15,53211,53213,53271,53290,53311,53332],{"className":53212,"ariaHidden":48},[47],[15,53214,53216,53220,53262,53265,53268],{"className":53215},[52],[15,53217],{"className":53218,"style":53219},[56],"height:0.6813em;",[15,53221,53224],{"className":53222},[61,53223],"accent",[15,53225,53227],{"className":53226},[342],[15,53228,53230],{"className":53229},[346],[15,53231,53233,53248],{"className":53232,"style":53219},[350],[15,53234,53236,53239],{"style":53235},"top:-3em;",[15,53237],{"className":53238,"style":3857},[357],[15,53240,53242],{"className":53241},[61],[15,53243,53245],{"className":53244},[61],[15,53246,3102],{"className":53247},[61,3162],[15,53249,53251,53254],{"style":53250},"top:-3.3634em;",[15,53252],{"className":53253,"style":3857},[357],[15,53255,53259],{"className":53256,"style":53258},[53257],"accent-body","left:-0.25em;",[15,53260,53185],{"className":53261},[61],[15,53263],{"className":53264,"style":128},[127],[15,53266,244],{"className":53267},[132],[15,53269],{"className":53270,"style":128},[127],[15,53272,53274,53278,53281,53284,53287],{"className":53273},[52],[15,53275],{"className":53276,"style":53277},[56],"height:0.7778em;vertical-align:-0.1944em;",[15,53279,79],{"className":53280,"style":29388},[61,411],[15,53282],{"className":53283,"style":772},[127],[15,53285,7756],{"className":53286},[375],[15,53288],{"className":53289,"style":772},[127],[15,53291,53293,53296,53299,53302,53305,53308],{"className":53292},[52],[15,53294],{"className":53295,"style":142},[56],[15,53297,741],{"className":53298},[146],[15,53300,91],{"className":53301},[61,411],[15,53303],{"className":53304,"style":772},[127],[15,53306,239],{"className":53307},[375],[15,53309],{"className":53310,"style":772},[127],[15,53312,53314,53317,53320,53323,53326,53329],{"className":53313},[52],[15,53315],{"className":53316,"style":142},[56],[15,53318,79],{"className":53319,"style":29388},[61,411],[15,53321,750],{"className":53322},[178],[15,53324],{"className":53325,"style":772},[127],[15,53327,53204],{"className":53328},[375],[15,53330],{"className":53331,"style":772},[127],[15,53333,53335,53338],{"className":53334},[52],[15,53336],{"className":53337,"style":2706},[56],[15,53339,53341],{"className":53340},[61],[15,53342,53344],{"className":53343},[61],[15,53345,3102],{"className":53346},[61,3162],"，即把真实类别处的参数置为 ",[15,53349,53351,53364],{"className":53350,"translate":19},[18],[15,53352,53354],{"className":53353},[23],[25,53355,53356],{"xmlns":27},[29,53357,53358,53362],{},[32,53359,53360],{},[89,53361,91],{},[40,53363,91],{"encoding":42},[15,53365,53367],{"className":53366,"ariaHidden":48},[47],[15,53368,53370,53373],{"className":53369},[52],[15,53371],{"className":53372,"style":859},[56],[15,53374,91],{"className":53375},[61],"、其余保持不变。证据正则项定义为\n",[15,53378,53380,53442],{"className":53379,"translate":19},[18],[15,53381,53383],{"className":53382},[23],[25,53384,53385],{"xmlns":27},[29,53386,53387,53439],{},[32,53388,53389,53396,53398,53401,53403],{},[227,53390,53391,53393],{},[35,53392,47453],{"mathvariant":37},[282,53394,53395],{},"reg",[81,53397,244],{},[35,53399,53400],{"mathvariant":231},"KL",[81,53402,1545],{},[32,53404,53405,53407,53409,53411,53413,53419,53421,53423,53425,53427,53429,53431,53433,53435,53437],{},[81,53406,14810],{"fence":48},[35,53408,4182],{"mathvariant":231},[81,53410,1545],{},[81,53412,741],{"stretchy":86},[53179,53414,53415,53417],{"accent":48},[35,53416,3102],{"mathvariant":3101},[81,53418,53185],{},[81,53420,750],{"stretchy":86},[282,53422,4796],{},[81,53424,21278],{"fence":86,"stretchy":48,"minsize":44812,"maxsize":44812},[282,53426,4796],{},[35,53428,4182],{"mathvariant":231},[81,53430,1545],{},[81,53432,741],{"stretchy":86},[89,53434,91],{"mathvariant":251},[81,53436,750],{"stretchy":86},[81,53438,14819],{"fence":48},[40,53440,53441],{"encoding":42},"\\mathcal{L}^{\\text{reg}} = \\operatorname{KL}\\left[\\operatorname{Dir}(\\tilde{\\boldsymbol{\\alpha}}) \\,\\big\\|\\, \\operatorname{Dir}(\\mathbf{1})\\right]",[15,53443,53445,53495],{"className":53444,"ariaHidden":48},[47],[15,53446,53448,53451,53486,53489,53492],{"className":53447},[52],[15,53449],{"className":53450,"style":57},[56],[15,53452,53454,53457],{"className":53453},[61],[15,53455,47453],{"className":53456},[61,62],[15,53458,53460],{"className":53459},[338],[15,53461,53463],{"className":53462},[342],[15,53464,53466],{"className":53465},[346],[15,53467,53469],{"className":53468,"style":37580},[350],[15,53470,53471,53474],{"style":716},[15,53472],{"className":53473,"style":358},[357],[15,53475,53477],{"className":53476},[362,363,364,365],[15,53478,53480],{"className":53479},[61,365],[15,53481,53483],{"className":53482},[61,13539,365],[15,53484,53395],{"className":53485},[61,365],[15,53487],{"className":53488,"style":128},[127],[15,53490,244],{"className":53491},[132],[15,53493],{"className":53494,"style":128},[127],[15,53496,53498,53501,53507,53510],{"className":53497},[52],[15,53499],{"className":53500,"style":45239},[56],[15,53502,53504],{"className":53503},[531],[15,53505,53400],{"className":53506},[61,1620],[15,53508],{"className":53509,"style":157},[127],[15,53511,53513,53519,53525,53528,53565,53568,53571,53617,53620,53623,53629,53632,53635,53638],{"className":53512},[161],[15,53514,53516],{"className":53515,"style":402},[146,401],[15,53517,14810],{"className":53518},[406,4063],[15,53520,53522],{"className":53521},[531],[15,53523,4182],{"className":53524},[61,1620],[15,53526,741],{"className":53527},[146],[15,53529,53531],{"className":53530},[61,53223],[15,53532,53534],{"className":53533},[342],[15,53535,53537],{"className":53536},[346],[15,53538,53540,53554],{"className":53539,"style":53219},[350],[15,53541,53542,53545],{"style":53235},[15,53543],{"className":53544,"style":3857},[357],[15,53546,53548],{"className":53547},[61],[15,53549,53551],{"className":53550},[61],[15,53552,3102],{"className":53553},[61,3162],[15,53555,53556,53559],{"style":53250},[15,53557],{"className":53558,"style":3857},[357],[15,53560,53562],{"className":53561,"style":53258},[53257],[15,53563,53185],{"className":53564},[61],[15,53566,750],{"className":53567},[178],[15,53569],{"className":53570,"style":157},[127],[15,53572,53574],{"className":53573},[61],[15,53575,53578],{"className":53576},[406,53577],"mult",[15,53579,53581,53608],{"className":53580},[342,472],[15,53582,53584,53605],{"className":53583},[346],[15,53585,53588],{"className":53586,"style":53587},[350],"height:0.85em;",[15,53589,53591,53595],{"style":53590},"top:-2.85em;",[15,53592],{"className":53593,"style":53594},[357],"height:3.2em;",[15,53596,53598],{"style":53597},"width:0.556em;height:1.2em;",[27683,53599,53602],{"xmlns":27685,"width":53600,"height":44812,"viewBox":53601},"0.556em","0 0 556 1200",[27691,53603],{"d":53604},"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",[15,53606,497],{"className":53607},[496],[15,53609,53611],{"className":53610},[346],[15,53612,53615],{"className":53613,"style":53614},[350],"height:0.35em;",[15,53616],{},[15,53618],{"className":53619,"style":157},[127],[15,53621],{"className":53622,"style":157},[127],[15,53624,53626],{"className":53625},[531],[15,53627,4182],{"className":53628},[61,1620],[15,53630,741],{"className":53631},[146],[15,53633,91],{"className":53634},[61,411],[15,53636,750],{"className":53637},[178],[15,53639,53641],{"className":53640,"style":402},[178,401],[15,53642,14819],{"className":53643},[406,4063],[11,53645,53646,53647,53696,53697,53784],{},"注意 ",[15,53648,53650,53672],{"className":53649,"translate":19},[18],[15,53651,53653],{"className":53652},[23],[25,53654,53655],{"xmlns":27},[29,53656,53657,53669],{},[32,53658,53659,53661,53663,53665,53667],{},[35,53660,4182],{"mathvariant":231},[81,53662,1545],{},[81,53664,741],{"stretchy":86},[89,53666,91],{"mathvariant":251},[81,53668,750],{"stretchy":86},[40,53670,53671],{"encoding":42},"\\operatorname{Dir}(\\mathbf{1})",[15,53673,53675],{"className":53674,"ariaHidden":48},[47],[15,53676,53678,53681,53687,53690,53693],{"className":53677},[52],[15,53679],{"className":53680,"style":142},[56],[15,53682,53684],{"className":53683},[531],[15,53685,4182],{"className":53686},[61,1620],[15,53688,741],{"className":53689},[146],[15,53691,91],{"className":53692},[61,411],[15,53694,750],{"className":53695},[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，",[15,53698,53700,53726],{"className":53699,"translate":19},[18],[15,53701,53703],{"className":53702},[23],[25,53704,53705],{"xmlns":27},[29,53706,53707,53723],{},[32,53708,53709,53711,53713,53715,53721],{},[35,53710,4182],{"mathvariant":231},[81,53712,1545],{},[81,53714,741],{"stretchy":86},[53179,53716,53717,53719],{"accent":48},[35,53718,3102],{"mathvariant":3101},[81,53720,53185],{},[81,53722,750],{"stretchy":86},[40,53724,53725],{"encoding":42},"\\operatorname{Dir}(\\tilde{\\boldsymbol{\\alpha}})",[15,53727,53729],{"className":53728,"ariaHidden":48},[47],[15,53730,53732,53735,53741,53744,53781],{"className":53731},[52],[15,53733],{"className":53734,"style":142},[56],[15,53736,53738],{"className":53737},[531],[15,53739,4182],{"className":53740},[61,1620],[15,53742,741],{"className":53743},[146],[15,53745,53747],{"className":53746},[61,53223],[15,53748,53750],{"className":53749},[342],[15,53751,53753],{"className":53752},[346],[15,53754,53756,53770],{"className":53755,"style":53219},[350],[15,53757,53758,53761],{"style":53235},[15,53759],{"className":53760,"style":3857},[357],[15,53762,53764],{"className":53763},[61],[15,53765,53767],{"className":53766},[61],[15,53768,3102],{"className":53769},[61,3162],[15,53771,53772,53775],{"style":53250},[15,53773],{"className":53774,"style":3857},[357],[15,53776,53778],{"className":53777,"style":53258},[53257],[15,53779,53185],{"className":53780},[61],[15,53782,750],{"className":53783},[178]," 将显著偏离均匀分布，KL 随之增大。",[882,53786,53788],{"title":53787},"命题 4.5（Dirichlet 的 KL 散度）",[11,53789,53790,53791,4732,53931,54617,54618,54675,54676,54762,54763],{},"对 ",[15,53792,53794,53826],{"className":53793,"translate":19},[18],[15,53795,53797],{"className":53796},[23],[25,53798,53799],{"xmlns":27},[29,53800,53801,53823],{},[32,53802,53803,53805,53807,53809,53811],{},[35,53804,3102],{"mathvariant":3101},[81,53806,94],{"separator":48},[35,53808,17120],{"mathvariant":3101},[81,53810,83],{},[289,53812,53813,53815,53821],{},[35,53814,259],{"mathvariant":258},[32,53816,53817,53819],{},[81,53818,1588],{},[89,53820,278],{},[35,53822,102],{},[40,53824,53825],{"encoding":42},"\\boldsymbol{\\alpha}, \\boldsymbol{\\beta} \\in \\mathbb{R}_{>0}^K",[15,53827,53829,53868],{"className":53828,"ariaHidden":48},[47],[15,53830,53832,53835,53844,53847,53850,53859,53862,53865],{"className":53831},[52],[15,53833],{"className":53834,"style":14301},[56],[15,53836,53838],{"className":53837},[61],[15,53839,53841],{"className":53840},[61],[15,53842,3102],{"className":53843},[61,3162],[15,53845,94],{"className":53846},[153],[15,53848],{"className":53849,"style":157},[127],[15,53851,53853],{"className":53852},[61],[15,53854,53856],{"className":53855},[61],[15,53857,17120],{"className":53858,"style":17363},[61,3162],[15,53860],{"className":53861,"style":128},[127],[15,53863,83],{"className":53864},[132],[15,53866],{"className":53867,"style":128},[127],[15,53869,53871,53874],{"className":53870},[52],[15,53872],{"className":53873,"style":3299},[56],[15,53875,53877,53880],{"className":53876},[61],[15,53878,259],{"className":53879},[61,427],[15,53881,53883],{"className":53882},[338],[15,53884,53886,53923],{"className":53885},[342,472],[15,53887,53889,53920],{"className":53888},[346],[15,53890,53892,53909],{"className":53891,"style":695},[350],[15,53893,53894,53897],{"style":3320},[15,53895],{"className":53896,"style":358},[357],[15,53898,53900],{"className":53899},[362,363,364,365],[15,53901,53903,53906],{"className":53902},[61,365],[15,53904,1588],{"className":53905},[132,365],[15,53907,278],{"className":53908},[61,365],[15,53910,53911,53914],{"style":716},[15,53912],{"className":53913,"style":358},[357],[15,53915,53917],{"className":53916},[362,363,364,365],[15,53918,102],{"className":53919,"style":174},[61,122,365],[15,53921,497],{"className":53922},[496],[15,53924,53926],{"className":53925},[346],[15,53927,53929],{"className":53928,"style":3356},[350],[15,53930],{},[15,53932,53934,54074],{"className":53933,"translate":19},[18],[15,53935,53937],{"className":53936},[23],[25,53938,53939],{"xmlns":27},[29,53940,53941,54071],{},[32,53942,53943,53945,53947,53979,53981,53983,53985,54007,54009,54023,54025,54031,54033,54039,54041,54043,54045,54047,54053,54055,54057,54059,54061,54067,54069],{},[35,53944,53400],{"mathvariant":231},[81,53946,1545],{},[32,53948,53949,53951,53953,53955,53957,53959,53961,53963,53965,53967,53969,53971,53973,53975,53977],{},[81,53950,14810],{"fence":48},[35,53952,4182],{"mathvariant":231},[81,53954,1545],{},[81,53956,741],{"stretchy":86},[35,53958,3102],{"mathvariant":3101},[81,53960,750],{"stretchy":86},[282,53962,4796],{},[81,53964,21278],{"fence":86,"stretchy":48,"minsize":44812,"maxsize":44812},[282,53966,4796],{},[35,53968,4182],{"mathvariant":231},[81,53970,1545],{},[81,53972,741],{"stretchy":86},[35,53974,17120],{"mathvariant":3101},[81,53976,750],{"stretchy":86},[81,53978,14819],{"fence":48},[81,53980,244],{},[35,53982,22645],{},[81,53984,1545],{},[3718,53986,53987,53997],{},[32,53988,53989,53991,53993,53995],{},[35,53990,3726],{},[81,53992,741],{"stretchy":86},[35,53994,17120],{"mathvariant":3101},[81,53996,750],{"stretchy":86},[32,53998,53999,54001,54003,54005],{},[35,54000,3726],{},[81,54002,741],{"stretchy":86},[35,54004,3102],{"mathvariant":3101},[81,54006,750],{"stretchy":86},[81,54008,7756],{},[289,54010,54011,54013,54021],{},[81,54012,293],{},[32,54014,54015,54017,54019],{},[35,54016,272],{},[81,54018,244],{},[89,54020,91],{},[35,54022,102],{},[81,54024,741],{"stretchy":86},[266,54026,54027,54029],{},[35,54028,3102],{},[35,54030,272],{},[81,54032,239],{},[266,54034,54035,54037],{},[35,54036,17120],{},[35,54038,272],{},[81,54040,750],{"stretchy":86},[81,54042,741],{"fence":86,"stretchy":48,"minsize":44812,"maxsize":44812},[35,54044,22676],{},[81,54046,741],{"stretchy":86},[266,54048,54049,54051],{},[35,54050,3102],{},[35,54052,272],{},[81,54054,750],{"stretchy":86},[81,54056,239],{},[35,54058,22676],{},[81,54060,741],{"stretchy":86},[266,54062,54063,54065],{},[35,54064,3102],{},[89,54066,278],{},[81,54068,750],{"stretchy":86},[81,54070,750],{"fence":86,"stretchy":48,"minsize":44812,"maxsize":44812},[40,54072,54073],{"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)",[15,54075,54077,54204,54325,54443,54556],{"className":54076,"ariaHidden":48},[47],[15,54078,54080,54083,54089,54092,54195,54198,54201],{"className":54079},[52],[15,54081],{"className":54082,"style":45239},[56],[15,54084,54086],{"className":54085},[531],[15,54087,53400],{"className":54088},[61,1620],[15,54090],{"className":54091,"style":157},[127],[15,54093,54095,54101,54107,54110,54119,54122,54125,54162,54165,54168,54174,54177,54186,54189],{"className":54094},[161],[15,54096,54098],{"className":54097,"style":402},[146,401],[15,54099,14810],{"className":54100},[406,4063],[15,54102,54104],{"className":54103},[531],[15,54105,4182],{"className":54106},[61,1620],[15,54108,741],{"className":54109},[146],[15,54111,54113],{"className":54112},[61],[15,54114,54116],{"className":54115},[61],[15,54117,3102],{"className":54118},[61,3162],[15,54120,750],{"className":54121},[178],[15,54123],{"className":54124,"style":157},[127],[15,54126,54128],{"className":54127},[61],[15,54129,54131],{"className":54130},[406,53577],[15,54132,54134,54154],{"className":54133},[342,472],[15,54135,54137,54151],{"className":54136},[346],[15,54138,54140],{"className":54139,"style":53587},[350],[15,54141,54142,54145],{"style":53590},[15,54143],{"className":54144,"style":53594},[357],[15,54146,54147],{"style":53597},[27683,54148,54149],{"xmlns":27685,"width":53600,"height":44812,"viewBox":53601},[27691,54150],{"d":53604},[15,54152,497],{"className":54153},[496],[15,54155,54157],{"className":54156},[346],[15,54158,54160],{"className":54159,"style":53614},[350],[15,54161],{},[15,54163],{"className":54164,"style":157},[127],[15,54166],{"className":54167,"style":157},[127],[15,54169,54171],{"className":54170},[531],[15,54172,4182],{"className":54173},[61,1620],[15,54175,741],{"className":54176},[146],[15,54178,54180],{"className":54179},[61],[15,54181,54183],{"className":54182},[61],[15,54184,17120],{"className":54185,"style":17363},[61,3162],[15,54187,750],{"className":54188},[178],[15,54190,54192],{"className":54191,"style":402},[178,401],[15,54193,14819],{"className":54194},[406,4063],[15,54196],{"className":54197,"style":128},[127],[15,54199,244],{"className":54200},[132],[15,54202],{"className":54203,"style":128},[127],[15,54205,54207,54210,54215,54218,54316,54319,54322],{"className":54206},[52],[15,54208],{"className":54209,"style":17647},[56],[15,54211,22774,54213],{"className":54212},[531],[15,54214,22777],{"style":15608},[15,54216],{"className":54217,"style":157},[127],[15,54219,54221,54224,54313],{"className":54220},[61],[15,54222],{"className":54223},[146,3837],[15,54225,54227],{"className":54226},[3718],[15,54228,54230,54305],{"className":54229},[342,472],[15,54231,54233,54302],{"className":54232},[346],[15,54234,54236,54265,54273],{"className":54235,"style":15310},[350],[15,54237,54238,54241],{"style":3853},[15,54239],{"className":54240,"style":3857},[357],[15,54242,54244],{"className":54243},[362,363,364,365],[15,54245,54247,54250,54253,54262],{"className":54246},[61,365],[15,54248,3726],{"className":54249,"style":3867},[61,122,365],[15,54251,741],{"className":54252},[146,365],[15,54254,54256],{"className":54255},[61,365],[15,54257,54259],{"className":54258},[61,365],[15,54260,3102],{"className":54261},[61,3162,365],[15,54263,750],{"className":54264},[178,365],[15,54266,54267,54270],{"style":3885},[15,54268],{"className":54269,"style":3857},[357],[15,54271],{"className":54272,"style":3893},[3892],[15,54274,54275,54278],{"style":15439},[15,54276],{"className":54277,"style":3857},[357],[15,54279,54281],{"className":54280},[362,363,364,365],[15,54282,54284,54287,54290,54299],{"className":54283},[61,365],[15,54285,3726],{"className":54286,"style":3867},[61,122,365],[15,54288,741],{"className":54289},[146,365],[15,54291,54293],{"className":54292},[61,365],[15,54294,54296],{"className":54295},[61,365],[15,54297,17120],{"className":54298,"style":17363},[61,3162,365],[15,54300,750],{"className":54301},[178,365],[15,54303,497],{"className":54304},[496],[15,54306,54308],{"className":54307},[346],[15,54309,54311],{"className":54310,"style":3918},[350],[15,54312],{},[15,54314],{"className":54315},[178,3837],[15,54317],{"className":54318,"style":772},[127],[15,54320,7756],{"className":54321},[375],[15,54323],{"className":54324,"style":772},[127],[15,54326,54328,54331,54391,54394,54434,54437,54440],{"className":54327},[52],[15,54329],{"className":54330,"style":3467},[56],[15,54332,54334,54337],{"className":54333},[531],[15,54335,293],{"className":54336,"style":537},[531,535,536],[15,54338,54340],{"className":54339},[338],[15,54341,54343,54383],{"className":54342},[342,472],[15,54344,54346,54380],{"className":54345},[346],[15,54347,54349,54369],{"className":54348,"style":550},[350],[15,54350,54351,54354],{"style":553},[15,54352],{"className":54353,"style":358},[357],[15,54355,54357],{"className":54356},[362,363,364,365],[15,54358,54360,54363,54366],{"className":54359},[61,365],[15,54361,272],{"className":54362,"style":492},[61,122,365],[15,54364,244],{"className":54365},[132,365],[15,54367,91],{"className":54368},[61,365],[15,54370,54371,54374],{"style":574},[15,54372],{"className":54373,"style":358},[357],[15,54375,54377],{"className":54376},[362,363,364,365],[15,54378,102],{"className":54379,"style":174},[61,122,365],[15,54381,497],{"className":54382},[496],[15,54384,54386],{"className":54385},[346],[15,54387,54389],{"className":54388,"style":593},[350],[15,54390],{},[15,54392,741],{"className":54393},[146],[15,54395,54397,54400],{"className":54396},[61],[15,54398,3102],{"className":54399,"style":3187},[61,122],[15,54401,54403],{"className":54402},[338],[15,54404,54406,54426],{"className":54405},[342,472],[15,54407,54409,54423],{"className":54408},[346],[15,54410,54412],{"className":54411,"style":479},[350],[15,54413,54414,54417],{"style":3202},[15,54415],{"className":54416,"style":358},[357],[15,54418,54420],{"className":54419},[362,363,364,365],[15,54421,272],{"className":54422,"style":492},[61,122,365],[15,54424,497],{"className":54425},[496],[15,54427,54429],{"className":54428},[346],[15,54430,54432],{"className":54431,"style":504},[350],[15,54433],{},[15,54435],{"className":54436,"style":772},[127],[15,54438,239],{"className":54439},[375],[15,54441],{"className":54442,"style":772},[127],[15,54444,54446,54449,54489,54492,54498,54501,54504,54544,54547,54550,54553],{"className":54445},[52],[15,54447],{"className":54448,"style":45239},[56],[15,54450,54452,54455],{"className":54451},[61],[15,54453,17120],{"className":54454,"style":17259},[61,122],[15,54456,54458],{"className":54457},[338],[15,54459,54461,54481],{"className":54460},[342,472],[15,54462,54464,54478],{"className":54463},[346],[15,54465,54467],{"className":54466,"style":479},[350],[15,54468,54469,54472],{"style":17443},[15,54470],{"className":54471,"style":358},[357],[15,54473,54475],{"className":54474},[362,363,364,365],[15,54476,272],{"className":54477,"style":492},[61,122,365],[15,54479,497],{"className":54480},[496],[15,54482,54484],{"className":54483},[346],[15,54485,54487],{"className":54486,"style":504},[350],[15,54488],{},[15,54490,750],{"className":54491},[178],[15,54493,54495],{"className":54494},[61],[15,54496,741],{"className":54497},[406,4063],[15,54499,22676],{"className":54500,"style":123},[61,122],[15,54502,741],{"className":54503},[146],[15,54505,54507,54510],{"className":54506},[61],[15,54508,3102],{"className":54509,"style":3187},[61,122],[15,54511,54513],{"className":54512},[338],[15,54514,54516,54536],{"className":54515},[342,472],[15,54517,54519,54533],{"className":54518},[346],[15,54520,54522],{"className":54521,"style":479},[350],[15,54523,54524,54527],{"style":3202},[15,54525],{"className":54526,"style":358},[357],[15,54528,54530],{"className":54529},[362,363,364,365],[15,54531,272],{"className":54532,"style":492},[61,122,365],[15,54534,497],{"className":54535},[496],[15,54537,54539],{"className":54538},[346],[15,54540,54542],{"className":54541,"style":504},[350],[15,54543],{},[15,54545,750],{"className":54546},[178],[15,54548],{"className":54549,"style":772},[127],[15,54551,239],{"className":54552},[375],[15,54554],{"className":54555,"style":772},[127],[15,54557,54559,54562,54565,54568,54608,54611],{"className":54558},[52],[15,54560],{"className":54561,"style":45239},[56],[15,54563,22676],{"className":54564,"style":123},[61,122],[15,54566,741],{"className":54567},[146],[15,54569,54571,54574],{"className":54570},[61],[15,54572,3102],{"className":54573,"style":3187},[61,122],[15,54575,54577],{"className":54576},[338],[15,54578,54580,54600],{"className":54579},[342,472],[15,54581,54583,54597],{"className":54582},[346],[15,54584,54586],{"className":54585,"style":1212},[350],[15,54587,54588,54591],{"style":3202},[15,54589],{"className":54590,"style":358},[357],[15,54592,54594],{"className":54593},[362,363,364,365],[15,54595,278],{"className":54596},[61,365],[15,54598,497],{"className":54599},[496],[15,54601,54603],{"className":54602},[346],[15,54604,54606],{"className":54605,"style":504},[350],[15,54607],{},[15,54609,750],{"className":54610},[178],[15,54612,54614],{"className":54613},[61],[15,54615,750],{"className":54616},[406,4063],"\n特别地，取 ",[15,54619,54621,54639],{"className":54620,"translate":19},[18],[15,54622,54624],{"className":54623},[23],[25,54625,54626],{"xmlns":27},[29,54627,54628,54636],{},[32,54629,54630,54632,54634],{},[35,54631,17120],{"mathvariant":3101},[81,54633,244],{},[89,54635,91],{"mathvariant":251},[40,54637,54638],{"encoding":42},"\\boldsymbol{\\beta} = \\mathbf{1}",[15,54640,54642,54666],{"className":54641,"ariaHidden":48},[47],[15,54643,54645,54648,54657,54660,54663],{"className":54644},[52],[15,54646],{"className":54647,"style":14301},[56],[15,54649,54651],{"className":54650},[61],[15,54652,54654],{"className":54653},[61],[15,54655,17120],{"className":54656,"style":17363},[61,3162],[15,54658],{"className":54659,"style":128},[127],[15,54661,244],{"className":54662},[132],[15,54664],{"className":54665,"style":128},[127],[15,54667,54669,54672],{"className":54668},[52],[15,54670],{"className":54671,"style":859},[56],[15,54673,91],{"className":54674},[61,411],"（此时 ",[15,54677,54679,54713],{"className":54678,"translate":19},[18],[15,54680,54682],{"className":54681},[23],[25,54683,54684],{"xmlns":27},[29,54685,54686,54710],{},[32,54687,54688,54690,54692,54694,54696,54698,54700,54702,54704,54706,54708],{},[35,54689,3726],{},[81,54691,741],{"stretchy":86},[89,54693,91],{"mathvariant":251},[81,54695,750],{"stretchy":86},[81,54697,244],{},[89,54699,91],{},[35,54701,3018],{"mathvariant":231},[35,54703,4288],{"mathvariant":231},[81,54705,741],{"stretchy":86},[35,54707,102],{},[81,54709,750],{"stretchy":86},[40,54711,54712],{"encoding":42},"B(\\mathbf{1}) = 1\u002F\\Gamma(K)",[15,54714,54716,54743],{"className":54715,"ariaHidden":48},[47],[15,54717,54719,54722,54725,54728,54731,54734,54737,54740],{"className":54718},[52],[15,54720],{"className":54721,"style":142},[56],[15,54723,3726],{"className":54724,"style":3867},[61,122],[15,54726,741],{"className":54727},[146],[15,54729,91],{"className":54730},[61,411],[15,54732,750],{"className":54733},[178],[15,54735],{"className":54736,"style":128},[127],[15,54738,244],{"className":54739},[132],[15,54741],{"className":54742,"style":128},[127],[15,54744,54746,54749,54753,54756,54759],{"className":54745},[52],[15,54747],{"className":54748,"style":142},[56],[15,54750,54752],{"className":54751},[61],"1\u002FΓ",[15,54754,741],{"className":54755},[146],[15,54757,102],{"className":54758,"style":174},[61,122],[15,54760,750],{"className":54761},[178],"）得\n",[15,54764,54766,54928],{"className":54765,"translate":19},[18],[15,54767,54769],{"className":54768},[23],[25,54770,54771],{"xmlns":27},[29,54772,54773,54925],{},[32,54774,54775,54781,54783,54785,54787,54789,54791,54801,54803,54805,54819,54821,54823,54825,54827,54837,54839,54841,54843,54845,54847,54849,54851,54853,54855,54869,54871,54881,54883,54885,54887,54889,54891,54893,54903,54905,54907,54909,54911,54921,54923],{},[227,54776,54777,54779],{},[35,54778,47453],{"mathvariant":37},[282,54780,53395],{},[81,54782,244],{},[35,54784,22645],{},[81,54786,1545],{},[35,54788,4288],{"mathvariant":231},[81,54790,741],{"stretchy":86},[266,54792,54793,54799],{},[53179,54794,54795,54797],{"accent":48},[35,54796,3102],{},[81,54798,53185],{},[89,54800,278],{},[81,54802,750],{"stretchy":86},[81,54804,239],{},[289,54806,54807,54809,54817],{},[81,54808,293],{},[32,54810,54811,54813,54815],{},[35,54812,272],{},[81,54814,244],{},[89,54816,91],{},[35,54818,102],{},[35,54820,22645],{},[81,54822,1545],{},[35,54824,4288],{"mathvariant":231},[81,54826,741],{"stretchy":86},[266,54828,54829,54835],{},[53179,54830,54831,54833],{"accent":48},[35,54832,3102],{},[81,54834,53185],{},[35,54836,272],{},[81,54838,750],{"stretchy":86},[81,54840,239],{},[35,54842,22645],{},[81,54844,1545],{},[35,54846,4288],{"mathvariant":231},[81,54848,741],{"stretchy":86},[35,54850,102],{},[81,54852,750],{"stretchy":86},[81,54854,7756],{},[289,54856,54857,54859,54867],{},[81,54858,293],{},[32,54860,54861,54863,54865],{},[35,54862,272],{},[81,54864,244],{},[89,54866,91],{},[35,54868,102],{},[81,54870,741],{"stretchy":86},[266,54872,54873,54879],{},[53179,54874,54875,54877],{"accent":48},[35,54876,3102],{},[81,54878,53185],{},[35,54880,272],{},[81,54882,239],{},[89,54884,91],{},[81,54886,750],{"stretchy":86},[81,54888,741],{"fence":86,"stretchy":48,"minsize":44812,"maxsize":44812},[35,54890,22676],{},[81,54892,741],{"stretchy":86},[266,54894,54895,54901],{},[53179,54896,54897,54899],{"accent":48},[35,54898,3102],{},[81,54900,53185],{},[35,54902,272],{},[81,54904,750],{"stretchy":86},[81,54906,239],{},[35,54908,22676],{},[81,54910,741],{"stretchy":86},[266,54912,54913,54919],{},[53179,54914,54915,54917],{"accent":48},[35,54916,3102],{},[81,54918,53185],{},[89,54920,278],{},[81,54922,750],{"stretchy":86},[81,54924,750],{"fence":86,"stretchy":48,"minsize":44812,"maxsize":44812},[40,54926,54927],{"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)",[15,54929,54931,54981,55084,55247,55282,55428,55532],{"className":54930,"ariaHidden":48},[47],[15,54932,54934,54937,54972,54975,54978],{"className":54933},[52],[15,54935],{"className":54936,"style":57},[56],[15,54938,54940,54943],{"className":54939},[61],[15,54941,47453],{"className":54942},[61,62],[15,54944,54946],{"className":54945},[338],[15,54947,54949],{"className":54948},[342],[15,54950,54952],{"className":54951},[346],[15,54953,54955],{"className":54954,"style":37580},[350],[15,54956,54957,54960],{"style":716},[15,54958],{"className":54959,"style":358},[357],[15,54961,54963],{"className":54962},[362,363,364,365],[15,54964,54966],{"className":54965},[61,365],[15,54967,54969],{"className":54968},[61,13539,365],[15,54970,53395],{"className":54971},[61,365],[15,54973],{"className":54974,"style":128},[127],[15,54976,244],{"className":54977},[132],[15,54979],{"className":54980,"style":128},[127],[15,54982,54984,54987,54992,54995,54998,55001,55072,55075,55078,55081],{"className":54983},[52],[15,54985],{"className":54986,"style":142},[56],[15,54988,22774,54990],{"className":54989},[531],[15,54991,22777],{"style":15608},[15,54993],{"className":54994,"style":157},[127],[15,54996,4288],{"className":54997},[61],[15,54999,741],{"className":55000},[146],[15,55002,55004,55038],{"className":55003},[61],[15,55005,55007],{"className":55006},[61,53223],[15,55008,55010],{"className":55009},[342],[15,55011,55013],{"className":55012},[346],[15,55014,55017,55025],{"className":55015,"style":55016},[350],"height:0.6679em;",[15,55018,55019,55022],{"style":53235},[15,55020],{"className":55021,"style":3857},[357],[15,55023,3102],{"className":55024,"style":3187},[61,122],[15,55026,55028,55031],{"style":55027},"top:-3.35em;",[15,55029],{"className":55030,"style":3857},[357],[15,55032,55035],{"className":55033,"style":55034},[53257],"left:-0.2222em;",[15,55036,53185],{"className":55037},[61],[15,55039,55041],{"className":55040},[338],[15,55042,55044,55064],{"className":55043},[342,472],[15,55045,55047,55061],{"className":55046},[346],[15,55048,55050],{"className":55049,"style":1212},[350],[15,55051,55052,55055],{"style":3202},[15,55053],{"className":55054,"style":358},[357],[15,55056,55058],{"className":55057},[362,363,364,365],[15,55059,278],{"className":55060},[61,365],[15,55062,497],{"className":55063},[496],[15,55065,55067],{"className":55066},[346],[15,55068,55070],{"className":55069,"style":504},[350],[15,55071],{},[15,55073,750],{"className":55074},[178],[15,55076],{"className":55077,"style":772},[127],[15,55079,239],{"className":55080},[375],[15,55082],{"className":55083,"style":772},[127],[15,55085,55087,55090,55150,55153,55158,55161,55164,55167,55235,55238,55241,55244],{"className":55086},[52],[15,55088],{"className":55089,"style":3467},[56],[15,55091,55093,55096],{"className":55092},[531],[15,55094,293],{"className":55095,"style":537},[531,535,536],[15,55097,55099],{"className":55098},[338],[15,55100,55102,55142],{"className":55101},[342,472],[15,55103,55105,55139],{"className":55104},[346],[15,55106,55108,55128],{"className":55107,"style":550},[350],[15,55109,55110,55113],{"style":553},[15,55111],{"className":55112,"style":358},[357],[15,55114,55116],{"className":55115},[362,363,364,365],[15,55117,55119,55122,55125],{"className":55118},[61,365],[15,55120,272],{"className":55121,"style":492},[61,122,365],[15,55123,244],{"className":55124},[132,365],[15,55126,91],{"className":55127},[61,365],[15,55129,55130,55133],{"style":574},[15,55131],{"className":55132,"style":358},[357],[15,55134,55136],{"className":55135},[362,363,364,365],[15,55137,102],{"className":55138,"style":174},[61,122,365],[15,55140,497],{"className":55141},[496],[15,55143,55145],{"className":55144},[346],[15,55146,55148],{"className":55147,"style":593},[350],[15,55149],{},[15,55151],{"className":55152,"style":157},[127],[15,55154,22774,55156],{"className":55155},[531],[15,55157,22777],{"style":15608},[15,55159],{"className":55160,"style":157},[127],[15,55162,4288],{"className":55163},[61],[15,55165,741],{"className":55166},[146],[15,55168,55170,55201],{"className":55169},[61],[15,55171,55173],{"className":55172},[61,53223],[15,55174,55176],{"className":55175},[342],[15,55177,55179],{"className":55178},[346],[15,55180,55182,55190],{"className":55181,"style":55016},[350],[15,55183,55184,55187],{"style":53235},[15,55185],{"className":55186,"style":3857},[357],[15,55188,3102],{"className":55189,"style":3187},[61,122],[15,55191,55192,55195],{"style":55027},[15,55193],{"className":55194,"style":3857},[357],[15,55196,55198],{"className":55197,"style":55034},[53257],[15,55199,53185],{"className":55200},[61],[15,55202,55204],{"className":55203},[338],[15,55205,55207,55227],{"className":55206},[342,472],[15,55208,55210,55224],{"className":55209},[346],[15,55211,55213],{"className":55212,"style":479},[350],[15,55214,55215,55218],{"style":3202},[15,55216],{"className":55217,"style":358},[357],[15,55219,55221],{"className":55220},[362,363,364,365],[15,55222,272],{"className":55223,"style":492},[61,122,365],[15,55225,497],{"className":55226},[496],[15,55228,55230],{"className":55229},[346],[15,55231,55233],{"className":55232,"style":504},[350],[15,55234],{},[15,55236,750],{"className":55237},[178],[15,55239],{"className":55240,"style":772},[127],[15,55242,239],{"className":55243},[375],[15,55245],{"className":55246,"style":772},[127],[15,55248,55250,55253,55258,55261,55264,55267,55270,55273,55276,55279],{"className":55249},[52],[15,55251],{"className":55252,"style":142},[56],[15,55254,22774,55256],{"className":55255},[531],[15,55257,22777],{"style":15608},[15,55259],{"className":55260,"style":157},[127],[15,55262,4288],{"className":55263},[61],[15,55265,741],{"className":55266},[146],[15,55268,102],{"className":55269,"style":174},[61,122],[15,55271,750],{"className":55272},[178],[15,55274],{"className":55275,"style":772},[127],[15,55277,7756],{"className":55278},[375],[15,55280],{"className":55281,"style":772},[127],[15,55283,55285,55288,55348,55351,55419,55422,55425],{"className":55284},[52],[15,55286],{"className":55287,"style":3467},[56],[15,55289,55291,55294],{"className":55290},[531],[15,55292,293],{"className":55293,"style":537},[531,535,536],[15,55295,55297],{"className":55296},[338],[15,55298,55300,55340],{"className":55299},[342,472],[15,55301,55303,55337],{"className":55302},[346],[15,55304,55306,55326],{"className":55305,"style":550},[350],[15,55307,55308,55311],{"style":553},[15,55309],{"className":55310,"style":358},[357],[15,55312,55314],{"className":55313},[362,363,364,365],[15,55315,55317,55320,55323],{"className":55316},[61,365],[15,55318,272],{"className":55319,"style":492},[61,122,365],[15,55321,244],{"className":55322},[132,365],[15,55324,91],{"className":55325},[61,365],[15,55327,55328,55331],{"style":574},[15,55329],{"className":55330,"style":358},[357],[15,55332,55334],{"className":55333},[362,363,364,365],[15,55335,102],{"className":55336,"style":174},[61,122,365],[15,55338,497],{"className":55339},[496],[15,55341,55343],{"className":55342},[346],[15,55344,55346],{"className":55345,"style":593},[350],[15,55347],{},[15,55349,741],{"className":55350},[146],[15,55352,55354,55385],{"className":55353},[61],[15,55355,55357],{"className":55356},[61,53223],[15,55358,55360],{"className":55359},[342],[15,55361,55363],{"className":55362},[346],[15,55364,55366,55374],{"className":55365,"style":55016},[350],[15,55367,55368,55371],{"style":53235},[15,55369],{"className":55370,"style":3857},[357],[15,55372,3102],{"className":55373,"style":3187},[61,122],[15,55375,55376,55379],{"style":55027},[15,55377],{"className":55378,"style":3857},[357],[15,55380,55382],{"className":55381,"style":55034},[53257],[15,55383,53185],{"className":55384},[61],[15,55386,55388],{"className":55387},[338],[15,55389,55391,55411],{"className":55390},[342,472],[15,55392,55394,55408],{"className":55393},[346],[15,55395,55397],{"className":55396,"style":479},[350],[15,55398,55399,55402],{"style":3202},[15,55400],{"className":55401,"style":358},[357],[15,55403,55405],{"className":55404},[362,363,364,365],[15,55406,272],{"className":55407,"style":492},[61,122,365],[15,55409,497],{"className":55410},[496],[15,55412,55414],{"className":55413},[346],[15,55415,55417],{"className":55416,"style":504},[350],[15,55418],{},[15,55420],{"className":55421,"style":772},[127],[15,55423,239],{"className":55424},[375],[15,55426],{"className":55427,"style":772},[127],[15,55429,55431,55434,55437,55440,55446,55449,55452,55520,55523,55526,55529],{"className":55430},[52],[15,55432],{"className":55433,"style":45239},[56],[15,55435,91],{"className":55436},[61],[15,55438,750],{"className":55439},[178],[15,55441,55443],{"className":55442},[61],[15,55444,741],{"className":55445},[406,4063],[15,55447,22676],{"className":55448,"style":123},[61,122],[15,55450,741],{"className":55451},[146],[15,55453,55455,55486],{"className":55454},[61],[15,55456,55458],{"className":55457},[61,53223],[15,55459,55461],{"className":55460},[342],[15,55462,55464],{"className":55463},[346],[15,55465,55467,55475],{"className":55466,"style":55016},[350],[15,55468,55469,55472],{"style":53235},[15,55470],{"className":55471,"style":3857},[357],[15,55473,3102],{"className":55474,"style":3187},[61,122],[15,55476,55477,55480],{"style":55027},[15,55478],{"className":55479,"style":3857},[357],[15,55481,55483],{"className":55482,"style":55034},[53257],[15,55484,53185],{"className":55485},[61],[15,55487,55489],{"className":55488},[338],[15,55490,55492,55512],{"className":55491},[342,472],[15,55493,55495,55509],{"className":55494},[346],[15,55496,55498],{"className":55497,"style":479},[350],[15,55499,55500,55503],{"style":3202},[15,55501],{"className":55502,"style":358},[357],[15,55504,55506],{"className":55505},[362,363,364,365],[15,55507,272],{"className":55508,"style":492},[61,122,365],[15,55510,497],{"className":55511},[496],[15,55513,55515],{"className":55514},[346],[15,55516,55518],{"className":55517,"style":504},[350],[15,55519],{},[15,55521,750],{"className":55522},[178],[15,55524],{"className":55525,"style":772},[127],[15,55527,239],{"className":55528},[375],[15,55530],{"className":55531,"style":772},[127],[15,55533,55535,55538,55541,55544,55612,55615],{"className":55534},[52],[15,55536],{"className":55537,"style":45239},[56],[15,55539,22676],{"className":55540,"style":123},[61,122],[15,55542,741],{"className":55543},[146],[15,55545,55547,55578],{"className":55546},[61],[15,55548,55550],{"className":55549},[61,53223],[15,55551,55553],{"className":55552},[342],[15,55554,55556],{"className":55555},[346],[15,55557,55559,55567],{"className":55558,"style":55016},[350],[15,55560,55561,55564],{"style":53235},[15,55562],{"className":55563,"style":3857},[357],[15,55565,3102],{"className":55566,"style":3187},[61,122],[15,55568,55569,55572],{"style":55027},[15,55570],{"className":55571,"style":3857},[357],[15,55573,55575],{"className":55574,"style":55034},[53257],[15,55576,53185],{"className":55577},[61],[15,55579,55581],{"className":55580},[338],[15,55582,55584,55604],{"className":55583},[342,472],[15,55585,55587,55601],{"className":55586},[346],[15,55588,55590],{"className":55589,"style":1212},[350],[15,55591,55592,55595],{"style":3202},[15,55593],{"className":55594,"style":358},[357],[15,55596,55598],{"className":55597},[362,363,364,365],[15,55599,278],{"className":55600},[61,365],[15,55602,497],{"className":55603},[496],[15,55605,55607],{"className":55606},[346],[15,55608,55610],{"className":55609,"style":504},[350],[15,55611],{},[15,55613,750],{"className":55614},[178],[15,55616,55618],{"className":55617},[61],[15,55619,750],{"className":55620},[406,4063],[11,55622,55623,55625,55626,55893,55894,9342,56197,56624],{},[661,55624,5117],{}," 由指数族的一般 KL 公式：",[15,55627,55629,55679],{"className":55628,"translate":19},[18],[15,55630,55632],{"className":55631},[23],[25,55633,55634],{"xmlns":27},[29,55635,55636,55676],{},[32,55637,55638,55640,55642,55644,55650,55652,55654,55656,55662,55664,55666,55668,55674],{},[35,55639,53400],{"mathvariant":231},[81,55641,1545],{},[81,55643,244],{},[266,55645,55646,55648],{},[35,55647,14807],{"mathvariant":258},[35,55649,3102],{"mathvariant":3101},[81,55651,14810],{"stretchy":86},[35,55653,22645],{},[81,55655,1545],{},[266,55657,55658,55660],{},[35,55659,11],{},[35,55661,3102],{"mathvariant":3101},[81,55663,239],{},[35,55665,22645],{},[81,55667,1545],{},[266,55669,55670,55672],{},[35,55671,11],{},[35,55673,17120],{"mathvariant":3101},[81,55675,14819],{"stretchy":86},[40,55677,55678],{"encoding":42},"\\operatorname{KL} = \\mathbb{E}_{\\boldsymbol{\\alpha}}[\\log p_{\\boldsymbol{\\alpha}} - \\log p_{\\boldsymbol{\\beta}}]",[15,55680,55682,55703,55827],{"className":55681,"ariaHidden":48},[47],[15,55683,55685,55688,55694,55697,55700],{"className":55684},[52],[15,55686],{"className":55687,"style":57},[56],[15,55689,55691],{"className":55690},[531],[15,55692,53400],{"className":55693},[61,1620],[15,55695],{"className":55696,"style":128},[127],[15,55698,244],{"className":55699},[132],[15,55701],{"className":55702,"style":128},[127],[15,55704,55706,55709,55758,55761,55766,55769,55818,55821,55824],{"className":55705},[52],[15,55707],{"className":55708,"style":142},[56],[15,55710,55712,55715],{"className":55711},[61],[15,55713,14807],{"className":55714},[61,427],[15,55716,55718],{"className":55717},[338],[15,55719,55721,55750],{"className":55720},[342,472],[15,55722,55724,55747],{"className":55723},[346],[15,55725,55727],{"className":55726,"style":20953},[350],[15,55728,55729,55732],{"style":482},[15,55730],{"className":55731,"style":358},[357],[15,55733,55735],{"className":55734},[362,363,364,365],[15,55736,55738],{"className":55737},[61,365],[15,55739,55741],{"className":55740},[61,365],[15,55742,55744],{"className":55743},[61,365],[15,55745,3102],{"className":55746},[61,3162,365],[15,55748,497],{"className":55749},[496],[15,55751,55753],{"className":55752},[346],[15,55754,55756],{"className":55755,"style":504},[350],[15,55757],{},[15,55759,14810],{"className":55760},[146],[15,55762,22774,55764],{"className":55763},[531],[15,55765,22777],{"style":15608},[15,55767],{"className":55768,"style":157},[127],[15,55770,55772,55775],{"className":55771},[61],[15,55773,11],{"className":55774},[61,122],[15,55776,55778],{"className":55777},[338],[15,55779,55781,55810],{"className":55780},[342,472],[15,55782,55784,55807],{"className":55783},[346],[15,55785,55787],{"className":55786,"style":20953},[350],[15,55788,55789,55792],{"style":482},[15,55790],{"className":55791,"style":358},[357],[15,55793,55795],{"className":55794},[362,363,364,365],[15,55796,55798],{"className":55797},[61,365],[15,55799,55801],{"className":55800},[61,365],[15,55802,55804],{"className":55803},[61,365],[15,55805,3102],{"className":55806},[61,3162,365],[15,55808,497],{"className":55809},[496],[15,55811,55813],{"className":55812},[346],[15,55814,55816],{"className":55815,"style":504},[350],[15,55817],{},[15,55819],{"className":55820,"style":772},[127],[15,55822,239],{"className":55823},[375],[15,55825],{"className":55826,"style":772},[127],[15,55828,55830,55833,55838,55841,55890],{"className":55829},[52],[15,55831],{"className":55832,"style":17976},[56],[15,55834,22774,55836],{"className":55835},[531],[15,55837,22777],{"style":15608},[15,55839],{"className":55840,"style":157},[127],[15,55842,55844,55847],{"className":55843},[61],[15,55845,11],{"className":55846},[61,122],[15,55848,55850],{"className":55849},[338],[15,55851,55853,55882],{"className":55852},[342,472],[15,55854,55856,55879],{"className":55855},[346],[15,55857,55859],{"className":55858,"style":479},[350],[15,55860,55861,55864],{"style":482},[15,55862],{"className":55863,"style":358},[357],[15,55865,55867],{"className":55866},[362,363,364,365],[15,55868,55870],{"className":55869},[61,365],[15,55871,55873],{"className":55872},[61,365],[15,55874,55876],{"className":55875},[61,365],[15,55877,17120],{"className":55878,"style":17363},[61,3162,365],[15,55880,497],{"className":55881},[496],[15,55883,55885],{"className":55884},[346],[15,55886,55888],{"className":55887,"style":15696},[350],[15,55889],{},[15,55891,14819],{"className":55892},[178],"。代入 1.2 节的密度并利用 ",[15,55895,55897,55957],{"className":55896,"translate":19},[18],[15,55898,55900],{"className":55899},[23],[25,55901,55902],{"xmlns":27},[29,55903,55904,55954],{},[32,55905,55906,55912,55914,55916,55918,55924,55926,55928,55930,55932,55938,55940,55942,55944,55946,55952],{},[266,55907,55908,55910],{},[35,55909,14807],{"mathvariant":258},[35,55911,3102],{"mathvariant":3101},[81,55913,14810],{"stretchy":86},[35,55915,22645],{},[81,55917,1545],{},[266,55919,55920,55922],{},[35,55921,1027],{},[35,55923,272],{},[81,55925,14819],{"stretchy":86},[81,55927,244],{},[35,55929,22676],{},[81,55931,741],{"stretchy":86},[266,55933,55934,55936],{},[35,55935,3102],{},[35,55937,272],{},[81,55939,750],{"stretchy":86},[81,55941,239],{},[35,55943,22676],{},[81,55945,741],{"stretchy":86},[266,55947,55948,55950],{},[35,55949,3102],{},[89,55951,278],{},[81,55953,750],{"stretchy":86},[40,55955,55956],{"encoding":42},"\\mathbb{E}_{\\boldsymbol{\\alpha}}[\\log x_k] = \\psi(\\alpha_k) - \\psi(\\alpha_0)",[15,55958,55960,56078,56142],{"className":55959,"ariaHidden":48},[47],[15,55961,55963,55966,56015,56018,56023,56026,56066,56069,56072,56075],{"className":55962},[52],[15,55964],{"className":55965,"style":142},[56],[15,55967,55969,55972],{"className":55968},[61],[15,55970,14807],{"className":55971},[61,427],[15,55973,55975],{"className":55974},[338],[15,55976,55978,56007],{"className":55977},[342,472],[15,55979,55981,56004],{"className":55980},[346],[15,55982,55984],{"className":55983,"style":20953},[350],[15,55985,55986,55989],{"style":482},[15,55987],{"className":55988,"style":358},[357],[15,55990,55992],{"className":55991},[362,363,364,365],[15,55993,55995],{"className":55994},[61,365],[15,55996,55998],{"className":55997},[61,365],[15,55999,56001],{"className":56000},[61,365],[15,56002,3102],{"className":56003},[61,3162,365],[15,56005,497],{"className":56006},[496],[15,56008,56010],{"className":56009},[346],[15,56011,56013],{"className":56012,"style":504},[350],[15,56014],{},[15,56016,14810],{"className":56017},[146],[15,56019,22774,56021],{"className":56020},[531],[15,56022,22777],{"style":15608},[15,56024],{"className":56025,"style":157},[127],[15,56027,56029,56032],{"className":56028},[61],[15,56030,1027],{"className":56031},[61,122],[15,56033,56035],{"className":56034},[338],[15,56036,56038,56058],{"className":56037},[342,472],[15,56039,56041,56055],{"className":56040},[346],[15,56042,56044],{"className":56043,"style":479},[350],[15,56045,56046,56049],{"style":482},[15,56047],{"className":56048,"style":358},[357],[15,56050,56052],{"className":56051},[362,363,364,365],[15,56053,272],{"className":56054,"style":492},[61,122,365],[15,56056,497],{"className":56057},[496],[15,56059,56061],{"className":56060},[346],[15,56062,56064],{"className":56063,"style":504},[350],[15,56065],{},[15,56067,14819],{"className":56068},[178],[15,56070],{"className":56071,"style":128},[127],[15,56073,244],{"className":56074},[132],[15,56076],{"className":56077,"style":128},[127],[15,56079,56081,56084,56087,56090,56130,56133,56136,56139],{"className":56080},[52],[15,56082],{"className":56083,"style":142},[56],[15,56085,22676],{"className":56086,"style":123},[61,122],[15,56088,741],{"className":56089},[146],[15,56091,56093,56096],{"className":56092},[61],[15,56094,3102],{"className":56095,"style":3187},[61,122],[15,56097,56099],{"className":56098},[338],[15,56100,56102,56122],{"className":56101},[342,472],[15,56103,56105,56119],{"className":56104},[346],[15,56106,56108],{"className":56107,"style":479},[350],[15,56109,56110,56113],{"style":3202},[15,56111],{"className":56112,"style":358},[357],[15,56114,56116],{"className":56115},[362,363,364,365],[15,56117,272],{"className":56118,"style":492},[61,122,365],[15,56120,497],{"className":56121},[496],[15,56123,56125],{"className":56124},[346],[15,56126,56128],{"className":56127,"style":504},[350],[15,56129],{},[15,56131,750],{"className":56132},[178],[15,56134],{"className":56135,"style":772},[127],[15,56137,239],{"className":56138},[375],[15,56140],{"className":56141,"style":772},[127],[15,56143,56145,56148,56151,56154,56194],{"className":56144},[52],[15,56146],{"className":56147,"style":142},[56],[15,56149,22676],{"className":56150,"style":123},[61,122],[15,56152,741],{"className":56153},[146],[15,56155,56157,56160],{"className":56156},[61],[15,56158,3102],{"className":56159,"style":3187},[61,122],[15,56161,56163],{"className":56162},[338],[15,56164,56166,56186],{"className":56165},[342,472],[15,56167,56169,56183],{"className":56168},[346],[15,56170,56172],{"className":56171,"style":1212},[350],[15,56173,56174,56177],{"style":3202},[15,56175],{"className":56176,"style":358},[357],[15,56178,56180],{"className":56179},[362,363,364,365],[15,56181,278],{"className":56182},[61,365],[15,56184,497],{"className":56185},[496],[15,56187,56189],{"className":56188},[346],[15,56190,56192],{"className":56191,"style":504},[350],[15,56193],{},[15,56195,750],{"className":56196},[178],[15,56198,56200,56290],{"className":56199,"translate":19},[18],[15,56201,56203],{"className":56202},[23],[25,56204,56205],{"xmlns":27},[29,56206,56207,56287],{},[32,56208,56209,56215,56217,56219,56221,56223,56229,56231,56233,56235,56237,56243,56245,56247,56249,56251,56253,56259,56261,56263,56265,56267,56269,56271,56273,56275,56277,56279,56285],{},[266,56210,56211,56213],{},[81,56212,293],{},[35,56214,272],{},[35,56216,14807],{"mathvariant":258},[81,56218,14810],{"stretchy":86},[35,56220,22645],{},[81,56222,1545],{},[266,56224,56225,56227],{},[35,56226,1027],{},[35,56228,272],{},[81,56230,14819],{"stretchy":86},[81,56232,244],{},[35,56234,22676],{},[81,56236,741],{"stretchy":86},[266,56238,56239,56241],{},[35,56240,3102],{},[89,56242,278],{},[81,56244,750],{"stretchy":86},[81,56246,239],{},[35,56248,102],{},[35,56250,22676],{},[81,56252,741],{"stretchy":86},[266,56254,56255,56257],{},[35,56256,3102],{},[89,56258,278],{},[81,56260,750],{"stretchy":86},[81,56262,244],{},[81,56264,239],{},[81,56266,741],{"stretchy":86},[35,56268,102],{},[81,56270,239],{},[89,56272,91],{},[81,56274,750],{"stretchy":86},[35,56276,22676],{},[81,56278,741],{"stretchy":86},[266,56280,56281,56283],{},[35,56282,3102],{},[89,56284,278],{},[81,56286,750],{"stretchy":86},[40,56288,56289],{"encoding":42},"\\sum_k \\mathbb{E}[\\log x_k] = \\psi(\\alpha_0) - K\\psi(\\alpha_0) = -(K-1)\\psi(\\alpha_0)",[15,56291,56293,56408,56472,56539,56563],{"className":56292,"ariaHidden":48},[47],[15,56294,56296,56299,56339,56342,56345,56348,56353,56356,56396,56399,56402,56405],{"className":56295},[52],[15,56297],{"className":56298,"style":6045},[56],[15,56300,56302,56305],{"className":56301},[531],[15,56303,293],{"className":56304,"style":537},[531,535,536],[15,56306,56308],{"className":56307},[338],[15,56309,56311,56331],{"className":56310},[342,472],[15,56312,56314,56328],{"className":56313},[346],[15,56315,56317],{"className":56316,"style":29964},[350],[15,56318,56319,56322],{"style":553},[15,56320],{"className":56321,"style":358},[357],[15,56323,56325],{"className":56324},[362,363,364,365],[15,56326,272],{"className":56327,"style":492},[61,122,365],[15,56329,497],{"className":56330},[496],[15,56332,56334],{"className":56333},[346],[15,56335,56337],{"className":56336,"style":593},[350],[15,56338],{},[15,56340],{"className":56341,"style":157},[127],[15,56343,14807],{"className":56344},[61,427],[15,56346,14810],{"className":56347},[146],[15,56349,22774,56351],{"className":56350},[531],[15,56352,22777],{"style":15608},[15,56354],{"className":56355,"style":157},[127],[15,56357,56359,56362],{"className":56358},[61],[15,56360,1027],{"className":56361},[61,122],[15,56363,56365],{"className":56364},[338],[15,56366,56368,56388],{"className":56367},[342,472],[15,56369,56371,56385],{"className":56370},[346],[15,56372,56374],{"className":56373,"style":479},[350],[15,56375,56376,56379],{"style":482},[15,56377],{"className":56378,"style":358},[357],[15,56380,56382],{"className":56381},[362,363,364,365],[15,56383,272],{"className":56384,"style":492},[61,122,365],[15,56386,497],{"className":56387},[496],[15,56389,56391],{"className":56390},[346],[15,56392,56394],{"className":56393,"style":504},[350],[15,56395],{},[15,56397,14819],{"className":56398},[178],[15,56400],{"className":56401,"style":128},[127],[15,56403,244],{"className":56404},[132],[15,56406],{"className":56407,"style":128},[127],[15,56409,56411,56414,56417,56420,56460,56463,56466,56469],{"className":56410},[52],[15,56412],{"className":56413,"style":142},[56],[15,56415,22676],{"className":56416,"style":123},[61,122],[15,56418,741],{"className":56419},[146],[15,56421,56423,56426],{"className":56422},[61],[15,56424,3102],{"className":56425,"style":3187},[61,122],[15,56427,56429],{"className":56428},[338],[15,56430,56432,56452],{"className":56431},[342,472],[15,56433,56435,56449],{"className":56434},[346],[15,56436,56438],{"className":56437,"style":1212},[350],[15,56439,56440,56443],{"style":3202},[15,56441],{"className":56442,"style":358},[357],[15,56444,56446],{"className":56445},[362,363,364,365],[15,56447,278],{"className":56448},[61,365],[15,56450,497],{"className":56451},[496],[15,56453,56455],{"className":56454},[346],[15,56456,56458],{"className":56457,"style":504},[350],[15,56459],{},[15,56461,750],{"className":56462},[178],[15,56464],{"className":56465,"style":772},[127],[15,56467,239],{"className":56468},[375],[15,56470],{"className":56471,"style":772},[127],[15,56473,56475,56478,56481,56484,56487,56527,56530,56533,56536],{"className":56474},[52],[15,56476],{"className":56477,"style":142},[56],[15,56479,102],{"className":56480,"style":174},[61,122],[15,56482,22676],{"className":56483,"style":123},[61,122],[15,56485,741],{"className":56486},[146],[15,56488,56490,56493],{"className":56489},[61],[15,56491,3102],{"className":56492,"style":3187},[61,122],[15,56494,56496],{"className":56495},[338],[15,56497,56499,56519],{"className":56498},[342,472],[15,56500,56502,56516],{"className":56501},[346],[15,56503,56505],{"className":56504,"style":1212},[350],[15,56506,56507,56510],{"style":3202},[15,56508],{"className":56509,"style":358},[357],[15,56511,56513],{"className":56512},[362,363,364,365],[15,56514,278],{"className":56515},[61,365],[15,56517,497],{"className":56518},[496],[15,56520,56522],{"className":56521},[346],[15,56523,56525],{"className":56524,"style":504},[350],[15,56526],{},[15,56528,750],{"className":56529},[178],[15,56531],{"className":56532,"style":128},[127],[15,56534,244],{"className":56535},[132],[15,56537],{"className":56538,"style":128},[127],[15,56540,56542,56545,56548,56551,56554,56557,56560],{"className":56541},[52],[15,56543],{"className":56544,"style":142},[56],[15,56546,239],{"className":56547},[61],[15,56549,741],{"className":56550},[146],[15,56552,102],{"className":56553,"style":174},[61,122],[15,56555],{"className":56556,"style":772},[127],[15,56558,239],{"className":56559},[375],[15,56561],{"className":56562,"style":772},[127],[15,56564,56566,56569,56572,56575,56578,56581,56621],{"className":56565},[52],[15,56567],{"className":56568,"style":142},[56],[15,56570,91],{"className":56571},[61],[15,56573,750],{"className":56574},[178],[15,56576,22676],{"className":56577,"style":123},[61,122],[15,56579,741],{"className":56580},[146],[15,56582,56584,56587],{"className":56583},[61],[15,56585,3102],{"className":56586,"style":3187},[61,122],[15,56588,56590],{"className":56589},[338],[15,56591,56593,56613],{"className":56592},[342,472],[15,56594,56596,56610],{"className":56595},[346],[15,56597,56599],{"className":56598,"style":1212},[350],[15,56600,56601,56604],{"style":3202},[15,56602],{"className":56603,"style":358},[357],[15,56605,56607],{"className":56606},[362,363,364,365],[15,56608,278],{"className":56609},[61,365],[15,56611,497],{"className":56612},[496],[15,56614,56616],{"className":56615},[346],[15,56617,56619],{"className":56618,"style":504},[350],[15,56620],{},[15,56622,750],{"className":56623},[178]," 即可化简。",[11,56626,56627,56628,42855],{},"总损失为两项的加权和，其中权重按训练步",[661,56629,56630],{},"退火",[15,56632,56634],{"className":56633,"translate":19},[212],[15,56635,56637,56718],{"className":56636,"translate":19},[18],[15,56638,56640],{"className":56639},[23],[25,56641,56642],{"xmlns":27,"display":221},[29,56643,56644,56715],{},[32,56645,56646,56648,56650,56657,56659,56666,56668,56674,56676,56678,56684,56686,56689,56691,56693],{},[35,56647,47453],{"mathvariant":37},[81,56649,244],{},[227,56651,56652,56654],{},[35,56653,47453],{"mathvariant":37},[282,56655,56656],{},"fit",[81,56658,7756],{},[266,56660,56661,56664],{},[35,56662,56663],{},"λ",[35,56665,5291],{},[282,56667,4796],{},[227,56669,56670,56672],{},[35,56671,47453],{"mathvariant":37},[282,56673,53395],{},[81,56675,94],{"separator":48},[127,56677],{"width":9838},[266,56679,56680,56682],{},[35,56681,56663],{},[35,56683,5291],{},[81,56685,244],{},[35,56687,56688],{},"min",[81,56690,1545],{},[282,56692,6793],{},[32,56694,56695,56697,56699,56701,56713],{},[81,56696,741],{"fence":48},[89,56698,91],{},[81,56700,94],{"separator":48},[3718,56702,56703,56705],{},[35,56704,5291],{},[266,56706,56707,56710],{},[35,56708,56709],{},"T",[282,56711,56712],{},"anneal",[81,56714,750],{"fence":48},[40,56716,56717],{"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)",[15,56719,56721,56739,56790,56935],{"className":56720,"ariaHidden":48},[47],[15,56722,56724,56727,56730,56733,56736],{"className":56723},[52],[15,56725],{"className":56726,"style":57},[56],[15,56728,47453],{"className":56729},[61,62],[15,56731],{"className":56732,"style":128},[127],[15,56734,244],{"className":56735},[132],[15,56737],{"className":56738,"style":128},[127],[15,56740,56742,56746,56781,56784,56787],{"className":56741},[52],[15,56743],{"className":56744,"style":56745},[56],"height:0.9824em;vertical-align:-0.0833em;",[15,56747,56749,56752],{"className":56748},[61],[15,56750,47453],{"className":56751},[61,62],[15,56753,56755],{"className":56754},[338],[15,56756,56758],{"className":56757},[342],[15,56759,56761],{"className":56760},[346],[15,56762,56764],{"className":56763,"style":18849},[350],[15,56765,56766,56769],{"style":353},[15,56767],{"className":56768,"style":358},[357],[15,56770,56772],{"className":56771},[362,363,364,365],[15,56773,56775],{"className":56774},[61,365],[15,56776,56778],{"className":56777},[61,13539,365],[15,56779,56656],{"className":56780},[61,365],[15,56782],{"className":56783,"style":772},[127],[15,56785,7756],{"className":56786},[375],[15,56788],{"className":56789,"style":772},[127],[15,56791,56793,56797,56838,56841,56877,56880,56883,56886,56926,56929,56932],{"className":56792},[52],[15,56794],{"className":56795,"style":56796},[56],"height:0.9088em;vertical-align:-0.1944em;",[15,56798,56800,56803],{"className":56799},[61],[15,56801,56663],{"className":56802},[61,122],[15,56804,56806],{"className":56805},[338],[15,56807,56809,56830],{"className":56808},[342,472],[15,56810,56812,56827],{"className":56811},[346],[15,56813,56816],{"className":56814,"style":56815},[350],"height:0.2806em;",[15,56817,56818,56821],{"style":482},[15,56819],{"className":56820,"style":358},[357],[15,56822,56824],{"className":56823},[362,363,364,365],[15,56825,5291],{"className":56826},[61,122,365],[15,56828,497],{"className":56829},[496],[15,56831,56833],{"className":56832},[346],[15,56834,56836],{"className":56835,"style":504},[350],[15,56837],{},[15,56839],{"className":56840,"style":157},[127],[15,56842,56844,56847],{"className":56843},[61],[15,56845,47453],{"className":56846},[61,62],[15,56848,56850],{"className":56849},[338],[15,56851,56853],{"className":56852},[342],[15,56854,56856],{"className":56855},[346],[15,56857,56860],{"className":56858,"style":56859},[350],"height:0.7144em;",[15,56861,56862,56865],{"style":353},[15,56863],{"className":56864,"style":358},[357],[15,56866,56868],{"className":56867},[362,363,364,365],[15,56869,56871],{"className":56870},[61,365],[15,56872,56874],{"className":56873},[61,13539,365],[15,56875,53395],{"className":56876},[61,365],[15,56878,94],{"className":56879},[153],[15,56881],{"className":56882,"style":10026},[127],[15,56884],{"className":56885,"style":157},[127],[15,56887,56889,56892],{"className":56888},[61],[15,56890,56663],{"className":56891},[61,122],[15,56893,56895],{"className":56894},[338],[15,56896,56898,56918],{"className":56897},[342,472],[15,56899,56901,56915],{"className":56900},[346],[15,56902,56904],{"className":56903,"style":56815},[350],[15,56905,56906,56909],{"style":482},[15,56907],{"className":56908,"style":358},[357],[15,56910,56912],{"className":56911},[362,363,364,365],[15,56913,5291],{"className":56914},[61,122,365],[15,56916,497],{"className":56917},[496],[15,56919,56921],{"className":56920},[346],[15,56922,56924],{"className":56923,"style":504},[350],[15,56925],{},[15,56927],{"className":56928,"style":128},[127],[15,56930,244],{"className":56931},[132],[15,56933],{"className":56934,"style":128},[127],[15,56936,56938,56941,56944,56947,56950],{"className":56937},[52],[15,56939],{"className":56940,"style":6508},[56],[15,56942,56688],{"className":56943},[531],[15,56945],{"className":56946,"style":7099},[127],[15,56948],{"className":56949,"style":157},[127],[15,56951,56953,56959,56962,56965,56968,57074],{"className":56952},[161],[15,56954,56956],{"className":56955,"style":402},[146,401],[15,56957,741],{"className":56958},[406,364],[15,56960,91],{"className":56961},[61],[15,56963,94],{"className":56964},[153],[15,56966],{"className":56967,"style":157},[127],[15,56969,56971,56974,57071],{"className":56970},[61],[15,56972],{"className":56973},[146,3837],[15,56975,56977],{"className":56976},[3718],[15,56978,56980,57063],{"className":56979},[342,472],[15,56981,56983,57060],{"className":56982},[346],[15,56984,56987,57041,57049],{"className":56985,"style":56986},[350],"height:1.2921em;",[15,56988,56989,56992],{"style":11393},[15,56990],{"className":56991,"style":3857},[357],[15,56993,56995],{"className":56994},[61],[15,56996,56998,57001],{"className":56997},[61],[15,56999,56709],{"className":57000,"style":34338},[61,122],[15,57002,57004],{"className":57003},[338],[15,57005,57007,57033],{"className":57006},[342,472],[15,57008,57010,57030],{"className":57009},[346],[15,57011,57013],{"className":57012,"style":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",[15,57084,57086,57109],{"className":57085,"translate":19},[18],[15,57087,57089],{"className":57088},[23],[25,57090,57091],{"xmlns":27},[29,57092,57093,57106],{},[32,57094,57095,57101,57104],{},[266,57096,57097,57099],{},[35,57098,56663],{},[35,57100,5291],{},[81,57102,57103],{},"≈",[89,57105,278],{},[40,57107,57108],{"encoding":42},"\\lambda_t \\approx 0",[15,57110,57112,57167],{"className":57111,"ariaHidden":48},[47],[15,57113,57115,57118,57158,57161,57164],{"className":57114},[52],[15,57116],{"className":57117,"style":34389},[56],[15,57119,57121,57124],{"className":57120},[61],[15,57122,56663],{"className":57123},[61,122],[15,57125,57127],{"className":57126},[338],[15,57128,57130,57150],{"className":57129},[342,472],[15,57131,57133,57147],{"className":57132},[346],[15,57134,57136],{"className":57135,"style":56815},[350],[15,57137,57138,57141],{"style":482},[15,57139],{"className":57140,"style":358},[357],[15,57142,57144],{"className":57143},[362,363,364,365],[15,57145,5291],{"className":57146},[61,122,365],[15,57148,497],{"className":57149},[496],[15,57151,57153],{"className":57152},[346],[15,57154,57156],{"className":57155,"style":504},[350],[15,57157],{},[15,57159],{"className":57160,"style":128},[127],[15,57162,57103],{"className":57163},[132],[15,57165],{"className":57166,"style":128},[127],[15,57168,57170,57173],{"className":57169},[52],[15,57171],{"className":57172,"style":859},[56],[15,57174,278],{"className":57175},[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 ",[15,57178,57180,57202],{"className":57179,"translate":19},[18],[15,57181,57183],{"className":57182},[23],[25,57184,57185],{"xmlns":27},[29,57186,57187,57199],{},[32,57188,57189,57195,57197],{},[266,57190,57191,57193],{},[35,57192,56663],{},[35,57194,5291],{},[81,57196,20691],{},[89,57198,91],{},[40,57200,57201],{"encoding":42},"\\lambda_t \\to 1",[15,57203,57205,57260],{"className":57204,"ariaHidden":48},[47],[15,57206,57208,57211,57251,57254,57257],{"className":57207},[52],[15,57209],{"className":57210,"style":34389},[56],[15,57212,57214,57217],{"className":57213},[61],[15,57215,56663],{"className":57216},[61,122],[15,57218,57220],{"className":57219},[338],[15,57221,57223,57243],{"className":57222},[342,472],[15,57224,57226,57240],{"className":57225},[346],[15,57227,57229],{"className":57228,"style":56815},[350],[15,57230,57231,57234],{"style":482},[15,57232],{"className":57233,"style":358},[357],[15,57235,57237],{"className":57236},[362,363,364,365],[15,57238,5291],{"className":57239},[61,122,365],[15,57241,497],{"className":57242},[496],[15,57244,57246],{"className":57245},[346],[15,57247,57249],{"className":57248,"style":504},[350],[15,57250],{},[15,57252],{"className":57253,"style":128},[127],[15,57255,20691],{"className":57256},[132],[15,57258],{"className":57259,"style":128},[127],[15,57261,57263,57266],{"className":57262},[52],[15,57264],{"className":57265,"style":859},[56],[15,57267,91],{"className":57268},[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 ",[15,57271,57273,57287],{"className":57272,"translate":19},[18],[15,57274,57276],{"className":57275},[23],[25,57277,57278],{"xmlns":27},[29,57279,57280,57284],{},[32,57281,57282],{},[35,57283,56663],{},[40,57285,57286],{"encoding":42},"\\lambda",[15,57288,57290],{"className":57289,"ariaHidden":48},[47],[15,57291,57293,57296],{"className":57292},[52],[15,57294],{"className":57295,"style":10443},[56],[15,57297,56663],{"className":57298},[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。",[877,57301,57303],{"id":57302},"_44-统一视角与变体","4.4 统一视角与变体",[11,57305,57306],{},"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",[15,57308,57310],{"className":57309,"translate":19},[212],[15,57311,57313,57372],{"className":57312,"translate":19},[18],[15,57314,57316],{"className":57315},[23],[25,57317,57318],{"xmlns":27,"display":221},[29,57319,57320,57369],{},[32,57321,57322,57328,57330,57350],{},[227,57323,57324,57326],{},[35,57325,47453],{"mathvariant":37},[282,57327,56656],{},[81,57329,244],{},[266,57331,57332,57334],{},[35,57333,14807],{"mathvariant":258},[32,57335,57336,57338,57340,57342,57344,57346,57348],{},[35,57337,24661],{"mathvariant":3101},[81,57339,4179],{},[35,57341,4182],{"mathvariant":231},[81,57343,1545],{},[81,57345,741],{"stretchy":86},[35,57347,3102],{"mathvariant":3101},[81,57349,750],{"stretchy":86},[32,57351,57352,57354,57357,57359,57361,57363,57365,57367],{},[81,57353,14810],{"fence":48},[35,57355,57356],{"mathvariant":231},"ℓ",[81,57358,741],{"stretchy":86},[35,57360,79],{"mathvariant":251},[81,57362,94],{"separator":48},[35,57364,24661],{"mathvariant":3101},[81,57366,750],{"stretchy":86},[81,57368,14819],{"fence":48},[40,57370,57371],{"encoding":42},"\\mathcal{L}^{\\text{fit}} = \\mathbb{E}_{\\boldsymbol{\\pi} \\sim \\operatorname{Dir}(\\boldsymbol{\\alpha})}\\left[\\ell(\\mathbf{y}, \\boldsymbol{\\pi})\\right]",[15,57373,57375,57425],{"className":57374,"ariaHidden":48},[47],[15,57376,57378,57381,57416,57419,57422],{"className":57377},[52],[15,57379],{"className":57380,"style":18849},[56],[15,57382,57384,57387],{"className":57383},[61],[15,57385,47453],{"className":57386},[61,62],[15,57388,57390],{"className":57389},[338],[15,57391,57393],{"className":57392},[342],[15,57394,57396],{"className":57395},[346],[15,57397,57399],{"className":57398,"style":18849},[350],[15,57400,57401,57404],{"style":353},[15,57402],{"className":57403,"style":358},[357],[15,57405,57407],{"className":57406},[362,363,364,365],[15,57408,57410],{"className":57409},[61,365],[15,57411,57413],{"className":57412},[61,13539,365],[15,57414,56656],{"className":57415},[61,365],[15,57417],{"className":57418,"style":128},[127],[15,57420,244],{"className":57421},[132],[15,57423],{"className":57424,"style":128},[127],[15,57426,57428,57432,57505,57508],{"className":57427},[52],[15,57429],{"className":57430,"style":57431},[56],"height:1.1052em;vertical-align:-0.3552em;",[15,57433,57435,57438],{"className":57434},[61],[15,57436,14807],{"className":57437},[61,427],[15,57439,57441],{"className":57440},[338],[15,57442,57444,57497],{"className":57443},[342,472],[15,57445,57447,57494],{"className":57446},[346],[15,57448,57450],{"className":57449,"style":22389},[350],[15,57451,57452,57455],{"style":47578},[15,57453],{"className":57454,"style":358},[357],[15,57456,57458],{"className":57457},[362,363,364,365],[15,57459,57461,57470,57473,57479,57482,57491],{"className":57460},[61,365],[15,57462,57464],{"className":57463},[61,365],[15,57465,57467],{"className":57466},[61,365],[15,57468,24661],{"className":57469,"style":24695},[61,3162,365],[15,57471,4179],{"className":57472},[132,365],[15,57474,57476],{"className":57475},[531,365],[15,57477,4182],{"className":57478},[61,1620,365],[15,57480,741],{"className":57481},[146,365],[15,57483,57485],{"className":57484},[61,365],[15,57486,57488],{"className":57487},[61,365],[15,57489,3102],{"className":57490},[61,3162,365],[15,57492,750],{"className":57493},[178,365],[15,57495,497],{"className":57496},[496],[15,57498,57500],{"className":57499},[346],[15,57501,57503],{"className":57502,"style":47630},[350],[15,57504],{},[15,57506],{"className":57507,"style":157},[127],[15,57509,57511,57514,57517,57520,57523,57526,57529,57538,57541],{"className":57510},[161],[15,57512,14810],{"className":57513,"style":402},[146,401],[15,57515,57356],{"className":57516},[61],[15,57518,741],{"className":57519},[146],[15,57521,79],{"className":57522,"style":29388},[61,411],[15,57524,94],{"className":57525},[153],[15,57527],{"className":57528,"style":157},[127],[15,57530,57532],{"className":57531},[61],[15,57533,57535],{"className":57534},[61],[15,57536,24661],{"className":57537,"style":24695},[61,3162],[15,57539,750],{"className":57540},[178],[15,57542,14819],{"className":57543,"style":402},[178,401],[11,57545,12917,57546,57579,57580,57692,57693,57812],{},[661,57547,57548,57549,57578],{},"对某个一阶损失 ",[15,57550,57552,57566],{"className":57551,"translate":19},[18],[15,57553,57555],{"className":57554},[23],[25,57556,57557],{"xmlns":27},[29,57558,57559,57563],{},[32,57560,57561],{},[35,57562,57356],{"mathvariant":231},[40,57564,57565],{"encoding":42},"\\ell",[15,57567,57569],{"className":57568,"ariaHidden":48},[47],[15,57570,57572,57575],{"className":57571},[52],[15,57573],{"className":57574,"style":10443},[56],[15,57576,57356],{"className":57577},[61]," 的先验期望","。取 ",[15,57581,57583,57611],{"className":57582,"translate":19},[18],[15,57584,57586],{"className":57585},[23],[25,57587,57588],{"xmlns":27},[29,57589,57590,57608],{},[32,57591,57592,57594,57596,57598,57600,57602],{},[35,57593,57356],{"mathvariant":231},[81,57595,244],{},[81,57597,239],{},[35,57599,22645],{},[81,57601,1545],{},[266,57603,57604,57606],{},[35,57605,24661],{},[35,57607,79],{},[40,57609,57610],{"encoding":42},"\\ell = -\\log \\pi_y",[15,57612,57614,57632],{"className":57613,"ariaHidden":48},[47],[15,57615,57617,57620,57623,57626,57629],{"className":57616},[52],[15,57618],{"className":57619,"style":10443},[56],[15,57621,57356],{"className":57622},[61],[15,57624],{"className":57625,"style":128},[127],[15,57627,244],{"className":57628},[132],[15,57630],{"className":57631,"style":128},[127],[15,57633,57635,57638,57641,57644,57649,57652],{"className":57634},[52],[15,57636],{"className":57637,"style":20933},[56],[15,57639,239],{"className":57640},[61],[15,57642],{"className":57643,"style":157},[127],[15,57645,22774,57647],{"className":57646},[531],[15,57648,22777],{"style":15608},[15,57650],{"className":57651,"style":157},[127],[15,57653,57655,57658],{"className":57654},[61],[15,57656,24661],{"className":57657,"style":123},[61,122],[15,57659,57661],{"className":57660},[338],[15,57662,57664,57684],{"className":57663},[342,472],[15,57665,57667,57681],{"className":57666},[346],[15,57668,57670],{"className":57669,"style":27010},[350],[15,57671,57672,57675],{"style":27013},[15,57673],{"className":57674,"style":358},[357],[15,57676,57678],{"className":57677},[362,363,364,365],[15,57679,79],{"className":57680,"style":123},[61,122,365],[15,57682,497],{"className":57683},[496],[15,57685,57687],{"className":57686},[346],[15,57688,57690],{"className":57689,"style":15696},[350],[15,57691],{}," 得命题 4.1 的 digamma 形式。取 ",[15,57694,57696,57726],{"className":57695,"translate":19},[18],[15,57697,57699],{"className":57698},[23],[25,57700,57701],{"xmlns":27},[29,57702,57703,57723],{},[32,57704,57705,57707,57709,57711,57713,57715,57717],{},[35,57706,57356],{"mathvariant":231},[81,57708,244],{},[35,57710,21278],{"mathvariant":231},[35,57712,79],{"mathvariant":251},[81,57714,239],{},[35,57716,24661],{"mathvariant":3101},[227,57718,57719,57721],{},[35,57720,21278],{"mathvariant":231},[89,57722,906],{},[40,57724,57725],{"encoding":42},"\\ell = \\|\\mathbf{y} - \\boldsymbol{\\pi}\\|^2",[15,57727,57729,57747,57768],{"className":57728,"ariaHidden":48},[47],[15,57730,57732,57735,57738,57741,57744],{"className":57731},[52],[15,57733],{"className":57734,"style":10443},[56],[15,57736,57356],{"className":57737},[61],[15,57739],{"className":57740,"style":128},[127],[15,57742,244],{"className":57743},[132],[15,57745],{"className":57746,"style":128},[127],[15,57748,57750,57753,57756,57759,57762,57765],{"className":57749},[52],[15,57751],{"className":57752,"style":142},[56],[15,57754,21278],{"className":57755},[61],[15,57757,79],{"className":57758,"style":29388},[61,411],[15,57760],{"className":57761,"style":772},[127],[15,57763,239],{"className":57764},[375],[15,57766],{"className":57767,"style":772},[127],[15,57769,57771,57774,57783],{"className":57770},[52],[15,57772],{"className":57773,"style":21402},[56],[15,57775,57777],{"className":57776},[61],[15,57778,57780],{"className":57779},[61],[15,57781,24661],{"className":57782,"style":24695},[61,3162],[15,57784,57786,57789],{"className":57785},[61],[15,57787,21278],{"className":57788},[61],[15,57790,57792],{"className":57791},[338],[15,57793,57795],{"className":57794},[342],[15,57796,57798],{"className":57797},[346],[15,57799,57801],{"className":57800,"style":7899},[350],[15,57802,57803,57806],{"style":716},[15,57804],{"className":57805,"style":358},[357],[15,57807,57809],{"className":57808},[362,363,364,365],[15,57810,906],{"className":57811},[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。",[11,57814,57815],{},"在此基础上，文献中出现的主要变体可归入三类：",[31520,57817,57818,57924,57934],{},[31523,57819,57820,57823,57824,57894,57895,57923],{},[661,57821,57822],{},"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 ",[15,57825,57827,57845],{"className":57826,"translate":19},[18],[15,57828,57830],{"className":57829},[23],[25,57831,57832],{"xmlns":27},[29,57833,57834,57842],{},[32,57835,57836],{},[266,57837,57838,57840],{},[35,57839,47453],{},[89,57841,906],{},[40,57843,57844],{"encoding":42},"L_2",[15,57846,57848],{"className":57847,"ariaHidden":48},[47],[15,57849,57851,57854],{"className":57850},[52],[15,57852],{"className":57853,"style":34331},[56],[15,57855,57857,57860],{"className":57856},[61],[15,57858,47453],{"className":57859},[61,122],[15,57861,57863],{"className":57862},[338],[15,57864,57866,57886],{"className":57865},[342,472],[15,57867,57869,57883],{"className":57868},[346],[15,57870,57872],{"className":57871,"style":1212},[350],[15,57873,57874,57877],{"style":482},[15,57875],{"className":57876,"style":358},[357],[15,57878,57880],{"className":57879},[362,363,364,365],[15,57881,906],{"className":57882},[61,365],[15,57884,497],{"className":57885},[496],[15,57887,57889],{"className":57888},[346],[15,57890,57892],{"className":57891,"style":504},[350],[15,57893],{}," 惩罚。其共同目标都是阻止 ",[15,57896,57898,57911],{"className":57897,"translate":19},[18],[15,57899,57901],{"className":57900},[23],[25,57902,57903],{"xmlns":27},[29,57904,57905,57909],{},[32,57906,57907],{},[35,57908,40230],{},[40,57910,40230],{"encoding":42},[15,57912,57914],{"className":57913,"ariaHidden":48},[47],[15,57915,57917,57920],{"className":57916},[52],[15,57918],{"className":57919,"style":57},[56],[15,57921,40230],{"className":57922,"style":40302},[61,122]," 的退化。",[31523,57925,57926,57929,57930,57933],{},[661,57927,57928],{},"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",[661,57931,57932],{},"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。",[31523,57935,57936,57939],{},[661,57937,57938],{},"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）。",[11,57941,57942,57943,57946,57947,57950],{},"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",[661,57944,57945],{},"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",[661,57948,57949],{},"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。",[877,57952,57954],{"id":57953},"_45-使用边界","4.5 使用边界",[11,57956,57957],{},"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：",[31520,57959,57960,58871,59001,59267,59328],{},[31523,57961,57962,57965,57966,58276,58277,58347,58348,867,58563,58566,58567,58750,58751,9342,58779,58841,58842,58870],{},[661,57963,57964],{},"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",[15,57967,57969,58055],{"className":57968,"translate":19},[18],[15,57970,57972],{"className":57971},[23],[25,57973,57974],{"xmlns":27},[29,57975,57976,58052],{},[32,57977,57978,57980,57982,57984,57986,57988,57990,57993,57995,57997,58000,58002,58004,58006,58008,58010,58012,58014,58016,58018,58020,58022,58024,58026,58028,58030,58032,58034,58036,58038,58040,58042,58044,58046,58048,58050],{},[35,57979,11],{},[81,57981,741],{"stretchy":86},[35,57983,79],{},[81,57985,3710],{},[35,57987,1027],{"mathvariant":251},[81,57989,94],{"separator":48},[35,57991,57992],{"mathvariant":37},"D",[81,57994,750],{"stretchy":86},[81,57996,244],{},[81,57998,57999],{},"∬",[35,58001,11],{},[81,58003,741],{"stretchy":86},[35,58005,79],{},[81,58007,3710],{},[35,58009,24661],{"mathvariant":3101},[81,58011,750],{"stretchy":86},[282,58013,4796],{},[35,58015,11],{},[81,58017,741],{"stretchy":86},[35,58019,24661],{"mathvariant":3101},[81,58021,3710],{},[35,58023,37117],{"mathvariant":3101},[81,58025,750],{"stretchy":86},[282,58027,4796],{},[35,58029,11],{},[81,58031,741],{"stretchy":86},[35,58033,37117],{"mathvariant":3101},[81,58035,3710],{},[35,58037,57992],{"mathvariant":37},[81,58039,750],{"stretchy":86},[282,58041,4796],{},[35,58043,2786],{"mathvariant":231},[35,58045,24661],{"mathvariant":3101},[282,58047,4796],{},[35,58049,2786],{"mathvariant":231},[35,58051,37117],{"mathvariant":3101},[40,58053,58054],{"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}",[15,58056,58058,58082,58112,58144,58189,58234],{"className":58057,"ariaHidden":48},[47],[15,58059,58061,58064,58067,58070,58073,58076,58079],{"className":58060},[52],[15,58062],{"className":58063,"style":142},[56],[15,58065,11],{"className":58066},[61,122],[15,58068,741],{"className":58069},[146],[15,58071,79],{"className":58072,"style":123},[61,122],[15,58074],{"className":58075,"style":128},[127],[15,58077,3710],{"className":58078},[132],[15,58080],{"className":58081,"style":128},[127],[15,58083,58085,58088,58091,58094,58097,58100,58103,58106,58109],{"className":58084},[52],[15,58086],{"className":58087,"style":142},[56],[15,58089,1027],{"className":58090},[61,411],[15,58092,94],{"className":58093},[153],[15,58095],{"className":58096,"style":157},[127],[15,58098,57992],{"className":58099,"style":12278},[61,62],[15,58101,750],{"className":58102},[178],[15,58104],{"className":58105,"style":128},[127],[15,58107,244],{"className":58108},[132],[15,58110],{"className":58111,"style":128},[127],[15,58113,58115,58119,58123,58126,58129,58132,58135,58138,58141],{"className":58114},[52],[15,58116],{"className":58117,"style":58118},[56],"height:1.111em;vertical-align:-0.306em;",[15,58120,57999],{"className":58121,"style":58122},[531,535,536],"margin-right:0.1945em;position:relative;top:-0.0005em;",[15,58124],{"className":58125,"style":157},[127],[15,58127,11],{"className":58128},[61,122],[15,58130,741],{"className":58131},[146],[15,58133,79],{"className":58134,"style":123},[61,122],[15,58136],{"className":58137,"style":128},[127],[15,58139,3710],{"className":58140},[132],[15,58142],{"className":58143,"style":128},[127],[15,58145,58147,58150,58159,58162,58165,58168,58171,58180,58183,58186],{"className":58146},[52],[15,58148],{"className":58149,"style":142},[56],[15,58151,58153],{"className":58152},[61],[15,58154,58156],{"className":58155},[61],[15,58157,24661],{"className":58158,"style":24695},[61,3162],[15,58160,750],{"className":58161},[178],[15,58163],{"className":58164,"style":157},[127],[15,58166,11],{"className":58167},[61,122],[15,58169,741],{"className":58170},[146],[15,58172,58174],{"className":58173},[61],[15,58175,58177],{"className":58176},[61],[15,58178,24661],{"className":58179,"style":24695},[61,3162],[15,58181],{"className":58182,"style":128},[127],[15,58184,3710],{"className":58185},[132],[15,58187],{"className":58188,"style":128},[127],[15,58190,58192,58195,58204,58207,58210,58213,58216,58225,58228,58231],{"className":58191},[52],[15,58193],{"className":58194,"style":142},[56],[15,58196,58198],{"className":58197},[61],[15,58199,58201],{"className":58200},[61],[15,58202,37117],{"className":58203,"style":37181},[61,3162],[15,58205,750],{"className":58206},[178],[15,58208],{"className":58209,"style":157},[127],[15,58211,11],{"className":58212},[61,122],[15,58214,741],{"className":58215},[146],[15,58217,58219],{"className":58218},[61],[15,58220,58222],{"className":58221},[61],[15,58223,37117],{"className":58224,"style":37181},[61,3162],[15,58226],{"className":58227,"style":128},[127],[15,58229,3710],{"className":58230},[132],[15,58232],{"className":58233,"style":128},[127],[15,58235,58237,58240,58243,58246,58249,58252,58261,58264,58267],{"className":58236},[52],[15,58238],{"className":58239,"style":142},[56],[15,58241,57992],{"className":58242,"style":12278},[61,62],[15,58244,750],{"className":58245},[178],[15,58247],{"className":58248,"style":157},[127],[15,58250,2786],{"className":58251},[61,1620],[15,58253,58255],{"className":58254},[61],[15,58256,58258],{"className":58257},[61],[15,58259,24661],{"className":58260,"style":24695},[61,3162],[15,58262],{"className":58263,"style":157},[127],[15,58265,2786],{"className":58266},[61,1620],[15,58268,58270],{"className":58269},[61],[15,58271,58273],{"className":58272},[61],[15,58274,37117],{"className":58275,"style":37181},[61,3162],"\n而 EDL 以点估计 ",[15,58278,58280,58299],{"className":58279,"translate":19},[18],[15,58281,58283],{"className":58282},[23],[25,58284,58285],{"xmlns":27},[29,58286,58287,58296],{},[32,58288,58289],{},[53179,58290,58291,58293],{"accent":48},[35,58292,37117],{"mathvariant":3101},[81,58294,58295],{},"^",[40,58297,58298],{"encoding":42},"\\hat{\\boldsymbol{\\theta}}",[15,58300,58302],{"className":58301,"ariaHidden":48},[47],[15,58303,58305,58309],{"className":58304},[52],[15,58306],{"className":58307,"style":58308},[56],"height:0.9579em;",[15,58310,58312],{"className":58311},[61,53223],[15,58313,58315],{"className":58314},[342],[15,58316,58318],{"className":58317},[346],[15,58319,58321,58335],{"className":58320,"style":58308},[350],[15,58322,58323,58326],{"style":53235},[15,58324],{"className":58325,"style":3857},[357],[15,58327,58329],{"className":58328},[61],[15,58330,58332],{"className":58331},[61],[15,58333,37117],{"className":58334,"style":37181},[61,3162],[15,58336,58338,58341],{"style":58337},"top:-3.2634em;",[15,58339],{"className":58340,"style":3857},[357],[15,58342,58344],{"className":58343,"style":53258},[53257],[15,58345,58295],{"className":58346},[61]," 替换外层积分，取 ",[15,58349,58351,58405],{"className":58350,"translate":19},[18],[15,58352,58354],{"className":58353},[23],[25,58355,58356],{"xmlns":27},[29,58357,58358,58402],{},[32,58359,58360,58362,58364,58366,58368,58370,58372,58374,58376,58378,58380,58382,58384,58386,58388,58390,58392,58398,58400],{},[35,58361,11],{},[81,58363,741],{"stretchy":86},[35,58365,24661],{"mathvariant":3101},[81,58367,3710],{},[35,58369,1027],{"mathvariant":251},[81,58371,750],{"stretchy":86},[81,58373,244],{},[35,58375,4182],{"mathvariant":231},[81,58377,1545],{},[81,58379,741],{"fence":86,"stretchy":48,"minsize":44812,"maxsize":44812},[35,58381,24661],{"mathvariant":3101},[81,58383,11294],{"separator":48},[35,58385,3102],{"mathvariant":3101},[81,58387,741],{"stretchy":86},[35,58389,1027],{"mathvariant":251},[81,58391,11294],{"separator":48},[53179,58393,58394,58396],{"accent":48},[35,58395,37117],{"mathvariant":3101},[81,58397,58295],{},[81,58399,750],{"stretchy":86},[81,58401,750],{"fence":86,"stretchy":48,"minsize":44812,"maxsize":44812},[40,58403,58404],{"encoding":42},"p(\\boldsymbol{\\pi} \\mid \\mathbf{x}) = \\operatorname{Dir}\\big(\\boldsymbol{\\pi}; \\boldsymbol{\\alpha}(\\mathbf{x}; \\hat{\\boldsymbol{\\theta}})\\big)",[15,58406,58408,58438,58459],{"className":58407,"ariaHidden":48},[47],[15,58409,58411,58414,58417,58420,58429,58432,58435],{"className":58410},[52],[15,58412],{"className":58413,"style":142},[56],[15,58415,11],{"className":58416},[61,122],[15,58418,741],{"className":58419},[146],[15,58421,58423],{"className":58422},[61],[15,58424,58426],{"className":58425},[61],[15,58427,24661],{"className":58428,"style":24695},[61,3162],[15,58430],{"className":58431,"style":128},[127],[15,58433,3710],{"className":58434},[132],[15,58436],{"className":58437,"style":128},[127],[15,58439,58441,58444,58447,58450,58453,58456],{"className":58440},[52],[15,58442],{"className":58443,"style":142},[56],[15,58445,1027],{"className":58446},[61,411],[15,58448,750],{"className":58449},[178],[15,58451],{"className":58452,"style":128},[127],[15,58454,244],{"className":58455},[132],[15,58457],{"className":58458,"style":128},[127],[15,58460,58462,58466,58472,58475,58481,58490,58493,58496,58505,58508,58511,58514,58517,58554,58557],{"className":58461},[52],[15,58463],{"className":58464,"style":58465},[56],"height:1.3079em;vertical-align:-0.35em;",[15,58467,58469],{"className":58468},[531],[15,58470,4182],{"className":58471},[61,1620],[15,58473],{"className":58474,"style":157},[127],[15,58476,58478],{"className":58477},[61],[15,58479,741],{"className":58480},[406,4063],[15,58482,58484],{"className":58483},[61],[15,58485,58487],{"className":58486},[61],[15,58488,24661],{"className":58489,"style":24695},[61,3162],[15,58491,11294],{"className":58492},[153],[15,58494],{"className":58495,"style":157},[127],[15,58497,58499],{"className":58498},[61],[15,58500,58502],{"className":58501},[61],[15,58503,3102],{"className":58504},[61,3162],[15,58506,741],{"className":58507},[146],[15,58509,1027],{"className":58510},[61,411],[15,58512,11294],{"className":58513},[153],[15,58515],{"className":58516,"style":157},[127],[15,58518,58520],{"className":58519},[61,53223],[15,58521,58523],{"className":58522},[342],[15,58524,58526],{"className":58525},[346],[15,58527,58529,58543],{"className":58528,"style":58308},[350],[15,58530,58531,58534],{"style":53235},[15,58532],{"className":58533,"style":3857},[357],[15,58535,58537],{"className":58536},[61],[15,58538,58540],{"className":58539},[61],[15,58541,37117],{"className":58542,"style":37181},[61,3162],[15,58544,58545,58548],{"style":58337},[15,58546],{"className":58547,"style":3857},[357],[15,58549,58551],{"className":58550,"style":53258},[53257],[15,58552,58295],{"className":58553},[61],[15,58555,750],{"className":58556},[178],[15,58558,58560],{"className":58559},[61],[15,58561,750],{"className":58562},[406,4063],[661,58564,58565],{},"结果是一个插值（plug-in）估计","，其内层积分给出 ",[15,58568,58570,58603],{"className":58569,"translate":19},[18],[15,58571,58573],{"className":58572},[23],[25,58574,58575],{"xmlns":27},[29,58576,58577,58601],{},[32,58578,58579,58585,58587,58593,58595],{},[266,58580,58581,58583],{},[35,58582,11],{},[35,58584,272],{},[81,58586,244],{},[266,58588,58589,58591],{},[35,58590,3102],{},[35,58592,272],{},[35,58594,3018],{"mathvariant":231},[266,58596,58597,58599],{},[35,58598,3102],{},[89,58600,278],{},[40,58602,43798],{"encoding":42},[15,58604,58606,58661],{"className":58605,"ariaHidden":48},[47],[15,58607,58609,58612,58652,58655,58658],{"className":58608},[52],[15,58610],{"className":58611,"style":9428},[56],[15,58613,58615,58618],{"className":58614},[61],[15,58616,11],{"className":58617},[61,122],[15,58619,58621],{"className":58620},[338],[15,58622,58624,58644],{"className":58623},[342,472],[15,58625,58627,58641],{"className":58626},[346],[15,58628,58630],{"className":58629,"style":479},[350],[15,58631,58632,58635],{"style":482},[15,58633],{"className":58634,"style":358},[357],[15,58636,58638],{"className":58637},[362,363,364,365],[15,58639,272],{"className":58640,"style":492},[61,122,365],[15,58642,497],{"className":58643},[496],[15,58645,58647],{"className":58646},[346],[15,58648,58650],{"className":58649,"style":504},[350],[15,58651],{},[15,58653],{"className":58654,"style":128},[127],[15,58656,244],{"className":58657},[132],[15,58659],{"className":58660,"style":128},[127],[15,58662,58664,58667,58707,58710],{"className":58663},[52],[15,58665],{"className":58666,"style":142},[56],[15,58668,58670,58673],{"className":58669},[61],[15,58671,3102],{"className":58672,"style":3187},[61,122],[15,58674,58676],{"className":58675},[338],[15,58677,58679,58699],{"className":58678},[342,472],[15,58680,58682,58696],{"className":58681},[346],[15,58683,58685],{"className":58684,"style":479},[350],[15,58686,58687,58690],{"style":3202},[15,58688],{"className":58689,"style":358},[357],[15,58691,58693],{"className":58692},[362,363,364,365],[15,58694,272],{"className":58695,"style":492},[61,122,365],[15,58697,497],{"className":58698},[496],[15,58700,58702],{"className":58701},[346],[15,58703,58705],{"className":58704,"style":504},[350],[15,58706],{},[15,58708,3018],{"className":58709},[61],[15,58711,58713,58716],{"className":58712},[61],[15,58714,3102],{"className":58715,"style":3187},[61,122],[15,58717,58719],{"className":58718},[338],[15,58720,58722,58742],{"className":58721},[342,472],[15,58723,58725,58739],{"className":58724},[346],[15,58726,58728],{"className":58727,"style":1212},[350],[15,58729,58730,58733],{"style":3202},[15,58731],{"className":58732,"style":358},[357],[15,58734,58736],{"className":58735},[362,363,364,365],[15,58737,278],{"className":58738},[61,365],[15,58740,497],{"className":58741},[496],[15,58743,58745],{"className":58744},[346],[15,58746,58748],{"className":58747,"style":504},[350],[15,58749],{},"（由聚合性），确实正确。但外层积分被抛弃，",[15,58752,58754,58767],{"className":58753,"translate":19},[18],[15,58755,58757],{"className":58756},[23],[25,58758,58759],{"xmlns":27},[29,58760,58761,58765],{},[32,58762,58763],{},[35,58764,33401],{},[40,58766,33401],{"encoding":42},[15,58768,58770],{"className":58769,"ariaHidden":48},[47],[15,58771,58773,58776],{"className":58772},[52],[15,58774],{"className":58775,"style":6026},[56],[15,58777,33401],{"className":58778},[61,122],[15,58780,58782,58805],{"className":58781,"translate":19},[18],[15,58783,58785],{"className":58784},[23],[25,58786,58787],{"xmlns":27},[29,58788,58789,58803],{},[32,58790,58791,58793,58795,58797,58799,58801],{},[35,58792,28898],{},[81,58794,741],{"stretchy":86},[35,58796,79],{},[81,58798,11294],{"separator":48},[35,58800,24661],{"mathvariant":3101},[81,58802,750],{"stretchy":86},[40,58804,46847],{"encoding":42},[15,58806,58808],{"className":58807,"ariaHidden":48},[47],[15,58809,58811,58814,58817,58820,58823,58826,58829,58838],{"className":58810},[52],[15,58812],{"className":58813,"style":142},[56],[15,58815,28898],{"className":58816,"style":28937},[61,122],[15,58818,741],{"className":58819},[146],[15,58821,79],{"className":58822,"style":123},[61,122],[15,58824,11294],{"className":58825},[153],[15,58827],{"className":58828,"style":157},[127],[15,58830,58832],{"className":58831},[61],[15,58833,58835],{"className":58834},[61],[15,58836,24661],{"className":58837,"style":24695},[61,3162],[15,58839,750],{"className":58840},[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 ",[15,58843,58845,58858],{"className":58844,"translate":19},[18],[15,58846,58848],{"className":58847},[23],[25,58849,58850],{"xmlns":27},[29,58851,58852,58856],{},[32,58853,58854],{},[35,58855,33401],{},[40,58857,33401],{"encoding":42},[15,58859,58861],{"className":58860,"ariaHidden":48},[47],[15,58862,58864,58867],{"className":58863},[52],[15,58865],{"className":58866,"style":6026},[56],[15,58868,33401],{"className":58869},[61,122]," 输出一个错误的类别。",[31523,58872,58873,58876,58877,58937,58938,59000],{},[661,58874,58875],{},"不确定性分数丢弃了证据的分布信息。"," 如 3.5 节所述，",[15,58878,58880,58901],{"className":58879,"translate":19},[18],[15,58881,58883],{"className":58882},[23],[25,58884,58885],{"xmlns":27},[29,58886,58887,58899],{},[32,58888,58889,58891,58893,58895,58897],{},[35,58890,33401],{},[81,58892,244],{},[35,58894,102],{},[35,58896,3018],{"mathvariant":231},[35,58898,40230],{},[40,58900,42949],{"encoding":42},[15,58902,58904,58922],{"className":58903,"ariaHidden":48},[47],[15,58905,58907,58910,58913,58916,58919],{"className":58906},[52],[15,58908],{"className":58909,"style":6026},[56],[15,58911,33401],{"className":58912},[61,122],[15,58914],{"className":58915,"style":128},[127],[15,58917,244],{"className":58918},[132],[15,58920],{"className":58921,"style":128},[127],[15,58923,58925,58928,58931,58934],{"className":58924},[52],[15,58926],{"className":58927,"style":142},[56],[15,58929,102],{"className":58930,"style":174},[61,122],[15,58932,3018],{"className":58933},[61],[15,58935,40230],{"className":58936,"style":40302},[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。",[15,58939,58941,58964],{"className":58940,"translate":19},[18],[15,58942,58944],{"className":58943},[23],[25,58945,58946],{"xmlns":27},[29,58947,58948,58962],{},[32,58949,58950,58952,58954,58956,58958,58960],{},[35,58951,28898],{},[81,58953,741],{"stretchy":86},[35,58955,79],{},[81,58957,11294],{"separator":48},[35,58959,24661],{"mathvariant":3101},[81,58961,750],{"stretchy":86},[40,58963,46847],{"encoding":42},[15,58965,58967],{"className":58966,"ariaHidden":48},[47],[15,58968,58970,58973,58976,58979,58982,58985,58988,58997],{"className":58969},[52],[15,58971],{"className":58972,"style":142},[56],[15,58974,28898],{"className":58975,"style":28937},[61,122],[15,58977,741],{"className":58978},[146],[15,58980,79],{"className":58981,"style":123},[61,122],[15,58983,11294],{"className":58984},[153],[15,58986],{"className":58987,"style":157},[127],[15,58989,58991],{"className":58990},[61],[15,58992,58994],{"className":58993},[61],[15,58995,24661],{"className":58996,"style":24695},[61,3162],[15,58998,750],{"className":58999},[178]," 缓解了这一点，但仍不引入任何关于模型参数的信息。",[31523,59002,59003,59006,59007,59262,59263,59266],{},[661,59004,59005],{},"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 ",[15,59008,59010,59052],{"className":59009,"translate":19},[18],[15,59011,59013],{"className":59012},[23],[25,59014,59015],{"xmlns":27},[29,59016,59017,59049],{},[32,59018,59019,59021,59035,59037,59047],{},[81,59020,741],{"stretchy":86},[3718,59022,59023,59025],{},[89,59024,91],{},[32,59026,59027,59029],{},[89,59028,906],{},[266,59030,59031,59033],{},[35,59032,3102],{},[35,59034,79],{},[81,59036,239],{},[3718,59038,59039,59041],{},[89,59040,91],{},[32,59042,59043,59045],{},[89,59044,906],{},[35,59046,40230],{},[81,59048,750],{"stretchy":86},[40,59050,59051],{"encoding":42},"(\\frac{1}{2\\alpha_y} - \\frac{1}{2S})",[15,59053,59055,59182],{"className":59054,"ariaHidden":48},[47],[15,59056,59058,59062,59065,59173,59176,59179],{"className":59057},[52],[15,59059],{"className":59060,"style":59061},[56],"height:1.3874em;vertical-align:-0.5423em;",[15,59063,741],{"className":59064},[146],[15,59066,59068,59071,59170],{"className":59067},[61],[15,59069],{"className":59070},[146,3837],[15,59072,59074],{"className":59073},[3718],[15,59075,59077,59162],{"className":59076},[342,472],[15,59078,59080,59159],{"className":59079},[346],[15,59081,59083,59137,59145],{"className":59082,"style":3850},[350],[15,59084,59085,59088],{"style":3853},[15,59086],{"className":59087,"style":3857},[357],[15,59089,59091],{"className":59090},[362,363,364,365],[15,59092,59094,59097],{"className":59093},[61,365],[15,59095,906],{"className":59096},[61,365],[15,59098,59100,59103],{"className":59099},[61,365],[15,59101,3102],{"className":59102,"style":3187},[61,122,365],[15,59104,59106],{"className":59105},[338],[15,59107,59109,59129],{"className":59108},[342,472],[15,59110,59112,59126],{"className":59111},[346],[15,59113,59115],{"className":59114,"style":29267},[350],[15,59116,59117,59120],{"style":4054},[15,59118],{"className":59119,"style":4058},[357],[15,59121,59123],{"className":59122},[362,4062,4063,365],[15,59124,79],{"className":59125,"style":123},[61,122,365],[15,59127,497],{"className":59128},[496],[15,59130,59132],{"className":59131},[346],[15,59133,59135],{"className":59134,"style":15952},[350],[15,59136],{},[15,59138,59139,59142],{"style":3885},[15,59140],{"className":59141,"style":3857},[357],[15,59143],{"className":59144,"style":3893},[3892],[15,59146,59147,59150],{"style":3896},[15,59148],{"className":59149,"style":3857},[357],[15,59151,59153],{"className":59152},[362,363,364,365],[15,59154,59156],{"className":59155},[61,365],[15,59157,91],{"className":59158},[61,365],[15,59160,497],{"className":59161},[496],[15,59163,59165],{"className":59164},[346],[15,59166,59168],{"className":59167,"style":49837},[350],[15,59169],{},[15,59171],{"className":59172},[178,3837],[15,59174],{"className":59175,"style":772},[127],[15,59177,239],{"className":59178},[375],[15,59180],{"className":59181,"style":772},[127],[15,59183,59185,59188,59259],{"className":59184},[52],[15,59186],{"className":59187,"style":42536},[56],[15,59189,59191,59194,59256],{"className":59190},[61],[15,59192],{"className":59193},[146,3837],[15,59195,59197],{"className":59196},[3718],[15,59198,59200,59248],{"className":59199},[342,472],[15,59201,59203,59245],{"className":59202},[346],[15,59204,59206,59223,59231],{"className":59205,"style":3850},[350],[15,59207,59208,59211],{"style":3853},[15,59209],{"className":59210,"style":3857},[357],[15,59212,59214],{"className":59213},[362,363,364,365],[15,59215,59217,59220],{"className":59216},[61,365],[15,59218,906],{"className":59219},[61,365],[15,59221,40230],{"className":59222,"style":40302},[61,122,365],[15,59224,59225,59228],{"style":3885},[15,59226],{"className":59227,"style":3857},[357],[15,59229],{"className":59230,"style":3893},[3892],[15,59232,59233,59236],{"style":3896},[15,59234],{"className":59235,"style":3857},[357],[15,59237,59239],{"className":59238},[362,363,364,365],[15,59240,59242],{"className":59241},[61,365],[15,59243,91],{"className":59244},[61,365],[15,59246,497],{"className":59247},[496],[15,59249,59251],{"className":59250},[346],[15,59252,59254],{"className":59253,"style":10189},[350],[15,59255],{},[15,59257],{"className":59258},[178,3837],[15,59260,750],{"className":59261},[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",[661,59264,59265],{},"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。",[31523,59268,59269,59272,59273,59327],{},[661,59270,59271],{},"「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 ",[15,59274,59276,59297],{"className":59275,"translate":19},[18],[15,59277,59279],{"className":59278},[23],[25,59280,59281],{"xmlns":27},[29,59282,59283,59295],{},[32,59284,59285,59287,59289,59291,59293],{},[35,59286,4182],{"mathvariant":231},[81,59288,1545],{},[81,59290,741],{"stretchy":86},[35,59292,3102],{"mathvariant":3101},[81,59294,750],{"stretchy":86},[40,59296,39039],{"encoding":42},[15,59298,59300],{"className":59299,"ariaHidden":48},[47],[15,59301,59303,59306,59312,59315,59324],{"className":59302},[52],[15,59304],{"className":59305,"style":142},[56],[15,59307,59309],{"className":59308},[531],[15,59310,4182],{"className":59311},[61,1620],[15,59313,741],{"className":59314},[146],[15,59316,59318],{"className":59317},[61],[15,59319,59321],{"className":59320},[61],[15,59322,3102],{"className":59323},[61,3162],[15,59325,750],{"className":59326},[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。",[31523,59329,59330,59333,59334,59450,59451,59479,59480,867],{},[661,59331,59332],{},"它对主干特征的可分性有隐含依赖。"," 证据由 ",[15,59335,59337,59365],{"className":59336,"translate":19},[18],[15,59338,59340],{"className":59339},[23],[25,59341,59342],{"xmlns":27},[29,59343,59344,59362],{},[32,59345,59346,59348,59350,59356,59358,59360],{},[35,59347,1980],{"mathvariant":251},[81,59349,244],{},[266,59351,59352,59354],{},[35,59353,22777],{},[35,59355,37117],{"mathvariant":3101},[81,59357,741],{"stretchy":86},[35,59359,1027],{"mathvariant":251},[81,59361,750],{"stretchy":86},[40,59363,59364],{"encoding":42},"\\mathbf{e} = g_{\\boldsymbol{\\theta}}(\\mathbf{x})",[15,59366,59368,59386],{"className":59367,"ariaHidden":48},[47],[15,59369,59371,59374,59377,59380,59383],{"className":59370},[52],[15,59372],{"className":59373,"style":2706},[56],[15,59375,1980],{"className":59376},[61,411],[15,59378],{"className":59379,"style":128},[127],[15,59381,244],{"className":59382},[132],[15,59384],{"className":59385,"style":128},[127],[15,59387,59389,59392,59441,59444,59447],{"className":59388},[52],[15,59390],{"className":59391,"style":142},[56],[15,59393,59395,59398],{"className":59394},[61],[15,59396,22777],{"className":59397,"style":123},[61,122],[15,59399,59401],{"className":59400},[338],[15,59402,59404,59433],{"className":59403},[342,472],[15,59405,59407,59430],{"className":59406},[346],[15,59408,59410],{"className":59409,"style":479},[350],[15,59411,59412,59415],{"style":27013},[15,59413],{"className":59414,"style":358},[357],[15,59416,59418],{"className":59417},[362,363,364,365],[15,59419,59421],{"className":59420},[61,365],[15,59422,59424],{"className":59423},[61,365],[15,59425,59427],{"className":59426},[61,365],[15,59428,37117],{"className":59429,"style":37181},[61,3162,365],[15,59431,497],{"className":59432},[496],[15,59434,59436],{"className":59435},[346],[15,59437,59439],{"className":59438,"style":504},[350],[15,59440],{},[15,59442,741],{"className":59443},[146],[15,59445,1027],{"className":59446},[61,411],[15,59448,750],{"className":59449},[178]," 给出，因此 ",[15,59452,59454,59467],{"className":59453,"translate":19},[18],[15,59455,59457],{"className":59456},[23],[25,59458,59459],{"xmlns":27},[29,59460,59461,59465],{},[32,59462,59463],{},[35,59464,1027],{"mathvariant":251},[40,59466,2696],{"encoding":42},[15,59468,59470],{"className":59469,"ariaHidden":48},[47],[15,59471,59473,59476],{"className":59472},[52],[15,59474],{"className":59475,"style":2706},[56],[15,59477,1027],{"className":59478},[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——",[661,59481,59482],{},"它把 OOD 问题从输出层转移到了特征层，而非解决它",[11,59484,59485,59486,59489],{},"综合以上点，恰当的使用立场是：",[661,59487,59488],{},"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。",[872,59491,59493],{"id":59492},"五应用","五、应用",[877,59495,59497],{"id":59496},"_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",[11,59499,59500,59501,42986,59561,59564,59565,59593,59594,59625],{},"由命题 3.4，",[15,59502,59504,59525],{"className":59503,"translate":19},[18],[15,59505,59507],{"className":59506},[23],[25,59508,59509],{"xmlns":27},[29,59510,59511,59523],{},[32,59512,59513,59515,59517,59519,59521],{},[35,59514,33401],{},[81,59516,244],{},[35,59518,102],{},[35,59520,3018],{"mathvariant":231},[35,59522,40230],{},[40,59524,42949],{"encoding":42},[15,59526,59528,59546],{"className":59527,"ariaHidden":48},[47],[15,59529,59531,59534,59537,59540,59543],{"className":59530},[52],[15,59532],{"className":59533,"style":6026},[56],[15,59535,33401],{"className":59536},[61,122],[15,59538],{"className":59539,"style":128},[127],[15,59541,244],{"className":59542},[132],[15,59544],{"className":59545,"style":128},[127],[15,59547,59549,59552,59555,59558],{"className":59548},[52],[15,59550],{"className":59551,"style":142},[56],[15,59553,102],{"className":59554,"style":174},[61,122],[15,59556,3018],{"className":59557},[61],[15,59559,40230],{"className":59560,"style":40302},[61,122],[661,59562,59563],{},"单调反比于总证据","的量，因此可以直接充当分布外（OOD）分数：",[15,59566,59568,59581],{"className":59567,"translate":19},[18],[15,59569,59571],{"className":59570},[23],[25,59572,59573],{"xmlns":27},[29,59574,59575,59579],{},[32,59576,59577],{},[35,59578,33401],{},[40,59580,33401],{"encoding":42},[15,59582,59584],{"className":59583,"ariaHidden":48},[47],[15,59585,59587,59590],{"className":59586},[52],[15,59588],{"className":59589,"style":6026},[56],[15,59591,33401],{"className":59592},[61,122]," 高意味着输入未能激活任何类别的证据。以阈值 ",[15,59595,59597,59612],{"className":59596,"translate":19},[18],[15,59598,59600],{"className":59599},[23],[25,59601,59602],{"xmlns":27},[29,59603,59604,59609],{},[32,59605,59606],{},[35,59607,59608],{},"τ",[40,59610,59611],{"encoding":42},"\\tau",[15,59613,59615],{"className":59614,"ariaHidden":48},[47],[15,59616,59618,59621],{"className":59617},[52],[15,59619],{"className":59620,"style":6026},[56],[15,59622,59608],{"className":59623,"style":59624},[61,122],"margin-right:0.1132em;"," 作拒识判决",[15,59627,59629],{"className":59628,"translate":19},[212],[15,59630,59632,59747],{"className":59631,"translate":19},[18],[15,59633,59635],{"className":59634},[23],[25,59636,59637],{"xmlns":27,"display":221},[29,59638,59639,59744],{},[32,59640,59641,59647,59649,59651,59653,59655],{},[53179,59642,59643,59645],{"accent":48},[35,59644,79],{},[81,59646,58295],{},[81,59648,741],{"stretchy":86},[35,59650,1027],{"mathvariant":251},[81,59652,750],{"stretchy":86},[81,59654,244],{},[32,59656,59657,59659],{},[81,59658,87],{"fence":48},[6441,59660,59663,59713],{"rowspacing":59661,"columnalign":59662,"columnspacing":6445},"0.36em","left left",[6447,59664,59665,59695],{},[6450,59666,59667],{},[286,59668,59669],{"scriptlevel":278,"displaystyle":86},[32,59670,59671,59674,59676,59687,59693],{},[35,59672,59673],{},"arg",[81,59675,1545],{},[266,59677,59678,59685],{},[32,59679,59680,59683],{},[35,59681,59682],{},"max",[81,59684,1545],{},[35,59686,272],{},[266,59688,59689,59691],{},[35,59690,11],{},[35,59692,272],{},[81,59694,94],{"separator":48},[6450,59696,59697],{},[286,59698,59699],{"scriptlevel":278,"displaystyle":86},[32,59700,59701,59703,59705,59707,59709,59711],{},[35,59702,33401],{},[81,59704,741],{"stretchy":86},[35,59706,1027],{"mathvariant":251},[81,59708,750],{"stretchy":86},[81,59710,21295],{},[35,59712,59608],{},[6447,59714,59715,59726],{},[6450,59716,59717],{},[286,59718,59719],{"scriptlevel":278,"displaystyle":86},[32,59720,59721,59724],{},[282,59722,59723],{},"reject",[81,59725,94],{"separator":48},[6450,59727,59728],{},[286,59729,59730],{"scriptlevel":278,"displaystyle":86},[32,59731,59732,59734,59736,59738,59740,59742],{},[35,59733,33401],{},[81,59735,741],{"stretchy":86},[35,59737,1027],{"mathvariant":251},[81,59739,750],{"stretchy":86},[81,59741,1588],{},[35,59743,59608],{},[40,59745,59746],{"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}",[15,59748,59750,59818],{"className":59749,"ariaHidden":48},[47],[15,59751,59753,59756,59800,59803,59806,59809,59812,59815],{"className":59752},[52],[15,59754],{"className":59755,"style":142},[56],[15,59757,59759],{"className":59758},[61,53223],[15,59760,59762,59791],{"className":59761},[342,472],[15,59763,59765,59788],{"className":59764},[346],[15,59766,59768,59776],{"className":59767,"style":10443},[350],[15,59769,59770,59773],{"style":53235},[15,59771],{"className":59772,"style":3857},[357],[15,59774,79],{"className":59775,"style":123},[61,122],[15,59777,59778,59781],{"style":53235},[15,59779],{"className":59780,"style":3857},[357],[15,59782,59785],{"className":59783,"style":59784},[53257],"left:-0.1944em;",[15,59786,58295],{"className":59787},[61],[15,59789,497],{"className":59790},[496],[15,59792,59794],{"className":59793},[346],[15,59795,59798],{"className":59796,"style":59797},[350],"height:0.1944em;",[15,59799],{},[15,59801,741],{"className":59802},[146],[15,59804,1027],{"className":59805},[61,411],[15,59807,750],{"className":59808},[178],[15,59810],{"className":59811,"style":128},[127],[15,59813,244],{"className":59814},[132],[15,59816],{"className":59817,"style":128},[127],[15,59819,59821,59825],{"className":59820},[52],[15,59822],{"className":59823,"style":59824},[56],"height:3em;vertical-align:-1.25em;",[15,59826,59828,59834,60080],{"className":59827},[161],[15,59829,59831],{"className":59830,"style":402},[146,401],[15,59832,87],{"className":59833},[406,8640],[15,59835,59837],{"className":59836},[61],[15,59838,59840,59989,59993],{"className":59839},[6441],[15,59841,59844],{"className":59842},[59843],"col-align-l",[15,59845,59847,59980],{"className":59846},[342,472],[15,59848,59850,59977],{"className":59849},[346],[15,59851,59854,59959],{"className":59852,"style":59853},[350],"height:1.69em;",[15,59855,59857,59861],{"style":59856},"top:-3.69em;",[15,59858],{"className":59859,"style":59860},[357],"height:3.008em;",[15,59862,59864,59870,59873,59913,59916,59956],{"className":59863},[61],[15,59865,59867,59868],{"className":59866},[531],"ar",[15,59869,22777],{"style":15608},[15,59871],{"className":59872,"style":157},[127],[15,59874,59876,59879],{"className":59875},[531],[15,59877,59682],{"className":59878},[531],[15,59880,59882],{"className":59881},[338],[15,59883,59885,59905],{"className":59884},[342,472],[15,59886,59888,59902],{"className":59887},[346],[15,59889,59891],{"className":59890,"style":479},[350],[15,59892,59893,59896],{"style":30109},[15,59894],{"className":59895,"style":358},[357],[15,59897,59899],{"className":59898},[362,363,364,365],[15,59900,272],{"className":59901,"style":492},[61,122,365],[15,59903,497],{"className":59904},[496],[15,59906,59908],{"className":59907},[346],[15,59909,59911],{"className":59910,"style":504},[350],[15,59912],{},[15,59914],{"className":59915,"style":157},[127],[15,59917,59919,59922],{"className":59918},[61],[15,59920,11],{"className":59921},[61,122],[15,59923,59925],{"className":59924},[338],[15,59926,59928,59948],{"className":59927},[342,472],[15,59929,59931,59945],{"className":59930},[346],[15,59932,59934],{"className":59933,"style":479},[350],[15,59935,59936,59939],{"style":482},[15,59937],{"className":59938,"style":358},[357],[15,59940,59942],{"className":59941},[362,363,364,365],[15,59943,272],{"className":59944,"style":492},[61,122,365],[15,59946,497],{"className":59947},[496],[15,59949,59951],{"className":59950},[346],[15,59952,59954],{"className":59953,"style":504},[350],[15,59955],{},[15,59957,94],{"className":59958},[153],[15,59960,59962,59965],{"style":59961},"top:-2.25em;",[15,59963],{"className":59964,"style":59860},[357],[15,59966,59968,59974],{"className":59967},[61],[15,59969,59971],{"className":59970},[61,13539],[15,59972,59723],{"className":59973},[61],[15,59975,94],{"className":59976},[153],[15,59978,497],{"className":59979},[496],[15,59981,59983],{"className":59982},[346],[15,59984,59987],{"className":59985,"style":59986},[350],"height:1.19em;",[15,59988],{},[15,59990],{"className":59991,"style":59992},[6657],"width:1em;",[15,59994,59996],{"className":59995},[59843],[15,59997,59999,60072],{"className":59998},[342,472],[15,60000,60002,60069],{"className":60001},[346],[15,60003,60005,60037],{"className":60004,"style":59853},[350],[15,60006,60007,60010],{"style":59856},[15,60008],{"className":60009,"style":59860},[357],[15,60011,60013,60016,60019,60022,60025,60028,60031,60034],{"className":60012},[61],[15,60014,33401],{"className":60015},[61,122],[15,60017,741],{"className":60018},[146],[15,60020,1027],{"className":60021},[61,411],[15,60023,750],{"className":60024},[178],[15,60026],{"className":60027,"style":128},[127],[15,60029,21295],{"className":60030},[132],[15,60032],{"className":60033,"style":128},[127],[15,60035,59608],{"className":60036,"style":59624},[61,122],[15,60038,60039,60042],{"style":59961},[15,60040],{"className":60041,"style":59860},[357],[15,60043,60045,60048,60051,60054,60057,60060,60063,60066],{"className"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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 ",[15,60087,60089,60134],{"className":60088,"translate":19},[18],[15,60090,60092],{"className":60091},[23],[25,60093,60094],{"xmlns":27},[29,60095,60096,60131],{},[32,60097,60098,60101,60103,60105,60107,60109,60111,60113,60115,60121],{},[35,60099,60100],{"mathvariant":231},"LSE",[81,60102,1545],{},[81,60104,741],{"stretchy":86},[35,60106,37270],{"mathvariant":251},[81,60108,750],{"stretchy":86},[81,60110,244],{},[35,60112,22645],{},[81,60114,1545],{},[266,60116,60117,60119],{},[81,60118,293],{},[35,60120,272],{},[227,60122,60123,60125],{},[35,60124,1980],{},[266,60126,60127,60129],{},[35,60128,37270],{},[35,60130,272],{},[40,60132,60133],{"encoding":42},"\\operatorname{LSE}(\\mathbf{z}) = \\log\\sum_k e^{z_k}",[15,60135,60137,60167],{"className":60136,"ariaHidden":48},[47],[15,60138,60140,60143,60149,60152,60155,60158,60161,60164],{"className":60139},[52],[15,60141],{"className":60142,"style":142},[56],[15,60144,60146],{"className":60145},[531],[15,60147,60100],{"className":60148},[61,1620],[15,60150,741],{"className":60151},[146],[15,60153,37270],{"className":60154},[61,411],[15,60156,750],{"className":60157},[178],[15,60159],{"className":60160,"style":128},[127],[15,60162,244],{"className":60163},[132],[15,60165],{"className":60166,"style":128},[127],[15,60168,60170,60173,60178,60181,60221,60224],{"className":60169},[52],[15,60171],{"className":60172,"style":6045},[56],[15,60174,22774,60176],{"className":60175},[531],[15,60177,22777],{"style":15608},[15,60179],{"className":60180,"style":157},[127],[15,60182,60184,60187],{"className":60183},[531],[15,60185,293],{"className":60186,"style":537},[531,535,536],[15,60188,60190],{"className":60189},[338],[15,60191,60193,60213],{"className":60192},[342,472],[15,60194,60196,60210],{"className":60195},[346],[15,60197,60199],{"className":60198,"style":29964},[350],[15,60200,60201,60204],{"style":553},[15,60202],{"className":60203,"style":358},[357],[15,60205,60207],{"className":60206},[362,363,364,365],[15,60208,272],{"className":60209,"style":492},[61,122,365],[15,60211,497],{"className":60212},[496],[15,60214,60216],{"className":60215},[346],[15,60217,60219],{"className":60218,"style":593},[350],[15,60220],{},[15,60222],{"className":60223,"style":157},[127],[15,60225,60227,60230],{"className":60226},[61],[15,60228,1980],{"className":60229},[61,122],[15,60231,60233],{"className":60232},[338],[15,60234,60236],{"className":60235},[342],[15,60237,60239],{"className":60238},[346],[15,60240,60242],{"className":60241,"style":37580},[350],[15,60243,60244,60247],{"style":716},[15,60245],{"className":60246,"style":358},[357],[15,60248,60250],{"className":60249},[362,363,364,365],[15,60251,60253],{"className":60252},[61,365],[15,60254,60256,60259],{"className":60255},[61,365],[15,60257,37270],{"className":60258,"style":37598},[61,122,365],[15,60260,60262],{"className":60261},[338],[15,60263,60265,60285],{"className":60264},[342,472],[15,60266,60268,60282],{"className":60267},[346],[15,60269,60271],{"className":60270,"style":22389},[350],[15,60272,60273,60276],{"style":37613},[15,60274],{"className":60275,"style":4058},[357],[15,60277,60279],{"className":60278},[362,4062,4063,365],[15,60280,272],{"className":60281,"style":492},[61,122,365],[15,60283,497],{"className":60284},[496],[15,60286,60288],{"className":60287},[346],[15,60289,60291],{"className":60290,"style":27230},[350],[15,60292],{},"。将它们在性质上加以对照：",[32234,60295,60296,60312],{},[32237,60297,60298],{},[21298,60299,60300,60303,60306,60309],{},[32242,60301,60302],{"align":32244},"分数",[32242,60304,60305],{"align":32244},"依赖的结构",[32242,60307,60308],{"align":32244},"是否需要 OOD 数据",[32242,60310,60311],{"align":32244},"不确定性语汇",[32256,60313,60314,60328,60341,60354],{},[21298,60315,60316,60319,60322,60325],{},[32261,60317,60318],{"align":32244},"MSP \u002F 熵",[32261,60320,60321],{"align":32244},"softmax 输出",[32261,60323,60324],{"align":32244},"否",[32261,60326,60327],{"align":32244},"无（受 3.3 节缺陷支配）",[21298,60329,60330,60333,60336,60338],{},[32261,60331,60332],{"align":32244},"Mahalanobis",[32261,60334,60335],{"align":32244},"各特征层的类条件高斯",[32261,60337,60324],{"align":32244},[32261,60339,60340],{"align":32244},"距离，几何语义",[21298,60342,60343,60346,60349,60351],{},[32261,60344,60345],{"align":32244},"能量分数",[32261,60347,60348],{"align":32244},"logits 的对数配分",[32261,60350,60324],{"align":32244},[32261,60352,60353],{"align":32244},"无显式不确定性",[21298,60355,60356,60389,60420,60424],{},[32261,60357,60358],{"align":32244},[661,60359,60360,60361],{},"EDL 的 ",[15,60362,60364,60377],{"className":60363,"translate":19},[18],[15,60365,60367],{"className":60366},[23],[25,60368,60369],{"xmlns":27},[29,60370,60371,60375],{},[32,60372,60373],{},[35,60374,33401],{},[40,60376,33401],{"encoding":42},[15,60378,60380],{"className":60379,"ariaHidden":48},[47],[15,60381,60383,60386],{"className":60382},[52],[15,60384],{"className":60385,"style":6026},[56],[15,60387,33401],{"className":60388},[61,122],[32261,60390,60391,60392],{"align":32244},"迪利克雷总强度 ",[15,60393,60395,60408],{"className":60394,"translate":19},[18],[15,60396,60398],{"className":60397},[23],[25,60399,60400],{"xmlns":27},[29,60401,60402,60406],{},[32,60403,60404],{},[35,60405,40230],{},[40,60407,40230],{"encoding":42},[15,60409,60411],{"className":60410,"ariaHidden":48},[47],[15,60412,60414,60417],{"className":60413},[52],[15,60415],{"className":60416,"style":57},[56],[15,60418,40230],{"className":60419,"style":40302},[61,122],[32261,60421,60422],{"align":32244},[661,60423,60324],{},[32261,60425,60426],{"align":32244},"证据缺失量，可解释",[11,60428,60429,60430,60433],{},"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",[661,60431,60432],{},"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）。",[11,60435,60436],{},"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。",[877,60438,60440],{"id":60439},"_52-校准与选择性预测","5.2 校准与选择性预测",[11,60442,60443,60446,60447,60667],{},[661,60444,60445],{},"校准","关心的是 ",[15,60448,60450,60495],{"className":60449,"translate":19},[18],[15,60451,60453],{"className":60452},[23],[25,60454,60455],{"xmlns":27},[29,60456,60457,60492],{},[32,60458,60459,60461,60463,60469,60471,60473,60475,60481,60483,60486,60488,60490],{},[35,60460,21273],{"mathvariant":258},[81,60462,741],{"stretchy":86},[53179,60464,60465,60467],{"accent":48},[35,60466,79],{},[81,60468,58295],{},[81,60470,244],{},[35,60472,79],{},[81,60474,3710],{},[53179,60476,60477,60479],{"accent":48},[35,60478,11],{},[81,60480,58295],{},[81,60482,244],{},[35,60484,60485],{},"q",[81,60487,750],{"stretchy":86},[81,60489,244],{},[35,60491,60485],{},[40,60493,60494],{"encoding":42},"\\mathbb{P}(\\hat{y} = y \\mid \\hat{p} = q) = q",[15,60496,60498,60561,60579,60637,60658],{"className":60497,"ariaHidden":48},[47],[15,60499,60501,60504,60507,60510,60552,60555,60558],{"className":60500},[52],[15,60502],{"className":60503,"style":142},[56],[15,60505,21273],{"className":60506},[61,427],[15,60508,741],{"className":60509},[146],[15,60511,60513],{"className":60512},[61,53223],[15,60514,60516,60544],{"className":60515},[342,472],[15,60517,60519,60541],{"className":60518},[346],[15,60520,60522,60530],{"className":60521,"style":10443},[350],[15,60523,60524,60527],{"style":53235},[15,60525],{"className":60526,"style":3857},[357],[15,60528,79],{"className":60529,"style":123},[61,122],[15,60531,60532,60535],{"style":53235},[15,60533],{"className":60534,"style":3857},[357],[15,60536,60538],{"className":60537,"style":59784},[53257],[15,60539,58295],{"className":60540},[61],[15,60542,497],{"className":60543},[496],[15,60545,60547],{"className":60546},[346],[15,60548,60550],{"className":60549,"style":59797},[350],[15,60551],{},[15,60553],{"className":60554,"style":128},[127],[15,60556,244],{"className":60557},[132],[15,60559],{"className":60560,"style":128},[127],[15,60562,60564,60567,60570,60573,60576],{"className":60563},[52],[15,60565],{"className":60566,"style":142},[56],[15,60568,79],{"className":60569,"style":123},[61,122],[15,60571],{"className":60572,"style":128},[127],[15,60574,3710],{"className":60575},[132],[15,60577],{"className":60578,"style":128},[127],[15,60580,60582,60585,60628,60631,60634],{"className":60581},[52],[15,60583],{"className":60584,"style":14301},[56],[15,60586,60588],{"className":60587},[61,53223],[15,60589,60591,60620],{"className":60590},[342,472],[15,60592,60594,60617],{"className":60593},[346],[15,60595,60597,60605],{"className":60596,"style":10443},[350],[15,60598,60599,60602],{"style":53235},[15,60600],{"className":60601,"style":3857},[357],[15,60603,11],{"className":60604},[61,122],[15,60606,60607,60610],{"style":53235},[15,60608],{"className":60609,"style":3857},[357],[15,60611,60614],{"className":60612,"style":60613},[53257],"left:-0.1667em;",[15,60615,58295],{"className":60616},[61],[15,60618,497],{"className":60619},[496],[15,60621,60623],{"className":60622},[346],[15,60624,60626],{"className":60625,"style":59797},[350],[15,60627],{},[15,60629],{"className":60630,"style":128},[127],[15,60632,244],{"className":60633},[132],[15,60635],{"className":60636,"style":128},[127],[15,60638,60640,60643,60646,60649,60652,60655],{"className":60639},[52],[15,60641],{"className":60642,"style":142},[56],[15,60644,60485],{"className":60645,"style":123},[61,122],[15,60647,750],{"className":60648},[178],[15,60650],{"className":60651,"style":128},[127],[15,60653,244],{"className":60654},[132],[15,60656],{"className":60657,"style":128},[127],[15,60659,60661,60664],{"className":60660},[52],[15,60662],{"className":60663,"style":9428},[56],[15,60665,60485],{"className":60666,"style":123},[61,122]," 是否成立，常用期望校准误差",[15,60669,60671],{"className":60670,"translate":19},[212],[15,60672,60674,60760],{"className":60673,"translate":19},[18],[15,60675,60677],{"className":60676},[23],[25,60678,60679],{"xmlns":27,"display":221},[29,60680,60681,60757],{},[32,60682,60683,60686,60688,60702,60718,60720,60723,60726,60728,60730,60736,60738,60740,60743,60745,60747,60753,60755],{},[282,60684,60685],{},"ECE",[81,60687,244],{},[5229,60689,60690,60692,60700],{},[81,60691,293],{},[32,60693,60694,60696,60698],{},[35,60695,29121],{},[81,60697,244],{},[89,60699,91],{},[35,60701,28913],{},[3718,60703,60704,60716],{},[32,60705,60706,60708,60714],{},[35,60707,3710],{"mathvariant":231},[266,60709,60710,60712],{},[35,60711,3726],{"mathvariant":37},[35,60713,29121],{},[35,60715,3710],{"mathvariant":231},[35,60717,14551],{},[282,60719,4796],{},[81,60721,3710],{"fence":86,"stretchy":48,"minsize":60722,"maxsize":60722},"1.8em",[35,60724,60725],{"mathvariant":231},"acc",[81,60727,1545],{},[81,60729,741],{"stretchy":86},[266,60731,60732,60734],{},[35,60733,3726],{"mathvariant":37},[35,60735,29121],{},[81,60737,750],{"stretchy":86},[81,60739,239],{},[35,60741,60742],{"mathvariant":231},"conf",[81,60744,1545],{},[81,60746,741],{"stretchy":86},[266,60748,60749,60751],{},[35,60750,3726],{"mathvariant":37},[35,60752,29121],{},[81,60754,750],{"stretchy":86},[81,60756,3710],{"fence":86,"stretchy":48,"minsize":60722,"maxsize":60722},[40,60758,60759],{"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|",[15,60761,60763,60784,61083],{"className":60762,"ariaHidden":48},[47],[15,60764,60766,60769,60775,60778,60781],{"className":60765},[52],[15,60767],{"className":60768,"style":57},[56],[15,60770,60772],{"className":60771},[61,13539],[15,60773,60685],{"className":60774},[61],[15,60776],{"className":60777,"style":128},[127],[15,60779,244],{"className":60780},[132],[15,60782],{"className":60783,"style":128},[127],[15,60785,60787,60791,60860,60863,60971,60974,61019,61022,61028,61031,61071,61074,61077,61080],{"className":60786},[52],[15,60788],{"className":60789,"style":60790},[56],"height:3.0954em;vertical-align:-1.2671em;",[15,60792,60794],{"className":60793},[531,5342],[15,60795,60797,60851],{"className":60796},[342,472],[15,60798,60800,60848],{"className":60799},[346],[15,60801,60803,60824,60834],{"className":60802,"style":5352},[350],[15,60804,60806,60809],{"style":60805},"top:-1.8829em;margin-left:0em;",[15,60807],{"className":60808,"style":5359},[357],[15,60810,60812],{"className":60811},[362,363,364,365],[15,60813,60815,60818,60821],{"className":60814},[61,365],[15,60816,29121],{"className":60817},[61,122,365],[15,60819,244],{"className":60820},[132,365],[15,60822,91],{"className":60823},[61,365],[15,60825,60826,60829],{"style":5377},[15,60827],{"className":60828,"style":5359},[357],[15,60830,60831],{},[15,60832,293],{"className":60833},[531,535,5386],[15,60835,60836,60839],{"style":5389},[15,60837],{"className":60838,"style":5359},[357],[15,60840,60842],{"className":60841},[362,363,364,365],[15,60843,60845],{"className":60844},[61,365],[15,60846,28913],{"className":60847,"style":25078},[61,122,365],[15,60849,497],{"className":60850},[496],[15,60852,60854],{"className":60853},[346],[15,60855,60858],{"className":60856,"style":60857},[350],"height:1.2671em;",[15,60859],{},[15,60861],{"className":60862,"style":157},[127],[15,60864,60866,60869,60968],{"className":60865},[61],[15,60867],{"className":60868},[146,3837],[15,60870,60872],{"className":60871},[3718],[15,60873,60875,60960],{"className":60874},[342,472],[15,60876,60878,60957],{"className":60877},[346],[15,60879,60882,60893,60901],{"className":60880,"style":60881},[350],"height:1.427em;",[15,60883,60884,60887],{"style":11393},[15,60885],{"className":60886,"style":3857},[357],[15,60888,60890],{"className":60889},[61],[15,60891,14551],{"className":60892},[61,122],[15,60894,60895,60898],{"style":3885},[15,60896],{"className":60897,"style":3857},[357],[15,60899],{"className":60900,"style":3893},[3892],[15,60902,60903,60906],{"style":11459},[15,60904],{"className":60905,"style":3857},[357],[15,60907,60909,60912,60954],{"className":60908},[61],[15,60910,3710],{"className":60911},[61],[15,60913,60915,60919],{"className":60914},[61],[15,60916,3726],{"className":60917,"style":60918},[61,62],"margin-right:0.0304em;",[15,60920,60922],{"className":60921},[338],[15,60923,60925,60946],{"className":60924},[342,472],[15,60926,60928,60943],{"className":60927},[346],[15,60929,60931],{"className":60930,"style":27010},[350],[15,60932,60934,60937],{"style":60933},"top:-2.55em;margin-left:-0.0304em;margin-right:0.05em;",[15,60935],{"className":60936,"style":358},[357],[15,60938,60940],{"className":60939},[362,363,364,365],[15,60941,29121],{"className":60942},[61,122,365],[15,60944,497],{"className":60945},[496],[15,60947,60949],{"className":60948},[346],[15,60950,60952],{"className":60951,"style":504},[350],[15,60953],{},[15,60955,3710],{"className":60956},[61],[15,60958,497],{"className":60959},[496],[15,60961,60963],{"className":60962},[346],[15,60964,60966],{"className":60965,"style":36583},[350],[15,60967],{},[15,60969],{"className":60970},[178,3837],[15,60972],{"className":60973,"style":157},[127],[15,60975,60977],{"className":60976},[61],[15,60978,60980],{"className":60979},[406,53577],[15,60981,60983,61010],{"className":60982},[342,472],[15,60984,60986,61007],{"className":60985},[346],[15,60987,60990],{"className":60988,"style":60989},[350],"height:1.15em;",[15,60991,60993,60997],{"style":60992},"top:-3.15em;",[15,60994],{"className":60995,"style":60996},[357],"height:3.8em;",[15,60998,61000],{"style":60999},"width:0.333em;height:1.8em;",[27683,61001,61004],{"xmlns":27685,"width":61002,"height":60722,"viewBox":61003},"0.333em","0 0 333 1800",[27691,61005],{"d":61006},"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",[15,61008,497],{"className":61009},[496],[15,61011,61013],{"className":61012},[346],[15,61014,61017],{"className":61015,"style":61016},[350],"height:0.65em;",[15,61018],{},[15,61020],{"className":61021,"style":157},[127],[15,61023,61025],{"className":61024},[531],[15,61026,60725],{"className":61027},[61,1620],[15,61029,741],{"className":61030},[146],[15,61032,61034,61037],{"className":61033},[61],[15,61035,3726],{"className":61036,"style":60918},[61,62],[15,61038,61040],{"className":61039},[338],[15,61041,61043,61063],{"className":61042},[342,472],[15,61044,61046,61060],{"className":61045},[346],[15,61047,61049],{"className":61048,"style":27010},[350],[15,61050,61051,61054],{"style":60933},[15,61052],{"className":61053,"style":358},[357],[15,61055,61057],{"className":61056},[362,363,364,365],[15,61058,29121],{"className":61059},[61,122,365],[15,61061,497],{"className":61062},[496],[15,61064,61066],{"className":61065},[346],[15,61067,61069],{"className":61068,"style":504},[350],[15,61070],{},[15,61072,750],{"className":61073},[178],[15,61075],{"className":61076,"style":772},[127],[15,61078,239],{"className":61079},[375],[15,61081],{"className":61082,"style":772},[127],[15,61084,61086,61089,61096,61099,61139,61142],{"className":61085},[52],[15,61087],{"className":61088,"style":394},[56],[15,61090,61092],{"className":61091},[531],[15,61093,60742],{"className":61094,"style":61095},[61,1620],"margin-right:0.0778em;",[15,61097,741],{"className":61098},[146],[15,61100,61102,61105],{"className":61101},[61],[15,61103,3726],{"className":61104,"style":60918},[61,62],[15,61106,61108],{"className":61107},[338],[15,61109,61111,61131],{"className":61110},[342,472],[15,61112,61114,61128],{"className":61113},[346],[15,61115,61117],{"className":61116,"style":27010},[350],[15,61118,61119,61122],{"style":60933},[15,61120],{"className":61121,"style":358},[357],[15,61123,61125],{"className":61124},[362,363,364,365],[15,61126,29121],{"className":61127},[61,122,365],[15,61129,497],{"className":61130},[496],[15,61132,61134],{"className":61133},[346],[15,61135,61137],{"className":61136,"style":504},[350],[15,61138],{},[15,61140,750],{"className":61141},[178],[15,61143,61145],{"className":61144},[61],[15,61146,61148],{"className":61147},[406,53577],[15,61149,61151,61171],{"className":61150},[342,472],[15,61152,61154,61168],{"className":61153},[346],[15,61155,61157],{"className":61156,"style":60989},[350],[15,61158,61159,61162],{"style":60992},[15,61160],{"className":61161,"style":60996},[357],[15,61163,61164],{"style":60999},[27683,61165,61166],{"xmlns":27685,"width":61002,"height":60722,"viewBox":61003},[27691,61167],{"d":61006},[15,61169,497],{"className":61170},[496],[15,61172,61174],{"className":61173},[346],[15,61175,61177],{"className":61176,"style":61016},[350],[15,61178],{},[11,61180,61181,61182,61252],{},"度量，其中 ",[15,61183,61185,61203],{"className":61184,"translate":19},[18],[15,61186,61188],{"className":61187},[23],[25,61189,61190],{"xmlns":27},[29,61191,61192,61200],{},[32,61193,61194],{},[266,61195,61196,61198],{},[35,61197,3726],{"mathvariant":37},[35,61199,29121],{},[40,61201,61202],{"encoding":42},"\\mathcal{B}_m",[15,61204,61206],{"className":61205,"ariaHidden":48},[47],[15,61207,61209,61212],{"className":61208},[52],[15,61210],{"className":61211,"style":34331},[56],[15,61213,61215,61218],{"className":61214},[61],[15,61216,3726],{"className":61217,"style":60918},[61,62],[15,61219,61221],{"className":61220},[338],[15,61222,61224,61244],{"className":61223},[342,472],[15,61225,61227,61241],{"className":61226},[346],[15,61228,61230],{"className":61229,"style":27010},[350],[15,61231,61232,61235],{"style":60933},[15,61233],{"className":61234,"style":358},[357],[15,61236,61238],{"className":61237},[362,363,364,365],[15,61239,29121],{"className":61240},[61,122,365],[15,61242,497],{"className":61243},[496],[15,61245,61247],{"className":61246},[346],[15,61248,61250],{"className":61249,"style":504},[350],[15,61251],{}," 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。",[11,61254,61255,1904,61258,61286,61287,61353,61354,61386,61387,61415,61416,61478],{},[661,61256,61257],{},"选择性预测（selective prediction）",[15,61259,61261,61274],{"className":61260,"translate":19},[18],[15,61262,61264],{"className":61263},[23],[25,61265,61266],{"xmlns":27},[29,61267,61268,61272],{},[32,61269,61270],{},[35,61271,33401],{},[40,61273,33401],{"encoding":42},[15,61275,61277],{"className":61276,"ariaHidden":48},[47],[15,61278,61280,61283],{"className":61279},[52],[15,61281],{"className":61282,"style":6026},[56],[15,61284,33401],{"className":61285},[61,122]," 的更自然的落点。经典 Chow 规则指出，在已知真实后验 ",[15,61288,61290,61314],{"className":61289,"translate":19},[18],[15,61291,61293],{"className":61292},[23],[25,61294,61295],{"xmlns":27},[29,61296,61297,61311],{},[32,61298,61299,61301,61303,61305,61307,61309],{},[35,61300,11],{},[81,61302,741],{"stretchy":86},[35,61304,79],{},[81,61306,3710],{},[35,61308,1027],{"mathvariant":251},[81,61310,750],{"stretchy":86},[40,61312,61313],{"encoding":42},"p(y \\mid \\mathbf{x})",[15,61315,61317,61341],{"className":61316,"ariaHidden":48},[47],[15,61318,61320,61323,61326,61329,61332,61335,61338],{"className":61319},[52],[15,61321],{"className":61322,"style":142},[56],[15,61324,11],{"className":61325},[61,122],[15,61327,741],{"className":61328},[146],[15,61330,79],{"className":61331,"style":123},[61,122],[15,61333],{"className":61334,"style":128},[127],[15,61336,3710],{"className":61337},[132],[15,61339],{"className":61340,"style":128},[127],[15,61342,61344,61347,61350],{"className":61343},[52],[15,61345],{"className":61346,"style":142},[56],[15,61348,1027],{"className":61349},[61,411],[15,61351,750],{"className":61352},[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",[661,61355,61356,61357,61385],{},"依赖于 ",[15,61358,61360,61373],{"className":61359,"translate":19},[18],[15,61361,61363],{"className":61362},[23],[25,61364,61365],{"xmlns":27},[29,61366,61367,61371],{},[32,61368,61369],{},[35,61370,11],{},[40,61372,11],{"encoding":42},[15,61374,61376],{"className":61375,"ariaHidden":48},[47],[15,61377,61379,61382],{"className":61378},[52],[15,61380],{"className":61381,"style":9428},[56],[15,61383,11],{"className":61384},[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 ",[15,61388,61390,61403],{"className":61389,"translate":19},[18],[15,61391,61393],{"className":61392},[23],[25,61394,61395],{"xmlns":27},[29,61396,61397,61401],{},[32,61398,61399],{},[35,61400,33401],{},[40,61402,33401],{"encoding":42},[15,61404,61406],{"className":61405,"ariaHidden":48},[47],[15,61407,61409,61412],{"className":61408},[52],[15,61410],{"className":61411,"style":6026},[56],[15,61413,33401],{"className":61414},[61,122]," 或 ",[15,61417,61419,61442],{"className":61418,"translate":19},[18],[15,61420,61422],{"className":61421},[23],[25,61423,61424],{"xmlns":27},[29,61425,61426,61440],{},[32,61427,61428,61430,61432,61434,61436,61438],{},[35,61429,28898],{},[81,61431,741],{"stretchy":86},[35,61433,79],{},[81,61435,11294],{"separator":48},[35,61437,24661],{"mathvariant":3101},[81,61439,750],{"stretchy":86},[40,61441,46847],{"encoding":42},[15,61443,61445],{"className":61444,"ariaHidden":48},[47],[15,61446,61448,61451,61454,61457,61460,61463,61466,61475],{"className":61447},[52],[15,61449],{"className":61450,"style":142},[56],[15,61452,28898],{"className":61453,"style":28937},[61,122],[15,61455,741],{"className":61456},[146],[15,61458,79],{"className":61459,"style":123},[61,122],[15,61461,11294],{"className":61462},[153],[15,61464],{"className":61465,"style":157},[127],[15,61467,61469],{"className":61468},[61],[15,61470,61472],{"className":61471},[61],[15,61473,24661],{"className":61474,"style":24695},[61,3162],[15,61476,750],{"className":61477},[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）。",[877,61480,61482],{"id":61481},"_53-主动学习与半监督学习","5.3 主动学习与半监督学习",[11,61484,61485,61488,61489,61415,61533,61611,61612,61640,61641,61669,61670,61673,61674,61702],{},[661,61486,61487],{},"主动学习。"," 采集函数可以直接取 ",[15,61490,61492,61512],{"className":61491,"translate":19},[18],[15,61493,61495],{"className":61494},[23],[25,61496,61497],{"xmlns":27},[29,61498,61499,61509],{},[32,61500,61501,61503,61505,61507],{},[35,61502,33401],{},[81,61504,741],{"stretchy":86},[35,61506,1027],{"mathvariant":251},[81,61508,750],{"stretchy":86},[40,61510,61511],{"encoding":42},"u(\\mathbf{x})",[15,61513,61515],{"className":61514,"ariaHidden":48},[47],[15,61516,61518,61521,61524,61527,61530],{"className":61517},[52],[15,61519],{"className":61520,"style":142},[56],[15,61522,33401],{"className":61523},[61,122],[15,61525,741],{"className":61526},[146],[15,61528,1027],{"className":61529},[61,411],[15,61531,750],{"className":61532},[178],[15,61534,61536,61566],{"className":61535,"translate":19},[18],[15,61537,61539],{"className":61538},[23],[25,61540,61541],{"xmlns":27},[29,61542,61543,61563],{},[32,61544,61545,61547,61549,61551,61553,61555,61557,61559,61561],{},[35,61546,28898],{},[81,61548,741],{"stretchy":86},[35,61550,79],{},[81,61552,11294],{"separator":48},[35,61554,24661],{"mathvariant":3101},[81,61556,750],{"stretchy":86},[81,61558,741],{"stretchy":86},[35,61560,1027],{"mathvariant":251},[81,61562,750],{"stretchy":86},[40,61564,61565],{"encoding":42},"I(y;\\boldsymbol{\\pi})(\\mathbf{x})",[15,61567,61569],{"className":61568,"ariaHidden":48},[47],[15,61570,61572,61575,61578,61581,61584,61587,61590,61599,61602,61605,61608],{"className":61571},[52],[15,61573],{"className":61574,"style":142},[56],[15,61576,28898],{"className":61577,"style":28937},[61,122],[15,61579,741],{"className":61580},[146],[15,61582,79],{"className":61583,"style":123},[61,122],[15,61585,11294],{"className":61586},[153],[15,61588],{"className":61589,"style":157},[127],[15,61591,61593],{"className":61592},[61],[15,61594,61596],{"className":61595},[61],[15,61597,24661],{"className":61598,"style":24695},[61,3162],[15,61600,750],{"className":61601},[178],[15,61603,741],{"className":61604},[146],[15,61606,1027],{"className":61607},[61,411],[15,61609,750],{"className":61610},[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 ",[15,61613,61615,61628],{"className":61614,"translate":19},[18],[15,61616,61618],{"className":61617},[23],[25,61619,61620],{"xmlns":27},[29,61621,61622,61626],{},[32,61623,61624],{},[35,61625,28898],{},[40,61627,28898],{"encoding":42},[15,61629,61631],{"className":61630,"ariaHidden":48},[47],[15,61632,61634,61637],{"className":61633},[52],[15,61635],{"className":61636,"style":57},[56],[15,61638,28898],{"className":61639,"style":28937},[61,122]," 而非 ",[15,61642,61644,61657],{"className":61643,"translate":19},[18],[15,61645,61647],{"className":61646},[23],[25,61648,61649],{"xmlns":27},[29,61650,61651,61655],{},[32,61652,61653],{},[35,61654,33401],{},[40,61656,33401],{"encoding":42},[15,61658,61660],{"className":61659,"ariaHidden":48},[47],[15,61661,61663,61666],{"className":61662},[52],[15,61664],{"className":61665,"style":6026},[56],[15,61667,33401],{"className":61668},[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",[661,61671,61672],{},"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 ",[15,61675,61677,61690],{"className":61676,"translate":19},[18],[15,61678,61680],{"className":61679},[23],[25,61681,61682],{"xmlns":27},[29,61683,61684,61688],{},[32,61685,61686],{},[35,61687,33401],{},[40,61689,33401],{"encoding":42},[15,61691,61693],{"className":61692,"ariaHidden":48},[47],[15,61694,61696,61699],{"className":61695},[52],[15,61697],{"className":61698,"style":6026},[56],[15,61700,33401],{"className":61701},[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。",[11,61704,61705,61708,61709,61737,61738,61789,61790,61818],{},[661,61706,61707],{},"半监督与伪标签。"," 伪标签流程中，",[15,61710,61712,61725],{"className":61711,"translate":19},[18],[15,61713,61715],{"className":61714},[23],[25,61716,61717],{"xmlns":27},[29,61718,61719,61723],{},[32,61720,61721],{},[35,61722,33401],{},[40,61724,33401],{"encoding":42},[15,61726,61728],{"className":61727,"ariaHidden":48},[47],[15,61729,61731,61734],{"className":61730},[52],[15,61732],{"className":61733,"style":6026},[56],[15,61735,33401],{"className":61736},[61,122]," 提供了比「最大概率」更细的过滤准则：取 ",[15,61739,61741,61759],{"className":61740,"translate":19},[18],[15,61742,61744],{"className":61743},[23],[25,61745,61746],{"xmlns":27},[29,61747,61748,61756],{},[32,61749,61750,61752,61754],{},[35,61751,33401],{},[81,61753,2488],{},[35,61755,59608],{},[40,61757,61758],{"encoding":42},"u \u003C \\tau",[15,61760,61762,61780],{"className":61761,"ariaHidden":48},[47],[15,61763,61765,61768,61771,61774,61777],{"className":61764},[52],[15,61766],{"className":61767,"style":4650},[56],[15,61769,33401],{"className":61770},[61,122],[15,61772],{"className":61773,"style":128},[127],[15,61775,2488],{"className":61776},[132],[15,61778],{"className":61779,"style":128},[127],[15,61781,61783,61786],{"className":61782},[52],[15,61784],{"className":61785,"style":6026},[56],[15,61787,59608],{"className":61788,"style":59624},[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 ",[15,61791,61793,61806],{"className":61792,"translate":19},[18],[15,61794,61796],{"className":61795},[23],[25,61797,61798],{"xmlns":27},[29,61799,61800,61804],{},[32,61801,61802],{},[35,61803,59608],{},[40,61805,59611],{"encoding":42},[15,61807,61809],{"className":61808,"ariaHidden":48},[47],[15,61810,61812,61815],{"className":61811},[52],[15,61813],{"className":61814,"style":6026},[56],[15,61816,59608],{"className":61817,"style":59624},[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。",[877,61820,61822],{"id":61821},"_54-安全关键场景","5.4 安全关键场景",[11,61824,61825,61826,42855],{},"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",[661,61827,61828],{},"可嵌入决策规则的二阶量",[33468,61830,61831,61921,61979,62014],{},[31523,61832,61833,61836,61837,61891,61892,61920],{},[661,61834,61835],{},"分割与体素级不确定性","：对每个体素输出 ",[15,61838,61840,61861],{"className":61839,"translate":19},[18],[15,61841,61843],{"className":61842},[23],[25,61844,61845],{"xmlns":27},[29,61846,61847,61859],{},[32,61848,61849,61851,61853,61855,61857],{},[35,61850,4182],{"mathvariant":231},[81,61852,1545],{},[81,61854,741],{"stretchy":86},[35,61856,3102],{"mathvariant":3101},[81,61858,750],{"stretchy":86},[40,61860,39039],{"encoding":42},[15,61862,61864],{"className":61863,"ariaHidden":48},[47],[15,61865,61867,61870,61876,61879,61888],{"className":61866},[52],[15,61868],{"className":61869,"style":142},[56],[15,61871,61873],{"className":61872},[531],[15,61874,4182],{"className":61875},[61,1620],[15,61877,741],{"className":61878},[146],[15,61880,61882],{"className":61881},[61],[15,61883,61885],{"className":61884},[61],[15,61886,3102],{"className":61887},[61,3162],[15,61889,750],{"className":61890},[178],"，得到逐像素的 ",[15,61893,61895,61908],{"className":61894,"translate":19},[18],[15,61896,61898],{"className":61897},[23],[25,61899,61900],{"xmlns":27},[29,61901,61902,61906],{},[32,61903,61904],{},[35,61905,33401],{},[40,61907,33401],{"encoding":42},[15,61909,61911],{"className":61910,"ariaHidden":48},[47],[15,61912,61914,61917],{"className":61913},[52],[15,61915],{"className":61916,"style":6026},[56],[15,61918,33401],{"className":61919},[61,122],"，可据此生成「需人工复核区域」的掩码。",[31523,61922,61923,61926,61927,61978],{},[661,61924,61925],{},"转诊（referral）机制","：将 ",[15,61928,61930,61948],{"className":61929,"translate":19},[18],[15,61931,61933],{"className":61932},[23],[25,61934,61935],{"xmlns":27},[29,61936,61937,61945],{},[32,61938,61939,61941,61943],{},[35,61940,33401],{},[81,61942,1588],{},[35,61944,59608],{},[40,61946,61947],{"encoding":42},"u > \\tau",[15,61949,61951,61969],{"className":61950,"ariaHidden":48},[47],[15,61952,61954,61957,61960,61963,61966],{"className":61953},[52],[15,61955],{"className":61956,"style":4650},[56],[15,61958,33401],{"className":61959},[61,122],[15,61961],{"className":61962,"style":128},[127],[15,61964,1588],{"className":61965},[132],[15,61967],{"className":61968,"style":128},[127],[15,61970,61972,61975],{"className":61971},[52],[15,61973],{"className":61974,"style":6026},[56],[15,61976,59608],{"className":61977,"style":59624},[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。",[31523,61980,61981,61984,61985,62013],{},[661,61982,61983],{},"开放集动作识别","：视频中存在未见动作时，",[15,61986,61988,62001],{"className":61987,"translate":19},[18],[15,61989,61991],{"className":61990},[23],[25,61992,61993],{"xmlns":27},[29,61994,61995,61999],{},[32,61996,61997],{},[35,61998,33401],{},[40,62000,33401],{"encoding":42},[15,62002,62004],{"className":62003,"ariaHidden":48},[47],[15,62005,62007,62010],{"className":62006},[52],[15,62008],{"className":62009,"style":6026},[56],[15,62011,33401],{"className":62012},[61,122]," 提供拒绝信号。",[31523,62015,62016,62019],{},[661,62017,62018],{},"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。",[11,62021,62022,62023,62026,62027,62055],{},"这些应用的共同点是：",[661,62024,62025],{},"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，",[15,62028,62030,62043],{"className":62029,"translate":19},[18],[15,62031,62033],{"className":62032},[23],[25,62034,62035],{"xmlns":27},[29,62036,62037,62041],{},[32,62038,62039],{},[35,62040,33401],{},[40,62042,33401],{"encoding":42},[15,62044,62046],{"className":62045,"ariaHidden":48},[47],[15,62047,62049,62052],{"className":62048},[52],[15,62050],{"className":62051,"style":6026},[56],[15,62053,33401],{"className":62054},[61,122]," 至多是一个监控指标。",[877,62057,62059],{"id":62058},"_55-回归情形的推广","5.5 回归情形的推广",[11,62061,62062,62063,42855],{},"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",[661,62064,62065],{},"正态—逆 Gamma（Normal-Inverse-Gamma, NIG）先验",[15,62067,62069],{"className":62068,"translate":19},[212],[15,62070,62072,62146],{"className":62071,"translate":19},[18],[15,62073,62075],{"className":62074},[23],[25,62076,62077],{"xmlns":27,"display":221},[29,62078,62079,62143],{},[32,62080,62081,62083,62085,62087,62089,62111,62113,62115,62121,62123,62133,62135,62137,62139,62141],{},[35,62082,18335],{},[81,62084,4179],{},[35,62086,24963],{"mathvariant":37},[282,62088,6793],{},[32,62090,62091,62093,62095,62097,62109],{},[81,62092,741],{"fence":48},[35,62094,9401],{},[81,62096,94],{"separator":48},[3718,62098,62099,62106],{},[227,62100,62101,62104],{},[35,62102,62103],{},"σ",[89,62105,906],{},[35,62107,62108],{},"υ",[81,62110,750],{"fence":48},[81,62112,94],{"separator":48},[127,62114],{"width":9838},[227,62116,62117,62119],{},[35,62118,62103],{},[89,62120,906],{},[81,62122,4179],{},[227,62124,62125,62127],{},[35,62126,4288],{"mathvariant":231},[32,62128,62129,62131],{},[81,62130,239],{},[89,62132,91],{},[81,62134,741],{"stretchy":86},[35,62136,3102],{},[81,62138,94],{"separator":48},[35,62140,17120],{},[81,62142,750],{"stretchy":86},[40,62144,62145],{"encoding":42},"\\mu \\sim \\mathcal{N}\\!\\left(\\gamma, \\frac{\\sigma^2}{\\upsilon}\\right),\n\\qquad\n\\sigma^2 \\sim \\Gamma^{-1}(\\alpha, \\beta)",[15,62147,62149,62167,62346],{"className":62148,"ariaHidden":48},[47],[15,62150,62152,62155,62158,62161,62164],{"className":62151},[52],[15,62153],{"className":62154,"style":9428},[56],[15,62156,18335],{"className":62157},[61,122],[15,62159],{"className":62160,"style":128},[127],[15,62162,4179],{"className":62163},[132],[15,62165],{"className":62166,"style":128},[127],[15,62168,62170,62174,62178,62181,62184,62296,62299,62302,62305,62308,62337,62340,62343],{"className":62169},[52],[15,62171],{"className":62172,"style":62173},[56],"height:2.4411em;vertical-align:-0.95em;",[15,62175,24963],{"className":62176,"style":62177},[61,62],"margin-right:0.1474em;",[15,62179],{"className":62180,"style":7099},[127],[15,62182],{"className":62183,"style":157},[127],[15,62185,62187,62193,62196,62199,62202,62290],{"className":62186},[161],[15,62188,62190],{"className":62189,"style":402},[146,401],[15,62191,741],{"className":62192},[406,364],[15,62194,9401],{"className":62195,"style":9435},[61,122],[15,62197,94],{"className":62198},[153],[15,62200],{"className":62201,"style":157},[127],[15,62203,62205,62208,62287],{"className":62204},[61],[15,62206],{"className":62207},[146,3837],[15,62209,62211],{"className":62210},[3718],[15,62212,62214,62279],{"className":62213},[342,472],[15,62215,62217,62276],{"className":62216},[346],[15,62218,62220,62231,62239],{"className":62219,"style":11390},[350],[15,62221,62222,62225],{"style":11393},[15,62223],{"className":62224,"style":3857},[357],[15,62226,62228],{"className":62227},[61],[15,62229,62108],{"className":62230,"style":123},[61,122],[15,62232,62233,62236],{"style":3885},[15,62234],{"className":62235,"style":3857},[357],[15,62237],{"className":62238,"style":3893},[3892],[15,62240,62241,62244],{"style":11459},[15,62242],{"className":62243,"style":3857},[357],[15,62245,62247],{"className":62246},[61],[15,62248,62250,62253],{"className":62249},[61],[15,62251,62103],{"className":62252,"style":123},[61,122],[15,62254,62256],{"className":62255},[338],[15,62257,62259],{"className":62258},[342],[15,62260,62262],{"className":62261},[346],[15,62263,62265],{"className":62264,"style":7899},[350],[15,62266,62267,62270],{"style":716},[15,62268],{"className":62269,"style":358},[357],[15,62271,62273],{"className":62272},[362,363,364,365],[15,62274,906],{"className":62275},[61,365],[15,62277,497],{"className":62278},[496],[15,62280,62282],{"className":62281},[346],[15,62283,62285],{"className":62284,"style":36583},[350],[15,62286],{},[15,62288],{"className":62289},[178,3837],[15,62291,62293],{"className":62292,"style":402},[178,401],[15,62294,750],{"className":62295},[406,364],[15,62297],{"className":62298,"style":157},[127],[15,62300,94],{"className":62301},[153],[15,62303],{"className":62304,"style":10026},[127],[15,62306],{"className":62307,"style":157},[127],[15,62309,62311,62314],{"className":62310},[61],[15,62312,62103],{"className":62313,"style":123},[61,122],[15,62315,62317],{"className":62316},[338],[15,62318,62320],{"className":62319},[342],[15,62321,62323],{"className":62322},[346],[15,62324,62326],{"className":62325,"style":7357},[350],[15,62327,62328,62331],{"style":353},[15,62329],{"className":62330,"style":358},[357],[15,62332,62334],{"className":62333},[362,363,364,365],[15,62335,906],{"className":62336},[61,365],[15,62338],{"className":62339,"style":128},[127],[15,62341,4179],{"className":62342},[132],[15,62344],{"className":62345,"style":128},[127],[15,62347,62349,62353,62388,62391,62394,62397,62400,62403],{"className":62348},[52],[15,62350],{"className":62351,"style":62352},[56],"height:1.1141em;vertical-align:-0.25em;",[15,62354,62356,62359],{"className":62355},[61],[15,62357,4288],{"className":62358},[61],[15,62360,62362],{"className":62361},[338],[15,62363,62365],{"className":62364},[342],[15,62366,62368],{"className":62367},[346],[15,62369,62371],{"className":62370,"style":7357},[350],[15,62372,62373,62376],{"style":353},[15,62374],{"className":62375,"style":358},[357],[15,62377,62379],{"className":62378},[362,363,364,365],[15,62380,62382,62385],{"className":62381},[61,365],[15,62383,239],{"className":62384},[61,365],[15,62386,91],{"className":62387},[61,365],[15,62389,741],{"className":62390},[146],[15,62392,3102],{"className":62393,"style":3187},[61,122],[15,62395,94],{"className":62396},[153],[15,62398],{"className":62399,"style":157},[127],[15,62401,17120],{"className":62402,"style":17259},[61,122],[15,62404,750],{"className":62405},[178],[11,62407,62408,62409,62487,62488,62522,62523,62552,62553,62624,62625,62708,62709,62829,62830,62522,62863,62891],{},"网络对每个输入输出四元组 ",[15,62410,62412,62442],{"className":62411,"translate":19},[18],[15,62413,62415],{"className":62414},[23],[25,62416,62417],{"xmlns":27},[29,62418,62419,62439],{},[32,62420,62421,62423,62425,62427,62429,62431,62433,62435,62437],{},[81,62422,741],{"stretchy":86},[35,62424,9401],{},[81,62426,94],{"separator":48},[35,62428,62108],{},[81,62430,94],{"separator":48},[35,62432,3102],{},[81,62434,94],{"separator":48},[35,62436,17120],{},[81,62438,750],{"stretchy":86},[40,62440,62441],{"encoding":42},"(\\gamma, \\upsilon, \\alpha, \\beta)",[15,62443,62445],{"className":62444,"ariaHidden":48},[47],[15,62446,62448,62451,62454,62457,62460,62463,62466,62469,62472,62475,62478,62481,62484],{"className":62447},[52],[15,62449],{"className":62450,"style":142},[56],[15,62452,741],{"className":62453},[146],[15,62455,9401],{"className":62456,"style":9435},[61,122],[15,62458,94],{"className":62459},[153],[15,62461],{"className":62462,"style":157},[127],[15,62464,62108],{"className":62465,"style":123},[61,122],[15,62467,94],{"className":62468},[153],[15,62470],{"className":62471,"style":157},[127],[15,62473,3102],{"className":62474,"style":3187},[61,122],[15,62476,94],{"className":62477},[153],[15,62479],{"className":62480,"style":157},[127],[15,62482,17120],{"className":62483,"style":17259},[61,122],[15,62485,750],{"className":62486},[178],"，边缘预测为自由度 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分布。其预测方差可精确分解为偶然部分 ",[15,62554,62556,62582],{"className":62555,"translate":19},[18],[15,62557,62559],{"className":62558},[23],[25,62560,62561],{"xmlns":27},[29,62562,62563,62579],{},[32,62564,62565,62567,62569,62571,62573,62575,62577],{},[35,62566,17120],{},[35,62568,3018],{"mathvariant":231},[81,62570,741],{"stretchy":86},[35,62572,3102],{},[81,62574,239],{},[89,62576,91],{},[81,62578,750],{"stretchy":86},[40,62580,62581],{"encoding":42},"\\beta\u002F(\\alpha - 1)",[15,62583,62585,62612],{"className":62584,"ariaHidden":48},[47],[15,62586,62588,62591,62594,62597,62600,62603,62606,62609],{"className":62587},[52],[15,62589],{"className":62590,"style":142},[56],[15,62592,17120],{"className":62593,"style":17259},[61,122],[15,62595,3018],{"className":62596},[61],[15,62598,741],{"className":62599},[146],[15,62601,3102],{"className":62602,"style":3187},[61,122],[15,62604],{"className":62605,"style":772},[127],[15,62607,239],{"className":62608},[375],[15,62610],{"className":62611,"style":772},[127],[15,62613,62615,62618,62621],{"className":62614},[52],[15,62616],{"className":62617,"style":142},[56],[15,62619,91],{"className":62620},[61],[15,62622,750],{"className":62623},[178]," 与认知部分 ",[15,62626,62628,62660],{"className":62627,"translate":19},[18],[15,62629,62631],{"className":62630},[23],[25,62632,62633],{"xmlns":27},[29,62634,62635,62657],{},[32,62636,62637,62639,62641,62643,62645,62647,62649,62651,62653,62655],{},[35,62638,17120],{},[35,62640,3018],{"mathvariant":231},[81,62642,14810],{"stretchy":86},[35,62644,62108],{},[81,62646,741],{"stretchy":86},[35,62648,3102],{},[81,62650,239],{},[89,62652,91],{},[81,62654,750],{"stretchy":86},[81,62656,14819],{"stretchy":86},[40,62658,62659],{"encoding":42},"\\beta\u002F[\\upsilon(\\alpha - 1)]",[15,62661,62663,62696],{"className":62662,"ariaHidden":48},[47],[15,62664,62666,62669,62672,62675,62678,62681,62684,62687,62690,62693],{"className":62665},[52],[15,62667],{"className":62668,"style":142},[56],[15,62670,17120],{"className":62671,"style":17259},[61,122],[15,62673,3018],{"className":62674},[61],[15,62676,14810],{"className":62677},[146],[15,62679,62108],{"className":62680,"style":123},[61,122],[15,62682,741],{"className":62683},[146],[15,62685,3102],{"className":62686,"style":3187},[61,122],[15,62688],{"className":62689,"style":772},[127],[15,62691,239],{"className":62692},[375],[15,62694],{"className":62695,"style":772},[127],[15,62697,62699,62702,62705],{"className":62698},[52],[15,62700],{"className":62701,"style":142},[56],[15,62703,91],{"className":62704},[61],[15,62706,43268],{"className":62707},[178],"，二者之和 ",[15,62710,62712,62754],{"className":62711,"translate":19},[18],[15,62713,62715],{"className":62714},[23],[25,62716,62717],{"xmlns":27},[29,62718,62719,62751],{},[32,62720,62721,62723,62725,62727,62729,62731,62733,62735,62737,62739,62741,62743,62745,62747,62749],{},[35,62722,17120],{},[81,62724,741],{"stretchy":86},[35,62726,62108],{},[81,62728,7756],{},[89,62730,91],{},[81,62732,750],{"stretchy":86},[35,62734,3018],{"mathvariant":231},[81,62736,14810],{"stretchy":86},[35,62738,62108],{},[81,62740,741],{"stretchy":86},[35,62742,3102],{},[81,62744,239],{},[89,62746,91],{},[81,62748,750],{"stretchy":86},[81,62750,14819],{"stretchy":86},[40,62752,62753],{"encoding":42},"\\beta(\\upsilon + 1)\u002F[\\upsilon(\\alpha - 1)]",[15,62755,62757,62781,62817],{"className":62756,"ariaHidden":48},[47],[15,62758,62760,62763,62766,62769,62772,62775,62778],{"className":62759},[52],[15,62761],{"className":62762,"style":142},[56],[15,62764,17120],{"className":62765,"style":17259},[61,122],[15,62767,741],{"className":62768},[146],[15,62770,62108],{"className":62771,"style":123},[61,122],[15,62773],{"className":62774,"style":772},[127],[15,62776,7756],{"className":62777},[375],[15,62779],{"className":62780,"style":772},[127],[15,62782,62784,62787,62790,62793,62796,62799,62802,62805,62808,62811,62814],{"className":62783},[52],[15,62785],{"className":62786,"style":142},[56],[15,62788,91],{"className":62789},[61],[15,62791,750],{"className":62792},[178],[15,62794,3018],{"className":62795},[61],[15,62797,14810],{"className":62798},[146],[15,62800,62108],{"className":62801,"style":123},[61,122],[15,62803,741],{"className":62804},[146],[15,62806,3102],{"className":62807,"style":3187},[61,122],[15,62809],{"className":62810,"style":772},[127],[15,62812,239],{"className":62813},[375],[15,62815],{"className":62816,"style":772},[127],[15,62818,62820,62823,62826],{"className":62819},[52],[15,62821],{"className":62822,"style":142},[56],[15,62824,91],{"className":62825},[61],[15,62827,43268],{"className":62828},[178]," 即自由度 ",[15,62831,62833,62848],{"className":62832,"translate":19},[18],[15,62834,62836],{"className":62835},[23],[25,62837,62838],{"xmlns":27},[29,62839,62840,62846],{},[32,62841,62842,62844],{},[89,62843,906],{},[35,62845,3102],{},[40,62847,62506],{"encoding":42},[15,62849,62851],{"className":62850,"ariaHidden":48},[47],[15,62852,62854,62857,62860],{"className":62853},[52],[15,62855],{"className":62856,"style":859},[56],[15,62858,906],{"className":62859},[61],[15,62861,3102],{"className":62862,"style":3187},[61,122],[15,62864,62866,62879],{"className":62865,"translate":19},[18],[15,62867,62869],{"className":62868},[23],[25,62870,62871],{"xmlns":27},[29,62872,62873,62877],{},[32,62874,62875],{},[35,62876,5291],{},[40,62878,5291],{"encoding":42},[15,62880,62882],{"className":62881,"ariaHidden":48},[47],[15,62883,62885,62888],{"className":62884},[52],[15,62886],{"className":62887,"style":62548},[56],[15,62889,5291],{"className":62890},[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。",[11,62893,62894],{},"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。",[877,62896,62898],{"id":62897},"_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",[11,62900,62901,62902,63064,63065,63214,63215,63373,63374,63514],{},"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 ",[15,62903,62905,62943],{"className":62904,"translate":19},[18],[15,62906,62908],{"className":62907},[23],[25,62909,62910],{"xmlns":27},[29,62911,62912,62940],{},[32,62913,62914,62920,62922,62924,62926,62928],{},[266,62915,62916,62918],{},[35,62917,3703],{"mathvariant":251},[35,62919,37117],{"mathvariant":3101},[81,62921,741],{"stretchy":86},[35,62923,1027],{"mathvariant":251},[81,62925,750],{"stretchy":86},[81,62927,83],{},[227,62929,62930,62932],{},[35,62931,40230],{"mathvariant":258},[32,62933,62934,62936,62938],{},[35,62935,2786],{},[81,62937,239],{},[89,62939,91],{},[40,62941,62942],{"encoding":42},"\\mathbf{f}_{\\boldsymbol{\\theta}}(\\mathbf{x}) \\in \\mathbb{S}^{d-1}",[15,62944,62946,63020],{"className":62945,"ariaHidden":48},[47],[15,62947,62949,62952,63002,63005,63008,63011,63014,63017],{"className":62948},[52],[15,62950],{"className":62951,"style":142},[56],[15,62953,62955,62958],{"className":62954},[61],[15,62956,3703],{"className":62957,"style":25078},[61,411],[15,62959,62961],{"className":62960},[338],[15,62962,62964,62994],{"className":62963},[342,472],[15,62965,62967,62991],{"className":62966},[346],[15,62968,62970],{"className":62969,"style":479},[350],[15,62971,62973,62976],{"style":62972},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[15,62974],{"className":62975,"style":358},[357],[15,62977,62979],{"className":62978},[362,363,364,365],[15,62980,62982],{"className":62981},[61,365],[15,62983,62985],{"className":62984},[61,365],[15,62986,62988],{"className":62987},[61,365],[15,62989,37117],{"className":62990,"style":37181},[61,3162,365],[15,62992,497],{"className":62993},[496],[15,62995,62997],{"className":62996},[346],[15,62998,63000],{"className":62999,"style":504},[350],[15,63001],{},[15,63003,741],{"className":63004},[146],[15,63006,1027],{"className":63007},[61,411],[15,63009,750],{"className":63010},[178],[15,63012],{"className":63013,"style":128},[127],[15,63015,83],{"className":63016},[132],[15,63018],{"className":63019,"style":128},[127],[15,63021,63023,63026],{"className":63022},[52],[15,63024],{"className":63025,"style":18959},[56],[15,63027,63029,63032],{"className":63028},[61],[15,63030,40230],{"className":63031},[61,427],[15,63033,63035],{"className":63034},[338],[15,63036,63038],{"className":63037},[342],[15,63039,63041],{"className":63040},[346],[15,63042,63044],{"className":63043,"style":18959},[350],[15,63045,63046,63049],{"style":716},[15,63047],{"className":63048,"style":358},[357],[15,63050,63052],{"className":63051},[362,363,364,365],[15,63053,63055,63058,63061],{"className":63054},[61,365],[15,63056,2786],{"className":63057},[61,122,365],[15,63059,239],{"className":63060},[375,365],[15,63062,91],{"className":63063},[61,365],"，各类原型为 ",[15,63066,63068,63105],{"className":63067,"translate":19},[18],[15,63069,63071],{"className":63070},[23],[25,63072,63073],{"xmlns":27},[29,63074,63075,63102],{},[32,63076,63077,63079,63085,63087,63090],{},[81,63078,87],{"stretchy":86},[266,63080,63081,63083],{},[35,63082,30432],{"mathvariant":251},[35,63084,272],{},[81,63086,105],{"stretchy":86},[81,63088,63089],{},"⊂",[227,63091,63092,63094],{},[35,63093,40230],{"mathvariant":258},[32,63095,63096,63098,63100],{},[35,63097,2786],{},[81,63099,239],{},[89,63101,91],{},[40,63103,63104],{"encoding":42},"\\{\\mathbf{v}_k\\} \\subset \\mathbb{S}^{d-1}",[15,63106,63108,63170],{"className":63107,"ariaHidden":48},[47],[15,63109,63111,63114,63117,63158,63161,63164,63167],{"className":63110},[52],[15,63112],{"className":63113,"style":142},[56],[15,63115,87],{"className":63116},[146],[15,63118,63120,63123],{"className":63119},[61],[15,63121,30432],{"className":63122,"style":29388},[61,411],[15,63124,63126],{"className":63125},[338],[15,63127,63129,63150],{"className":63128},[342,472],[15,63130,63132,63147],{"className":63131},[346],[15,63133,63135],{"className":63134,"style":479},[350],[15,63136,63138,63141],{"style":63137},"top:-2.55em;margin-left:-0.016em;margin-right:0.05em;",[15,63139],{"className":63140,"style":358},[357],[15,63142,63144],{"className":63143},[362,363,364,365],[15,63145,272],{"className":63146,"style":492},[61,122,365],[15,63148,497],{"className":63149},[496],[15,63151,63153],{"className":63152},[346],[15,63154,63156],{"className":63155,"style":504},[350],[15,63157],{},[15,63159,105],{"className":63160},[178],[15,63162],{"className":63163,"style":128},[127],[15,63165,63089],{"className":63166},[132],[15,63168],{"className":63169,"style":128},[127],[15,63171,63173,63176],{"className":63172},[52],[15,63174],{"className":63175,"style":18959},[56],[15,63177,63179,63182],{"className":63178},[61],[15,63180,40230],{"className":63181},[61,427],[15,63183,63185],{"className":63184},[338],[15,63186,63188],{"className":63187},[342],[15,63189,63191],{"className":63190},[346],[15,63192,63194],{"className":63193,"style":18959},[350],[15,63195,63196,63199],{"style":716},[15,63197],{"className":63198,"style":358},[357],[15,63200,63202],{"className":63201},[362,363,364,365],[15,63203,63205,63208,63211],{"className":63204},[61,365],[15,63206,2786],{"className":63207},[61,122,365],[15,63209,239],{"className":63210},[375,365],[15,63212,91],{"className":63213},[61,365],"，相似度 ",[15,63216,63218,63254],{"className":63217,"translate":19},[18],[15,63219,63221],{"className":63220},[23],[25,63222,63223],{"xmlns":27},[29,63224,63225,63251],{},[32,63226,63227,63233,63235,63238,63240,63242,63248],{},[266,63228,63229,63231],{},[35,63230,5858],{},[35,63232,272],{},[81,63234,244],{},[81,63236,63237],{"stretchy":86},"⟨",[35,63239,3703],{"mathvariant":251},[81,63241,94],{"separator":48},[266,63243,63244,63246],{},[35,63245,30432],{"mathvariant":251},[35,63247,272],{},[81,63249,63250],{"stretchy":86},"⟩",[40,63252,63253],{"encoding":42},"s_k = \\langle \\mathbf{f}, \\mathbf{v}_k \\rangle",[15,63255,63257,63312],{"className":63256,"ariaHidden":48},[47],[15,63258,63260,63263,63303,63306,63309],{"className":63259},[52],[15,63261],{"className":63262,"style":2509},[56],[15,63264,63266,63269],{"className":63265},[61],[15,63267,5858],{"className":63268},[61,122],[15,63270,63272],{"className":63271},[338],[15,63273,63275,63295],{"className":63274},[342,472],[15,63276,63278,63292],{"className":63277},[346],[15,63279,63281],{"className":63280,"style":479},[350],[15,63282,63283,63286],{"style":482},[15,63284],{"className":63285,"style":358},[357],[15,63287,63289],{"className":63288},[362,363,364,365],[15,63290,272],{"className":63291,"style":492},[61,122,365],[15,63293,497],{"className":63294},[496],[15,63296,63298],{"className":63297},[346],[15,63299,63301],{"className":63300,"style":504},[350],[15,63302],{},[15,63304],{"className":63305,"style":128},[127],[15,63307,244],{"className":63308},[132],[15,63310],{"className":63311,"style":128},[127],[15,63313,63315,63318,63321,63324,63327,63330,63370],{"className":63314},[52],[15,63316],{"className":63317,"style":142},[56],[15,63319,63237],{"className":63320},[146],[15,63322,3703],{"className":63323,"style":25078},[61,411],[15,63325,94],{"className":63326},[153],[15,63328],{"className":63329,"style":157},[127],[15,63331,63333,63336],{"className":63332},[61],[15,63334,30432],{"className":63335,"style":29388},[61,411],[15,63337,63339],{"className":63338},[338],[15,63340,63342,63362],{"className":63341},[342,472],[15,63343,63345,63359],{"className":63344},[346],[15,63346,63348],{"className":63347,"style":479},[350],[15,63349,63350,63353],{"style":63137},[15,63351],{"className":63352,"style":358},[357],[15,63354,63356],{"className":63355},[362,363,364,365],[15,63357,272],{"className":63358,"style":492},[61,122,365],[15,63360,497],{"className":63361},[496],[15,63363,63365],{"className":63364},[346],[15,63366,63368],{"className":63367,"style":504},[350],[15,63369],{},[15,63371,63250],{"className":63372},[178],"。传统原型方法以 ",[15,63375,63377,63406],{"className":63376,"translate":19},[18],[15,63378,63380],{"className":63379},[23],[25,63381,63382],{"xmlns":27},[29,63383,63384,63403],{},[32,63385,63386,63392,63394,63397],{},[266,63387,63388,63390],{},[35,63389,37270],{"mathvariant":251},[35,63391,272],{},[81,63393,244],{},[35,63395,63396],{},"κ",[266,63398,63399,63401],{},[35,63400,5858],{},[35,63402,272],{},[40,63404,63405],{"encoding":42},"\\mathbf{z}_k = \\kappa s_k",[15,63407,63409,63465],{"className":63408,"ariaHidden":48},[47],[15,63410,63412,63416,63456,63459,63462],{"className":63411},[52],[15,63413],{"className":63414,"style":63415},[56],"height:0.5944em;vertical-align:-0.15em;",[15,63417,63419,63422],{"className":63418},[61],[15,63420,37270],{"className":63421},[61,411],[15,63423,63425],{"className":63424},[338],[15,63426,63428,63448],{"className":63427},[342,472],[15,63429,63431,63445],{"className":63430},[346],[15,63432,63434],{"className":63433,"style":479},[350],[15,63435,63436,63439],{"style":482},[15,63437],{"className":63438,"style":358},[357],[15,63440,63442],{"className":63441},[362,363,364,365],[15,63443,272],{"className":63444,"style":492},[61,122,365],[15,63446,497],{"className":63447},[496],[15,63449,63451],{"className":63450},[346],[15,63452,63454],{"className":63453,"style":504},[350],[15,63455],{},[15,63457],{"className":63458,"style":128},[127],[15,63460,244],{"className":63461},[132],[15,63463],{"className":63464,"style":128},[127],[15,63466,63468,63471,63474],{"className":63467},[52],[15,63469],{"className":63470,"style":2509},[56],[15,63472,63396],{"className":63473},[61,122],[15,63475,63477,63480],{"className":63476},[61],[15,63478,5858],{"className":63479},[61,122],[15,63481,63483],{"className":63482},[338],[15,63484,63486,63506],{"className":63485},[342,472],[15,63487,63489,63503],{"className":63488},[346],[15,63490,63492],{"className":63491,"style":479},[350],[15,63493,63494,63497],{"style":482},[15,63495],{"className":63496,"style":358},[357],[15,63498,63500],{"className":63499},[362,363,364,365],[15,63501,272],{"className":63502,"style":492},[61,122,365],[15,63504,497],{"className":63505},[496],[15,63507,63509],{"className":63508},[346],[15,63510,63512],{"className":63511,"style":504},[350],[15,63513],{}," 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。",[11,63516,63517],{},"一个直接的融合方案是把相似度送入证据函数：",[15,63519,63521],{"className":63520,"translate":19},[212],[15,63522,63524,63634],{"className":63523,"translate":19},[18],[15,63525,63527],{"className":63526},[23],[25,63528,63529],{"xmlns":27,"display":221},[29,63530,63531,63631],{},[32,63532,63533,63539,63541,63543,63545,63547,63549,63555,63557,63559,63561,63567,63569,63571,63573,63575,63577,63579,63581,63587,63589,63591,63593,63595,63597,63599,63601,63615,63617,63619,63621,63623,63629],{},[266,63534,63535,63537],{},[35,63536,1980],{},[35,63538,272],{},[81,63540,244],{},[35,63542,22777],{},[81,63544,741],{"stretchy":86},[35,63546,63396],{},[282,63548,4796],{},[266,63550,63551,63553],{},[35,63552,5858],{},[35,63554,272],{},[81,63556,750],{"stretchy":86},[81,63558,94],{"separator":48},[127,63560],{"width":9838},[266,63562,63563,63565],{},[35,63564,3102],{},[35,63566,272],{},[81,63568,244],{},[89,63570,91],{},[81,63572,7756],{},[35,63574,22777],{},[81,63576,741],{"stretchy":86},[35,63578,63396],{},[282,63580,4796],{},[266,63582,63583,63585],{},[35,63584,5858],{},[35,63586,272],{},[81,63588,750],{"stretchy":86},[81,63590,94],{"separator":48},[127,63592],{"width":9838},[35,63594,40230],{},[81,63596,244],{},[35,63598,102],{},[81,63600,7756],{},[5229,63602,63603,63605,63613],{},[81,63604,293],{},[32,63606,63607,63609,63611],{},[35,63608,272],{},[81,63610,244],{},[89,63612,91],{},[35,63614,102],{},[35,63616,22777],{},[81,63618,741],{"stretchy":86},[35,63620,63396],{},[282,63622,4796],{},[266,63624,63625,63627],{},[35,63626,5858],{},[35,63628,272],{},[81,63630,750],{"stretchy":86},[40,63632,63633],{"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)",[15,63635,63637,63692,63811,63829,63911,63929],{"className":63636,"ariaHidden":48},[47],[15,63638,63640,63643,63683,63686,63689],{"className":63639},[52],[15,63641],{"className":63642,"style":2509},[56],[15,63644,63646,63649],{"className":63645},[61],[15,63647,1980],{"className":63648},[61,122],[15,63650,63652],{"className":63651},[338],[15,63653,63655,63675],{"className":63654},[342,472],[15,63656,63658,63672],{"className":63657},[346],[15,63659,63661],{"className":63660,"style":479},[350],[15,63662,63663,63666],{"style":482},[15,63664],{"className":63665,"style":358},[357],[15,63667,63669],{"className":63668},[362,363,364,365],[15,63670,272],{"className":63671,"style":492},[61,122,365],[15,63673,497],{"className":63674},[496],[15,63676,63678],{"className":63677},[346],[15,63679,63681],{"className":63680,"style":504},[350],[15,63682],{},[15,63684],{"className":63685,"style":128},[127],[15,63687,244],{"className":63688},[132],[15,63690],{"className":63691,"style":128},[127],[15,63693,63695,63698,63701,63704,63707,63710,63750,63753,63756,63759,63762,63802,63805,63808],{"className":63694},[52],[15,63696],{"className":63697,"style":142},[56],[15,63699,22777],{"className":63700,"style":123},[61,122],[15,63702,741],{"className":63703},[146],[15,63705,63396],{"className":63706},[61,122],[15,63708],{"className":63709,"style":157},[127],[15,63711,63713,63716],{"className":63712},[61],[15,63714,5858],{"className":63715},[61,122],[15,63717,63719],{"className":63718},[338],[15,63720,63722,63742],{"className":63721},[342,472],[15,63723,63725,63739],{"className":63724},[346],[15,63726,63728],{"className":63727,"style":479},[350],[15,63729,63730,63733],{"style":482},[15,63731],{"className":63732,"style":358},[357],[15,63734,63736],{"className":63735},[362,363,364,365],[15,63737,272],{"className":63738,"style":492},[61,122,365],[15,63740,497],{"className":63741},[496],[15,63743,63745],{"className":63744},[346],[15,63746,63748],{"className":63747,"style":504},[350],[15,63749],{},[15,63751,750],{"className":63752},[178],[15,63754,94],{"className":63755},[153],[15,63757],{"className":63758,"style":10026},[127],[15,63760],{"className":63761,"style":157},[127],[15,63763,63765,63768],{"className":63764},[61],[15,63766,3102],{"className":63767,"style":3187},[61,122],[15,63769,63771],{"className":63770},[338],[15,63772,63774,63794],{"className":63773},[342,472],[15,63775,63777,63791],{"className":63776},[346],[15,63778,63780],{"className":63779,"style":479},[350],[15,63781,63782,63785],{"style":3202},[15,63783],{"className":63784,"style":358},[357],[15,63786,63788],{"className":63787},[362,363,364,365],[15,63789,272],{"className":63790,"style":492},[61,122,365],[15,63792,497],{"className":63793},[496],[15,63795,63797],{"className":63796},[346],[15,63798,63800],{"className":63799,"style":504},[350],[15,63801],{},[15,63803],{"className":63804,"style":128},[127],[15,63806,244],{"className":63807},[132],[15,63809],{"className":63810,"style":128},[127],[15,63812,63814,63817,63820,63823,63826],{"className":63813},[52],[15,63815],{"className":63816,"style":2565},[56],[15,63818,91],{"className":63819},[61],[15,63821],{"className":63822,"style":772},[127],[15,63824,7756],{"className":63825},[375],[15,63827],{"className":63828,"style":772},[127],[15,63830,63832,63835,63838,63841,63844,63847,63887,63890,63893,63896,63899,63902,63905,63908],{"className":63831},[52],[15,63833],{"className":63834,"style":142},[56],[15,63836,22777],{"className":63837,"style":123},[61,122],[15,63839,741],{"className":63840},[146],[15,63842,63396],{"className":63843},[61,122],[15,63845],{"className":63846,"style":157},[127],[15,63848,63850,63853],{"className":63849},[61],[15,63851,5858],{"className":63852},[61,122],[15,63854,63856],{"className":63855},[338],[15,63857,63859,63879],{"className":63858},[342,472],[15,63860,63862,63876],{"className":63861},[346],[15,63863,63865],{"className":63864,"style":479},[350],[15,63866,63867,63870],{"style":482},[15,63868],{"className":63869,"style":358},[357],[15,63871,63873],{"className":63872},[362,363,364,365],[15,63874,272],{"className":63875,"style":492},[61,122,365],[15,63877,497],{"className":63878},[496],[15,63880,63882],{"className":63881},[346],[15,63883,63885],{"className":63884,"style":504},[350],[15,63886],{},[15,63888,750],{"className":63889},[178],[15,63891,94],{"className":63892},[153],[15,63894],{"className":63895,"style":10026},[127],[15,63897],{"className":63898,"style":157},[127],[15,63900,40230],{"className":63901,"style":40302},[61,122],[15,63903],{"className":63904,"style":128},[127],[15,63906,244],{"className":63907},[132],[15,63909],{"className":63910,"style":128},[127],[15,63912,63914,63917,63920,63923,63926],{"className":63913},[52],[15,63915],{"className":63916,"style":972},[56],[15,63918,102],{"className":63919,"style":174},[61,122],[15,63921],{"className":63922,"style":772},[127],[15,63924,7756],{"className":63925},[375],[15,63927],{"className":63928,"style":772},[127],[15,63930,63932,63936,64002,64005,64008,64011,64014,64017,64057],{"className":63931},[52],[15,63933],{"className":63934,"style":63935},[56],"height:3.1304em;vertical-align:-1.3021em;",[15,63937,63939],{"className":63938},[531,5342],[15,63940,63942,63993],{"className":63941},[342,472],[15,63943,63945,63990],{"className":63944},[346],[15,63946,63948,63969,63979],{"className":63947,"style":5352},[350],[15,63949,63951,63954],{"style":63950},"top:-1.8479em;margin-left:0em;",[15,63952],{"className":63953,"style":5359},[357],[15,63955,63957],{"className":63956},[362,363,364,365],[15,63958,63960,63963,63966],{"className":63959},[61,365],[15,63961,272],{"className":63962,"style":492},[61,122,365],[15,63964,244],{"className":63965},[132,365],[15,63967,91],{"className":63968},[61,365],[15,63970,63971,63974],{"style":5377},[15,63972],{"className":63973,"style":5359},[357],[15,63975,63976],{},[15,63977,293],{"className":63978},[531,535,5386],[15,63980,63981,63984],{"style":5389},[15,63982],{"className":63983,"style":5359},[357],[15,63985,63987],{"className":63986},[362,363,364,365],[15,63988,102],{"className":63989,"style":174},[61,122,365],[15,63991,497],{"className":63992},[496],[15,63994,63996],{"className":63995},[346],[15,63997,64000],{"className":63998,"style":63999},[350],"height:1.3021em;",[15,64001],{},[15,64003],{"className":64004,"style":157},[127],[15,64006,22777],{"className":64007,"style":123},[61,122],[15,64009,741],{"className":64010},[146],[15,64012,63396],{"className":64013},[61,122],[15,64015],{"className":64016,"style":157},[127],[15,64018,64020,64023],{"className":64019},[61],[15,64021,5858],{"className":64022},[61,122],[15,64024,64026],{"className":64025},[338],[15,64027,64029,64049],{"className":64028},[342,472],[15,64030,64032,64046],{"className":64031},[346],[15,64033,64035],{"className":64034,"style":479},[350],[15,64036,64037,64040],{"style":482},[15,64038],{"className":64039,"style":358},[357],[15,64041,64043],{"className":64042},[362,363,364,365],[15,64044,272],{"className":64045,"style":492},[61,122,365],[15,64047,497],{"className":64048},[496],[15,64050,64052],{"className":64051},[346],[15,64053,64055],{"className":64054,"style":504},[350],[15,64056],{},[15,64058,750],{"className":64059},[178],[11,64061,49009,64062,64090,64091,64120,64121,64191,64192,64220,64221,64249,64250,64319,64320,867,64370],{},[15,64063,64065,64078],{"className":64064,"translate":19},[18],[15,64066,64068],{"className":64067},[23],[25,64069,64070],{"xmlns":27},[29,64071,64072,64076],{},[32,64073,64074],{},[35,64075,22777],{},[40,64077,22777],{"encoding":42},[15,64079,64081],{"className":64080,"ariaHidden":48},[47],[15,64082,64084,64087],{"className":64083},[52],[15,64085],{"className":64086,"style":9428},[56],[15,64088,22777],{"className":64089,"style":123},[61,122]," 取 3.4 节的非负激活，",[15,64092,64094,64108],{"className":64093,"translate":19},[18],[15,64095,64097],{"className":64096},[23],[25,64098,64099],{"xmlns":27},[29,64100,64101,64105],{},[32,64102,64103],{},[35,64104,63396],{},[40,64106,64107],{"encoding":42},"\\kappa",[15,64109,64111],{"className":64110,"ariaHidden":48},[47],[15,64112,64114,64117],{"className":64113},[52],[15,64115],{"className":64116,"style":6026},[56],[15,64118,63396],{"className":64119},[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，",[15,64122,64124,64142],{"className":64123,"translate":19},[18],[15,64125,64127],{"className":64126},[23],[25,64128,64129],{"xmlns":27},[29,64130,64131,64139],{},[32,64132,64133],{},[266,64134,64135,64137],{},[35,64136,5858],{},[35,64138,272],{},[40,64140,64141],{"encoding":42},"s_k",[15,64143,64145],{"className":64144,"ariaHidden":48},[47],[15,64146,64148,64151],{"className":64147},[52],[15,64149],{"className":64150,"style":2509},[56],[15,64152,64154,64157],{"className":64153},[61],[15,64155,5858],{"className":64156},[61,122],[15,64158,64160],{"className":64159},[338],[15,64161,64163,64183],{"className":64162},[342,472],[15,64164,64166,64180],{"className":64165},[346],[15,64167,64169],{"className":64168,"style":479},[350],[15,64170,64171,64174],{"style":482},[15,64172],{"className":64173,"style":358},[357],[15,64175,64177],{"className":64176},[362,363,364,365],[15,64178,272],{"className":64179,"style":492},[61,122,365],[15,64181,497],{"className":64182},[496],[15,64184,64186],{"className":64185},[346],[15,64187,64189],{"className":64188,"style":504},[350],[15,64190],{}," 接近 ",[15,64193,64195,64208],{"className":64194,"translate":19},[18],[15,64196,64198],{"className":64197},[23],[25,64199,64200],{"xmlns":27},[29,64201,64202,64206],{},[32,64203,64204],{},[89,64205,91],{},[40,64207,91],{"encoding":42},[15,64209,64211],{"className":64210,"ariaHidden":48},[47],[15,64212,64214,64217],{"className":64213},[52],[15,64215],{"className":64216,"style":859},[56],[15,64218,91],{"className":64219},[61],"、证据充分、",[15,64222,64224,64237],{"className":64223,"translate":19},[18],[15,64225,64227],{"className":64226},[23],[25,64228,64229],{"xmlns":27},[29,64230,64231,64235],{},[32,64232,64233],{},[35,64234,33401],{},[40,64236,33401],{"encoding":42},[15,64238,64240],{"className":64239,"ariaHidden":48},[47],[15,64241,64243,64246],{"className":64242},[52],[15,64244],{"className":64245,"style":6026},[56],[15,64247,33401],{"className":64248},[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 ",[15,64251,64253,64270],{"className":64252,"translate":19},[18],[15,64254,64256],{"className":64255},[23],[25,64257,64258],{"xmlns":27},[29,64259,64260,64268],{},[32,64261,64262],{},[266,64263,64264,64266],{},[35,64265,1980],{},[35,64267,272],{},[40,64269,39275],{"encoding":42},[15,64271,64273],{"className":64272,"ariaHidden":48},[47],[15,64274,64276,64279],{"className":64275},[52],[15,64277],{"className":64278,"style":2509},[56],[15,64280,64282,64285],{"className":64281},[61],[15,64283,1980],{"className":64284},[61,122],[15,64286,64288],{"className":64287},[338],[15,64289,64291,64311],{"className":64290},[342,472],[15,64292,64294,64308],{"className":64293},[346],[15,64295,64297],{"className":64296,"style":479},[350],[15,64298,64299,64302],{"style":482},[15,64300],{"className":64301,"style":358},[357],[15,64303,64305],{"className":64304},[362,363,364,365],[15,64306,272],{"className":64307,"style":492},[61,122,365],[15,64309,497],{"className":64310},[496],[15,64312,64314],{"className":64313},[346],[15,64315,64317],{"className":64316,"style":504},[350],[15,64318],{}," 同时趋零、",[15,64321,64323,64340],{"className":64322,"translate":19},[18],[15,64324,64326],{"className":64325},[23],[25,64327,64328],{"xmlns":27},[29,64329,64330,64338],{},[32,64331,64332,64334,64336],{},[35,64333,33401],{},[81,64335,20691],{},[89,64337,91],{},[40,64339,42264],{"encoding":42},[15,64341,64343,64361],{"className":64342,"ariaHidden":48},[47],[15,64344,64346,64349,64352,64355,64358],{"className":64345},[52],[15,64347],{"className":64348,"style":6026},[56],[15,64350,33401],{"className":64351},[61,122],[15,64353],{"className":64354,"style":128},[127],[15,64356,20691],{"className":64357},[132],[15,64359],{"className":64360,"style":128},[127],[15,64362,64364,64367],{"className":64363},[52],[15,64365],{"className":64366,"style":859},[56],[15,64368,91],{"className":64369},[61],[661,64371,64372],{},"「离原型远」与「证据不足」在此被同一个机制表达。",[11,64374,64375,64376,64379,64380,64383,64384,64456,64457,64485,64486,64489,64490,64524],{},"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",[661,64377,64378],{},"类概率单纯形","上，原型方法定义在",[661,64381,64382],{},"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。",[15,64385,64387,64411],{"className":64386,"translate":19},[18],[15,64388,64390],{"className":64389},[23],[25,64391,64392],{"xmlns":27},[29,64393,64394,64408],{},[32,64395,64396,64398,64400,64402,64404,64406],{},[35,64397,1027],{"mathvariant":251},[81,64399,14584],{},[35,64401,3102],{"mathvariant":3101},[81,64403,741],{"stretchy":86},[35,64405,1027],{"mathvariant":251},[81,64407,750],{"stretchy":86},[40,64409,64410],{"encoding":42},"\\mathbf{x} \\mapsto \\boldsymbol{\\alpha}(\\mathbf{x})",[15,64412,64414,64432],{"className":64413,"ariaHidden":48},[47],[15,64415,64417,64420,64423,64426,64429],{"className":64416},[52],[15,64418],{"className":64419,"style":14603},[56],[15,64421,1027],{"className":64422},[61,411],[15,64424],{"className":64425,"style":128},[127],[15,64427,14584],{"className":64428},[132],[15,64430],{"className":64431,"style":128},[127],[15,64433,64435,64438,64447,64450,64453],{"className":64434},[52],[15,64436],{"className":64437,"style":142},[56],[15,64439,64441],{"className":64440},[61],[15,64442,64444],{"className":64443},[61],[15,64445,3102],{"className":64446},[61,3162],[15,64448,741],{"className":64449},[146],[15,64451,1027],{"className":64452},[61,411],[15,64454,750],{"className":64455},[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——",[15,64458,64460,64473],{"className":64459,"translate":19},[18],[15,64461,64463],{"className":64462},[23],[25,64464,64465],{"xmlns":27},[29,64466,64467,64471],{},[32,64468,64469],{},[35,64470,33401],{},[40,64472,33401],{"encoding":42},[15,64474,64476],{"className":64475,"ariaHidden":48},[47],[15,64477,64479,64482],{"className":64478},[52],[15,64480],{"className":64481,"style":6026},[56],[15,64483,33401],{"className":64484},[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",[661,64487,64488],{},"联合","模型，例如令 ",[15,64491,64493,64506],{"className":64492,"translate":19},[18],[15,64494,64496],{"className":64495},[23],[25,64497,64498],{"xmlns":27},[29,64499,64500,64504],{},[32,64501,64502],{},[35,64503,3102],{"mathvariant":3101},[40,64505,4143],{"encoding":42},[15,64507,64509],{"className":64508,"ariaHidden":48},[47],[15,64510,64512,64515],{"className":64511},[52],[15,64513],{"className":64514,"style":2706},[56],[15,64516,64518],{"className":64517},[61],[15,64519,64521],{"className":64520},[61],[15,64522,3102],{"className":64523},[61,3162]," 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。",[11,64526,64527],{},"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：",[31520,64529,64530,64594],{},[31523,64531,64532,64535,64536,64564,64565,64593],{},[661,64533,64534],{},"长尾与稀有类。"," 稀有类的原型天然缺乏证据支撑，",[15,64537,64539,64552],{"className":64538,"translate":19},[18],[15,64540,64542],{"className":64541},[23],[25,64543,64544],{"xmlns":27},[29,64545,64546,64550],{},[32,64547,64548],{},[35,64549,33401],{},[40,64551,33401],{"encoding":42},[15,64553,64555],{"className":64554,"ariaHidden":48},[47],[15,64556,64558,64561],{"className":64557},[52],[15,64559],{"className":64560,"style":6026},[56],[15,64562,33401],{"className":64563},[61,122]," 更高。这既可以被用作保护机制（在损失中对高 ",[15,64566,64568,64581],{"className":64567,"translate":19},[18],[15,64569,64571],{"className":64570},[23],[25,64572,64573],{"xmlns":27},[29,64574,64575,64579],{},[32,64576,64577],{},[35,64578,33401],{},[40,64580,33401],{"encoding":42},[15,64582,64584],{"className":64583,"ariaHidden":48},[47],[15,64585,64587,64590],{"className":64586},[52],[15,64588],{"className":64589,"style":6026},[56],[15,64591,33401],{"className":64592},[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。",[31523,64595,64596,64599,64600,61415,64628,64690,64691],{},[661,64597,64598],{},"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。",[15,64601,64603,64616],{"className":64602,"translate":19},[18],[15,64604,64606],{"className":64605},[23],[25,64607,64608],{"xmlns":27},[29,64609,64610,64614],{},[32,64611,64612],{},[35,64613,33401],{},[40,64615,33401],{"encoding":42},[15,64617,64619],{"className":64618,"ariaHidden":48},[47],[15,64620,64622,64625],{"className":64621},[52],[15,64623],{"className":64624,"style":6026},[56],[15,64626,33401],{"className":64627},[61,122],[15,64629,64631,64654],{"className":64630,"translate":19},[18],[15,64632,64634],{"className":64633},[23],[25,64635,64636],{"xmlns":27},[29,64637,64638,64652],{},[32,64639,64640,64642,64644,64646,64648,64650],{},[35,64641,28898],{},[81,64643,741],{"stretchy":86},[35,64645,79],{},[81,64647,11294],{"separator":48},[35,64649,24661],{"mathvariant":3101},[81,64651,750],{"stretchy":86},[40,64653,46847],{"encoding":42},[15,64655,64657],{"className":64656,"ariaHidden":48},[47],[15,64658,64660,64663,64666,64669,64672,64675,64678,64687],{"className":64659},[52],[15,64661],{"className":64662,"style":142},[56],[15,64664,28898],{"className":64665,"style":28937},[61,122],[15,64667,741],{"className":64668},[146],[15,64670,79],{"className":64671,"style":123},[61,122],[15,64673,11294],{"className":64674},[153],[15,64676],{"className":64677,"style":157},[127],[15,64679,64681],{"className":64680},[61],[15,64682,64684],{"className":64683},[61],[15,64685,24661],{"className":64686,"style":24695},[61,3162],[15,64688,750],{"className":64689},[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：",[661,64692,64693],{},"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。",[872,64695,64696],{"id":64696},"结语",[11,64698,64699],{},"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：",[33468,64701,64702,64812,64818],{},[31523,64703,64704,64707,64708,31703,64742,64811],{},[661,64705,64706],{},"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 ",[15,64709,64711,64724],{"className":64710,"translate":19},[18],[15,64712,64714],{"className":64713},[23],[25,64715,64716],{"xmlns":27},[29,64717,64718,64722],{},[32,64719,64720],{},[35,64721,18335],{"mathvariant":3101},[40,64723,19440],{"encoding":42},[15,64725,64727],{"className":64726,"ariaHidden":48},[47],[15,64728,64730,64733],{"className":64729},[52],[15,64731],{"className":64732,"style":829},[56],[15,64734,64736],{"className":64735},[61],[15,64737,64739],{"className":64738},[61],[15,64740,18335],{"className":64741},[61,3162],[15,64743,64745,64762],{"className":64744,"translate":19},[18],[15,64746,64748],{"className":64747},[23],[25,64749,64750],{"xmlns":27},[29,64751,64752,64760],{},[32,64753,64754],{},[266,64755,64756,64758],{},[35,64757,3102],{},[89,64759,278],{},[40,64761,19479],{"encoding":42},[15,64763,64765],{"className":64764,"ariaHidden":48},[47],[15,64766,64768,64771],{"className":64767},[52],[15,64769],{"className":64770,"style":2509},[56],[15,64772,64774,64777],{"className":64773},[61],[15,64775,3102],{"className":64776,"style":3187},[61,122],[15,64778,64780],{"className":64779},[338],[15,64781,64783,64803],{"className":64782},[342,472],[15,64784,64786,64800],{"className":64785},[346],[15,64787,64789],{"className":64788,"style":1212},[350],[15,64790,64791,64794],{"style":3202},[15,64792],{"className":64793,"style":358},[357],[15,64795,64797],{"className":64796},[362,363,364,365],[15,64798,278],{"className":64799},[61,365],[15,64801,497],{"className":64802},[496],[15,64804,64806],{"className":64805},[346],[15,64807,64809],{"className":64808,"style":504},[350],[15,64810],{}," 两个正交分量构成（命题 2.2）。",[31523,64813,64814,64817],{},[661,64815,64816],{},"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）。",[31523,64819,64820,64823],{},[661,64821,64822],{},"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。",[11,64825,64826,64827,64830,64831,64891],{},"证据深度学习的核心洞见可以压缩为一句话：",[661,64828,64829],{},"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 ",[15,64832,64834,64855],{"className":64833,"translate":19},[18],[15,64835,64837],{"className":64836},[23],[25,64838,64839],{"xmlns":27},[29,64840,64841,64853],{},[32,64842,64843,64845,64847,64849,64851],{},[35,64844,33401],{},[81,64846,244],{},[35,64848,102],{},[35,64850,3018],{"mathvariant":231},[35,64852,40230],{},[40,64854,42949],{"encoding":42},[15,64856,64858,64876],{"className":64857,"ariaHidden":48},[47],[15,64859,64861,64864,64867,64870,64873],{"className":64860},[52],[15,64862],{"className":64863,"style":6026},[56],[15,64865,33401],{"className":64866},[61,122],[15,64868],{"className":64869,"style":128},[127],[15,64871,244],{"className":64872},[132],[15,64874],{"className":64875,"style":128},[127],[15,64877,64879,64882,64885,64888],{"className":64878},[52],[15,64880],{"className":64881,"style":142},[56],[15,64883,102],{"className":64884,"style":174},[61,122],[15,64886,3018],{"className":64887},[61],[15,64889,40230],{"className":64890,"style":40302},[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。",[11,64893,64894],{},"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」。",[11,64896,64897,64898,64900,64901,64929,64930],{},"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",[661,64899,42989],{},"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）。",[15,64902,64904,64917],{"className":64903,"translate":19},[18],[15,64905,64907],{"className":64906},[23],[25,64908,64909],{"xmlns":27},[29,64910,64911,64915],{},[32,64912,64913],{},[35,64914,33401],{},[40,64916,33401],{"encoding":42},[15,64918,64920],{"className":64919,"ariaHidden":48},[47],[15,64921,64923,64926],{"className":64922},[52],[15,64924],{"className":64925,"style":6026},[56],[15,64927,33401],{"className":64928},[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：",[661,64931,64932],{},"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。",[11,64934,64935,64936,64939],{},"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，",[661,64937,64938],{},"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":64941,"searchDepth":64942,"depth":64942,"links":64943},"",2,[64944,64951,64959,64967,64974,64982],{"id":874,"depth":64942,"text":875,"children":64945},[64946,64948,64949,64950],{"id":879,"depth":64947,"text":880},3,{"id":3079,"depth":64947,"text":3080},{"id":9378,"depth":64947,"text":9379},{"id":12507,"depth":64947,"text":12508},{"id":14660,"depth":64942,"text":14661,"children":64952},[64953,64954,64955,64956,64957,64958],{"id":14664,"depth":64947,"text":14665},{"id":19419,"depth":64947,"text":19420},{"id":21673,"depth":64947,"text":21674},{"id":24639,"depth":64947,"text":24640},{"id":28797,"depth":64947,"text":28798},{"id":31479,"depth":64947,"text":31480},{"id":33262,"depth":64942,"text":33263,"children":64960},[64961,64962,64963,64964,64965,64966],{"id":33266,"depth":64947,"text":33267},{"id":34924,"depth":64947,"text":34925},{"id":37094,"depth":64947,"text":37095},{"id":38310,"depth":64947,"text":38311},{"id":40208,"depth":64947,"text":40209},{"id":42920,"depth":64947,"text":42921},{"id":47334,"depth":64942,"text":47335,"children":64968},[64969,64970,64971,64972,64973],{"id":47425,"depth":64947,"text":47426},{"id":50878,"depth":64947,"text":50879},{"id":53066,"depth":64947,"text":53067},{"id":57302,"depth":64947,"text":57303},{"id":57953,"depth":64947,"text":57954},{"id":59492,"depth":64942,"text":59493,"children":64975},[64976,64977,64978,64979,64980,64981],{"id":59496,"depth":64947,"text":59497},{"id":60439,"depth":64947,"text":60440},{"id":61481,"depth":64947,"text":61482},{"id":61821,"depth":64947,"text":61822},{"id":62058,"depth":64947,"text":62059},{"id":62897,"depth":64947,"text":62898},{"id":64696,"depth":64942,"text":64696},"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":64983},"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。",[64994,64995],"statistics","machine-learning","2026-10-01T18:52:06+08:00","PklX79kbgmXizMsWGOJ3PcHWRm9JgKQrfILpGImulBA",{"id":64999,"title":65000,"body":65001,"description":65005,"draft":64984,"enableCodeHighlight":64984,"enableCodeHightlight":64984,"enableComment":64985,"enableD2":64984,"enableKatex":64984,"extension":64986,"image":64941,"important":64984,"location":64987,"meta":65330,"navigation":64985,"ogImage":64987,"path":65331,"seo":65332,"stem":65333,"summary":65334,"tags":65335,"time":65337,"__hash__":65338},"blog\u002Fblog\u002F2026\u002F0010-biography-of-claude-shannon.md","克劳德·香农小传",{"type":8,"value":65002,"toc":65320},[65003,65006,65009,65012,65015,65018,65021,65024,65027,65030,65033,65036,65039,65042,65045,65048,65051,65054,65057,65060,65067,65070,65073,65076,65079,65082,65085,65088,65091,65094,65097,65100,65103,65106,65109,65112,65115,65118,65121,65124,65127,65130,65133,65139,65142,65145,65148,65151,65154,65157,65160,65163,65166,65170,65173,65176,65179,65182,65185,65188,65191,65194,65197,65200,65203,65206,65209,65212,65217,65220,65223,65227,65233,65236,65239,65242,65245,65248,65251,65254,65257,65260,65263,65266,65270,65276,65279,65282,65285,65288,65291,65294,65297,65300,65303,65306,65309,65312,65315],[11,65004,65005],{},"美国 20 世纪 50 年代的某个深夜，贝尔实验室的走廊里很安静。一个瘦高的中年人骑着独轮车，从长长的走廊那头慢悠悠地滑过来，手里还抛着几个球。偶尔有加班的同事推开门，看一眼，笑一笑，又把头缩回去继续算他们的方程——大家早就习惯了。",[11,65007,65008],{},"克劳德·艾尔伍德·香农（Claude Elwood Shannon）最有分量的成就，是用一篇论文给「信息」下了数学定义，由此奠定了整个数字时代的理论根基。可是如果你问他的同事、家人和朋友他是个什么样的人，他们说的多半不是公式，而是独轮车、玩具、小机器和一个永远带着笑意的安静身影。",[872,65010,65011],{"id":65011},"密歇根小镇的修理少年",[11,65013,65014],{},"1916 年 4 月 30 日，香农出生在美国密歇根州的佩托斯基，不久后全家搬到北边一个叫盖洛德的小镇，他在那里长大。父亲老克劳德做过商人，也当过遗嘱认证法官；母亲梅布尔是语言教师，后来做了盖洛德中学的校长。姐姐凯瑟琳比他大几岁，日后成了数学教授。家里的气氛是安稳、务实的：父亲有生意头脑，母亲爱读书，孩子们被允许把时间花在自己感兴趣的东西上。",[11,65016,65017],{},"香农从小就是个「动手的孩子」。他爱做航模，也做过无线电遥控的小船。最著名的一桩少年轶事，是他和一个住在大约半英里外的朋友之间架设的电报线：两个孩子没有现成的电线，就利用沿途的铁丝围栏，把它们接起来当作通信线路，晚上各自守在家里，用莫尔斯电码互相「发报」。对一个中西部小镇的男孩来说，这既是游戏，也是他第一次亲手搭起「通信系统」。",[11,65019,65020],{},"他还干过一份差事：给西联电报公司送电报。镇上的人看见他骑着车来来往往，也许想不到，这个孩子日后会为「通信」写下一套完整的数学理论。他修收音机的本事在镇上小有名气，据说当地一家百货商店的收音机坏了，也会请他去看看。",[11,65022,65023],{},"这里得补一点背景：香农的少年时代，正好赶上 1920 年代美国的无线电热。广播电台一座座冒出来，杂志上登着教人自己绕线圈、装矿石收音机的图纸，成千上万个男孩在自家阁楼上折腾「无线电」。香农赶上了这个浪潮的浪尖，而且他不是照着图纸装，是喜欢把东西拆开，弄清里面到底怎么回事。回头看，他一生的路数大概就是这个：感兴趣的不是「怎么用」，而是「它是怎么被装起来的」。",[11,65025,65026],{},"他的偶像是爱迪生。后来他发现自己和爱迪生居然是远亲，高兴了好一阵子。少年时代他还读到爱伦·坡的小说《金甲虫》，书里主人公靠破译密码找到宝藏，这个故事在他心里种下了对密码的迷恋，多年以后，战时的密码工作和他的信息论，都与这份兴趣有着隐秘的联系。另外，坡还写过一篇《被窃的信》，讲的是「猜对方下一手会怎么出」的心理游戏——二十多年后，香农在一份机器设计的备忘录里，又把这篇小说翻出来引用了一次。",[11,65028,65029],{},"这个阶段的香农，已经显出日后的底色：不爱张扬，不太合群，但遇到「有意思的东西」就会安静地一头扎进去，一玩就是几个小时。",[872,65031,65032],{"id":65032},"一门选修课和一篇论文",[11,65034,65035],{},"1932 年，香农进入密歇根大学。他最后同时拿到了电气工程和数学两个学士学位，这个组合在当时并不常见，却像是为他日后的工作预先搭好的脚手架。",[11,65037,65038],{},"大学里有一件看起来很小的事，影响却极大：他在一门课上接触到了乔治·布尔的逻辑代数——一套只用「真」和「假」、「是」和「非」来做运算的符号系统。大多数同学把它当成一个哲学趣闻，香农却盯着它看了很久，因为他同时懂得电路：开关只有「开」和「关」两种状态，这不正好对应「真」和「假」吗？",[11,65040,65041],{},"毕业后，他去了麻省理工学院，做万尼瓦尔·布什的助手，管理一台著名的庞然大物——微分分析机。这是一台由轴、齿轮和大量继电器组成的机械模拟计算机，占满一间屋子。香农在它旁边一待就是很长时间，对着一堆纠缠的继电器电路琢磨：能不能找到一套规则，让人不靠反复试错就能设计出正确的开关电路？",[11,65043,65044],{},"1937 年，他交出了答案——硕士论文《继电器与开关电路的符号分析》。他证明了布尔代数可以用来描述和简化开关电路，也就是说，逻辑可以被「造」出来。",[11,65046,65047],{},"这篇论文在评价史上的位置，后来被反复提起。历史学家赫尔曼·戈德斯坦 1972 年写下了可能是最实在的一句：它「把数字电路设计从一门手艺变成了一门科学」。数学家所罗门·戈隆布则在 2001 年的悼文里说，这是「有史以来最伟大的硕士论文之一」。流传最广的一版出自哈佛大学的认知心理学家霍华德·加德纳，1985 年他写道，这或许「是这个世纪最重要、也最著名的硕士论文」。这句话后来被到处引用，有时还被人升格成「科学史家的定评」——其实加德纳是心理学家，不是科学史家，这一点值得记一笔。",[11,65049,65050],{},"还有一个评价来自他的老师本人。布什后来谈到这篇论文时说：「我当时就认为它是一篇经典之作——这个评价我很少给出。」",[11,65052,65053],{},"比论文本身更能说明问题的，是布什给人写推荐信时的措辞。1938 年，他在一封信里这样描述这个年轻人：「一个相当不合常规的年轻人……他非常害羞内向，极为谦逊，很容易被带偏。」在另一封推荐信里，布什又补了一句：「他腼腆、讨人喜欢，是需要被小心对待的那种人。」一位导师要一边极力推荐、一边提醒别人别把他吓跑，这件事本身就说明了香农是个什么样的人。",[11,65055,65056],{},"当时的他才二十出头，说起那篇论文却相当轻描淡写，更像是把两样好玩的东西拼到了一起。",[11,65058,65059],{},"其间他还在贝尔实验室度过了一个夏天，那是他与这家公司缘分的开始。布什后来建议他换个领域，于是他的博士论文竟是关于理论遗传学的，题目叫《理论遗传学的代数学》。为此他去了冷泉港，和遗传学家一起工作。他用数学处理孟德尔遗传，这篇论文并没有像硕士论文那样引起轰动，却再次显示了他的习惯：哪里有「结构」，就想用代数去看一看。",[11,65061,65062],{},[65063,65064],"img",{"alt":65065,"src":65066},"Claude Shannon in 1932","https:\u002F\u002Fimage-assets.dreams.plus\u002F202610162230267.png",[11,65068,65069],{},"1940 年，他拿到了博士学位。同一年，他与诺玛·莱沃结婚，但这段婚姻很短，两人很快就分开了。香农在感情上一向内敛，很少对外人谈起这些。",[872,65071,65072],{"id":65072},"普林斯顿的拜访与贝尔实验室的夜晚",[11,65074,65075],{},"博士毕业后，香农去了普林斯顿高等研究院，在那里的几个月里，他见到了爱因斯坦、哥德尔和冯·诺依曼等人。普林斯顿的气氛很学院，学者们常在林荫道上散步、低声争论。香农既敬佩他们，又觉得自己并不真正属于这里：他更愿意在一间堆满零件的屋子里琢磨东西。",[11,65077,65078],{},"倒是有两位年长的数学家早早看出了他的分量。诺伯特·维纳在 1940 年前后这样评价这个年轻人：「一个才华与智力都极为出众的人……他已经做出了极具原创性的工作，毫无疑问是正在升起的一颗星。」而冯·诺依曼后来跟他的一次闲谈，给这门学科留下了一个沿用至今的词——这件事我们放到 1948 年再说。",[11,65080,65081],{},"1941 年，他加入了贝尔实验室，那个年代的贝尔实验室是一座奇特的「自由王国」，管理层相信让聪明人自己找事做，终会有收获。香农正好适合这样的环境。战争期间，他参与了防空火力控制系统的研究，也接触了保密通信和密码学方面的工作，其中包括著名的 SIGSALY 加密话音系统的相关研究。",[11,65083,65084],{},"他在贝尔实验室交到了几个真正投契的朋友，其中两个往后会反复出现。",[11,65086,65087],{},"一位是约翰·皮尔斯，日后被称作「通信卫星之父」的工程师。皮尔斯差不多是香农在实验室里最好的思想对手，两个人互相扔点子、合写论文、交换成堆的书。皮尔斯留下的说法里有一条后来被引用得最多：香农谈起自己的想法时，常常是从听别人抱怨某个系统开始的——「他漫不经心地听着，然后发明出一个新系统。」皮尔斯三番五次跟他说「这个想法你应该写出来」，香农的回答通常是：「‘应该’是什么意思？」",[11,65089,65090],{},"另一位是巴尼·奥利弗，后来去了惠普，也是「搜寻地外文明」计划最早的推动者之一。奥利弗与香农、皮尔斯三人合写过那篇《脉码调制的哲学》，还一起申请了专利。奥利弗晚年这样回忆自己在这个圈子里的位置：「我们成了朋友，所以很多理论可以说是我替他接生的。他把想法扔给我……我比谁都更早知道信息论。」",[11,65092,65093],{},"他的生活节奏在同事眼里有些古怪：不太爱开会，常常关上办公室的门独自思考，下班时间也不固定。他当时住在纽约格林威治村，每天要往返于曼哈顿与实验室之间。据说他爱在深夜工作，走廊里的独轮车也是从这个时期开始出名的。他一边骑，一边思考难题，对他而言，玩耍和思考本是同一件事的两面。",[11,65095,65096],{},"1943 年初，英国数学家艾伦·图灵访问贝尔实验室。两人在食堂里一起喝茶聊天，由于图灵的密码工作属于机密，他们没法谈具体项目，聊的主要是更宽泛的话题：机器能不能思考？能不能教机器学东西？",[11,65098,65099],{},"图灵后来在饭桌上向人复述过他印象深刻的一点：「香农不只想往大脑里喂数据，他还想喂文化层面的东西！他想给它放音乐。」（这句话是旁人的转述，见于安德鲁·霍奇斯的图灵传记，不是两人当时的通信。）图灵自己则傲慢得很可爱，他说他并不想造一个「强大的大脑」：「我想要的只是一个平庸的大脑，差不多像美国电话电报公司总裁那样的。」",[11,65101,65102],{},"有意思的是，两个人对这次交谈的记忆并不一样。近四十年后，香农接受口述史访问时回忆说，他们那次「完全没谈密码」，「谈得更多的是人脑和计算机器」这类问题；至于他自己关于信息的那套想法，图灵当时「并不认为方向是对的」，他「得到的差不多是负面的反馈」。一次被后人追认为「两个天才难得交会」的谈话，在当事人那里，其实还带着一点被质疑的味道。",[11,65104,65105],{},"这是两个隐居于各自世界里的天才难得的一次交会：一个刚从布莱切利园的机密里走出来，一个正准备写下通信的数学理论，而他们在饭桌上真正关心的，是造一个会想东西的机器。",[11,65107,65108],{},"在这些年里，香农一边为战争做事，一边在心里慢慢酝酿一个更大的问题：通信的本质到底是什么？在 1945 年的一份内部保密备忘录里，他已经用概率论的语言分析密码系统，战后这份工作以《保密系统的通信理论》为题公开发表。密码学与通信理论，在他脑中是同一枚硬币的两面。",[872,65110,65111],{"id":65111},"一篇论文与一个安静的家",[11,65113,65114],{},"1948 年，《贝尔系统技术杂志》分两期连载了香农的论文《通信的数学理论》。这篇论文提出了一整套全新的概念：用「比特」衡量信息量；把信息源的不确定性定义为「熵」；证明了在噪声信道中，只要传输速率低于某个「信道容量」，就存在办法让错误率任意小。",[11,65116,65117],{},"「比特」这个词是同事约翰·图基提出来的，意思是「二进制数字」。图基是普林斯顿的统计学家，战争期间常驻贝尔实验室，据一份 1947 年 1 月的实验室备忘录，这个说法最迟在那时已经定下来了。香农在论文里最早把「bit」用作信息的单位，并且明确注明「此词由 J. W. 图基建议」。据说命名是在实验室的午餐桌上敲定的，有人提议叫「binit」，有人提议叫「bigit」，最后「bit」胜出。一个今天人人都在用的单位，诞生过程就是这么随便。",[11,65119,65120],{},"关于「熵」这个名字，有一个流传很广的故事：据说冯·诺依曼建议他用这个词，理由是反正没有人真正懂熵，辩论的时候他总能占上风。这个故事细节不一，可信度看起来也难以考证。",[11,65122,65123],{},"但值得补充的是，这个故事最早的出处其实是香农自己。1971 年，他在《科学美国人》上的一篇文章里回忆道：「我最头疼的是该叫它什么。我想过叫‘信息’，但这个词被用得太滥，于是决定叫‘不确定性’。后来我跟约翰·冯·诺依曼讨论，他有更好的主意。他对我说：‘你就叫它熵吧，有两个理由。第一，你的不确定性函数在统计力学里已经用这个名字了，它本来就有名字。第二，而且更重要：没有人真正知道熵到底是什么，所以辩论的时候你总是占上风。’」——所以这不是后人编的段子，而是他本人的追述；只是「1940 年前后的一次谈话、1971 年才写下来」这条时间线，本身也提醒我们：细节不必太当真。",[11,65125,65126],{},"论文发表之后的反响，皮尔斯有过一个很好的说法：这篇文章「像一颗炸弹，而且是一颗延时炸弹」。他后来在另一篇回顾里说，这件事「像一道凭空劈下来的闪电」，而且「我不知道还有哪套理论是这样一整套、几乎没什么前身和历史地冒出来的」。皮尔斯也留下了两句让同行记了一辈子的实话：1950 年以前发表的信息论文章，「大多数跟香农的工作没什么关系」；以及「我从没遇到过一个真正弄懂了信息论的物理学家」。",[11,65128,65129],{},"当时也不是没有冷脸。统计学家约瑟夫·杜布给 1949 年那本书写的书评就不客气，说它更像「有启发性」而非数学性的著作，「而且作者在数学上的意图并不总是体面的」。",[11,65131,65132],{},"真正把这篇论文推向公众的，是另一个人：洛克菲勒基金会的沃伦·韦弗。韦弗在战时管过科学动员，眼光极好。1949 年，他把自己的通俗介绍与香农的论文合在一起，出了《通信的数学理论》这本书。在给基金会主席的信里，韦弗把香农做的事比作「吉布斯为物理化学做过的事」。而在他自己那篇导言里，他写道：",[65134,65135,65136],"blockquote",{},[11,65137,65138],{},"「communication（通信）这个词在这里将取一个非常宽泛的含义，包括一个头脑影响另一个头脑的全部过程。这当然不限于书面和口头的言语，还包括音乐、绘画、戏剧、芭蕾，事实上包括人的一切行为。」",[11,65140,65141],{},"韦弗还给出了那个著名的三层划分——技术的层次（符号传得多准）、语义的层次（符号多准确地传达了意思）、效果的层次（收到的意思多有效地改变了人的行动）。他最后甚至写下这样一句近乎诗的话：在这套理论里，「熵不只说算术的语言，它还说语言的语言」。",[11,65143,65144],{},"这就是「信息论热」的起点：一篇极其克制的数学论文，配上一篇极其热情的导言。往后几十年里，各种拿「信息」「熵」「冗余」去解释生物、心理、语言、经济、管理的文章层出不穷——而这桩公案，我们后面还会看到香农本人的反应。",[11,65146,65147],{},"对同行来讲，这篇论文的分量是不需要解释的。多年后，大卫·斯莱皮安在自己编的《信息论关键论文选》前言里写道：「本世纪大概没有哪一件工作，比香农 1948 年那篇文章更深刻地改变了人对通信的理解。」1998 年，在论文发表五十周年的综述里，塞尔希奥·贝尔杜第一次用了那个后来被反复引用的比喻：它是「信息时代的《大宪章》」。顺便说一句，《科学美国人》1990 年那篇专访里其实写的是「数字时代的《大宪章》」——两个版本后来常被混在一起，都算在《科学美国人》头上。",[11,65149,65150],{},"有意思的是，这样一篇改变世界的论文，作者本人几乎没有因此改变生活。他在发表之后依旧是那个爱摆弄东西、不爱应酬的人。",[11,65152,65153],{},"1949 年 3 月，他与贝尔实验室里的数值分析员贝蒂（玛丽·伊丽莎白·摩尔）结婚。据说牵线的人是皮尔斯——皮尔斯当时是贝蒂的上司。贝蒂是罗格斯大学的数学毕业生，聪明、幽默、有主见，既是他生活上的伙伴，也是工作上的搭档：他的一些论文手稿甚至是她的字迹抄写的，这让后来的研究者困惑了很久。她会吹钢琴，他会单簧管，两人下班后会找一些二声部的谱子合着弹。两人后来有了三个孩子。",[11,65155,65156],{},"还有一件圈里人都知道的小事。香农这篇论文出来的时候，诺伯特·维纳其实也在同一片水域里作业——他 1948 年出的《控制论》同样掀起了巨浪。两人的关系始终客气，维纳却在 1946 到 1947 年间私下抱怨过：「我的一个竞争者，贝尔电话公司的香农，要抢在我前面发表文章了。」香农这边对维纳一直很尊敬，说他是「我年轻时候的偶像」；至于两个人气质上的差别，麻省理工的托马斯·凯拉思后来总结得很直白：「维纳喜欢出风头，香农则完全不在乎这些。」",[11,65158,65159],{},"1949 年到 1950 年，他还发表了关于计算机下棋的论文，提出了让机器评估棋局、搜索走法的基本思路，这些想法后来成了计算机博弈的源头。",[11,65161,65162],{},"这件事值得多说两句。1950 年 3 月，他在《哲学杂志》上发表了《为计算机下棋编制程序》——这是这个题目上的第一篇技术论文；同年 2 月，《科学美国人》上还有一篇面向大众的《一台会下棋的机器》。文章中他把下棋程序分成两类，一类是硬算到底的「A 型」，一类是只搜有意思的分支的「B 型」，他自己更看好后者；他还估算出棋局的变化量大约是 10 的 120 次方，这个数字后来被叫作「香农数」。更早一点，1949 年他就在家里做出了一台用继电器控制的下棋机器，并且把它演示给国际象棋大师爱德华·拉斯克看过。有意思的是图灵当时也在做类似的事：他和戴维·钱珀瑙恩在 1948 年前后设计过一个下棋程序，因为没机器可用，只能自己拿纸笔一步一步地「手动运行」。图灵 1950 年那篇《计算机器与智能》里，还引用了香农的棋类文章。从香农的 A 型\u002FB 型分类往下，洛斯阿拉莫斯 1956 年的 MANIAC、伯恩斯坦 1958 年在 IBM 704 上的程序、麦卡锡他们在麻省理工写的程序，一直到 1967 年格林布拉特的棋程序，用的都是这套架子。",[11,65164,65165],{},"香农夫妇很爱玩杂耍、下棋，家里也总有一些稀奇古怪的小发明。对孩子们来说，这位爸爸并不严厉，更像一个爱和他们一起折腾的大朋友。",[872,65167,65169],{"id":65168},"玩具机器与赌场","玩具、机器与赌场",[11,65171,65172],{},"如果说香农的理论成就是他的「正业」，那么他造的那些机器，就是他生活里最鲜活的部分。",[11,65174,65175],{},"1950 年，他做了一只名叫「忒修斯」的电动机械鼠，能够在迷宫里摸索前进，一旦走过一遍，它就能记住路线，第二次直接走到出口。名字来自那个用线团走出迷宫的希腊英雄。这只木老鼠本身只是个装着磁铁和几根铜丝触须的壳子，真正的「脑子」藏在地板下面：一整排从电话交换机上拆下来的继电器，驱动着一台带电磁铁的滑车，把老鼠拖着往四个方向走。触须碰到墙，对应的继电器就记下这一格有墙，老鼠拐九十度继续试，直到撞上一块铜做的「奶酪」，蜂鸣器响起来。二十五格的迷宫，大约两个比特记一格，全部记忆加起来也就几十个比特——而它第二次就能一路直着跑到出口。迷宫改了，它还能把学到的忘掉重学。这是最早的「学习机器」之一。香农自己后来开玩笑说：严格讲，「那只老鼠并没有在解迷宫，是迷宫在解那只老鼠。」",[11,65177,65178],{},"1951 年 3 月，他在梅西基金会的控制论会议上做了题为《一台迷宫求解机的展示》的报告（会议记录到 1952 年才出版，贝尔实验室同年还拍了一部短片，报纸杂志跟着报道了一轮，所以很多书上把年份写成 1952）。据说现场有位神经生理学家拉尔夫·杰拉德，看见老鼠在一个圈里绕来绕去，喊了一声「这是神经症！」，香农于是又给它加了一个「抗神经症回路」。",[11,65180,65181],{},"他造过一台「读心机器」，能根据人以前猜硬币正反的习惯来预测下一次选择。这件事有个具体的对手。贝尔实验室的同事戴夫·哈格尔巴格先做了一台专门猜硬币的机器，叫 SEER——「序列外推机器人」，机器里面记着八种局面（上一轮自己赢还是输、对手换没换招），每种局面配一个可以上下加减的计数器，靠这点统计去押下一次；据他那篇 1956 年的论文，这台机器跟人对打了九千多局，赢了大约五成三。香农 1953 年 3 月写了一份备忘录，标题就叫《一台读心的（？）机器》，开门见山地说，这是「对哈格尔巴格的机器的一个简化模型」。两人的机器后来被接在一起互相对打，因为没人按按钮，还得再串一台「裁判」机器来产生随机数。谁赢的说法不一：多数说法是香农那台简单些的机器小胜，也有说它「赢得相当轻松」的——但具体比分，恐怕已经没人说得清了。香农在备忘录里还写了一句有趣的话：这类游戏，冯·诺依曼和摩根施特恩用博弈论处理过，爱伦·坡在《被窃的信》里用心理学处理过，而他这台机器「更接近坡的玩法，而不是冯·诺依曼的」。",[11,65183,65184],{},"他还造过一台只有一个功能的「终极机器」：盒子上有一个开关，你把它拨到「开」，盒盖就会打开，伸出一只机械手，把开关拨回「关」，然后缩回去，盖上盖子。这个点子最初来自他的朋友马文·明斯基——1952 年夏天明斯基在贝尔实验室跟他共事，后来回忆说：「我提议做这个机器，香农喜欢它，就让公司做了一批，送给几位高管。」香农在自己的桌上留了一台。它是被作家阿瑟·克拉克写下来的，克拉克在贝尔实验室看见过它，1958 年这样描述：「它不过是一只小木匣，雪茄盒大小，一面装着开关。你一扳开关，里面就响起一阵愤怒而一本正经的嗡嗡声。盒盖慢慢抬起，一只手从下面伸出来，把开关拨回去，然后缩回盒里。盒盖啪地合上，像棺材合盖一样决绝……一台什么也不做、只把自己关掉的机器，让人有一种说不出的不祥感。」——半个多世纪后，这个点子在网上重新火了一遍，人们叫它「最没用的机器」，做的人多半并不知道它出自明斯基的玩笑和香农的手。",[11,65186,65187],{},"他还设计了一台只用罗马数字运算的计算机，起名 THROBAC，说的是「节俭的罗马数字回溯式计算机」，名字本身就像个玩笑。这台机器 1953 年 4 月留下了两份技术备忘录，实物今天是麻省理工博物馆的藏品。",[11,65189,65190],{},"他的「玩具房」是家里最神奇的地方。那里有各式各样的独轮车，其中包括一些特别古怪的设计：有一辆没有座垫，有一辆没有脚蹬，有一辆是双人的，还有一辆的轮毂是偏心的，踩起来人会一上一下地起伏——他说这是头一辆这么做的车，帮他调试的助手查理·曼宁一边担心他的安全，一边为它第一次骑成鼓掌。车库里的独轮车和「大小轮」堆得落了一层灰。有会杂耍的机器人，其中一个被装扮成喜剧演员 W.C. 菲尔兹的形象：那是用儿童建筑玩具搭出来的，用三个钢球，球在一个绷紧的鼓面上弹跳，靠两片摇摆的桨「接一下、抛一下」，出手的误差则用短短的沟槽去纠正——全机没有一个传感器。他还做过三个小丑的「永不掉球」杂耍装置，起因是贝蒂从蛋糕装饰店带回一个四英寸的小丑，「才一块九毛八」。他的得意是：「有史以来最厉害的杂耍家也没法把纪录保持几分钟，可我的小丑整夜都在抛，一次也不掉！」",[11,65192,65193],{},"据说他还造过会喷火的小号——一吹，喇叭口就喷出火苗；还有火箭动力的飞盘、烧汽油的弹跳杆、一双巨大的泡沫「水上行走鞋」、一根从厨房用缆线连到地下实验室的机械手指（一拉，手指就勾一勾，等于叫人）、一台能设定翻转次数的机械硬币、从门廊下滑到湖边的简易缆车。爱德华·索普第一次走进他家地下室时的感想，被传记作者记了下来：「一个搞小玩意的人的天堂……我这是遇到了终极小玩意爱好者。」",[11,65195,65196],{},"他对杂耍尤其着迷，甚至总结出了一个「杂耍定理」。它把五个量用一行等式联系起来：设每只手接球到把球抛出的时间为 D，球在空中滞留的时间为 F，一只手空着的比例为 V，那么 (F + D) × 手数 = (V + D) × 球数。这个式子直到今天还在杂耍圈里被引用。他也写过一篇《杂耍的科学》，投给了《科学美国人》，但那份稿子改了又改，最后没发表——他 1987 年接受采访时还说：「我是给《科学美国人》写了一篇，至今没发出来。」文章要等到他去世后，被收进 1993 年的《香农论文集》里才和读者见面。对别人来说，这些都是不务正业；对他来说，这是理解世界的另一种方式。",[11,65198,65199],{},"他的兴趣还延伸到了钱的世界。1961 年，他与数学家爱德华·索普合作，做了一台可穿戴的轮盘赌预测装置：一个香烟盒大小的模拟计算机，开关藏在鞋里用大脚趾操作，一只脚负责给轮盘计时，另一只负责给球计时，结果则用几个音调通过塞进一只耳朵的小喇叭报出来；连导线都用胶水粘上颜色，尽量跟皮肤和头发混在一起。它预测的不是具体哪一个格子，而是一段扇区。那年 8 月，他们带着它真的去了拉斯维加斯：香农在轮盘旁边计时，索普在赌台另一头下注，两位太太负责望风。机器是灵的，小堆的筹码慢慢变大堆；但麻烦在于那根比头发还细的耳机线老是断，一断就得停半天。有一次同桌一位女士「惊恐地看过来」，索普赶紧走开，才发现「那个小喇叭像一只外星虫子一样从我的耳道里探出头来」。香农自己事后总结得很平：「我们碰到一堆实际问题，实际上从来没赚到钱。」这件事他们守口如瓶到 1966 年才披露；今天它被追认为「第一台可穿戴计算机」，不过这个称呼是后人的追认，不是当时的说法。",[11,65201,65202],{},"同一件事还有个前传，主角是索普的黑杰克工作。索普那篇论文本来想叫《二十一点的制胜策略》，是香农劝他把标题改得平淡些，最后成了《二十一点的一种有利策略》，1961 年发表在《美国科学院院刊》上——投这篇稿子需要一位院士背书，香农正是那位数学部的院士，据索普回忆，香农在签字前把他「盘问」了一遍才放过他。后来那本《击败庄家》把算牌术传遍了全世界。至于去拉斯维加斯那次，是轮盘赌，不是黑杰克，很多人会把这两件事混在一起。",[11,65204,65205],{},"他对股票的态度同样如此。香农并不迷恋盯盘，他会买入自己看好的少数几家技术公司并长期持有，比如惠普、摩托罗拉，还有他的老朋友亨利·辛格尔顿掌舵的泰利达因——辛格尔顿从大学时代起就跟他相识，后来还把他请进了董事会。他 1987 年跟采访者算过这笔账：「泰利达因我们拿了二十五年，每年复合增长百分之二十七。」这个数字在当时的圈子里是件稀罕事。至于那个更著名的「年化百分之二十八、跑赢一千多家基金」的说法，出处是庞德斯通 2005 年的书，依据的是一份未曾发表的访谈笔记，听个大概即可，别当成审计过的账目。他自己对这套本事的总结倒是简单：「我们是基本面派，不是技术派。」他对那些画图形的人毫不客气，把「头肩顶」「深 V」之类称作「重要数据的一种非常粗糙的复制品」；至于经济学家说的有效市场，他直说「我根本不信」。他还留下一句在麻省理工流传很广的俏皮话：有人问他为什么不用自己那套选股方法，他说：「算了吧，手续费会把你吃光。」——另有一次他对人说：「我是靠股市挣钱的，不是靠证定理挣钱的。」家里的账目，其实主要是贝蒂在管。女儿佩吉后来说：「这从来都是他们两个人的项目……他们两个都是赌徒。」他们甚至用一台早期的苹果 II 型电脑，每晚算一遍持仓的增长速度。",[11,65207,65208],{},"同一时期，他还是 1956 年达特茅斯人工智能夏季研讨会的发起者之一，与麦卡锡、明斯基、罗切斯特一起，把「人工智能」这个话题带到了学界的桌面上。有个细节值得记下：他自己并不喜欢「人工智能」这个词，更愿意用老派的「自动机研究」——那年他和麦卡锡合编的那本论文集，名字就叫《自动机研究》。至于那场为期一个多月的讨论会，他本人到底待了多久，史料上并不清楚：会议没有留下正式的出席名单，有回忆说他到场聊过一两天就走了，也有同时代人的笔记把他记成在场者。能确定的是，他不是那个从头住到尾的人。",[11,65210,65211],{},"然而他又是个清醒的人。1956 年，他在《IRE 信息论汇刊》上写了一篇很短、不到六百词的社论，题目叫《跟风》。这篇文章今天读起来依然刺人：",[65134,65213,65214],{},[11,65215,65216],{},"「信息论在最近几年变成了一股科学上的跟风热潮。它本来只是通信工程师的一件技术工具，却在通俗媒体和学术媒体上都得到了异乎寻常的关注……它大概已经被吹到了超出它实际成就的高度。」「大自然的秘密很少会一次让出好几条。」「人们会很容易发现，用上几个带劲的词，比如信息、熵、冗余，并不能解决我们所有的问题。」「这个行当无疑是被卖出去的，如果不是被卖过头的话……我们该做的是研究，而不是宣传。几篇第一流的论文，胜过一大批构思拙劣或者半途而废的文章。」",[11,65218,65219],{},"一个创立了一门学科的人，反而最先劝大家别神化这门学科——这种低调，几乎成了他性格的标签。",[11,65221,65222],{},"这篇社论不是没人回应，恰恰相反。维纳在三个月后同一份刊物上写了一篇《什么是信息论？》，客气地反驳说，不应该把信息论缩得那么窄；彼得·埃利亚斯则在 1958 年写了一篇讽刺社论，题目叫《两篇著名的论文》，虚构了一篇题为《信息论、光合作用与宗教》的文章来嘲笑滥用，末尾写：「我建议我们别再写这种东西了，把大量人力释放出来，去做那些真正需要研究的、激动人心的重要问题。」它常被后人称作学界最早的一次「索卡尔式恶作剧」的前身。罗伯特·法诺对此的感受很直接：「我不喜欢‘信息论’这个名字。克劳德也不喜欢。因为它听起来像是一门关于‘信息’的理论——可它不是，它是关于信息传输的，不是关于信息的。太多人没弄明白这一点。」贝蒂说得更家常：「那些人把他的话拉来拉去，他有点被惹烦了。他们不明白他想做的是什么。」",[872,65224,65226],{"id":65225},"麻省理工的怪教授","麻省理工的「怪教授」",[11,65228,65229],{},[65063,65230],{"alt":65231,"src":65232},"Shannon SM ’40, PhD ’40, who taught at MIT from 1956 until his retirement in 1978, was known to many as the father of information theory and of modern digital communications.","https:\u002F\u002Fimage-assets.dreams.plus\u002F202610162232493.png",[11,65234,65235],{},"1956 年 1 月，香农向贝尔实验室请了长假，回到麻省理工做客座教授，头衔是「电气通信访问教授」；1957 年他正式留下，同时挂两个教授头衔——电气工程系的通信科学教授，以及数学系教授。1958 年 10 月，学校宣布他出任新设立的「唐纳科学讲席教授」，这笔讲席由费城的唐纳基金会捐资五十万美元设立，是当时麻省理工收到过的最大一笔讲席捐赠。他和家人搬到了马萨诸塞州的温彻斯特，住进一栋靠近上神秘湖的大房子。房子 1858 年为一位名叫艾伦·德怀特的女士所建，她是托马斯·杰斐逊的曾外孙女，房子的设计据说参考了蒙蒂塞洛。他们后来给它起了个名字：「熵之屋」。",[11,65237,65238],{},"在学生眼里，香农是个特别的老师。他不爱讲课，更不爱写文章，上课风格随意，有时偏离主题，把话题带到他自己最近在摆弄的玩意上。",[11,65240,65241],{},"莱昂纳德·克莱因罗克形容他的课「像一顿好饭：你去了，他给你的东西是干净的、直观的」。罗伯特·加拉格记得他讲课时的那种神情：「他不会说‘这一章的要点是什么什么’。他会说：‘昨天晚上我在琢磨这个，想出来一个挺有意思的看法。’说的时候嘴角还带着一点坏笑。」大卫·福尼则是被他的研究讨论班拉进这行的，福尼说，他的教学方法就是讲自己正在做的问题、半成品的结果、猜想；有些问题他有漂亮的结果，有些问题他除了把问题本身说清楚以外毫无进展——「而这对一个正在找论文题目的研究生来说，简直太好了。」赫尔曼·豪斯也记得一件小事：他去旁听香农的讨论班，香农「把我当作和他平等的人对待」，并且对他这个晚辈提了一个关键问题，后来长成了豪斯那本书的第四章。香农的课还有个技术性细节常被提起：他一边回答问题，一边用一只手把一截粉笔抛上抛下，从来没有失手过。有位学生的总结很干脆：「我们都像敬神一样敬香农。」",[11,65243,65244],{},"做导师这件事，他却躲得很彻底。他一生带出的博士生极少——按后来的统计，只有三个；同一时期，麦卡锡、明斯基那些人各自都带出了一整代人。加拉格的解释很实在：「他的博士生非常少……你要是待在麻省理工，身边站着香农这么个庞然大物，你得有相当大的自我，才敢开口请他带你。」有人劝他多带几个学生，他的回答是：「我当不了导师。我没办法给任何人建议。我不觉得自己有资格去建议别人。」",[11,65246,65247],{},"他的办公室很乱，邮件和文稿堆得老高，据说他很多信件长期无人回复。这件事有个物证：他给自己存信件的文件夹起了个名字，叫「我拖得太久还没回的信」。记者乔恩·格特纳有一句形容极为传神：「信号可以抵达克劳德·香农。然后就再也传不下去了。」有位 1969 年去麻省理工访问的苏联学者回忆，他在走廊里看见了「香农教授」的门牌，然后过了一年多才意识到，这扇门几乎没开过——香农已经很少来学校了。倒是有位同事借用他那间空掉的办公室时，在里面翻出一张一年前开出、还没兑现的支票。人们还常提到，他在教工俱乐部的午饭桌上算他的股票想法，餐巾纸写满就换一张。",[11,65249,65250],{},"他对学术的虚名、头衔、荣誉兴趣平平，常常对别人的称赞报以腼腆一笑。他的名头则越来越大：1966 年 12 月，美国国家科学奖章的消息公布，次年 2 月由约翰逊总统在白宫东厅颁发，同一年他还拿到了 IEEE 荣誉奖章。颁奖那天，七岁的女儿佩吉为了穿哪条裙子跟母亲争执了一路，最后是副总统汉弗莱的大笑把她吓到躲在母亲腿后面。贝蒂说：「他是个非常谦逊的人。他得过很多奖，但奖从没让他头脑发热，他也从不谈这些。」他那些荣誉博士的兜帽太多，于是自己做了一个可以旋转的架子挂起来，跟挂领带的架子差不多。",[11,65252,65253],{},"领奖和演讲则一直是他的软肋。1973 年，第一届「香农讲座」在以色列的阿什凯隆举行，埃尔温·伯利坎普后来回忆：「我从没见过那么严重的怯场。我根本想不到，一个人在朋友面前可以怕成这样。」他需要有人在后台替他做很久的心理建设，还得由一位朋友陪着才肯上台。他后来推掉类似的邀约时，理由是一句玩笑：「退休以后，贝蒂不擦窗户，我不做演讲。」",[11,65255,65256],{},"这些年里，他的另一桩爱好是音乐：他会吹单簧管，喜欢迪克西兰爵士乐，也常参加小型即兴演奏。楼下客厅里，他吹单簧管、贝蒂弹钢琴的二重奏，是这个家的日常。爵士乐里的即兴与结构，与他的科学趣味可谓同源。",[11,65258,65259],{},"他爱下棋，但并不在乎输赢；他自己做的下棋机器能赢一般水平的对手，他却更乐意站在旁边看机器怎么「思考」。他骑着独轮车穿过麻省理工的走廊，边走边杂耍，让路过的学生又惊又喜。",[11,65261,65262],{},"家里那三个孩子，是在一堆零件中间长大的。儿子罗伯特生于 1952 年，后来做计算机，1998 年先于父亲去世；安德鲁生于 1954 年，成了钢琴家和作曲家；女儿佩吉生于 1959 年，后来当了老师。家里那间被报纸叫作「玩具房」的两层附楼，孩子们只叫它「爸爸的房间」。佩吉回忆说，父亲在家工作的时间很多，去学校只是为了上课和见学生，「所以我从小到大觉得他老在身边，跟很多上班的人家不一样」。晚饭后谁洗碗，家里甚至有台会走的发条机器老鼠来决定：把它放在餐桌中间，看它往哪边掉下去，掉到谁那边就谁洗。有一次佩吉把一盒牙签撒在了门廊上，父亲没有生气，而是说：「你知道吗，用这个可以算出圆周率。」——他说的是投针问题。家里饭桌上的话题常常是股票，孩子们很早就被教着读《华尔街日报》。至于辅导作业，他会因为「新数学」的教材顺序不对而发一顿牢骚，然后一路讲到虚数上去，讲到孩子已经完全不指望他把题讲完。",[11,65264,65265],{},"1978 年，他从教授岗位上退休，此后更多地待在家中，把时间交给他的工坊、他的玩具和他的书。",[872,65267,65269],{"id":65268},"熵之屋里的晚年","「熵之屋」里的晚年",[11,65271,65272],{},[65063,65273],{"alt":65274,"src":65275},"Claude Shannon's electromechanical mouse, which he called Theseus, was one of the earliest attempts to \"teach\" a machine to \"learn\" and one of the first experiments in artificial intelligence.","https:\u002F\u002Fimage-assets.dreams.plus\u002F202610162229686.png",[11,65277,65278],{},"退休后的香农依然享受生活：造机器，玩杂耍，和贝蒂一起旅行、散步，在家里招待来访的朋友。他不太谈自己的成就，也不爱做回顾。很多年后，来访者发现他的家里很少有显眼的荣誉陈列，反倒到处是工作台、零件和各式玩具。",[11,65280,65281],{},"他的动手能力没有退化，反而更放得开了。他做了一台解魔方的机器——这是真的，不是传说，那台机器今天也在麻省理工博物馆里；他还为魔方写过一首七十来行、带脚注的打油诗，标明要用《Ta-Ra-Ra Boom-De-Ay》的调子唱。1985 年，他获得了第一届京都奖，去京都领奖时做了一场题为《通信与计算的发展，以及我的爱好》的演讲，那是他晚年最长、也最完整的一次公开表述。他在演讲里说，历史上真正重要的不是帝王将相，而是「达尔文、牛顿、贝多芬这样的思想者和创造者」。",[11,65283,65284],{},"进入 80 年代，他的记忆开始衰退。1983 年，他去看医生，结论是「很可能处在阿尔茨海默病非常非常早期的阶段」（这是贝蒂的原话）。起初是忘电话号码、忘名字、忘面孔，出门去趟杂货店会忘了怎么回来，字迹也开始发抖。",[11,65286,65287],{},"1985 年，在英国布莱顿举办的国际信息论研讨会上，出现了一件令人难忘的事。他和贝蒂没有事先打招呼就出现在会场，在几个报告厅里进出，直到有人认出他来，消息才传开。晚宴上主办方把他请上台，他只简短地讲了几句，就被自己的掌声淹没了，最后说了一句：「这——太荒唐了。」然后就没什么可说的了，于是从口袋里掏出三个球，当场抛了起来。事后有人问主持人该怎么形容这件事，得到的回答是：「就像牛顿突然出现在一场物理学会议上。」参会者排起长队请他签名，很多年轻研究者是平生第一次见到这位读了一辈子其论文的人。当时的信息论学会会长也回忆说，那天在场的人里，大部分其实并不认得他，「但有一种嗡嗡声：香农来了，在哪儿，在哪儿？」——那是他最后几次公开露面之一。第二年，学会正式邀请他出席了在密歇根安娜堡举行的研讨会，那时病情已经相当明显：他做了一场十几分钟的晚宴讲话，里面还塞满了趣闻和玩笑，可贝蒂替他挡掉了大部分寒暄。",[11,65289,65290],{},"此后，病情逐渐加重。女儿佩吉说：「他会到处乱走。我们住在一条繁忙的街边……看着自己的亲人得这种病，是件很可怕的事。」有一阵子，家里参加了当地医院的阿尔茨海默病研究，贝蒂充当对照组。佩吉说：「有些日子他认得人，有些日子不认……有时候还是原来那个爸爸，有时候完全不知道自己在哪儿。」1992 年，佩吉还跟他有过一次关于研究生院的正常交谈；到 1993 年，罗伯特·法诺问他一件从前的事，香农只是回答：「我不记得了。」",[11,65292,65293],{},"1993 年他摔了一跤，髋骨骨折，康复得很慢。贝蒂本想让他在家里，还在「熵之屋」里布置了一间放得下病床的房间，最后在佩吉的劝说下，把他送进了梅德福的一家护理机构。贝蒂几乎每天去两次，佩吉说她「非常尽心」。贝蒂自己回忆：「中午我过去，护士们会排队坐在一张长凳上等我，因为我一进门，他脸上就会亮起来，咧嘴笑。」他在那里还会把助行架拆开来，琢磨更好的设计；听到音乐就用手指打拍子；护理人员会给他出些简单的算术题；他也会朝着楼梯和门外走。90 年代，他渐渐认不出身边的人。2001 年 2 月 24 日，克劳德·香农在这家护理机构去世，享年 84 岁。他的大脑捐给了阿尔茨海默病研究。",[11,65295,65296],{},"贝蒂比他多活了十六年，2017 年 5 月去世。他们结婚六十八年，从他还在贝尔实验室算信道容量，一直到「熵之屋」里的人渐渐走完。",[11,65298,65299],{},"他早年还给自己画过一场想象中葬礼的速写，那张图一点也不悲伤：一支爵士乐队在前面开道，六位独轮车手抬着棺材（图上写着「6 个骑独轮车的人 \u002F 1 位亲人」），后面跟着哭泣的寡妇、八人杂耍队和一台八条腿的杂耍机器，一辆「棋局彩车」上，棋手戴维·利维正在跟电脑对局，再往后是四只按斯金纳的方法训练过的猫、各种小机器、慢跑的人和一支四百一十七人的乐队。一个人能这样安排自己的葬礼，说明他对「正经」和「不正经」的分界，从一开始就没怎么放在心上。",[872,65301,65302],{"id":65302},"一个把工作当成游戏的人",[11,65304,65305],{},"香农的一生没有戏剧性的冲突，没有激烈的论战，也没有惊人的野心，很多时候他就是在一间堆满零件的屋子里，对着一个有趣的问题发呆。他提出的理论，为今天的手机通信、互联网、数据压缩、纠错码和密码学等整个数字世界打下了地基；而他自己的日子，却过得像一个永远不肯长大的孩子。",[11,65307,65308],{},"同行对他的评价，几乎都带着一点无从下手的敬意。大卫·斯莱皮安说：「他真的是个天才。我认识的人里，只有他配得上这个词。」又说：「他数学并不算懂得深，但他能现造出他需要的数学。」罗伯特·加拉格的版本是：「他的眼光清澈得惊人。爱因斯坦也是这样。」至于他到底做了什么，加拉格还有一句更朴素的话：「是香农看出，二进制位是一切通信的基本元素。那是他真正的发现，整个通信革命都是从这里长出来的。」托马斯·科弗说：「香农把三个领域炸了出来。」詹姆斯·马西在密码学上补了一句：「毫不夸张地说，香农的文章标志着密码学从一门手艺变成了一门科学。」而马文·明斯基记住的却是另一面：「不管出现什么题目，他都带着兴致扑上去，用某种出人意料的办法去攻它。对他而言，问题看上去越难，找到新东西的机会就越大。」",[11,65310,65311],{},"他自己对这一切的解释，比任何评论都更实在。1987 年，他对采访者说：「我不认为我做事是为了得奖——虽然另一间屋子里放着两打奖章。我更主要是被好奇心驱动，从来不是为了钱。我只是想知道，东西到底是怎么被装起来的。」「答案找到之后，把它们写下来、发表出来，总是让我很痛苦……我做过很多事，从来没写出来过。太懒了吧。我楼上有一个文件夹，装的全是没写完的论文。」三年后，他又说了一句后来被引用得最多的话：「我一直在追逐自己感兴趣的东西，不太在意它有多少金钱价值，或者对世界有多少价值。我在完全没用的东西上花了很多时间。」",[11,65313,65314],{},"回望那些日常的小事：少年时在围栏上架起的电报线，独自在走廊里骑独轮车的深夜，家里摆满的各种怪机器，晚年杂耍时那一点羞涩的笑容，你会发现，这些并不是他「正业之外」的点缀，而恰恰是他创造力的来源。他相信好奇心，相信玩耍，也相信，好玩的东西往往藏着深刻的真理。",[65134,65316,65317],{},[11,65318,65319],{},"卡德班克和斯隆在《自然》上的讣文里写过一句很准的话：香农大概能排进二十世纪科学主要人物名单的前列，「尽管他的名字对公众来说相对陌生」。这大概正是他本人会满意的位置。",{"title":64941,"searchDepth":64942,"depth":64942,"links":65321},[65322,65323,65324,65325,65326,65327,65328,65329],{"id":65011,"depth":64942,"text":65011},{"id":65032,"depth":64942,"text":65032},{"id":65072,"depth":64942,"text":65072},{"id":65111,"depth":64942,"text":65111},{"id":65168,"depth":64942,"text":65169},{"id":65225,"depth":64942,"text":65226},{"id":65268,"depth":64942,"text":65269},{"id":65302,"depth":64942,"text":65302},{},"\u002Fblog\u002F2026\u002F0010-biography-of-claude-shannon",{"title":65000,"description":65005},"blog\u002F2026\u002F0010-biography-of-claude-shannon","从密歇根小镇那根架在铁丝围栏上的电报线，到贝尔实验室走廊里的独轮车，再到麻省理工那间堆满零件的办公室。他给「信息」下了数学定义，自己却一辈子只对有意思的东西负责。",[65336],"thoughts","2026-10-16T22:03:15+08:00","FVIfEic5VrK7HmvsYttp69z46Fc79opnUYBtJpxN80w",{"id":65340,"title":65341,"body":65342,"description":65346,"draft":64984,"enableCodeHighlight":64985,"enableCodeHightlight":64984,"enableComment":64985,"enableD2":64985,"enableKatex":64985,"extension":64986,"image":64941,"important":64984,"location":67044,"meta":67045,"navigation":64985,"ogImage":64987,"path":67046,"seo":67047,"stem":67048,"summary":67049,"tags":67050,"time":67051,"__hash__":67052},"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":65343,"toc":67032},[65344,65347,65350,65358,65361,65364,65369,65375,65378,65381,65384,65390,65393,65398,65401,65404,65407,65410,65413,65416,65419,65422,65425,65428,65434,65437,65440,65443,65446,65484,65487,65490,65496,65499,65502,65505,65508,65511,65514,65517,65520,65523,65526,65532,65535,65654,65976,65987,65990,65993,65996,65999,66002,66005,66008,66014,66125,66132,66139,66142,66145,66148,66155,66162,66165,66168,66171,66189,66192,66307,66314,66317,66320,66430,66454,66460,66466,66475,66478,66484,66487,66494,66608,66715,66719,66722,66725,66736,66853,66856,66860,66863,66869,66872,66882,66885,66892,66894,66897,66900,66903,66906,66909,66916,67023,67028],[11,65345,65346],{},"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。",[11,65348,65349],{},"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。",[11,65351,65352,65353,65357],{},"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，",[33406,65354,65356],{"href":65355},"\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。",[11,65359,65360],{},"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。",[11,65362,65363],{},"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。",[11,65365,65366],{},[661,65367,65368],{},"一般来说，写作分为几种类型：日常写作、科学写作、技术写作。",[11,65370,65371,65372],{},"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。",[661,65373,65374],{},"使用的认知模型完全不同。",[11,65376,65377],{},"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。",[11,65379,65380],{},"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」。",[872,65382,65383],{"id":65383},"日常写作",[11,65385,65386,65389],{},[661,65387,65388],{},"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。",[11,65391,65392],{},"日常写作是「线性的」。比如：",[65134,65394,65395],{},[11,65396,65397],{},"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。",[11,65399,65400],{},"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。",[11,65402,65403],{},"它的基本结构就是：想法 → 句子 → 段落 → 文章。",[11,65405,65406],{},"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。",[11,65408,65409],{},"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。",[11,65411,65412],{},"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。",[11,65414,65415],{},"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。",[11,65417,65418],{},"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。",[11,65420,65421],{},"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。",[11,65423,65424],{},"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。",[872,65426,65427],{"id":65427},"技术写作",[11,65429,65430,65433],{},[661,65431,65432],{},"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。",[11,65435,65436],{},"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。",[11,65438,65439],{},"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。",[11,65441,65442],{},"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。",[11,65444,65445],{},"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：",[65447,65448,65452],"pre",{"className":65449,"code":65450,"language":65451,"meta":64941,"style":64941},"language-markdown shiki shiki-themes github-light github-light github-dark","# Installation\n## Requirements\n## Install\n# Usage\n# Troubleshooting\n","markdown",[65453,65454,65455,65462,65467,65472,65478],"code",{"__ignoreMap":64941},[15,65456,65459],{"class":65457,"line":65458},"line",1,[15,65460,65461],{},"# Installation\n",[15,65463,65464],{"class":65457,"line":64942},[15,65465,65466],{},"## Requirements\n",[15,65468,65469],{"class":65457,"line":64947},[15,65470,65471],{},"## Install\n",[15,65473,65475],{"class":65457,"line":65474},4,[15,65476,65477],{},"# Usage\n",[15,65479,65481],{"class":65457,"line":65480},5,[15,65482,65483],{},"# Troubleshooting\n",[11,65485,65486],{},"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。",[11,65488,65489],{},"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：",[11,65491,65492],{},[65063,65493],{"alt":65494,"src":65495},"Markdown 层次写作模型","https:\u002F\u002Fimage-assets.dreams.plus\u002F202609222216870.png",[11,65497,65498],{},"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。",[11,65500,65501],{},"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。",[872,65503,65504],{"id":65504},"科学写作",[11,65506,65507],{},"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。",[11,65509,65510],{},"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。",[11,65512,65513],{},"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。",[11,65515,65516],{},"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。",[11,65518,65519],{},"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。",[11,65521,65522],{},"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。",[11,65524,65525],{},"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",[11,65527,65528],{},[65063,65529],{"alt":65530,"src":65531},"科学写作的写作模型","https:\u002F\u002Fimage-assets.dreams.plus\u002F202609222126754.png",[11,65533,65534],{},"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。",[11,65536,65537,65538,65653],{},"这也是为什么 ",[15,65539,65541,65556],{"className":65540,"translate":19},[18],[15,65542,65544],{"className":65543},[23],[25,65545,65546],{"xmlns":27},[29,65547,65548,65553],{},[32,65549,65550],{},[282,65551,65552],{},"LaTeX",[40,65554,65555],{"encoding":42},"\\LaTeX",[15,65557,65559],{"className":65558,"ariaHidden":48},[47],[15,65560,65562,65566],{"className":65561},[52],[15,65563],{"className":65564,"style":65565},[56],"height:0.8988em;vertical-align:-0.2155em;",[15,65567,65569,65573,65577,65599,65603],{"className":65568},[61,13539],[15,65570,47453],{"className":65571},[61,65572],"textrm",[15,65574],{"className":65575,"style":65576},[127],"margin-right:-0.36em;",[15,65578,65580],{"className":65579},[342],[15,65581,65583],{"className":65582},[346],[15,65584,65586],{"className":65585,"style":57},[350],[15,65587,65589,65592],{"style":65588},"top:-2.905em;",[15,65590],{"className":65591,"style":358},[357],[15,65593,65595],{"className":65594},[61],[15,65596,65598],{"className":65597},[61,65572,365,362,363,364],"A",[15,65600],{"className":65601,"style":65602},[127],"margin-right:-0.15em;",[15,65604,65606,65609,65612,65646,65650],{"className":65605},[61,13539],[15,65607,56709],{"className":65608},[61,65572],[15,65610],{"className":65611,"style":7099},[127],[15,65613,65615,65637],{"className":65614},[342,472],[15,65616,65618,65634],{"className":65617},[346],[15,65619,65622],{"className":65620,"style":65621},[350],"height:0.4678em;",[15,65623,65625,65628],{"style":65624},"top:-2.7845em;",[15,65626],{"className":65627,"style":3857},[357],[15,65629,65631],{"className":65630},[61],[15,65632,14807],{"className":65633},[61,65572],[15,65635,497],{"className":65636},[496],[15,65638,65640],{"className":65639},[346],[15,65641,65644],{"className":65642,"style":65643},[350],"height:0.2155em;",[15,65645],{},[15,65647],{"className":65648,"style":65649},[127],"margin-right:-0.125em;",[15,65651,38],{"className":65652},[61,65572]," 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。",[11,65655,65656,65759,65760,65763,65764,65867,65868,65971,65972,65975],{},[15,65657,65659,65672],{"className":65658,"translate":19},[18],[15,65660,65662],{"className":65661},[23],[25,65663,65664],{"xmlns":27},[29,65665,65666,65670],{},[32,65667,65668],{},[282,65669,65552],{},[40,65671,65555],{"encoding":42},[15,65673,65675],{"className":65674,"ariaHidden":48},[47],[15,65676,65678,65681],{"className":65677},[52],[15,65679],{"className":65680,"style":65565},[56],[15,65682,65684,65687,65690,65710,65713],{"className":65683},[61,13539],[15,65685,47453],{"className":65686},[61,65572],[15,65688],{"className":65689,"style":65576},[127],[15,65691,65693],{"className":65692},[342],[15,65694,65696],{"className":65695},[346],[15,65697,65699],{"className":65698,"style":57},[350],[15,65700,65701,65704],{"style":65588},[15,65702],{"className":65703,"style":358},[357],[15,65705,65707],{"className":65706},[61],[15,65708,65598],{"className":65709},[61,65572,365,362,363,364],[15,65711],{"className":65712,"style":65602},[127],[15,65714,65716,65719,65722,65753,65756],{"className":65715},[61,13539],[15,65717,56709],{"className":65718},[61,65572],[15,65720],{"className":65721,"style":7099},[127],[15,65723,65725,65745],{"className":65724},[342,472],[15,65726,65728,65742],{"className":65727},[346],[15,65729,65731],{"className":65730,"style":65621},[350],[15,65732,65733,65736],{"style":65624},[15,65734],{"className":65735,"style":3857},[357],[15,65737,65739],{"className":65738},[61],[15,65740,14807],{"className":65741},[61,65572],[15,65743,497],{"className":65744},[496],[15,65746,65748],{"className":65747},[346],[15,65749,65751],{"className":65750,"style":65643},[350],[15,65752],{},[15,65754],{"className":65755,"style":65649},[127],[15,65757,38],{"className":65758},[61,65572]," 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 ",[661,65761,65762],{},"科学对象是一等公民"," 这样一种思想。",[15,65765,65767,65780],{"className":65766,"translate":19},[18],[15,65768,65770],{"className":65769},[23],[25,65771,65772],{"xmlns":27},[29,65773,65774,65778],{},[32,65775,65776],{},[282,65777,65552],{},[40,65779,65555],{"encoding":42},[15,65781,65783],{"className":65782,"ariaHidden":48},[47],[15,65784,65786,65789],{"className":65785},[52],[15,65787],{"className":65788,"style":65565},[56],[15,65790,65792,65795,65798,65818,65821],{"className":65791},[61,13539],[15,65793,47453],{"className":65794},[61,65572],[15,65796],{"className":65797,"style":65576},[127],[15,65799,65801],{"className":65800},[342],[15,65802,65804],{"className":65803},[346],[15,65805,65807],{"className":65806,"style":57},[350],[15,65808,65809,65812],{"style":65588},[15,65810],{"className":65811,"style":358},[357],[15,65813,65815],{"className":65814},[61],[15,65816,65598],{"className":65817},[61,65572,365,362,363,364],[15,65819],{"className":65820,"style":65602},[127],[15,65822,65824,65827,65830,65861,65864],{"className":65823},[61,13539],[15,65825,56709],{"className":65826},[61,65572],[15,65828],{"className":65829,"style":7099},[127],[15,65831,65833,65853],{"className":65832},[342,472],[15,65834,65836,65850],{"className":65835},[346],[15,65837,65839],{"className":65838,"style":65621},[350],[15,65840,65841,65844],{"style":65624},[15,65842],{"className":65843,"style":3857},[357],[15,65845,65847],{"className":65846},[61],[15,65848,14807],{"className":65849},[61,65572],[15,65851,497],{"className":65852},[496],[15,65854,65856],{"className":65855},[346],[15,65857,65859],{"className":65858,"style":65643},[350],[15,65860],{},[15,65862],{"className":65863,"style":65649},[127],[15,65865,38],{"className":65866},[61,65572]," 内置的 equation、theorem、lemma、proof、definition、figure、table、citation、cross-reference、appendix 等等组件，这些东西在 ",[15,65869,65871,65884],{"className":65870,"translate":19},[18],[15,65872,65874],{"className":65873},[23],[25,65875,65876],{"xmlns":27},[29,65877,65878,65882],{},[32,65879,65880],{},[282,65881,65552],{},[40,65883,65555],{"encoding":42},[15,65885,65887],{"className":65886,"ariaHidden":48},[47],[15,65888,65890,65893],{"className":65889},[52],[15,65891],{"className":65892,"style":65565},[56],[15,65894,65896,65899,65902,65922,65925],{"className":65895},[61,13539],[15,65897,47453],{"className":65898},[61,65572],[15,65900],{"className":65901,"style":65576},[127],[15,65903,65905],{"className":65904},[342],[15,65906,65908],{"className":65907},[346],[15,65909,65911],{"className":65910,"style":57},[350],[15,65912,65913,65916],{"style":65588},[15,65914],{"className":65915,"style":358},[357],[15,65917,65919],{"className":65918},[61],[15,65920,65598],{"className":65921},[61,65572,365,362,363,364],[15,65923],{"className":65924,"style":65602},[127],[15,65926,65928,65931,65934,65965,65968],{"className":65927},[61,13539],[15,65929,56709],{"className":65930},[61,65572],[15,65932],{"className":65933,"style":7099},[127],[15,65935,65937,65957],{"className":65936},[342,472],[15,65938,65940,65954],{"className":65939},[346],[15,65941,65943],{"className":65942,"style":65621},[350],[15,65944,65945,65948],{"style":65624},[15,65946],{"className":65947,"style":3857},[357],[15,65949,65951],{"className":65950},[61],[15,65952,14807],{"className":65953},[61,65572],[15,65955,497],{"className":65956},[496],[15,65958,65960],{"className":65959},[346],[15,65961,65963],{"className":65962,"style":65643},[350],[15,65964],{},[15,65966],{"className":65967,"style":65649},[127],[15,65969,38],{"className":65970},[61,65572]," 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 ",[65453,65973,65974],{},"\\ref"," 各个科学对象。",[11,65977,65978,65979,65982,65983,65986],{},"写一个公式并加上 ",[65453,65980,65981],{},"\\label{eq:loss}","，再用 ",[65453,65984,65985],{},"\\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。",[11,65988,65989],{},"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。",[11,65991,65992],{},"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。",[11,65994,65995],{},"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。",[11,65997,65998],{},"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。",[11,66000,66001],{},"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。",[872,66003,66004],{"id":66004},"压扁的代价",[11,66006,66007],{},"上面说的「压扁」不是一个比喻，它是一件有具体成本的事。",[11,66009,66010,66011],{},"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。",[661,66012,66013],{},"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。",[11,66015,66016,66017,66120,66121,66124],{},"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 ",[15,66018,66020,66033],{"className":66019,"translate":19},[18],[15,66021,66023],{"className":66022},[23],[25,66024,66025],{"xmlns":27},[29,66026,66027,66031],{},[32,66028,66029],{},[282,66030,65552],{},[40,66032,65555],{"encoding":42},[15,66034,66036],{"className":66035,"ariaHidden":48},[47],[15,66037,66039,66042],{"className":66038},[52],[15,66040],{"className":66041,"style":65565},[56],[15,66043,66045,66048,66051,66071,66074],{"className":66044},[61,13539],[15,66046,47453],{"className":66047},[61,65572],[15,66049],{"className":66050,"style":65576},[127],[15,66052,66054],{"className":66053},[342],[15,66055,66057],{"className":66056},[346],[15,66058,66060],{"className":66059,"style":57},[350],[15,66061,66062,66065],{"style":65588},[15,66063],{"className":66064,"style":358},[357],[15,66066,66068],{"className":66067},[61],[15,66069,65598],{"className":66070},[61,65572,365,362,363,364],[15,66072],{"className":66073,"style":65602},[127],[15,66075,66077,66080,66083,66114,66117],{"className":66076},[61,13539],[15,66078,56709],{"className":66079},[61,65572],[15,66081],{"className":66082,"style":7099},[127],[15,66084,66086,66106],{"className":66085},[342,472],[15,66087,66089,66103],{"className":66088},[346],[15,66090,66092],{"className":66091,"style":65621},[350],[15,66093,66094,66097],{"style":65624},[15,66095],{"className":66096,"style":3857},[357],[15,66098,66100],{"className":66099},[61],[15,66101,14807],{"className":66102},[61,65572],[15,66104,497],{"className":66105},[496],[15,66107,66109],{"className":66108},[346],[15,66110,66112],{"className":66111,"style":65643},[350],[15,66113],{},[15,66115],{"className":66116,"style":65649},[127],[15,66118,38],{"className":66119},[61,65572]," 里这是一个局部操作：加一个 ",[65453,66122,66123],{},"\\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。",[11,66126,66127,66128,66131],{},"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",[661,66129,66130],{},"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。",[11,66133,66134,66135,66138],{},"这带来一个更隐蔽的后果：",[661,66136,66137],{},"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。",[11,66140,66141],{},"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。",[872,66143,66144],{"id":66144},"读者也在走这张图",[11,66146,66147],{},"线性化的成本不只由写作者承担，读者也在付。",[11,66149,66150,66151,66154],{},"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。",[661,66152,66153],{},"阅读本身就是在图上做遍历","，而且是带随机跳转的遍历。",[11,66156,66157,66158,66161],{},"IMRaD 与其说是一种逻辑结构，不如说是",[661,66159,66160],{},"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。",[11,66163,66164],{},"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。",[11,66166,66167],{},"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。",[11,66169,66170],{},"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？",[11,66172,66173,66174,66177,66178,867,66181,66184,66185,66188],{},"是图，但粒度不对。这些工具的节点是",[661,66175,66176],{},"笔记","，不是",[661,66179,66180],{},"命题",[65453,66182,66183],{},"[[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",[661,66186,66187],{},"边的类型","：支持、反驳、假设、推导、检验。",[11,66190,66191],{},"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。",[11,66193,66194,66195,66198,66199,66306],{},"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 ",[65453,66196,66197],{},"@fig:xxx"," 这样的交叉引用。但它们做的是",[661,66200,66201,66202,66305],{},"把 ",[15,66203,66205,66218],{"className":66204,"translate":19},[18],[15,66206,66208],{"className":66207},[23],[25,66209,66210],{"xmlns":27},[29,66211,66212,66216],{},[32,66213,66214],{},[282,66215,65552],{},[40,66217,65555],{"encoding":42},[15,66219,66221],{"className":66220,"ariaHidden":48},[47],[15,66222,66224,66227],{"className":66223},[52],[15,66225],{"className":66226,"style":65565},[56],[15,66228,66230,66233,66236,66256,66259],{"className":66229},[61,13539],[15,66231,47453],{"className":66232},[61,65572],[15,66234],{"className":66235,"style":65576},[127],[15,66237,66239],{"className":66238},[342],[15,66240,66242],{"className":66241},[346],[15,66243,66245],{"className":66244,"style":57},[350],[15,66246,66247,66250],{"style":65588},[15,66248],{"className":66249,"style":358},[357],[15,66251,66253],{"className":66252},[61],[15,66254,65598],{"className":66255},[61,65572,365,362,363,364],[15,66257],{"className":66258,"style":65602},[127],[15,66260,66262,66265,66268,66299,66302],{"className":66261},[61,13539],[15,66263,56709],{"className":66264},[61,65572],[15,66266],{"className":66267,"style":7099},[127],[15,66269,66271,66291],{"className":66270},[342,472],[15,66272,66274,66288],{"className":66273},[346],[15,66275,66277],{"className":66276,"style":65621},[350],[15,66278,66279,66282],{"style":65624},[15,66280],{"className":66281,"style":3857},[357],[15,66283,66285],{"className":66284},[61],[15,66286,14807],{"className":66287},[61,65572],[15,66289,497],{"className":66290},[496],[15,66292,66294],{"className":66293},[346],[15,66295,66297],{"className":66296,"style":65643},[350],[15,66298],{},[15,66300],{"className":66301,"style":65649},[127],[15,66303,38],{"className":66304},[61,65572]," 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。",[11,66308,66309,66310,66313],{},"两个方向的工具缺的是同一样东西：",[661,66311,66312],{},"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。",[872,66315,66316],{"id":66316},"科学写作需要什么",[11,66318,66319],{},"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。",[11,66321,66322,66325,66326,66429],{},[661,66323,66324],{},"第一，科学对象是一等公民。"," 这一点其实 ",[15,66327,66329,66342],{"className":66328,"translate":19},[18],[15,66330,66332],{"className":66331},[23],[25,66333,66334],{"xmlns":27},[29,66335,66336,66340],{},[32,66337,66338],{},[282,66339,65552],{},[40,66341,65555],{"encoding":42},[15,66343,66345],{"className":66344,"ariaHidden":48},[47],[15,66346,66348,66351],{"className":66347},[52],[15,66349],{"className":66350,"style":65565},[56],[15,66352,66354,66357,66360,66380,66383],{"className":66353},[61,13539],[15,66355,47453],{"className":66356},[61,65572],[15,66358],{"className":66359,"style":65576},[127],[15,66361,66363],{"className":66362},[342],[15,66364,66366],{"className":66365},[346],[15,66367,66369],{"className":66368,"style":57},[350],[15,66370,66371,66374],{"style":65588},[15,66372],{"className":66373,"style":358},[357],[15,66375,66377],{"className":66376},[61],[15,66378,65598],{"className":66379},[61,65572,365,362,363,364],[15,66381],{"className":66382,"style":65602},[127],[15,66384,66386,66389,66392,66423,66426],{"className":66385},[61,13539],[15,66387,56709],{"className":66388},[61,65572],[15,66390],{"className":66391,"style":7099},[127],[15,66393,66395,66415],{"className":66394},[342,472],[15,66396,66398,66412],{"className":66397},[346],[15,66399,66401],{"className":66400,"style":65621},[350],[15,66402,66403,66406],{"style":65624},[15,66404],{"className":66405,"style":3857},[357],[15,66407,66409],{"className":66408},[61],[15,66410,14807],{"className":66411},[61,65572],[15,66413,497],{"className":66414},[496],[15,66416,66418],{"className":66417},[346],[15,66419,66421],{"className":66420,"style":65643},[350],[15,66422],{},[15,66424],{"className":66425,"style":65649},[127],[15,66427,38],{"className":66428},[61,65572]," 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。",[11,66431,66432,802,66435,6148,66438,6148,66441,6148,66444,6148,66447,6148,66450,66453],{},[661,66433,66434],{},"第二，边是有类型的。",[65453,66436,66437],{},"supports",[65453,66439,66440],{},"contradicts",[65453,66442,66443],{},"assumes",[65453,66445,66446],{},"derives",[65453,66448,66449],{},"tests",[65453,66451,66452],{},"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。",[11,66455,66456,66459],{},[661,66457,66458],{},"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。",[11,66461,66462,66465],{},[661,66463,66464],{},"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。",[11,66467,66468,66471,66472,66474],{},[661,66469,66470],{},"第五，一致性检查应该上升到语义层。"," 编译器检查 ",[65453,66473,65974],{}," 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。",[872,66476,66477],{"id":66477},"为什么这件事一直没做成",[11,66479,66480,66481,867],{},"既然需求这么明确，为什么没有这样的工具？因为",[661,66482,66483],{},"读一张图很便宜，写一张图很贵",[11,66485,66486],{},"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。",[11,66488,66489,66490,66493],{},"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：",[661,66491,66492],{},"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。",[11,66495,66496,66497,66600,66601,66603,66604,66607],{},"这反过来解释了一件我一直觉得有意思的事：",[15,66498,66500,66513],{"className":66499,"translate":19},[18],[15,66501,66503],{"className":66502},[23],[25,66504,66505],{"xmlns":27},[29,66506,66507,66511],{},[32,66508,66509],{},[282,66510,65552],{},[40,66512,65555],{"encoding":42},[15,66514,66516],{"className":66515,"ariaHidden":48},[47],[15,66517,66519,66522],{"className":66518},[52],[15,66520],{"className":66521,"style":65565},[56],[15,66523,66525,66528,66531,66551,66554],{"className":66524},[61,13539],[15,66526,47453],{"className":66527},[61,65572],[15,66529],{"className":66530,"style":65576},[127],[15,66532,66534],{"className":66533},[342],[15,66535,66537],{"className":66536},[346],[15,66538,66540],{"className":66539,"style":57},[350],[15,66541,66542,66545],{"style":65588},[15,66543],{"className":66544,"style":358},[357],[15,66546,66548],{"className":66547},[61],[15,66549,65598],{"className":66550},[61,65572,365,362,363,364],[15,66552],{"className":66553,"style":65602},[127],[15,66555,66557,66560,66563,66594,66597],{"className":66556},[61,13539],[15,66558,56709],{"className":66559},[61,65572],[15,66561],{"className":66562,"style":7099},[127],[15,66564,66566,66586],{"className":66565},[342,472],[15,66567,66569,66583],{"className":66568},[346],[15,66570,66572],{"className":66571,"style":65621},[350],[15,66573,66574,66577],{"style":65624},[15,66575],{"className":66576,"style":3857},[357],[15,66578,66580],{"className":66579},[61],[15,66581,14807],{"className":66582},[61,65572],[15,66584,497],{"className":66585},[496],[15,66587,66589],{"className":66588},[346],[15,66590,66592],{"className":66591,"style":65643},[350],[15,66593],{},[15,66595],{"className":66596,"style":65649},[127],[15,66598,38],{"className":66599},[61,65572]," 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 ",[65453,66602,66123],{},"。它是一个",[661,66605,66606],{},"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。",[11,66609,66610,66611,66714],{},"所以 ",[15,66612,66614,66627],{"className":66613,"translate":19},[18],[15,66615,66617],{"className":66616},[23],[25,66618,66619],{"xmlns":27},[29,66620,66621,66625],{},[32,66622,66623],{},[282,66624,65552],{},[40,66626,65555],{"encoding":42},[15,66628,66630],{"className":66629,"ariaHidden":48},[47],[15,66631,66633,66636],{"className":66632},[52],[15,66634],{"className":66635,"style":65565},[56],[15,66637,66639,66642,66645,66665,66668],{"className":66638},[61,13539],[15,66640,47453],{"className":66641},[61,65572],[15,66643],{"className":66644,"style":65576},[127],[15,66646,66648],{"className":66647},[342],[15,66649,66651],{"className":66650},[346],[15,66652,66654],{"className":66653,"style":57},[350],[15,66655,66656,66659],{"style":65588},[15,66657],{"className":66658,"style":358},[357],[15,66660,66662],{"className":66661},[61],[15,66663,65598],{"className":66664},[61,65572,365,362,363,364],[15,66666],{"className":66667,"style":65602},[127],[15,66669,66671,66674,66677,66708,66711],{"className":66670},[61,13539],[15,66672,56709],{"className":66673},[61,65572],[15,66675],{"className":66676,"style":7099},[127],[15,66678,66680,66700],{"className":66679},[342,472],[15,66681,66683,66697],{"className":66682},[346],[15,66684,66686],{"className":66685,"style":65621},[350],[15,66687,66688,66691],{"style":65624},[15,66689],{"className":66690,"style":3857},[357],[15,66692,66694],{"className":66693},[61],[15,66695,14807],{"className":66696},[61,65572],[15,66698,497],{"className":66699},[496],[15,66701,66703],{"className":66702},[346],[15,66704,66706],{"className":66705,"style":65643},[350],[15,66707],{},[15,66709],{"className":66710,"style":65649},[127],[15,66712,38],{"className":66713},[61,65572]," 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。",[872,66716,66718],{"id":66717},"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",[11,66720,66721],{},"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。",[11,66723,66724],{},"草图层不管顺序，只管对象和关系。给每个对象一个短 ID：",[66726,66727,66728],"div",{},[65063,66729],{"height":66730,"width":66731,"style":66732,"data-d2-theme":66733,"alt":66734,"src":66735},679,1255,"max-width:100%;height:auto","default","以《因果结构模型文章》为例，论证的依赖关系图","data:image\u002Fsvg+xml;base64,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\u002Fmm7\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",[11,66737,66738,66739,66842,66843,66845,66846,66848,66849,66852],{},"然后在成文层，把这些 ID 落成 ",[15,66740,66742,66755],{"className":66741,"translate":19},[18],[15,66743,66745],{"className":66744},[23],[25,66746,66747],{"xmlns":27},[29,66748,66749,66753],{},[32,66750,66751],{},[282,66752,65552],{},[40,66754,65555],{"encoding":42},[15,66756,66758],{"className":66757,"ariaHidden":48},[47],[15,66759,66761,66764],{"className":66760},[52],[15,66762],{"className":66763,"style":65565},[56],[15,66765,66767,66770,66773,66793,66796],{"className":66766},[61,13539],[15,66768,47453],{"className":66769},[61,65572],[15,66771],{"className":66772,"style":65576},[127],[15,66774,66776],{"className":66775},[342],[15,66777,66779],{"className":66778},[346],[15,66780,66782],{"className":66781,"style":57},[350],[15,66783,66784,66787],{"style":65588},[15,66785],{"className":66786,"style":358},[357],[15,66788,66790],{"className":66789},[61],[15,66791,65598],{"className":66792},[61,65572,365,362,363,364],[15,66794],{"className":66795,"style":65602},[127],[15,66797,66799,66802,66805,66836,66839],{"className":66798},[61,13539],[15,66800,56709],{"className":66801},[61,65572],[15,66803],{"className":66804,"style":7099},[127],[15,66806,66808,66828],{"className":66807},[342,472],[15,66809,66811,66825],{"className":66810},[346],[15,66812,66814],{"className":66813,"style":65621},[350],[15,66815,66816,66819],{"style":65624},[15,66817],{"className":66818,"style":3857},[357],[15,66820,66822],{"className":66821},[61],[15,66823,14807],{"className":66824},[61,65572],[15,66826,497],{"className":66827},[496],[15,66829,66831],{"className":66830},[346],[15,66832,66834],{"className":66833,"style":65643},[350],[15,66835],{},[15,66837],{"className":66838,"style":65649},[127],[15,66840,38],{"className":66841},[61,65572]," 的 ",[65453,66844,66123],{}," 和 ",[65453,66847,65974],{},"。这样做的好处是：",[661,66850,66851],{},"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。",[11,66854,66855],{},"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。",[872,66857,66859],{"id":66858},"当-llm-进入这张图","当 LLM 进入这张图",[11,66861,66862],{},"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。",[11,66864,66865,66866],{},"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：",[661,66867,66868],{},"LLM 是历史上第一个有可能承担「图维护」这件事的东西。",[11,66870,66871],{},"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。",[11,66873,66874,66875,66878,66879,867],{},"但这里有一个前提，而且是致命的前提：",[661,66876,66877],{},"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",[661,66880,66881],{},"作者建图、机器维护图",[11,66883,66884],{},"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。",[11,66886,66887,66888,66891],{},"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。",[661,66889,66890],{},"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。",[872,66893,64696],{"id":64696},[11,66895,66896],{},"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。",[11,66898,66899],{},"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。",[11,66901,66902],{},"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。",[11,66904,66905],{},"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。",[11,66907,66908],{},"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。",[11,66910,66911,66912,66915],{},"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",[661,66913,66914],{},"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。",[11,66917,66918,66919,67022],{},"所以在那个系统出现之前，我大概还会继续用 Markdown 写博客、用 ",[15,66920,66922,66935],{"className":66921,"translate":19},[18],[15,66923,66925],{"className":66924},[23],[25,66926,66927],{"xmlns":27},[29,66928,66929,66933],{},[32,66930,66931],{},[282,66932,65552],{},[40,66934,65555],{"encoding":42},[15,66936,66938],{"className":66937,"ariaHidden":48},[47],[15,66939,66941,66944],{"className":66940},[52],[15,66942],{"className":66943,"style":65565},[56],[15,66945,66947,66950,66953,66973,66976],{"className":66946},[61,13539],[15,66948,47453],{"className":66949},[61,65572],[15,66951],{"className":66952,"style":65576},[127],[15,66954,66956],{"className":66955},[342],[15,66957,66959],{"className":66958},[346],[15,66960,66962],{"className":66961,"style":57},[350],[15,66963,66964,66967],{"style":65588},[15,66965],{"className":66966,"style":358},[357],[15,66968,66970],{"className":66969},[61],[15,66971,65598],{"className":66972},[61,65572,365,362,363,364],[15,66974],{"className":66975,"style":65602},[127],[15,66977,66979,66982,66985,67016,67019],{"className":66978},[61,13539],[15,66980,56709],{"className":66981},[61,65572],[15,66983],{"className":66984,"style":7099},[127],[15,66986,66988,67008],{"className":66987},[342,472],[15,66989,66991,67005],{"className":66990},[346],[15,66992,66994],{"className":66993,"style":65621},[350],[15,66995,66996,66999],{"style":65624},[15,66997],{"className":66998,"style":3857},[357],[15,67000,67002],{"className":67001},[61],[15,67003,14807],{"className":67004},[61,65572],[15,67006,497],{"className":67007},[496],[15,67009,67011],{"className":67010},[346],[15,67012,67014],{"className":67013,"style":65643},[350],[15,67015],{},[15,67017],{"className":67018,"style":65649},[127],[15,67020,38],{"className":67021},[61,65572]," 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。",[65134,67024,67025],{},[11,67026,67027],{},"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。",[67029,67030,67031],"style",{},"html .light .shiki span {color: var(--shiki-light);background: var(--shiki-light-bg);font-style: var(--shiki-light-font-style);font-weight: var(--shiki-light-font-weight);text-decoration: var(--shiki-light-text-decoration);}html.light .shiki span {color: var(--shiki-light);background: var(--shiki-light-bg);font-style: var(--shiki-light-font-style);font-weight: var(--shiki-light-font-weight);text-decoration: var(--shiki-light-text-decoration);}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: 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var(--shiki-dark-text-decoration);}",{"title":64941,"searchDepth":64942,"depth":64942,"links":67033},[67034,67035,67036,67037,67038,67039,67040,67041,67042,67043],{"id":65383,"depth":64942,"text":65383},{"id":65427,"depth":64942,"text":65427},{"id":65504,"depth":64942,"text":65504},{"id":66004,"depth":64942,"text":66004},{"id":66144,"depth":64942,"text":66144},{"id":66316,"depth":64942,"text":66316},{"id":66477,"depth":64942,"text":66477},{"id":66717,"depth":64942,"text":66718},{"id":66858,"depth":64942,"text":66859},{"id":64696,"depth":64942,"text":64696},"甘肃兰州",{},"\u002Fblog\u002F2026\u002F0006-markdown-ke-xue-xie-zuo-tong-dian-wen-ti",{"title":65341,"description":65346},"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」。",[65336],"2026-09-22T22:53:33+08:00","bdCBbBDqfmLRWSRlu3duJ1udbcTOKHpeg3xAyvxH5EM",1794483698456]