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KL、JS、全变差、卡方、Hellinger 等众多散度，具有非负性、联合凸性、变分对偶表示和数据处理不等式等核心性质，是信息论和生成模型理论中的基础工具。",[528,529,531],"h2",{"id":530},"_1-定义","1. 定义",[11,533,534,535,580,581,638,639,668,669,697,698,751],{},"设 ",[15,536,538,557],{"className":537,"translate":19},[18],[15,539,541],{"className":540},[23],[25,542,543],{"xmlns":27},[29,544,545,554],{},[32,546,547,549,552],{},[35,548,206],{},[110,550,551],{"separator":46},",",[35,553,213],{},[39,555,556],{"encoding":41},"P,Q",[15,558,560],{"className":559,"ariaHidden":46},[45],[15,561,563,567,570,574,577],{"className":562},[50],[15,564],{"className":565,"style":566},[54],"height:0.8778em;vertical-align:-0.1944em;",[15,568,206],{"className":569,"style":334},[59,60],[15,571,551],{"className":572},[573],"mpunct",[15,575],{"className":576,"style":406},[157],[15,578,213],{"className":579},[59,60]," 是同一个可测空间 ",[15,582,584,609],{"className":583,"translate":19},[18],[15,585,587],{"className":586},[23],[25,588,589],{"xmlns":27},[29,590,591,606],{},[32,592,593,595,599,601,604],{},[110,594,113],{"stretchy":112},[35,596,598],{"mathvariant":597},"script","X",[110,600,551],{"separator":46},[35,602,603],{"mathvariant":597},"A",[110,605,120],{"stretchy":112},[39,607,608],{"encoding":41},"(\\mathcal X,\\mathcal A)",[15,610,612],{"className":611,"ariaHidden":46},[45],[15,613,615,618,621,626,629,632,635],{"className":614},[50],[15,616],{"className":617,"style":139},[54],[15,619,113],{"className":620},[146],[15,622,598],{"className":623,"style":625},[59,624],"mathcal","margin-right:0.1464em;",[15,627,551],{"className":628},[573],[15,630],{"className":631,"style":406},[157],[15,633,603],{"className":634},[59,624],[15,636,120],{"className":637},[153]," 上的两个概率分布，且 ",[15,640,642,655],{"className":641,"translate":19},[18],[15,643,645],{"className":644},[23],[25,646,647],{"xmlns":27},[29,648,649,653],{},[32,650,651],{},[35,652,206],{},[39,654,206],{"encoding":41},[15,656,658],{"className":657,"ariaHidden":46},[45],[15,659,661,665],{"className":660},[50],[15,662],{"className":663,"style":664},[54],"height:0.6833em;",[15,666,206],{"className":667,"style":334},[59,60]," 关于 ",[15,670,672,685],{"className":671,"translate":19},[18],[15,673,675],{"className":674},[23],[25,676,677],{"xmlns":27},[29,678,679,683],{},[32,680,681],{},[35,682,213],{},[39,684,213],{"encoding":41},[15,686,688],{"className":687,"ariaHidden":46},[45],[15,689,691,694],{"className":690},[50],[15,692],{"className":693,"style":566},[54],[15,695,213],{"className":696},[59,60]," 绝对连续（即 ",[15,699,701,720],{"className":700,"translate":19},[18],[15,702,704],{"className":703},[23],[25,705,706],{"xmlns":27},[29,707,708,717],{},[32,709,710,712,715],{},[35,711,206],{},[110,713,714],{},"≪",[35,716,213],{},[39,718,719],{"encoding":41},"P\\ll 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Radon–Nikodym 导数（在离散情形就是概率质量之比，在密度存在时就是密度之比 ",[15,920,922,954],{"className":921,"translate":19},[18],[15,923,925],{"className":924},[23],[25,926,927],{"xmlns":27},[29,928,929,951],{},[32,930,931,933,935,937,939,942,945,947,949],{},[35,932,11],{},[110,934,113],{"stretchy":112},[35,936,774],{},[110,938,120],{"stretchy":112},[35,940,941],{"mathvariant":209},"\u002F",[35,943,944],{},"q",[110,946,113],{"stretchy":112},[35,948,774],{},[110,950,120],{"stretchy":112},[39,952,953],{"encoding":41},"p(x)\u002Fq(x)",[15,955,957],{"className":956,"ariaHidden":46},[45],[15,958,960,963,966,969,972,975,978,982,985,988],{"className":959},[50],[15,961],{"className":962,"style":139},[54],[15,964,11],{"className":965},[59,60],[15,967,113],{"className":968},[146],[15,970,774],{"className":971},[59,60],[15,973,120],{"className":974},[153],[15,976,941],{"className":977},[59],[15,979,944],{"className":980,"style":981},[59,60],"margin-right:0.0359em;",[15,983,113],{"className":984},[146],[15,986,774],{"className":987},[59,60],[15,989,120],{"className":990},[153],"）。",[993,994,995,1231,1638,1670,2250],"blockquote",{},[11,996,997,1001,1002,1103,1104,1169,1170,668,1198,1226,1227,1230],{},[998,999,1000],"strong",{},"定义 1.1（f-散度）"," 给定一个凸函数 ",[15,1003,1005,1039],{"className":1004,"translate":19},[18],[15,1006,1008],{"className":1007},[23],[25,1009,1010],{"xmlns":27},[29,1011,1012,1036],{},[32,1013,1014,1016,1019,1021,1023,1025,1028,1030,1033],{},[35,1015,37],{},[110,1017,1018],{},":",[110,1020,230],{"stretchy":112},[115,1022,126],{},[110,1024,551],{"separator":46},[35,1026,1027],{"mathvariant":209},"∞",[110,1029,120],{"stretchy":112},[110,1031,1032],{},"→",[35,1034,1035],{"mathvariant":222},"R",[39,1037,1038],{"encoding":41},"f:[0,\\infty)\\to\\mathbb R",[15,1040,1042,1060,1093],{"className":1041,"ariaHidden":46},[45],[15,1043,1045,1048,1051,1054,1057],{"className":1044},[50],[15,1046],{"className":1047,"style":55},[54],[15,1049,37],{"className":1050,"style":61},[59,60],[15,1052],{"className":1053,"style":158},[157],[15,1055,1018],{"className":1056},[162],[15,1058],{"className":1059,"style":158},[157],[15,1061,1063,1066,1069,1072,1075,1078,1081,1084,1087,1090],{"className":1062},[50],[15,1064],{"className":1065,"style":139},[54],[15,1067,230],{"className":1068},[146],[15,1070,126],{"className":1071},[59],[15,1073,551],{"className":1074},[573],[15,1076],{"className":1077,"style":406},[157],[15,1079,1027],{"className":1080},[59],[15,1082,120],{"className":1083},[153],[15,1085],{"className":1086,"style":158},[157],[15,1088,1032],{"className":1089},[162],[15,1091],{"className":1092,"style":158},[157],[15,1094,1096,1100],{"className":1095},[50],[15,1097],{"className":1098,"style":1099},[54],"height:0.6889em;",[15,1101,1035],{"className":1102},[59,366],"，且满足 ",[15,1105,1107,1130],{"className":1106,"translate":19},[18],[15,1108,1110],{"className":1109},[23],[25,1111,1112],{"xmlns":27},[29,1113,1114,1128],{},[32,1115,1116,1118,1120,1122,1124,1126],{},[35,1117,37],{},[110,1119,113],{"stretchy":112},[115,1121,117],{},[110,1123,120],{"stretchy":112},[110,1125,123],{},[115,1127,126],{},[39,1129,129],{"encoding":41},[15,1131,1133,1160],{"className":1132,"ariaHidden":46},[45],[15,1134,1136,1139,1142,1145,1148,1151,1154,1157],{"className":1135},[50],[15,1137],{"className":1138,"style":139},[54],[15,1140,37],{"className":1141,"style":61},[59,60],[15,1143,113],{"className":1144},[146],[15,1146,117],{"className":1147},[59],[15,1149,120],{"className":1150},[153],[15,1152],{"className":1153,"style":158},[157],[15,1155,123],{"className":1156},[162],[15,1158],{"className":1159,"style":158},[157],[15,1161,1163,1166],{"className":1162},[50],[15,1164],{"className":1165,"style":172},[54],[15,1167,126],{"className":1168},[59],"，则 ",[15,1171,1173,1186],{"className":1172,"translate":19},[18],[15,1174,1176],{"className":1175},[23],[25,1177,1178],{"xmlns":27},[29,1179,1180,1184],{},[32,1181,1182],{},[35,1183,206],{},[39,1185,206],{"encoding":41},[15,1187,1189],{"className":1188,"ariaHidden":46},[45],[15,1190,1192,1195],{"className":1191},[50],[15,1193],{"className":1194,"style":664},[54],[15,1196,206],{"className":1197,"style":334},[59,60],[15,1199,1201,1214],{"className":1200,"translate":19},[18],[15,1202,1204],{"className":1203},[23],[25,1205,1206],{"xmlns":27},[29,1207,1208,1212],{},[32,1209,1210],{},[35,1211,213],{},[39,1213,213],{"encoding":41},[15,1215,1217],{"className":1216,"ariaHidden":46},[45],[15,1218,1220,1223],{"className":1219},[50],[15,1221],{"className":1222,"style":566},[54],[15,1224,213],{"className":1225},[59,60]," 的 ",[998,1228,1229],{},"f-散度"," 定义为",[15,1232,1234],{"className":1233,"translate":19},[180],[15,1235,1237,1344],{"className":1236,"translate":19},[18],[15,1238,1240],{"className":1239},[23],[25,1241,1242],{"xmlns":27,"display":189},[29,1243,1244,1341],{},[1245,1246,1250],"mtable",{"rowspacing":1247,"columnalign":1248,"columnspacing":1249},"0.25em","right left","0em",[1251,1252,1253,1277],"mtr",{},[1254,1255,1256],"mtd",{},[1257,1258,1259],"mstyle",{"scriptlevel":126,"displaystyle":46},[32,1260,1261,1267,1269,1271,1273,1275],{},[195,1262,1263,1265],{},[35,1264,199],{},[35,1266,37],{},[110,1268,113],{"stretchy":112},[35,1270,206],{},[35,1272,210],{"mathvariant":209},[35,1274,213],{},[110,1276,120],{"stretchy":112},[1254,1278,1279],{},[1257,1280,1281],{"scriptlevel":126,"displaystyle":46},[32,1282,1283,1285,1287,1290,1292,1294,1296,1298,1302,1304,1307,1335,1337,1339],{},[32,1284],{},[110,1286,123],{},[110,1288,1289],{},"∫",[35,1291,944],{},[110,1293,113],{"stretchy":112},[35,1295,774],{},[110,1297,120],{"stretchy":112},[1299,1300,1301],"mtext",{}," ",[35,1303,37],{},[1299,1305,1306],{}," ⁣",[32,1308,1309,1311,1333],{},[110,1310,113],{"fence":46},[238,1312,1313,1323],{},[32,1314,1315,1317,1319,1321],{},[35,1316,11],{},[110,1318,113],{"stretchy":112},[35,1320,774],{},[110,1322,120],{"stretchy":112},[32,1324,1325,1327,1329,1331],{},[35,1326,944],{},[110,1328,113],{"stretchy":112},[35,1330,774],{},[110,1332,120],{"stretchy":112},[110,1334,120],{"fence":46},[35,1336,244],{},[35,1338,774],{},[110,1340,551],{"separator":46},[39,1342,1343],{"encoding":41},"\\begin{aligned}\nD_f(P\\|Q)\n&=\\int q(x)\\,\nf\\!\\left(\\frac{p(x)}{q(x)}\\right)dx,\n\\end{aligned}",[15,1345,1347],{"className":1346,"ariaHidden":46},[45],[15,1348,1350,1353],{"className":1349},[50],[15,1351],{"className":1352,"style":359},[54],[15,1354,1356],{"className":1355},[59],[15,1357,1359,1450],{"className":1358},[1245],[15,1360,1363],{"className":1361},[1362],"col-align-r",[15,1364,1366,1441],{"className":1365},[285,286],[15,1367,1369,1438],{"className":1368},[290],[15,1370,1373],{"className":1371,"style":1372},[294],"height:1.45em;",[15,1374,1376,1380],{"style":1375},"top:-3.45em;",[15,1377],{"className":1378,"style":1379},[302],"height:3.45em;",[15,1381,1383,1423,1426,1429,1432,1435],{"className":1382},[59],[15,1384,1386,1389],{"className":1385},[59],[15,1387,199],{"className":1388,"style":277},[59,60],[15,1390,1392],{"className":1391},[281],[15,1393,1395,1415],{"className":1394},[285,286],[15,1396,1398,1412],{"className":1397},[290],[15,1399,1401],{"className":1400,"style":295},[294],[15,1402,1403,1406],{"style":298},[15,1404],{"className":1405,"style":303},[302],[15,1407,1409],{"className":1408},[307,308,309,310],[15,1410,37],{"className":1411,"style":61},[59,60,310],[15,1413,318],{"className":1414},[317],[15,1416,1418],{"className":1417},[290],[15,1419,1421],{"className":1420,"style":325},[294],[15,1422],{},[15,1424,113],{"className":1425},[146],[15,1427,206],{"className":1428,"style":334},[59,60],[15,1430,210],{"className":1431},[59],[15,1433,213],{"className":1434},[59,60],[15,1436,120],{"className":1437},[153],[15,1439,318],{"className":1440},[317],[15,1442,1444],{"className":1443},[290],[15,1445,1448],{"className":1446,"style":1447},[294],"height:0.95em;",[15,1449],{},[15,1451,1454],{"className":1452},[1453],"col-align-l",[15,1455,1457,1630],{"className":1456},[285,286],[15,1458,1460,1627],{"className":1459},[290],[15,1461,1463],{"className":1462,"style":1372},[294],[15,1464,1465,1468],{"style":1375},[15,1466],{"className":1467,"style":1379},[302],[15,1469,1471,1474,1477,1480,1483,1490,1493,1496,1499,1502,1505,1508,1511,1515,1518,1615,1618,1621,1624],{"className":1470},[59],[15,1472],{"className":1473},[59],[15,1475],{"className":1476,"style":158},[157],[15,1478,123],{"className":1479},[162],[15,1481],{"className":1482,"style":158},[157],[15,1484,1289],{"className":1485,"style":1489},[1486,1487,1488],"mop","op-symbol","large-op","margin-right:0.4445em;position:relative;top:-0.0011em;",[15,1491],{"className":1492,"style":406},[157],[15,1494,944],{"className":1495,"style":981},[59,60],[15,1497,113],{"className":1498},[146],[15,1500,774],{"className":1501},[59,60],[15,1503,120],{"className":1504},[153],[15,1506],{"className":1507,"style":406},[157],[15,1509,37],{"className":1510,"style":61},[59,60],[15,1512],{"className":1513,"style":1514},[157],"margin-right:-0.1667em;",[15,1516],{"className":1517,"style":406},[157],[15,1519,1521,1527,1609],{"className":1520},[410],[15,1522,1524],{"className":1523,"style":415},[146,414],[15,1525,113],{"className":1526},[419,309],[15,1528,1530,1533,1606],{"className":1529},[59],[15,1531],{"className":1532},[146,441],[15,1534,1536],{"className":1535},[238],[15,1537,1539,1597],{"className":1538},[285,286],[15,1540,1542,1594],{"className":1541},[290],[15,1543,1546,1566,1574],{"className":1544,"style":1545},[294],"height:1.427em;",[15,1547,1548,1551],{"style":457},[15,1549],{"className":1550,"style":461},[302],[15,1552,1554,1557,1560,1563],{"className":1553},[59],[15,1555,944],{"className":1556,"style":981},[59,60],[15,1558,113],{"className":1559},[146],[15,1561,774],{"className":1562},[59,60],[15,1564,120],{"className":1565},[153],[15,1567,1568,1571],{"style":473},[15,1569],{"className":1570,"style":461},[302],[15,1572],{"className":1573,"style":481},[480],[15,1575,1576,1579],{"style":484},[15,1577],{"className":1578,"style":461},[302],[15,1580,1582,1585,1588,1591],{"className":1581},[59],[15,1583,11],{"className":1584},[59,60],[15,1586,113],{"className":1587},[146],[15,1589,774],{"className":1590},[59,60],[15,1592,120],{"className":1593},[153],[15,1595,318],{"className":1596},[317],[15,1598,1600],{"className":1599},[290],[15,1601,1604],{"className":1602,"style":1603},[294],"height:0.936em;",[15,1605],{},[15,1607],{"className":1608},[153,441],[15,1610,1612],{"className":1611,"style":415},[153,414],[15,1613,120],{"className":1614},[419,309],[15,1616],{"className":1617,"style":406},[157],[15,1619,244],{"className":1620},[59,60],[15,1622,774],{"className":1623},[59,60],[15,1625,551],{"className":1626},[573],[15,1628,318],{"className":1629},[317],[15,1631,1633],{"className":1632},[290],[15,1634,1636],{"className":1635,"style":1447},[294],[15,1637],{},[11,1639,1640,1641,1669],{},"更一般地写成关于 ",[15,1642,1644,1657],{"className":1643,"translate":19},[18],[15,1645,1647],{"className":1646},[23],[25,1648,1649],{"xmlns":27},[29,1650,1651,1655],{},[32,1652,1653],{},[35,1654,213],{},[39,1656,213],{"encoding":41},[15,1658,1660],{"className":1659,"ariaHidden":46},[45],[15,1661,1663,1666],{"className":1662},[50],[15,1664],{"className":1665,"style":566},[54],[15,1667,213],{"className":1668},[59,60]," 的期望形式",[15,1671,1673],{"className":1672,"translate":19},[180],[15,1674,1676,1810],{"className":1675,"translate":19},[18],[15,1677,1679],{"className":1678},[23],[25,1680,1681],{"xmlns":27,"display":189},[29,1682,1683,1807],{},[1245,1684,1685,1757],{"rowspacing":1247,"columnalign":1248,"columnspacing":1249},[1251,1686,1687,1709],{},[1254,1688,1689],{},[1257,1690,1691],{"scriptlevel":126,"displaystyle":46},[32,1692,1693,1699,1701,1703,1705,1707],{},[195,1694,1695,1697],{},[35,1696,199],{},[35,1698,37],{},[110,1700,113],{"stretchy":112},[35,1702,206],{},[35,1704,210],{"mathvariant":209},[35,1706,213],{},[110,1708,120],{"stretchy":112},[1254,1710,1711],{},[1257,1712,1713],{"scriptlevel":126,"displaystyle":46},[32,1714,1715,1717,1719,1725,1727],{},[32,1716],{},[110,1718,123],{},[195,1720,1721,1723],{},[35,1722,223],{"mathvariant":222},[35,1724,213],{},[1299,1726,1306],{},[32,1728,1729,1731,1733,1735,1755],{},[110,1730,230],{"fence":46},[35,1732,37],{},[1299,1734,1306],{},[32,1736,1737,1739,1753],{},[110,1738,113],{"fence":46},[238,1740,1741,1747],{},[32,1742,1743,1745],{},[35,1744,244],{},[35,1746,206],{},[32,1748,1749,1751],{},[35,1750,244],{},[35,1752,213],{},[110,1754,120],{"fence":46},[110,1756,257],{"fence":46},[1251,1758,1759,1765],{},[1254,1760,1761],{},[1257,1762,1763],{"scriptlevel":126,"displaystyle":46},[32,1764],{},[1254,1766,1767],{},[1257,1768,1769],{"scriptlevel":126,"displaystyle":46},[32,1770,1771,1773,1775,1777,1779,1781,1801,1803,1805],{},[32,1772],{},[110,1774,123],{},[110,1776,1289],{},[35,1778,37],{},[1299,1780,1306],{},[32,1782,1783,1785,1799],{},[110,1784,113],{"fence":46},[238,1786,1787,1793],{},[32,1788,1789,1791],{},[35,1790,244],{},[35,1792,206],{},[32,1794,1795,1797],{},[35,1796,244],{},[35,1798,213],{},[110,1800,120],{"fence":46},[35,1802,244],{},[35,1804,213],{},[110,1806,551],{"separator":46},[39,1808,1809],{"encoding":41},"\\begin{aligned}\nD_f(P\\|Q)\n&=\\mathbb E_Q\\!\\left[\nf\\!\\left(\\frac{dP}{dQ}\\right)\\right]\\\\\n&=\\int f\\!\\left(\\frac{dP}{dQ}\\right)dQ,\n\\end{aligned}",[15,1811,1813],{"className":1812,"ariaHidden":46},[45],[15,1814,1816,1820],{"className":1815},[50],[15,1817],{"className":1818,"style":1819},[54],"height:5.1001em;vertical-align:-2.3em;",[15,1821,1823],{"className":1822},[59],[15,1824,1826,1924],{"className":1825},[1245],[15,1827,1829],{"className":1828},[1362],[15,1830,1832,1915],{"className":1831},[285,286],[15,1833,1835,1912],{"className":1834},[290],[15,1836,1839,1903],{"className":1837,"style":1838},[294],"height:2.8em;",[15,1840,1842,1845],{"style":1841},"top:-4.8em;",[15,1843],{"className":1844,"style":1379},[302],[15,1846,1848,1888,1891,1894,1897,1900],{"className":1847},[59],[15,1849,1851,1854],{"className":1850},[59],[15,1852,199],{"className":1853,"style":277},[59,60],[15,1855,1857],{"className":1856},[281],[15,1858,1860,1880],{"className":1859},[285,286],[15,1861,1863,1877],{"className":1862},[290],[15,1864,1866],{"className":1865,"style":295},[294],[15,1867,1868,1871],{"style":298},[15,1869],{"className":1870,"style":303},[302],[15,1872,1874],{"className":1873},[307,308,309,310],[15,1875,37],{"className":1876,"style":61},[59,60,310],[15,1878,318],{"className":1879},[317],[15,1881,1883],{"className":1882},[290],[15,1884,1886],{"className":1885,"style":325},[294],[15,1887],{},[15,1889,113],{"className":1890},[146],[15,1892,206],{"className":1893,"style":334},[59,60],[15,1895,210],{"className":1896},[59],[15,1898,213],{"className":1899},[59,60],[15,1901,120],{"className":1902},[153],[15,1904,1906,1909],{"style":1905},"top:-2.1em;",[15,1907],{"className":1908,"style":1379},[302],[15,1910],{"className":1911},[59],[15,1913,318],{"className":1914},[317],[15,1916,1918],{"className":1917},[290],[15,1919,1922],{"className":1920,"style":1921},[294],"height:2.3em;",[15,1923],{},[15,1925,1927],{"className":1926},[1453],[15,1928,1930,2242],{"className":1929},[285,286],[15,1931,1933,2239],{"className":1932},[290],[15,1934,1936,2109],{"className":1935,"style":1838},[294],[15,1937,1938,1941],{"style":1841},[15,1939],{"className":1940,"style":1379},[302],[15,1942,1944,1947,1950,1953,1956,1996,1999,2002],{"className":1943},[59],[15,1945],{"className":1946},[59],[15,1948],{"className":1949,"style":158},[157],[15,1951,123],{"className":1952},[162],[15,1954],{"className":1955,"style":158},[157],[15,1957,1959,1962],{"className":1958},[59],[15,1960,223],{"className":1961},[59,366],[15,1963,1965],{"className":1964},[281],[15,1966,1968,1988],{"className":1967},[285,286],[15,1969,1971,1985],{"className":1970},[290],[15,1972,1974],{"className":1973,"style":379},[294],[15,1975,1976,1979],{"style":382},[15,1977],{"className":1978,"style":303},[302],[15,1980,1982],{"className":1981},[307,308,309,310],[15,1983,213],{"className":1984},[59,60,310],[15,1986,318],{"className":1987},[317],[15,1989,1991],{"className":1990},[290],[15,1992,1994],{"className":1993,"style":325},[294],[15,1995],{},[15,1997],{"className":1998,"style":1514},[157],[15,2000],{"className":2001,"style":406},[157],[15,2003,2005,2011,2014,2017,2020,2103],{"className":2004},[410],[15,2006,2008],{"className":2007,"style":415},[146,414],[15,2009,230],{"className":2010},[419,309],[15,2012,37],{"className":2013,"style":61},[59,60],[15,2015],{"className":2016,"style":1514},[157],[15,2018],{"className":2019,"style":406},[157],[15,2021,2023,2029,2097],{"className":2022},[410],[15,2024,2026],{"className":2025,"style":415},[146,414],[15,2027,113],{"className":2028},[419,309],[15,2030,2032,2035,2094],{"className":2031},[59],[15,2033],{"className":2034},[146,441],[15,2036,2038],{"className":2037},[238],[15,2039,2041,2086],{"className":2040},[285,286],[15,2042,2044,2083],{"className":2043},[290],[15,2045,2047,2061,2069],{"className":2046,"style":454},[294],[15,2048,2049,2052],{"style":457},[15,2050],{"className":2051,"style":461},[302],[15,2053,2055,2058],{"className":2054},[59],[15,2056,244],{"className":2057},[59,60],[15,2059,213],{"className":2060},[59,60],[15,2062,2063,2066],{"style":473},[15,2064],{"className":2065,"style":461},[302],[15,2067],{"className":2068,"style":481},[480],[15,2070,2071,2074],{"style":484},[15,2072],{"className":2073,"style":461},[302],[15,2075,2077,2080],{"className":2076},[59],[15,2078,244],{"className":2079},[59,60],[15,2081,206],{"className":2082,"style":334},[59,60],[15,2084,318],{"className":2085},[317],[15,2087,2089],{"className":2088},[290],[15,2090,2092],{"className":2091,"style":506},[294],[15,2093],{},[15,2095],{"className":2096},[153,441],[15,2098,2100],{"className":2099,"style":415},[153,414],[15,2101,120],{"className":2102},[419,309],[15,2104,2106],{"className":2105,"style":415},[153,414],[15,2107,257],{"className":2108},[419,309],[15,2110,2111,2114],{"style":1905},[15,2112],{"className":2113,"style":1379},[302],[15,2115,2117,2120,2123,2126,2129,2132,2135,2138,2141,2144,2227,2230,2233,2236],{"className":2116},[59],[15,2118],{"className":2119},[59],[15,2121],{"className":2122,"style":158},[157],[15,2124,123],{"className":2125},[1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",[15,2254,2256,2269],{"className":2255,"translate":19},[18],[15,2257,2259],{"className":2258},[23],[25,2260,2261],{"xmlns":27},[29,2262,2263,2267],{},[32,2264,2265],{},[35,2266,944],{},[39,2268,944],{"encoding":41},[15,2270,2272],{"className":2271,"ariaHidden":46},[45],[15,2273,2275,2279],{"className":2274},[50],[15,2276],{"className":2277,"style":2278},[54],"height:0.625em;vertical-align:-0.1944em;",[15,2280,944],{"className":2281,"style":981},[59,60]," 是 ",[15,2284,2286,2299],{"className":2285,"translate":19},[18],[15,2287,2289],{"className":2288},[23],[25,2290,2291],{"xmlns":27},[29,2292,2293,2297],{},[32,2294,2295],{},[35,2296,213],{},[39,2298,213],{"encoding":41},[15,2300,2302],{"className":2301,"ariaHidden":46},[45],[15,2303,2305,2308],{"className":2304},[50],[15,2306],{"className":2307,"style":566},[54],[15,2309,213],{"className":2310},[59,60]," 的密度。",[2313,2314,2316],"h3",{"id":2315},"_11-不依赖参考测度","1.1 不依赖参考测度",[11,2318,2319,2320,2351,2352,2382,2383,2411,2412,2440,2441,2609,2610,2747,2748,2885,2886,2950,2951,3156],{},"上面的定义里出现了「密度」，这会让读者误以为它依赖 Lebesgue 测度。其实 f-散度只依赖两个分布本身。取任一 ",[15,2321,2323,2338],{"className":2322,"translate":19},[18],[15,2324,2326],{"className":2325},[23],[25,2327,2328],{"xmlns":27},[29,2329,2330,2335],{},[32,2331,2332],{},[35,2333,2334],{},"σ",[39,2336,2337],{"encoding":41},"\\sigma",[15,2339,2341],{"className":2340,"ariaHidden":46},[45],[15,2342,2344,2348],{"className":2343},[50],[15,2345],{"className":2346,"style":2347},[54],"height:0.4306em;",[15,2349,2334],{"className":2350,"style":981},[59,60],"-有限测度 ",[15,2353,2355,2370],{"className":2354,"translate":19},[18],[15,2356,2358],{"className":2357},[23],[25,2359,2360],{"xmlns":27},[29,2361,2362,2367],{},[32,2363,2364],{},[35,2365,2366],{},"μ",[39,2368,2369],{"encoding":41},"\\mu",[15,2371,2373],{"className":2372,"ariaHidden":46},[45],[15,2374,2376,2379],{"className":2375},[50],[15,2377],{"className":2378,"style":2278},[54],[15,2380,2366],{"className":2381},[59,60]," 同时控制 ",[15,2384,2386,2399],{"className":2385,"translate":19},[18],[15,2387,2389],{"className":2388},[23],[25,2390,2391],{"xmlns":27},[29,2392,2393,2397],{},[32,2394,2395],{},[35,2396,206],{},[39,2398,206],{"encoding":41},[15,2400,2402],{"className":2401,"ariaHidden":46},[45],[15,2403,2405,2408],{"className":2404},[50],[15,2406],{"className":2407,"style":664},[54],[15,2409,206],{"className":2410,"style":334},[59,60]," 与 ",[15,2413,2415,2428],{"className":2414,"translate":19},[18],[15,2416,2418],{"className":2417},[23],[25,2419,2420],{"xmlns":27},[29,2421,2422,2426],{},[32,2423,2424],{},[35,2425,213],{},[39,2427,213],{"encoding":41},[15,2429,2431],{"className":2430,"ariaHidden":46},[45],[15,2432,2434,2437],{"className":2433},[50],[15,2435],{"className":2436,"style":566},[54],[15,2438,213],{"className":2439},[59,60],"（例如 ",[15,2442,2444,2480],{"className":2443,"translate":19},[18],[15,2445,2447],{"className":2446},[23],[25,2448,2449],{"xmlns":27},[29,2450,2451,2477],{},[32,2452,2453,2455,2457,2466,2468,2470,2473,2475],{},[35,2454,2366],{},[110,2456,123],{},[1257,2458,2459],{"scriptlevel":126,"displaystyle":112},[238,2460,2461,2463],{},[115,2462,117],{},[115,2464,2465],{},"2",[110,2467,113],{"stretchy":112},[35,2469,206],{},[110,2471,2472],{},"+",[35,2474,213],{},[110,2476,120],{"stretchy":112},[39,2478,2479],{"encoding":41},"\\mu=\\tfrac12(P+Q)",[15,2481,2483,2501,2597],{"className":2482,"ariaHidden":46},[45],[15,2484,2486,2489,2492,2495,2498],{"className":2485},[50],[15,2487],{"className":2488,"style":2278},[54],[15,2490,2366],{"className":2491},[59,60],[15,2493],{"className":2494,"style":158},[157],[15,2496,123],{"className":2497},[162],[15,2499],{"className":2500,"style":158},[157],[15,2502,2504,2508,2580,2583,2586,2590,2594],{"className":2503},[50],[15,2505],{"className":2506,"style":2507},[54],"height:1.1901em;vertical-align:-0.345em;",[15,2509,2511,2514,2577],{"className":2510},[59],[15,2512],{"className":2513},[146,441],[15,2515,2517],{"className":2516},[238],[15,2518,2520,2568],{"className":2519},[285,286],[15,2521,2523,2565],{"className":2522},[290],[15,2524,2527,2542,2550],{"className":2525,"style":2526},[294],"height:0.8451em;",[15,2528,2530,2533],{"style":2529},"top:-2.655em;",[15,2531],{"className":2532,"style":461},[302],[15,2534,2536],{"className":2535},[307,308,309,310],[15,2537,2539],{"className":2538},[59,310],[15,2540,2465],{"className":2541},[59,310],[15,2543,2544,2547],{"style":473},[15,2545],{"className":2546,"style":461},[302],[15,2548],{"className":2549,"style":481},[480],[15,2551,2553,2556],{"style":2552},"top:-3.394em;",[15,2554],{"className":2555,"style":461},[302],[15,2557,2559],{"className":2558},[307,308,309,310],[15,2560,2562],{"className":2561},[59,310],[15,2563,117],{"className":2564},[59,310],[15,2566,318],{"className":2567},[317],[15,2569,2571],{"className":2570},[290],[15,2572,2575],{"className":2573,"style":2574},[294],"height:0.345em;",[15,2576],{},[15,2578],{"className":2579},[153,441],[15,2581,113],{"className":2582},[146],[15,2584,206],{"className":2585,"style":334},[59,60],[15,2587],{"className":2588,"style":2589},[157],"margin-right:0.2222em;",[15,2591,2472],{"className":2592},[2593],"mbin",[15,2595],{"className":2596,"style":2589},[157],[15,2598,2600,2603,2606],{"className":2599},[50],[15,2601],{"className":2602,"style":139},[54],[15,2604,213],{"className":2605},[59,60],[15,2607,120],{"className":2608},[153],"），令 ",[15,2611,2613,2643],{"className":2612,"translate":19},[18],[15,2614,2616],{"className":2615},[23],[25,2617,2618],{"xmlns":27},[29,2619,2620,2640],{},[32,2621,2622,2624,2626],{},[35,2623,11],{},[110,2625,123],{},[238,2627,2628,2634],{},[32,2629,2630,2632],{},[35,2631,244],{},[35,2633,206],{},[32,2635,2636,2638],{},[35,2637,244],{},[35,2639,2366],{},[39,2641,2642],{"encoding":41},"p=\\frac{dP}{d\\mu}",[15,2644,2646,2664],{"className":2645,"ariaHidden":46},[45],[15,2647,2649,2652,2655,2658,2661],{"className":2648},[50],[15,2650],{"className":2651,"style":2278},[54],[15,2653,11],{"className":2654},[59,60],[15,2656],{"className":2657,"style":158},[157],[15,2659,123],{"className":2660},[162],[15,2662],{"className":2663,"style":158},[157],[15,2665,2667,2671],{"className":2666},[50],[15,2668],{"className":2669,"style":2670},[54],"height:1.3612em;vertical-align:-0.4811em;",[15,2672,2674,2677,2744],{"className":2673},[59],[15,2675],{"className":2676},[146,441],[15,2678,2680],{"className":2679},[238],[15,2681,2683,2735],{"className":2682},[285,286],[15,2684,2686,2732],{"className":2685},[290],[15,2687,2690,2707,2715],{"className":2688,"style":2689},[294],"height:0.8801em;",[15,2691,2692,2695],{"style":2529},[15,2693],{"className":2694,"style":461},[302],[15,2696,2698],{"className":2697},[307,308,309,310],[15,2699,2701,2704],{"className":2700},[59,310],[15,2702,244],{"className":2703},[59,60,310],[15,2705,2366],{"className":2706},[59,60,310],[15,2708,2709,2712],{"style":473},[15,2710],{"className":2711,"style":461},[302],[15,2713],{"className":2714,"style":481},[480],[15,2716,2717,2720],{"style":2552},[15,2718],{"className":2719,"style":461},[302],[15,2721,2723],{"className":2722},[307,308,309,310],[15,2724,2726,2729],{"className":2725},[59,310],[15,2727,244],{"className":2728},[59,60,310],[15,2730,206],{"className":2731,"style":334},[59,60,310],[15,2733,318],{"className":2734},[317],[15,2736,2738],{"className":2737},[290],[15,2739,2742],{"className":2740,"style":2741},[294],"height:0.4811em;",[15,2743],{},[15,2745],{"className":2746},[153,441],"，",[15,2749,2751,2781],{"className":2750,"translate":19},[18],[15,2752,2754],{"className":2753},[23],[25,2755,2756],{"xmlns":27},[29,2757,2758,2778],{},[32,2759,2760,2762,2764],{},[35,2761,944],{},[110,2763,123],{},[238,2765,2766,2772],{},[32,2767,2768,2770],{},[35,2769,244],{},[35,2771,213],{},[32,2773,2774,2776],{},[35,2775,244],{},[35,2777,2366],{},[39,2779,2780],{"encoding":41},"q=\\frac{dQ}{d\\mu}",[15,2782,2784,2802],{"className":2783,"ariaHidden":46},[45],[15,2785,2787,2790,2793,2796,2799],{"className":2786},[50],[15,2788],{"className":2789,"style":2278},[54],[15,2791,944],{"className":2792,"style":981},[59,60],[15,2794],{"className":2795,"style":158},[157],[15,2797,123],{"className":2798},[162],[15,2800],{"className":2801,"style":158},[157],[15,2803,2805,2809],{"className":2804},[50],[15,2806],{"className":2807,"style":2808},[54],"height:1.4133em;vertical-align:-0.4811em;",[15,2810,2812,2815,2882],{"className":2811},[59],[15,2813],{"className":2814},[146,441],[15,2816,2818],{"className":2817},[238],[15,2819,2821,2874],{"className":2820},[285,286],[15,2822,2824,2871],{"className":2823},[290],[15,2825,2828,2845,2853],{"className":2826,"style":2827},[294],"height:0.9322em;",[15,2829,2830,2833],{"style":2529},[15,2831],{"className":2832,"style":461},[302],[15,2834,2836],{"className":2835},[307,308,309,310],[15,2837,2839,2842],{"className":2838},[59,310],[15,2840,244],{"className":2841},[59,60,310],[15,2843,2366],{"className":2844},[59,60,310],[15,2846,2847,2850],{"style":473},[15,2848],{"className":2849,"style":461},[302],[15,2851],{"className":2852,"style":481},[480],[15,2854,2856,2859],{"style":2855},"top:-3.4461em;",[15,2857],{"className":2858,"style":461},[302],[15,2860,2862],{"className":2861},[307,308,309,310],[15,2863,2865,2868],{"className":2864},[59,310],[15,2866,244],{"className":2867},[59,60,310],[15,2869,213],{"className":2870},[59,60,310],[15,2872,318],{"className":2873},[317],[15,2875,2877],{"className":2876},[290],[15,2878,2880],{"className":2879,"style":2741},[294],[15,2881],{},[15,2883],{"className":2884},[153,441],"。在集合 ",[15,2887,2889,2914],{"className":2888,"translate":19},[18],[15,2890,2892],{"className":2891},[23],[25,2893,2894],{"xmlns":27},[29,2895,2896,2911],{},[32,2897,2898,2901,2903,2906,2908],{},[110,2899,2900],{"stretchy":112},"{",[35,2902,944],{},[110,2904,2905],{},">",[115,2907,126],{},[110,2909,2910],{"stretchy":112},"}",[39,2912,2913],{"encoding":41},"\\{q>0\\}",[15,2915,2917,2938],{"className":2916,"ariaHidden":46},[45],[15,2918,2920,2923,2926,2929,2932,2935],{"className":2919},[50],[15,2921],{"className":2922,"style":139},[54],[15,2924,2900],{"className":2925},[146],[15,2927,944],{"className":2928,"style":981},[59,60],[15,2930],{"className":2931,"style":158},[157],[15,2933,2905],{"className":2934},[162],[15,2936],{"className":2937,"style":158},[157],[15,2939,2941,2944,2947],{"className":2940},[50],[15,2942],{"className":2943,"style":139},[54],[15,2945,126],{"className":2946},[59],[15,2948,2910],{"className":2949},[153]," 上有 ",[15,2952,2954,2988],{"className":2953,"translate":19},[18],[15,2955,2957],{"className":2956},[23],[25,2958,2959],{"xmlns":27},[29,2960,2961,2985],{},[32,2962,2963,2977,2979],{},[238,2964,2965,2971],{},[32,2966,2967,2969],{},[35,2968,244],{},[35,2970,206],{},[32,2972,2973,2975],{},[35,2974,244],{},[35,2976,213],{},[110,2978,123],{},[238,2980,2981,2983],{},[35,2982,11],{},[35,2984,944],{},[39,2986,2987],{"encoding":41},"\\frac{dP}{dQ}=\\frac{p}{q}",[15,2989,2991,3080],{"className":2990,"ariaHidden":46},[45],[15,2992,2994,2997,3071,3074,3077],{"className":2993},[50],[15,2995],{"className":2996,"style":2670},[54],[15,2998,3000,3003,3068],{"className":2999},[59],[15,3001],{"className":3002},[146,441],[15,3004,3006],{"className":3005},[238],[15,3007,3009,3060],{"className":3008},[285,286],[15,3010,3012,3057],{"className":3011},[290],[15,3013,3015,3032,3040],{"className":3014,"style":2689},[294],[15,3016,3017,3020],{"style":2529},[15,3018],{"className":3019,"style":461},[302],[15,3021,3023],{"className":3022},[307,308,309,310],[15,3024,3026,3029],{"className":3025},[59,310],[15,3027,244],{"className":3028},[59,60,310],[15,3030,213],{"className":3031},[59,60,310],[15,3033,3034,3037],{"style":473},[15,3035],{"className":3036,"style":461},[302],[15,3038],{"className":3039,"style":481},[480],[15,3041,3042,3045],{"style":2552},[15,3043],{"className":3044,"style":461},[302],[15,3046,3048],{"className":3047},[307,308,309,310],[15,3049,3051,3054],{"className":3050},[59,310],[15,3052,244],{"className":3053},[59,60,310],[15,3055,206],{"className":3056,"style":334},[59,60,310],[15,3058,318],{"className":3059},[317],[15,3061,3063],{"className":3062},[290],[15,3064,3066],{"className":3065,"style":2741},[294],[15,3067],{},[15,3069],{"className":3070},[153,441],[15,3072],{"className":3073,"style":158},[157],[15,3075,123],{"className":3076},[162],[15,3078],{"className":3079,"style":158},[157],[15,3081,3083,3087],{"className":3082},[50],[15,3084],{"className":3085,"style":3086},[54],"height:1.2286em;vertical-align:-0.4811em;",[15,3088,3090,3093,3153],{"className":3089},[59],[15,3091],{"className":3092},[146,441],[15,3094,3096],{"className":3095},[238],[15,3097,3099,3145],{"className":3098},[285,286],[15,3100,3102,3142],{"className":3101},[290],[15,3103,3106,3120,3128],{"className":3104,"style":3105},[294],"height:0.7475em;",[15,3107,3108,3111],{"style":2529},[15,3109],{"className":3110,"style":461},[302],[15,3112,3114],{"className":3113},[307,308,309,310],[15,3115,3117],{"className":3116},[59,310],[15,3118,944],{"className":3119,"style":981},[59,60,310],[15,3121,3122,3125],{"style":473},[15,3123],{"className":3124,"style":461},[302],[15,3126],{"className":3127,"style":481},[480],[15,3129,3130,3133],{"style":2855},[15,3131],{"className":3132,"style":461},[302],[15,3134,3136],{"className":3135},[307,308,309,310],[15,3137,3139],{"className":3138},[59,310],[15,3140,11],{"className":3141},[59,60,310],[15,3143,318],{"className":3144},[317],[15,3146,3148],{"className":3147},[290],[15,3149,3151],{"className":3150,"style":2741},[294],[15,3152],{},[15,3154],{"className":3155},[153,441],"，于是",[15,3158,3160],{"className":3159,"translate":19},[180],[15,3161,3163,3313],{"className":3162,"translate":19},[18],[15,3164,3166],{"className":3165},[23],[25,3167,3168],{"xmlns":27,"display":189},[29,3169,3170,3310],{},[1245,3171,3172,3246],{"rowspacing":1247,"columnalign":1248,"columnspacing":1249},[1251,3173,3174,3196],{},[1254,3175,3176],{},[1257,3177,3178],{"scriptlevel":126,"displaystyle":46},[32,3179,3180,3186,3188,3190,3192,3194],{},[195,3181,3182,3184],{},[35,3183,199],{},[35,3185,37],{},[110,3187,113],{"stretchy":112},[35,3189,206],{},[35,3191,210],{"mathvariant":209},[35,3193,213],{},[110,3195,120],{"stretchy":112},[1254,3197,3198],{},[1257,3199,3200],{"scriptlevel":126,"displaystyle":46},[32,3201,3202,3204,3206,3222,3224,3226,3238,3240,3242,3244],{},[32,3203],{},[110,3205,123],{},[195,3207,3208,3210],{},[110,3209,1289],{},[32,3211,3212,3214,3216,3218,3220],{},[110,3213,2900],{"stretchy":112},[35,3215,944],{},[110,3217,2905],{},[115,3219,126],{},[110,3221,2910],{"stretchy":112},[35,3223,37],{},[1299,3225,1306],{},[32,3227,3228,3230,3236],{},[110,3229,113],{"fence":46},[238,3231,3232,3234],{},[35,3233,11],{},[35,3235,944],{},[110,3237,120],{"fence":46},[35,3239,944],{},[1299,3241,1301],{},[35,3243,244],{},[35,3245,2366],{},[1251,3247,3248,3254],{},[1254,3249,3250],{},[1257,3251,3252],{"scriptlevel":126,"displaystyle":46},[32,3253],{},[1254,3255,3256],{},[1257,3257,3258],{"scriptlevel":126,"displaystyle":46},[32,3259,3260,3262,3265,3267,3283,3285,3287,3299,3302,3304,3306,3308],{},[32,3261],{},[157,3263],{"width":3264},"2em",[110,3266,2472],{},[195,3268,3269,3271],{},[110,3270,1289],{},[32,3272,3273,3275,3277,3279,3281],{},[110,3274,2900],{"stretchy":112},[35,3276,944],{},[110,3278,123],{},[115,3280,126],{},[110,3282,2910],{"stretchy":112},[35,3284,37],{},[1299,3286,1306],{},[32,3288,3289,3291,3297],{},[110,3290,113],{"fence":46},[238,3292,3293,3295],{},[35,3294,11],{},[115,3296,126],{},[110,3298,120],{"fence":46},[110,3300,3301],{},"⋅",[115,3303,126],{},[1299,3305,1301],{},[35,3307,244],{},[35,3309,2366],{},[39,3311,3312],{"encoding":41},"\\begin{aligned}\nD_f(P\\|Q)\n&=\\int_{\\{q>0\\}}\nf\\!\\left(\\frac{p}{q}\\right)q\\,d\\mu\\\\\n&\\qquad+\\int_{\\{q=0\\}}\nf\\!\\left(\\frac{p}{0}\\right)\\cdot 0\\,d\\mu\n\\end{aligned}",[15,3314,3316],{"className":3315,"ariaHidden":46},[45],[15,3317,3319,3323],{"className":3318},[50],[15,3320],{"className":3321,"style":3322},[54],"height:5.2839em;vertical-align:-2.392em;",[15,3324,3326],{"className":3325},[59],[15,3327,3329,3427],{"className":3328},[1245],[15,3330,3332],{"className":3331},[1362],[15,3333,3335,3418],{"className":3334},[285,286],[15,3336,3338,3415],{"className":3337},[290],[15,3339,3342,3406],{"className":3340,"style":3341},[294],"height:2.892em;",[15,3343,3345,3348],{"style":3344},"top:-4.892em;",[15,3346],{"className":3347,"style":1379},[302],[15,3349,3351,3391,3394,3397,3400,3403],{"className":3350},[59],[15,3352,3354,3357],{"className":3353},[59],[15,3355,199],{"className":3356,"style":277},[59,60],[15,3358,3360],{"className":3359},[281],[15,3361,3363,3383],{"className":3362},[285,286],[15,3364,3366,3380],{"className":3365},[290],[15,3367,3369],{"className":3368,"style":295},[294],[15,3370,3371,3374],{"style":298},[15,3372],{"className":3373,"style":303},[302],[15,3375,3377],{"className":3376},[307,308,309,310],[15,3378,37],{"className":3379,"style":61},[59,60,310],[15,3381,318],{"className":3382},[317],[15,3384,3386],{"className":3385},[290],[15,3387,3389],{"className":3388,"style":325},[294],[15,3390],{},[15,3392,113],{"className":3393},[146],[15,3395,206],{"className":3396,"style":334},[59,60],[15,3398,210],{"className":3399},[59],[15,3401,213],{"className":3402},[59,60],[15,3404,120],{"className":3405},[153],[15,3407,3409,3412],{"style":3408},"top:-2.145em;",[15,3410],{"className":3411,"style":1379},[302],[15,3413],{"className":3414},[59],[15,3416,318],{"className":3417},[317],[15,3419,3421],{"className":3420},[290],[15,3422,3425],{"className":3423,"style":3424},[294],"height:2.392em;",[15,3426],{},[15,3428,3430],{"className":3429},[1453],[15,3431,3433,3816],{"className":3432},[285,286],[15,3434,3436,3813],{"className":3435},[290],[15,3437,3439,3622],{"className":3438,"style":3341},[294],[15,3440,3441,3444],{"style":3344},[15,3442],{"className":3443,"style":1379},[302],[15,3445,3447,3450,3453,3456,3459,3517,3520,3523,3526,3529,3607,3610,3613,3616,3619],{"className":3446},[59],[15,3448],{"className":3449},[59],[15,3451],{"className":3452,"style":158},[157],[15,3454,123],{"className":3455},[162],[15,3457],{"className":3458,"style":158},[157],[15,3460,3462,3465],{"className":3461},[1486],[15,3463,1289],{"className":3464,"style":1489},[1486,1487,1488],[15,3466,3468],{"className":3467},[281],[15,3469,3471,3508],{"className":3470},[285,286],[15,3472,3474,3505],{"className":3473},[290],[15,3475,3478],{"className":3476,"style":3477},[294],"height:-0.3869em;",[15,3479,3481,3484],{"style":3480},"top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;",[15,3482],{"className":3483,"style":303},[302],[15,3485,3487],{"className":3486},[307,308,309,310],[15,3488,3490,3493,3496,3499,3502],{"className":3489},[59,310],[15,3491,2900],{"className":3492},[146,310],[15,3494,944],{"className":3495,"style":981},[59,60,310],[15,3497,2905],{"className":3498},[162,310],[15,3500,126],{"className":3501},[59,310],[15,3503,2910],{"className":3504},[153,310],[15,3506,318],{"className":3507},[317],[15,3509,3511],{"className":3510},[290],[15,3512,3515],{"className":3513,"style":3514},[294],"height:1.0869em;",[15,3516],{},[15,3518],{"className":3519,"style":406},[157],[15,3521,37],{"className":3522,"style":61},[59,60],[15,3524],{"className":3525,"style":1514},[157],[15,3527],{"className":3528,"style":406},[157],[15,3530,3532,3538,3601],{"className":3531},[410],[15,3533,3535],{"className":3534,"style":415},[146,414],[15,3536,113],{"className":3537},[419,309],[15,3539,3541,3544,3598],{"className":3540},[59],[15,3542],{"className":3543},[146,441],[15,3545,3547],{"className":3546},[238],[15,3548,3550,3590],{"className":3549},[285,286],[15,3551,3553,3587],{"className":3552},[290],[15,3554,3557,3568,3576],{"className":3555,"style":3556},[294],"height:1.1076em;",[15,3558,3559,3562],{"style":457},[15,3560],{"className":3561,"style":461},[302],[15,3563,3565],{"className":3564},[59],[15,3566,944],{"className":3567,"style":981},[59,60],[15,3569,3570,3573],{"style":473},[15,3571],{"className":3572,"style":461},[302],[15,3574],{"className":3575,"style":481},[480],[15,3577,3578,3581],{"style":484},[15,3579],{"className":3580,"style":461},[302],[15,3582,3584],{"className":3583},[59],[15,3585,11],{"className":3586},[59,60],[15,3588,318],{"className":3589},[317],[15,3591,3593],{"className":3592},[290],[15,3594,3596],{"className":3595,"style":506},[294],[15,3597],{},[15,3599],{"className":3600},[153,441],[15,3602,3604],{"className":3603,"style":415},[153,414],[15,3605,120],{"className":3606},[419,309],[15,3608],{"className":3609,"style":406},[157],[15,3611,944],{"className":3612,"style":981},[59,60],[15,3614],{"className":3615,"style":406},[157],[15,3617,244],{"className":3618},[59,60],[15,3620,2366],{"className":3621},[59,60],[15,3623,3624,3627],{"style":3408},[15,3625],{"className":3626,"s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",[15,4005,4007,4025],{"className":4006,"translate":19},[18],[15,4008,4010],{"className":4009},[23],[25,4011,4012],{"xmlns":27},[29,4013,4014,4022],{},[32,4015,4016],{},[195,4017,4018,4020],{},[35,4019,199],{},[35,4021,37],{},[39,4023,4024],{"encoding":41},"D_f",[15,4026,4028],{"className":4027,"ariaHidden":46},[45],[15,4029,4031,4035],{"className":4030},[50],[15,4032],{"className":4033,"style":4034},[54],"height:0.9694em;vertical-align:-0.2861em;",[15,4036,4038,4041],{"className":4037},[59],[15,4039,199],{"className":4040,"style":277},[59,60],[15,4042,4044],{"className":4043},[281],[15,4045,4047,4067],{"className":4046},[285,286],[15,4048,4050,4064],{"className":4049},[290],[15,4051,4053],{"className":4052,"style":295},[294],[15,4054,4055,4058],{"style":298},[15,4056],{"className":4057,"style":303},[302],[15,4059,4061],{"className":4060},[307,308,309,310],[15,4062,37],{"className":4063,"style":61},[59,60,310],[15,4065,318],{"className":4066},[317],[15,4068,4070],{"className":4069},[290],[15,4071,4073],{"className":4072,"style":325},[294],[15,4074],{}," 是否有限。例如对 KL 的 ",[15,4077,4079,4111],{"className":4078,"translate":19},[18],[15,4080,4082],{"className":4081},[23],[25,4083,4084],{"xmlns":27},[29,4085,4086,4108],{},[32,4087,4088,4090,4092,4094,4096,4098,4100,4103,4106],{},[35,4089,37],{},[110,4091,113],{"stretchy":112},[35,4093,769],{},[110,4095,120],{"stretchy":112},[110,4097,123],{},[35,4099,769],{},[35,4101,4102],{},"log",[110,4104,4105],{},"⁡",[35,4107,769],{},[39,4109,4110],{"encoding":41},"f(t)=t\\log t",[15,4112,4114,4141],{"className":4113,"ariaHidden":46},[45],[15,4115,4117,4120,4123,4126,4129,4132,4135,4138],{"className":4116},[50],[15,4118],{"className":4119,"style":139},[54],[15,4121,37],{"className":4122,"style":61},[59,60],[15,4124,113],{"className":4125},[146],[15,4127,769],{"className":4128},[59,60],[15,4130,120],{"className":4131},[153],[15,4133],{"className":4134,"style":158},[157],[15,4136,123],{"className":4137},[162],[15,4139],{"className":4140,"style":158},[157],[15,4142,4144,4147,4150,4153,4161,4164],{"className":4143},[50],[15,4145],{"className":4146,"style":55},[54],[15,4148,769],{"className":4149},[59,60],[15,4151],{"className":4152,"style":406},[157],[15,4154,4156,4157],{"className":4155},[1486],"lo",[15,4158,4160],{"style":4159},"margin-right:0.0139em;","g",[15,4162],{"className":4163,"style":406},[157],[15,4165,769],{"className":4166},[59,60],"，当 ",[15,4169,4171,4184],{"className":4170,"translate":19},[18],[15,4172,4174],{"className":4173},[23],[25,4175,4176],{"xmlns":27},[29,4177,4178,4182],{},[32,4179,4180],{},[35,4181,206],{},[39,4183,206],{"encoding":41},[15,4185,4187],{"className":4186,"ariaHidden":46},[45],[15,4188,4190,4193],{"className":4189},[50],[15,4191],{"className":4192,"style":664},[54],[15,4194,206],{"className":4195,"style":334},[59,60],[15,4197,4199,4212],{"className":4198,"translate":19},[18],[15,4200,4202],{"className":4201},[23],[25,4203,4204],{"xmlns":27},[29,4205,4206,4210],{},[32,4207,4208],{},[35,4209,213],{},[39,4211,213],{"encoding":41},[15,4213,4215],{"className":4214,"ariaHidden":46},[45],[15,4216,4218,4221],{"className":4217},[50],[15,4219],{"className":4220,"style":566},[54],[15,4222,213],{"className":4223},[59,60]," 的零测集上放质量时这一项为 ",[15,4226,4228,4244],{"className":4227,"translate":19},[18],[15,4229,4231],{"className":4230},[23],[25,4232,4233],{"xmlns":27},[29,4234,4235,4241],{},[32,4236,4237,4239],{},[110,4238,2472],{},[35,4240,1027],{"mathvariant":209},[39,4242,4243],{"encoding":41},"+\\infty",[15,4245,4247],{"className":4246,"ariaHidden":46},[45],[15,4248,4250,4254,4257],{"className":4249},[50],[15,4251],{"className":4252,"style":4253},[54],"height:0.6667em;vertical-align:-0.0833em;",[15,4255,2472],{"className":4256},[59],[15,4258,1027],{"className":4259},[59],"。因此约定：",[15,4262,4264],{"className":4263,"translate":19},[180],[15,4265,4267,4381],{"className":4266,"translate":19},[18],[15,4268,4270],{"className":4269},[23],[25,4271,4272],{"xmlns":27,"display":189},[29,4273,4274,4378],{},[32,4275,4276,4278,4281,4283,4286,4289,4291,4297,4299,4301,4303,4305,4307,4309],{},[35,4277,206],{},[110,4279,4280],{},"≪̸",[35,4282,213],{},[1299,4284,4285],{}," ",[110,4287,4288],{},"⟹",[1299,4290,4285],{},[195,4292,4293,4295],{},[35,4294,199],{},[35,4296,37],{},[110,4298,113],{"stretchy":112},[35,4300,206],{},[35,4302,210],{"mathvariant":209},[35,4304,213],{},[110,4306,120],{"stretchy":112},[110,4308,123],{},[32,4310,4311,4313],{},[110,4312,2900],{"fence":46},[1245,4314,4318,4353],{"rowspacing":4315,"columnalign":4316,"columnspacing":4317},"0.36em","left left","1em",[1251,4319,4320,4332],{},[1254,4321,4322],{},[1257,4323,4324],{"scriptlevel":126,"displaystyle":112},[32,4325,4326,4328,4330],{},[110,4327,2472],{},[35,4329,1027],{"mathvariant":209},[110,4331,551],{"separator":46},[1254,4333,4334],{},[1257,4335,4336],{"scriptlevel":126,"displaystyle":112},[32,4337,4338,4341,4343,4346,4348,4351],{},[1299,4339,4340],{},"若 ",[35,4342,37],{},[1299,4344,4345],{}," 在 ",[35,4347,1027],{"mathvariant":209},[1299,4349,4350],{}," 处增长",[110,4352,551],{"separator":46},[1251,4354,4355,4366],{},[1254,4356,4357],{},[1257,4358,4359],{"scriptlevel":126,"displaystyle":112},[32,4360,4361,4364],{},[1299,4362,4363],{},"有限极限值",[110,4365,551],{"separator":46},[1254,4367,4368],{},[1257,4369,4370],{"scriptlevel":126,"displaystyle":112},[32,4371,4372,4375],{},[1299,4373,4374],{},"否则",[35,4376,4377],{"mathvariant":209},".",[39,4379,4380],{"encoding":41},"P\\not\\ll Q\\ \\Longrightarrow\\\nD_f(P\\|Q)=\n\\begin{cases}\n+\\infty, & \\text{若 } f \\text{ 在 } \\infty\n\\text{ 处增长},\\\\[2pt]\n\\text{有限极限值}, & \\text{否则}.\n\\end{cases}",[15,4382,4384,4439,4463,4533],{"className":4383,"ariaHidden":46},[45],[15,4385,4387,4390,4393,4396,4429,4433,4436],{"className":4386},[50],[15,4388],{"className":4389,"style":55},[54],[15,4391,206],{"className":4392,"style":334},[59,60],[15,4394],{"className":4395,"style":158},[157],[15,4397,4399],{"className":4398},[162],[15,4400,4403],{"className":4401},[59,4402],"vbox",[15,4404,4407],{"className":4405},[4406],"thinbox",[15,4408,4411,4414,4425],{"className":4409},[4410],"rlap",[15,4412],{"className":4413,"style":55},[54],[15,4415,4418],{"className":4416},[4417],"inner",[15,4419,4421],{"className":4420},[59],[15,4422,4424],{"className":4423},[162],"",[15,4426],{"className":4427},[4428],"fix",[15,4430],{"className":4431},[157,4432],"nobreak",[15,4434,714],{"className":4435},[162],[15,4437],{"className":4438,"style":158},[157],[15,4440,4442,4445,4448,4451,4454,4457,4460],{"className":4441},[50],[15,4443],{"className":4444,"style":566},[54],[15,4446,213],{"className":4447},[59,60],[15,4449,4285],{"className":4450},[157],[15,4452],{"className":4453,"style":158},[157],[15,4455,4288],{"className":4456},[162],[15,4458,4285],{"className":4459},[157],[15,4461],{"className":4462,"style":158},[157],[15,4464,4466,4469,4509,4512,4515,4518,4521,4524,4527,4530],{"className":4465},[50],[15,4467],{"className":4468,"style":270},[54],[15,4470,4472,4475],{"className":4471},[59],[15,4473,199],{"className":4474,"style":277},[59,60],[15,4476,4478],{"className":4477},[281],[15,4479,4481,4501],{"className":4480},[285,286],[15,4482,4484,4498],{"className":4483},[290],[15,4485,4487],{"className":4486,"style":295},[294],[15,4488,4489,4492],{"style":298},[15,4490],{"className":4491,"style":303},[302],[15,4493,4495],{"className":4494},[307,308,309,310],[15,4496,37],{"className":4497,"style":61},[59,60,310],[15,4499,318],{"className":4500},[317],[15,4502,4504],{"className":4503},[290],[15,4505,4507],{"className":4506,"style":325},[294],[15,4508],{},[15,4510,113],{"className":4511},[146],[15,4513,206],{"className":4514,"style":334},[59,60],[15,4516,210],{"className":4517},[59],[15,4519,213],{"className":4520},[59,60],[15,4522,120],{"className":4523},[153],[15,4525],{"className":4526,"style":158},[157],[15,4528,123],{"className":4529},[162],[15,4531],{"className":4532,"style":158},[157],[15,4534,4536,4540],{"className":4535},[50],[15,4537],{"className":4538,"style":4539},[54],"height:3.08em;vertical-align:-1.29em;",[15,4541,4543,4550,4715],{"className":4542},[410],[15,4544,4546],{"className":4545,"style":415},[146,414],[15,4547,2900],{"className":4548},[419,4549],"size4",[15,4551,4553],{"className":4552},[59],[15,4554,4556,4620,4625],{"className":4555},[1245],[15,4557,4559],{"className":4558},[1453],[15,4560,4562,4611],{"className":4561},[285,286],[15,4563,4565,4608],{"className":4564},[290],[15,4566,4569,4588],{"className":4567,"style":4568},[294],"height:1.79em;",[15,4570,4572,4576],{"style":4571},"top:-3.79em;",[15,4573],{"className":4574,"style":4575},[302],"height:3.008em;",[15,4577,4579,4582,4585],{"className":4578},[59],[15,4580,2472],{"className":4581},[59],[15,4583,1027],{"className":4584},[59],[15,4586,551],{"className":4587},[573],[15,4589,4591,4594],{"style":4590},"top:-2.15em;",[15,4592],{"className":4593,"style":4575},[302],[15,4595,4597,4605],{"className":4596},[59],[15,4598,4601],{"className":4599},[59,4600],"text",[15,4602,4363],{"className":4603},[59,4604],"cjk_fallback",[15,4606,551],{"className":4607},[573],[15,4609,318],{"className":4610},[317],[15,4612,4614],{"className":4613},[290],[15,4615,4618],{"className":4616,"style":4617},[294],"height:1.29em;",[15,4619],{},[15,4621],{"className":4622,"style":4624},[4623],"arraycolsep","width:1em;",[15,4626,4628],{"className":4627},[1453],[15,4629,4631,4707],{"className":4630},[285,286],[15,4632,4634,4704],{"className":4633},[290],[15,4635,4637,4687],{"className":4636,"style":4568},[294],[15,4638,4639,4642],{"style":4571},[15,4640],{"className":4641,"style":4575},[302],[15,4643,4645,4655,4658,4671,4674,4684],{"className":4644},[59],[15,4646,4648,4652],{"className":4647},[59,4600],[15,4649,4651],{"className":4650},[59,4604],"若",[15,4653,4285],{"className":4654},[59],[15,4656,37],{"className":4657,"style":61},[59,60],[15,4659,4661,4664,4668],{"className":4660},[59,4600],[15,4662,4285],{"className":4663},[59],[15,4665,4667],{"className":4666},[59,4604],"在",[15,4669,4285],{"className":4670},[59],[15,4672,1027],{"className":4673},[59],[15,4675,4677,4680],{"className":4676},[59,4600],[15,4678,4285],{"className":4679},[59],[15,4681,4683],{"className":4682},[59,4604],"处增长",[15,4685,551],{"className":4686},[573],[15,4688,4689,4692],{"style":4590},[15,4690],{"className":4691,"style":4575},[302],[15,4693,4695,4701],{"className":4694},[59],[15,4696,4698],{"className":4697},[59,4600],[15,4699,4374],{"className":4700},[59,4604],[15,4702,4377],{"className":4703},[59],[15,4705,318],{"className":4706},[317],[15,4708,4710],{"className":4709},[290],[15,4711,4713],{"className":4712,"style":4617},[294],[15,4714],{},[15,4716],{"className":4717},[153,441],[11,4719,4720,4721,3855,4749,4777,4778,4954,4955,5024,5025,5075,5076,5145,5146,5203],{},"其中「否则」指 ",[15,4722,4724,4737],{"className":4723,"translate":19},[18],[15,4725,4727],{"className":4726},[23],[25,4728,4729],{"xmlns":27},[29,4730,4731,4735],{},[32,4732,4733],{},[35,4734,37],{},[39,4736,37],{"encoding":41},[15,4738,4740],{"className":4739,"ariaHidden":46},[45],[15,4741,4743,4746],{"className":4742},[50],[15,4744],{"className":4745,"style":55},[54],[15,4747,37],{"className":4748,"style":61},[59,60],[15,4750,4752,4765],{"className":4751,"translate":19},[18],[15,4753,4755],{"className":4754},[23],[25,4756,4757],{"xmlns":27},[29,4758,4759,4763],{},[32,4760,4761],{},[35,4762,1027],{"mathvariant":209},[39,4764,3872],{"encoding":41},[15,4766,4768],{"className":4767,"ariaHidden":46},[45],[15,4769,4771,4774],{"className":4770},[50],[15,4772],{"className":4773,"style":2347},[54],[15,4775,1027],{"className":4776},[59]," 处有界（如 ",[15,4779,4781,4823],{"className":4780,"translate":19},[18],[15,4782,4784],{"className":4783},[23],[25,4785,4786],{"xmlns":27},[29,4787,4788,4820],{},[32,4789,4790,4792,4794,4796,4798,4800,4808,4811,4813,4816,4818],{},[35,4791,37],{},[110,4793,113],{"stretchy":112},[35,4795,769],{},[110,4797,120],{"stretchy":112},[110,4799,123],{},[1257,4801,4802],{"scriptlevel":126,"displaystyle":112},[238,4803,4804,4806],{},[115,4805,117],{},[115,4807,2465],{},[110,4809,4810],{"stretchy":112},"∣",[35,4812,769],{},[110,4814,4815],{},"−",[115,4817,117],{},[110,4819,4810],{"stretchy":112},[39,4821,4822],{"encoding":41},"f(t)=\\tfrac12\\lvert t-1\\rvert",[15,4824,4826,4853,4942],{"className":4825,"ariaHidden":46},[45],[15,4827,4829,4832,4835,4838,4841,4844,4847,4850],{"className":4828},[50],[15,4830],{"className":4831,"style":139},[54],[15,4833,37],{"className":4834,"style":61},[59,60],[15,4836,113],{"className":4837},[146],[15,4839,769],{"className":4840},[59,60],[15,4842,120],{"className":4843},[153],[15,4845],{"className":4846,"style":158},[157],[15,4848,123],{"className":4849},[162],[15,4851],{"className":4852,"style":158},[157],[15,4854,4856,4859,4927,4930,4933,4936,4939],{"className":4855},[50],[15,4857],{"className":4858,"style":2507},[54],[15,4860,4862,4865,4924],{"className":4861},[59],[15,4863],{"className":4864},[146,441],[15,4866,4868],{"className":4867},[238],[15,4869,4871,4916],{"className":4870},[285,286],[15,4872,4874,4913],{"className":4873},[290],[15,4875,4877,4891,4899],{"className":4876,"style":2526},[294],[15,4878,4879,4882],{"style":2529},[15,4880],{"className":4881,"style":461},[302],[15,4883,4885],{"className":4884},[307,308,309,310],[15,4886,4888],{"className":4887},[59,310],[15,4889,2465],{"className":4890},[59,310],[15,4892,4893,4896],{"style":473},[15,4894],{"className":4895,"style":461},[302],[15,4897],{"className":4898,"style":481},[480],[15,4900,4901,4904],{"style":2552},[15,4902],{"className":4903,"style":461},[302],[15,4905,4907],{"className":4906},[307,308,309,310],[15,4908,4910],{"className":4909},[59,310],[15,4911,117],{"className":4912},[59,310],[15,4914,318],{"className":4915},[317],[15,4917,4919],{"className":4918},[290],[15,4920,4922],{"className":4921,"style":2574},[294],[15,4923],{},[15,4925],{"className":4926},[153,441],[15,4928,4810],{"className":4929},[146],[15,4931,769],{"className":4932},[59,60],[15,4934],{"className":4935,"style":2589},[157],[15,4937,4815],{"className":4938},[2593],[15,4940],{"className":4941,"style":2589},[157],[15,4943,4945,4948,4951],{"className":4944},[50],[15,4946],{"className":4947,"style":139},[54],[15,4949,117],{"className":4950},[59],[15,4952,4810],{"className":4953},[153],"），此时 ",[15,4956,4958,4975],{"className":4957,"translate":19},[18],[15,4959,4961],{"className":4960},[23],[25,4962,4963],{"xmlns":27},[29,4964,4965,4973],{},[32,4966,4967],{},[195,4968,4969,4971],{},[35,4970,199],{},[35,4972,37],{},[39,4974,4024],{"encoding":41},[15,4976,4978],{"className":4977,"ariaHidden":46},[45],[15,4979,4981,4984],{"className":4980},[50],[15,4982],{"className":4983,"style":4034},[54],[15,4985,4987,4990],{"className":4986},[59],[15,4988,199],{"className":4989,"style":277},[59,60],[15,4991,4993],{"className":4992},[281],[15,4994,4996,5016],{"className":4995},[285,286],[15,4997,4999,5013],{"className":4998},[290],[15,5000,5002],{"className":5001,"style":295},[294],[15,5003,5004,5007],{"style":298},[15,5005],{"className":5006,"style":303},[302],[15,5008,5010],{"className":5009},[307,308,309,310],[15,5011,37],{"className":5012,"style":61},[59,60,310],[15,5014,318],{"className":5015},[317],[15,5017,5019],{"className":5018},[290],[15,5020,5022],{"className":5021,"style":325},[294],[15,5023],{}," 取有限极限值。下文如无特别说明，默认 ",[15,5026,5028,5045],{"className":5027,"translate":19},[18],[15,5029,5031],{"className":5030},[23],[25,5032,5033],{"xmlns":27},[29,5034,5035,5043],{},[32,5036,5037,5039,5041],{},[35,5038,206],{},[110,5040,714],{},[35,5042,213],{},[39,5044,719],{"encoding":41},[15,5046,5048,5066],{"className":5047,"ariaHidden":46},[45],[15,5049,5051,5054,5057,5060,5063],{"className":5050},[50],[15,5052],{"className":5053,"style":729},[54],[15,5055,206],{"className":5056,"style":334},[59,60],[15,5058],{"className":5059,"style":158},[157],[15,5061,714],{"className":5062},[162],[15,5064],{"className":5065,"style":158},[157],[15,5067,5069,5072],{"className":5068},[50],[15,5070],{"className":5071,"style":566},[54],[15,5073,213],{"className":5074},[59,60],"，并把 ",[15,5077,5079,5096],{"className":5078,"translate":19},[18],[15,5080,5082],{"className":5081},[23],[25,5083,5084],{"xmlns":27},[29,5085,5086,5094],{},[32,5087,5088],{},[195,5089,5090,5092],{},[35,5091,199],{},[35,5093,37],{},[39,5095,4024],{"encoding":41},[15,5097,5099],{"className":5098,"ariaHidden":46},[45],[15,5100,5102,5105],{"className":5101},[50],[15,5103],{"className":5104,"style":4034},[54],[15,5106,5108,5111],{"className":5107},[59],[15,5109,199],{"className":5110,"style":277},[59,60],[15,5112,5114],{"className":5113},[281],[15,5115,5117,5137],{"className":5116},[285,286],[15,5118,5120,5134],{"className":5119},[290],[15,5121,5123],{"className":5122,"style":295},[294],[15,5124,5125,5128],{"style":298},[15,5126],{"className":5127,"style":303},[302],[15,5129,5131],{"className":5130},[307,308,309,310],[15,5132,37],{"className":5133,"style":61},[59,60,310],[15,5135,318],{"className":5136},[317],[15,5138,5140],{"className":5139},[290],[15,5141,5143],{"className":5142,"style":325},[294],[15,5144],{}," 看成 ",[15,5147,5149,5173],{"className":5148,"translate":19},[18],[15,5150,5152],{"className":5151},[23],[25,5153,5154],{"xmlns":27},[29,5155,5156,5170],{},[32,5157,5158,5160,5162,5164,5166,5168],{},[110,5159,230],{"stretchy":112},[115,5161,126],{},[110,5163,551],{"separator":46},[110,5165,2472],{},[35,5167,1027],{"mathvariant":209},[110,5169,257],{"stretchy":112},[39,5171,5172],{"encoding":41},"[0,+\\infty]",[15,5174,5176],{"className":5175,"ariaHidden":46},[45],[15,5177,5179,5182,5185,5188,5191,5194,5197,5200],{"className":5178},[50],[15,5180],{"className":5181,"style":139},[54],[15,5183,230],{"className":5184},[146],[15,5186,126],{"className":5187},[59],[15,5189,551],{"className":5190},[573],[15,5192],{"className":5193,"style":406},[157],[15,5195,2472],{"className":5196},[59],[15,5198,1027],{"className":5199},[59],[15,5201,257],{"className":5202},[153]," 中的值。",[2313,5205,5207],{"id":5206},"_12-只依赖似然比","1.2 只依赖似然比",[11,5209,5210,5211,5306,5307,5379,5380,5457,5458,5486,5487,5529],{},"由定义立刻看出：",[15,5212,5214,5242],{"className":5213,"translate":19},[18],[15,5215,5217],{"className":5216},[23],[25,5218,5219],{"xmlns":27},[29,5220,5221,5239],{},[32,5222,5223,5229,5231,5233,5235,5237],{},[195,5224,5225,5227],{},[35,5226,199],{},[35,5228,37],{},[110,5230,113],{"stretchy":112},[35,5232,206],{},[35,5234,210],{"mathvariant":209},[35,5236,213],{},[110,5238,120],{"stretchy":112},[39,5240,5241],{"encoding":41},"D_f(P\\|Q)",[15,5243,5245],{"className":5244,"ariaHidden":46},[45],[15,5246,5248,5251,5291,5294,5297,5300,5303],{"className":5247},[50],[15,5249],{"className":5250,"style":270},[54],[15,5252,5254,5257],{"className":5253},[59],[15,5255,199],{"className":5256,"style":277},[59,60],[15,5258,5260],{"className":5259},[281],[15,5261,5263,5283],{"className":5262},[285,286],[15,5264,5266,5280],{"className":5265},[290],[15,5267,5269],{"className":5268,"style":295},[294],[15,5270,5271,5274],{"style":298},[15,5272],{"className":5273,"style":303},[302],[15,5275,5277],{"className":5276},[307,308,309,310],[15,5278,37],{"className":5279,"style":61},[59,60,310],[15,5281,318],{"className":5282},[317],[15,5284,5286],{"className":5285},[290],[15,5287,5289],{"className":5288,"style":325},[294],[15,5290],{},[15,5292,113],{"className":5293},[146],[15,5295,206],{"className":5296,"style":334},[59,60],[15,5298,210],{"className":5299},[59],[15,5301,213],{"className":5302},[59,60],[15,5304,120],{"className":5305},[153]," 只通过似然比 ",[15,5308,5310,5336],{"className":5309,"translate":19},[18],[15,5311,5313],{"className":5312},[23],[25,5314,5315],{"xmlns":27},[29,5316,5317,5333],{},[32,5318,5319,5321,5323,5325,5327,5329,5331],{},[35,5320,769],{},[110,5322,123],{},[35,5324,244],{},[35,5326,206],{},[35,5328,941],{"mathvariant":209},[35,5330,244],{},[35,5332,213],{},[39,5334,5335],{"encoding":41},"t=dP\u002FdQ",[15,5337,5339,5358],{"className":5338,"ariaHidden":46},[45],[15,5340,5342,5346,5349,5352,5355],{"className":5341},[50],[15,5343],{"className":5344,"style":5345},[54],"height:0.6151em;",[15,5347,769],{"className":5348},[59,60],[15,5350],{"className":5351,"style":158},[157],[15,5353,123],{"className":5354},[162],[15,5356],{"className":5357,"style":158},[157],[15,5359,5361,5364,5367,5370,5373,5376],{"className":5360},[50],[15,5362],{"className":5363,"style":139},[54],[15,5365,244],{"className":5366},[59,60],[15,5368,206],{"className":5369,"style":334},[59,60],[15,5371,941],{"className":5372},[59],[15,5374,244],{"className":5375},[59,60],[15,5377,213],{"className":5378},[59,60]," 起作用。也就是说，对任意可测映射 ",[15,5381,5383,5407],{"className":5382,"translate":19},[18],[15,5384,5386],{"className":5385},[23],[25,5387,5388],{"xmlns":27},[29,5389,5390,5404],{},[32,5391,5392,5395,5397,5399,5401],{},[35,5393,5394],{},"T",[110,5396,1018],{},[35,5398,598],{"mathvariant":597},[110,5400,1032],{},[35,5402,5403],{"mathvariant":597},"Y",[39,5405,5406],{"encoding":41},"T:\\mathcal X\\to\\mathcal Y",[15,5408,5410,5428,5446],{"className":5409,"ariaHidden":46},[45],[15,5411,5413,5416,5419,5422,5425],{"className":5412},[50],[15,5414],{"className":5415,"style":664},[54],[15,5417,5394],{"className":5418,"style":334},[59,60],[15,5420],{"className":5421,"style":158},[157],[15,5423,1018],{"className":5424},[162],[15,5426],{"className":5427,"style":158},[157],[15,5429,5431,5434,5437,5440,5443],{"className":5430},[50],[15,5432],{"className":5433,"style":664},[54],[15,5435,598],{"className":5436,"style":625},[59,624],[15,5438],{"className":5439,"style":158},[157],[15,5441,1032],{"className":5442},[162],[15,5444],{"className":5445,"style":158},[157],[15,5447,5449,5453],{"className":5448},[50],[15,5450],{"className":5451,"style":5452},[54],"height:0.7805em;vertical-align:-0.0972em;",[15,5454,5403],{"className":5455,"style":5456},[59,624],"margin-right:0.0822em;","，它是 ",[15,5459,5461,5474],{"className":5460,"translate":19},[18],[15,5462,5464],{"className":5463},[23],[25,5465,5466],{"xmlns":27},[29,5467,5468,5472],{},[32,5469,5470],{},[35,5471,769],{},[39,5473,769],{"encoding":41},[15,5475,5477],{"className":5476,"ariaHidden":46},[45],[15,5478,5480,5483],{"className":5479},[50],[15,5481],{"className":5482,"style":5345},[54],[15,5484,769],{"className":5485},[59,60]," 的函数，而不是 ",[15,5488,5490,5508],{"className":5489,"translate":19},[18],[15,5491,5493],{"className":5492},[23],[25,5494,5495],{"xmlns":27},[29,5496,5497,5505],{},[32,5498,5499,5501,5503],{},[35,5500,11],{},[110,5502,551],{"separator":46},[35,5504,944],{},[39,5506,5507],{"encoding":41},"p,q",[15,5509,5511],{"className":5510,"ariaHidden":46},[45],[15,5512,5514,5517,5520,5523,5526],{"className":5513},[50],[15,5515],{"className":5516,"style":2278},[54],[15,5518,11],{"className":5519},[59,60],[15,5521,551],{"className":5522},[573],[15,5524],{"className":5525,"style":406},[157],[15,5527,944],{"className":5528,"style":981},[59,60]," 各自的函数。",[11,5531,5532,5533,5572],{},"这一点是 f-散度「缺少几何」的根源，也是它与 Wasserstein 距离分道扬镳的地方（见第 6 节）：只要保持比值 ",[15,5534,5536,5554],{"className":5535,"translate":19},[18],[15,5537,5539],{"className":5538},[23],[25,5540,5541],{"xmlns":27},[29,5542,5543,5551],{},[32,5544,5545,5547,5549],{},[35,5546,11],{},[35,5548,941],{"mathvariant":209},[35,5550,944],{},[39,5552,5553],{"encoding":41},"p\u002Fq",[15,5555,5557],{"className":5556,"ariaHidden":46},[45],[15,5558,5560,5563,5566,5569],{"className":5559},[50],[15,5561],{"className":5562,"style":139},[54],[15,5564,11],{"className":5565},[59,60],[15,5567,941],{"className":5568},[59],[15,5570,944],{"className":5571,"style":981},[59,60]," 不变，把两个分布同时「搬到别处」，f-散度的值完全不动。",[2313,5574,5576],{"id":5575},"_13-非单射与仿射自由度","1.3 非单射与仿射自由度",[11,5578,5579,5580,5647,5648,5712],{},"注意 ",[15,5581,5583,5605],{"className":5582,"translate":19},[18],[15,5584,5586],{"className":5585},[23],[25,5587,5588],{"xmlns":27},[29,5589,5590,5602],{},[32,5591,5592,5594,5596,5598,5600],{},[110,5593,1289],{},[35,5595,244],{},[35,5597,213],{},[110,5599,123],{},[115,5601,117],{},[39,5603,5604],{"encoding":41},"\\int dQ=1",[15,5606,5608,5638],{"className":5607,"ariaHidden":46},[45],[15,5609,5611,5615,5620,5623,5626,5629,5632,5635],{"className":5610},[50],[15,5612],{"className":5613,"style":5614},[54],"height:1.1111em;vertical-align:-0.3061em;",[15,5616,1289],{"className":5617,"style":5619},[1486,1487,5618],"small-op","margin-right:0.1945em;position:relative;top:-0.0006em;",[15,5621],{"className":5622,"style":406},[157],[15,5624,244],{"className":5625},[59,60],[15,5627,213],{"className":5628},[59,60],[15,5630],{"className":5631,"style":158},[157],[15,5633,123],{"className":5634},[162],[15,5636],{"className":5637,"style":158},[157],[15,5639,5641,5644],{"className":5640},[50],[15,5642],{"className":5643,"style":172},[54],[15,5645,117],{"className":5646},[59],"、",[15,5649,5651,5673],{"className":5650,"translate":19},[18],[15,5652,5654],{"className":5653},[23],[25,5655,5656],{"xmlns":27},[29,5657,5658,5670],{},[32,5659,5660,5662,5664,5666,5668],{},[110,5661,1289],{},[35,5663,244],{},[35,5665,206],{},[110,5667,123],{},[115,5669,117],{},[39,5671,5672],{"encoding":41},"\\int dP=1",[15,5674,5676,5703],{"className":5675,"ariaHidden":46},[45],[15,5677,5679,5682,5685,5688,5691,5694,5697,5700],{"className":5678},[50],[15,5680],{"className":5681,"style":5614},[54],[15,5683,1289],{"className":5684,"style":5619},[1486,1487,5618],[15,5686],{"className":5687,"style":406},[157],[15,5689,244],{"className":5690},[59,60],[15,5692,206],{"className":5693,"style":334},[59,60],[15,5695],{"className":5696,"style":158},[157],[15,5698,123],{"className":5699},[162],[15,5701],{"className":5702,"style":158},[157],[15,5704,5706,5709],{"className":5705},[50],[15,5707],{"className":5708,"style":172},[54],[15,5710,117],{"className":5711},[59],"，因此",[15,5714,5716],{"className":5715,"translate":19},[180],[15,5717,5719,5770],{"className":5718,"translate":19},[18],[15,5720,5722],{"className":5721},[23],[25,5723,5724],{"xmlns":27,"display":189},[29,5725,5726,5767],{},[32,5727,5728,5730,5732,5734,5736,5738,5740,5742,5744,5746,5748,5750,5752,5754,5756,5758,5760,5762,5764],{},[110,5729,1289],{},[110,5731,113],{"stretchy":112},[35,5733,769],{},[110,5735,4815],{},[115,5737,117],{},[110,5739,120],{"stretchy":112},[1299,5741,1301],{},[35,5743,244],{},[35,5745,213],{},[110,5747,123],{},[110,5749,1289],{},[35,5751,244],{},[35,5753,206],{},[110,5755,4815],{},[110,5757,1289],{},[35,5759,244],{},[35,5761,213],{},[110,5763,123],{},[115,5765,5766],{},"0.",[39,5768,5769],{"encoding":41},"\\int (t-1)\\,dQ=\\int dP-\\int dQ=0 .",[15,5771,5773,5798,5828,5855,5882],{"className":5772,"ariaHidden":46},[45],[15,5774,5776,5780,5783,5786,5789,5792,5795],{"className":5775},[50],[15,5777],{"className":5778,"style":5779},[54],"height:2.2222em;vertical-align:-0.8622em;",[15,5781,1289],{"className":5782,"style":1489},[1486,1487,1488],[15,5784,113],{"className":5785},[146],[15,5787,769],{"className":5788},[59,60],[15,5790],{"className":5791,"style":2589},[157],[15,5793,4815],{"className":5794},[2593],[15,5796],{"className":5797,"style":2589},[157],[15,5799,5801,5804,5807,5810,5813,5816,5819,5822,5825],{"className":5800},[50],[15,5802],{"className":5803,"style":139},[54],[15,5805,117],{"className":5806},[59],[15,5808,120],{"className":5809},[153],[15,5811],{"className":5812,"style":406},[157],[15,5814,244],{"className":5815},[59,60],[15,5817,213],{"className":5818},[59,60],[15,5820],{"className":5821,"style":158},[157],[15,5823,123],{"className":5824},[162],[15,5826],{"className":5827,"style":158},[157],[15,5829,5831,5834,5837,5840,5843,5846,5849,5852],{"className":5830},[50],[15,5832],{"className":5833,"style":5779},[54],[15,5835,1289],{"className":5836,"style":1489},[1486,1487,1488],[15,5838],{"className":5839,"style":406},[157],[15,5841,244],{"className":5842},[59,60],[15,5844,206],{"className":5845,"style":334},[59,60],[15,5847],{"className":5848,"style":2589},[157],[15,5850,4815],{"className":5851},[2593],[15,5853],{"className":5854,"style":2589},[157],[15,5856,5858,5861,5864,5867,5870,5873,5876,5879],{"className":5857},[50],[15,5859],{"className":5860,"style":5779},[54],[15,5862,1289],{"className":5863,"style":1489},[1486,1487,1488],[15,5865],{"className":5866,"style":406},[157],[15,5868,244],{"className":5869},[59,60],[15,5871,213],{"className":5872},[59,60],[15,5874],{"className":5875,"style":158},[157],[15,5877,123],{"className":5878},[162],[15,5880],{"className":5881,"style":158},[157],[15,5883,5885,5888],{"className":5884},[50],[15,5886],{"className":5887,"style":172},[54],[15,5889,5766],{"className":5890},[59],[11,5892,5893,5894,2747],{},"于是对任意常数 ",[15,5895,5897,5917],{"className":5896,"translate":19},[18],[15,5898,5900],{"className":5899},[23],[25,5901,5902],{"xmlns":27},[29,5903,5904,5914],{},[32,5905,5906,5909,5912],{},[35,5907,5908],{},"c",[110,5910,5911],{},"∈",[35,5913,1035],{"mathvariant":222},[39,5915,5916],{"encoding":41},"c\\in\\mathbb 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",[15,6498,6500,6517],{"className":6499,"translate":19},[18],[15,6501,6503],{"className":6502},[23],[25,6504,6505],{"xmlns":27},[29,6506,6507,6515],{},[32,6508,6509],{},[195,6510,6511,6513],{},[35,6512,199],{},[35,6514,37],{},[39,6516,4024],{"encoding":41},[15,6518,6520],{"className":6519,"ariaHidden":46},[45],[15,6521,6523,6526],{"className":6522},[50],[15,6524],{"className":6525,"style":4034},[54],[15,6527,6529,6532],{"className":6528},[59],[15,6530,199],{"className":6531,"style":277},[59,60],[15,6533,6535],{"className":6534},[281],[15,6536,6538,6558],{"className":6537},[285,286],[15,6539,6541,6555],{"className":6540},[290],[15,6542,6544],{"className":6543,"style":295},[294],[15,6545,6546,6549],{"style":298},[15,6547],{"className":6548,"style":303},[302],[15,6550,6552],{"className":6551},[307,308,309,310],[15,6553,37],{"className":6554,"style":61},[59,60,310],[15,6556,318],{"className":6557},[317],[15,6559,6561],{"className":6560},[290],[15,6562,6564],{"className":6563,"style":325},[294],[15,6565],{}," 只依赖 ",[15,6568,6570,6583],{"className":6569,"translate":19},[18],[15,6571,6573],{"className":6572},[23],[25,6574,6575],{"xmlns":27},[29,6576,6577,6581],{},[32,6578,6579],{},[35,6580,37],{},[39,6582,37],{"encoding":41},[15,6584,6586],{"className":6585,"ariaHidden":46},[45],[15,6587,6589,6592],{"className":6588},[50],[15,6590],{"className":6591,"style":55},[54],[15,6593,37],{"className":6594,"style":61},[59,60]," 在「模去 ",[15,6597,6599,6623],{"className":6598,"translate":19},[18],[15,6600,6602],{"className":6601},[23],[25,6603,6604],{"xmlns":27},[29,6605,6606,6620],{},[32,6607,6608,6610,6612,6614,6616,6618],{},[35,6609,5908],{},[110,6611,113],{"stretchy":112},[35,6613,769],{},[110,6615,4815],{},[115,6617,117],{},[110,6619,120],{"stretchy":112},[39,6621,6622],{"encoding":41},"c(t-1)",[15,6624,6626,6650],{"className":6625,"ariaHidden":46},[45],[15,6627,6629,6632,6635,6638,6641,6644,6647],{"className":6628},[50],[15,6630],{"className":6631,"style":139},[54],[15,6633,5908],{"className":6634},[59,60],[15,6636,113],{"className":6637},[146],[15,6639,769],{"className":6640},[59,60],[15,6642],{"className":6643,"style":2589},[157],[15,6645,4815],{"className":6646},[2593],[15,6648],{"className":6649,"style":2589},[157],[15,6651,6653,6656,6659],{"className":6652},[50],[15,6654],{"className":6655,"style":139},[54],[15,6657,117],{"className":6658},[59],[15,6660,120],{"className":6661},[153],"」之后的等价类。两个常用规范化是：",[6664,6665,6666,6735],"ol",{},[6667,6668,6669,6734],"li",{},[15,6670,6672,6695],{"className":6671,"translate":19},[18],[15,6673,6675],{"className":6674},[23],[25,6676,6677],{"xmlns":27},[29,6678,6679,6693],{},[32,6680,6681,6683,6685,6687,6689,6691],{},[35,6682,37],{},[110,6684,113],{"stretchy":112},[115,6686,117],{},[110,6688,120],{"stretchy":112},[110,6690,123],{},[115,6692,126],{},[39,6694,129],{"encoding":41},[15,6696,6698,6725],{"className":6697,"ariaHidden":46},[45],[15,6699,6701,6704,6707,6710,6713,6716,6719,6722],{"className":6700},[50],[15,6702],{"className":6703,"style":139},[54],[15,6705,37],{"className":6706,"style":61},[59,60],[15,6708,113],{"className":6709},[146],[15,6711,117],{"className":6712},[59],[15,6714,120],{"className":6715},[153],[15,6717],{"className":6718,"style":158},[157],[15,6720,123],{"className":6721},[162],[15,6723],{"className":6724,"style":158},[157],[15,6726,6728,6731],{"className":6727},[50],[15,6729],{"className":6730,"style":172},[54],[15,6732,126],{"className":6733},[59],"（本文采用）；",[6667,6736,6737,6802,6803,6907,6908,2747,7119,7147],{},[15,6738,6740,6763],{"className":6739,"translate":19},[18],[15,6741,6743],{"className":6742},[23],[25,6744,6745],{"xmlns":27},[29,6746,6747,6761],{},[32,6748,6749,6751,6753,6755,6757,6759],{},[35,6750,37],{},[110,6752,113],{"stretchy":112},[115,6754,117],{},[110,6756,120],{"stretchy":112},[110,6758,123],{},[115,6760,126],{},[39,6762,129],{"encoding":41},[15,6764,6766,6793],{"className":6765,"ariaHidden":46},[45],[15,6767,6769,6772,6775,6778,6781,6784,6787,6790],{"className":6768},[50],[15,6770],{"className":6771,"style":139},[54],[15,6773,37],{"className":6774,"style":61},[59,60],[15,6776,113],{"className":6777},[146],[15,6779,117],{"className":6780},[59],[15,6782,120],{"className":6783},[153],[15,6785],{"className":6786,"style":158},[157],[15,6788,123],{"className":6789},[162],[15,6791],{"className":6792,"style":158},[157],[15,6794,6796,6799],{"className":6795},[50],[15,6797],{"className":6798,"style":172},[54],[15,6800,126],{"className":6801},[59]," 且 ",[15,6804,6806,6836],{"className":6805,"translate":19},[18],[15,6807,6809],{"className":6808},[23],[25,6810,6811],{"xmlns":27},[29,6812,6813,6833],{},[32,6814,6815,6823,6825,6827,6829,6831],{},[6816,6817,6818,6820],"msup",{},[35,6819,37],{},[110,6821,6822],{"mathvariant":209,"lspace":1249,"rspace":1249},"′",[110,6824,113],{"stretchy":112},[115,6826,117],{},[110,6828,120],{"stretchy":112},[110,6830,123],{},[115,6832,126],{},[39,6834,6835],{"encoding":41},"f'(1)=0",[15,6837,6839,6898],{"className":6838,"ariaHidden":46},[45],[15,6840,6842,6846,6880,6883,6886,6889,6892,6895],{"className":6841},[50],[15,6843],{"className":6844,"style":6845},[54],"height:1.0019em;vertical-align:-0.25em;",[15,6847,6849,6852],{"className":6848},[59],[15,6850,37],{"className":6851,"style":61},[59,60],[15,6853,6855],{"className":6854},[281],[15,6856,6858],{"className":6857},[285],[15,6859,6861],{"className":6860},[290],[15,6862,6865],{"className":6863,"style":6864},[294],"height:0.7519em;",[15,6866,6868,6871],{"style":6867},"top:-3.063em;margin-right:0.05em;",[15,6869],{"className":6870,"style":303},[302],[15,6872,6874],{"className":6873},[307,308,309,310],[15,6875,6877],{"className":6876},[59,310],[15,6878,6822],{"className":6879},[59,310],[15,6881,113],{"className":6882},[146],[15,6884,117],{"className":6885},[59],[15,6887,120],{"className":6888},[153],[15,6890],{"className":6891,"style":158},[157],[15,6893,123],{"className":6894},[162],[15,6896],{"className":6897,"style":158},[157],[15,6899,6901,6904],{"className":6900},[50],[15,6902],{"className":6903,"style":172},[54],[15,6905,126],{"className":6906},[59],"——此时由凸性 ",[15,6909,6911,6970],{"className":6910,"translate":19},[18],[15,6912,6914],{"className":6913},[23],[25,6915,6916],{"xmlns":27},[29,6917,6918,6967],{},[32,6919,6920,6922,6924,6926,6928,6931,6933,6935,6937,6939,6941,6947,6949,6951,6953,6955,6957,6959,6961,6963,6965],{},[35,6921,37],{},[110,6923,113],{"stretchy":112},[35,6925,769],{},[110,6927,120],{"stretchy":112},[110,6929,6930],{},"≥",[35,6932,37],{},[110,6934,113],{"stretchy":112},[115,6936,117],{},[110,6938,120],{"stretchy":112},[110,6940,2472],{},[6816,6942,6943,6945],{},[35,6944,37],{},[110,6946,6822],{"mathvariant":209,"lspace":1249,"rspace":1249},[110,6948,113],{"stretchy":112},[115,6950,117],{},[110,6952,120],{"stretchy":112},[110,6954,113],{"stretchy":112},[35,6956,769],{},[110,6958,4815],{},[115,6960,117],{},[110,6962,120],{"stretchy":112},[110,6964,123],{},[115,6966,126],{},[39,6968,6969],{"encoding":41},"f(t)\\ge f(1)+f'(1)(t-1)=0",[15,6971,6973,7000,7027,7089,7110],{"className":6972,"ariaHidden":46},[45],[15,6974,6976,6979,6982,6985,6988,6991,6994,6997],{"className":6975},[50],[15,6977],{"className":6978,"style":139},[54],[15,6980,37],{"className":6981,"style":61},[59,60],[15,6983,113],{"className":6984},[146],[15,6986,769],{"className":6987},[59,60],[15,6989,120],{"className":6990},[153],[15,6992],{"className":6993,"style":158},[157],[15,6995,6930],{"className":6996},[162],[15,6998],{"className":6999,"style":158},[157],[15,7001,7003,7006,7009,7012,7015,7018,7021,7024],{"className":7002},[50],[15,7004],{"className":7005,"style":139},[54],[15,7007,37],{"className":7008,"style":61},[59,60],[15,7010,113],{"className":7011},[146],[15,7013,117],{"className":7014},[59],[15,7016,120],{"className":7017},[153],[15,7019],{"className":7020,"style":2589},[157],[15,7022,2472],{"className":7023},[2593],[15,7025],{"className":7026,"style":2589},[157],[15,7028,7030,7033,7065,7068,7071,7074,7077,7080,7083,7086],{"className":7029},[50],[15,7031],{"className":7032,"style":6845},[54],[15,7034,7036,7039],{"className":7035},[59],[15,7037,37],{"className":7038,"style":61},[59,60],[15,7040,7042],{"className":7041},[281],[15,7043,7045],{"className":7044},[285],[15,7046,7048],{"className":7047},[290],[15,7049,7051],{"className":7050,"style":6864},[294],[15,7052,7053,7056],{"style":6867},[15,7054],{"className":7055,"style":303},[302],[15,7057,7059],{"className":7058},[307,308,309,310],[15,7060,7062],{"className":7061},[59,310],[15,7063,6822],{"className":7064},[59,310],[15,7066,113],{"className":7067},[146],[15,7069,117],{"className":7070},[59],[15,7072,120],{"className":7073},[153],[15,7075,113],{"className":7076},[146],[15,7078,769],{"className":7079},[59,60],[15,7081],{"className":7082,"style":2589},[157],[15,7084,4815],{"className":7085},[2593],[15,7087],{"className":7088,"style":2589},[157],[15,7090,7092,7095,7098,7101,7104,7107],{"className":7091},[50],[15,7093],{"className":7094,"style":139},[54],[15,7096,117],{"className":7097},[59],[15,7099,120],{"className":7100},[153],[15,7102],{"className":7103,"style":158},[157],[15,7105,123],{"className":7106},[162],[15,7108],{"className":7109,"style":158},[157],[15,7111,7113,7116],{"className":7112},[50],[15,7114],{"className":7115,"style":172},[54],[15,7117,126],{"className":7118},[59],[15,7120,7122,7135],{"className":7121,"translate":19},[18],[15,7123,7125],{"className":7124},[23],[25,7126,7127],{"xmlns":27},[29,7128,7129,7133],{},[32,7130,7131],{},[35,7132,37],{},[39,7134,37],{"encoding":41},[15,7136,7138],{"className":7137,"ariaHidden":46},[45],[15,7139,7141,7144],{"className":7140},[50],[15,7142],{"className":7143,"style":55},[54],[15,7145,37],{"className":7146,"style":61},[59,60]," 在整个定义域上非负。",[11,7149,7150,7151,7236,7237,7290,7291,7360],{},"第 3 节里的 KL 的 ",[15,7152,7154,7183],{"className":7153,"translate":19},[18],[15,7155,7157],{"className":7156},[23],[25,7158,7159],{"xmlns":27},[29,7160,7161,7181],{},[32,7162,7163,7165,7167,7169,7171,7173,7175,7177,7179],{},[35,7164,37],{},[110,7166,113],{"stretchy":112},[35,7168,769],{},[110,7170,120],{"stretchy":112},[110,7172,123],{},[35,7174,769],{},[35,7176,4102],{},[110,7178,4105],{},[35,7180,769],{},[39,7182,4110],{"encoding":41},[15,7184,7186,7213],{"className":7185,"ariaHidden":46},[45],[15,7187,7189,7192,7195,7198,7201,7204,7207,7210],{"className":7188},[50],[15,7190],{"className":7191,"style":139},[54],[15,7193,37],{"className":7194,"style":61},[59,60],[15,7196,113],{"className":7197},[146],[15,7199,769],{"className":7200},[59,60],[15,7202,120],{"className":7203},[153],[15,7205],{"className":7206,"style":158},[157],[15,7208,123],{"className":7209},[162],[15,7211],{"className":7212,"style":158},[157],[15,7214,7216,7219,7222,7225,7230,7233],{"className":7215},[50],[15,7217],{"className":7218,"style":55},[54],[15,7220,769],{"className":7221},[59,60],[15,7223],{"className":7224,"style":406},[157],[15,7226,4156,7228],{"className":7227},[1486],[15,7229,4160],{"style":4159},[15,7231],{"className":7232,"style":406},[157],[15,7234,769],{"className":7235},[59,60]," 不满足第 2 条（它在 ",[15,7238,7240,7259],{"className":7239,"translate":19},[18],[15,7241,7243],{"className":7242},[23],[25,7244,7245],{"xmlns":27},[29,7246,7247,7256],{},[32,7248,7249,7251,7254],{},[35,7250,769],{},[110,7252,7253],{},"\u003C",[115,7255,117],{},[39,7257,7258],{"encoding":41},"t\u003C1",[15,7260,7262,7281],{"className":7261,"ariaHidden":46},[45],[15,7263,7265,7269,7272,7275,7278],{"className":7264},[50],[15,7266],{"className":7267,"style":7268},[54],"height:0.6542em;vertical-align:-0.0391em;",[15,7270,769],{"className":7271},[59,60],[15,7273],{"className":7274,"style":158},[157],[15,7276,7253],{"className":7277},[162],[15,7279],{"className":7280,"style":158},[157],[15,7282,7284,7287],{"className":7283},[50],[15,7285],{"className":7286,"style":172},[54],[15,7288,117],{"className":7289},[59]," 时取负值），但这无妨：",[15,7292,7294,7311],{"className":7293,"translate":19},[18],[15,7295,7297],{"className":7296},[23],[25,7298,7299],{"xmlns":27},[29,7300,7301,7309],{},[32,7302,7303],{},[195,7304,7305,7307],{},[35,7306,199],{},[35,7308,37],{},[39,7310,4024],{"encoding":41},[15,7312,7314],{"className":7313,"ariaHidden":46},[45],[15,7315,7317,7320],{"className":7316},[50],[15,7318],{"className":7319,"style":4034},[54],[15,7321,7323,7326],{"className":7322},[59],[15,7324,199],{"className":7325,"style":277},[59,60],[15,7327,7329],{"className":7328},[281],[15,7330,7332,7352],{"className":7331},[285,286],[15,7333,7335,7349],{"className":7334},[290],[15,7336,7338],{"className":7337,"style":295},[294],[15,7339,7340,7343],{"style":298},[15,7341],{"className":7342,"style":303},[302],[15,7344,7346],{"className":7345},[307,308,309,310],[15,7347,37],{"className":7348,"style":61},[59,60,310],[15,7350,318],{"className":7351},[317],[15,7353,7355],{"className":7354},[290],[15,7356,7358],{"className":7357,"style":325},[294],[15,7359],{}," 仍然非负（见 4.1），因为「逐点非负」从来不是 f-散度非负性的必要条件。",[2313,7362,7364],{"id":7363},"_14-等价书写形式","1.4 等价书写形式",[15,7366,7368],{"className":7367,"translate":19},[180],[15,7369,7371,7605],{"className":7370,"translate":19},[18],[15,7372,7374],{"className":7373},[23],[25,7375,7376],{"xmlns":27,"display":189},[29,7377,7378,7602],{},[1245,7379,7380,7452,7524],{"rowspacing":1247,"columnalign":1248,"columnspacing":1249},[1251,7381,7382,7404],{},[1254,7383,7384],{},[1257,7385,7386],{"scriptlevel":126,"displaystyle":46},[32,7387,7388,7394,7396,7398,7400,7402],{},[195,7389,7390,7392],{},[35,7391,199],{},[35,7393,37],{},[110,7395,113],{"stretchy":112},[35,7397,206],{},[35,7399,210],{"mathvariant":209},[35,7401,213],{},[110,7403,120],{"stretchy":112},[1254,7405,7406],{},[1257,7407,7408],{"scriptlevel":126,"displaystyle":46},[32,7409,7410,7412,7414,7420,7422],{},[32,7411],{},[110,7413,123],{},[195,7415,7416,7418],{},[35,7417,223],{"mathvariant":222},[35,7419,213],{},[1299,7421,1306],{},[32,7423,7424,7426,7428,7430,7450],{},[110,7425,230],{"fence":46},[35,7427,37],{},[1299,7429,1306],{},[32,7431,7432,7434,7448],{},[110,7433,113],{"fence":46},[238,7435,7436,7442],{},[32,7437,7438,7440],{},[35,7439,244],{},[35,7441,206],{},[32,7443,7444,7446],{},[35,7445,244],{},[35,7447,213],{},[110,7449,120],{"fence":46},[110,7451,257],{"fence":46},[1251,7453,7454,7460],{},[1254,7455,7456],{},[1257,7457,7458],{"scriptlevel":126,"displaystyle":46},[32,7459],{},[1254,7461,7462],{},[1257,7463,7464],{"scriptlevel":126,"displaystyle":46},[32,7465,7466,7468,7470,7476,7478],{},[32,7467],{},[110,7469,123],{},[195,7471,7472,7474],{},[35,7473,223],{"mathvariant":222},[35,7475,206],{},[1299,7477,1306],{},[32,7479,7480,7482,7496,7498,7500,7502,7522],{},[110,7481,230],{"fence":46},[238,7483,7484,7490],{},[32,7485,7486,7488],{},[35,7487,244],{},[35,7489,213],{},[32,7491,7492,7494],{},[35,7493,244],{},[35,7495,206],{},[1299,7497,1301],{},[35,7499,37],{},[1299,7501,1306],{},[32,7503,7504,7506,7520],{},[110,7505,113],{"fence":46},[238,7507,7508,7514],{},[32,7509,7510,7512],{},[35,7511,244],{},[35,7513,206],{},[32,7515,7516,7518],{},[35,7517,244],{},[35,7519,213],{},[110,7521,120],{"fence":46},[110,7523,257],{"fence":46},[1251,7525,7526,7532],{},[1254,7527,7528],{},[1257,7529,7530],{"scriptlevel":126,"displaystyle":46},[32,7531],{},[1254,7533,7534],{},[1257,7535,7536],{"scriptlevel":126,"displaystyle":46},[32,7537,7538,7540,7542,7548,7550,7600],{},[32,7539],{},[110,7541,123],{},[195,7543,7544,7546],{},[35,7545,223],{"mathvariant":222},[35,7547,206],{},[1299,7549,1306],{},[32,7551,7552,7554,7576,7578,7598],{},[110,7553,230],{"fence":46},[7555,7556,7557,7559,7573],"msubsup",{},[35,7558,37],{},[32,7560,7561,7564,7567,7570],{},[35,7562,7563],{},"s",[35,7565,7566],{},"k",[35,7568,7569],{},"e",[35,7571,7572],{},"w",[110,7574,7575],{"lspace":1249,"rspace":1249},"∗",[1299,7577,1306],{},[32,7579,7580,7582,7596],{},[110,7581,113],{"fence":46},[238,7583,7584,7590],{},[32,7585,7586,7588],{},[35,7587,244],{},[35,7589,213],{},[32,7591,7592,7594],{},[35,7593,244],{},[35,7595,206],{},[110,7597,120],{"fence":46},[110,7599,257],{"fence":46},[110,7601,551],{"separator":46},[39,7603,7604],{"encoding":41},"\\begin{aligned}\nD_f(P\\|Q)\n&=\\mathbb E_Q\\!\\left[\nf\\!\\left(\\frac{dP}{dQ}\\right)\\right]\\\\[2pt]\n&=\\mathbb E_P\\!\\left[\\frac{dQ}{dP}\\,\nf\\!\\left(\\frac{dP}{dQ}\\right)\\right]\\\\\n&=\\mathbb 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是后面 4.3 节的「斜共轭」。最后一个等式说明：交换 ",[15,8600,8602,8619],{"className":8601,"translate":19},[18],[15,8603,8605],{"className":8604},[23],[25,8606,8607],{"xmlns":27},[29,8608,8609,8617],{},[32,8610,8611,8613,8615],{},[35,8612,206],{},[110,8614,551],{"separator":46},[35,8616,213],{},[39,8618,556],{"encoding":41},[15,8620,8622],{"className":8621,"ariaHidden":46},[45],[15,8623,8625,8628,8631,8634,8637],{"className":8624},[50],[15,8626],{"className":8627,"style":566},[54],[15,8629,206],{"className":8630,"style":334},[59,60],[15,8632,551],{"className":8633},[573],[15,8635],{"className":8636,"style":406},[157],[15,8638,213],{"className":8639},[59,60]," 只是把 ",[15,8642,8644,8657],{"className":8643,"translate":19},[18],[15,8645,8647],{"className":8646},[23],[25,8648,8649],{"xmlns":27},[29,8650,8651,8655],{},[32,8652,8653],{},[35,8654,37],{},[39,8656,37],{"encoding":41},[15,8658,8660],{"className":8659,"ariaHidden":46},[45],[15,8661,8663,8666],{"className":8662},[50],[15,8664],{"className":8665,"style":55},[54],[15,8667,37],{"className":8668,"style":61},[59,60]," 换成 ",[15,8671,8673,8704],{"className":8672,"translate":19},[18],[15,8674,8676],{"className":8675},[23],[25,8677,8678],{"xmlns":27},[29,8679,8680,8701],{},[32,8681,8682,8684,8687,8689,8691,8693,8695,8697,8699],{},[35,8683,769],{},[110,8685,8686],{},"↦",[35,8688,769],{},[35,8690,37],{},[110,8692,113],{"stretchy":112},[115,8694,117],{},[35,8696,941],{"mathvariant":209},[35,8698,769],{},[110,8700,120],{"stretchy":112},[39,8702,8703],{"encoding":41},"t\\mapsto t f(1\u002Ft)",[15,8705,8707,8726],{"className":8706,"ariaHidden":46},[45],[15,8708,8710,8714,8717,8720,8723],{"className":8709},[50],[15,8711],{"className":8712,"style":8713},[54],"height:0.6261em;vertical-align:-0.011em;",[15,8715,769],{"className":8716},[59,60],[15,8718],{"className":8719,"style":158},[157],[15,8721,8686],{"className":8722},[162],[15,8724],{"className":8725,"style":158},[157],[15,8727,8729,8732,8735,8738,8741,8744,8747],{"className":8728},[50],[15,8730],{"className":8731,"style":139},[54],[15,8733,769],{"className":8734},[59,60],[15,8736,37],{"className":8737,"style":61},[59,60],[15,8739,113],{"className":8740},[146],[15,8742,8591],{"className":8743},[59],[15,8745,769],{"className":8746},[59,60],[15,8748,120],{"className":8749},[153],"，这对推导「反向 KL」之类的对称性非常方便。",[528,8752,8754,8755,8783,8784],{"id":8753},"_2-为何-fff-凸且-f10f10f10","2. 为何 ",[15,8756,8758,8771],{"className":8757,"translate":19},[18],[15,8759,8761],{"className":8760},[23],[25,8762,8763],{"xmlns":27},[29,8764,8765,8769],{},[32,8766,8767],{},[35,8768,37],{},[39,8770,37],{"encoding":41},[15,8772,8774],{"className":8773,"ariaHidden":46},[45],[15,8775,8777,8780],{"className":8776},[50],[15,8778],{"className":8779,"style":55},[54],[15,8781,37],{"className":8782,"style":61},[59,60]," 凸且 ",[15,8785,8787,8810],{"className":8786,"translate":19},[18],[15,8788,8790],{"className":8789},[23],[25,8791,8792],{"xmlns":27},[29,8793,8794,8808],{},[32,8795,8796,8798,8800,8802,8804,8806],{},[35,8797,37],{},[110,8799,113],{"stretchy":112},[115,8801,117],{},[110,8803,120],{"stretchy":112},[110,8805,123],{},[115,8807,126],{},[39,8809,129],{"encoding":41},[15,8811,8813,8840],{"className":8812,"ariaHidden":46},[45],[15,8814,8816,8819,8822,8825,8828,8831,8834,8837],{"className":8815},[50],[15,8817],{"className":8818,"style":139},[54],[15,8820,37],{"className":8821,"style":61},[59,60],[15,8823,113],{"className":8824},[146],[15,8826,117],{"className":8827},[59],[15,8829,120],{"className":8830},[153],[15,8832],{"className":8833,"style":158},[157],[15,8835,123],{"className":8836},[162],[15,8838],{"className":8839,"style":158},[157],[15,8841,8843,8846],{"className":8842},[50],[15,8844],{"className":8845,"style":172},[54],[15,8847,126],{"className":8848},[59],[2313,8850,8852],{"id":8851},"_21-非负性与-jensen","2.1 非负性与 Jensen",[11,8854,534,8855,1169,8908,8952,8953,9022,9023,9093],{},[15,8856,8858,8877],{"className":8857,"translate":19},[18],[15,8859,8861],{"className":8860},[23],[25,8862,8863],{"xmlns":27},[29,8864,8865,8874],{},[32,8866,8867,8869,8872],{},[35,8868,598],{},[110,8870,8871],{},"∼",[35,8873,213],{},[39,8875,8876],{"encoding":41},"X\\sim Q",[15,8878,8880,8899],{"className":8879,"ariaHidden":46},[45],[15,8881,8883,8886,8890,8893,8896],{"className":8882},[50],[15,8884],{"className":8885,"style":664},[54],[15,8887,598],{"className":8888,"style":8889},[59,60],"margin-right:0.0785em;",[15,8891],{"className":8892,"style":158},[157],[15,8894,8871],{"className":8895},[162],[15,8897],{"className":8898,"style":158},[157],[15,8900,8902,8905],{"className":8901},[50],[15,8903],{"className":8904,"style":566},[54],[15,8906,213],{"className":8907},[59,60],[15,8909,8911,8931],{"className":8910,"translate":19},[18],[15,8912,8914],{"className":8913},[23],[25,8915,8916],{"xmlns":27},[29,8917,8918,8928],{},[32,8919,8920,8922,8924,8926],{},[35,8921,37],{},[110,8923,113],{"stretchy":112},[35,8925,598],{},[110,8927,120],{"stretchy":112},[39,8929,8930],{"encoding":41},"f(X)",[15,8932,8934],{"className":8933,"ariaHidden":46},[45],[15,8935,8937,8940,8943,8946,8949],{"className":8936},[50],[15,8938],{"className":8939,"style":139},[54],[15,8941,37],{"className":8942,"style":61},[59,60],[15,8944,113],{"className":8945},[146],[15,8947,598],{"className":8948,"style":8889},[59,60],[15,8950,120],{"className":8951},[153]," 的期望就是 ",[15,8954,8956,8973],{"className":8955,"translate":19},[18],[15,8957,8959],{"className":8958},[23],[25,8960,8961],{"xmlns":27},[29,8962,8963,8971],{},[32,8964,8965],{},[195,8966,8967,8969],{},[35,8968,199],{},[35,8970,37],{},[39,8972,4024],{"encoding":41},[15,8974,8976],{"className":8975,"ariaHidden":46},[45],[15,8977,8979,8982],{"className":8978},[50],[15,8980],{"className":8981,"style":4034},[54],[15,8983,8985,8988],{"className":8984},[59],[15,8986,199],{"className":8987,"style":277},[59,60],[15,8989,8991],{"className":8990},[281],[15,8992,8994,9014],{"className":8993},[285,286],[15,8995,8997,9011],{"className":8996},[290],[15,8998,9000],{"className":8999,"style":295},[294],[15,9001,9002,9005],{"style":298},[15,9003],{"className":9004,"style":303},[302],[15,9006,9008],{"className":9007},[307,308,309,310],[15,9009,37],{"className":9010,"style":61},[59,60,310],[15,9012,318],{"className":9013},[317],[15,9015,9017],{"className":9016},[290],[15,9018,9020],{"className":9019,"style":325},[294],[15,9021],{},"。因为 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满足",[15,9095,9097],{"className":9096,"translate":19},[180],[15,9098,9100,9162],{"className":9099,"translate":19},[18],[15,9101,9103],{"className":9102},[23],[25,9104,9105],{"xmlns":27,"display":189},[29,9106,9107,9159],{},[32,9108,9109,9115,9117,9119,9121,9123,9125,9139,9141,9143,9145,9147,9149,9151,9153,9155,9157],{},[195,9110,9111,9113],{},[35,9112,223],{"mathvariant":222},[35,9114,213],{},[110,9116,230],{"stretchy":112},[35,9118,769],{},[110,9120,257],{"stretchy":112},[110,9122,123],{},[110,9124,1289],{},[238,9126,9127,9133],{},[32,9128,9129,9131],{},[35,9130,244],{},[35,9132,206],{},[32,9134,9135,9137],{},[35,9136,244],{},[35,9138,213],{},[1299,9140,1301],{},[35,9142,244],{},[35,9144,213],{},[110,9146,123],{},[110,9148,1289],{},[35,9150,244],{},[35,9152,206],{},[110,9154,123],{},[115,9156,117],{},[110,9158,551],{"separator":46},[39,9160,9161],{"encoding":41},"\\mathbb E_Q[t]=\\int \\frac{dP}{dQ}\\,dQ=\\int dP=1,",[15,9163,9165,9229,9327,9354],{"className":9164,"ariaHidden":46},[45],[15,9166,9168,9171,9211,9214,9217,9220,9223,9226],{"className":9167},[50],[15,9169],{"className":9170,"style":270},[54],[15,9172,9174,9177],{"className":9173},[59],[15,9175,223],{"className":9176},[59,366],[15,9178,9180],{"className":9179},[281],[15,9181,9183,9203],{"className":9182},[285,286],[15,9184,9186,9200],{"className":9185},[290],[15,9187,9189],{"className":9188,"style":379},[294],[15,9190,9191,9194],{"style":382},[15,9192],{"className":9193,"style":303},[302],[15,9195,9197],{"className":9196},[307,308,309,310],[15,9198,213],{"className":9199},[59,60,310],[15,9201,318],{"className":9202},[317],[15,9204,9206],{"className":9205},[290],[15,9207,9209],{"className":9208,"style":325},[294],[15,9210],{},[15,9212,230],{"className":9213},[146],[15,9215,769],{"className":9216},[59,60],[15,9218,257],{"className":9219},[153],[15,9221],{"className":9222,"style":158},[157],[15,9224,123],{"className":9225},[162],[15,9227],{"className":9228,"style":158},[157],[15,9230,9232,9235,9238,9241,9309,9312,9315,9318,9321,9324],{"className":9231},[50],[15,9233],{"className":9234,"style":838},[54],[15,9236,1289],{"className":9237,"style":1489},[1486,1487,1488],[15,9239],{"className":9240,"style":406},[157],[15,9242,9244,9247,9306],{"className":9243},[59],[15,9245],{"className":9246},[146,441],[15,9248,9250],{"className":9249},[238],[15,9251,9253,9298],{"className":9252},[285,286],[15,9254,9256,9295],{"className":9255},[290],[15,9257,9259,9273,9281],{"className":9258,"style":454},[294],[15,9260,9261,9264],{"style":457},[15,9262],{"className":9263,"style":461},[302],[15,9265,9267,9270],{"className":9266},[59],[15,9268,244],{"className":9269},[59,60],[15,9271,213],{"className":9272},[59,60],[15,9274,9275,9278],{"style":473},[15,9276],{"className":9277,"style":461},[302],[15,9279],{"className":9280,"style":481},[480],[15,9282,9283,9286],{"style":484},[15,9284],{"className":9285,"style":461},[302],[15,9287,9289,9292],{"className":9288},[59],[15,9290,244],{"className":9291},[59,60],[15,9293,206],{"className":9294,"style":334},[59,60],[15,9296,318],{"className":9297},[317],[15,9299,9301],{"className":9300},[290],[15,9302,9304],{"className":9303,"style":506},[294],[15,9305],{},[15,9307],{"className":9308},[153,441],[15,9310],{"className":9311,"style":406},[157],[15,9313,244],{"className":9314},[59,60],[15,9316,213],{"className":9317},[59,60],[15,9319],{"className":9320,"style":158},[157],[15,9322,123],{"className":9323},[162],[15,9325],{"className":9326,"style":158},[157],[15,9328,9330,9333,9336,9339,9342,9345,9348,9351],{"className":9329},[50],[15,9331],{"className":9332,"style":5779},[54],[15,9334,1289],{"className":9335,"style":1489},[1486,1487,1488],[15,9337],{"className":9338,"style":406},[157],[15,9340,244],{"className":9341},[59,60],[15,9343,206],{"className":9344,"style":334},[59,60],[15,9346],{"className":9347,"style":158},[157],[15,9349,123],{"className":9350},[162],[15,9352],{"className":9353,"style":158},[157],[15,9355,9357,9361,9364],{"className":9356},[50],[15,9358],{"className":9359,"style":9360},[54],"height:0.8389em;vertical-align:-0.1944em;",[15,9362,117],{"className":9363},[59],[15,9365,551],{"className":9366},[573],[11,9368,9369,9370,9398,9399,9402],{},"对凸函数 ",[15,9371,9373,9386],{"className":9372,"translate":19},[18],[15,9374,9376],{"className":9375},[23],[25,9377,9378],{"xmlns":27},[29,9379,9380,9384],{},[32,9381,9382],{},[35,9383,37],{},[39,9385,37],{"encoding":41},[15,9387,9389],{"className":9388,"ariaHidden":46},[45],[15,9390,9392,9395],{"className":9391},[50],[15,9393],{"className":9394,"style":55},[54],[15,9396,37],{"className":9397,"style":61},[59,60]," 用 ",[998,9400,9401],{},"Jensen 不等式","：",[15,9404,9406],{"className":9405,"translate":19},[180],[15,9407,9409,9528],{"className":9408,"translate":19},[18],[15,9410,9412],{"className":9411},[23],[25,9413,9414],{"xmlns":27,"display":189},[29,9415,9416,9525],{},[1245,9417,9418,9495],{"rowspacing":1247,"columnalign":1248,"columnspacing":1249},[1251,9419,9420,9463],{},[1254,9421,9422],{},[1257,9423,9424],{"scriptlevel":126,"displaystyle":46},[32,9425,9426,9432,9434,9436,9438,9440,9442,9444,9450,9453,9455,9457,9459,9461],{},[195,9427,9428,9430],{},[35,9429,199],{},[35,9431,37],{},[110,9433,113],{"stretchy":112},[35,9435,206],{},[35,9437,210],{"mathvariant":209},[35,9439,213],{},[110,9441,120],{"stretchy":112},[110,9443,123],{},[195,9445,9446,9448],{},[35,9447,223],{"mathvariant":222},[35,9449,213],{},[110,9451,230],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},"1.2em",[35,9454,37],{},[110,9456,113],{"stretchy":112},[35,9458,769],{},[110,9460,120],{"stretchy":112},[110,9462,257],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[1254,9464,9465],{},[1257,9466,9467],{"scriptlevel":126,"displaystyle":46},[32,9468,9469,9471,9473,9475,9477,9479,9481,9487,9489,9491,9493],{},[32,9470],{},[1299,9472,4285],{},[110,9474,6930],{},[1299,9476,4285],{},[35,9478,37],{},[110,9480,113],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[195,9482,9483,9485],{},[35,9484,223],{"mathvariant":222},[35,9486,213],{},[110,9488,230],{"stretchy":112},[35,9490,769],{},[110,9492,257],{"stretchy":112},[110,9494,120],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[1251,9496,9497,9503],{},[1254,9498,9499],{},[1257,9500,9501],{"scriptlevel":126,"displaystyle":46},[32,9502],{},[1254,9504,9505],{},[1257,9506,9507],{"scriptlevel":126,"displaystyle":46},[32,9508,9509,9511,9513,9515,9517,9519,9521,9523],{},[32,9510],{},[110,9512,123],{},[35,9514,37],{},[110,9516,113],{"stretchy":112},[115,9518,117],{},[110,9520,120],{"stretchy":112},[110,9522,123],{},[115,9524,5766],{},[39,9526,9527],{"encoding":41},"\\begin{aligned}\nD_f(P\\|Q)=\\mathbb E_Q\\big[f(t)\\big]\n&\\ \\ge\\ f\\big(\\mathbb E_Q[t]\\big)\\\\[2pt]\n&=f(1)=0 .\n\\end{aligned}",[15,9529,9531],{"className":9530,"ariaHidden":46},[45],[15,9532,9534,9538],{"className":9533},[50],[15,9535],{"className":9536,"style":9537},[54],"height:2.91em;vertical-align:-1.205em;",[15,9539,9541],{"className":9540},[59],[15,9542,9544,9716],{"className":9543},[1245],[15,9545,9547],{"className":9546},[1362],[15,9548,9550,9707],{"className":9549},[285,286],[15,9551,9553,9704],{"className":9552},[290],[15,9554,9557,9695],{"className":9555,"style":9556},[294],"height:1.705em;",[15,9558,9560,9563],{"style":9559},"top:-3.855em;",[15,9561],{"className":9562,"style":461},[302],[15,9564,9566,9606,9609,9612,9615,9618,9621,9624,9627,9630,9670,9677,9680,9683,9686,9689],{"className":9565},[59],[15,9567,9569,9572],{"className":9568},[59],[15,9570,199],{"className":9571,"style":277},[59,60],[15,9573,9575],{"className":9574},[281],[15,9576,9578,9598],{"className":9577},[285,286],[15,9579,9581,9595],{"className":9580},[290],[15,9582,9584],{"className":9583,"style":295},[294],[15,9585,9586,9589],{"style":298},[15,9587],{"className":9588,"style":303},[302],[15,9590,9592],{"className":9591},[307,308,309,310],[15,9593,37],{"className":9594,"style":61},[59,60,310],[15,9596,318],{"className":9597},[317],[15,9599,9601],{"className":9600},[290],[15,9602,9604],{"className":9603,"style":325},[294],[15,9605],{},[15,9607,113],{"className":9608},[146],[15,9610,206],{"className":9611,"style":334},[59,60],[15,9613,210],{"className":9614},[59],[15,9616,213],{"className":9617},[59,60],[15,9619,120],{"className":9620},[153],[15,9622],{"className":9623,"style":158},[157],[15,9625,123],{"className":9626},[162],[15,9628],{"className":9629,"style":158},[157],[15,9631,9633,9636],{"className":9632},[59],[15,9634,223],{"className":9635},[59,366],[15,9637,9639],{"className":9638},[281],[15,9640,9642,9662],{"className":9641},[285,286],[15,9643,9645,9659],{"className":9644},[290],[15,9646,9648],{"className":9647,"style":379},[294],[15,9649,9650,9653],{"style":382},[15,9651],{"className":9652,"style":303},[302],[15,9654,9656],{"className":9655},[307,308,309,310],[15,9657,213],{"className":9658},[59,60,310],[15,9660,318],{"className":9661},[317],[15,9663,9665],{"className":9664},[290],[15,9666,9668],{"className":9667,"style":325},[294],[15,9669],{},[15,9671,9673],{"className":9672},[59],[15,9674,230],{"className":9675},[419,9676],"size1",[15,9678,37],{"className":9679,"style":61},[59,60],[15,9681,113],{"className":9682},[146],[15,9684,769],{"className":9685},[59,60],[15,9687,120],{"className":9688},[153],[15,9690,9692],{"className":9691},[59],[15,9693,257],{"className":9694},[419,9676],[15,9696,9698,9701],{"style":9697},"top:-2.155em;",[15,9699],{"className":9700,"style":461},[302],[15,9702],{"className":9703},[59],[15,9705,318],{"className":9706},[317],[15,9708,9710],{"className":9709},[290],[15,9711,9714],{"className":9712,"style":9713},[294],"height:1.205em;",[15,9715],{},[15,9717,9719],{"className":9718},[1453],[15,9720,9722,9865],{"className":9721},[285,286],[15,9723,9725,9862],{"className":9724},[290],[15,9726,9728,9818],{"className":9727,"style":9556},[294],[15,9729,9730,9733],{"style":9559},[15,9731],{"className":9732,"style":461},[302],[15,9734,9736,9739,9742,9745,9748,9751,9754,9757,9763,9803,9806,9809,9812],{"className":9735},[59],[15,9737],{"className":9738},[59],[15,9740,4285],{"className":9741},[157],[15,9743],{"className":9744,"style":158},[157],[15,9746,6930],{"className":9747},[162],[15,9749,4285],{"className":9750},[157],[15,9752],{"className":9753,"style":158},[157],[15,9755,37],{"className":9756,"style":61},[59,60],[15,9758,9760],{"className":9759},[59],[15,9761,113],{"className":9762},[419,9676],[15,9764,9766,9769],{"className":9765},[59],[15,9767,223],{"className":9768},[59,366],[15,9770,9772],{"className":9771},[281],[15,9773,9775,9795],{"className":9774},[285,286],[15,9776,9778,9792],{"className":9777},[290],[15,9779,9781],{"className":9780,"style":379},[294],[15,9782,9783,9786],{"style":382},[15,9784],{"className":9785,"style":303},[302],[15,9787,9789],{"className":9788},[307,308,309,310],[15,9790,213],{"className":9791},[59,60,310],[15,9793,318],{"className":9794},[317],[15,9796,9798],{"className":9797},[290],[15,9799,9801],{"className":9800,"style":325},[294],[15,9802],{},[15,9804,230],{"className":9805},[146],[15,9807,769],{"className":9808},[59,60],[15,9810,257],{"className":9811},[153],[15,9813,9815],{"className":9814},[59],[15,9816,120],{"className":9817},[419,9676],[15,9819,9820,9823],{"style":9697},[15,9821],{"className":9822,"style":461},[302],[15,9824,9826,9829,9832,9835,9838,9841,9844,9847,9850,9853,9856,9859],{"className":9825},[59],[15,9827],{"className":9828},[59],[15,9830],{"className":9831,"style":158},[157],[15,9833,123],{"className":9834},[162],[15,9836],{"className":9837,"style":158},[157],[15,9839,37],{"className":9840,"style":61},[59,60],[15,9842,113],{"className":9843},[146],[15,9845,117],{"className":9846},[59],[15,9848,120],{"className":9849},[153],[15,9851],{"className":9852,"style":158},[157],[15,9854,123],{"className":9855},[162],[15,9857],{"className":9858,"style":158},[157],[15,9860,5766],{"className":9861},[59],[15,9863,318],{"className":9864},[317],[15,9866,9868],{"className":9867},[290],[15,9869,9871],{"className":9870,"style":9713},[294],[15,9872],{},[11,9874,9875,9876,9904,9905,9933,9934,10027],{},"这就是非负性。",[15,9877,9879,9892],{"className":9878,"translate":19},[18],[15,9880,9882],{"className":9881},[23],[25,9883,9884],{"xmlns":27},[29,9885,9886,9890],{},[32,9887,9888],{},[35,9889,37],{},[39,9891,37],{"encoding":41},[15,9893,9895],{"className":9894,"ariaHidden":46},[45],[15,9896,9898,9901],{"className":9897},[50],[15,9899],{"className":9900,"style":55},[54],[15,9902,37],{"className":9903,"style":61},[59,60]," 的凸性是关键：若 ",[15,9906,9908,9921],{"className":9907,"translate":19},[18],[15,9909,9911],{"className":9910},[23],[25,9912,9913],{"xmlns":27},[29,9914,9915,9919],{},[32,9916,9917],{},[35,9918,37],{},[39,9920,37],{"encoding":41},[15,9922,9924],{"className":9923,"ariaHidden":46},[45],[15,9925,9927,9930],{"className":9926},[50],[15,9928],{"className":9929,"style":55},[54],[15,9931,37],{"className":9932,"style":61},[59,60]," 是凹函数，Jensen 方向反转，",[15,9935,9937,9960],{"className":9936,"translate":19},[18],[15,9938,9940],{"className":9939},[23],[25,9941,9942],{"xmlns":27},[29,9943,9944,9957],{},[32,9945,9946,9952,9955],{},[195,9947,9948,9950],{},[35,9949,199],{},[35,9951,37],{},[110,9953,9954],{},"≤",[115,9956,126],{},[39,9958,9959],{"encoding":41},"D_f\\le 0",[15,9961,9963,10018],{"className":9962,"ariaHidden":46},[45],[15,9964,9966,9969,10009,10012,10015],{"className":9965},[50],[15,9967],{"className":9968,"style":4034},[54],[15,9970,9972,9975],{"className":9971},[59],[15,9973,199],{"className":9974,"style":277},[59,60],[15,9976,9978],{"className":9977},[281],[15,9979,9981,10001],{"className":9980},[285,286],[15,9982,9984,9998],{"className":9983},[290],[15,9985,9987],{"className":9986,"style":295},[294],[15,9988,9989,9992],{"style":298},[15,9990],{"className":9991,"style":303},[302],[15,9993,9995],{"className":9994},[307,308,309,310],[15,9996,37],{"className":9997,"style":61},[59,60,310],[15,9999,318],{"className":10000},[317],[15,10002,10004],{"className":10003},[290],[15,10005,10007],{"className":10006,"style":325},[294],[15,10008],{},[15,10010],{"className":10011,"style":158},[157],[15,10013,9954],{"className":10014},[162],[15,10016],{"className":10017,"style":158},[157],[15,10019,10021,10024],{"className":10020},[50],[15,10022],{"className":10023,"style":172},[54],[15,10025,126],{"className":10026},[59],"，那就不是「散度」了。",[10029,10030,10031,10183],"ul",{},[6667,10032,10033,10064,10065,10182],{},[998,10034,10035,10063],{},[15,10036,10038,10051],{"className":10037,"translate":19},[18],[15,10039,10041],{"className":10040},[23],[25,10042,10043],{"xmlns":27},[29,10044,10045,10049],{},[32,10046,10047],{},[35,10048,37],{},[39,10050,37],{"encoding":41},[15,10052,10054],{"className":10053,"ariaHidden":46},[45],[15,10055,10057,10060],{"className":10056},[50],[15,10058],{"className":10059,"style":55},[54],[15,10061,37],{"className":10062,"style":61},[59,60]," 凸","：保证 ",[15,10066,10068,10100],{"className":10067,"translate":19},[18],[15,10069,10071],{"className":10070},[23],[25,10072,10073],{"xmlns":27},[29,10074,10075,10097],{},[32,10076,10077,10083,10085,10087,10089,10091,10093,10095],{},[195,10078,10079,10081],{},[35,10080,199],{},[35,10082,37],{},[110,10084,113],{"stretchy":112},[35,10086,206],{},[35,10088,210],{"mathvariant":209},[35,10090,213],{},[110,10092,120],{"stretchy":112},[110,10094,6930],{},[115,10096,126],{},[39,10098,10099],{"encoding":41},"D_f(P\\|Q)\\ge 0",[15,10101,10103,10173],{"className":10102,"ariaHidden":46},[45],[15,10104,10106,10109,10149,10152,10155,10158,10161,10164,10167,10170],{"className":10105},[50],[15,10107],{"className":10108,"style":270},[54],[15,10110,10112,10115],{"className":10111},[59],[15,10113,199],{"className":10114,"style":277},[59,60],[15,10116,10118],{"className":10117},[281],[15,10119,10121,10141],{"className":10120},[285,286],[15,10122,10124,10138],{"className":10123},[290],[15,10125,10127],{"className":10126,"style":295},[294],[15,10128,10129,10132],{"style":298},[15,10130],{"className":10131,"style":303},[302],[15,10133,10135],{"className":10134},[307,308,309,310],[15,10136,37],{"className":10137,"style":61},[59,60,310],[15,10139,318],{"className":10140},[317],[15,10142,10144],{"className":10143},[290],[15,10145,10147],{"className":10146,"style":325},[294],[15,10148],{},[15,10150,113],{"className":10151},[146],[15,10153,206],{"className":10154,"style":334},[59,60],[15,10156,210],{"className":10157},[59],[15,10159,213],{"className":10160},[59,60],[15,10162,120],{"className":10163},[153],[15,10165],{"className":10166,"style":158},[157],[15,10168,6930],{"className":10169},[162],[15,10171],{"className":10172,"style":158},[157],[15,10174,10176,10179],{"className":10175},[50],[15,10177],{"className":10178,"style":172},[54],[15,10180,126],{"className":10181},[59],"。",[6667,10184,10185,10252,10253,10304,10305,10439,10440,10182],{},[998,10186,10187],{},[15,10188,10190,10213],{"className":10189,"translate":19},[18],[15,10191,10193],{"className":10192},[23],[25,10194,10195],{"xmlns":27},[29,10196,10197,10211],{},[32,10198,10199,10201,10203,10205,10207,10209],{},[35,10200,37],{},[110,10202,113],{"stretchy":112},[115,10204,117],{},[110,10206,120],{"stretchy":112},[110,10208,123],{},[115,10210,126],{},[39,10212,129],{"encoding":41},[15,10214,10216,10243],{"className":10215,"ariaHidden":46},[45],[15,10217,10219,10222,10225,10228,10231,10234,10237,10240],{"className":10218},[50],[15,10220],{"className":10221,"style":139},[54],[15,10223,37],{"className":10224,"style":61},[59,60],[15,10226,113],{"className":10227},[146],[15,10229,117],{"className":10230},[59],[15,10232,120],{"className":10233},[153],[15,10235],{"className":10236,"style":158},[157],[15,10238,123],{"className":10239},[162],[15,10241],{"className":10242,"style":158},[157],[15,10244,10246,10249],{"className":10245},[50],[15,10247],{"className":10248,"style":172},[54],[15,10250,126],{"className":10251},[59],"：保证当 ",[15,10254,10256,10274],{"className":10255,"translate":19},[18],[15,10257,10259],{"className":10258},[23],[25,10260,10261],{"xmlns":27},[29,10262,10263,10271],{},[32,10264,10265,10267,10269],{},[35,10266,206],{},[110,10268,123],{},[35,10270,213],{},[39,10272,10273],{"encoding":41},"P=Q",[15,10275,10277,10295],{"className":10276,"ariaHidden":46},[45],[15,10278,10280,10283,10286,10289,10292],{"className":10279},[50],[15,10281],{"className":10282,"style":664},[54],[15,10284,206],{"className":10285,"style":334},[59,60],[15,10287],{"className":10288,"style":158},[157],[15,10290,123],{"className":10291},[162],[15,10293],{"className":10294,"style":158},[157],[15,10296,10298,10301],{"className":10297},[50],[15,10299],{"className":10300,"style":566},[54],[15,10302,213],{"className":10303},[59,60]," 时，",[15,10306,10308,10338],{"className":10307,"translate":19},[18],[15,10309,10311],{"className":10310},[23],[25,10312,10313],{"xmlns":27},[29,10314,10315,10335],{},[32,10316,10317,10331,10333],{},[238,10318,10319,10325],{},[32,10320,10321,10323],{},[35,10322,244],{},[35,10324,206],{},[32,10326,10327,10329],{},[35,10328,244],{},[35,10330,213],{},[110,10332,123],{},[115,10334,117],{},[39,10336,10337],{"encoding":41},"\\frac{dP}{dQ}=1",[15,10339,10341,10430],{"className":10340,"ariaHidden":46},[45],[15,10342,10344,10347,10421,10424,10427],{"className":10343},[50],[15,10345],{"className":10346,"style":2670},[54],[15,10348,10350,10353,10418],{"className":10349},[59],[15,10351],{"className":10352},[146,441],[15,10354,10356],{"className":10355},[238],[15,10357,10359,10410],{"className":10358},[285,286],[15,10360,10362,10407],{"className":10361},[290],[15,10363,10365,10382,10390],{"className":10364,"style":2689},[294],[15,10366,10367,10370],{"style":2529},[15,10368],{"className":10369,"style":461},[302],[15,10371,10373],{"className":10372},[307,308,309,310],[15,10374,10376,10379],{"className":10375},[59,310],[15,10377,244],{"className":10378},[59,60,310],[15,10380,213],{"className":10381},[59,60,310],[15,10383,10384,10387],{"style":473},[15,10385],{"className":10386,"style":461},[302],[15,10388],{"className":10389,"style":481},[480],[15,10391,10392,10395],{"style":2552},[15,10393],{"className":10394,"style":461},[302],[15,10396,10398],{"className":10397},[307,308,309,310],[15,10399,10401,10404],{"className":10400},[59,310],[15,10402,244],{"className":10403},[59,60,310],[15,10405,206],{"className":10406,"style":334},[59,60,310],[15,10408,318],{"className":10409},[317],[15,10411,10413],{"className":10412},[290],[15,10414,10416],{"className":10415,"style":2741},[294],[15,10417],{},[15,10419],{"className":10420},[153,441],[15,10422],{"className":10423,"style":158},[157],[15,10425,123],{"className":10426},[162],[15,10428],{"className":10429,"style":158},[157],[15,10431,10433,10436],{"className":10432},[50],[15,10434],{"className":10435,"style":172},[54],[15,10437,117],{"className":10438},[59],"，从而 ",[15,10441,10443,10475],{"className":10442,"translate":19},[18],[15,10444,10446],{"className":10445},[23],[25,10447,10448],{"xmlns":27},[29,10449,10450,10472],{},[32,10451,10452,10458,10460,10462,10464,10466,10468,10470],{},[195,10453,10454,10456],{},[35,10455,199],{},[35,10457,37],{},[110,10459,113],{"stretchy":112},[35,10461,206],{},[35,10463,210],{"mathvariant":209},[35,10465,206],{},[110,10467,120],{"stretchy":112},[110,10469,123],{},[115,10471,126],{},[39,10473,10474],{"encoding":41},"D_f(P\\|P)=0",[15,10476,10478,10548],{"className":10477,"ariaHidden":46},[45],[15,10479,10481,10484,10524,10527,10530,10533,10536,10539,10542,10545],{"className":10480},[50],[15,10482],{"className":10483,"style":270},[54],[15,10485,10487,10490],{"className":10486},[59],[15,10488,199],{"className":10489,"style":277},[59,60],[15,10491,10493],{"className":10492},[281],[15,10494,10496,10516],{"className":10495},[285,286],[15,10497,10499,10513],{"className":10498},[290],[15,10500,10502],{"className":10501,"style":295},[294],[15,10503,10504,10507],{"style":298},[15,10505],{"className":10506,"style":303},[302],[15,10508,10510],{"className":10509},[307,308,309,310],[15,10511,37],{"className":10512,"style":61},[59,60,310],[15,10514,318],{"className":10515},[317],[15,10517,10519],{"className":10518},[290],[15,10520,10522],{"className":10521,"style":325},[294],[15,10523],{},[15,10525,113],{"className":10526},[146],[15,10528,206],{"className":10529,"style":334},[59,60],[15,10531,210],{"className":10532},[59],[15,10534,206],{"className":10535,"style":334},[59,60],[15,10537,120],{"className":10538},[153],[15,10540],{"className":10541,"style":158},[157],[15,10543,123],{"className":10544},[162],[15,10546],{"className":10547,"style":158},[157],[15,10549,10551,10554],{"className":10550},[50],[15,10552],{"className":10553,"style":172},[54],[15,10555,126],{"className":10556},[59],[11,10558,10559],{},"因此 f-散度天然具有非负性和「相同分布散度为零」的性质。",[2313,10561,10563],{"id":10562},"_22-凸性为何必需","2.2 凸性为何必需",[11,10565,10566,10567,10636,10637,10665],{},"有人会问：凸性是不是一个过强的技术条件？答案是否定的。若要求 ",[15,10568,10570,10587],{"className":10569,"translate":19},[18],[15,10571,10573],{"className":10572},[23],[25,10574,10575],{"xmlns":27},[29,10576,10577,10585],{},[32,10578,10579],{},[195,10580,10581,10583],{},[35,10582,199],{},[35,10584,37],{},[39,10586,4024],{"encoding":41},[15,10588,10590],{"className":10589,"ariaHidden":46},[45],[15,10591,10593,10596],{"className":10592},[50],[15,10594],{"className":10595,"style":4034},[54],[15,10597,10599,10602],{"className":10598},[59],[15,10600,199],{"className":10601,"style":277},[59,60],[15,10603,10605],{"className":10604},[281],[15,10606,10608,10628],{"className":10607},[285,286],[15,10609,10611,10625],{"className":10610},[290],[15,10612,10614],{"className":10613,"style":295},[294],[15,10615,10616,10619],{"style":298},[15,10617],{"className":10618,"style":303},[302],[15,10620,10622],{"className":10621},[307,308,309,310],[15,10623,37],{"className":10624,"style":61},[59,60,310],[15,10626,318],{"className":10627},[317],[15,10629,10631],{"className":10630},[290],[15,10632,10634],{"className":10633,"style":325},[294],[15,10635],{}," 对一切「随机变换」（马尔可夫核）单调不增（即第 4.5 节的数据处理不等式），那么 ",[15,10638,10640,10653],{"className":10639,"translate":19},[18],[15,10641,10643],{"className":10642},[23],[25,10644,10645],{"xmlns":27},[29,10646,10647,10651],{},[32,10648,10649],{},[35,10650,37],{},[39,10652,37],{"encoding":41},[15,10654,10656],{"className":10655,"ariaHidden":46},[45],[15,10657,10659,10662],{"className":10658},[50],[15,10660],{"className":10661,"style":55},[54],[15,10663,37],{"className":10664,"style":61},[59,60]," 必须凸。二元分布上就能看到这一点。",[993,10667,10668],{},[11,10669,10670,10673,10674,10869,10870,10911,10912,10940,10941,10969],{},[998,10671,10672],{},"命题 2.1（Csiszár 刻画）"," 若数据处理不等式 ",[15,10675,10677,10728],{"className":10676,"translate":19},[18],[15,10678,10680],{"className":10679},[23],[25,10681,10682],{"xmlns":27},[29,10683,10684,10725],{},[32,10685,10686,10692,10694,10696,10699,10701,10703,10705,10707,10709,10715,10717,10719,10721,10723],{},[195,10687,10688,10690],{},[35,10689,199],{},[35,10691,37],{},[110,10693,113],{"stretchy":112},[35,10695,206],{},[35,10697,10698],{},"K",[35,10700,210],{"mathvariant":209},[35,10702,213],{},[35,10704,10698],{},[110,10706,120],{"stretchy":112},[110,10708,9954],{},[195,10710,10711,10713],{},[35,10712,199],{},[35,10714,37],{},[110,10716,113],{"stretchy":112},[35,10718,206],{},[35,10720,210],{"mathvariant":209},[35,10722,213],{},[110,10724,120],{"stretchy":112},[39,10726,10727],{"encoding":41},"D_f(PK\\|QK)\\le D_f(P\\|Q)",[15,10729,10731,10808],{"className":10730,"ariaHidden":46},[45],[15,10732,10734,10737,10777,10780,10783,10787,10790,10793,10796,10799,10802,10805],{"className":10733},[50],[15,10735],{"className":10736,"style":270},[54],[15,10738,10740,10743],{"className":10739},[59],[15,10741,199],{"className":10742,"style":277},[59,60],[15,10744,10746],{"className":10745},[281],[15,10747,10749,10769],{"className":10748},[285,286],[15,10750,10752,10766],{"className":10751},[290],[15,10753,10755],{"className":10754,"style":295},[294],[15,10756,10757,10760],{"style":298},[15,10758],{"className":10759,"style":303},[302],[15,10761,10763],{"className":10762},[307,308,309,310],[15,10764,37],{"className":10765,"style":61},[59,60,310],[15,10767,318],{"className":10768},[317],[15,10770,10772],{"className":10771},[290],[15,10773,10775],{"className":10774,"style":325},[294],[15,10776],{},[15,10778,113],{"className":10779},[146],[15,10781,206],{"className":10782,"style":334},[59,60],[15,10784,10698],{"className":10785,"style":10786},[59,60],"margin-right:0.0715em;",[15,10788,210],{"className":10789},[59],[15,10791,213],{"className":10792},[59,60],[15,10794,10698],{"className":10795,"style":10786},[59,60],[15,10797,120],{"className":10798},[153],[15,10800],{"className":10801,"style":158},[157],[15,10803,9954],{"className":10804},[162],[15,10806],{"className":10807,"style":158},[157],[15,10809,10811,10814,10854,10857,10860,10863,10866],{"className":10810},[50],[15,10812],{"className":10813,"style":270},[54],[15,10815,10817,10820],{"className":10816},[59],[15,10818,199],{"className":10819,"style":277},[59,60],[15,10821,10823],{"className":10822},[281],[15,10824,10826,10846],{"className":10825},[285,286],[15,10827,10829,10843],{"className":10828},[290],[15,10830,10832],{"className":10831,"style":295},[294],[15,10833,10834,10837],{"style":298},[15,10835],{"className":10836,"style":303},[302],[15,10838,10840],{"className":10839},[307,308,309,310],[15,10841,37],{"className":10842,"style":61},[59,60,310],[15,10844,318],{"className":10845},[317],[15,10847,10849],{"className":10848},[290],[15,10850,10852],{"className":10851,"style":325},[294],[15,10853],{},[15,10855,113],{"className":10856},[146],[15,10858,206],{"className":10859,"style":334},[59,60],[15,10861,210],{"className":10862},[59],[15,10864,213],{"className":10865},[59,60],[15,10867,120],{"className":10868},[153]," 对一切概率对 ",[15,10871,10873,10890],{"className":10872,"translate":19},[18],[15,10874,10876],{"className":10875},[23],[25,10877,10878],{"xmlns":27},[29,10879,10880,10888],{},[32,10881,10882,10884,10886],{},[35,10883,206],{},[110,10885,551],{"separator":46},[35,10887,213],{},[39,10889,556],{"encoding":41},[15,10891,10893],{"className":10892,"ariaHidden":46},[45],[15,10894,10896,10899,10902,10905,10908],{"className":10895},[50],[15,10897],{"className":10898,"style":566},[54],[15,10900,206],{"className":10901,"style":334},[59,60],[15,10903,551],{"className":10904},[573],[15,10906],{"className":10907,"style":406},[157],[15,10909,213],{"className":10910},[59,60]," 与一切马尔可夫核 ",[15,10913,10915,10928],{"className":10914,"translate":19},[18],[15,10916,10918],{"className":10917},[23],[25,10919,10920],{"xmlns":27},[29,10921,10922,10926],{},[32,10923,10924],{},[35,10925,10698],{},[39,10927,10698],{"encoding":41},[15,10929,10931],{"className":10930,"ariaHidden":46},[45],[15,10932,10934,10937],{"className":10933},[50],[15,10935],{"className":10936,"style":664},[54],[15,10938,10698],{"className":10939,"style":10786},[59,60]," 成立，则 ",[15,10942,10944,10957],{"className":10943,"translate":19},[18],[15,10945,10947],{"className":10946},[23],[25,10948,10949],{"xmlns":27},[29,10950,10951,10955],{},[32,10952,10953],{},[35,10954,37],{},[39,10956,37],{"encoding":41},[15,10958,10960],{"className":10959,"ariaHidden":46},[45],[15,10961,10963,10966],{"className":10962},[50],[15,10964],{"className":10965,"style":55},[54],[15,10967,37],{"className":10968,"style":61},[59,60]," 凸。",[11,10971,10972],{},[998,10973,10974],{},"证明（两步）。",[11,10976,10977,11064,11065,2282,11299,11328,11329,11405,11406,5647,11604,11801,11802,11939],{},[998,10978,10979,10980,6496,11011,11063],{},"第一步：DPI ",[15,10981,10983,10998],{"className":10982,"translate":19},[18],[15,10984,10986],{"className":10985},[23],[25,10987,10988],{"xmlns":27},[29,10989,10990,10995],{},[32,10991,10992],{},[110,10993,10994],{},"⇒",[39,10996,10997],{"encoding":41},"\\Rightarrow",[15,10999,11001],{"className":11000,"ariaHidden":46},[45],[15,11002,11004,11008],{"className":11003},[50],[15,11005],{"className":11006,"style":11007},[54],"height:0.3669em;",[15,11009,10994],{"className":11010},[162],[15,11012,11014,11036],{"className":11013,"translate":19},[18],[15,11015,11017],{"className":11016},[23],[25,11018,11019],{"xmlns":27},[29,11020,11021,11033],{},[32,11022,11023,11025,11027,11029,11031],{},[110,11024,113],{"stretchy":112},[35,11026,206],{},[110,11028,551],{"separator":46},[35,11030,213],{},[110,11032,120],{"stretchy":112},[39,11034,11035],{"encoding":41},"(P,Q)",[15,11037,11039],{"className":11038,"ariaHidden":46},[45],[15,11040,11042,11045,11048,11051,11054,11057,11060],{"className":11041},[50],[15,11043],{"className":11044,"style":139},[54],[15,11046,113],{"className":11047},[146],[15,11049,206],{"className":11050,"style":334},[59,60],[15,11052,551],{"className":11053},[573],[15,11055],{"className":11056,"style":406},[157],[15,11058,213],{"className":11059},[59,60],[15,11061,120],{"className":11062},[153]," 联合凸。"," 设 ",[15,11066,11068,11110],{"className":11067,"translate":19},[18],[15,11069,11071],{"className":11070},[23],[25,11072,11073],{"xmlns":27},[29,11074,11075,11107],{},[32,11076,11077,11083,11085,11091,11093,11099,11101],{},[195,11078,11079,11081],{},[35,11080,206],{},[115,11082,117],{},[110,11084,551],{"separator":46},[195,11086,11087,11089],{},[35,11088,206],{},[115,11090,2465],{},[110,11092,551],{"separator":46},[195,11094,11095,11097],{},[35,11096,213],{},[115,11098,117],{},[110,11100,551],{"separator":46},[195,11102,11103,11105],{},[35,11104,213],{},[115,11106,2465],{},[39,11108,11109],{"encoding":41},"P_1,P_2,Q_1,Q_2",[15,11111,11113],{"className":11112,"ariaHidden":46},[45],[15,11114,11116,11119,11161,11164,11167,11207,11210,11213,11253,11256,11259],{"className":11115},[50],[15,11117],{"className":11118,"style":566},[54],[15,11120,11122,11125],{"className":11121},[59],[15,11123,206],{"className":11124,"style":334},[59,60],[15,11126,11128],{"className":11127},[281],[15,11129,11131,11153],{"className":11130},[285,286],[15,11132,11134,11150],{"className":11133},[290],[15,11135,11138],{"className":11136,"style":11137},[294],"height:0.3011em;",[15,11139,11141,11144],{"style":11140},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[15,11142],{"className":11143,"style":303},[302],[15,11145,11147],{"className":11146},[307,308,309,310],[15,11148,117],{"className":11149},[59,310],[15,11151,318],{"className":11152},[317],[15,11154,11156],{"className":11155},[290],[15,11157,11159],{"className":11158,"style":7970},[294],[15,11160],{},[15,11162,551],{"className":11163},[573],[15,11165],{"className":11166,"style":406},[157],[15,11168,11170,11173],{"className":11169},[59],[15,11171,206],{"className":11172,"style":334},[59,60],[15,11174,11176],{"className":11175},[281],[15,11177,11179,11199],{"className":11178},[285,286],[15,11180,11182,11196],{"className":11181},[290],[15,11183,11185],{"className":11184,"style":11137},[294],[15,11186,11187,11190],{"style":11140},[15,11188],{"className":11189,"style":303},[302],[15,11191,11193],{"className":11192},[307,308,309,310],[15,11194,2465],{"className":11195},[59,310],[15,11197,318],{"className":11198},[317],[15,11200,11202],{"className":11201},[290],[15,11203,11205],{"className":11204,"style":7970},[294],[15,11206],{},[15,11208,551],{"className":11209},[573],[15,11211],{"className":11212,"style":406},[157],[15,11214,11216,11219],{"className":11215},[59],[15,11217,213],{"className":11218},[59,60],[15,11220,11222],{"className":11221},[281],[15,11223,11225,11245],{"className":11224},[285,286],[15,11226,11228,11242],{"className":11227},[290],[15,11229,11231],{"className":11230,"style":11137},[294],[15,11232,11233,11236],{"style":382},[15,11234],{"className":11235,"style":303},[302],[15,11237,11239],{"className":11238},[307,308,309,310],[15,11240,117],{"className":11241},[59,310],[15,11243,318],{"className":11244},[317],[15,11246,11248],{"className":11247},[290],[15,11249,11251],{"className":11250,"style":7970},[294],[15,11252],{},[15,11254,551],{"className":11255},[573],[15,11257],{"className":11258,"style":406},[157],[15,11260,11262,11265],{"className":11261},[59],[15,11263,213],{"className":11264},[59,60],[15,11266,11268],{"className":11267},[281],[15,11269,11271,11291],{"className":11270},[285,286],[15,11272,11274,11288],{"className":11273},[290],[15,11275,11277],{"className":11276,"style":11137},[294],[15,11278,11279,11282],{"style":382},[15,11280],{"className":11281,"style":303},[302],[15,11283,11285],{"className":11284},[307,308,309,310],[15,11286,2465],{"className":11287},[59,310],[15,11289,318],{"className":11290},[317],[15,11292,11294],{"className":11293},[290],[15,11295,11297],{"className":11296,"style":7970},[294],[15,11298],{},[15,11300,11302,11316],{"className":11301,"translate":19},[18],[15,11303,11305],{"className":11304},[23],[25,11306,11307],{"xmlns":27},[29,11308,11309,11313],{},[32,11310,11311],{},[35,11312,598],{"mathvariant":597},[39,11314,11315],{"encoding":41},"\\mathcal X",[15,11317,11319],{"className":11318,"ariaHidden":46},[45],[15,11320,11322,11325],{"className":11321},[50],[15,11323],{"className":11324,"style":664},[54],[15,11326,598],{"className":11327,"style":625},[59,624]," 上任给的四对分布，",[15,11330,11332,11359],{"className":11331,"translate":19},[18],[15,11333,11335],{"className":11334},[23],[25,11336,11337],{"xmlns":27},[29,11338,11339,11356],{},[32,11340,11341,11344,11346,11348,11350,11352,11354],{},[35,11342,11343],{},"λ",[110,11345,5911],{},[110,11347,113],{"stretchy":112},[115,11349,126],{},[110,11351,551],{"separator":46},[115,11353,117],{},[110,11355,120],{"stretchy":112},[39,11357,11358],{"encoding":41},"\\lambda\\in(0,1)",[15,11360,11362,11381],{"className":11361,"ariaHidden":46},[45],[15,11363,11365,11369,11372,11375,11378],{"className":11364},[50],[15,11366],{"className":11367,"style":11368},[54],"height:0.7335em;vertical-align:-0.0391em;",[15,11370,11343],{"className":11371},[59,60],[15,11373],{"className":11374,"style":158},[157],[15,11376,5911],{"className":11377},[162],[15,11379],{"className":11380,"style":158},[157],[15,11382,11384,11387,11390,11393,11396,11399,11402],{"className":11383},[50],[15,11385],{"className":11386,"style":139},[54],[15,11388,113],{"className":11389},[146],[15,11391,126],{"className":11392},[59],[15,11394,551],{"className":11395},[573],[15,11397],{"className":11398,"style":406},[157],[15,11400,117],{"className":11401},[59],[15,11403,120],{"className":11404},[153],"，记 ",[15,11407,11409,11451],{"className":11408,"translate":19},[18],[15,11410,11412],{"className":11411},[23],[25,11413,11414],{"xmlns":27},[29,11415,11416,11448],{},[32,11417,11418,11420,11422,11424,11430,11432,11434,11436,11438,11440,11442],{},[35,11419,206],{},[110,11421,123],{},[35,11423,11343],{},[195,11425,11426,11428],{},[35,11427,206],{},[115,11429,117],{},[110,11431,2472],{},[110,11433,113],{"stretchy":112},[115,11435,117],{},[110,11437,4815],{},[35,11439,11343],{},[110,11441,120],{"stretchy":112},[195,11443,11444,11446],{},[35,11445,206],{},[115,11447,2465],{},[39,11449,11450],{"encoding":41},"P=\\lambda P_1+(1-\\lambda)P_2",[15,11452,11454,11472,11531,11552],{"className":11453,"ariaHidden":46},[45],[15,11455,11457,11460,11463,11466,11469],{"className":11456},[50],[15,11458],{"className":11459,"style":664},[54],[15,11461,206],{"className":11462,"style":334},[59,60],[15,11464],{"className":11465,"style":158},[157],[15,11467,123],{"className":11468},[162],[15,11470],{"className":11471,"style":158},[157],[15,11473,11475,11479,11482,11522,11525,11528],{"className":11474},[50],[15,11476],{"className":11477,"style":11478},[54],"height:0.8444em;vertical-align:-0.15em;",[15,11480,11343],{"className":11481},[59,60],[15,11483,11485,11488],{"className":11484},[59],[15,11486,206],{"className":11487,"style":334},[59,60],[15,11489,11491],{"className":11490},[281],[15,11492,11494,11514],{"className":11493},[285,286],[15,11495,11497,11511],{"className":11496},[290],[15,11498,11500],{"className":11499,"style":11137},[294],[15,11501,11502,11505],{"style":11140},[15,11503],{"className":11504,"style":303},[302],[15,11506,11508],{"className":11507},[307,308,309,310],[15,11509,117],{"className":11510},[59,310],[15,11512,318],{"className":11513},[317],[15,11515,11517],{"className":11516},[290],[15,11518,11520],{"className":11519,"style":7970},[294],[15,11521],{},[15,11523],{"className":11524,"style":2589},[157],[15,11526,2472],{"className":11527},[2593],[15,11529],{"className":11530,"style":2589},[157],[15,11532,11534,11537,11540,11543,11546,11549],{"className":11533},[50],[15,11535],{"className":11536,"style":139},[54],[15,11538,113],{"className":11539},[146],[15,11541,117],{"className":11542},[59],[15,11544],{"className":11545,"style":2589},[157],[15,11547,4815],{"className":11548},[2593],[15,11550],{"className":11551,"style":2589},[157],[15,11553,11555,11558,11561,11564],{"className":11554},[50],[15,11556],{"className":11557,"style":139},[54],[15,11559,11343],{"className":11560},[59,60],[15,11562,120],{"className":11563},[153],[15,11565,11567,11570],{"className":11566},[59],[15,11568,206],{"className":11569,"style":334},[59,60],[15,11571,11573],{"className":11572},[281],[15,11574,11576,11596],{"className":11575},[285,286],[15,11577,11579,11593],{"className":11578},[290],[15,11580,11582],{"className":11581,"style":11137},[294],[15,11583,11584,11587],{"style":11140},[15,11585],{"className":11586,"style":303},[302],[15,11588,11590],{"className":11589},[307,308,309,310],[15,11591,2465],{"className":11592},[59,310],[15,11594,318],{"className":11595},[317],[15,11597,11599],{"className":11598},[290],[15,11600,11602],{"className":11601,"style":7970},[294],[15,11603],{},[15,11605,11607,11649],{"className":11606,"translate":19},[18],[15,11608,11610],{"className":11609},[23],[25,11611,11612],{"xmlns":27},[29,11613,11614,11646],{},[32,11615,11616,11618,11620,11622,11628,11630,11632,11634,11636,11638,11640],{},[35,11617,213],{},[110,11619,123],{},[35,11621,11343],{},[195,11623,11624,11626],{},[35,11625,213],{},[115,11627,117],{},[110,11629,2472],{},[110,11631,113],{"stretchy":112},[115,11633,117],{},[110,11635,4815],{},[35,11637,11343],{},[110,11639,120],{"stretchy":112},[195,11641,11642,11644],{},[35,11643,213],{},[115,11645,2465],{},[39,11647,11648],{"encoding":41},"Q=\\lambda Q_1+(1-\\lambda)Q_2",[15,11650,11652,11670,11728,11749],{"className":11651,"ariaHidden":46},[45],[15,11653,11655,11658,11661,11664,11667],{"className":11654},[50],[15,11656],{"className":11657,"style":566},[54],[15,11659,213],{"className":11660},[59,60],[15,11662],{"className":11663,"style":158},[157],[15,11665,123],{"className":11666},[162],[15,11668],{"className":11669,"style":158},[157],[15,11671,11673,11676,11679,11719,11722,11725],{"className":11672},[50],[15,11674],{"className":11675,"style":55},[54],[15,11677,11343],{"className":11678},[59,60],[15,11680,11682,11685],{"className":11681},[59],[15,11683,213],{"className":11684},[59,60],[15,11686,11688],{"className":11687},[281],[15,11689,11691,11711],{"className":11690},[285,286],[15,11692,11694,11708],{"className":11693},[290],[15,11695,11697],{"className":11696,"style":11137},[294],[15,11698,11699,11702],{"style":382},[15,11700],{"className":11701,"style":303},[302],[15,11703,11705],{"className":11704},[307,308,309,310],[15,11706,117],{"className":11707},[59,310],[15,11709,318],{"className":11710},[317],[15,11712,11714],{"className":11713},[290],[15,11715,11717],{"className":11716,"style":7970},[294],[15,11718],{},[15,11720],{"className":11721,"style":2589},[157],[15,11723,2472],{"className":11724},[2593],[15,11726],{"className":11727,"style":2589},[157],[15,11729,11731,11734,11737,11740,11743,11746],{"className":11730},[50],[15,11732],{"className":11733,"style":139},[54],[15,11735,113],{"className":11736},[146],[15,11738,117],{"className":11739},[59],[15,11741],{"className":11742,"style":2589},[157],[15,11744,4815],{"className":11745},[2593],[15,11747],{"className":11748,"style":2589},[157],[15,11750,11752,11755,11758,11761],{"className":11751},[50],[15,11753],{"className":11754,"style":139},[54],[15,11756,11343],{"className":11757},[59,60],[15,11759,120],{"className":11760},[153],[15,11762,11764,11767],{"className":11763},[59],[15,11765,213],{"className":11766},[59,60],[15,11768,11770],{"className":11769},[281],[15,11771,11773,11793],{"className":11772},[285,286],[15,11774,11776,11790],{"className":11775},[290],[15,11777,11779],{"className":11778,"style":11137},[294],[15,11780,11781,11784],{"style":382},[15,11782],{"className":11783,"style":303},[302],[15,11785,11787],{"className":11786},[307,308,309,310],[15,11788,2465],{"className":11789},[59,310],[15,11791,318],{"className":11792},[317],[15,11794,11796],{"className":11795},[290],[15,11797,11799],{"className":11798,"style":7970},[294],[15,11800],{},"。构造扩大一倍的样本空间 ",[15,11803,11805,11842],{"className":11804,"translate":19},[18],[15,11806,11808],{"className":11807},[23],[25,11809,11810],{"xmlns":27},[29,11811,11812,11839],{},[32,11813,11814,11822,11824,11826,11828,11830,11832,11834,11837],{},[11815,11816,11817,11819],"mover",{"accent":46},[35,11818,598],{"mathvariant":597},[110,11820,11821],{},"~",[110,11823,123],{},[110,11825,2900],{"stretchy":112},[115,11827,117],{},[110,11829,551],{"separator":46},[115,11831,2465],{},[110,11833,2910],{"stretchy":112},[110,11835,11836],{},"×",[35,11838,598],{"mathvariant":597},[39,11840,11841],{"encoding":41},"\\tilde{\\mathcal X}=\\{1,2\\}\\times\\mathcal X",[15,11843,11845,11897,11930],{"className":11844,"ariaHidden":46},[45],[15,11846,11848,11852,11888,11891,11894],{"className":11847},[50],[15,11849],{"className":11850,"style":11851},[54],"height:0.9202em;",[15,11853,11856],{"className":11854},[59,11855],"accent",[15,11857,11859],{"className":11858},[285],[15,11860,11862],{"className":11861},[290],[15,11863,11865,11874],{"className":11864,"style":11851},[294],[15,11866,11868,11871],{"style":11867},"top:-3em;",[15,11869],{"className":11870,"style":461},[302],[15,11872,598],{"className":11873,"style":625},[59,624],[15,11875,11877,11880],{"style":11876},"top:-3.6023em;",[15,11878],{"className":11879,"style":461},[302],[15,11881,11885],{"className":11882,"style":11884},[11883],"accent-body","left:-0.1111em;",[15,11886,11821],{"className":11887},[59],[15,11889],{"className":11890,"style":158},[157],[15,11892,123],{"className":11893},[162],[15,11895],{"className":11896,"style":158},[157],[15,11898,11900,11903,11906,11909,11912,11915,11918,11921,11924,11927],{"className":11899},[50],[15,11901],{"className":11902,"style":139},[54],[15,11904,2900],{"className":11905},[146],[15,11907,117],{"className":11908},[59],[15,11910,551],{"className":11911},[573],[15,11913],{"className":11914,"style":406},[157],[15,11916,2465],{"className":11917},[59],[15,11919,2910],{"className":11920},[153],[15,11922],{"className":11923,"style":2589},[157],[15,11925,11836],{"className":11926},[2593],[15,11928],{"className":11929,"style":2589},[157],[15,11931,11933,11936],{"className":11932},[50],[15,11934],{"className":11935,"style":664},[54],[15,11937,598],{"className":11938,"style":625},[59,624]," 及其上的两个分块分布：",[15,11941,11943],{"className":11942,"translate":19},[180],[15,11944,11946,12081],{"className":11945,"translate":19},[18],[15,11947,11949],{"className":11948},[23],[25,11950,11951],{"xmlns":27,"display":189},[29,11952,11953,12078],{},[32,11954,11955,11961,11963,11965,11968,11970,11972,11974,11976,11978,11984,11986,11992,11994,11996,11998,12000,12002,12008,12010,12012,12014,12016,12018,12020,12022,12024,12030,12032,12038,12040,12042,12044,12046,12048,12054,12056,12058,12060,12062,12068,12070,12072,12074,12076],{},[11815,11956,11957,11959],{"accent":46},[35,11958,206],{},[110,11960,11821],{},[110,11962,113],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[110,11964,113],{"stretchy":112},[35,11966,11967],{},"i",[110,11969,551],{"separator":46},[35,11971,774],{},[110,11973,120],{"stretchy":112},[110,11975,120],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[110,11977,123],{},[195,11979,11980,11982],{},[35,11981,7572],{},[35,11983,11967],{},[1299,11985,1301],{},[195,11987,11988,11990],{},[35,11989,206],{},[35,11991,11967],{},[110,11993,113],{"stretchy":112},[35,11995,774],{},[110,11997,120],{"stretchy":112},[110,11999,551],{"separator":46},[157,12001],{"width":3264},[11815,12003,12004,12006],{"accent":46},[35,12005,213],{},[110,12007,11821],{},[110,12009,113],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[110,12011,113],{"stretchy":112},[35,12013,11967],{},[110,12015,551],{"separator":46},[35,12017,774],{},[110,12019,120],{"stretchy":112},[110,12021,120],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[110,12023,123],{},[195,12025,12026,12028],{},[35,12027,7572],{},[35,12029,11967],{},[1299,12031,1301],{},[195,12033,12034,12036],{},[35,12035,213],{},[35,12037,11967],{},[110,12039,113],{"stretchy":112},[35,12041,774],{},[110,12043,120],{"stretchy":112},[110,12045,551],{"separator":46},[157,12047],{"width":3264},[195,12049,12050,12052],{},[35,12051,7572],{},[115,12053,117],{},[110,12055,123],{},[35,12057,11343],{},[110,12059,551],{"separator":46},[1299,12061,4285],{},[195,12063,12064,12066],{},[35,12065,7572],{},[115,12067,2465],{},[110,12069,123],{},[115,12071,117],{},[110,12073,4815],{},[35,12075,11343],{},[35,12077,4377],{"mathvariant":209},[39,12079,12080],{"encoding":41},"\\tilde P\\big((i,x)\\big)=w_i\\,P_i(x),\\qquad\n\\tilde Q\\big((i,x)\\big)=w_i\\,Q_i(x),\\qquad\nw_1=\\lambda,\\ w_2=1-\\lambda .",[15,12082,12084,12162,12353,12509,12576,12595],{"className":12083,"ariaHidden":46},[45],[15,12085,12087,12091,12123,12129,12132,12135,12138,12141,12144,12147,12153,12156,12159],{"className":12086},[50],[15,12088],{"className":12089,"style":12090},[54],"height:1.2702em;vertical-align:-0.35em;",[15,12092,12094],{"className":12093},[59,11855],[15,12095,12097],{"className":12096},[285],[15,12098,12100],{"className":12099},[290],[15,12101,12103,12111],{"className":12102,"style":11851},[294],[15,12104,12105,12108],{"style":11867},[15,12106],{"className":12107,"style":461},[302],[15,12109,206],{"className":12110,"style":334},[59,60],[15,12112,12113,12116],{"style":11876},[15,12114],{"className":12115,"style":461},[302],[15,12117,12120],{"className":12118,"style":12119},[11883],"left:-0.1667em;",[15,12121,11821],{"className":12122},[59],[15,12124,12126],{"className":12125},[59],[15,12127,113],{"className":12128},[419,9676],[15,12130,113],{"className":12131},[146],[15,12133,11967],{"className":12134},[59,60],[15,12136,551],{"className":12137},[573],[15,12139],{"className":12140,"style":406},[157],[15,12142,774],{"className":12143},[59,60],[15,12145,120],{"className":12146},[153],[15,12148,12150],{"className":12149},[59],[15,12151,120],{"className":12152},[419,9676],[15,12154],{"className":12155,"style":158},[157],[15,12157,123],{"className":12158},[162],[15,12160],{"className":12161,"style":158},[157],[15,12163,12165,12168,12210,12213,12253,12256,12259,12262,12265,12268,12271,12314,12320,12323,12326,12329,12332,12335,12338,12344,12347,12350],{"className":12164},[50],[15,12166],{"className":12167,"style":12090},[54],[15,12169,12171,12174],{"className":12170},[59],[15,12172,7572],{"className":12173,"style":8276},[59,60],[15,12175,12177],{"className":12176},[281],[15,12178,12180,12202],{"className":12179},[285,286],[15,12181,12183,12199],{"className":12182},[290],[15,12184,12187],{"className":12185,"style":12186},[294],"height:0.3117em;",[15,12188,12190,12193],{"style":12189},"top:-2.55em;margin-left:-0.0269em;margin-right:0.05em;",[15,12191],{"className":12192,"style":303},[302],[15,12194,12196],{"className":12195},[307,308,309,310],[15,12197,11967],{"className":12198},[59,60,310],[15,12200,318],{"className":12201},[317],[15,12203,12205],{"className":12204},[290],[15,12206,12208],{"className":12207,"style":7970},[294],[15,12209],{},[15,12211],{"className":12212,"style":406},[157],[15,12214,12216,12219],{"className":12215},[59],[15,12217,206],{"className":12218,"style":334},[59,60],[15,12220,12222],{"className":12221},[281],[15,12223,12225,12245],{"className":12224},[285,286],[15,12226,12228,12242],{"className":12227},[290],[15,12229,12231],{"className":12230,"style":12186},[294],[15,12232,12233,12236],{"style":11140},[15,12234],{"className":12235,"style":303},[302],[15,12237,12239],{"className":12238},[307,308,309,310],[15,12240,11967],{"className":12241},[59,60,310],[15,12243,318],{"className":12244},[317],[15,12246,12248],{"className":12247},[290],[15,12249,12251],{"className":12250,"style":7970},[294],[15,12252],{},[15,12254,113],{"className":12255},[146],[15,12257,774],{"className":12258},[59,60],[15,12260,120],{"className":12261},[153],[15,12263,551],{"className":12264},[573],[15,12266],{"className":12267,"style":3636},[157],[15,12269],{"className":12270,"style":406},[157],[15,12272,12274],{"className":12273},[59,11855],[15,12275,12277,12305],{"className":12276},[285,286],[15,12278,12280,12302],{"className":12279},[290],[15,12281,12283,12291],{"className":12282,"style":11851},[294],[15,12284,12285,12288],{"style":11867},[15,12286],{"className":12287,"style":461},[302],[15,12289,213],{"className":12290},[59,60],[15,12292,12293,12296],{"style":11876},[15,12294],{"className":12295,"style":461},[302],[15,12297,12299],{"className":12298,"style":12119},[11883],[15,12300,11821],{"className":12301},[59],[15,12303,318],{"className":12304},[317],[15,12306,12308],{"className":12307},[290],[15,12309,12312],{"className":12310,"style":12311},[294],"height:0.1944em;",[15,12313],{},[15,12315,12317],{"className":12316},[59],[15,12318,113],{"className":12319},[419,9676],[15,12321,113],{"className":12322},[146],[15,12324,11967],{"className":12325},[59,60],[15,12327,551],{"className":12328},[573],[15,12330],{"className":12331,"style":406},[157],[15,12333,774],{"className":12334},[59,60],[15,12336,120],{"className":12337},[153],[15,12339,12341],{"className":12340},[59],[15,12342,120],{"className":12343},[419,9676],[15,12345],{"className":12346,"style":158},[157],[15,12348,123],{"className":12349},[162],[15,12351],{"className":12352,"style":158},[157],[15,12354,12356,12359,12399,12402,12442,12445,12448,12451,12454,12457,12460,12500,12503,12506],{"className":12355},[50],[15,12357],{"className":12358,"style":139},[54],[15,12360,12362,12365],{"className":12361},[59],[15,12363,7572],{"className":12364,"style":8276},[59,60],[15,12366,12368],{"className":12367},[281],[15,12369,12371,12391],{"className":12370},[285,286],[15,12372,12374,12388],{"className":12373},[290],[15,12375,12377],{"className":12376,"style":12186},[294],[15,12378,12379,12382],{"style":12189},[15,12380],{"className":12381,"style":303},[302],[15,12383,12385],{"className":12384},[307,308,309,310],[15,12386,11967],{"className":12387},[59,60,310],[15,12389,318],{"className":12390},[317],[15,12392,12394],{"className":12393},[290],[15,12395,12397],{"className":12396,"style":7970},[294],[15,12398],{},[15,12400],{"className":12401,"style":406},[157],[15,12403,12405,12408],{"className":12404},[59],[15,12406,213],{"className":12407},[59,60],[15,12409,12411],{"className":12410},[281],[15,12412,12414,12434],{"className":12413},[285,286],[15,12415,12417,12431],{"className":12416},[290],[15,12418,12420],{"className":12419,"style":12186},[294],[15,12421,12422,12425],{"style":382},[15,12423],{"className":12424,"style":303},[302],[15,12426,12428],{"className":12427},[307,308,309,310],[15,12429,11967],{"className":12430},[59,60,310],[15,12432,318],{"className":12433},[317],[15,12435,12437],{"className":12436},[290],[15,12438,12440],{"className":12439,"style":7970},[294],[15,12441],{},[15,12443,113],{"className":12444},[146],[15,12446,774],{"className":12447},[59,60],[15,12449,120],{"className":12450},[153],[15,12452,551],{"className":12453},[573],[15,12455],{"className":12456,"style":3636},[157],[15,12458],{"className":12459,"style":406},[157],[15,12461,12463,12466],{"className":12462},[59],[15,12464,7572],{"className":12465,"style":8276},[59,60],[15,12467,12469],{"className":12468},[281],[15,12470,12472,12492],{"className":12471},[285,286],[15,12473,12475,12489],{"className":12474},[290],[15,12476,12478],{"className":12477,"style":11137},[294],[15,12479,12480,12483],{"style":12189},[15,12481],{"className":12482,"style":303},[302],[15,12484,12486],{"className":12485},[307,308,309,310],[15,12487,117],{"className":12488},[59,310],[15,12490,318],{"className":12491},[317],[15,12493,12495],{"className":12494},[290],[15,12496,12498],{"className":12497,"style":7970},[294],[15,12499],{},[15,12501],{"className":12502,"style":158},[157],[15,12504,123],{"className":12505},[162],[15,12507],{"className":12508,"style":158},[157],[15,12510,12512,12515,12518,12521,12524,12527,12567,12570,12573],{"className":12511},[50],[15,12513],{"className":12514,"style":55},[54],[15,12516,11343],{"className":12517},[59,60],[15,12519,551],{"className":12520},[573],[15,12522,4285],{"className":12523},[157],[15,12525],{"className":12526,"style":406},[157],[15,12528,12530,12533],{"className":12529},[59],[15,12531,7572],{"className":12532,"style":8276},[59,60],[15,12534,12536],{"className":12535},[281],[15,12537,12539,12559],{"className":12538},[285,286],[15,12540,12542,12556],{"className":12541},[290],[15,12543,12545],{"className":12544,"style":11137},[294],[15,12546,12547,12550],{"style":12189},[15,12548],{"className":12549,"style":303},[302],[15,12551,12553],{"className":12552},[307,308,309,310],[15,12554,2465],{"className":12555},[59,310],[15,12557,318],{"className":12558},[317],[15,12560,12562],{"className":12561},[290],[15,12563,12565],{"className":12564,"style":7970},[294],[15,12566],{},[15,12568],{"className":12569,"style":158},[157],[15,12571,123],{"className":12572},[162],[15,12574],{"className":12575,"style":158},[157],[15,12577,12579,12583,12586,12589,12592],{"className":12578},[50],[15,12580],{"className":12581,"style":12582},[54],"height:0.7278em;vertical-align:-0.0833em;",[15,12584,117],{"className":12585},[59],[15,12587],{"className":12588,"style":2589},[157],[15,12590,4815],{"className":12591},[2593],[15,12593],{"className":12594,"style":2589},[157],[15,12596,12598,12602,12605],{"className":12597},[50],[15,12599],{"className":12600,"style":12601},[54],"height:0.6944em;",[15,12603,11343],{"className":12604},[59,60],[15,12606,4377],{"className":12607},[59],[11,12609,12610,12611,6496,12614,12715,12716,12744,12745,5647,12833,12933,12934,13347,13348,13419],{},"取",[998,12612,12613],{},"确定性核",[15,12615,12617,12649],{"className":12616,"translate":19},[18],[15,12618,12620],{"className":12619},[23],[25,12621,12622],{"xmlns":27},[29,12623,12624,12646],{},[32,12625,12626,12628,12630,12632,12634,12636,12638,12640,12642,12644],{},[35,12627,10698],{},[110,12629,1018],{},[1299,12631,1301],{},[110,12633,113],{"stretchy":112},[35,12635,11967],{},[110,12637,551],{"separator":46},[35,12639,774],{},[110,12641,120],{"stretchy":112},[110,12643,8686],{},[35,12645,774],{},[39,12647,12648],{"encoding":41},"K:\\,(i,x)\\mapsto x",[15,12650,12652,12673,12706],{"className":12651,"ariaHidden":46},[45],[15,12653,12655,12658,12661,12664,12667,12670],{"className":12654},[50],[15,12656],{"className":12657,"style":664},[54],[15,12659,10698],{"className":12660,"style":10786},[59,60],[15,12662],{"className":12663,"style":158},[157],[15,12665,1018],{"className":12666},[162],[15,12668],{"className":12669,"style":406},[157],[15,12671],{"className":12672,"style":158},[157],[15,12674,12676,12679,12682,12685,12688,12691,12694,12697,12700,12703],{"className":12675},[50],[15,12677],{"className":12678,"style":139},[54],[15,12680,113],{"className":12681},[146],[15,12683,11967],{"className":12684},[59,60],[15,12686,551],{"className":12687},[573],[15,12689],{"className":12690,"style":406},[157],[15,12692,774],{"className":12693},[59,60],[15,12695,120],{"className":12696},[153],[15,12698],{"className":12699,"style":158},[157],[15,12701,8686],{"className":12702},[162],[15,12704],{"className":12705,"style":158},[157],[15,12707,12709,12712],{"className":12708},[50],[15,12710],{"className":12711,"style":2347},[54],[15,12713,774],{"className":12714},[59,60],"（\"擦掉\"分量下标，只保留 ",[15,12717,12719,12732],{"className":12718,"translate":19},[18],[15,12720,12722],{"className":12721},[23],[25,12723,12724],{"xmlns":27},[29,12725,12726,12730],{},[32,12727,12728],{},[35,12729,774],{},[39,12731,774],{"encoding":41},[15,12733,12735],{"className":12734,"ariaHidden":46},[45],[15,12736,12738,12741],{"className":12737},[50],[15,12739],{"className":12740,"style":2347},[54],[15,12742,774],{"className":12743},[59,60],"）。由分块结构立刻算出 ",[15,12746,12748,12772],{"className":12747,"translate":19},[18],[15,12749,12751],{"className":12750},[23],[25,12752,12753],{"xmlns":27},[29,12754,12755,12769],{},[32,12756,12757,12763,12765,12767],{},[11815,12758,12759,12761],{"accent":46},[35,12760,206],{},[110,12762,11821],{},[35,12764,10698],{},[110,12766,123],{},[35,12768,206],{},[39,12770,12771],{"encoding":41},"\\tilde PK=P",[15,12773,12775,12824],{"className":12774,"ariaHidden":46},[45],[15,12776,12778,12781,12812,12815,12818,12821],{"className":12777},[50],[15,12779],{"className":12780,"style":11851},[54],[15,12782,12784],{"className":12783},[59,11855],[15,12785,12787],{"className":12786},[285],[15,12788,12790],{"className":12789},[290],[15,12791,12793,12801],{"className":12792,"style":11851},[294],[15,12794,12795,12798],{"style":11867},[15,12796],{"className":12797,"style":461},[302],[15,12799,206],{"className":12800,"style":334},[59,60],[15,12802,12803,12806],{"style":11876},[15,12804],{"className":12805,"style":461},[302],[15,12807,12809],{"className":12808,"style":12119},[11883],[15,12810,11821],{"className":12811},[59],[15,12813,10698],{"className":12814,"style":10786},[59,60],[15,12816],{"className":12817,"style":158},[157],[15,12819,123],{"className":12820},[162],[15,12822],{"className":12823,"style":158},[157],[15,12825,12827,12830],{"className":12826},[50],[15,12828],{"className":12829,"style":664},[54],[15,12831,206],{"className":12832,"style":334},[59,60],[15,12834,12836,12860],{"className":12835,"translate":19},[18],[15,12837,12839],{"className":12838},[23],[25,12840,12841],{"xmlns":27},[29,12842,12843,12857],{},[32,12844,12845,12851,12853,12855],{},[11815,12846,12847,12849],{"accent":46},[35,12848,213],{},[110,12850,11821],{},[35,12852,10698],{},[110,12854,123],{},[35,12856,213],{},[39,12858,12859],{"encoding":41},"\\tilde QK=Q",[15,12861,12863,12924],{"className":12862,"ariaHidden":46},[45],[15,12864,12866,12870,12912,12915,12918,12921],{"className":12865},[50],[15,12867],{"className":12868,"style":12869},[54],"height:1.1146em;vertical-align:-0.1944em;",[15,12871,12873],{"className":12872},[59,11855],[15,12874,12876,12904],{"className":12875},[285,286],[15,12877,12879,12901],{"className":12878},[290],[15,12880,12882,12890],{"className":12881,"style":11851},[294],[15,12883,12884,12887],{"style":11867},[15,12885],{"className":12886,"style":461},[302],[15,12888,213],{"className":12889},[59,60],[15,12891,12892,12895],{"style":11876},[15,12893],{"className":12894,"style":461},[302],[15,12896,12898],{"className":12897,"style":12119},[11883],[15,12899,11821],{"className":12900},[59],[15,12902,318],{"className":12903},[317],[15,12905,12907],{"className":12906},[290],[15,12908,12910],{"className":12909,"style":12311},[294],[15,12911],{},[15,12913,10698],{"className":12914,"style":10786},[59,60],[15,12916],{"className":12917,"style":158},[157],[15,12919,123],{"className":12920},[162],[15,12922],{"className":12923,"style":158},[157],[15,12925,12927,12930],{"className":12926},[50],[15,12928],{"className":12929,"style":566},[54],[15,12931,213],{"className":12932},[59,60],"，且似然比是分块恒定的：",[15,12935,12937,13015],{"className":12936,"translate":19},[18],[15,12938,12940],{"className":12939},[23],[25,12941,12942],{"xmlns":27},[29,12943,12944,13012],{},[32,12945,12946,12970,12972,12974,12976,12978,12980,12982,13006,13008,13010],{},[1257,12947,12948],{"scriptlevel":126,"displaystyle":46},[238,12949,12950,12960],{},[32,12951,12952,12954],{},[35,12953,244],{},[11815,12955,12956,12958],{"accent":46},[35,12957,206],{},[110,12959,11821],{},[32,12961,12962,12964],{},[35,12963,244],{},[11815,12965,12966,12968],{"accent":46},[35,12967,213],{},[110,12969,11821],{},[110,12971,113],{"stretchy":112},[35,12973,11967],{},[110,12975,551],{"separator":46},[35,12977,774],{},[110,12979,120],{"stretchy":112},[110,12981,123],{},[1257,12983,12984],{"scriptlevel":126,"displaystyle":46},[238,12985,12986,12996],{},[32,12987,12988,12990],{},[35,12989,244],{},[195,12991,12992,12994],{},[35,12993,206],{},[35,12995,11967],{},[32,12997,12998,13000],{},[35,12999,244],{},[195,13001,13002,13004],{},[35,13003,213],{},[35,13005,11967],{},[110,13007,113],{"stretchy":112},[35,13009,774],{},[110,13011,120],{"stretchy":112},[39,13013,13014],{"encoding":41},"\\dfrac{d\\tilde P}{d\\tilde Q}(i,x)=\\dfrac{dP_i}{dQ_i}(x)",[15,13016,13018,13190],{"className":13017,"ariaHidden":46},[45],[15,13019,13021,13025,13163,13166,13169,13172,13175,13178,13181,13184,13187],{"className":13020},[50],[15,13022],{"className":13023,"style":13024},[54],"height:2.6018em;vertical-align:-1.0046em;",[15,13026,13028,13031,13160],{"className":13027},[59],[15,13029],{"className":13030},[146,441],[15,13032,13034],{"className":13033},[238],[15,13035,13037,13151],{"className":13036},[285,286],[15,13038,13040,13148],{"className":13039},[290],[15,13041,13044,13098,13106],{"className":13042,"style":13043},[294],"height:1.5972em;",[15,13045,13047,13050],{"style":13046},"top:-2.1898em;",[15,13048],{"className":13049,"style":461},[302],[15,13051,13053,13056],{"className":13052},[59],[15,13054,244],{"className":13055},[59,60],[15,13057,13059],{"className":13058},[59,11855],[15,13060,13062,13090],{"className":13061},[285,286],[15,13063,13065,13087],{"className":13064},[290],[15,13066,13068,13076],{"className":13067,"style":11851},[294],[15,13069,13070,13073],{"style":11867},[15,13071],{"className":13072,"style":461},[302],[15,13074,213],{"className":13075},[59,60],[15,13077,13078,13081],{"style":11876},[15,13079],{"className":13080,"style":461},[302],[15,13082,13084],{"className":13083,"style":12119},[11883],[15,13085,11821],{"className":13086},[59],[15,13088,318],{"className":13089},[317],[15,13091,13093],{"className":13092},[290],[15,13094,13096],{"className":13095,"style":12311},[294],[15,13097],{},[15,13099,13100,13103],{"style":473},[15,13101],{"className":13102,"style":461},[302],[15,13104],{"className":13105,"style":481},[480],[15,13107,13108,13111],{"style":484},[15,13109],{"className":13110,"style":461},[302],[15,13112,13114,13117],{"className":13113},[59],[15,13115,244],{"className":13116},[59,60],[15,13118,13120],{"className":13119},[59,11855],[15,13121,13123],{"className":13122},[285],[15,13124,13126],{"className":13125},[290],[15,13127,13129,13137],{"className":13128,"style":11851},[294],[15,13130,13131,13134],{"style":11867},[15,13132],{"className":13133,"style":461},[302],[15,13135,206],{"className":13136,"style":334},[59,60],[15,13138,13139,13142],{"style":11876},[15,13140],{"className":13141,"style":461},[302],[15,13143,13145],{"className":13144,"style":12119},[11883],[15,13146,11821],{"className":13147},[59],[15,13149,318],{"className":13150},[317],[15,13152,13154],{"className":13153},[290],[15,13155,13158],{"className":13156,"style":13157},[294],"height:1.0046em;",[15,13159],{},[15,13161],{"className":13162},[153,441],[15,13164,113],{"className":13165},[146],[15,13167,11967],{"className":13168},[59,60],[15,13170,551],{"className":13171},[573],[15,13173],{"className":13174,"style":406},[157],[15,13176,774],{"className":13177},[59,60],[15,13179,120],{"className":13180},[153],[15,13182],{"className":13183,"style":158},[157],[15,13185,123],{"className":13186},[162],[15,13188],{"className":13189,"style":158},[157],[15,13191,13193,13196,13338,13341,13344],{"className":13192},[50],[15,13194],{"className":13195,"style":838},[54],[15,13197,13199,13202,13335],{"className":13198},[59],[15,13200],{"className":13201},[146,441],[15,13203,13205],{"className":13204},[238],[15,13206,13208,13327],{"className":13207},[285,286],[15,13209,13211,13324],{"className":13210},[290],[15,13212,13214,13265,13273],{"className":13213,"style":454},[294],[15,13215,13216,13219],{"style":457},[15,13217],{"className":13218,"style":461},[302],[15,13220,13222,13225],{"className":13221},[59],[15,13223,244],{"className":13224},[59,60],[15,13226,13228,13231],{"className":13227},[59],[15,13229,213],{"className":13230},[59,60],[15,13232,13234],{"className":13233},[281],[15,13235,13237,13257],{"className":13236},[285,286],[15,13238,13240,13254],{"className":13239},[290],[15,13241,13243],{"className":13242,"style":12186},[294],[15,13244,13245,13248],{"style":382},[15,13246],{"className":13247,"style":303},[302],[15,13249,13251],{"className":13250},[307,308,309,310],[15,13252,11967],{"className":13253},[59,60,310],[15,13255,318],{"className":13256},[317],[15,13258,13260],{"className":13259},[290],[15,13261,13263],{"className":13262,"style":7970},[294],[15,13264],{},[15,13266,13267,13270],{"style":473},[15,13268],{"className":13269,"style":461},[302],[15,13271],{"className":13272,"style":481},[480],[15,13274,13275,13278],{"style":484},[15,13276],{"className":13277,"style":461},[302],[15,13279,13281,13284],{"className":13280},[59],[15,13282,244],{"className":13283},[59,60],[15,13285,13287,13290],{"className":13286},[59],[15,13288,206],{"className":13289,"style":334},[59,60],[15,13291,13293],{"className":13292},[281],[15,13294,13296,13316],{"className":13295},[285,286],[15,13297,13299,13313],{"className":13298},[290],[15,13300,13302],{"className":13301,"style":12186},[294],[15,13303,13304,13307],{"style":11140},[15,13305],{"className":13306,"style":303},[302],[15,13308,13310],{"className":13309},[307,308,309,310],[15,13311,11967],{"className":13312},[59,60,310],[15,13314,318],{"className":13315},[317],[15,13317,13319],{"className":13318},[290],[15,13320,13322],{"className":13321,"style":7970},[294],[15,13323],{},[15,13325,318],{"className":13326},[317],[15,13328,13330],{"className":13329},[290],[15,13331,13333],{"className":13332,"style":506},[294],[15,13334],{},[15,13336],{"className":13337},[153,441],[15,13339,113],{"className":13340},[146],[15,13342,774],{"className":13343},[59,60],[15,13345,120],{"className":13346},[153],"（权重 ",[15,13349,13351,13369],{"className":13350,"translate":19},[18],[15,13352,13354],{"className":13353},[23],[25,13355,13356],{"xmlns":27},[29,13357,13358,13366],{},[32,13359,13360],{},[195,13361,13362,13364],{},[35,13363,7572],{},[35,13365,11967],{},[39,13367,13368],{"encoding":41},"w_i",[15,13370,13372],{"className":13371,"ariaHidden":46},[45],[15,13373,13375,13379],{"className":13374},[50],[15,13376],{"className":13377,"style":13378},[54],"height:0.5806em;vertical-align:-0.15em;",[15,13380,13382,13385],{"className":13381},[59],[15,13383,7572],{"className":13384,"style":8276},[59,60],[15,13386,13388],{"className":13387},[281],[15,13389,13391,13411],{"className":13390},[285,286],[15,13392,13394,13408],{"className":13393},[290],[15,13395,13397],{"className":13396,"style":12186},[294],[15,13398,13399,13402],{"style":12189},[15,13400],{"className":13401,"style":303},[302],[15,13403,13405],{"className":13404},[307,308,309,310],[15,13406,11967],{"className":13407},[59,60,310],[15,13409,318],{"className":13410},[317],[15,13412,13414],{"className":13413},[290],[15,13415,13417],{"className":13416,"style":7970},[294],[15,13418],{}," 在分子分母中约去）。代入定义：",[15,13421,13423],{"className":13422,"translate":19},[180],[15,13424,13426,13569],{"className":13425,"translate":19},[18],[15,13427,13429],{"className":13428},[23],[25,13430,13431],{"xmlns":27,"display":189},[29,13432,13433,13566],{},[32,13434,13435,13441,13443,13449,13451,13457,13459,13461,13468,13474,13476,13482,13484,13490,13492,13498,13500,13502,13504,13510,13512,13518,13520,13526,13528,13530,13532,13534,13536,13538,13540,13546,13548,13554,13556,13562,13564],{},[195,13436,13437,13439],{},[35,13438,199],{},[35,13440,37],{},[110,13442,113],{"stretchy":112},[11815,13444,13445,13447],{"accent":46},[35,13446,206],{},[110,13448,11821],{},[35,13450,210],{"mathvariant":209},[11815,13452,13453,13455],{"accent":46},[35,13454,213],{},[110,13456,11821],{},[110,13458,120],{"stretchy":112},[110,13460,123],{},[6018,13462,13463,13466],{},[110,13464,13465],{},"∑",[35,13467,11967],{},[195,13469,13470,13472],{},[35,13471,7572],{},[35,13473,11967],{},[1299,13475,1301],{},[195,13477,13478,13480],{},[35,13479,199],{},[35,13481,37],{},[110,13483,113],{"stretchy":112},[195,13485,13486,13488],{},[35,13487,206],{},[35,13489,11967],{},[35,13491,210],{"mathvariant":209},[195,13493,13494,13496],{},[35,13495,213],{},[35,13497,11967],{},[110,13499,120],{"stretchy":112},[110,13501,123],{},[35,13503,11343],{},[195,13505,13506,13508],{},[35,13507,199],{},[35,13509,37],{},[110,13511,113],{"stretchy":112},[195,13513,13514,13516],{},[35,13515,206],{},[115,13517,117],{},[35,13519,210],{"mathvariant":209},[195,13521,13522,13524],{},[35,13523,213],{},[115,13525,117],{},[110,13527,120],{"stretchy":112},[110,13529,2472],{},[110,13531,113],{"stretchy":112},[115,13533,117],{},[110,13535,4815],{},[35,13537,11343],{},[110,13539,120],{"stretchy":112},[195,13541,13542,13544],{},[35,13543,199],{},[35,13545,37],{},[110,13547,113],{"stretchy":112},[195,13549,13550,13552],{},[35,13551,206],{},[115,13553,2465],{},[35,13555,210],{"mathvariant":209},[195,13557,13558,13560],{},[35,13559,213],{},[115,13561,2465],{},[110,13563,120],{"stretchy":112},[35,13565,4377],{"mathvariant":209},[39,13567,13568],{"encoding":41},"D_f(\\tilde P\\|\\tilde Q)\n=\\sum_i w_i\\,D_f(P_i\\|Q_i)\n=\\lambda D_f(P_1\\|Q_1)+(1-\\lambda)D_f(P_2\\|Q_2).",[15,13570,13572,13710,13951,14098,14119],{"className":13571,"ariaHidden":46},[45],[15,13573,13575,13579,13619,13622,13653,13656,13698,13701,13704,13707],{"className":13574},[50],[15,13576],{"className":13577,"style":13578},[54],"height:1.2063em;vertical-align:-0.2861em;",[15,13580,13582,13585],{"className":13581},[59],[15,13583,199],{"className":13584,"style":277},[59,60],[15,13586,13588],{"className":13587},[281],[15,13589,13591,13611],{"className":13590},[285,286],[15,13592,13594,13608],{"className":13593},[290],[15,13595,13597],{"className":13596,"style":295},[294],[15,13598,13599,13602],{"style":298},[15,13600],{"className":13601,"style":303},[302],[15,13603,13605],{"className":13604},[307,308,309,310],[15,13606,37],{"className":13607,"style":61},[59,60,310],[15,13609,318],{"className":13610},[317],[15,13612,13614],{"className":13613},[290],[15,13615,13617],{"className":13616,"style":325},[294],[15,13618],{},[15,13620,113],{"className":13621},[146],[15,13623,13625],{"className":13624},[59,11855],[15,13626,13628],{"className":13627},[285],[15,13629,13631],{"className":13630},[290],[15,13632,13634,13642],{"className":13633,"style":11851},[294],[15,13635,13636,13639],{"style":11867},[15,13637],{"className":13638,"style":461},[302],[15,13640,206],{"className":13641,"style":334},[59,60],[15,13643,13644,13647],{"style":11876},[15,13645],{"className":13646,"style":461},[302],[15,13648,13650],{"className":13649,"style":12119},[11883],[15,13651,11821],{"className":13652},[59],[15,13654,210],{"className":13655},[59],[15,13657,13659],{"className":13658},[59,11855],[15,13660,13662,13690],{"className":13661},[285,286],[15,13663,13665,13687],{"className":13664},[290],[15,13666,13668,13676],{"className":13667,"style":11851},[294],[15,13669,13670,13673],{"style":11867},[15,13671],{"className":13672,"style":461},[302],[15,13674,213],{"className":13675},[59,60],[15,13677,13678,13681],{"style":11876},[15,13679],{"className":13680,"style":461},[302],[15,13682,13684],{"className":13683,"style":12119},[11883],[15,13685,11821],{"className":13686},[59],[15,13688,318],{"className":13689},[317],[15,13691,13693],{"className":13692},[290],[15,13694,13696],{"className":13695,"style":12311},[294],[15,13697],{},[15,13699,120],{"className":13700},[153],[15,13702],{"className":13703,"style":158},[157],[15,13705,123],{"className":13706},[162],[15,13708],{"className":13709,"style":158},[157],[15,13711,13713,13717,13767,13770,13810,13813,13853,13856,13896,13899,13939,13942,13945,13948],{"className":13712},[50],[15,13714],{"className":13715,"style":13716},[54],"height:2.3277em;vertical-align:-1.2777em;",[15,13718,13721],{"className":13719},[1486,13720],"op-limits",[15,13722,13724,13758],{"className":13723},[285,286],[15,13725,13727,13755],{"className":13726},[290],[15,13728,13731,13744],{"className":13729,"style":13730},[294],"height:1.05em;",[15,13732,13734,13738],{"style":13733},"top:-1.8723em;margin-left:0em;",[15,13735],{"className":13736,"style":13737},[302],"height:3.05em;",[15,13739,13741],{"className":13740},[307,308,309,310],[15,13742,11967],{"className":13743},[59,60,310],[15,13745,13747,13750],{"style":13746},"top:-3.05em;",[15,13748],{"className":13749,"style":13737},[302],[15,13751,13752],{},[15,13753,13465],{"className":13754},[1486,1487,1488],[15,13756,318],{"className":13757},[317],[15,13759,13761],{"className":13760},[290],[15,13762,13765],{"className":13763,"style":13764},[294],"height:1.2777em;",[15,13766],{},[15,13768],{"className":13769,"style":406},[157],[15,13771,13773,13776],{"className":13772},[59],[15,13774,7572],{"className":13775,"style":8276},[59,60],[15,13777,13779],{"className":13778},[281],[15,13780,13782,13802],{"className":13781},[285,286],[15,13783,13785,13799],{"className":13784},[290],[15,13786,13788],{"className":13787,"style":12186},[294],[15,13789,13790,13793],{"style":12189},[15,13791],{"className":13792,"style":303},[302],[15,13794,13796],{"className":13795},[307,308,309,310],[15,13797,11967],{"className":13798},[59,60,310],[15,13800,318],{"className":13801},[317],[15,13803,13805],{"className":13804},[290],[15,13806,13808],{"className":13807,"style":7970},[294],[15,13809],{},[15,13811],{"className":13812,"style":406},[157],[15,13814,13816,13819],{"className":13815},[59],[15,13817,199],{"className":13818,"style":277},[59,60],[15,13820,13822],{"className":13821},[281],[15,13823,13825,13845],{"className":13824},[285,286],[15,13826,13828,13842],{"className":13827},[290],[15,13829,13831],{"className":13830,"style":295},[294],[15,13832,13833,13836],{"style":298},[15,13834],{"className":13835,"style":303},[302],[15,13837,13839],{"className":13838},[307,308,309,310],[15,13840,37],{"className":13841,"style":61},[59,60,310],[15,13843,318],{"className":13844},[317],[15,13846,13848],{"className":13847},[290],[15,13849,13851],{"className":13850,"style":325},[294],[15,13852],{},[15,13854,113],{"className":13855},[146],[15,13857,13859,13862],{"className":13858},[59],[15,13860,206],{"className":13861,"style":334},[59,60],[15,13863,13865],{"className":13864},[281],[15,13866,13868,13888],{"className":13867},[285,286],[15,13869,13871,13885],{"className":13870},[290],[15,13872,13874],{"className":13873,"style":12186},[294],[15,13875,13876,13879],{"style":11140},[15,13877],{"className":13878,"style":303},[302],[15,13880,13882],{"className":13881},[307,308,309,310],[15,13883,11967],{"className":13884},[59,60,310],[15,13886,318],{"className":13887},[317],[15,13889,13891],{"className":13890},[290],[15,13892,13894],{"className":13893,"style":7970},[294],[15,13895],{},[15,13897,210],{"className":13898},[59],[15,13900,13902,13905],{"className":13901},[59],[15,13903,213],{"className":13904},[59,60],[15,13906,13908],{"className":13907},[281],[15,13909,13911,13931],{"className":13910},[285,286],[15,13912,13914,13928],{"className":13913},[290],[15,13915,13917],{"className":13916,"style":12186},[294],[15,13918,13919,13922],{"style":382},[15,13920],{"className":13921,"style":303},[302],[15,13923,13925],{"className":13924},[307,308,309,310],[15,13926,11967],{"className":13927},[59,60,310],[15,13929,318],{"className":13930},[317],[15,13932,13934],{"className":13933},[290],[15,13935,13937],{"className":13936,"style":7970},[294],[15,13938],{},[15,13940,120],{"className":13941},[153],[15,13943],{"className":13944,"style":158},[157],[15,13946,123],{"className":13947},[162],[15,13949],{"className":13950,"style":158},[157],[15,13952,13954,13957,13960,14000,14003,14043,14046,14086,14089,14092,14095],{"className":13953},[50],[15,13955],{"className":13956,"style":270},[54],[15,13958,11343],{"className":13959},[59,60],[15,13961,13963,13966],{"className":13962},[59],[15,13964,199],{"className":13965,"style":277},[59,60],[15,13967,13969],{"className":13968},[281],[15,13970,13972,13992],{"className":13971},[285,286],[15,13973,13975,13989],{"className":13974},[290],[15,13976,13978],{"className":13977,"style":295},[294],[15,13979,13980,13983],{"style":298},[15,13981],{"className":13982,"style":303},[302],[15,13984,13986],{"className":13985},[307,308,309,310],[15,13987,37],{"className":13988,"style":61},[59,60,310],[15,13990,318],{"className":13991},[317],[15,13993,13995],{"className":13994},[290],[15,13996,13998],{"className":13997,"style":325},[294],[15,13999],{},[15,14001,113],{"className":14002},[146],[15,14004,14006,14009],{"className":14005},[59],[15,14007,206],{"className":14008,"style":334},[59,60],[15,14010,14012],{"className":14011},[281],[15,14013,14015,14035],{"className":14014},[285,286],[15,14016,14018,14032],{"className":14017},[290],[15,14019,14021],{"className":14020,"style":11137},[294],[15,14022,14023,14026],{"style":11140},[15,14024],{"className":14025,"style":303},[302],[15,14027,14029],{"className":14028},[307,308,309,310],[15,14030,117],{"className":14031},[59,310],[15,14033,318],{"className":14034},[317],[15,14036,14038],{"className":14037},[290],[15,14039,14041],{"className":14040,"style":7970},[294],[15,14042],{},[15,14044,210],{"className":14045},[59],[15,14047,14049,14052],{"className":14048},[59],[15,14050,213],{"className":14051},[59,60],[15,14053,14055],{"className":14054},[281],[15,14056,14058,14078],{"className":14057},[285,286],[15,14059,14061,14075],{"className":14060},[290],[15,14062,14064],{"className":14063,"style":11137},[294],[15,14065,14066,14069],{"style":382},[15,14067],{"className":14068,"style":303},[302],[15,14070,14072],{"className":14071},[307,308,309,310],[15,14073,117],{"className":14074},[59,310],[15,14076,318],{"className":14077},[317],[15,14079,14081],{"className":14080},[290],[15,14082,14084],{"className":14083,"style":7970},[294],[15,14085],{},[15,14087,120],{"className":14088},[153],[15,14090],{"className":14091,"style":2589},[157],[15,14093,2472],{"className":14094},[2593],[15,14096],{"className":14097,"style":2589},[157],[15,14099,14101,14104,14107,14110,14113,14116],{"className":14100},[50],[15,14102],{"className":14103,"style":139},[54],[15,14105,113],{"className":14106},[146],[15,14108,117],{"className":14109},[59],[15,14111],{"className":14112,"style":2589},[157],[15,14114,4815],{"className":14115},[2593],[15,14117],{"className":14118,"style":2589},[157],[15,14120,14122,14125,14128,14131,14171,14174,14214,14217,14257,14260],{"className":14121},[50],[15,14123],{"className":14124,"style":270},[54],[15,14126,11343],{"className":14127},[59,60],[15,14129,120],{"className":14130},[153],[15,14132,14134,14137],{"className":14133},[59],[15,14135,199],{"className":14136,"style":277},[59,60],[15,14138,14140],{"className":14139},[281],[15,14141,14143,14163],{"className":14142},[285,286],[15,14144,14146,14160],{"className":14145},[290],[15,14147,14149],{"className":14148,"style":295},[294],[15,14150,14151,14154],{"style":298},[15,14152],{"className":14153,"style":303},[302],[15,14155,14157],{"className":14156},[307,308,309,310],[15,14158,37],{"className":14159,"style":61},[59,60,310],[15,14161,318],{"className":14162},[317],[15,14164,14166],{"className":14165},[290],[15,14167,14169],{"className":14168,"style":325},[294],[15,14170],{},[15,14172,113],{"className":14173},[146],[15,14175,14177,14180],{"className":14176},[59],[15,14178,206],{"className":14179,"style":334},[59,60],[15,14181,14183],{"className":14182},[281],[15,14184,14186,14206],{"className":14185},[285,286],[15,14187,14189,14203],{"className":14188},[290],[15,14190,14192],{"className":14191,"style":11137},[294],[15,14193,14194,14197],{"style":11140},[15,14195],{"className":14196,"style":303},[302],[15,14198,14200],{"className":14199},[307,308,309,310],[15,14201,2465],{"className":14202},[59,310],[15,14204,318],{"className":14205},[317],[15,14207,14209],{"className":14208},[290],[15,14210,14212],{"className":14211,"style":7970},[294],[15,14213],{},[15,14215,210],{"className":14216},[59],[15,14218,14220,14223],{"className":14219},[59],[15,14221,213],{"className":14222},[59,60],[15,14224,14226],{"className":14225},[281],[15,14227,14229,14249],{"className":14228},[285,286],[15,14230,14232,14246],{"className":14231},[290],[15,14233,14235],{"className":14234,"style":11137},[294],[15,14236,14237,14240],{"style":382},[15,14238],{"className":14239,"style":303},[302],[15,14241,14243],{"className":14242},[307,308,309,310],[15,14244,2465],{"className":14245},[59,310],[15,14247,318],{"className":14248},[317],[15,14250,14252],{"className":14251},[290],[15,14253,14255],{"className":14254,"style":7970},[294],[15,14256],{},[15,14258,120],{"className":14259},[153],[15,14261,4377],{"className":14262},[59],[11,14264,14265,14266,14394],{},"对 ",[15,14267,14269,14299],{"className":14268,"translate":19},[18],[15,14270,14272],{"className":14271},[23],[25,14273,14274],{"xmlns":27},[29,14275,14276,14296],{},[32,14277,14278,14280,14286,14288,14294],{},[110,14279,113],{"stretchy":112},[11815,14281,14282,14284],{"accent":46},[35,14283,206],{},[110,14285,11821],{},[110,14287,551],{"separator":46},[11815,14289,14290,14292],{"accent":46},[35,14291,213],{},[110,14293,11821],{},[110,14295,120],{"stretchy":112},[39,14297,14298],{"encoding":41},"(\\tilde P,\\tilde Q)",[15,14300,14302],{"className":14301,"ariaHidden":46},[45],[15,14303,14305,14309,14312,14343,14346,14349,14391],{"className":14304},[50],[15,14306],{"className":14307,"style":14308},[54],"height:1.1702em;vertical-align:-0.25em;",[15,14310,113],{"className":14311},[146],[15,14313,14315],{"className":14314},[59,11855],[15,14316,14318],{"className":14317},[285],[15,14319,14321],{"className":14320},[290],[15,14322,14324,14332],{"className":14323,"style":11851},[294],[15,14325,14326,14329],{"style":11867},[15,14327],{"className":14328,"style":461},[302],[15,14330,206],{"className":14331,"style":334},[59,60],[15,14333,14334,14337],{"style":11876},[15,14335],{"className":14336,"style":461},[302],[15,14338,14340],{"className":14339,"style":12119},[11883],[15,14341,11821],{"className":14342},[59],[15,14344,551],{"className":14345},[573],[15,14347],{"className":14348,"style":406},[157],[15,14350,14352],{"className":14351},[59,11855],[15,14353,14355,14383],{"className":14354},[285,286],[15,14356,14358,14380],{"className":14357},[290],[15,14359,14361,14369],{"className":14360,"style":11851},[294],[15,14362,14363,14366],{"style":11867},[15,14364],{"className":14365,"style":461},[302],[15,14367,213],{"className":14368},[59,60],[15,14370,14371,14374],{"style":11876},[15,14372],{"className":14373,"style":461},[302],[15,14375,14377],{"className":14376,"style":12119},[11883],[15,14378,11821],{"className":14379},[59],[15,14381,318],{"className":14382},[317],[15,14384,14386],{"className":14385},[290],[15,14387,14389],{"className":14388,"style":12311},[294],[15,14390],{},[15,14392,120],{"className":14393},[153]," 使用 DPI，即得到联合凸性：",[15,14396,14398],{"className":14397,"translate":19},[180],[15,14399,14401,14581],{"className":14400,"translate":19},[18],[15,14402,14404],{"className":14403},[23],[25,14405,14406],{"xmlns":27,"display":189},[29,14407,14408,14578],{},[1245,14409,14410,14494],{"rowspacing":1247,"columnalign":1248,"columnspacing":1249},[1251,14411,14412,14418],{},[1254,14413,14414],{},[1257,14415,14416],{"scriptlevel":126,"displaystyle":46},[32,14417],{},[1254,14419,14420],{},[1257,14421,14422],{"scriptlevel":126,"displaystyle":46},[32,14423,14424,14426,14432,14434,14436,14442,14444,14446,14448,14450,14452,14454,14460,14462,14464,14466,14468,14474,14476,14478,14480,14482,14484,14486,14492],{},[32,14425],{},[195,14427,14428,14430],{},[35,14429,199],{},[35,14431,37],{},[110,14433,113],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[35,14435,11343],{},[195,14437,14438,14440],{},[35,14439,206],{},[115,14441,117],{},[110,14443,2472],{},[110,14445,113],{"stretchy":112},[115,14447,117],{},[110,14449,4815],{},[35,14451,11343],{},[110,14453,120],{"stretchy":112},[195,14455,14456,14458],{},[35,14457,206],{},[115,14459,2465],{},[1299,14461,1301],{},[110,14463,210],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[1299,14465,1301],{},[35,14467,11343],{},[195,14469,14470,14472],{},[35,14471,213],{},[115,14473,117],{},[110,14475,2472],{},[110,14477,113],{"stretchy":112},[115,14479,117],{},[110,14481,4815],{},[35,14483,11343],{},[110,14485,120],{"stretchy":112},[195,14487,14488,14490],{},[35,14489,213],{},[115,14491,2465],{},[110,14493,120],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[1251,14495,14496,14502],{},[1254,14497,14498],{},[1257,14499,14500],{"scriptlevel":126,"displaystyle":46},[32,14501],{},[1254,14503,14504],{},[1257,14505,14506],{"scriptlevel":126,"displaystyle":46},[32,14507,14508,14510,14512,14514,14516,14522,14524,14530,14532,14538,14540,14542,14544,14546,14548,14550,14552,14558,14560,14566,14568,14574,14576],{},[32,14509],{},[157,14511],{"width":3264},[110,14513,9954],{},[35,14515,11343],{},[195,14517,14518,14520],{},[35,14519,199],{},[35,14521,37],{},[110,14523,113],{"stretchy":112},[195,14525,14526,14528],{},[35,14527,206],{},[115,14529,117],{},[35,14531,210],{"mathvariant":209},[195,14533,14534,14536],{},[35,14535,213],{},[115,14537,117],{},[110,14539,120],{"stretchy":112},[110,14541,2472],{},[110,14543,113],{"stretchy":112},[115,14545,117],{},[110,14547,4815],{},[35,14549,11343],{},[110,14551,120],{"stretchy":112},[195,14553,14554,14556],{},[35,14555,199],{},[35,14557,37],{},[110,14559,113],{"stretchy":112},[195,14561,14562,14564],{},[35,14563,206],{},[115,14565,2465],{},[35,14567,210],{"mathvariant":209},[195,14569,14570,14572],{},[35,14571,213],{},[115,14573,2465],{},[110,14575,120],{"stretchy":112},[35,14577,4377],{"mathvariant":209},[39,14579,14580],{"encoding":41},"\\begin{aligned}\n&D_f\\big(\\lambda P_1+(1-\\lambda)P_2\n\\,\\big\\|\\,\\lambda Q_1+(1-\\lambda)Q_2\\big)\\\\\n&\\qquad\\le \\lambda D_f(P_1\\|Q_1)\n+(1-\\lambda)D_f(P_2\\|Q_2).\n\\end{aligned}",[15,14582,14584],{"className":14583,"ariaHidden":46},[45],[15,14585,14587,14591],{"className":14586},[50],[15,14588],{"className":14589,"style":14590},[54],"height:2.71em;vertical-align:-1.105em;",[15,14592,14594],{"className":14593},[59],[15,14595,14597,14641],{"className":14596},[1245],[15,14598,14600],{"className":14599},[1362],[15,14601,14603,14632],{"className":14602},[285,286],[15,14604,14606,14629],{"className":14605},[290],[15,14607,14610,14620],{"className":14608,"style":14609},[294],"height:1.605em;",[15,14611,14613,14617],{"style":14612},"top:-3.605em;",[15,14614],{"className":14615,"style":14616},[302],"height:2.85em;",[15,14618],{"className":14619},[59],[15,14621,14623,14626],{"style":14622},"top:-2.105em;",[15,14624],{"className":14625,"style":14616},[302],[15,14627],{"className":14628},[59],[15,14630,318],{"className":14631},[317],[15,14633,14635],{"className":14634},[290],[15,14636,14639],{"className":14637,"style":14638},[294],"height:1.105em;",[15,14640],{},[15,14642,14644],{"className":14643},[1453],[15,14645,14647,15316],{"className":14646},[285,286],[15,14648,14650,15313],{"className":14649},[290],[15,14651,14653,14995],{"className":14652,"style":14609},[294],[15,14654,14656,14659],{"style":14655},"top:-3.755em;",[15,14657],{"className":14658,"style":461},[302],[15,14660,14662,14665,14705,14711,14714,14754,14757,14760,14763,14766,14769,14772,14775,14778,14781,14784,14824,14827,14873,14876,14879,14919,14922,14925,14928,14931,14934,14937,14940,14943,14946,14949,14989],{"className":14661},[59],[15,14663],{"className":14664},[59],[15,14666,14668,14671],{"className":14667},[59],[15,14669,199],{"className":14670,"style":277},[59,60],[15,14672,14674],{"className":14673},[281],[15,14675,14677,14697],{"className":14676},[285,286],[15,14678,14680,14694],{"className":14679},[290],[15,14681,14683],{"className":14682,"style":295},[294],[15,14684,14685,14688],{"style":298},[15,14686],{"className":14687,"style":303},[302],[15,14689,14691],{"className":14690},[307,308,309,310],[15,14692,37],{"className":14693,"style":61},[59,60,310],[15,14695,318],{"className":14696},[317],[15,14698,14700],{"className":14699},[290],[15,14701,14703],{"className":14702,"style":325},[294],[15,14704],{},[15,14706,14708],{"className":14707},[59],[15,14709,113],{"className":14710},[419,9676],[15,14712,11343],{"className":14713},[59,60],[15,14715,14717,14720],{"className":14716},[59],[15,14718,206],{"className":14719,"style":334},[59,60],[15,14721,14723],{"className":14722},[281],[15,14724,14726,14746],{"className":14725},[285,286],[15,14727,14729,14743],{"className":14728},[290],[15,14730,14732],{"className":14731,"style":11137},[294],[15,14733,14734,14737],{"style":11140},[15,14735],{"className":14736,"style":303},[302],[15,14738,14740],{"className":14739},[307,308,309,310],[15,14741,117],{"className":14742},[59,310],[15,14744,318],{"className":14745},[317],[15,14747,14749],{"className":14748},[290],[15,14750,14752],{"className":14751,"style":7970},[294],[15,14753],{},[15,14755],{"className":14756,"style":2589},[157],[15,14758,2472],{"className":14759},[2593],[15,14761],{"className":14762,"style":2589},[157],[15,14764,113],{"className":14765},[146],[15,14767,117],{"className":14768},[59],[15,14770],{"className":14771,"style":2589},[157],[15,14773,4815],{"className":14774},[2593],[15,14776],{"className":14777,"style":2589},[157],[15,14779,11343],{"className":14780},[59,60],[15,14782,120],{"className":14783},[153],[15,14785,14787,14790],{"className":14786},[59],[15,14788,206],{"className":14789,"style":334},[59,60],[15,14791,14793],{"className":14792},[281],[15,14794,14796,14816],{"className":14795},[285,286],[15,14797,14799,14813],{"className":14798},[290],[15,14800,14802],{"className":14801,"style":11137},[294],[15,14803,14804,14807],{"style":11140},[15,14805],{"className":14806,"style":303},[302],[15,14808,14810],{"className":14809},[307,308,309,310],[15,14811,2465],{"className":14812},[59,310],[15,14814,318],{"className":14815},[317],[15,14817,14819],{"className":14818},[290],[15,14820,14822],{"className":14821,"style":7970},[294],[15,14823],{},[15,14825],{"className":14826,"style":406},[157],[15,14828,14830],{"className":14829},[59],[15,14831,14834],{"className":14832},[419,14833],"mult",[15,14835,14837,14864],{"className":14836},[285,286],[15,14838,14840,14861],{"className":14839},[290],[15,14841,14844],{"className":14842,"style":14843},[294],"height:0.85em;",[15,14845,14847,14851],{"style":14846},"top:-2.85em;",[15,14848],{"className":14849,"style":14850},[302],"height:3.2em;",[15,14852,14854],{"style":14853},"width:0.556em;height:1.2em;",[6325,14855,14858],{"xmlns":6327,"width":14856,"height":9452,"viewBox":14857},"0.556em","0 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h43z",[15,14862,318],{"className":14863},[317],[15,14865,14867],{"className":14866},[290],[15,14868,14871],{"className":14869,"style":14870},[294],"height:0.35em;",[15,14872],{},[15,14874],{"className":14875,"style":406},[157],[15,14877,11343],{"className":14878},[59,60],[15,14880,14882,14885],{"className":14881},[59],[15,14883,213],{"className":14884},[59,60],[15,14886,14888],{"className":14887},[281],[15,14889,14891,14911],{"className":14890},[285,286],[15,14892,14894,14908],{"className":14893},[290],[15,14895,14897],{"className":14896,"style":11137},[294],[15,14898,14899,14902],{"style":382},[15,14900],{"className":14901,"style":303},[302],[15,14903,14905],{"className":14904},[307,308,309,310],[15,14906,117],{"className":14907},[59,310],[15,14909,318],{"className":14910},[317],[15,14912,14914],{"className":14913},[290],[15,14915,14917],{"className":14916,"style":7970},[294],[15,14918],{},[15,14920],{"className":14921,"style":2589},[157],[15,14923,2472],{"className":14924},[2593],[15,14926],{"className":14927,"style":2589},[157],[15,14929,113],{"className":14930},[146],[15,14932,117],{"className":14933},[59],[15,14935],{"className":14936,"style":2589},[157],[15,14938,4815],{"className":14939},[2593],[15,14941],{"className":14942,"style":2589},[157],[15,14944,11343],{"className":14945},[59,60],[15,14947,120],{"className":14948},[153],[15,14950,14952,14955],{"className":14951},[59],[15,14953,213],{"className":14954},[59,60],[15,14956,14958],{"className":14957},[281],[15,14959,14961,14981],{"className":14960},[285,286],[15,14962,14964,14978],{"className":14963},[290],[15,14965,14967],{"className":14966,"style":11137},[294],[15,14968,14969,14972],{"style":382},[15,14970],{"className":14971,"style":303},[302],[15,14973,14975],{"className":14974},[307,308,309,310],[15,14976,2465],{"className":14977},[59,310],[15,14979,318],{"className":14980},[317],[15,14982,14984],{"className":14983},[290],[15,14985,14987],{"className":14986,"style":7970},[294],[15,14988],{},[15,14990,14992],{"className":14991},[59],[15,14993,120],{"className":14994},[419,9676],[15,14996,14998,15001],{"style":14997},"top:-2.255em;",[15,14999],{"className":15000,"style":461},[302],[15,15002,15004,15007,15010,15013,15016,15019,15022,15062,15065,15105,15108,15148,15151,15154,15157,15160,15163,15166,15169,15172,15175,15178,15181,15221,15224,15264,15267,15307,15310],{"className":15003},[59],[15,15005],{"className":15006},[59],[15,15008],{"className":15009,"style":3636},[157],[15,15011],{"className":15012,"style":158},[157],[15,15014,9954],{"className":15015},[162],[15,15017],{"className":15018,"style":158},[157],[15,15020,11343],{"className":15021},[59,60],[15,15023,15025,15028],{"className":15024},[59],[15,15026,199],{"className":15027,"style":277},[59,60],[15,15029,15031],{"className":15030},[281],[15,15032,15034,15054],{"className":15033},[285,286],[15,15035,15037,15051],{"className":15036},[290],[15,15038,15040],{"className":15039,"style":295},[294],[15,15041,15042,15045],{"style":298},[15,15043],{"className":15044,"style":303},[302],[15,15046,15048],{"className":15047},[307,308,309,310],[15,15049,37],{"className":15050,"style":61},[59,60,310],[15,15052,318],{"className":15053},[317],[15,15055,15057],{"className":15056},[290],[15,15058,15060],{"className":15059,"style":325},[294],[15,15061],{},[15,15063,113],{"className":15064},[146],[15,15066,15068,15071],{"className":15067},[59],[15,15069,206],{"className":15070,"style":334},[59,60],[15,15072,15074],{"className":15073},[281],[15,15075,15077,15097],{"className":15076},[285,286],[15,15078,15080,15094],{"className":15079},[290],[15,15081,15083],{"className":15082,"style":11137},[294],[15,15084,15085,15088],{"style":11140},[15,15086],{"className":15087,"style":303},[302],[15,15089,15091],{"className":15090},[307,308,309,310],[15,15092,117],{"className":15093},[59,310],[15,15095,318],{"className":15096},[317],[15,15098,15100],{"className":15099},[290],[15,15101,15103],{"className":15102,"style":7970},[294],[15,15104],{},[15,15106,210],{"className":15107},[59],[15,15109,15111,15114],{"className":15110},[59],[15,15112,213],{"className":15113},[59,60],[15,15115,15117],{"className":15116},[281],[15,15118,15120,15140],{"className":15119},[285,286],[15,15121,15123,15137],{"className":15122},[290],[15,15124,15126],{"className":15125,"style":11137},[294],[15,15127,15128,15131],{"style":382},[15,15129],{"className":15130,"style":303},[302],[15,15132,15134],{"className":15133},[307,308,309,310],[15,15135,117],{"className":15136},[59,310],[15,15138,318],{"className":15139},[317],[15,15141,15143],{"className":15142},[290],[15,15144,15146],{"className":15145,"style":7970},[294],[15,15147],{},[15,15149,120],{"className":15150},[153],[15,15152],{"className":15153,"style":2589},[157],[15,15155,2472],{"className":15156},[2593],[15,15158],{"className":15159,"style":2589},[157],[15,15161,113],{"className":15162},[146],[15,15164,117],{"className":15165},[59],[15,15167],{"className":15168,"style":2589},[157],[15,15170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",[15,15330,15332,15345],{"className":15331,"translate":19},[18],[15,15333,15335],{"className":15334},[23],[25,15336,15337],{"xmlns":27},[29,15338,15339,15343],{},[32,15340,15341],{},[110,15342,10994],{},[39,15344,10997],{"encoding":41},[15,15346,15348],{"className":15347,"ariaHidden":46},[45],[15,15349,15351,15354],{"className":15350},[50],[15,15352],{"className":15353,"style":11007},[54],[15,15355,10994],{"className":15356},[162],[15,15358,15360,15373],{"className":15359,"translate":19},[18],[15,15361,15363],{"className":15362},[23],[25,15364,15365],{"xmlns":27},[29,15366,15367,15371],{},[32,15368,15369],{},[35,15370,37],{},[39,15372,37],{"encoding":41},[15,15374,15376],{"className":15375,"ariaHidden":46},[45],[15,15377,15379,15382],{"className":15378},[50],[15,15380],{"className":15381,"style":55},[54],[15,15383,37],{"className":15384,"style":61},[59,60]," 固定 ",[15,15387,15389,15412],{"className":15388,"translate":19},[18],[15,15390,15392],{"className":15391},[23],[25,15393,15394],{"xmlns":27},[29,15395,15396,15409],{},[32,15397,15398,15400,15402,15405,15407],{},[35,15399,7563],{},[110,15401,551],{"separator":46},[35,15403,15404],{},"u",[110,15406,6930],{},[115,15408,126],{},[39,15410,15411],{"encoding":41},"s,u\\ge0",[15,15413,15415,15443],{"className":15414,"ariaHidden":46},[45],[15,15416,15418,15422,15425,15428,15431,15434,15437,15440],{"className":15417},[50],[15,15419],{"className":15420,"style":15421},[54],"height:0.8304em;vertical-align:-0.1944em;",[15,15423,7563],{"className":15424},[59,60],[15,15426,551],{"className":15427},[573],[15,15429],{"className":15430,"style":406},[157],[15,15432,15404],{"className":15433},[59,60],[15,15435],{"className":15436,"style":158},[157],[15,15438,6930],{"className":15439},[162],[15,15441],{"className":15442,"style":158},[157],[15,15444,15446,15449],{"className":15445},[50],[15,15447],{"className":15448,"style":172},[54],[15,15450,126],{"className":15451},[59],[15,15453,15455,15480],{"className":15454,"translate":19},[18],[15,15456,15458],{"className":15457},[23],[25,15459,15460],{"xmlns":27},[29,15461,15462,15478],{},[32,15463,15464,15466,15468,15470,15472,15474,15476],{},[35,15465,11343],{},[110,15467,5911],{},[110,15469,113],{"stretchy":112},[115,15471,126],{},[110,15473,551],{"separator":46},[115,15475,117],{},[110,15477,120],{"stretchy":112},[39,15479,11358],{"encoding":41},[15,15481,15483,15501],{"className":15482,"ariaHidden":46},[45],[15,15484,15486,15489,15492,15495,15498],{"className":15485},[50],[15,15487],{"className":15488,"style":11368},[54],[15,15490,11343],{"className":15491},[59,60],[15,15493],{"className":15494,"style":158},[157],[15,15496,5911],{"className":15497},[162],[15,15499],{"className":15500,"style":158},[157],[15,15502,15504,15507,15510,15513,15516,15519,15522],{"className":15503},[50],[15,15505],{"className":15506,"style":139},[54],[15,15508,113],{"className":15509},[146],[15,15511,126],{"className":15512},[59],[15,15514,551],{"className":15515},[573],[15,15517],{"className":15518,"style":406},[157],[15,15520,117],{"className":15521},[59],[15,15523,120],{"className":15524},[153],"，记目标值 ",[15,15527,15529,15563],{"className":15528,"translate":19},[18],[15,15530,15532],{"className":15531},[23],[25,15533,15534],{"xmlns":27},[29,15535,15536,15560],{},[32,15537,15538,15540,15542,15544,15546,15548,15550,15552,15554,15556,15558],{},[35,15539,7572],{},[110,15541,123],{},[35,15543,11343],{},[35,15545,7563],{},[110,15547,2472],{},[110,15549,113],{"stretchy":112},[115,15551,117],{},[110,15553,4815],{},[35,15555,11343],{},[110,15557,120],{"stretchy":112},[35,15559,15404],{},[39,15561,15562],{"encoding":41},"w=\\lambda s+(1-\\lambda)u",[15,15564,15566,15584,15606,15627],{"className":15565,"ariaHidden":46},[45],[15,15567,15569,15572,15575,15578,15581],{"className":15568},[50],[15,15570],{"className":15571,"style":2347},[54],[15,15573,7572],{"className":15574,"style":8276},[59,60],[15,15576],{"className":15577,"style":158},[157],[15,15579,123],{"className":15580},[162],[15,15582],{"className":15583,"style":158},[157],[15,15585,15587,15591,15594,15597,15600,15603],{"className":15586},[50],[15,15588],{"className":15589,"style":15590},[54],"height:0.7778em;vertical-align:-0.0833em;",[15,15592,11343],{"className":15593},[59,60],[15,15595,7563],{"className":15596},[59,60],[15,15598],{"className":15599,"style":2589},[157],[15,15601,2472],{"className":15602},[2593],[15,15604],{"className":15605,"style":2589},[157],[15,15607,15609,15612,15615,15618,15621,15624],{"className":15608},[50],[15,15610],{"className":15611,"style":139},[54],[15,15613,113],{"className":15614},[146],[15,15616,117],{"className":15617},[59],[15,15619],{"className":15620,"style":2589},[157],[15,15622,4815],{"className":15623},[2593],[15,15625],{"className":15626,"style":2589},[157],[15,15628,15630,15633,15636,15639],{"className":15629},[50],[15,15631],{"className":15632,"style":139},[54],[15,15634,11343],{"className":15635},[59,60],[15,15637,120],{"className":15638},[153],[15,15640,15404],{"className":15641},[59,60],"。在两点空间 ",[15,15644,15646,15668],{"className":15645,"translate":19},[18],[15,15647,15649],{"className":15648},[23],[25,15650,15651],{"xmlns":27},[29,15652,15653,15665],{},[32,15654,15655,15657,15659,15661,15663],{},[110,15656,2900],{"stretchy":112},[115,15658,117],{},[110,15660,551],{"separator":46},[115,15662,2465],{},[110,15664,2910],{"stretchy":112},[39,15666,15667],{"encoding":41},"\\{1,2\\}",[15,15669,15671],{"className":15670,"ariaHidden":46},[45],[15,15672,15674,15677,15680,15683,15686,15689,15692],{"className":15673},[50],[15,15675],{"className":15676,"style":139},[54],[15,15678,2900],{"className":15679},[146],[15,15681,117],{"className":15682},[59],[15,15684,551],{"className":15685},[573],[15,15687],{"className":15688,"style":406},[157],[15,15690,2465],{"className":15691},[59],[15,15693,2910],{"className":15694},[153]," 上取同一个底分布 ",[15,15697,15699,15732],{"className":15698,"translate":19},[18],[15,15700,15702],{"className":15701},[23],[25,15703,15704],{"xmlns":27},[29,15705,15706,15729],{},[32,15707,15708,15710,15712,15714,15717,15719,15721,15723,15725,15727],{},[35,15709,213],{},[110,15711,123],{},[110,15713,113],{"stretchy":112},[35,15715,15716],{},"δ",[110,15718,551],{"separator":46},[1299,15720,1301],{},[115,15722,117],{},[110,15724,4815],{},[35,15726,15716],{},[110,15728,120],{"stretchy":112},[39,15730,15731],{"encoding":41},"Q=(\\delta,\\,1-\\delta)",[15,15733,15735,15753,15787],{"className":15734,"ariaHidden":46},[45],[15,15736,15738,15741,15744,15747,15750],{"className":15737},[50],[15,15739],{"className":15740,"style":566},[54],[15,15742,213],{"className":15743},[59,60],[15,15745],{"className":15746,"style":158},[157],[15,15748,123],{"className":15749},[162],[15,15751],{"className":15752,"style":158},[157],[15,15754,15756,15759,15762,15766,15769,15772,15775,15778,15781,15784],{"className":15755},[50],[15,15757],{"className":15758,"style":139},[54],[15,15760,113],{"className":15761},[146],[15,15763,15716],{"className":15764,"style":15765},[59,60],"margin-right:0.0379em;",[15,15767,551],{"className":15768},[573],[15,15770],{"className":15771,"style":406},[157],[15,15773],{"className":15774,"style":406},[157],[15,15776,117],{"className":15777},[59],[15,15779],{"className":15780,"style":2589},[157],[15,15782,4815],{"className":15783},[2593],[15,15785],{"className":15786,"style":2589},[157],[15,15788,15790,15793,15796],{"className":15789},[50],[15,15791],{"className":15792,"style":139},[54],[15,15794,15716],{"className":15795,"style":15765},[59,60],[15,15797,120],{"className":15798},[153],"（",[15,15801,15803,15829],{"className":15802,"translate":19},[18],[15,15804,15806],{"className":15805},[23],[25,15807,15808],{"xmlns":27},[29,15809,15810,15826],{},[32,15811,15812,15814,15816,15818,15820,15822,15824],{},[35,15813,15716],{},[110,15815,5911],{},[110,15817,113],{"stretchy":112},[115,15819,126],{},[110,15821,551],{"separator":46},[115,15823,117],{},[110,15825,120],{"stretchy":112},[39,15827,15828],{"encoding":41},"\\delta\\in(0,1)",[15,15830,15832,15850],{"className":15831,"ariaHidden":46},[45],[15,15833,15835,15838,15841,15844,15847],{"className":15834},[50],[15,15836],{"className":15837,"style":11368},[54],[15,15839,15716],{"className":15840,"style":15765},[59,60],[15,15842],{"className":15843,"style":158},[157],[15,15845,5911],{"className":15846},[162],[15,15848],{"className":15849,"style":158},[157],[15,15851,15853,15856,15859,15862,15865,15868,15871],{"className":15852},[50],[15,15854],{"className":15855,"style":139},[54],[15,15857,113],{"className":15858},[146],[15,15860,126],{"className":15861},[59],[15,15863,551],{"className":15864},[573],[15,15866],{"className":15867,"style":406},[157],[15,15869,117],{"className":15870},[59],[15,15872,120],{"className":15873},[153]," 充分小），并取似然比在第一坐标上为常数的分布：",[15,15876,15878],{"className":15877,"translate":19},[180],[15,15879,15881,15956],{"className":15880,"translate":19},[18],[15,15882,15884],{"className":15883},[23],[25,15885,15886],{"xmlns":27,"display":189},[29,15887,15888,15953],{},[32,15889,15890,15896,15898,15900,15902,15904,15906,15909,15911,15913,15915,15917,15919,15921,15923,15929,15931,15933,15935,15937,15939,15941,15943,15945,15947,15949,15951],{},[195,15891,15892,15894],{},[35,15893,206],{},[115,15895,117],{},[110,15897,123],{},[110,15899,113],{"stretchy":112},[35,15901,7563],{},[35,15903,15716],{},[110,15905,551],{"separator":46},[1299,15907,15908],{},"  ",[115,15910,117],{},[110,15912,4815],{},[35,15914,7563],{},[35,15916,15716],{},[110,15918,120],{"stretchy":112},[110,15920,551],{"separator":46},[157,15922],{"width":3264},[195,15924,15925,15927],{},[35,15926,206],{},[115,15928,2465],{},[110,15930,123],{},[110,15932,113],{"stretchy":112},[35,15934,15404],{},[35,15936,15716],{},[110,15938,551],{"separator":46},[1299,15940,15908],{},[115,15942,117],{},[110,15944,4815],{},[35,15946,15404],{},[35,15948,15716],{},[110,15950,120],{"stretchy":112},[110,15952,551],{"separator":46},[39,15954,15955],{"encoding":41},"P_1=(s\\delta,\\;1-s\\delta),\\qquad\nP_2=(u\\delta,\\;1-u\\delta),",[15,15957,15959,16015,16049,16119,16155],{"className":15958,"ariaHidden":46},[45],[15,15960,15962,15966,16006,16009,16012],{"className":15961},[50],[15,15963],{"className":15964,"style":15965},[54],"height:0.8333em;vertical-align:-0.15em;",[15,15967,15969,15972],{"className":15968},[59],[15,15970,206],{"className":15971,"style":334},[59,60],[15,15973,15975],{"className":15974},[281],[15,15976,15978,15998],{"className":15977},[285,286],[15,15979,15981,15995],{"className":15980},[290],[15,15982,15984],{"className":15983,"style":11137},[294],[15,15985,15986,15989],{"style":11140},[15,15987],{"className":15988,"style":303},[302],[15,15990,15992],{"className":15991},[307,308,309,310],[15,15993,117],{"className":15994},[59,310],[15,15996,318],{"className":15997},[317],[15,15999,16001],{"className":16000},[290],[15,16002,16004],{"className":16003,"style":7970},[294],[15,16005],{},[15,16007],{"className":16008,"style":158},[157],[15,16010,123],{"className":16011},[162],[15,16013],{"className":16014,"style":158},[157],[15,16016,16018,16021,16024,16028,16031,16034,16037,16040,16043,16046],{"className":16017},[50],[15,16019],{"className":16020,"style":139},[54],[15,16022,113],{"className":16023},[146],[15,16025,16027],{"className":16026,"style":15765},[59,60],"sδ",[15,16029,551],{"className":16030},[573],[15,16032],{"className":16033,"style":158},[157],[15,16035],{"className":16036,"style":406},[157],[15,16038,117],{"className":16039},[59],[15,16041],{"className":16042,"style":2589},[157],[15,16044,4815],{"className":16045},[2593],[15,16047],{"className":16048,"style":2589},[157],[15,16050,16052,16055,16058,16061,16064,16067,16070,16110,16113,16116],{"className":16051},[50],[15,16053],{"className":16054,"style":139},[54],[15,16056,16027],{"className":16057,"style":15765},[59,60],[15,16059,120],{"className":16060},[153],[15,16062,551],{"className":16063},[573],[15,16065],{"className":16066,"style":3636},[157],[15,16068],{"className":16069,"style":406},[157],[15,16071,16073,16076],{"className":16072},[59],[15,16074,206],{"className":16075,"style":334},[59,60],[15,16077,16079],{"className":16078},[281],[15,16080,16082,16102],{"className":16081},[285,286],[15,16083,16085,16099],{"className":16084},[290],[15,16086,16088],{"className":16087,"style":11137},[294],[15,16089,16090,16093],{"style":11140},[15,16091],{"className":16092,"style":303},[302],[15,16094,16096],{"className":16095},[307,308,309,310],[15,16097,2465],{"className":16098},[59,310],[15,16100,318],{"className":16101},[317],[15,16103,16105],{"className":16104},[290],[15,16106,16108],{"className":16107,"style":7970},[294],[15,16109],{},[15,16111],{"className":16112,"style":158},[157],[15,16114,123],{"className":16115},[162],[15,16117],{"className":16118,"style":158},[157],[15,16120,16122,16125,16128,16131,16134,16137,16140,16143,16146,16149,16152],{"className":16121},[50],[15,16123],{"className":16124,"style":139},[54],[15,16126,113],{"className":16127},[146],[15,16129,15404],{"className":16130},[59,60],[15,16132,15716],{"className":16133,"style":15765},[59,60],[15,16135,551],{"className":16136},[573],[15,16138],{"className":16139,"style":158},[157],[15,16141],{"className":16142,"style":406},[157],[15,16144,117],{"className":16145},[59],[15,16147],{"className":16148,"style":2589},[157],[15,16150,4815],{"className":16151},[2593],[15,16153],{"className":16154,"style":2589},[157],[15,16156,16158,16161,16164,16167,16170],{"className":16157},[50],[15,16159],{"className":16160,"style":139},[54],[15,16162,15404],{"className":16163},[59,60],[15,16165,15716],{"className":16166,"style":15765},[59,60],[15,16168,120],{"className":16169},[153],[15,16171,551],{"className":16172},[573],[11,16174,15799,16175,16204,16205,16247,16248,2411,16276,16439,16440,16469,16470,2747],{},[15,16176,16178,16192],{"className":16177,"translate":19},[18],[15,16179,16181],{"className":16180},[23],[25,16182,16183],{"xmlns":27},[29,16184,16185,16189],{},[32,16186,16187],{},[35,16188,15716],{},[39,16190,16191],{"encoding":41},"\\delta",[15,16193,16195],{"className":16194,"ariaHidden":46},[45],[15,16196,16198,16201],{"className":16197},[50],[15,16199],{"className":16200,"style":12601},[54],[15,16202,15716],{"className":16203,"style":15765},[59,60]," 足够小时这都是合法概率分布；",[15,16206,16208,16226],{"className":16207,"translate":19},[18],[15,16209,16211],{"className":16210},[23],[25,16212,16213],{"xmlns":27},[29,16214,16215,16223],{},[32,16216,16217,16219,16221],{},[115,16218,126],{},[110,16220,551],{"separator":46},[115,16222,117],{},[39,16224,16225],{"encoding":41},"0,1",[15,16227,16229],{"className":16228,"ariaHidden":46},[45],[15,16230,16232,16235,16238,16241,16244],{"className":16231},[50],[15,16233],{"className":16234,"style":9360},[54],[15,16236,126],{"className":16237},[59],[15,16239,551],{"className":16240},[573],[15,16242],{"className":16243,"style":406},[157],[15,16245,117],{"className":16246},[59]," 两点上的似然比分别为 ",[15,16249,16251,16264],{"className":16250,"translate":19},[18],[15,16252,16254],{"className":16253},[23],[25,16255,16256],{"xmlns":27},[29,16257,16258,16262],{},[32,16259,16260],{},[35,16261,7563],{},[39,16263,7563],{"encoding":41},[15,16265,16267],{"className":16266,"ariaHidden":46},[45],[15,16268,16270,16273],{"className":16269},[50],[15,16271],{"className":16272,"style":2347},[54],[15,16274,7563],{"className":16275},[59,60],[15,16277,16279,16321],{"className":16278,"translate":19},[18],[15,16280,16282],{"className":16281},[23],[25,16283,16284],{"xmlns":27},[29,16285,16286,16318],{},[32,16287,16288,16290,16292,16294,16296,16298],{},[35,16289,4160],{},[110,16291,113],{"stretchy":112},[35,16293,7563],{},[110,16295,120],{"stretchy":112},[110,16297,123],{},[238,16299,16300,16310],{},[32,16301,16302,16304,16306,16308],{},[115,16303,117],{},[110,16305,4815],{},[35,16307,7563],{},[35,16309,15716],{},[32,16311,16312,16314,16316],{},[115,16313,117],{},[110,16315,4815],{},[35,16317,15716],{},[39,16319,16320],{"encoding":41},"g(s)=\\frac{1-s\\delta}{1-\\delta}",[15,16322,16324,16351],{"className":16323,"ariaHidden":46},[45],[15,16325,16327,16330,16333,16336,16339,16342,16345,16348],{"className":16326},[50],[15,16328],{"className":16329,"style":139},[54],[15,16331,4160],{"className":16332,"style":981},[59,60],[15,16334,113],{"className":16335},[146],[15,16337,7563],{"className":16338},[59,60],[15,16340,120],{"className":16341},[153],[15,16343],{"className":16344,"style":158},[157],[15,16346,123],{"className":16347},[162],[15,16349],{"className":16350,"style":158},[157],[15,16352,16354,16358],{"className":16353},[50],[15,16355],{"className":16356,"style":16357},[54],"height:1.2834em;vertical-align:-0.4033em;",[15,16359,16361,16364,16436],{"className":16360},[59],[15,16362],{"className":16363},[146,441],[15,16365,16367],{"className":16366},[238],[15,16368,16370,16427],{"className":16369},[285,286],[15,16371,16373,16424],{"className":16372},[290],[15,16374,16376,16396,16404],{"className":16375,"style":2689},[294],[15,16377,16378,16381],{"style":2529},[15,16379],{"className":16380,"style":461},[302],[15,16382,16384],{"className":16383},[307,308,309,310],[15,16385,16387,16390,16393],{"className":16386},[59,310],[15,16388,117],{"className":16389},[59,310],[15,16391,4815],{"className":16392},[2593,310],[15,16394,15716],{"className":16395,"style":15765},[59,60,310],[15,16397,16398,16401],{"style":473},[15,16399],{"className":16400,"style":461},[302],[15,16402],{"className":16403,"style":481},[480],[15,16405,16406,16409],{"style":2552},[15,16407],{"className":16408,"style":461},[302],[15,16410,16412],{"className":16411},[307,308,309,310],[15,16413,16415,16418,16421],{"className":16414},[59,310],[15,16416,117],{"className":16417},[59,310],[15,16419,4815],{"className":16420},[2593,310],[15,16422,16027],{"className":16423,"style":15765},[59,60,310],[15,16425,318],{"className":16426},[317],[15,16428,16430],{"className":16429},[290],[15,16431,16434],{"className":16432,"style":16433},[294],"height:0.4033em;",[15,16435],{},[15,16437],{"className":16438},[153,441],"）。于是一般公式给出：对任何第一坐标似然比为常值 ",[15,16441,16443,16457],{"className":16442,"translate":19},[18],[15,16444,16446],{"className":16445},[23],[25,16447,16448],{"xmlns":27},[29,16449,16450,16455],{},[32,16451,16452],{},[35,16453,16454],{},"r",[39,16456,16454],{"encoding":41},[15,16458,16460],{"className":16459,"ariaHidden":46},[45],[15,16461,16463,16466],{"className":16462},[50],[15,16464],{"className":16465,"style":2347},[54],[15,16467,16454],{"className":16468,"style":277},[59,60]," 的分布 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+(1-\\delta)\\,\nf\\!\\left(\\frac{1-r\\delta}{1-\\delta}\\right).\n\\end{aligned}",[15,16653,16655],{"className":16654,"ariaHidden":46},[45],[15,16656,16658,16662],{"className":16657},[50],[15,16659],{"className":16660,"style":16661},[54],"height:3.91em;vertical-align:-1.705em;",[15,16663,16665],{"className":16664},[59],[15,16666,16668,16820],{"className":16667},[1245],[15,16669,16671],{"className":16670},[1362],[15,16672,16674,16812],{"className":16673},[285,286],[15,16675,16677,16809],{"className":16676},[290],[15,16678,16681,16800],{"className":16679,"style":16680},[294],"height:2.205em;",[15,16682,16684,16687],{"style":16683},"top:-4.805em;",[15,16685],{"className":16686,"style":1379},[302],[15,16688,16690,16730,16736,16739,16742,16745,16748,16751,16788,16791,16794],{"className":16689},[59],[15,16691,16693,16696],{"className":16692},[59],[15,16694,199],{"className":16695,"style":277},[59,60],[15,16697,16699],{"className":16698},[281],[15,16700,16702,16722],{"className":16701},[285,286],[15,16703,16705,16719],{"className":16704},[290],[15,16706,16708],{"className":16707,"style":295},[294],[15,16709,16710,16713],{"style":298},[15,16711],{"className":16712,"style":303},[302],[15,16714,16716],{"className":16715},[307,308,309,310],[15,16717,37],{"className":16718,"style":61},[59,60,310],[15,16720,318],{"className":16721},[317],[15,16723,16725],{"className":16724},[290],[15,16726,16728],{"className":16727,"style":325},[294],[15,16729],{},[15,16731,16733],{"className":16732},[59],[15,16734,113],{"className":16735},[419,9676],[15,16737,206],{"className":16738,"style":334},[59,60],[15,16740,113],{"className":16741},[146],[15,16743,16454],{"className":16744,"style":277},[59,60],[15,16746,120],{"className":16747},[153],[15,16749],{"className":16750,"style":406},[157],[15,16752,16754],{"className":16753},[59],[15,16755,16757],{"className":16756},[419,14833],[15,16758,16760,16780],{"className":16759},[285,286],[15,16761,16763,16777],{"className":16762},[290],[15,16764,16766],{"className":16765,"style":14843},[294],[15,16767,16768,16771],{"style":14846},[15,16769],{"className":16770,"style":14850},[302],[15,16772,16773],{"style":14853},[6325,16774,16775],{"xmlns":6327,"width":14856,"height":9452,"viewBox":14857},[6333,16776],{"d":14860},[15,16778,318],{"className":16779},[317],[15,16781,16783],{"className":16782},[290],[15,16784,16786],{"className":16785,"style":14870},[294],[15,16787],{},[15,16789],{"className":16790,"style":406},[157],[15,16792,213],{"className":16793},[59,60],[15,16795,16797],{"className":16796},[59],[15,16798,120],{"className":16799},[419,9676],[15,16801,16803,16806],{"style":16802},"top:-2.695em;",[15,16804],{"className":16805,"style":1379},[302],[15,16807],{"className":16808},[59],[15,16810,318],{"className":16811},[317],[15,16813,16815],{"className":16814},[290],[15,16816,16818],{"className":16817,"style":9556},[294],[15,16819],{},[15,16821,16823],{"className":16822},[1453],[15,16824,16826,17040],{"className":16825},[285,286],[15,16827,16829,17037],{"className":16828},[290],[15,16830,16832,16870],{"className":16831,"style":16680},[294],[15,16833,16834,16837],{"style":16683},[15,16835],{"className":16836,"style":1379},[302],[15,16838,16840,16843,16846,16849,16852,16855,16858,16861,16864,16867],{"className":16839},[59],[15,16841],{"className":16842},[59],[15,16844],{"className":16845,"style":158},[157],[15,16847,123],{"className":16848},[162],[15,16850],{"className":16851,"style":158},[157],[15,16853,15716],{"className":16854,"style":15765},[59,60],[15,16856],{"className":16857,"style":406},[157],[15,16859,37],{"className":16860,"style":61},[59,60],[15,16862,113],{"className":16863},[146],[15,16865,16454],{"className":16866,"style":277},[59,60],[15,16868,120],{"className":16869},[153],[15,16871,16872,16875],{"style":16802},[15,16873],{"className":16874,"style":1379},[302],[15,16876,16878,16881,16884,16887,16890,16893,16896,16899,16902,16905,16908,16911,16914,16917,16920,16923,16926,17031,17034],{"className":16877},[59],[15,16879],{"className":16880},[59],[15,16882],{"className":16883,"style":3636},[157],[15,16885],{"className":16886,"style":2589},[157],[15,16888,2472],{"className":16889},[2593],[15,16891],{"className":16892,"style":2589},[157],[15,16894,113],{"className":16895},[146],[15,16897,117],{"className":16898},[59],[15,16900],{"className":16901,"style":2589},[157],[15,16903,4815],{"className":16904},[2593],[15,16906],{"className":16907,"style":2589},[157],[15,16909,15716],{"className":16910,"style":15765},[59,60],[15,16912,120],{"className":16913},[153],[15,16915],{"className":16916,"style":406},[157],[15,16918,37],{"className":16919,"style":61},[59,60],[15,16921],{"className":16922,"style":1514},[157],[15,16924],{"className":16925,"style":406},[157],[15,16927,16929,16935,17025],{"className":16928},[410],[15,16930,16932],{"className":16931,"style":415},[146,414],[15,16933,113],{"className":16934},[419,309],[15,16936,16938,16941,17022],{"className":16937},[59],[15,16939],{"className":169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",[15,17052,17054,17072],{"className":17053,"translate":19},[18],[15,17055,17057],{"className":17056},[23],[25,17058,17059],{"xmlns":27},[29,17060,17061,17069],{},[32,17062,17063,17065,17067],{},[35,17064,16454],{},[110,17066,123],{},[35,17068,7563],{},[39,17070,17071],{"encoding":41},"r=s",[15,17073,17075,17093],{"className":17074,"ariaHidden":46},[45],[15,17076,17078,17081,17084,17087,17090],{"className":17077},[50],[15,17079],{"className":17080,"style":2347},[54],[15,17082,16454],{"className":17083,"style":277},[59,60],[15,17085],{"className":17086,"style":158},[157],[15,17088,123],{"className":17089},[162],[15,17091],{"className":17092,"style":158},[157],[15,17094,17096,17099],{"className":17095},[50],[15,17097],{"className":17098,"style":2347},[54],[15,17100,7563],{"className":17101},[59,60],[15,17103,17105,17123],{"className":17104,"translate":19},[18],[15,17106,17108],{"className":17107},[23],[25,17109,17110],{"xmlns":27},[29,17111,17112,17120],{},[32,17113,17114,17116,17118],{},[35,17115,16454],{},[110,17117,123],{},[35,17119,15404],{},[39,17121,17122],{"encoding":41},"r=u",[15,17124,17126,17144],{"className":17125,"ariaHidden":46},[45],[15,17127,17129,17132,17135,17138,17141],{"className":17128},[50],[15,17130],{"className":17131,"style":2347},[54],[15,17133,16454],{"className":17134,"style":277},[59,60],[15,17136],{"className":17137,"style":158},[157],[15,17139,123],{"className":17140},[162],[15,17142],{"className":17143,"style":158},[157],[15,17145,17147,17150],{"className":17146},[50],[15,17148],{"className":17149,"style":2347},[54],[15,17151,15404],{"className":17152},[59,60],"、以及混合后的 ",[15,17155,17157,17175],{"className":17156,"translate":19},[18],[15,17158,17160],{"className":17159},[23],[25,17161,17162],{"xmlns":27},[29,17163,17164,17172],{},[32,17165,17166,17168,17170],{},[35,17167,16454],{},[110,17169,123],{},[35,17171,7572],{},[39,17173,17174],{"encoding":41},"r=w",[15,17176,17178,17196],{"className":17177,"ariaHidden":46},[45],[15,17179,17181,17184,17187,17190,17193],{"className":17180},[50],[15,17182],{"className":17183,"style":2347},[54],[15,17185,16454],{"className":17186,"style":277},[59,60],[15,17188],{"className":17189,"style":158},[157],[15,17191,123],{"className":17192},[162],[15,17194],{"className":17195,"style":158},[157],[15,17197,17199,17202],{"className":17198},[50],[15,17200],{"className":17201,"style":2347},[54],[15,17203,7572],{"className":17204,"style":8276},[59,60],"（混合后第一坐标似然比恰为 ",[15,17207,17209,17243],{"className":17208,"translate":19},[18],[15,17210,17212],{"className":17211},[23],[25,17213,17214],{"xmlns":27},[29,17215,17216,17240],{},[32,17217,17218,17220,17222,17224,17226,17228,17230,17232,17234,17236,17238],{},[35,17219,11343],{},[35,17221,7563],{},[110,17223,2472],{},[110,17225,113],{"stretchy":112},[115,17227,117],{},[110,17229,4815],{},[35,17231,11343],{},[110,17233,120],{"stretchy":112},[35,17235,15404],{},[110,17237,123],{},[35,17239,7572],{},[39,17241,17242],{"encoding":41},"\\lambda s+(1-\\lambda)u=w",[15,17244,17246,17267,17288,17312],{"className":17245,"ariaHidden":46},[45],[15,17247,17249,17252,17255,17258,17261,17264],{"className":17248},[50],[15,17250],{"className":17251,"style":15590},[54],[15,17253,11343],{"className":17254},[59,60],[15,17256,7563],{"className":17257},[59,60],[15,17259],{"className":17260,"style":2589},[157],[15,17262,2472],{"className":17263},[2593],[15,17265],{"className":17266,"style":2589},[157],[15,17268,17270,17273,17276,17279,17282,17285],{"className":17269},[50],[15,17271],{"className":17272,"style":139},[54],[15,17274,113],{"className":17275},[146],[15,17277,117],{"className":17278},[59],[15,17280],{"className":17281,"style":2589},[157],[15,17283,4815],{"className":17284},[2593],[15,17286],{"className":17287,"style":2589},[157],[15,17289,17291,17294,17297,17300,17303,17306,17309],{"className":17290},[50],[15,17292],{"className":17293,"style":139},[54],[15,17295,11343],{"className":17296},[59,60],[15,17298,120],{"className":17299},[153],[15,17301,15404],{"className":17302},[59,60],[15,17304],{"className":17305,"style":158},[157],[15,17307,123],{"className":17308},[162],[15,17310],{"className":17311,"style":158},[157],[15,17313,17315,17318],{"className":17314},[50],[15,17316],{"className":17317,"style":2347},[54],[15,17319,7572],{"className":17320,"style":8276},[59,60],"，因为两个分布在",[998,17323,17324],{},"同一个",[15,17326,17328,17341],{"className":17327,"translate":19},[18],[15,17329,17331],{"className":17330},[23],[25,17332,17333],{"xmlns":27},[29,17334,17335,17339],{},[32,17336,17337],{},[35,17338,213],{},[39,17340,213],{"encoding":41},[15,17342,17344],{"className":17343,"ariaHidden":46},[45],[15,17345,17347,17350],{"className":17346},[50],[15,17348],{"className":17349,"style":566},[54],[15,17351,213],{"className":17352},[59,60]," 下混合，比值线性）代入第一步的联合凸性：",[15,17355,17357],{"className":17356,"translate":19},[180],[15,17358,17360,17557],{"className":17359,"translate":19},[18],[15,17361,17363],{"className":17362},[23],[25,17364,17365],{"xmlns":27,"display":189},[29,17366,17367,17554],{},[1245,17368,17369,17421,17482],{"rowspacing":1247,"columnalign":1248,"columnspacing":1249},[1251,17370,17371,17377],{},[1254,17372,17373],{},[1257,17374,17375],{"scriptlevel":126,"displaystyle":46},[32,17376],{},[1254,17378,17379],{},[1257,17380,17381],{"scriptlevel":126,"displaystyle":46},[32,17382,17383,17385,17387,17389,17391,17393,17395,17397,17399,17401,17403,17405,17407,17409,17411,17413,17415,17417,17419],{},[32,17384],{},[35,17386,15716],{},[35,17388,37],{},[110,17390,113],{"stretchy":112},[35,17392,7572],{},[110,17394,120],{"stretchy":112},[110,17396,2472],{},[110,17398,113],{"stretchy":112},[115,17400,117],{},[110,17402,4815],{},[35,17404,15716],{},[110,17406,120],{"stretchy":112},[35,17408,37],{},[110,17410,113],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[35,17412,4160],{},[110,17414,113],{"stretchy":112},[35,17416,7572],{},[110,17418,120],{"stretchy":112},[110,17420,120],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[1251,17422,17423,17429],{},[1254,17424,17425],{},[1257,17426,17427],{"scriptlevel":126,"displaystyle":46},[32,17428],{},[1254,17430,17431],{},[1257,17432,17433],{"scriptlevel":126,"displaystyle":46},[32,17434,17435,17437,17439,17441,17444,17446,17448,17450,17452,17454,17456,17458,17460,17462,17464,17466,17468,17470,17472,17474,17476,17478,17480],{},[32,17436],{},[110,17438,9954],{},[35,17440,11343],{},[110,17442,230],{"fence":112,"stretchy":46,"minsize":17443,"maxsize":17443},"1.8em",[35,17445,15716],{},[35,17447,37],{},[110,17449,113],{"stretchy":112},[35,17451,7563],{},[110,17453,120],{"stretchy":112},[110,17455,2472],{},[110,17457,113],{"stretchy":112},[115,17459,117],{},[110,17461,4815],{},[35,17463,15716],{},[110,17465,120],{"stretchy":112},[35,17467,37],{},[110,17469,113],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[35,17471,4160],{},[110,17473,113],{"stretchy":112},[35,17475,7563],{},[110,17477,120],{"stretchy":112},[110,17479,120],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[110,17481,257],{"fence":112,"stretchy":46,"minsize":17443,"maxsize":17443},[1251,17483,17484,17490],{},[1254,17485,17486],{},[1257,17487,17488],{"scriptlevel":126,"displaystyle":46},[32,17489],{},[1254,17491,17492],{},[1257,17493,17494],{"scriptlevel":126,"displaystyle":46},[32,17495,17496,17498,17500,17502,17504,17506,17508,17510,17512,17514,17516,17518,17520,17522,17524,17526,17528,17530,17532,17534,17536,17538,17540,17542,17544,17546,17548,17550,17552],{},[32,17497],{},[157,17499],{"width":3264},[110,17501,2472],{},[110,17503,113],{"stretchy":112},[115,17505,117],{},[110,17507,4815],{},[35,17509,11343],{},[110,17511,120],{"stretchy":112},[110,17513,230],{"fence":112,"stretchy":46,"minsize":17443,"maxsize":17443},[35,17515,15716],{},[35,17517,37],{},[110,17519,113],{"stretchy":112},[35,17521,15404],{},[110,17523,120],{"stretchy":112},[110,17525,2472],{},[110,17527,113],{"stretchy":112},[115,17529,117],{},[110,17531,4815],{},[35,17533,15716],{},[110,17535,120],{"stretchy":112},[35,17537,37],{},[110,17539,113],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[35,17541,4160],{},[110,17543,113],{"stretchy":112},[35,17545,15404],{},[110,17547,120],{"stretchy":112},[110,17549,120],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[110,17551,257],{"fence":112,"stretchy":46,"minsize":17443,"maxsize":17443},[110,17553,551],{"separator":46},[39,17555,17556],{"encoding":41},"\\begin{aligned}\n&\\delta f(w)+(1-\\delta)f\\big(g(w)\\big)\\\\\n&\\le \\lambda\\Big[\\delta f(s)\n+(1-\\delta)f\\big(g(s)\\big)\\Big]\\\\\n&\\qquad+(1-\\lambda)\\Big[\\delta f(u)\n+(1-\\delta)f\\big(g(u)\\big)\\Big],\n\\end{aligned}",[15,17558,17560],{"className":17559,"ariaHidden":46},[45],[15,17561,17563,17567],{"className":17562},[50],[15,17564],{"className":17565,"style":17566},[54],"height:5.41em;vertical-align:-2.455em;",[15,17568,17570],{"className":17569},[59],[15,17571,17573,17626],{"className":17572},[1245],[15,17574,17576],{"className":17575},[1362],[15,17577,17579,17617],{"className":17578},[285,286],[15,17580,17582,17614],{"className":17581},[290],[15,17583,17586,17596,17605],{"className":17584,"style":17585},[294],"height:2.955em;",[15,17587,17589,17593],{"style":17588},"top:-5.255em;",[15,17590],{"className":17591,"style":17592},[302],"height:3.15em;",[15,17594],{"className":17595},[59],[15,17597,17599,17602],{"style":17598},"top:-3.445em;",[15,17600],{"className":17601,"style":17592},[302],[15,17603],{"className":17604},[59],[15,17606,17608,17611],{"style":17607},"top:-1.345em;",[15,17609],{"className":17610,"style":17592},[302],[15,17612],{"className":17613},[59],[15,17615,318],{"className":17616},[317],[15,17618,17620],{"className":17619},[290],[15,17621,17624],{"className":17622,"style":17623},[294],"height:2.455em;",[15,17625],{},[15,17627,17629],{"className":17628},[1453],[15,17630,17632,17962],{"className":17631},[285,286],[15,17633,17635,17959],{"className":17634},[290],[15,17636,17638,17721,17828],{"className":17637,"style":17585},[294],[15,17639,17640,17643],{"style":17588},[15,17641],{"className":17642,"style":17592},[302],[15,17644,17646,17649,17652,17655,17658,17661,17664,17667,17670,17673,17676,17679,17682,17685,17688,17691,17694,17697,17703,17706,17709,17712,17715],{"className":17645},[59],[15,17647],{"className":17648},[59],[15,17650,15716],{"className":17651,"style":15765},[59,60],[15,17653,37],{"className":17654,"style":61},[59,60],[15,17656,113],{"className":17657},[146],[15,17659,7572],{"className":17660,"style":8276},[59,60],[15,17662,120],{"className":17663},[153],[15,17665],{"className":17666,"style":2589},[157],[15,17668,2472],{"className":17669},[2593],[15,17671],{"className":17672,"style":2589},[157],[15,17674,113],{"className":17675},[146],[15,17677,117],{"className":17678},[59],[15,17680],{"className":17681,"style":2589},[157],[15,17683,4815],{"className":17684},[2593],[15,17686],{"className":17687,"style":2589},[157],[15,17689,15716],{"className":17690,"style":15765},[59,60],[15,17692,120],{"className":17693},[153],[15,17695,37],{"className":17696,"style":61},[59,60],[15,17698,17700],{"className":17699},[59],[15,17701,113],{"className":17702},[419,9676],[15,17704,4160],{"className":17705,"style":981},[59,60],[15,17707,113],{"className":17708},[146],[15,17710,7572],{"className":17711,"style":8276},[59,60],[15,17713,120],{"className":17714},[153],[15,17716,17718],{"className":17717},[59],[15,17719,120],{"className":17720},[419,9676],[15,17722,17723,17726],{"style":17598},[15,17724],{"className":17725,"style":17592},[302],[15,17727,17729,17732,17735,17738,17741,17744,17750,17753,17756,17759,17762,17765,17768,17771,17774,17777,17780,17783,17786,17789,17792,17795,17798,17804,17807,17810,17813,17816,17822],{"className":17728},[59],[15,17730],{"className":17731},[59],[15,17733],{"className":17734,"style":158},[157],[15,17736,9954],{"className":17737},[162],[15,17739],{"className":17740,"style":158},[157],[15,17742,11343],{"className":17743},[59,60],[15,17745,17747],{"className":17746},[59],[15,17748,230],{"className":17749},[419,3722],[15,17751,15716],{"className":17752,"style":15765},[59,60],[15,17754,37],{"className":17755,"style":61},[59,60],[15,17757,113],{"className":17758},[146],[15,17760,7563],{"className":17761},[59,60],[15,17763,120],{"className":17764},[153],[15,17766],{"className":17767,"style":2589},[157],[15,17769,2472],{"className":17770},[2593],[15,17772],{"className":17773,"style":2589},[157],[15,17775,113],{"className":17776},[146],[15,17778,117],{"className":17779},[59],[15,17781],{"className":17782,"style":2589},[157],[15,17784,4815],{"className":17785},[2593],[15,17787],{"className":17788,"style":2589},[157],[15,17790,15716],{"className":17791,"style":15765},[59,60],[15,17793,120],{"className":17794},[153],[15,17796,37],{"className":17797,"style":61},[59,60],[15,17799,17801],{"className":17800},[59],[15,17802,113],{"className":17803},[419,9676],[15,17805,4160],{"className":17806,"style":981},[59,60],[15,17808,113],{"className":17809},[146],[15,17811,7563],{"className":17812},[59,60],[15,17814,120],{"className":17815},[153],[15,17817,17819],{"className":17818},[59],[15,17820,120],{"className":17821},[419,9676],[15,17823,17825],{"className":17824},[59],[15,17826,257],{"className":17827},[419,3722],[15,17829,17830,17833],{"style":17607},[15,17831],{"className":17832,"style":17592},[302],[15,17834,17836,17839,17842,17845,17848,17851,17854,17857,17860,17863,17866,17869,17872,17878,17881,17884,17887,17890,17893,17896,17899,17902,17905,17908,17911,17914,17917,17920,17923,17926,17932,17935,17938,17941,17944,17950,17956],{"classNa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19},[294],[15,18141],{},[15,18143],{"className":18144},[153,441],"。注意 ",[15,18147,18149,18162],{"className":18148,"translate":19},[18],[15,18150,18152],{"className":18151},[23],[25,18153,18154],{"xmlns":27},[29,18155,18156,18160],{},[32,18157,18158],{},[35,18159,4160],{},[39,18161,4160],{"encoding":41},[15,18163,18165],{"className":18164,"ariaHidden":46},[45],[15,18166,18168,18171],{"className":18167},[50],[15,18169],{"className":18170,"style":2278},[54],[15,18172,4160],{"className":18173,"style":981},[59,60]," 是仿射函数且 ",[15,18176,18178,18230],{"className":18177,"translate":19},[18],[15,18179,18181],{"className":18180},[23],[25,18182,18183],{"xmlns":27},[29,18184,18185,18227],{},[32,18186,18187,18189,18191,18193,18195,18197,18199,18201,18203,18205,18207,18209,18211,18213,18215,18217,18219,18221,18223,18225],{},[35,18188,4160],{},[110,18190,113],{"stretchy":112},[35,18192,7572],{},[110,18194,120],{"stretchy":112},[110,18196,123],{},[35,18198,11343],{},[35,18200,4160],{},[110,18202,113],{"stretchy":112},[35,18204,7563],{},[110,18206,120],{"stretchy":112},[110,18208,2472],{},[110,18210,113],{"stretchy":112},[115,18212,117],{},[110,18214,4815],{},[35,18216,11343],{},[110,18218,120],{"stretchy":112},[35,18220,4160],{},[110,18222,113],{"stretchy":112},[35,18224,15404],{},[110,18226,120],{"stretchy":112},[39,18228,18229],{"encoding":41},"g(w)=\\lambda g(s)+(1-\\lambda)g(u)",[15,18231,18233,18260,18290,18311],{"className":18232,"ariaHidden":46},[45],[15,18234,18236,18239,18242,18245,18248,18251,18254,18257],{"className":18235},[50],[15,18237],{"className":18238,"style":139},[54],[15,18240,4160],{"className":18241,"style":981},[59,60],[15,18243,113],{"className":18244},[146],[15,18246,7572],{"className":18247,"style":8276},[59,60],[15,18249,120],{"className":18250},[153],[15,18252],{"className":18253,"style":158},[157],[15,18255,123],{"className":18256},[162],[15,18258],{"className":18259,"style":158},[157],[15,18261,18263,18266,18269,18272,18275,18278,18281,18284,18287],{"className":18262},[50],[15,18264],{"className":18265,"style":139},[54],[15,18267,11343],{"className":18268},[59,60],[15,18270,4160],{"className":18271,"style":981},[59,60],[15,18273,113],{"className":18274},[146],[15,18276,7563],{"className":18277},[59,60],[15,18279,120],{"className":18280},[153],[15,18282],{"className":18283,"style":2589},[157],[15,18285,2472],{"className":18286},[2593],[15,18288],{"className":18289,"style":2589},[157],[15,18291,18293,18296,18299,18302,18305,18308],{"className":18292},[50],[15,18294],{"className":18295,"style":139},[54],[15,18297,113],{"className":18298},[146],[15,18300,117],{"className":18301},[59],[15,18303],{"className":18304,"style":2589},[157],[15,18306,4815],{"className":18307},[2593],[15,18309],{"className":18310,"style":2589},[157],[15,18312,18314,18317,18320,18323,18326,18329,18332],{"className":18313},[50],[15,18315],{"className":18316,"style":139},[54],[15,18318,11343],{"className":18319},[59,60],[15,18321,120],{"className":18322},[153],[15,18324,4160],{"className":18325,"style":981},[59,60],[15,18327,113],{"className":18328},[146],[15,18330,15404],{"className":18331},[59,60],[15,18333,120],{"className":18334},[153],"，用 ",[15,18337,18339,18352],{"className":18338,"translate":19},[18],[15,18340,18342],{"className":18341},[23],[25,18343,18344],{"xmlns":27},[29,18345,18346,18350],{},[32,18347,18348],{},[35,18349,37],{},[39,18351,37],{"encoding":41},[15,18353,18355],{"className":18354,"ariaHidden":46},[45],[15,18356,18358,18361],{"className":18357},[50],[15,18359],{"className":18360,"style":55},[54],[15,18362,37],{"className":18363,"style":61},[59,60],[15,18365,18367,18380],{"className":18366,"translate":19},[18],[15,18368,18370],{"className":18369},[23],[25,18371,18372],{"xmlns":27},[29,18373,18374,18378],{},[32,18375,18376],{},[115,18377,117],{},[39,18379,117],{"encoding":41},[15,18381,18383],{"className":18382,"ariaHidden":46},[45],[15,18384,18386,18389],{"className":18385},[50],[15,18387],{"className":18388,"style":172},[54],[15,18390,117],{"className":18391},[59]," 处的支撑线（凸函数在可微点处 ",[15,18394,18396,18466],{"className":18395,"translate":19},[18],[15,18397,18399],{"className":18398},[23],[25,18400,18401],{"xmlns":27},[29,18402,18403,18463],{},[32,18404,18405,18407,18409,18411,18413,18415,18417,18419,18421,18423,18425,18431,18433,18435,18437,18439,18441,18443,18445,18447,18449,18455,18457,18459,18461],{},[35,18406,37],{},[110,18408,113],{"stretchy":112},[35,18410,769],{},[110,18412,120],{"stretchy":112},[110,18414,6930],{},[35,18416,37],{},[110,18418,113],{"stretchy":112},[115,18420,117],{},[110,18422,120],{"stretchy":112},[110,18424,2472],{},[6816,18426,18427,18429],{},[35,18428,37],{},[110,18430,6822],{"mathvariant":209,"lspace":1249,"rspace":1249},[110,18432,113],{"stretchy":112},[115,18434,117],{},[110,18436,120],{"stretchy":112},[110,18438,113],{"stretchy":112},[35,18440,769],{},[110,18442,4815],{},[115,18444,117],{},[110,18446,120],{"stretchy":112},[110,18448,7575],{},[6816,18450,18451,18453],{},[35,18452,37],{},[110,18454,6822],{"mathvariant":209,"lspace":1249,"rspace":1249},[110,18456,113],{"stretchy":112},[115,18458,117],{},[110,18460,120],{"stretchy":112},[110,18462,7575],{},[39,18464,18465],{"encoding":41},"f(t)\\ge f(1)+f'(1)(t-1)*f'(1)*",[15,18467,18469,18496,18523,18585,18606],{"className":18468,"ariaHidden":46},[45],[15,18470,18472,18475,18478,18481,18484,18487,18490,18493],{"className":18471},[50],[15,18473],{"className":18474,"style":139},[54],[15,18476,37],{"className":18477,"style":61},[59,60],[15,18479,113],{"className":18480},[146],[15,18482,769],{"className":18483},[59,60],[15,18485,120],{"className":18486},[153],[15,18488],{"className":18489,"style":158},[157],[15,18491,6930],{"className":18492},[162],[15,18494],{"className":18495,"style":158},[157],[15,18497,18499,18502,18505,18508,18511,18514,18517,18520],{"className":18498},[50],[15,18500],{"className":18501,"style":139},[54],[15,18503,37],{"className":18504,"style":61},[59,60],[15,18506,113],{"className":18507},[146],[15,18509,117],{"className":18510},[59],[15,18512,120],{"className":18513},[153],[15,18515],{"className":18516,"style":2589},[157],[15,18518,2472],{"className":18519},[2593],[15,18521],{"className":18522,"style":2589},[157],[15,18524,18526,18529,18561,18564,18567,18570,18573,18576,18579,18582],{"className":18525},[50],[15,18527],{"className":18528,"style":6845},[54],[15,18530,18532,18535],{"className":18531},[59],[15,18533,37],{"className":18534,"style":61},[59,60],[15,18536,18538],{"className":18537},[281],[15,18539,18541],{"className":18540},[285],[15,18542,18544],{"className":18543},[290],[15,18545,18547],{"className":18546,"style":6864},[294],[15,18548,18549,18552],{"style":6867},[15,18550],{"className":18551,"style":303},[302],[15,18553,18555],{"className":18554},[307,308,309,310],[15,18556,18558],{"className":18557},[59,310],[15,18559,6822],{"className":18560},[59,310],[15,18562,113],{"className":18563},[146],[15,18565,117],{"className":18566},[59],[15,18568,120],{"className":18569},[153],[15,18571,113],{"className":18572},[146],[15,18574,769],{"className":18575},[59,60],[15,18577],{"className":18578,"style":2589},[157],[15,18580,4815],{"className":18581},[2593],[15,18583],{"className":18584,"style":2589},[157],[15,18586,18588,18591,18594,18597,18600,18603],{"className":18587},[50],[15,18589],{"className":18590,"style":139},[54],[15,18592,117],{"className":18593},[59],[15,18595,120],{"className":18596},[153],[15,18598],{"className":18599,"style":2589},[157],[15,18601,7575],{"className":18602},[2593],[15,18604],{"className":18605,"style":2589},[157],[15,18607,18609,18612,18644,18647,18650,18653],{"className":18608},[50],[15,18610],{"className":18611,"style":6845},[54],[15,18613,18615,18618],{"className":18614},[59],[15,18616,37],{"className":18617,"style":61},[59,60],[15,18619,18621],{"className":18620},[281],[15,18622,18624],{"className":18623},[285],[15,18625,18627],{"className":18626},[290],[15,18628,18630],{"className":18629,"style":6864},[294],[15,18631,18632,18635],{"style":6867},[15,18633],{"className":18634,"style":303},[302],[15,18636,18638],{"className":18637},[307,308,309,310],[15,18639,18641],{"className":18640},[59,310],[15,18642,6822],{"className":18643},[59,310],[15,18645,113],{"className":18646},[146],[15,18648,117],{"className":18649},[59],[15,18651,120],{"className":18652},[153],[15,18654,7575],{"className":18655},[59],"，此处只需次梯度 ",[15,18658,18660,18684],{"className":18659,"translate":19},[18],[15,18661,18663],{"className":18662},[23],[25,18664,18665],{"xmlns":27},[29,18666,18667,18681],{},[32,18668,18669,18675,18677,18679],{},[6816,18670,18671,18673],{},[35,18672,37],{},[110,18674,6822],{"mathvariant":209,"lspace":1249,"rspace":1249},[110,18676,113],{"stretchy":112},[115,18678,117],{},[110,18680,120],{"stretchy":112},[39,18682,18683],{"encoding":41},"f'(1)",[15,18685,18687],{"className":18686,"ariaHidden":46},[45],[15,18688,18690,18693,18725,18728,18731],{"className":18689},[50],[15,18691],{"className":18692,"style":6845},[54],[15,18694,18696,18699],{"className":18695},[59],[15,18697,37],{"className":18698,"style":61},[59,60],[15,18700,18702],{"className":18701},[281],[15,18703,18705],{"className":18704},[285],[15,18706,18708],{"className":18707},[290],[15,18709,18711],{"className":18710,"style":6864},[294],[15,18712,18713,18716],{"style":6867},[15,18714],{"className":18715,"style":303},[302],[15,18717,18719],{"className":18718},[307,308,309,310],[15,18720,18722],{"className":18721},[59,310],[15,18723,6822],{"className":18724},[59,310],[15,18726,113],{"className":18727},[146],[15,18729,117],{"className":18730},[59],[15,18732,120],{"className":18733},[153],"，即 ",[15,18736,18738,18786],{"className":18737,"translate":19},[18],[15,18739,18741],{"className":18740},[23],[25,18742,18743],{"xmlns":27},[29,18744,18745,18783],{},[32,18746,18747,18749,18751,18753,18755,18757,18759,18761,18763,18765,18767,18773,18775,18777,18779,18781],{},[35,18748,37],{},[110,18750,113],{"stretchy":112},[35,18752,769],{},[110,18754,120],{"stretchy":112},[110,18756,6930],{},[35,18758,37],{},[110,18760,113],{"stretchy":112},[115,18762,117],{},[110,18764,120],{"stretchy":112},[110,18766,2472],{},[195,18768,18769,18771],{},[35,18770,4160],{},[115,18772,117],{},[110,18774,113],{"stretchy":112},[35,18776,769],{},[110,18778,4815],{},[115,18780,117],{},[110,18782,120],{"stretchy":112},[39,18784,18785],{"encoding":41},"f(t)\\ge f(1)+g_1(t-1)",[15,18787,18789,18816,18843,18905],{"className":18788,"ariaHidden":46},[45],[15,18790,18792,18795,18798,18801,18804,18807,18810,18813],{"className":18791},[50],[15,18793],{"className":18794,"style":139},[54],[15,18796,37],{"className":18797,"style":61},[59,60],[15,18799,113],{"className":18800},[146],[15,18802,769],{"className":18803},[59,60],[15,18805,120],{"className":18806},[153],[15,18808],{"className":18809,"style":158},[157],[15,18811,6930],{"className":18812},[162],[15,18814],{"className":18815,"style":158},[157],[15,18817,18819,18822,18825,18828,18831,18834,18837,18840],{"className":18818},[50],[15,18820],{"className":18821,"style":139},[54],[15,18823,37],{"className":18824,"style":61},[59,60],[15,18826,113],{"className":18827},[146],[15,18829,117],{"className":18830},[59],[15,18832,120],{"className":18833},[153],[15,18835],{"className":18836,"style":2589},[157],[15,18838,2472],{"className":18839},[2593],[15,18841],{"className":18842,"style":2589},[157],[15,18844,18846,18849,18890,18893,18896,18899,18902],{"className":18845},[50],[15,18847],{"className":18848,"style":139},[54],[15,18850,18852,18855],{"className":18851},[59],[15,18853,4160],{"className":18854,"style":981},[59,60],[15,18856,18858],{"className":18857},[281],[15,18859,18861,18882],{"className":18860},[285,286],[15,18862,18864,18879],{"className":18863},[290],[15,18865,18867],{"className":18866,"style":11137},[294],[15,18868,18870,18873],{"style":18869},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[15,18871],{"className":18872,"style":303},[302],[15,18874,18876],{"className":18875},[307,308,309,310],[15,18877,117],{"className":18878},[59,310],[15,18880,318],{"className":18881},[317],[15,18883,18885],{"className":18884},[290],[15,18886,18888],{"className":18887,"style":7970},[294],[15,18889],{},[15,18891,113],{"className":18892},[146],[15,18894,769],{"className":18895},[59,60],[15,18897],{"className":18898,"style":2589},[157],[15,18900,4815],{"className":18901},[2593],[15,18903],{"className":18904,"style":2589},[157],[15,18906,18908,18911,18914],{"className":18907},[50],[15,18909],{"className":18910,"style":139},[54],[15,18912,117],{"className":18913},[59],[15,18915,120],{"className":18916},[153]," 对任一次梯度 ",[15,18919,18921,18939],{"className":18920,"translate":19},[18],[15,18922,18924],{"className":18923},[23],[25,18925,18926],{"xmlns":27},[29,18927,18928,18936],{},[32,18929,18930],{},[195,18931,18932,18934],{},[35,18933,4160],{},[115,18935,117],{},[39,18937,18938],{"encoding":41},"g_1",[15,18940,18942],{"className":18941,"ariaHidden":46},[45],[15,18943,18945,18948],{"className":18944},[50],[15,18946],{"className":18947,"style":2278},[54],[15,18949,18951,18954],{"className":18950},[59],[15,18952,4160],{"className":18953,"style":981},[59,60],[15,18955,18957],{"className":18956},[281],[15,18958,18960,18980],{"className":18959},[285,286],[15,18961,18963,18977],{"className":18962},[290],[15,18964,18966],{"className":18965,"style":11137},[294],[15,18967,18968,18971],{"style":18869},[15,18969],{"className":18970,"style":303},[302],[15,18972,18974],{"className":18973},[307,308,309,310],[15,18975,117],{"className":18976},[59,310],[15,18978,318],{"className":18979},[317],[15,18981,18983],{"className":18982},[290],[15,18984,18986],{"className":18985,"style":7970},[294],[15,18987],{},"）处理非主项：对凸的 ",[15,18990,18992,19005],{"className":18991,"translate":19},[18],[15,18993,18995],{"className":18994},[23],[25,18996,18997],{"xmlns":27},[29,18998,18999,19003],{},[32,19000,19001],{},[35,19002,37],{},[39,19004,37],{"encoding":41},[15,19006,19008],{"className":19007,"ariaHidden":46},[45],[15,19009,19011,19014],{"className":19010},[50],[15,19012],{"className":19013,"style":55},[54],[15,19015,37],{"className":19016,"style":61},[59,60],[15,19018,19020,19046],{"className":19019,"translate":19},[18],[15,19021,19023],{"className":19022},[23],[25,19024,19025],{"xmlns":27},[29,19026,19027,19043],{},[32,19028,19029,19031,19033,19035,19037,19039,19041],{},[35,19030,37],{},[110,19032,113],{"stretchy":112},[35,19034,4160],{},[110,19036,113],{"stretchy":112},[35,19038,16454],{},[110,19040,120],{"stretchy":112},[110,19042,120],{"stretchy":112},[39,19044,19045],{"encoding":41},"f(g(r))",[15,19047,19049],{"className":19048,"ariaHidden":46},[45],[15,19050,19052,19055,19058,19061,19064,19067,19070],{"className":19051},[50],[15,19053],{"className":19054,"style":139},[54],[15,19056,37],{"className":19057,"style":61},[59,60],[15,19059,113],{"className":19060},[146],[15,19062,4160],{"className":19063,"style":981},[59,60],[15,19065,113],{"className":19066},[146],[15,19068,16454],{"className":19069,"style":277},[59,60],[15,19071,19073],{"className":19072},[153],"))"," 中关于 ",[15,19076,19078,19091],{"className":19077,"translate":19},[18],[15,19079,19081],{"className":19080},[23],[25,19082,19083],{"xmlns":27},[29,19084,19085,19089],{},[32,19086,19087],{},[35,19088,16454],{},[39,19090,16454],{"encoding":41},[15,19092,19094],{"className":19093,"ariaHidden":46},[45],[15,19095,19097,19100],{"className":19096},[50],[15,19098],{"className":19099,"style":2347},[54],[15,19101,16454],{"className":19102,"style":277},[59,60]," 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f(1)+g_1\\cdot\\delta\\,\\tfrac{1-r}{1-\\delta}\n+O(\\delta^2)\\\\\n&= 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项在 ",[15,20071,20073,20086],{"className":20072,"translate":19},[18],[15,20074,20076],{"className":20075},[23],[25,20077,20078],{"xmlns":27},[29,20079,20080,20084],{},[32,20081,20082],{},[35,20083,16454],{},[39,20085,16454],{"encoding":41},[15,20087,20089],{"className":20088,"ariaHidden":46},[45],[15,20090,20092,20095],{"className":20091},[50],[15,20093],{"className":20094,"style":2347},[54],[15,20096,16454],{"className":20097,"style":277},[59,60]," 变化时一致有界，因为 ",[15,20100,20102,20136],{"className":20101,"translate":19},[18],[15,20103,20105],{"className":20104},[23],[25,20106,20107],{"xmlns":27},[29,20108,20109,20133],{},[32,20110,20111,20113,20115,20117,20119,20121,20123,20125,20127,20129,20131],{},[35,20112,4160],{},[110,20114,113],{"stretchy":112},[35,20116,16454],{},[110,20118,120],{"stretchy":112},[110,20120,123],{},[115,20122,117],{},[110,20124,2472],{},[35,20126,19247],{},[110,20128,113],{"stretchy":112},[35,20130,15716],{},[110,20132,120],{"stretchy":112},[39,20134,20135],{"encoding":41},"g(r)=1+O(\\delta)",[15,20137,20139,20166,20184],{"className":20138,"ariaHidden":46},[45],[15,20140,20142,20145,20148,20151,20154,20157,20160,20163],{"className":20141},[50],[15,20143],{"className":20144,"style":139},[54],[15,20146,4160],{"className":20147,"style":981},[59,60],[15,20149,113],{"className":20150},[146],[15,20152,16454],{"className":20153,"style":277},[59,60],[15,20155,120],{"className":20156},[153],[15,20158],{"className":20159,"style":158},[157],[15,20161,123],{"className":20162},[162],[15,20164],{"className":20165,"style":158},[157],[15,20167,20169,20172,20175,20178,20181],{"className":20168},[50],[15,20170],{"className":20171,"style":12582},[54],[15,20173,117],{"className":20174},[59],[15,20176],{"className":20177,"style":2589},[157],[15,20179,2472],{"className":20180},[2593],[15,20182],{"className":20183,"style":2589},[157],[15,20185,20187,20190,20193,20196,20199],{"className":20186},[50],[15,20188],{"className":20189,"style":139},[54],[15,20191,19247],{"className":20192,"style":277},[59,60],[15,20194,113],{"className":20195},[146],[15,20197,15716],{"className":20198,"style":15765},[59,60],[15,20200,120],{"className":20201},[153],[15,20203,20205,20218],{"className":20204,"translate":19},[18],[15,20206,20208],{"className":20207},[23],[25,20209,20210],{"xmlns":27},[29,20211,20212,20216],{},[32,20213,20214],{},[35,20215,37],{},[39,20217,37],{"encoding":41},[15,20219,20221],{"className":20220,"ariaHidden":46},[45],[15,20222,20224,20227],{"className":20223},[50],[15,20225],{"className":20226,"style":55},[54],[15,20228,37],{"className":20229,"style":61},[59,60]," 连续）。代回不等式：两边的主项（",[15,20232,20234,20256],{"className":20233,"translate":19},[18],[15,20235,20237],{"className":20236},[23],[25,20238,20239],{"xmlns":27},[29,20240,20241,20253],{},[32,20242,20243,20245,20247,20249,20251],{},[35,20244,15716],{},[35,20246,37],{},[110,20248,113],{"stretchy":112},[110,20250,3301],{},[110,20252,120],{"stretchy":112},[39,20254,20255],{"encoding":41},"\\delta f(\\cdot)",[15,20257,20259],{"className":20258,"ariaHidden":46},[45],[15,20260,20262,20265,20268,20271,20274,20277],{"className":20261},[50],[15,20263],{"className":20264,"style":139},[54],[15,20266,15716],{"className":20267,"style":15765},[59,60],[15,20269,37],{"className":20270,"style":61},[59,60],[15,20272,113],{"className":20273},[146],[15,20275,3301],{"className":20276},[59],[15,20278,120],{"className":20279},[153]," 部分）正是 ",[15,20282,20284,20332],{"className":20283,"translate":19},[18],[15,20285,20287],{"className":20286},[23],[25,20288,20289],{"xmlns":27},[29,20290,20291,20329],{},[32,20292,20293,20295,20297,20299,20301,20303,20305,20307,20309,20311,20313,20315,20317,20319,20321,20323,20325,20327],{},[35,20294,15716],{},[110,20296,230],{"stretchy":112},[35,20298,11343],{},[35,20300,37],{},[110,20302,113],{"stretchy":112},[35,20304,7563],{},[110,20306,120],{"stretchy":112},[110,20308,2472],{},[110,20310,113],{"stretchy":112},[115,20312,117],{},[110,20314,4815],{},[35,20316,11343],{},[110,20318,120],{"stretchy":112},[35,20320,37],{},[110,20322,113],{"stretchy":112},[35,20324,15404],{},[110,20326,120],{"stretchy":112},[110,20328,257],{"stretchy":112},[39,20330,20331],{"encoding":41},"\\delta[\\lambda f(s)+(1-\\lambda)f(u)]",[15,20333,20335,20371,20392],{"className":20334,"ariaHidden":46},[45],[15,20336,20338,20341,20344,20347,20350,20353,20356,20359,20362,20365,20368],{"className":20337},[50],[15,20339],{"className":20340,"style":139},[54],[15,20342,15716],{"className":20343,"style":15765},[59,60],[15,20345,230],{"className":20346},[146],[15,20348,11343],{"className":20349},[59,60],[15,20351,37],{"className":20352,"style":61},[59,60],[15,20354,113],{"className":20355},[146],[15,20357,7563],{"className":20358},[59,60],[15,20360,120],{"className":20361},[153],[15,20363],{"className":20364,"style":2589},[157],[15,20366,2472],{"className":20367},[2593],[15,20369],{"className":20370,"style":2589},[157],[15,20372,20374,20377,20380,20383,20386,20389],{"className":20373},[50],[15,20375],{"className":20376,"style":139},[54],[15,20378,113],{"className":20379},[146],[15,20381,117],{"className":20382},[59],[15,20384],{"className":20385,"style":2589},[157],[15,20387,4815],{"className":20388},[2593],[15,20390],{"className":20391,"style":2589},[157],[15,20393,20395,20398,20401,20404,20407,20410,20413],{"className":20394},[50],[15,20396],{"className":20397,"style":139},[54],[15,20399,11343],{"className":20400},[59,60],[15,20402,120],{"className":20403},[153],[15,20405,37],{"className":20406,"style":61},[59,60],[15,20408,113],{"className":20409},[146],[15,20411,15404],{"className":20412},[59,60],[15,20414,20416],{"className":20415},[153],")]",[15,20418,20420,20442],{"className":20419,"translate":19},[18],[15,20421,20423],{"className":20422},[23],[25,20424,20425],{"xmlns":27},[29,20426,20427,20439],{},[32,20428,20429,20431,20433,20435,20437],{},[35,20430,15716],{},[35,20432,37],{},[110,20434,113],{"stretchy":112},[35,20436,7572],{},[110,20438,120],{"stretchy":112},[39,20440,20441],{"encoding":41},"\\delta f(w)",[15,20443,20445],{"className":20444,"ariaHidden":46},[45],[15,20446,20448,20451,20454,20457,20460,20463],{"className":20447},[50],[15,20449],{"className":20450,"style":139},[54],[15,20452,15716],{"className":20453,"style":15765},[59,60],[15,20455,37],{"className":20456,"style":61},[59,60],[15,20458,113],{"className":20459},[146],[15,20461,7572],{"className":20462,"style":8276},[59,60],[15,20464,120],{"className":20465},[153],"；非主项部分",[15,20468,20470],{"className":20469,"translate":19},[180],[15,20471,20473,20559],{"className":20472,"translate":19},[18],[15,20474,20476],{"className":20475},[23],[25,20477,20478],{"xmlns":27,"display":189},[29,20479,20480,20556],{},[32,20481,20482,20484,20486,20488,20490,20492,20494,20500,20502,20504,20506,20508,20510,20512,20514,20516,20518,20520,20522,20524,20526,20528,20530,20532,20534,20536,20538,20540,20542,20544,20546,20548,20550,20552,20554],{},[110,20483,113],{"stretchy":112},[115,20485,117],{},[110,20487,4815],{},[35,20489,15716],{},[110,20491,120],{"stretchy":112},[1299,20493,1301],{},[195,20495,20496,20498],{},[35,20497,4160],{},[115,20499,117],{},[35,20501,15716],{},[110,20503,230],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[35,20505,11343],{},[110,20507,113],{"stretchy":112},[115,20509,117],{},[110,20511,4815],{},[35,20513,7563],{},[110,20515,120],{"stretchy":112},[110,20517,2472],{},[110,20519,113],{"stretchy":112},[115,20521,117],{},[110,20523,4815],{},[35,20525,11343],{},[110,20527,120],{"stretchy":112},[110,20529,113],{"stretchy":112},[115,20531,117],{},[110,20533,4815],{},[35,20535,15404],{},[110,20537,120],{"stretchy":112},[110,20539,4815],{},[110,20541,113],{"stretchy":112},[115,20543,117],{},[110,20545,4815],{},[35,20547,7572],{},[110,20549,120],{"stretchy":112},[110,20551,257],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[110,20553,123],{},[115,20555,126],{},[39,20557,20558],{"encoding":41},"(1-\\delta)\\,g_1\\delta\\big[\\lambda(1-s)\n+(1-\\lambda)(1-u)-(1-w)\\big]=0",[15,20560,20562,20583,20666,20687,20708,20735,20756,20777,20804],{"className":20561,"ariaHidden":46},[45],[15,20563,20565,20568,20571,20574,20577,20580],{"className":20564},[50],[15,20566],{"className":20567,"style":139},[54],[15,20569,113],{"className":20570},[146],[15,20572,117],{"className":20573},[59],[15,20575],{"className":20576,"style":2589},[157],[15,20578,4815],{"className":20579},[2593],[15,20581],{"className":20582,"style":2589},[157],[15,20584,20586,20590,20593,20596,20599,20639,20642,20648,20651,20654,20657,20660,20663],{"className":20585},[50],[15,20587],{"className":20588,"style":20589},[54],"height:1.2em;vertical-align:-0.35em;",[15,20591,15716],{"className":20592,"style":15765},[59,60],[15,20594,120],{"className":20595},[153],[15,20597],{"className":20598,"style":406},[157],[15,20600,20602,20605],{"className":20601},[59],[15,20603,4160],{"className":20604,"style":981},[59,60],[15,20606,20608],{"className":20607},[281],[15,20609,20611,20631],{"className":20610},[285,286],[15,20612,20614,20628],{"className":20613},[290],[15,20615,20617],{"className":20616,"style":11137},[294],[15,20618,20619,20622],{"style":18869},[15,20620],{"className":20621,"style":303},[302],[15,20623,20625],{"className":20624},[307,308,309,310],[15,20626,117],{"className":20627},[59,310],[15,20629,318],{"className":20630},[317],[15,20632,20634],{"className":20633},[290],[15,20635,20637],{"className":20636,"style":7970},[294],[15,20638],{},[15,20640,15716],{"className":20641,"style":15765},[59,60],[15,20643,20645],{"className":20644},[59],[15,20646,230],{"className":20647},[419,9676],[15,20649,11343],{"className":20650},[59,60],[15,20652,113],{"className":20653},[146],[15,20655,117],{"className":20656},[59],[15,20658],{"className":20659,"style":2589},[157],[15,20661,4815],{"className":20662},[2593],[15,20664],{"className":20665,"style":2589},[157],[15,20667,20669,20672,20675,20678,20681,20684],{"className":20668},[50],[15,20670],{"className":20671,"style":139},[54],[15,20673,7563],{"className":20674},[59,60],[15,20676,120],{"className":20677},[153],[15,20679],{"className":20680,"style":2589},[157],[15,20682,2472],{"className":20683},[2593],[15,20685],{"className":20686,"style":2589},[157],[15,20688,20690,20693,20696,20699,20702,20705],{"className":20689},[50],[15,20691],{"className":20692,"style":139},[54],[15,20694,113],{"className":20695},[146],[15,20697,117],{"className":20698},[59],[15,20700],{"className":20701,"style":2589},[157],[15,20703,4815],{"className":20704},[2593],[15,20706],{"className":20707,"style":2589},[157],[15,20709,20711,20714,20717,20720,20723,20726,20729,20732],{"className":20710},[50],[15,20712],{"className":20713,"style":139},[54],[15,20715,11343],{"className":20716},[59,60],[15,20718,120],{"className":20719},[153],[15,20721,113],{"className":20722},[146],[15,20724,117],{"className":20725},[59],[15,20727],{"className":20728,"style":2589},[157],[15,20730,4815],{"className":20731},[2593],[15,20733],{"className":20734,"style":2589},[157],[15,20736,20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0，因为 ",[15,20817,20819,20852],{"className":20818,"translate":19},[18],[15,20820,20822],{"className":20821},[23],[25,20823,20824],{"xmlns":27},[29,20825,20826,20850],{},[32,20827,20828,20830,20832,20834,20836,20838,20840,20842,20844,20846,20848],{},[35,20829,7572],{},[110,20831,123],{},[35,20833,11343],{},[35,20835,7563],{},[110,20837,2472],{},[110,20839,113],{"stretchy":112},[115,20841,117],{},[110,20843,4815],{},[35,20845,11343],{},[110,20847,120],{"stretchy":112},[35,20849,15404],{},[39,20851,15562],{"encoding":41},[15,20853,20855,20873,20894,20915],{"className":20854,"ariaHidden":46},[45],[15,20856,20858,20861,20864,20867,20870],{"className":20857},[50],[15,20859],{"className":20860,"style":2347},[54],[15,20862,7572],{"className":20863,"style":8276},[59,60],[15,20865],{"className":20866,"style":158},[157],[15,20868,123],{"className":20869},[162],[15,20871],{"className":20872,"style":158},[157],[15,20874,20876,20879,20882,20885,20888,20891],{"className":20875},[50],[15,20877],{"className":20878,"style":15590},[54],[15,20880,11343],{"className":20881},[59,60],[15,20883,7563],{"className":20884},[59,60],[15,20886],{"className":20887,"style":2589},[157],[15,20889,2472],{"className":20890},[2593],[15,20892],{"className":20893,"style":2589},[157],[15,20895,20897,20900,20903,20906,20909,20912],{"className":20896},[50],[15,20898],{"className":20899,"style":139},[54],[15,20901,113],{"className":20902},[146],[15,20904,117],{"className":20905},[59],[15,20907],{"className":20908,"style":2589},[157],[15,20910,4815],{"className":20911},[2593],[15,20913],{"className":20914,"style":2589},[157],[15,20916,20918,20921,20924,20927],{"className":20917},[50],[15,20919],{"className":20920,"style":139},[54],[15,20922,11343],{"className":20923},[59,60],[15,20925,120],{"className":20926},[153],[15,20928,15404],{"className":20929},[59,60],"）。于是不等式两边同除以 ",[15,20932,20934,20952],{"className":20933,"translate":19},[18],[15,20935,20937],{"className":20936},[23],[25,20938,20939],{"xmlns":27},[29,20940,20941,20949],{},[32,20942,20943,20945,20947],{},[35,20944,15716],{},[110,20946,2905],{},[115,20948,126],{},[39,20950,20951],{"encoding":41},"\\delta>0",[15,20953,20955,20973],{"className":20954,"ariaHidden":46},[45],[15,20956,20958,20961,20964,20967,20970],{"className":20957},[50],[15,20959],{"className":20960,"style":11368},[54],[15,20962,15716],{"className":20963,"style":15765},[59,60],[15,20965],{"className":20966,"style":158},[157],[15,20968,2905],{"className":20969},[162],[15,20971],{"className":20972,"style":158},[157],[15,20974,20976,20979],{"className":20975},[50],[15,20977],{"className":20978,"style":172},[54],[15,20980,126],{"className":20981},[59],"，令 ",[15,20984,20986,21008],{"className":20985,"translate":19},[18],[15,20987,20989],{"className":20988},[23],[25,20990,20991],{"xmlns":27},[29,20992,20993,21005],{},[32,20994,20995,20997,20999],{},[35,20996,15716],{},[110,20998,1032],{},[6816,21000,21001,21003],{},[115,21002,126],{},[110,21004,2472],{},[39,21006,21007],{"encoding":41},"\\delta\\to0^+",[15,21009,21011,21029],{"className":21010,"ariaHidden":46},[45],[15,21012,21014,21017,21020,21023,21026],{"className":21013},[50],[15,21015],{"className":21016,"style":12601},[54],[15,21018,15716],{"className":21019,"style":15765},[59,60],[15,21021],{"className":21022,"style":158},[157],[15,21024,1032],{"className":21025},[162],[15,21027],{"className":21028,"style":158},[157],[15,21030,21032,21036],{"className":21031},[50],[15,21033],{"className":21034,"style":21035},[54],"height:0.7713em;",[15,21037,21039,21042],{"className":21038},[59],[15,21040,126],{"className":21041},[59],[15,21043,21045],{"className":21044},[281],[15,21046,21048],{"className":21047},[285],[15,21049,21051],{"className":21050},[290],[15,21052,21054],{"className":21053,"style":21035},[294],[15,21055,21056,21059],{"style":6867},[15,21057],{"className":21058,"style":303},[302],[15,21060,21062],{"className":21061},[307,308,309,310],[15,21063,2472],{"className":21064},[2593,310],"，得",[15,21067,21069],{"className":21068,"translate":19},[180],[15,21070,21072,21142],{"className":21071,"translate":19},[18],[15,21073,21075],{"className":21074},[23],[25,21076,21077],{"xmlns":27,"display":189},[29,21078,21079,21139],{},[32,21080,21081,21083,21085,21087,21089,21091,21093,21095,21097,21099,21101,21103,21105,21107,21109,21111,21113,21115,21117,21119,21121,21123,21125,21127,21129,21131,21133,21135,21137],{},[35,21082,37],{},[110,21084,113],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[35,21086,11343],{},[35,21088,7563],{},[110,21090,2472],{},[110,21092,113],{"stretchy":112},[115,21094,117],{},[110,21096,4815],{},[35,21098,11343],{},[110,21100,120],{"stretchy":112},[35,21102,15404],{},[110,21104,120],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[110,21106,9954],{},[35,21108,11343],{},[35,21110,37],{},[110,21112,113],{"stretchy":112},[35,21114,7563],{},[110,21116,120],{"stretchy":112},[110,21118,2472],{},[110,21120,113],{"stretchy":112},[115,21122,117],{},[110,21124,4815],{},[35,21126,11343],{},[110,21128,120],{"stretchy":112},[35,21130,37],{},[110,21132,113],{"stretchy":112},[35,21134,15404],{},[110,21136,120],{"stretchy":112},[110,21138,551],{"separator":46},[39,21140,21141],{"encoding":41},"f\\big(\\lambda s+(1-\\lambda)u\\big)\n\\le\\lambda f(s)+(1-\\lambda)f(u),",[15,21143,21145,21175,21196,21226,21256,21277],{"className":21144,"ariaHidden":46},[45],[15,21146,21148,21151,21154,21160,21163,21166,21169,21172],{"className":21147},[50],[15,21149],{"className":21150,"style":20589},[54],[15,21152,37],{"className":21153,"style":61},[59,60],[15,21155,21157],{"className":21156},[59],[15,21158,113],{"className":21159},[419,9676],[15,21161,11343],{"className":21162},[59,60],[15,21164,7563],{"className":21165},[59,60],[15,21167],{"className":21168,"style":2589},[157],[15,21170,2472],{"className":21171},[2593],[15,21173],{"className":21174,"style":2589},[157],[15,21176,21178,21181,21184,21187,21190,21193],{"className":21177},[50],[15,21179],{"className":21180,"style":139},[54],[15,21182,113],{"className":21183},[146],[15,21185,117],{"className":21186},[59],[15,21188],{"className":21189,"style":2589},[157],[15,21191,4815],{"className":21192},[2593],[15,21194],{"className":21195,"style":2589},[157],[15,21197,21199,21202,21205,21208,21211,21217,21220,21223],{"className":21198},[50],[15,21200],{"className":21201,"style":20589},[54],[15,21203,11343],{"className":21204},[59,60],[15,21206,120],{"className":21207},[153],[15,21209,15404],{"className":21210},[59,60],[15,21212,21214],{"className":21213},[59],[15,21215,120],{"className":21216},[419,9676],[15,21218],{"className":21219,"style":158},[157],[15,21221,9954],{"className":21222},[162],[15,21224],{"className":21225,"style":158},[157],[15,21227,21229,21232,21235,21238,21241,21244,21247,21250,21253],{"className":21228},[50],[15,21230],{"className":21231,"style":139},[54],[15,21233,11343],{"className":21234},[59,60],[15,21236,37],{"className":21237,"style":61},[59,60],[15,21239,113],{"className":21240},[146],[15,21242,7563],{"className":21243},[59,60],[15,21245,120],{"className":21246},[153],[15,21248],{"className":21249,"style":2589},[157],[15,21251,2472],{"className":21252},[2593],[15,21254],{"className":21255,"style":2589},[157],[15,21257,21259,21262,21265,21268,21271,21274],{"className":21258},[50],[15,21260],{"className":21261,"style":139},[54],[15,21263,113],{"className":21264},[146],[15,21266,117],{"className":21267},[59],[15,21269],{"className":21270,"style":2589},[157],[15,21272,4815],{"className":21273},[2593],[15,21275],{"className":21276,"style":2589},[157],[15,21278,21280,21283,21286,21289,21292,21295,21298,21301],{"className":21279},[50],[15,21281],{"className":21282,"style":139},[54],[15,21284,11343],{"className":21285},[59,60],[15,21287,120],{"className":21288},[153],[15,21290,37],{"className":21291,"style":61},[59,60],[15,21293,113],{"className":21294},[146],[15,21296,15404],{"className":21297},[59,60],[15,21299,120],{"className":21300},[153],[15,21302,551],{"className":21303},[573],[11,21305,21306,21307,10969],{},"即 ",[15,21308,21310,21323],{"className":21309,"translate":19},[18],[15,21311,21313],{"className":21312},[23],[25,21314,21315],{"xmlns":27},[29,21316,21317,21321],{},[32,21318,21319],{},[35,21320,37],{},[39,21322,37],{"encoding":41},[15,21324,21326],{"className":21325,"ariaHidden":46},[45],[15,21327,21329,21332],{"className":21328},[50],[15,21330],{"className":21331,"style":55},[54],[15,21333,37],{"className":21334,"style":61},[59,60],[11,21336,21337,21338,21341,21342,21370,21371,21401,21402,21430],{},"所以\"凸\"与\"数据处理\"是同一件事的两面。这也是 f-散度族成为\"一切满足数据处理不等式的最一般散度\"的原因：它不是一堆例子的偶然汇总，而是一个有刻画定理支撑的完整族。换句话说，Csiszár–Ali–Silvey 定理给出一个完整的",[998,21339,21340],{},"刻画","：凸函数 ",[15,21343,21345,21358],{"className":21344,"translate":19},[18],[15,21346,21348],{"className":21347},[23],[25,21349,21350],{"xmlns":27},[29,21351,21352,21356],{},[32,21353,21354],{},[35,21355,37],{},[39,21357,37],{"encoding":41},[15,21359,21361],{"className":21360,"ariaHidden":46},[45],[15,21362,21364,21367],{"className":21363},[50],[15,21365],{"className":21366,"style":55},[54],[15,21368,37],{"className":21369,"style":61},[59,60],"（模去仿射项）",[15,21372,21374,21389],{"className":21373,"translate":19},[18],[15,21375,21377],{"className":21376},[23],[25,21378,21379],{"xmlns":27},[29,21380,21381,21386],{},[32,21382,21383],{},[110,21384,21385],{},"↔",[39,21387,21388],{"encoding":41},"\\leftrightarrow",[15,21390,21392],{"className":21391,"ariaHidden":46},[45],[15,21393,21395,21398],{"className":21394},[50],[15,21396],{"className":21397,"style":11007},[54],[15,21399,21385],{"className":21400},[162]," 联合凸泛函 ",[15,21403,21405,21418],{"className":21404,"translate":19},[18],[15,21406,21408],{"className":21407},[23],[25,21409,21410],{"xmlns":27},[29,21411,21412,21416],{},[32,21413,21414],{},[110,21415,21385],{},[39,21417,21388],{"encoding":41},[15,21419,21421],{"className":21420,"ariaHidden":46},[45],[15,21422,21424,21427],{"className":21423},[50],[15,21425],{"className":21426,"style":11007},[54],[15,21428,21385],{"className":21429},[162]," 满足数据处理不等式的分布差异度量，三者一一对应。",[528,21432,21434],{"id":21433},"_3-常见散度","3. 常见散度",[11,21436,21437,21438,21466],{},"不同的 ",[15,21439,21441,21454],{"className":21440,"translate":19},[18],[15,21442,21444],{"className":21443},[23],[25,21445,21446],{"xmlns":27},[29,21447,21448,21452],{},[32,21449,21450],{},[35,21451,37],{},[39,21453,37],{"encoding":41},[15,21455,21457],{"className":21456,"ariaHidden":46},[45],[15,21458,21460,21463],{"className":21459},[50],[15,21461],{"className":21462,"style":55},[54],[15,21464,37],{"className":21465,"style":61},[59,60]," 对应不同的散度。KL、JS、全变差、卡方、Hellinger、Wasserstein、MMD 等散度，表面上都是「衡量两个分布差多少」，但它们的根本区别可以归结为几个维度：",[11,21468,21469],{},"它们对「差异」的定义方式不同，对「支撑集不重叠」的处理不同，对「几何\u002F度量结构」的利用不同，因而导出的拓扑、梯度行为和优化性质也不同。",[11,21471,21472],{},"可以把不同散度的区别归纳为五个核心问题：",[6664,21474,21475,21478,21481,21484,21487],{},[6667,21476,21477],{},"差异定义在什么量上？是密度比值、密度差、累积分布差，还是搬运质量？",[6667,21479,21480],{},"是否对称？",[6667,21482,21483],{},"是否有界？",[6667,21485,21486],{},"是否利用样本空间的几何结构？",[6667,21488,21489],{},"诱导的拓扑强弱如何？",[11,21491,21492],{},"这五点决定了散度在理论性质和实际优化中的行为差异。下面逐个把常用散度「代入模板」并验算。",[2313,21494,21496],{"id":21495},"_31-kl-散度","3.1 KL 散度",[11,21498,21499],{},"取：",[15,21501,21503],{"className":21502,"translate":19},[180],[15,21504,21506,21535],{"className":21505,"translate":19},[18],[15,21507,21509],{"className":21508},[23],[25,21510,21511],{"xmlns":27,"display":189},[29,21512,21513,21533],{},[32,21514,21515,21517,21519,21521,21523,21525,21527,21529,21531],{},[35,21516,37],{},[110,21518,113],{"stretchy":112},[35,21520,769],{},[110,21522,120],{"stretchy":112},[110,21524,123],{},[35,21526,769],{},[35,21528,4102],{},[110,21530,4105],{},[35,21532,769],{},[39,21534,4110],{"encoding":41},[15,21536,21538,21565],{"className":21537,"ariaHidden":46},[45],[15,21539,21541,21544,21547,21550,21553,21556,21559,21562],{"className":21540},[50],[15,21542],{"className":21543,"style":139},[54],[15,21545,37],{"className":21546,"style":61},[59,60],[15,21548,113],{"className":21549},[146],[15,21551,769],{"className":21552},[59,60],[15,21554,120],{"className":21555},[153],[15,21557],{"className":21558,"style":158},[157],[15,21560,123],{"className":21561},[162],[15,21563],{"className":21564,"style":158},[157],[15,21566,21568,21571,21574,21577,21582,21585],{"className":21567},[50],[15,21569],{"className":21570,"style":55},[54],[15,21572,769],{"className":21573},[59,60],[15,21575],{"className":21576,"style":406},[157],[15,21578,4156,21580],{"className":21579},[1486],[15,21581,4160],{"style":4159},[15,21583],{"className":21584,"style":406},[157],[15,21586,769],{"className":21587},[59,60],[11,21589,21590,21591,21624,21625,2747,21690,21823],{},"（下面 ",[15,21592,21594,21610],{"className":21593,"translate":19},[18],[15,21595,21597],{"className":21596},[23],[25,21598,21599],{"xmlns":27},[29,21600,21601,21607],{},[32,21602,21603,21605],{},[35,21604,4102],{},[110,21606,4105],{},[39,21608,21609],{"encoding":41},"\\log",[15,21611,21613],{"className":21612,"ariaHidden":46},[45],[15,21614,21616,21619],{"className":21615},[50],[15,21617],{"className":21618,"style":55},[54],[15,21620,4156,21622],{"className":21621},[1486],[15,21623,4160],{"style":4159}," 均为自然对数。）则 ",[15,21626,21628,21651],{"className":21627,"translate":19},[18],[15,21629,21631],{"className":21630},[23],[25,21632,21633],{"xmlns":27},[29,21634,21635,21649],{},[32,21636,21637,21639,21641,21643,21645,21647],{},[35,21638,37],{},[110,21640,113],{"stretchy":112},[115,21642,117],{},[110,21644,120],{"stretchy":112},[110,21646,123],{},[115,21648,126],{},[39,21650,129],{"encoding":41},[15,21652,21654,21681],{"className":21653,"ariaHidden":46},[45],[15,21655,21657,21660,21663,21666,21669,21672,21675,21678],{"className":21656},[50],[15,21658],{"className":21659,"style":139},[54],[15,21661,37],{"className":21662,"style":61},[59,60],[15,21664,113],{"className":21665},[146],[15,21667,117],{"className":21668},[59],[15,21670,120],{"className":21671},[153],[15,21673],{"className":21674,"style":158},[157],[15,21676,123],{"className":21677},[162],[15,21679],{"className":21680,"style":158},[157],[15,21682,21684,21687],{"className":21683},[50],[15,21685],{"className":21686,"style":172},[54],[15,21688,126],{"className":21689},[59],[15,21691,21693,21733],{"className":21692,"translate":19},[18],[15,21694,21696],{"className":21695},[23],[25,21697,21698],{"xmlns":27},[29,21699,21700,21730],{},[32,21701,21702,21712,21714,21716,21718,21720,21722,21724,21726,21728],{},[6816,21703,21704,21706],{},[35,21705,37],{},[32,21707,21708,21710],{},[110,21709,6822],{"mathvariant":209},[110,21711,6822],{"mathvariant":209},[110,21713,113],{"stretchy":112},[35,21715,769],{},[110,21717,120],{"stretchy":112},[110,21719,123],{},[115,21721,117],{},[35,21723,941],{"mathvariant":209},[35,21725,769],{},[110,21727,2905],{},[115,21729,126],{},[39,21731,21732],{"encoding":41},"f''(t)=1\u002Ft>0",[15,21734,21736,21793,21814],{"className":21735,"ariaHidden":46},[45],[15,21737,21739,21742,21775,21778,21781,21784,21787,21790],{"className":21738},[50],[15,21740],{"className":21741,"style":6845},[54],[15,21743,21745,21748],{"className":21744},[59],[15,21746,37],{"className":21747,"style":61},[59,60],[15,21749,21751],{"className":21750},[281],[15,21752,21754],{"className":21753},[285],[15,21755,21757],{"className":21756},[290],[15,21758,21760],{"className":21759,"style":6864},[294],[15,21761,21762,21765],{"style":6867},[15,21763],{"className":21764,"style":303},[302],[15,21766,21768],{"className":21767},[307,308,309,310],[15,21769,21771],{"className":21770},[59,310],[15,21772,21774],{"className":21773},[59,310],"′′",[15,21776,113],{"className":21777},[146],[15,21779,769],{"className":21780},[59,60],[15,21782,120],{"className":21783},[153],[15,21785],{"className":21786,"style":158},[157],[15,21788,123],{"className":21789},[162],[15,21791],{"className":21792,"style":158},[157],[15,21794,21796,21799,21802,21805,21808,21811],{"className":21795},[50],[15,21797],{"className":21798,"style":139},[54],[15,21800,8591],{"className":21801},[59],[15,21803,769],{"className":21804},[59,60],[15,21806],{"className":21807,"style":158},[157],[15,21809,2905],{"className":21810},[162],[15,21812],{"className":21813,"style":158},[157],[15,21815,21817,21820],{"className":21816},[50],[15,21818],{"className":21819,"style":172},[54],[15,21821,126],{"className":21822},[59]," 凸。代入模板：",[15,21825,21827],{"className":21826,"translate":19},[180],[15,21828,21830,21933],{"className":21829,"translate":19},[18],[15,21831,21833],{"className":21832},[23],[25,21834,21835],{"xmlns":27,"display":189},[29,21836,21837,21930],{},[32,21838,21839,21845,21847,21849,21851,21853,21855,21857,21859,21861,21863,21869,21871,21873,21879,21881,21883,21885,21887,21889,21891,21893,21895,21901,21903,21905,21907,21909,21920,21922,21924,21926,21928],{},[195,21840,21841,21843],{},[35,21842,199],{},[35,21844,37],{},[110,21846,113],{"stretchy":112},[35,21848,206],{},[35,21850,210],{"mathvariant":209},[35,21852,213],{},[110,21854,120],{"stretchy":112},[110,21856,123],{},[110,21858,1289],{},[35,21860,944],{},[110,21862,3301],{},[238,21864,21865,21867],{},[35,21866,11],{},[35,21868,944],{},[35,21870,4102],{},[110,21872,4105],{},[238,21874,21875,21877],{},[35,21876,11],{},[35,21878,944],{},[1299,21880,1301],{},[35,21882,244],{},[35,21884,774],{},[110,21886,123],{},[110,21888,1289],{},[35,21890,11],{},[35,21892,4102],{},[110,21894,4105],{},[238,21896,21897,21899],{},[35,21898,11],{},[35,21900,944],{},[1299,21902,1301],{},[35,21904,244],{},[35,21906,774],{},[110,21908,123],{},[195,21910,21911,21913],{},[35,21912,199],{},[32,21914,21915,21917],{},[35,21916,10698],{},[35,21918,21919],{},"L",[110,21921,113],{"stretchy":112},[35,21923,206],{},[35,21925,210],{"mathvariant":209},[35,21927,213],{},[110,21929,120],{"stretchy":112},[39,21931,21932],{"encoding":41},"D_f(P\\|Q)\n=\\int q\\cdot \\frac{p}{q}\\log\\frac{p}{q}\\,dx\n=\\int p\\log\\frac{p}{q}\\,dx\n=D_{KL}(P\\|Q)",[15,21934,21936,22006,22030,22190,22297],{"className":21935,"ariaHidden":46},[45],[15,21937,21939,21942,21982,21985,21988,21991,21994,21997,22000,22003],{"className":21938},[50],[15,21940],{"className":21941,"style":270},[54],[15,21943,21945,21948],{"className":21944},[59],[15,21946,199],{"className":21947,"style":277},[59,60],[15,21949,21951],{"className":21950},[281],[15,21952,21954,21974],{"className":21953},[285,286],[15,21955,21957,21971],{"className":21956},[290],[15,21958,21960],{"className":21959,"style":295},[294],[15,21961,21962,21965],{"style":298},[15,21963],{"className":21964,"style":303},[302],[15,21966,21968],{"className":21967},[307,308,309,310],[15,21969,37],{"className":21970,"style":61},[59,60,310],[15,21972,318],{"className":21973},[317],[15,21975,21977],{"className":21976},[290],[15,21978,21980],{"className":21979,"style":325},[294],[15,21981],{},[15,21983,113],{"className":21984},[146],[15,21986,206],{"className":21987,"style":334},[59,60],[15,21989,210],{"className":21990},[59],[15,21992,213],{"className":21993},[59,60],[15,21995,120],{"className":21996},[153],[15,21998],{"className":21999,"style":158},[157],[15,22001,123],{"className":22002},[162],[15,22004],{"className":22005,"style":158},[157],[15,22007,22009,22012,22015,22018,22021,22024,22027],{"className":22008},[50],[15,22010],{"className":22011,"style":5779},[54],[15,22013,1289],{"className":22014,"style":1489},[1486,1487,1488],[15,22016],{"className":22017,"style":406},[157],[15,22019,944],{"className":22020,"style":981},[59,60],[15,22022],{"className":22023,"style":2589},[157],[15,22025,3301],{"className":22026},[2593],[15,22028],{"className":22029,"style":2589},[157],[15,22031,22033,22037,22099,22102,22107,22110,22172,22175,22178,22181,22184,22187],{"className":22032},[50],[15,22034],{"className":22035,"style":22036},[54],"height:1.988em;vertical-align:-0.8804em;",[15,22038,22040,22043,22096],{"className":22039},[59],[15,22041],{"className":22042},[146,441],[15,22044,22046],{"className":22045},[238],[15,22047,22049,22088],{"className":22048},[285,286],[15,22050,22052,22085],{"className":22051},[290],[15,22053,22055,22066,22074],{"className":22054,"style":3556},[294],[15,22056,22057,22060],{"style":457},[15,22058],{"className":22059,"style":461},[302],[15,22061,22063],{"className":22062},[59],[15,22064,944],{"className":22065,"style":981},[59,60],[15,22067,22068,22071],{"style":473},[15,22069],{"className":22070,"style":461},[302],[15,22072],{"className":22073,"style":481},[480],[15,22075,22076,22079],{"style":484},[15,22077],{"className":22078,"style":461},[302],[15,22080,22082],{"className":22081},[59],[15,22083,11],{"className":22084},[59,60],[15,22086,318],{"className":22087},[317],[15,22089,22091],{"className":22090},[290],[15,22092,22094],{"className":22093,"style":506},[294],[15,22095],{},[15,22097],{"className":22098},[153,441],[15,22100],{"className":22101,"style":406},[157],[15,22103,4156,22105],{"className":22104},[1486],[15,22106,4160],{"style":4159},[15,22108],{"className":22109,"style":406},[157],[15,22111,22113,22116,22169],{"className":22112},[59],[15,22114],{"className":22115},[146,441],[15,22117,22119],{"className":22118},[238],[15,22120,22122,22161],{"className":22121},[285,286],[15,22123,22125,22158],{"className":22124},[290],[15,22126,22128,22139,22147],{"className":22127,"style":3556},[294],[15,22129,22130,22133],{"style":457},[15,22131],{"className":22132,"style":461},[302],[15,22134,22136],{"className":22135},[59],[15,22137,944],{"className":22138,"style":981},[59,60],[15,22140,22141,22144],{"style":473},[15,22142],{"className":22143,"style":461},[302],[15,22145],{"className":22146,"style":481},[480],[15,22148,22149,22152],{"style":484},[15,22150],{"className":22151,"style":461},[302],[15,22153,22155],{"className":22154},[59],[15,22156,11],{"className":22157},[59,60],[15,22159,318],{"className":22160},[317],[15,22162,22164],{"className":22163},[290],[15,22165,22167],{"className":22166,"style":506},[294],[15,22168],{},[15,22170],{"className":22171},[153,441],[15,22173],{"className":22174,"style":406},[157],[15,22176,244],{"className":22177},[59,60],[15,22179,774],{"className":22180},[59,60],[15,22182],{"className":22183,"style":158},[157],[15,22185,123],{"className":22186},[162],[15,22188],{"className":22189,"style":158},[157],[15,22191,22193,22197,22200,22203,22206,22209,22214,22217,22279,22282,22285,22288,22291,22294],{"className":22192},[50],[15,22194],{"className":22195,"style":22196},[54],"height:2.2404em;vertical-align:-0.8804em;",[15,22198,1289],{"className":22199,"style":1489},[1486,1487,1488],[15,22201],{"className":22202,"style":406},[157],[15,22204,11],{"className":22205},[59,60],[15,22207],{"className":22208,"style":406},[157],[15,22210,4156,22212],{"className":22211},[1486],[15,22213,4160],{"style":4159},[15,22215],{"className":22216,"style":406},[157],[15,22218,22220,22223,22276],{"className":22219},[59],[15,22221],{"className":22222},[146,441],[15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KL 散度 ",[15,22368,22370,22402],{"className":22369,"translate":19},[18],[15,22371,22373],{"className":22372},[23],[25,22374,22375],{"xmlns":27},[29,22376,22377,22399],{},[32,22378,22379,22389,22391,22393,22395,22397],{},[195,22380,22381,22383],{},[35,22382,199],{},[32,22384,22385,22387],{},[35,22386,10698],{},[35,22388,21919],{},[110,22390,113],{"stretchy":112},[35,22392,213],{},[35,22394,210],{"mathvariant":209},[35,22396,206],{},[110,22398,120],{"stretchy":112},[39,22400,22401],{"encoding":41},"D_{KL}(Q\\|P)",[15,22403,22405],{"className":22404,"ariaHidden":46},[45],[15,22406,22408,22411,22457,22460,22463,22466,22469],{"className":22407},[50],[15,22409],{"className":22410,"style":139},[54],[15,22412,22414,22417],{"className":22413},[59],[15,22415,199],{"className":22416,"style":277},[59,60],[15,22418,22420],{"className":22419},[281],[15,22421,22423,22449],{"className":22422},[285,286],[15,22424,22426,22446],{"className":22425},[290],[15,22427,22429],{"className":22428,"style":379},[294],[15,22430,22431,22434],{"style":298},[15,22432],{"className":22433,"style":303},[302],[15,22435,22437],{"className":22436},[307,308,309,310],[15,22438,22440,22443],{"className":22439},[59,310],[15,22441,10698],{"className":22442,"style":10786},[59,60,310],[15,22444,21919],{"className":22445},[59,60,310],[15,22447,318],{"className":22448},[317],[15,22450,22452],{"className":22451},[290],[15,22453,22455],{"className":22454,"style":7970},[294],[15,22456],{},[15,22458,113],{"className":22459},[146],[15,22461,213],{"className":22462},[59,60],[15,22464,210],{"className":22465},[59],[15,22467,206],{"className":22468,"style":334},[59,60],[15,22470,120],{"className":22471},[153]," 对应斜共轭 ",[15,22474,22476,22502],{"className":22475,"translate":19},[18],[15,22477,22479],{"className":22478},[23],[25,22480,22481],{"xmlns":27},[29,22482,22483,22499],{},[32,22484,22485,22487,22489,22491,22493,22495,22497],{},[35,22486,769],{},[35,22488,37],{},[110,22490,113],{"stretchy":112},[115,22492,117],{},[35,22494,941],{"mathvariant":209},[35,22496,769],{},[110,22498,120],{"stretchy":112},[39,22500,22501],{"encoding":41},"t f(1\u002Ft)",[15,22503,22505],{"className":22504,"ariaHidden":46},[45],[15,22506,22508,22511,22514,22517,22520,22523,22526],{"className":22507},[50],[15,22509],{"className":22510,"style":139},[54],[15,22512,769],{"className":22513},[59,60],[15,22515,37],{"className":22516,"style":61},[59,60],[15,22518,113],{"className":22519},[146],[15,22521,8591],{"className":22522},[59],[15,22524,769],{"className":22525},[59,60],[15,22527,120],{"className":22528},[153],[15,22530,22532],{"className":22531,"translate":19},[180],[15,22533,22535,22657],{"className":22534,"translate":19},[18],[15,22536,22538],{"className":22537},[23],[25,22539,22540],{"xmlns":27,"display":189},[29,22541,22542,22654],{},[1245,22543,22544,22578],{"rowspacing":1247,"columnalign":1248,"columnspacing":1249},[1251,22545,22546,22560],{},[1254,22547,22548],{},[1257,22549,22550],{"scriptlevel":126,"displaystyle":46},[32,22551,22552,22554,22556,22558],{},[35,22553,37],{},[110,22555,113],{"stretchy":112},[35,22557,769],{},[110,22559,120],{"stretchy":112},[1254,22561,22562],{},[1257,22563,22564],{"scriptlevel":126,"displaystyle":46},[32,22565,22566,22568,22570,22572,22574,22576],{},[32,22567],{},[110,22569,123],{},[35,22571,769],{},[35,22573,4102],{},[110,22575,4105],{},[35,22577,769],{},[1251,22579,22580,22586],{},[1254,22581,22582],{},[1257,22583,22584],{"scriptlevel":126,"displaystyle":46},[32,22585],{},[1254,22587,22588],{},[1257,22589,22590],{"scriptlevel":126,"displaystyle":46},[32,22591,22592,22594,22596,22598,22614,22616,22618,22620,22622,22624,22626,22632,22634,22636,22642,22644,22646,22648,22650,22652],{},[32,22593],{},[110,22595,4288],{},[1299,22597,4285],{},[7555,22599,22600,22602,22612],{},[35,22601,37],{},[32,22603,22604,22606,22608,22610],{},[35,22605,7563],{},[35,22607,7566],{},[35,22609,7569],{},[35,22611,7572],{},[110,22613,7575],{"lspace":1249,"rspace":1249},[110,22615,113],{"stretchy":112},[35,22617,769],{},[110,22619,120],{"stretchy":112},[110,22621,123],{},[35,22623,769],{},[110,22625,3301],{},[238,22627,22628,22630],{},[115,22629,117],{},[35,22631,769],{},[35,22633,4102],{},[110,22635,4105],{},[238,22637,22638,22640],{},[115,22639,117],{},[35,22641,769],{},[110,22643,123],{},[110,22645,4815],{},[35,22647,4102],{},[110,22649,4105],{},[35,22651,769],{},[110,22653,551],{"separator":46},[39,22655,22656],{"encoding":41},"\\begin{aligned}\nf(t)&=t\\log t\\\\[2pt]\n&\\Longrightarrow\\ f^{*}_{skew}(t)\n=t\\cdot\\frac1t\\log\\frac1t=-\\log t,\n\\end{aligned}",[15,22658,22660],{"className":22659,"ariaHidden":46},[45],[15,22661,22663,22667],{"className":22662},[50],[15,22664],{"className":22665,"style":22666},[54],"height:3.7074em;vertical-align:-1.6037em;",[15,22668,22670],{"className":22669},[59],[15,22671,22673,22729],{"className":22672},[1245],[15,22674,22676],{"className":22675},[1362],[15,22677,22679,22720],{"className":22678},[285,286],[15,22680,22682,22717],{"className":22681},[290],[15,22683,22686,22708],{"className":22684,"style":22685},[294],"height:2.1037em;",[15,22687,22689,22693],{"style":22688},"top:-4.5852em;",[15,22690],{"className":22691,"style":22692},[302],"height:3.3214em;",[15,22694,22696,22699,22702,22705],{"className":22695},[59],[15,22697,37],{"className":22698,"style":61},[59,60],[15,22700,113],{"className":22701},[146],[15,22703,769],{"className":22704},[59,60],[15,22706,120],{"className":22707},[153],[15,22709,22711,22714],{"style":22710},"top:-2.4037em;",[15,22712],{"className":22713,"style":22692},[302],[15,22715],{"className":22716},[59],[15,22718,318],{"className":22719},[317],[15,22721,22723],{"className":22722},[290],[15,22724,22727],{"className":22725,"style":22726},[294],"height:1.6037em;",[15,22728],{},[15,22730,22732],{"className":22731},[1453],[15,22733,22735,23065],{"className":22734},[285,286],[15,22736,22738,23062],{"className":22737},[290],[15,22739,22741,22778],{"className":22740,"style":22685},[294],[15,22742,22743,22746],{"style":22688},[15,22744],{"className":22745,"style":22692},[302],[15,22747,22749,22752,22755,22758,22761,22764,22767,22772,22775],{"className":22748},[59],[15,22750],{"className":22751},[59],[15,22753],{"className":22754,"style":158},[157],[15,22756,123],{"className":22757},[162],[15,22759],{"className":22760,"style":158},[157],[15,22762,769],{"className":22763},[59,60],[15,22765],{"className":22766,"style":406},[157],[15,22768,4156,22770],{"className":22769},[1486],[15,22771,4160],{"style":4159},[15,22773],{"className":22774,"style":406},[157],[15,22776,769],{"className":22777},[59,60],[15,22779,22780,22783],{"style":22710},[15,22781],{"className":22782,"style":22692},[302],[15,22784,22786,22789,22792,22795,22798,22801,22867,22870,22873,22876,22879,22882,22885,22888,22891,22894,22897,22960,22963,22968,22971,23033,23036,23039,23042,23045,23048,23053,23056,23059],{"className":22785},[59],[15,22787],{"className":22788},[59],[15,22790],{"className":22791,"style":158},[157],[15,22793,4288],{"className":22794},[162],[15,22796,4285],{"className":22797},[157],[15,22799],{"className":22800,"style":158},[157],[15,22802,22804,22807],{"className":22803},[59],[15,22805,37],{"className":22806,"style":61},[59,60],[15,22808,22810],{"className":22809},[281],[15,22811,22813,22859],{"className":22812},[285,286],[15,22814,22816,22856],{"className":22815},[290],[15,22817,22819,22842],{"className":22818,"style":8250},[294],[15,22820,22821,22824],{"style":8253},[15,22822],{"className":22823,"style":303},[302],[15,22825,22827],{"className":22826},[307,308,309,310],[15,22828,22830,22833,22836,22839],{"className":22829},[59,310],[15,22831,7563],{"className":22832},[59,60,310],[15,22834,7566],{"className":22835,"style":8269},[59,60,310],[15,22837,7569],{"className":22838},[59,60,310],[15,22840,7572],{"className":22841,"style":8276},[59,60,310],[15,22843,22844,22847],{"style":8279},[15,22845],{"className":22846,"style":303},[302],[15,22848,22850],{"className":22849},[307,308,309,310],[15,22851,22853],{"className":22852},[59,310],[15,22854,7575],{"className":22855},[59,310],[15,22857,318],{"className":22858},[317],[15,22860,22862],{"className":22861},[290],[15,22863,22865],{"className":22864,"style":8301},[294],[15,22866],{},[15,22868,113],{"className":22869},[146],[15,22871,769],{"className":22872},[59,60],[15,22874,120],{"className":22875},[153],[15,22877],{"className":22878,"style":158},[157],[15,22880,123],{"className":22881},[162],[15,22883],{"className":22884,"style":158},[157],[15,22886,769],{"className":22887},[59,60],[15,22889],{"className":22890,"style":2589},[157],[15,22892,3301],{"className":22893},[2593],[15,22895],{"className":22896,"style":2589},[157],[15,22898,22900,22903,22957],{"className":22899},[59],[15,22901],{"className":22902},[146,441],[15,22904,22906],{"className":22905},[238],[15,22907,22909,22949],{"className":22908},[285,286],[15,22910,22912,22946],{"className":22911},[290],[15,22913,22916,22927,22935],{"className":22914,"style":22915},[294],"height:1.3214em;",[15,22917,22918,22921],{"style":457},[15,22919],{"className":22920,"style":461},[302],[15,22922,22924],{"className":22923},[59],[15,22925,769],{"className":22926},[59,60],[15,22928,22929,22932],{"style":473},[15,22930],{"className":22931,"style":461},[302],[15,22933],{"className":22934,"style":481},[480],[15,22936,22937,22940],{"style":484},[15,22938],{"className":22939,"style":461},[302],[15,22941,22943],{"className":22942},[59],[15,22944,117],{"className":22945},[59],[15,22947,318],{"className":22948},[317],[15,22950,22952],{"className":22951},[290],[15,22953,22955],{"className":22954,"style":3780},[294],[15,22956],{},[15,22958],{"className":22959},[153,441],[15,22961],{"className":22962,"style":406},[157],[15,22964,4156,22966],{"className":22965},[1486],[15,22967,4160],{"style":4159},[15,22969],{"className":22970,"style":406},[157],[15,22972,22974,22977,23030],{"className":22973},[59],[15,22975],{"className":22976},[146,441],[15,22978,22980],{"className":22979},[238],[15,22981,22983,23022],{"className":22982},[285,286],[15,22984,22986,23019],{"className":22985},[290],[15,22987,22989,23000,23008],{"className":22988,"style":22915},[294],[15,22990,22991,22994],{"style":457},[15,22992],{"className":22993,"style":461},[302],[15,22995,22997],{"className":22996},[59],[15,22998,769],{"className":22999},[59,60],[15,23001,23002,23005],{"style":473},[15,23003],{"className":23004,"style":461},[302],[15,23006],{"className":23007,"style":481},[480],[15,23009,23010,23013],{"style":484},[15,23011],{"className":23012,"style":461},[302],[15,23014,23016],{"className":23015},[59],[15,23017,117],{"className":23018},[59],[15,23020,318],{"className":23021},[317],[15,23023,23025],{"className":23024},[290],[15,23026,23028],{"className":23027,"style":3780},[294],[15,23029],{},[15,23031],{"className":23032},[153,441],[15,23034],{"className":23035,"style":158},[157],[15,23037,123],{"className":23038},[162],[15,23040],{"className":23041,"style":158},[157],[15,23043,4815],{"className":23044},[59],[15,23046],{"className":23047,"style":406},[157],[15,23049,4156,23051],{"className":23050},[1486],[15,23052,4160],{"style":4159},[15,23054],{"className":23055,"style":406},[157],[15,23057,769],{"className":23058},[59,60],[15,23060,551],{"className":23061},[573],[15,23063,318],{"className":23064},[317],[15,23066,23068],{"className":23067},[290],[15,23069,23071],{"className":23070,"style":22726},[294],[15,23072],{},[11,23074,23075],{},"所以",[15,23077,23079],{"className":23078,"translate":19},[180],[15,23080,23082,23158],{"className":23081,"translate":19},[18],[15,23083,23085],{"className":23084},[23],[25,23086,23087],{"xmlns":27,"display":189},[29,23088,23089,23155],{},[32,23090,23091,23093,23095,23097,23099,23101,23103,23105,23107,23109,23111,23113,23115,23121,23123,23125,23127,23129,23131,23133,23143,23145,23147,23149,23151,23153],{},[35,23092,37],{},[110,23094,113],{"stretchy":112},[35,23096,769],{},[110,23098,120],{"stretchy":112},[110,23100,123],{},[110,23102,4815],{},[35,23104,4102],{},[110,23106,4105],{},[35,23108,769],{},[1299,23110,4285],{},[110,23112,4288],{},[1299,23114,4285],{},[195,23116,23117,23119],{},[35,23118,199],{},[35,23120,37],{},[110,23122,113],{"stretchy":112},[35,23124,206],{},[35,23126,210],{"mathvariant":209},[35,23128,213],{},[110,23130,120],{"stretchy":112},[110,23132,123],{},[195,23134,23135,23137],{},[35,23136,199],{},[32,23138,23139,23141],{},[35,23140,10698],{},[35,23142,21919],{},[110,23144,113],{"stretchy":112},[35,23146,213],{},[35,23148,210],{"mathvariant":209},[35,23150,206],{},[110,23152,120],{"stretchy":112},[35,23154,4377],{"mathvariant":209},[39,23156,23157],{"encoding":41},"f(t)=-\\log t\n\\ \\Longrightarrow\\ D_f(P\\|Q)=D_{KL}(Q\\|P).",[15,23159,23161,23188,23226,23296],{"className":23160,"ariaHidden":46},[45],[15,23162,23164,23167,23170,23173,23176,23179,23182,23185],{"className":23163},[50],[15,23165],{"className":23166,"style":139},[54],[15,23168,37],{"className":23169,"style":61},[59,60],[15,23171,113],{"className":23172},[146],[15,23174,769],{"className":23175},[59,60],[15,23177,120],{"className":23178},[153],[15,23180],{"className":23181,"style":158},[157],[15,23183,123],{"className":23184},[162],[15,23186],{"className":23187,"style":158},[157],[15,23189,23191,23194,23197,23200,23205,23208,23211,23214,23217,23220,23223],{"className":23190},[50],[15,23192],{"className":23193,"style":55},[54],[15,23195,4815],{"className":23196},[59],[15,23198],{"className":23199,"style":406},[157],[15,23201,4156,23203],{"className":23202},[1486],[15,23204,4160],{"style":4159},[15,23206],{"className":23207,"style":406},[157],[15,23209,769],{"className":23210},[59,60],[15,23212,4285],{"className":23213},[157],[15,23215],{"className":23216,"style":158},[157],[15,23218,4288],{"className":23219},[162],[15,23221,4285],{"className":23222},[157],[15,23224],{"className":23225,"style":158},[157],[15,23227,23229,23232,23272,23275,23278,23281,23284,23287,23290,23293],{"className":23228},[50],[15,23230],{"className":23231,"style":270},[54],[15,23233,23235,23238],{"className":23234},[59],[15,23236,199],{"className":23237,"style":277},[59,60],[15,23239,23241],{"className":23240},[281],[15,23242,23244,23264],{"className":23243},[285,286],[15,23245,23247,23261],{"className":23246},[290],[15,23248,23250],{"className":23249,"style":295},[294],[15,23251,23252,23255],{"style":298},[15,23253],{"className":23254,"style":303},[302],[15,23256,23258],{"className":23257},[307,308,309,310],[15,23259,37],{"className":23260,"style":61},[59,60,310],[15,23262,318],{"className":23263},[317],[15,23265,23267],{"className":23266},[290],[15,23268,23270],{"className":23269,"style":325},[294],[15,23271],{},[15,23273,113],{"className":23274},[146],[15,23276,206],{"className":23277,"style":334},[59,60],[15,23279,210],{"className":23280},[59],[15,23282,213],{"className":23283},[59,60],[15,23285,120],{"className":23286},[153],[15,23288],{"className":23289,"style":158},[157],[15,23291,123],{"className":23292},[162],[15,23294],{"className":23295,"style":158},[157],[15,23297,23299,23302,23348,23351,23354,23357,23360,23363],{"className":23298},[50],[15,23300],{"className":23301,"style":139},[54],[15,23303,23305,23308],{"className":23304},[59],[15,23306,199],{"className":23307,"style":277},[59,60],[15,23309,23311],{"className":23310},[281],[15,23312,23314,23340],{"className":23313},[285,286],[15,23315,23317,23337],{"className":23316},[290],[15,23318,23320],{"className":23319,"style":379},[294],[15,23321,23322,23325],{"style":298},[15,23323],{"className":23324,"style":303},[302],[15,23326,23328],{"className":23327},[307,308,309,310],[15,23329,23331,23334],{"className":23330},[59,310],[15,23332,10698],{"className":23333,"style":10786},[59,60,310],[15,23335,21919],{"className":23336},[59,60,310],[15,23338,318],{"className":23339},[317],[15,23341,23343],{"className":23342},[290],[15,23344,23346],{"className":23345,"style":7970},[294],[15,23347],{},[15,23349,113],{"className":23350},[146],[15,23352,213],{"className":23353},[59,60],[15,23355,210],{"className":23356},[59],[15,23358,206],{"className":23359,"style":334},[59,60],[15,23361,120],{"className":23362},[153],[15,23364,4377],{"className":23365},[59],[11,23367,23368,23371,23372,23490,23491,23556,23557,23650],{},[998,23369,23370],{},"对称化：Jeffreys 散度。"," 取 ",[15,23373,23375,23413],{"className":23374,"translate":19},[18],[15,23376,23378],{"className":23377},[23],[25,23379,23380],{"xmlns":27},[29,23381,23382,23410],{},[32,23383,23384,23386,23388,23390,23392,23394,23396,23398,23400,23402,23404,23406,23408],{},[35,23385,37],{},[110,23387,113],{"stretchy":112},[35,23389,769],{},[110,23391,120],{"stretchy":112},[110,23393,123],{},[110,23395,113],{"stretchy":112},[35,23397,769],{},[110,23399,4815],{},[115,23401,117],{},[110,23403,120],{"stretchy":112},[35,23405,4102],{},[110,23407,4105],{},[35,23409,769],{},[39,23411,23412],{"encoding":41},"f(t)=(t-1)\\log t",[15,23414,23416,23443,23464],{"className":23415,"ariaHidden":46},[45],[15,23417,23419,23422,23425,23428,23431,23434,23437,23440],{"className":23418},[50],[15,23420],{"className":23421,"style":139},[54],[15,23423,37],{"className":23424,"style":61},[59,60],[15,23426,113],{"className":23427},[146],[15,23429,769],{"className":23430},[59,60],[15,23432,120],{"className":23433},[153],[15,23435],{"className":23436,"style":158},[157],[15,23438,123],{"className":23439},[162],[15,23441],{"className":23442,"style":158},[157],[15,23444,23446,23449,23452,23455,23458,23461],{"className":23445},[50],[15,23447],{"className":23448,"style":139},[54],[15,23450,113],{"className":23451},[146],[15,23453,769],{"className":23454},[59,60],[15,23456],{"className":23457,"style":2589},[157],[15,23459,4815],{"className":23460},[2593],[15,23462],{"className":23463,"style":2589},[157],[15,23465,23467,23470,23473,23476,23479,23484,23487],{"className":23466},[50],[15,23468],{"className":23469,"style":139},[54],[15,23471,117],{"className":23472},[59],[15,23474,120],{"className":23475},[153],[15,23477],{"className":23478,"style":406},[157],[15,23480,4156,23482],{"className":23481},[1486],[15,23483,4160],{"style":4159},[15,23485],{"className":23486,"style":406},[157],[15,23488,769],{"className":23489},[59,60],"（它满足 ",[15,23492,23494,23517],{"className":23493,"translate":19},[18],[15,23495,23497],{"className":23496},[23],[25,23498,23499],{"xmlns":27},[29,23500,23501,23515],{},[32,23502,23503,23505,23507,23509,23511,23513],{},[35,23504,37],{},[110,23506,113],{"stretchy":112},[115,23508,117],{},[110,23510,120],{"stretchy":112},[110,23512,123],{},[115,23514,126],{},[39,23516,129],{"encoding":41},[15,23518,23520,23547],{"className":23519,"ariaHidden":46},[45],[15,23521,23523,23526,23529,23532,23535,23538,23541,23544],{"className":23522},[50],[15,23524],{"className":23525,"style":139},[54],[15,23527,37],{"className":23528,"style":61},[59,60],[15,23530,113],{"className":23531},[146],[15,23533,117],{"className":23534},[59],[15,23536,120],{"className":23537},[153],[15,23539],{"className":23540,"style":158},[157],[15,23542,123],{"className":23543},[162],[15,23545],{"className":23546,"style":158},[157],[15,23548,23550,23553],{"className":23549},[50],[15,23551],{"className":23552,"style":172},[54],[15,23554,126],{"className":23555},[59],"，且 ",[15,23558,23560,23596],{"className":23559,"translate":19},[18],[15,23561,23563],{"className":23562},[23],[25,23564,23565],{"xmlns":27},[29,23566,23567,23593],{},[32,23568,23569,23571,23573,23575,23577,23579,23581,23583,23585,23587,23589,23591],{},[35,23570,769],{},[35,23572,37],{},[110,23574,113],{"stretchy":112},[115,23576,117],{},[35,23578,941],{"mathvariant":209},[35,23580,769],{},[110,23582,120],{"stretchy":112},[110,23584,123],{},[35,23586,37],{},[110,23588,113],{"stretchy":112},[35,23590,769],{},[110,23592,120],{"stretchy":112},[39,23594,23595],{"encoding":41},"t f(1\u002Ft)=f(t)",[15,23597,23599,23632],{"className":23598,"ariaHidden":46},[45],[15,23600,23602,23605,23608,23611,23614,23617,23620,23623,23626,23629],{"className":23601},[50],[15,23603],{"className":23604,"style":139},[54],[15,23606,769],{"className":23607},[59,60],[15,23609,37],{"className":23610,"style":61},[59,60],[15,23612,113],{"className":23613},[146],[15,23615,8591],{"className":23616},[59],[15,23618,769],{"className":23619},[59,60],[15,23621,120],{"className":23622},[153],[15,23624],{"className":23625,"style":158},[157],[15,23627,123],{"className":23628},[162],[15,23630],{"className":23631,"style":158},[157],[15,23633,23635,23638,23641,23644,23647],{"className":23634},[50],[15,23636],{"className":23637,"style":139},[54],[15,23639,37],{"className":23640,"style":61},[59,60],[15,23642,113],{"className":23643},[146],[15,23645,769],{"className":23646},[59,60],[15,23648,120],{"className":23649},[153],"），则",[15,23652,23654],{"className":23653,"translate":19},[180],[15,23655,23657,23785],{"className":23656,"translate":19},[18],[15,23658,23660],{"className":23659},[23],[25,23661,23662],{"xmlns":27,"display":189},[29,23663,23664,23782],{},[32,23665,23666,23672,23674,23676,23678,23680,23682,23684,23686,23688,23690,23692,23694,23696,23698,23700,23706,23708,23710,23712,23715,23717,23727,23729,23731,23733,23735,23737,23739,23749,23751,23753,23755,23757,23759,23761,23763,23770,23772,23774,23776,23778,23780],{},[195,23667,23668,23670],{},[35,23669,199],{},[35,23671,37],{},[110,23673,113],{"stretchy":112},[35,23675,206],{},[35,23677,210],{"mathvariant":209},[35,23679,213],{},[110,23681,120],{"stretchy":112},[110,23683,123],{},[110,23685,1289],{},[110,23687,113],{"stretchy":112},[35,23689,11],{},[110,23691,4815],{},[35,23693,944],{},[110,23695,120],{"stretchy":112},[35,23697,4102],{},[110,23699,4105],{},[238,23701,23702,23704],{},[35,23703,11],{},[35,23705,944],{},[1299,23707,1301],{},[35,23709,244],{},[35,23711,774],{},[157,23713],{"linebreak":23714},"newline",[110,23716,123],{},[195,23718,23719,23721],{},[35,23720,199],{},[32,23722,23723,23725],{},[35,23724,10698],{},[35,23726,21919],{},[110,23728,113],{"stretchy":112},[35,23730,206],{},[35,23732,210],{"mathvariant":209},[35,23734,213],{},[110,23736,120],{"stretchy":112},[110,23738,2472],{},[195,23740,23741,23743],{},[35,23742,199],{},[32,23744,23745,23747],{},[35,23746,10698],{},[35,23748,21919],{},[110,23750,113],{"stretchy":112},[35,23752,213],{},[35,23754,210],{"mathvariant":209},[35,23756,206],{},[110,23758,120],{"stretchy":112},[110,23760,123],{},[110,23762,1018],{},[195,23764,23765,23767],{},[35,23766,199],{},[35,23768,23769],{},"J",[110,23771,113],{"stretchy":112},[35,23773,206],{},[110,23775,551],{"separator":46},[35,23777,213],{},[110,23779,120],{"stretchy":112},[110,23781,551],{"separator":46},[39,23783,23784],{"encoding":41},"D_f(P\\|Q)=\\int (p-q)\\log\\frac{p}{q}\\,dx \\\\\n=D_{KL}(P\\|Q)+D_{KL}(Q\\|P)=:D_J(P,Q),",[15,23786,23788,23858,23882,23976,23979,23991,24067,24144],{"className":23787,"ariaHidden":46},[45],[15,23789,23791,23794,23834,23837,23840,23843,23846,23849,23852,23855],{"className":23790},[50],[15,23792],{"className":23793,"style":270},[54],[15,23795,23797,23800],{"className":23796},[59],[15,23798,199],{"className":23799,"style":277},[59,60],[15,23801,23803],{"className":23802},[281],[15,23804,23806,23826],{"className":23805},[285,286],[15,23807,23809,23823],{"className":23808},[290],[15,23810,23812],{"className":23811,"style":295},[294],[15,23813,23814,23817],{"style":298},[15,23815],{"className":23816,"style":303},[302],[15,23818,23820],{"className":23819},[307,308,309,310],[15,23821,37],{"className":23822,"style":61},[59,60,310],[15,23824,318],{"className":23825},[317],[15,23827,23829],{"className":23828},[290],[15,23830,23832],{"className":23831,"style":325},[294],[15,23833],{},[15,23835,113],{"className":23836},[146],[15,23838,206],{"className":23839,"style":334},[59,60],[15,23841,210],{"className":23842},[59],[15,23844,213],{"className":23845},[59,60],[15,23847,120],{"className":23848},[153],[15,23850],{"className":23851,"style":158},[157],[15,23853,123],{"className":23854},[162],[15,23856],{"className":23857,"style":158},[157],[15,23859,23861,23864,23867,23870,23873,23876,23879],{"className":23860},[50],[15,23862],{"className":23863,"style":5779},[54],[15,23865,1289],{"className":23866,"style":1489},[1486,1487,1488],[15,23868,113],{"className":23869},[146],[15,23871,11],{"className":23872},[59,60],[15,23874],{"className":23875,"style":2589},[157],[15,23877,4815],{"className":23878},[2593],[15,23880],{"className":23881,"style":2589},[157],[15,23883,23885,23888,23891,23894,23897,23902,23905,23967,23970,23973],{"className":23884},[50],[15,23886],{"className":23887,"style":22036},[54],[15,23889,944],{"className":23890,"style":981},[59,60],[15,23892,120],{"className":23893},[153],[15,23895],{"className":23896,"style":406},[157],[15,23898,4156,23900],{"className":23899},[1486],[15,23901,4160],{"style":4159},[15,23903],{"className":23904,"style":406},[157],[15,23906,23908,23911,23964],{"className":23907},[59],[15,23909],{"className":23910},[146,441],[15,23912,23914],{"className":23913},[238],[15,23915,23917,23956],{"className":23916},[285,286],[15,23918,23920,23953],{"className":23919},[290],[15,23921,23923,23934,23942],{"className":23922,"style":3556},[294],[15,23924,23925,23928],{"style":457},[15,23926],{"className":23927,"style":461},[302],[15,23929,23931],{"className":23930},[59],[15,23932,944],{"className":23933,"style":981},[59,60],[15,23935,23936,23939],{"style":473},[15,23937],{"className":23938,"style":461},[302],[15,23940],{"className":23941,"style":481},[480],[15,23943,23944,23947],{"style":484},[15,23945],{"className":23946,"style":461},[302],[15,23948,23950],{"className":23949},[59],[15,23951,11],{"className":23952},[59,60],[15,23954,318],{"className":23955},[317],[15,23957,23959],{"className":23958},[290],[15,23960,23962],{"className":23961,"style":506},[294],[15,23963],{},[15,23965],{"className":23966},[153,441],[15,23968],{"className":23969,"style":406},[157],[15,23971,244],{"className":23972},[59,60],[15,23974,774],{"className":23975},[59,60],[15,23977],{"className":23978},[157,23714],[15,23980,23982,23985,23988],{"className":23981},[50],[15,23983],{"className":23984,"style":11007},[54],[15,23986,123],{"className":23987},[162],[15,23989],{"className":23990,"style":158},[157],[15,23992,23994,23997,24043,24046,24049,24052,24055,24058,24061,24064],{"className":23993},[50],[15,23995],{"className":23996,"style":139},[54],[15,23998,24000,24003],{"className":23999},[59],[15,24001,199],{"className":24002,"style":277},[59,60],[15,24004,24006],{"className":24005},[281],[15,24007,24009,24035],{"className":24008},[285,286],[15,24010,24012,24032],{"className":24011},[290],[15,24013,24015],{"className":24014,"style":379},[294],[15,24016,24017,24020],{"style":298},[15,24018],{"className":24019,"style":303},[302],[15,24021,24023],{"className":24022},[307,308,309,310],[15,24024,24026,24029],{"className":24025},[59,310],[15,24027,10698],{"className":24028,"style":10786},[59,60,310],[15,24030,21919],{"className":24031},[59,60,310],[15,24033,318],{"className":24034},[317],[15,24036,24038],{"className":24037},[290],[15,24039,24041],{"className":24040,"style":7970},[294],[15,24042],{},[15,24044,113],{"className":24045},[146],[15,24047,206],{"className":24048,"style":334},[59,60],[15,24050,210],{"className":24051},[59],[15,24053,213],{"className":24054},[59,60],[15,24056,120],{"className":24057},[153],[15,24059],{"className":24060,"style":2589},[157],[15,24062,2472],{"className":24063},[2593],[15,24065],{"className":24066,"style":2589},[157],[15,24068,24070,24073,24119,24122,24125,24128,24131,24134,24137,24141],{"className":24069},[50],[15,24071],{"className":24072,"style":139},[54],[15,24074,24076,24079],{"className":24075},[59],[15,24077,199],{"className":24078,"style":277},[59,60],[15,24080,24082],{"className":24081},[281],[15,24083,24085,24111],{"className":24084},[285,286],[15,24086,24088,24108],{"className":24087},[290],[15,24089,24091],{"className":24090,"style":379},[294],[15,24092,24093,24096],{"style":298},[15,24094],{"className":24095,"style":303},[302],[15,24097,24099],{"className":24098},[307,308,309,310],[15,24100,24102,24105],{"className":24101},[59,310],[15,24103,10698],{"className":24104,"style":10786},[59,60,310],[15,24106,21919],{"className":24107},[59,60,310],[15,24109,318],{"className":24110},[317],[15,24112,24114],{"className":24113},[290],[15,24115,24117],{"className":24116,"style":7970},[294],[15,24118],{},[15,24120,113],{"className":24121},[146],[15,24123,213],{"className":24124},[59,60],[15,24126,210],{"className":24127},[59],[15,24129,206],{"className":24130,"style":334},[59,60],[15,24132,120],{"className":24133},[153],[15,24135],{"className":24136,"style":158},[157],[15,24138,24140],{"className":24139},[162],"=:",[15,24142],{"className":24143,"style":158},[157],[15,24145,24147,24150,24191,24194,24197,24200,24203,24206,24209],{"className":24146},[50],[15,24148],{"className":24149,"style":139},[54],[15,24151,24153,24156],{"className":24152},[59],[15,24154,199],{"className":24155,"style":277},[59,60],[15,24157,24159],{"className":24158},[281],[15,24160,24162,24183],{"className":24161},[285,286],[15,24163,24165,24180],{"className":24164},[290],[15,24166,24168],{"className":24167,"style":379},[294],[15,24169,24170,24173],{"style":298},[15,24171],{"className":24172,"style":303},[302],[15,24174,24176],{"className":24175},[307,308,309,310],[15,24177,23769],{"className":24178,"style":24179},[59,60,310],"margin-right:0.0962em;",[15,24181,318],{"className":24182},[317],[15,24184,24186],{"className":24185},[290],[15,24187,24189],{"className":24188,"style":7970},[294],[15,24190],{},[15,24192,113],{"className":24193},[146],[15,24195,206],{"className":24196,"style":334},[59,60],[15,24198,551],{"className":24199},[573],[15,24201],{"className":24202,"style":406},[157],[15,24204,213],{"className":24205},[59,60],[15,24207,120],{"className":24208},[153],[15,24210,551],{"className":24211},[573],[11,24213,24214],{},"这是 KL 的两向之和，也是一个人为构造的对称 f-散度。",[11,24216,24217,24220,24221,24224,24225,24253,24254,24282,24283,24311,24312,24344,24345,24450,24451,24479,24480,24508,24509,24512],{},[998,24218,24219],{},"意义。"," KL 是最基础的散度，等价于信息论中的",[998,24222,24223],{},"相对熵","：若真实分布是 ",[15,24226,24228,24241],{"className":24227,"translate":19},[18],[15,24229,24231],{"className":24230},[23],[25,24232,24233],{"xmlns":27},[29,24234,24235,24239],{},[32,24236,24237],{},[35,24238,206],{},[39,24240,206],{"encoding":41},[15,24242,24244],{"className":24243,"ariaHidden":46},[45],[15,24245,24247,24250],{"className":24246},[50],[15,24248],{"className":24249,"style":664},[54],[15,24251,206],{"className":24252,"style":334},[59,60],"、而用 ",[15,24255,24257,24270],{"className":24256,"translate":19},[18],[15,24258,24260],{"className":24259},[23],[25,24261,24262],{"xmlns":27},[29,24263,24264,24268],{},[32,24265,24266],{},[35,24267,213],{},[39,24269,213],{"encoding":41},[15,24271,24273],{"className":24272,"ariaHidden":46},[45],[15,24274,24276,24279],{"className":24275},[50],[15,24277],{"className":24278,"style":566},[54],[15,24280,213],{"className":24281},[59,60]," 去编码来自 ",[15,24284,24286,24299],{"className":24285,"translate":19},[18],[15,24287,24289],{"className":24288},[23],[25,24290,24291],{"xmlns":27},[29,24292,24293,24297],{},[32,24294,24295],{},[35,24296,206],{},[39,24298,206],{"encoding":41},[15,24300,24302],{"className":24301,"ariaHidden":46},[45],[15,24303,24305,24308],{"className":24304},[50],[15,24306],{"className":24307,"style":664},[54],[15,24309,206],{"className":24310,"style":334},[59,60]," 的样本，那么 ",[15,24313,24315,24330],{"className":24314,"translate":19},[18],[15,24316,24318],{"className":24317},[23],[25,24319,24320],{"xmlns":27},[29,24321,24322,24328],{},[32,24323,24324,24326],{},[35,24325,4102],{},[110,24327,4105],{},[39,24329,21609],{"encoding":41},[15,24331,24333],{"className":24332,"ariaHidden":46},[45],[15,24334,24336,24339],{"className":24335},[50],[15,24337],{"className":24338,"style":55},[54],[15,24340,4156,24342],{"className":24341},[1486],[15,24343,4160],{"style":4159}," 以 2 为底时 ",[15,24346,24348,24380],{"className":24347,"translate":19},[18],[15,24349,24351],{"className":24350},[23],[25,24352,24353],{"xmlns":27},[29,24354,24355,24377],{},[32,24356,24357,24367,24369,24371,24373,24375],{},[195,24358,24359,24361],{},[35,24360,199],{},[32,24362,24363,24365],{},[35,24364,10698],{},[35,24366,21919],{},[110,24368,113],{"stretchy":112},[35,24370,206],{},[35,24372,210],{"mathvariant":209},[35,24374,213],{},[110,24376,120],{"stretchy":112},[39,24378,24379],{"encoding":41},"D_{KL}(P\\|Q)",[15,24381,24383],{"className":24382,"ariaHidden":46},[45],[15,24384,24386,24389,24435,24438,24441,24444,24447],{"className":24385},[50],[15,24387],{"className":24388,"style":139},[54],[15,24390,24392,24395],{"className":24391},[59],[15,24393,199],{"className":24394,"style":277},[59,60],[15,24396,24398],{"className":24397},[281],[15,24399,24401,24427],{"className":24400},[285,286],[15,24402,24404,24424],{"className":24403},[290],[15,24405,24407],{"className":24406,"style":379},[294],[15,24408,24409,24412],{"style":298},[15,24410],{"className":24411,"style":303},[302],[15,24413,24415],{"className":24414},[307,308,309,310],[15,24416,24418,24421],{"className":24417},[59,310],[15,24419,10698],{"className":24420,"style":10786},[59,60,310],[15,24422,21919],{"className":24423},[59,60,310],[15,24425,318],{"className":24426},[317],[15,24428,24430],{"className":24429},[290],[15,24431,24433],{"className":24432,"style":7970},[294],[15,24434],{},[15,24436,113],{"className":24437},[146],[15,24439,206],{"className":24440,"style":334},[59,60],[15,24442,210],{"className":24443},[59],[15,24445,213],{"className":24446},[59,60],[15,24448,120],{"className":24449},[153]," 正是用 ",[15,24452,24454,24467],{"className":24453,"translate":19},[18],[15,24455,24457],{"className":24456},[23],[25,24458,24459],{"xmlns":27},[29,24460,24461,24465],{},[32,24462,24463],{},[35,24464,213],{},[39,24466,213],{"encoding":41},[15,24468,24470],{"className":24469,"ariaHidden":46},[45],[15,24471,24473,24476],{"className":24472},[50],[15,24474],{"className":24475,"style":566},[54],[15,24477,213],{"className":24478},[59,60]," 编码比用 ",[15,24481,24483,24496],{"className":24482,"translate":19},[18],[15,24484,24486],{"className":24485},[23],[25,24487,24488],{"xmlns":27},[29,24489,24490,24494],{},[32,24491,24492],{},[35,24493,206],{},[39,24495,206],{"encoding":41},[15,24497,24499],{"className":24498,"ariaHidden":46},[45],[15,24500,24502,24505],{"className":24501},[50],[15,24503],{"className":24504,"style":664},[54],[15,24506,206],{"className":24507,"style":334},[59,60]," 编码",[998,24510,24511],{},"多付出的比特数","，即",[15,24514,24516],{"className":24515,"translate":19},[180],[15,24517,24519,24661],{"className":24518,"translate":19},[18],[15,24520,24522],{"className":24521},[23],[25,24523,24524],{"xmlns":27,"display":189},[29,24525,24526,24658],{},[1245,24527,24528,24601],{"rowspacing":1247,"columnalign":1248,"columnspacing":1249},[1251,24529,24530,24556],{},[1254,24531,24532],{},[1257,24533,24534],{"scriptlevel":126,"displaystyle":46},[32,24535,24536,24546,24548,24550,24552,24554],{},[195,24537,24538,24540],{},[35,24539,199],{},[32,24541,24542,24544],{},[35,24543,10698],{},[35,24545,21919],{},[110,24547,113],{"stretchy":112},[35,24549,206],{},[35,24551,210],{"mathvariant":209},[35,24553,213],{},[110,24555,120],{"stretchy":112},[1254,24557,24558],{},[1257,24559,24560],{"scriptlevel":126,"displaystyle":46},[32,24561,24562,24564,24566],{},[32,24563],{},[110,24565,123],{},[6018,24567,24568,24598],{},[6018,24569,24570,24596],{},[32,24571,24572,24578,24580,24582,24584,24586,24588,24590,24592,24594],{},[195,24573,24574,24576],{},[35,24575,223],{"mathvariant":222},[35,24577,206],{},[110,24579,230],{"stretchy":112},[110,24581,4815],{},[35,24583,4102],{},[110,24585,4105],{},[35,24587,944],{},[110,24589,113],{"stretchy":112},[35,24591,598],{},[110,24593,120],{"stretchy":112},[110,24595,257],{"stretchy":112},[110,24597,6042],{"stretchy":46},[1299,24599,24600],{},"交叉熵",[1251,24602,24603,24609],{},[1254,24604,24605],{},[1257,24606,24607],{"scriptlevel":126,"displaystyle":46},[32,24608],{},[1254,24610,24611],{},[1257,24612,24613],{"scriptlevel":126,"displaystyle":46},[32,24614,24615,24617,24619,24621,24656],{},[32,24616],{},[157,24618],{"width":3264},[110,24620,4815],{},[6018,24622,24623,24653],{},[6018,24624,24625,24651],{},[32,24626,24627,24633,24635,24637,24639,24641,24643,24645,24647,24649],{},[195,24628,24629,24631],{},[35,24630,223],{"mathvariant":222},[35,24632,206],{},[110,24634,230],{"stretchy":112},[110,24636,4815],{},[35,24638,4102],{},[110,24640,4105],{},[35,24642,11],{},[110,24644,113],{"stretchy":112},[35,24646,598],{},[110,24648,120],{"stretchy":112},[110,24650,257],{"stretchy":112},[110,24652,6042],{"stretchy":46},[1299,24654,24655],{},"熵",[35,24657,4377],{"mathvariant":209},[39,24659,24660],{"encoding":41},"\\begin{aligned}\nD_{KL}(P\\|Q)\n&=\\underbrace{\\mathbb E_P[-\\log q(X)]}\n_{\\text{交叉熵}}\\\\[2pt]\n&\\qquad-\\underbrace{\\mathbb E_P[-\\log p(X)]}\n_{\\text{熵}} .\n\\end{aligned}",[15,24662,24664],{"className":24663,"ariaHidden":46},[45],[15,24665,24667,24671],{"className":24666},[50],[15,24668],{"className":24669,"style":24670},[54],"height:5.1327em;vertical-align:-2.3163em;",[15,24672,24674],{"className":24673},[59],[15,24675,24677,24781],{"className":24676},[1245],[15,24678,24680],{"className":24679},[1362],[15,24681,24683,24772],{"className":24682},[285,286],[15,24684,24686,24769],{"className":24685},[290],[15,24687,24690,24760],{"className":24688,"style":24689},[294],"height:2.8163em;",[15,24691,24693,24696],{"style":24692},"top:-4.9763em;",[15,24694],{"className":24695,"style":461},[302],[15,24697,24699,24745,24748,24751,24754,24757],{"className":24698},[59],[15,24700,24702,24705],{"className":24701},[59],[15,24703,199],{"className":24704,"style":277},[59,60],[15,24706,24708],{"className":24707},[281],[15,24709,24711,24737],{"className":24710},[285,286],[15,24712,24714,24734],{"className":24713},[290],[15,24715,24717],{"className":24716,"style":379},[294],[15,24718,24719,24722],{"style":298},[15,24720],{"className":24721,"style":303},[302],[15,24723,24725],{"className":24724},[307,308,309,310],[15,24726,24728,24731],{"className":24727},[59,310],[15,24729,10698],{"className":24730,"style":10786},[59,60,310],[15,24732,21919],{"className":24733},[59,60,310],[15,24735,318],{"className":24736},[317],[15,24738,24740],{"className":24739},[290],[15,24741,24743],{"className":24742,"style":7970},[294],[15,24744],{},[15,24746,113],{"className":24747},[146],[15,24749,206],{"className":24750,"style":334},[59,60],[15,24752,210],{"className":24753},[59],[15,24755,213],{"className":24756},[59,60],[15,24758,120],{"className":24759},[153],[15,24761,24763,24766],{"style":24762},"top:-2.26em;",[15,24764],{"className":24765,"style":461},[302],[15,24767],{"className":24768},[59],[15,24770,318],{"className":24771},[317],[15,24773,24775],{"className":24774},[290],[15,24776,24779],{"className":24777,"style":24778},[294],"height:2.3163em;",[15,24780],{},[15,24782,24784],{"className":24783},[1453],[15,24785,24787,25200],{"className":24786},[285,286],[15,24788,24790,25197],{"className":24789},[290],[15,24791,24793,24993],{"className":24792,"style":24689},[294],[15,24794,24795,24798],{"style":24692},[15,24796],{"className":24797,"style":461},[302],[15,24799,24801,24804,24807,24810,24813],{"className":24800},[59],[15,24802],{"className":24803},[59],[15,24805],{"className":24806,"style":158},[157],[15,24808,123],{"className":24809},[162],[15,24811],{"className":24812,"style":158},[157],[15,24814,24816],{"className":24815},[410,6018],[15,24817,24819,24984],{"className":24818},[285,286],[15,24820,24822,24981],{"className":24821},[290],[15,24823,24826,24844],{"className":24824,"style":24825},[294],"height:0.75em;",[15,24827,24829,24832],{"style":24828},"top:-1.4237em;",[15,24830],{"className":24831,"style":461},[302],[15,24833,24835],{"className":24834},[307,308,309,310],[15,24836,24838],{"className":24837},[59,310],[15,24839,24841],{"className":24840},[59,4600,310],[15,24842,24600],{"className":24843},[59,4604,310],[15,24845,24846,24849],{"style":11867},[15,24847],{"className":24848,"style":461},[302],[15,24850,24852],{"className":24851},[410,6018],[15,24853,24855,24972],{"className":24854},[285,286],[15,24856,24858,24969],{"className":24857},[290],[15,24859,24861,24892],{"className":24860,"style":24825},[294],[15,24862,24865,24868],{"className":24863,"style":24864},[6309],"top:-2.102em;",[15,24866],{"className":24867,"style":461},[302],[15,24869,24871,24878,24885],{"className":24870,"style":6318},[6317],[15,24872,24874],{"className":24873,"style":6323},[6322],[6325,24875,24876],{"xmlns":6327,"width":6328,"height":6329,"viewBox":6330,"preserveAspectRatio":6331},[6333,24877],{"d":6335},[15,24879,24881],{"className":24880,"style":6323},[6339],[6325,24882,24883],{"xmlns":6327,"width":6328,"height":6329,"viewBox":6330,"preserveAspectRatio":6342},[6333,24884],{"d":6345},[15,24886,24888],{"className":24887,"style":6323},[6349],[6325,24889,24890],{"xmlns":6327,"width":6328,"height":6329,"viewBox":6330,"preserveAspectRatio":6352},[6333,24891],{"d":6355},[15,24893,24894,24897],{"style":11867},[15,24895],{"className":24896,"style":461},[302],[15,24898,24900,24940,24943,24946,24949,24954,24957,24960,24963,24966],{"className":24899},[59],[15,24901,24903,24906],{"className":24902},[59],[15,24904,223],{"className":24905},[59,366],[15,24907,24909],{"className":24908},[281],[15,24910,24912,24932],{"className":24911},[285,286],[15,24913,24915,24929],{"className":24914},[290],[15,24916,24918],{"className":24917,"style":379},[294],[15,24919,24920,24923],{"style":382},[15,24921],{"className":24922,"style":303},[302],[15,24924,24926],{"className":24925},[307,308,309,310],[15,24927,206],{"className":24928,"style":334},[59,60,310],[15,24930,318],{"className":24931},[317],[15,24933,24935],{"className":24934},[290],[15,24936,24938],{"className":24937,"style":7970},[294],[15,24939],{},[15,24941,230],{"className":24942},[146],[15,24944,4815],{"className":24945},[59],[15,24947],{"className":24948,"style":406},[157],[15,24950,4156,24952],{"className":24951},[1486],[15,24953,4160],{"style":4159},[15,24955],{"className":24956,"style":406},[157],[15,24958,944],{"className":24959,"style":981},[59,60],[15,24961,113],{"className":24962},[146],[15,24964,598],{"className":24965,"style":8889},[59,60],[15,24967,20416],{"className":24968},[153],[15,24970,318],{"className":24971},[317],[15,24973,24975],{"className":24974},[290],[15,24976,24979],{"className":24977,"style":24978},[294],"height:0.898em;",[15,24980],{},[15,24982,318],{"className":24983},[317],[15,24985,24987],{"className":24986},[290],[15,24988,24991],{"className":24989,"style":24990},[294],"height:1.5763em;",[15,24992],{},[15,24994,24995,24998],{"style":24762},[15,24996],{"className":24997,"style":461},[302],[15,24999,25001,25004,25007,25010,25013,25016,25191,25194],{"className":25000},[59],[15,25002],{"className":25003},[59],[15,25005],{"className":25006,"style":3636},[157],[15,25008],{"className":25009,"style":2589},[157],[15,25011,4815],{"className":25012},[2593],[15,25014],{"className":25015,"style":2589},[157],[15,25017,25019],{"className":25018},[410,6018],[15,25020,25022,25183],{"className":25021},[285,286],[15,25023,25025,25180],{"className":25024},[290],[15,25026,25028,25045],{"className":25027,"style":24825},[294],[15,25029,25030,25033],{"style":24828},[15,25031],{"className":25032,"style":461},[302],[15,25034,25036],{"className":25035},[307,308,309,310],[15,25037,25039],{"className":25038},[59,310],[15,25040,25042],{"className":25041},[59,4600,310],[15,25043,24655],{"className":25044},[59,4604,310],[15,25046,25047,25050],{"style":11867},[15,25048],{"className":25049,"style":461},[302],[15,25051,25053],{"className":25052},[410,6018],[15,25054,25056,25172],{"className":25055},[285,286],[15,25057,25059,25169],{"className":25058},[290],[15,25060,25062,25092],{"className":25061,"style":24825},[294],[15,25063,25065,25068],{"className":25064,"style":24864},[6309],[15,25066],{"className":25067,"style":461},[302],[15,25069,25071,25078,25085],{"className":25070,"style":6318},[6317],[15,25072,25074],{"className":25073,"style":6323},[6322],[6325,25075,25076],{"xmlns":6327,"width":6328,"height":6329,"viewBox":6330,"preserveAspectRatio":6331},[6333,25077],{"d":6335},[15,25079,25081],{"className":25080,"style":6323},[6339],[6325,25082,25083],{"xmlns":6327,"width":6328,"height":6329,"viewBox":6330,"preserveAspectRatio":6342},[6333,25084],{"d":6345},[15,25086,25088],{"className":25087,"style":6323},[6349],[6325,25089,25090],{"xmlns":6327,"width":6328,"height":6329,"viewBox":6330,"preserveAspectRatio":6352},[6333,25091],{"d":6355},[15,25093,25094,25097],{"style":11867},[15,25095],{"className":25096,"style":461},[302],[15,25098,25100,25140,25143,25146,25149,25154,25157,25160,25163,25166],{"className":25099},[59],[15,25101,25103,25106],{"className":25102},[59],[15,25104,223],{"className":25105},[59,366],[15,25107,25109],{"className":25108},[281],[15,25110,25112,25132],{"className":25111},[285,286],[15,25113,25115,25129],{"className":25114},[290],[15,25116,25118],{"className":25117,"style":379},[294],[15,25119,25120,25123],{"style":382},[15,25121],{"className":25122,"style":303},[302],[15,25124,25126],{"className":25125},[307,308,309,310],[15,25127,206],{"className":25128,"style":334},[59,60,310],[15,25130,318],{"className":25131},[317],[15,25133,25135],{"className":25134},[290],[15,25136,25138],{"className":25137,"style":7970},[294],[15,25139],{},[15,25141,230],{"className":25142},[146],[15,25144,4815],{"className":25145},[59],[15,25147],{"className":25148,"style":406},[157],[15,25150,4156,25152],{"className":25151},[1486],[15,25153,4160],{"style":4159},[15,25155],{"className":25156,"style":406},[157],[15,25158,11],{"className":25159},[59,60],[15,25161,113],{"className":25162},[146],[15,25164,598],{"className":25165,"style":8889},[59,60],[15,25167,20416],{"className":25168},[153],[15,25170,318],{"className":25171},[317],[15,25173,25175],{"className":25174},[290],[15,25176,25178],{"className":25177,"style":24978},[294],[15,25179],{},[15,25181,318],{"className":25182},[317],[15,25184,25186],{"className":25185},[290],[15,25187,25189],{"className":25188,"style":24990},[294],[15,25190],{},[15,25192],{"className":25193,"style":406},[157],[15,25195,4377],{"className":25196},[59],[15,25198,318],{"className":25199},[317],[15,25201,25203],{"className":25202},[290],[15,25204,25206],{"className":25205,"style":24778},[294],[15,25207],{},[11,25209,25210,25211,25315,25316,3855,25344,25372,25373,25376,25377,25405,25406,25510,25511,3855,25539,25567,25568,25571],{},"在统计中它是最大似然估计的对数似然比的期望，决定了 Stein 引理里第二类错误的最优指数，也是 AIC\u002FBIC 这类模型选择的基石；在机器学习中它是最大似然训练、变分推断与知识蒸馏的目标。它的不对称性带来两种相反的行为：",[15,25212,25214,25245],{"className":25213,"translate":19},[18],[15,25215,25217],{"className":25216},[23],[25,25218,25219],{"xmlns":27},[29,25220,25221,25243],{},[32,25222,25223,25233,25235,25237,25239,25241],{},[195,25224,25225,25227],{},[35,25226,199],{},[32,25228,25229,25231],{},[35,25230,10698],{},[35,25232,21919],{},[110,25234,113],{"stretchy":112},[35,25236,206],{},[35,25238,210],{"mathvariant":209},[35,25240,213],{},[110,25242,120],{"stretchy":112},[39,25244,24379],{"encoding":41},[15,25246,25248],{"className":25247,"ariaHidden":46},[45],[15,25249,25251,25254,25300,25303,25306,25309,25312],{"className":25250},[50],[15,25252],{"className":25253,"style":139},[54],[15,25255,25257,25260],{"className":25256},[59],[15,25258,199],{"className":25259,"style":277},[59,60],[15,25261,25263],{"className":25262},[281],[15,25264,25266,25292],{"className":25265},[285,286],[15,25267,25269,25289],{"className":25268},[290],[15,25270,25272],{"className":25271,"style":379},[294],[15,25273,25274,25277],{"style":298},[15,25275],{"className":25276,"style":303},[302],[15,25278,25280],{"className":25279},[307,308,309,310],[15,25281,25283,25286],{"className":25282},[59,310],[15,25284,10698],{"className":25285,"style":10786},[59,60,310],[15,25287,21919],{"className":25288},[59,60,310],[15,25290,318],{"className":25291},[317],[15,25293,25295],{"className":25294},[290],[15,25296,25298],{"className":25297,"style":7970},[294],[15,25299],{},[15,25301,113],{"className":25302},[146],[15,25304,206],{"className":25305,"style":334},[59,60],[15,25307,210],{"className":25308},[59],[15,25310,213],{"className":25311},[59,60],[15,25313,120],{"className":25314},[153]," 要求 ",[15,25317,25319,25332],{"className":25318,"translate":19},[18],[15,25320,25322],{"className":25321},[23],[25,25323,25324],{"xmlns":27},[29,25325,25326,25330],{},[32,25327,25328],{},[35,25329,213],{},[39,25331,213],{"encoding":41},[15,25333,25335],{"className":25334,"ariaHidden":46},[45],[15,25336,25338,25341],{"className":25337},[50],[15,25339],{"className":25340,"style":566},[54],[15,25342,213],{"className":25343},[59,60],[15,25345,25347,25360],{"className":25346,"translate":19},[18],[15,25348,25350],{"className":25349},[23],[25,25351,25352],{"xmlns":27},[29,25353,25354,25358],{},[32,25355,25356],{},[35,25357,206],{},[39,25359,206],{"encoding":41},[15,25361,25363],{"className":25362,"ariaHidden":46},[45],[15,25364,25366,25369],{"className":25365},[50],[15,25367],{"className":25368,"style":664},[54],[15,25370,206],{"className":25371,"style":334},[59,60]," 有质量处不为零（",[998,25374,25375],{},"零回避","，mass-covering，倾向覆盖 ",[15,25378,25380,25393],{"className":25379,"translate":19},[18],[15,25381,25383],{"className":25382},[23],[25,25384,25385],{"xmlns":27},[29,25386,25387,25391],{},[32,25388,25389],{},[35,25390,206],{},[39,25392,206],{"encoding":41},[15,25394,25396],{"className":25395,"ariaHidden":46},[45],[15,25397,25399,25402],{"className":25398},[50],[15,25400],{"className":25401,"style":664},[54],[15,25403,206],{"className":25404,"style":334},[59,60]," 的每一个模式），而 ",[15,25407,25409,25440],{"className":25408,"translate":19},[18],[15,25410,25412],{"className":25411},[23],[25,25413,25414],{"xmlns":27},[29,25415,25416,25438],{},[32,25417,25418,25428,25430,25432,25434,25436],{},[195,25419,25420,25422],{},[35,25421,199],{},[32,25423,25424,25426],{},[35,25425,10698],{},[35,25427,21919],{},[110,25429,113],{"stretchy":112},[35,25431,213],{},[35,25433,210],{"mathvariant":209},[35,25435,206],{},[110,25437,120],{"stretchy":112},[39,25439,22401],{"encoding":41},[15,25441,25443],{"className":25442,"ariaHidden":46},[45],[15,25444,25446,25449,25495,25498,25501,25504,25507],{"className":25445},[50],[15,25447],{"className":25448,"style":139},[54],[15,25450,25452,25455],{"className":25451},[59],[15,25453,199],{"className":25454,"style":277},[59,60],[15,25456,25458],{"className":25457},[281],[15,25459,25461,25487],{"className":25460},[285,286],[15,25462,25464,25484],{"className":25463},[290],[15,25465,25467],{"className":25466,"style":379},[294],[15,25468,25469,25472],{"style":298},[15,25470],{"className":25471,"style":303},[302],[15,25473,25475],{"className":25474},[307,308,309,310],[15,25476,25478,25481],{"className":25477},[59,310],[15,25479,10698],{"className":25480,"style":10786},[59,60,310],[15,25482,21919],{"className":25483},[59,60,310],[15,25485,318],{"className":25486},[317],[15,25488,25490],{"className":25489},[290],[15,25491,25493],{"className":25492,"style":7970},[294],[15,25494],{},[15,25496,113],{"className":25497},[146],[15,25499,213],{"className":25500},[59,60],[15,25502,210],{"className":25503},[59],[15,25505,206],{"className":25506,"style":334},[59,60],[15,25508,120],{"className":25509},[153]," 会把 ",[15,25512,25514,25527],{"className":25513,"translate":19},[18],[15,25515,25517],{"className":25516},[23],[25,25518,25519],{"xmlns":27},[29,25520,25521,25525],{},[32,25522,25523],{},[35,25524,213],{},[39,25526,213],{"encoding":41},[15,25528,25530],{"className":25529,"ariaHidden":46},[45],[15,25531,25533,25536],{"className":25532},[50],[15,25534],{"className":25535,"style":566},[54],[15,25537,213],{"className":25538},[59,60],[15,25540,25542,25555],{"className":25541,"translate":19},[18],[15,25543,25545],{"className":25544},[23],[25,25546,25547],{"xmlns":27},[29,25548,25549,25553],{},[32,25550,25551],{},[35,25552,206],{},[39,25554,206],{"encoding":41},[15,25556,25558],{"className":25557,"ariaHidden":46},[45],[15,25559,25561,25564],{"className":25560},[50],[15,25562],{"className":25563,"style":664},[54],[15,25565,206],{"className":25566,"style":334},[59,60]," 为零处压到零（",[998,25569,25570],{},"零强制","，mode-seeking，倾向锁定单一模式）。GAN 的模式崩塌与变分推断的均值场近似，本质都由这一不对称性决定。",[2313,25573,25575],{"id":25574},"_32-js-散度","3.2 JS 散度",[11,25577,25578,25579,9402],{},"先回顾 JS 散度的定义，令 ",[15,25580,25582,25615],{"className":25581,"translate":19},[18],[15,25583,25585],{"className":25584},[23],[25,25586,25587],{"xmlns":27},[29,25588,25589,25612],{},[32,25590,25591,25594,25596,25602,25604,25606,25608,25610],{},[35,25592,25593],{},"M",[110,25595,123],{},[238,25597,25598,25600],{},[115,25599,117],{},[115,25601,2465],{},[110,25603,113],{"stretchy":112},[35,25605,206],{},[110,25607,2472],{},[35,25609,213],{},[110,25611,120],{"stretchy":112},[39,25613,25614],{"encoding":41},"M=\\frac12(P+Q)",[15,25616,25618,25637,25726],{"className":25617,"ariaHidden":46},[45],[15,25619,25621,25624,25628,25631,25634],{"className":25620},[50],[15,25622],{"className":25623,"style":664},[54],[15,25625,25593],{"className":25626,"style":25627},[59,60],"margin-right:0.109em;",[15,25629],{"className":25630,"style":158},[157],[15,25632,123],{"className":25633},[162],[15,25635],{"className":25636,"style":158},[157],[15,25638,25640,25643,25711,25714,25717,25720,25723],{"className":25639},[50],[15,25641],{"className":25642,"style":2507},[54],[15,25644,25646,25649,25708],{"className":25645},[59],[15,25647],{"className":25648},[146,441],[15,25650,25652],{"className":25651},[238],[15,25653,25655,25700],{"className":25654},[285,286],[15,25656,25658,25697],{"className":25657},[290],[15,25659,25661,25675,25683],{"className":25660,"style":2526},[294],[15,25662,25663,25666],{"style":2529},[15,25664],{"className":25665,"style":461},[302],[15,25667,25669],{"className":25668},[307,308,309,310],[15,25670,25672],{"className":25671},[59,310],[15,25673,2465],{"className":25674},[59,310],[15,25676,25677,25680],{"style":473},[15,25678],{"className":25679,"style":461},[302],[15,25681],{"className":25682,"style":481},[480],[15,25684,25685,25688],{"style":2552},[15,25686],{"className":25687,"style":461},[302],[15,25689,25691],{"className":25690},[307,308,309,310],[15,25692,25694],{"className":25693},[59,310],[15,25695,117],{"className":25696},[59,310],[15,25698,318],{"className":25699},[317],[15,25701,25703],{"className":25702},[290],[15,25704,25706],{"className":25705,"style":2574},[294],[15,25707],{},[15,25709],{"className":25710},[153,441],[15,25712,113],{"className":25713},[146],[15,25715,206],{"className":25716,"style":334},[59,60],[15,25718],{"className":25719,"style":2589},[157],[15,25721,2472],{"className":25722},[2593],[15,25724],{"className":25725,"style":2589},[157],[15,25727,25729,25732,25735],{"className":25728},[50],[15,25730],{"className":25731,"style":139},[54],[15,25733,213],{"className":25734},[59,60],[15,25736,120],{"className":25737},[153],[15,25739,25741],{"className":25740,"translate":19},[180],[15,25742,25744,25843],{"className":25743,"translate":19},[18],[15,25745,25747],{"className":25746},[23],[25,25748,25749],{"xmlns":27,"display":189},[29,25750,25751,25840],{},[32,25752,25753,25764,25766,25768,25770,25772,25774,25776,25782,25792,25794,25806,25808,25814,25824,25826,25838],{},[195,25754,25755,25757],{},[35,25756,199],{},[32,25758,25759,25761],{},[35,25760,23769],{},[35,25762,25763],{},"S",[110,25765,113],{"stretchy":112},[35,25767,206],{},[35,25769,210],{"mathvariant":209},[35,25771,213],{},[110,25773,120],{"stretchy":112},[110,25775,123],{},[238,25777,25778,25780],{},[115,25779,117],{},[115,25781,2465],{},[195,25783,25784,25786],{},[35,25785,199],{},[32,25787,25788,25790],{},[35,25789,10698],{},[35,25791,21919],{},[1299,25793,1306],{},[32,25795,25796,25798,25800,25802,25804],{},[110,25797,113],{"fence":46},[35,25799,206],{},[35,25801,210],{"mathvariant":209},[35,25803,25593],{},[110,25805,120],{"fence":46},[110,25807,2472],{},[238,25809,25810,25812],{},[115,25811,117],{},[115,25813,2465],{},[195,25815,25816,25818],{},[35,25817,199],{},[32,25819,25820,25822],{},[35,25821,10698],{},[35,25823,21919],{},[1299,25825,1306],{},[32,25827,25828,25830,25832,25834,25836],{},[110,25829,113],{"fence":46},[35,25831,213],{},[35,25833,210],{"mathvariant":209},[35,25835,25593],{},[110,25837,120],{"fence":46},[35,25839,4377],{"mathvariant":209},[39,25841,25842],{"encoding":41},"D_{JS}(P\\|Q)=\\frac12 D_{KL}\\!\\left(P\\|M\\right)\n+\\frac12 D_{KL}\\!\\left(Q\\|M\\right).",[15,25844,25846,25923,26071],{"className":25845,"ariaHidden":46},[45],[15,25847,25849,25852,25899,25902,25905,25908,25911,25914,25917,25920],{"className":25848},[50],[15,25850],{"className":25851,"style":139},[54],[15,25853,25855,25858],{"className":25854},[59],[15,25856,199],{"className":25857,"style":277},[59,60],[15,25859,25861],{"className":25860},[281],[15,25862,25864,25891],{"className":25863},[285,286],[15,25865,25867,25888],{"className":25866},[290],[15,25868,25870],{"className":25869,"style":379},[294],[15,25871,25872,25875],{"style":298},[15,25873],{"className":25874,"style":303},[302],[15,25876,25878],{"className":25877},[307,308,309,310],[15,25879,25881,25884],{"className":25880},[59,310],[15,25882,23769],{"className":25883,"style":24179},[59,60,310],[15,25885,25763],{"className":25886,"style":25887},[59,60,310],"margin-right:0.0576em;",[15,25889,318],{"className":25890},[317],[15,25892,25894],{"className":25893},[290],[15,25895,25897],{"className":25896,"style":7970},[294],[15,25898],{},[15,25900,113],{"className":25901},[146],[15,25903,206],{"className":25904,"style":334},[59,60],[15,25906,210],{"className":25907},[59],[15,25909,213],{"className":25910},[59,60],[15,25912,120],{"className":25913},[153],[15,25915],{"className":25916,"style":158},[157],[15,25918,123],{"className":25919},[162],[15,25921],{"className":25922,"style":158},[157],[15,25924,25926,25930,25992,26038,26041,26044,26062,26065,26068],{"className":25925},[50],[15,25927],{"className":25928,"style":25929},[54],"height:2.0074em;vertical-align:-0.686em;",[15,25931,25933,25936,25989],{"className":25932},[59],[15,25934],{"className":25935},[146,441],[15,25937,25939],{"className":25938},[238],[15,25940,25942,25981],{"className":25941},[285,286],[15,25943,25945,25978],{"className":25944},[290],[15,25946,25948,25959,25967],{"className":25947,"style":22915},[294],[15,25949,25950,25953],{"style":457},[15,25951],{"className":25952,"style":461},[302],[15,25954,25956],{"className":25955},[59],[15,25957,2465],{"className":25958},[59],[15,25960,25961,25964],{"style":473},[15,25962],{"className":25963,"style":461},[302],[15,25965],{"className":25966,"style":481},[480],[15,25968,25969,25972],{"style":484},[15,25970],{"className":25971,"style":461},[302],[15,25973,25975],{"className":25974},[59],[15,25976,117],{"className":25977},[59],[15,25979,318],{"className":25980},[317],[15,25982,25984],{"className":25983},[290],[15,25985,25987],{"className":25986,"style":3780},[294],[15,25988],{},[15,25990],{"className":25991},[153,441],[15,25993,25995,25998],{"className":25994},[59],[15,25996,199],{"className":25997,"style":277},[59,60],[15,25999,26001],{"className":26000},[281],[15,26002,26004,26030],{"className":26003},[285,286],[15,26005,26007,26027],{"className":26006},[290],[15,26008,26010],{"className":26009,"style":379},[294],[15,26011,26012,26015],{"style":298},[15,26013],{"className":26014,"style":303},[302],[15,26016,26018],{"className":26017},[307,308,309,310],[15,26019,26021,26024],{"className":26020},[59,310],[15,26022,10698],{"className":26023,"style":10786},[59,60,310],[15,26025,21919],{"className":26026},[59,60,310],[15,26028,318],{"className":26029},[317],[15,26031,26033],{"className":26032},[290],[15,26034,26036],{"className":26035,"style":7970},[294],[15,26037],{},[15,26039],{"className":26040,"style":1514},[157],[15,26042],{"className":26043,"style":406},[157],[15,26045,26047,26050,26053,26056,26059],{"className":26046},[410],[15,26048,113],{"className":26049,"style":415},[146,414],[15,26051,206],{"className":26052,"style":334},[59,60],[15,26054,210],{"className":26055},[59],[15,26057,25593],{"className":26058,"style":25627},[59,60],[15,26060,120],{"className":26061,"style":415},[153,414],[15,26063],{"className":26064,"style":2589},[157],[15,26066,2472],{"className":26067},[2593],[15,26069],{"className":26070,"style":2589},[157],[15,26072,26074,26077,26139,26185,26188,26191,26209,26212],{"className":26073},[50],[15,26075],{"className":26076,"style":25929},[54],[15,26078,26080,26083,26136],{"className":26079},[59],[15,26081],{"className":26082},[146,441],[15,26084,26086],{"className":26085},[238],[15,26087,26089,26128],{"className":26088},[285,286],[15,26090,26092,26125],{"className":26091},[290],[15,26093,26095,26106,26114],{"className":26094,"style":22915},[294],[15,26096,26097,26100],{"style":457},[15,26098],{"className":26099,"style":461},[302],[15,26101,26103],{"className":26102},[59],[15,26104,2465],{"className":26105},[59],[15,26107,26108,26111],{"style":473},[15,26109],{"className":26110,"style":461},[302],[15,26112],{"className":26113,"style":481},[480],[15,26115,26116,26119],{"style":484},[15,26117],{"className":26118,"style":461},[302],[15,26120,26122],{"className":26121},[59],[15,26123,117],{"className":26124},[59],[15,26126,318],{"className":26127},[317],[15,26129,26131],{"className":26130},[290],[15,26132,26134],{"className":26133,"style":3780},[294],[15,26135],{},[15,26137],{"className":26138},[153,441],[15,26140,26142,26145],{"className":26141},[59],[15,26143,199],{"className":26144,"style":277},[59,60],[15,26146,26148],{"className":26147},[281],[15,26149,26151,26177],{"className":26150},[285,286],[15,26152,26154,26174],{"className":26153},[290],[15,26155,26157],{"className":26156,"style":379},[294],[15,26158,26159,26162],{"style":298},[15,26160],{"className":26161,"style":303},[302],[15,26163,26165],{"className":26164},[307,308,309,310],[15,26166,26168,26171],{"className":26167},[59,310],[15,26169,10698],{"className":26170,"style":10786},[59,60,310],[15,26172,21919],{"className":26173},[59,60,310],[15,26175,318],{"className":26176},[317],[15,26178,26180],{"className":26179},[290],[15,26181,26183],{"className":26182,"style":7970},[294],[15,26184],{},[15,26186],{"className":26187,"style":1514},[157],[15,26189],{"className":26190,"style":406},[157],[15,26192,26194,26197,26200,26203,26206],{"className":26193},[410],[15,26195,113],{"className":26196,"style":415},[146,414],[15,26198,213],{"className":26199},[59,60],[15,26201,210],{"className":26202},[59],[15,26204,25593],{"className":26205,"style":25627},[59,60],[15,26207,120],{"className":26208,"style":415},[153,414],[15,26210],{"className":26211,"style":406},[157],[15,26213,4377],{"className":26214},[59],[11,26216,26217,26220,26221,26249,26250,26305,26306,26334,26335,26486],{},[998,26218,26219],{},"它是单个 f-散度。"," 这一点很容易被误解。我们直接构造 ",[15,26222,26224,26237],{"className":26223,"translate":19},[18],[15,26225,26227],{"className":26226},[23],[25,26228,26229],{"xmlns":27},[29,26230,26231,26235],{},[32,26232,26233],{},[35,26234,37],{},[39,26236,37],{"encoding":41},[15,26238,26240],{"className":26239,"ariaHidden":46},[45],[15,26241,26243,26246],{"className":26242},[50],[15,26244],{"className":26245,"style":55},[54],[15,26247,37],{"className":26248,"style":61},[59,60],"：代入 ",[15,26251,26253,26273],{"className":26252,"translate":19},[18],[15,26254,26256],{"className":26255},[23],[25,26257,26258],{"xmlns":27},[29,26259,26260,26270],{},[32,26261,26262,26264,26266,26268],{},[35,26263,11],{},[110,26265,123],{},[35,26267,769],{},[35,26269,944],{},[39,26271,26272],{"encoding":41},"p=tq",[15,26274,26276,26294],{"className":26275,"ariaHidden":46},[45],[15,26277,26279,26282,26285,26288,26291],{"className":26278},[50],[15,26280],{"className":26281,"style":2278},[54],[15,26283,11],{"className":26284},[59,60],[15,26286],{"className":26287,"style":158},[157],[15,26289,123],{"className":26290},[162],[15,26292],{"className":26293,"style":158},[157],[15,26295,26297,26301],{"className":26296},[50],[15,26298],{"className":26299,"style":26300},[54],"height:0.8095em;vertical-align:-0.1944em;",[15,26302,26304],{"className":26303,"style":981},[59,60],"tq","，于是 ",[15,26307,26309,26322],{"className":26308,"translate":19},[18],[15,26310,26312],{"className":26311},[23],[25,26313,26314],{"xmlns":27},[29,26315,26316,26320],{},[32,26317,26318],{},[35,26319,25593],{},[39,26321,25593],{"encoding":41},[15,26323,26325],{"className":26324,"ariaHidden":46},[45],[15,26326,26328,26331],{"className":26327},[50],[15,26329],{"className":26330,"style":664},[54],[15,26332,25593],{"className":26333,"style":25627},[59,60]," 的密度为 ",[15,26336,26338,26373],{"className":26337,"translate":19},[18],[15,26339,26341],{"className":26340},[23],[25,26342,26343],{"xmlns":27},[29,26344,26345,26370],{},[32,26346,26347,26350,26352],{},[35,26348,26349],{},"m",[110,26351,123],{},[238,26353,26354,26368],{},[32,26355,26356,26358,26360,26362,26364,26366],{},[110,26357,113],{"stretchy":112},[35,26359,769],{},[110,26361,2472],{},[115,26363,117],{},[110,26365,120],{"stretchy":112},[35,26367,944],{},[115,26369,2465],{},[39,26371,26372],{"encoding":41},"m=\\frac{(t+1)q}{2}",[15,26374,26376,26394],{"className":26375,"ariaHidden":46},[45],[15,26377,26379,26382,26385,26388,26391],{"className":26378},[50],[15,26380],{"className":26381,"style":2347},[54],[15,26383,26349],{"className":26384},[59,60],[15,26386],{"className":26387,"style":158},[157],[15,26389,123],{"className":26390},[162],[15,26392],{"className":26393,"style":158},[157],[15,26395,26397,26401],{"className":26396},[50],[15,26398],{"className":26399,"style":26400},[54],"height:1.355em;vertical-align:-0.345em;",[15,26402,26404,26407,26483],{"className":26403},[59],[15,26405],{"className":26406},[146,441],[15,26408,26410],{"className":26409},[238],[15,26411,26413,26475],{"className":26412},[285,286],[15,26414,26416,26472],{"className":26415},[290],[15,26417,26420,26434,26442],{"className":26418,"style":26419},[294],"height:1.01em;",[15,26421,26422,26425],{"style":2529},[15,26423],{"className":26424,"style":461},[302],[15,26426,26428],{"className":26427},[307,308,309,310],[15,26429,26431],{"className":26430},[59,310],[15,26432,2465],{"className":26433},[59,310],[15,26435,26436,26439],{"style":473},[15,26437],{"className":26438,"style":461},[302],[15,26440],{"className":26441,"style":481},[480],[15,26443,26445,26448],{"style":26444},"top:-3.485em;",[15,26446],{"className":26447,"style":461},[302],[15,26449,26451],{"className":26450},[307,308,309,310],[15,26452,26454,26457,26460,26463,26466,26469],{"className":26453},[59,310],[15,26455,113],{"className":26456},[146,310],[15,26458,769],{"className":26459},[59,60,310],[15,26461,2472],{"className":26462},[2593,310],[15,26464,117],{"className":26465},[59,310],[15,26467,120],{"className":26468},[153,310],[15,26470,944],{"className":26471,"style":981},[59,60,310],[15,26473,318],{"className":26474},[317],[15,26476,26478],{"className":26477},[290],[15,26479,26481],{"className":26480,"style":2574},[294],[15,26482],{},[15,26484],{"className":26485},[153,441],"，逐项算：",[15,26488,26490],{"className":26489,"translate":19},[180],[15,26491,26493,26555],{"className":26492,"translate":19},[18],[15,26494,26496],{"className":26495},[23],[25,26497,26498],{"xmlns":27,"display":189},[29,26499,26500,26552],{},[32,26501,26502,26508,26510,26526,26528,26530,26536,26538,26550],{},[238,26503,26504,26506],{},[35,26505,11],{},[35,26507,26349],{},[110,26509,123],{},[238,26511,26512,26518],{},[32,26513,26514,26516],{},[115,26515,2465],{},[35,26517,769],{},[32,26519,26520,26522,26524],{},[35,26521,769],{},[110,26523,2472],{},[115,26525,117],{},[110,26527,551],{"separator":46},[157,26529],{"width":3264},[238,26531,26532,26534],{},[35,26533,944],{},[35,26535,26349],{},[110,26537,123],{},[238,26539,26540,26542],{},[115,26541,2465],{},[32,26543,26544,26546,26548],{},[35,26545,769],{},[110,26547,2472],{},[115,26549,117],{},[110,26551,551],{"separator":46},[39,26553,26554],{"encoding":41},"\\frac{p}{m}=\\frac{2t}{t+1},\\qquad\n\\frac{q}{m}=\\frac{2}{t+1},",[15,26556,26558,26636,26800],{"className":26557,"ariaHidden":46},[45],[15,26559,26561,26565,26627,26630,26633],{"className":26560},[50],[15,26562],{"className":26563,"style":26564},[54],"height:1.7936em;vertical-align:-0.686em;",[15,26566,26568,26571,26624],{"className":26567},[59],[15,26569],{"className":26570},[146,441],[15,26572,26574],{"className":26573},[238],[15,26575,26577,26616],{"className":26576},[285,286],[15,26578,26580,26613],{"className":26579},[290],[15,26581,26583,26594,26602],{"className":26582,"style":3556},[294],[15,26584,26585,26588],{"style":457},[15,26586],{"className":26587,"style":461},[302],[15,26589,26591],{"className":26590},[59],[15,26592,26349],{"className":26593},[59,60],[15,26595,26596,26599],{"style":473},[15,26597],{"className":26598,"style":461},[302],[15,26600],{"className":26601,"style":481},[480],[15,26603,26604,26607],{"style":484},[15,26605],{"className":26606,"style":461},[302],[15,26608,26610],{"className":26609},[59],[15,26611,11],{"className":26612},[59,60],[15,26614,318],{"className":26615},[317],[15,26617,26619],{"className":26618},[290],[15,26620,26622],{"className":26621,"style":3780},[294],[15,26623],{},[15,26625],{"className":26626},[153,441],[15,26628],{"className":26629,"style":158},[157],[15,26631,123],{"className":26632},[162],[15,26634],{"className":26635,"style":158},[157],[15,26637,26639,26643,26720,26723,26726,26729,26791,26794,26797],{"className":26638},[50],[15,26640],{"className":26641,"style":26642},[54],"height:2.0908em;vertical-align:-0.7693em;",[15,26644,26646,26649,26717],{"className":26645},[59],[15,26647],{"className":26648},[146,441],[15,26650,26652],{"className":26651},[238],[15,26653,26655,26709],{"className":26654},[285,286],[15,26656,26658,26706],{"className":26657},[290],[15,26659,26661,26684,26692],{"className":26660,"style":22915},[294],[15,26662,26663,26666],{"style":457},[15,26664],{"className":26665,"style":461},[302],[15,26667,26669,26672,26675,26678,26681],{"className":26668},[59],[15,26670,769],{"className":26671},[59,60],[15,26673],{"className":26674,"style":2589},[157],[15,26676,2472],{"className":26677},[2593],[15,26679],{"className":26680,"style":2589},[157],[15,26682,117],{"className":26683},[59],[15,26685,26686,26689],{"style":473},[15,26687],{"className":26688,"style":461},[302],[15,26690],{"className":26691,"style":481},[480],[15,26693,26694,26697],{"style":484},[15,26695],{"className":26696,"style":461},[302],[15,26698,26700,26703],{"className":26699},[59],[15,26701,2465],{"className":26702},[59],[15,26704,769],{"className":26705},[59,60],[15,26707,318],{"className":26708},[317],[15,26710,26712],{"className":26711},[290],[15,26713,26715],{"className":26714,"style":17019},[294],[15,26716],{},[15,26718],{"className":26719},[153,441],[15,26721,551],{"className":26722},[573],[15,26724],{"className":26725,"style":3636},[157],[15,26727],{"className":26728,"style":406},[157],[15,26730,26732,26735,26788],{"className":26731},[59],[15,26733],{"className":26734},[146,441],[15,26736,26738],{"className":26737},[238],[15,26739,26741,26780],{"className":26740},[285,286],[15,26742,26744,26777],{"className":26743},[290],[15,26745,26747,26758,26766],{"className":26746,"style":3556},[294],[15,26748,26749,26752],{"style":457},[15,26750],{"className":26751,"style":461},[302],[15,26753,26755],{"className":26754},[59],[15,26756,26349],{"className":26757},[59,60],[15,26759,26760,26763],{"style":473},[15,26761],{"className":26762,"style":461},[302],[15,26764],{"className":26765,"style":481},[480],[15,26767,26768,26771],{"style":484},[15,26769],{"className":26770,"style":461},[302],[15,26772,26774],{"className":26773},[59],[15,26775,944],{"className":26776,"style":981},[59,60],[15,26778,318],{"className":26779},[317],[15,26781,26783],{"className":26782},[290],[15,26784,26786],{"className":26785,"style":3780},[294],[15,26787],{},[15,26789],{"className":26790},[153,441],[15,26792],{"className":26793,"style":158},[157],[15,26795,123],{"className":26796},[162],[15,26798],{"className":26799,"style":158},[157],[15,26801,26803,26806,26880],{"className":26802},[50],[15,26804],{"className":26805,"style":26642},[54],[15,26807,26809,26812,26877],{"className":26808},[59],[15,26810],{"className":26811},[146,441],[15,26813,26815],{"className":26814},[238],[15,26816,26818,26869],{"className":26817},[285,286],[15,26819,26821,26866],{"className":26820},[290],[15,26822,26824,26847,26855],{"className":26823,"style":22915},[294],[15,26825,26826,26829],{"style":457},[15,26827],{"className":26828,"style":461},[302],[15,26830,26832,26835,26838,26841,26844],{"className":26831},[59],[15,26833,769],{"className":26834},[59,60],[15,26836],{"className":26837,"style":2589},[157],[15,26839,2472],{"className":26840},[2593],[15,26842],{"className":26843,"style":2589},[157],[15,26845,117],{"className":26846},[59],[15,26848,26849,26852],{"style":473},[15,26850],{"className":26851,"style":461},[302],[15,26853],{"className":26854,"style":481},[480],[15,26856,26857,26860],{"style":484},[15,26858],{"className":26859,"style":461},[302],[15,26861,26863],{"className":26862},[59],[15,26864,2465],{"className":26865},[59],[15,26867,318],{"className":26868},[317],[15,26870,26872],{"className":26871},[290],[15,26873,26875],{"className":26874,"style":17019},[294],[15,26876],{},[15,26878],{"className":26879},[153,441],[15,26881,551],{"className":26882},[573],[15,26884,26886],{"className":26885,"translate":19},[180],[15,26887,26889,27029],{"className":26888,"translate":19},[18],[15,26890,26892],{"className":26891},[23],[25,26893,26894],{"xmlns":27,"display":189},[29,26895,26896,27026],{},[1245,26897,26898,26978],{"rowspacing":1247,"columnalign":1248,"columnspacing":1249},[1251,26899,26900,26926],{},[1254,26901,26902],{},[1257,26903,26904],{"scriptlevel":126,"displaystyle":46},[32,26905,26906,26916,26918,26920,26922,26924],{},[195,26907,26908,26910],{},[35,26909,199],{},[32,26911,26912,26914],{},[35,26913,23769],{},[35,26915,25763],{},[110,26917,113],{"stretchy":112},[35,26919,206],{},[35,26921,210],{"mathvariant":209},[35,26923,213],{},[110,26925,120],{"stretchy":112},[1254,26927,26928],{},[1257,26929,26930],{"scriptlevel":126,"displaystyle":46},[32,26931,26932,26934,26936,26942,26944,26946,26948,26950,26956,26958,26964,26966,26968,26970,26972],{},[32,26933],{},[110,26935,123],{},[238,26937,26938,26940],{},[115,26939,117],{},[115,26941,2465],{},[110,26943,1289],{},[35,26945,11],{},[35,26947,4102],{},[110,26949,4105],{},[238,26951,26952,26954],{},[35,26953,11],{},[35,26955,26349],{},[110,26957,2472],{},[238,26959,26960,26962],{},[115,26961,117],{},[115,26963,2465],{},[110,26965,1289],{},[35,26967,944],{},[35,26969,4102],{},[110,26971,4105],{},[238,26973,26974,26976],{},[35,26975,944],{},[35,26977,26349],{},[1251,26979,26980,26986],{},[1254,26981,26982],{},[1257,26983,26984],{"scriptlevel":126,"displaystyle":46},[32,26985],{},[1254,26987,26988],{},[1257,26989,26990],{"scriptlevel":126,"displaystyle":46},[32,26991,26992,26994,26996,26998,27000,27002,27012,27014,27016,27018,27020,27022,27024],{},[32,26993],{},[110,26995,123],{},[110,26997,1289],{},[35,26999,944],{},[1299,27001,1301],{},[195,27003,27004,27006],{},[35,27005,37],{},[32,27007,27008,27010],{},[35,27009,23769],{},[35,27011,25763],{},[110,27013,113],{"stretchy":112},[35,27015,769],{},[110,27017,120],{"stretchy":112},[1299,27019,1301],{},[35,27021,244],{},[35,27023,774],{},[110,27025,551],{"separator":46},[39,27027,27028],{"encoding":41},"\\begin{aligned}\nD_{JS}(P\\|Q)\n&=\\frac12\\int p\\log\\frac{p}{m}\n+\\frac12\\int q\\log\\frac{q}{m}\\\\[2pt]\n&=\\int q\\,f_{JS}(t)\\,dx,\n\\end{aligned}",[15,27030,27032],{"className":27031,"ariaHidden":46},[45],[15,27033,27035,27039],{"className":27034},[50],[15,27036],{"className":27037,"style":27038},[54],"height:4.7445em;vertical-align:-2.1223em;",[15,27040,27042],{"className":27041},[59],[15,27043,27045,27148],{"className":27044},[1245],[15,27046,27048],{"className":27047},[1362],[15,27049,27051,27139],{"className":27050},[285,286],[15,27052,27054,27136],{"className":27053},[290],[15,27055,27058,27128],{"className":27056,"style":27057},[294],"height:2.6223em;",[15,27059,27061,27064],{"style":27060},"top:-4.6223em;",[15,27062],{"className":27063,"style":6275},[302],[15,27065,27067,27113,27116,27119,27122,27125],{"className":27066},[59],[15,27068,27070,27073],{"className":27069},[59],[15,27071,199],{"className":27072,"style":277},[59,60],[15,27074,27076],{"className":27075},[281],[15,27077,27079,27105],{"className":27078},[285,286],[15,27080,27082,27102],{"className":27081},[290],[15,27083,27085],{"className":27084,"style":379},[294],[15,27086,27087,27090],{"style":298},[15,27088],{"className":27089,"style":303},[302],[15,27091,27093],{"className":27092},[307,308,309,310],[15,27094,27096,27099],{"className":27095},[59,310],[15,27097,23769],{"className":27098,"style":24179},[59,60,310],[15,27100,25763],{"className":27101,"style":25887},[59,60,310],[15,27103,318],{"className":27104},[317],[15,27106,27108],{"className":27107},[290],[15,27109,27111],{"className":27110,"style":7970},[294],[15,27112],{},[15,27114,113],{"className":27115},[146],[15,27117,206],{"className":27118,"style":334},[59,60],[15,27120,210],{"className":27121},[59],[15,27123,213],{"className":27124},[59,60],[15,27126,120],{"className":27127},[153],[15,27129,27130,27133],{"style":1905},[15,27131],{"className":27132,"style":6275},[302],[15,27134],{"className":27135},[59],[15,27137,318],{"className":27138},[317],[15,27140,27142],{"className":27141},[290],[15,27143,27146],{"className":27144,"style":27145},[294],"height:2.1223em;",[15,27147],{},[15,27149,27151],{"className":27150},[1453],[15,27152,27154,27586],{"className":27153},[285,286],[15,27155,27157,27583],{"className":27156},[290],[15,27158,27160,27483],{"className":27159,"style":27057},[294],[15,27161,27162,27165],{"style":27060},[15,27163],{"className":27164,"style":6275},[302],[15,27166,27168,27171,27174,27177,27180,27242,27245,27248,27251,27254,27257,27262,27265,27327,27330,27333,27336,27398,27401,27404,27407,27410,27413,27418,27421],{"className":27167},[59],[15,27169],{"className":27170},[59],[15,27172],{"className":27173,"style":158},[157],[15,27175,123],{"className":27176},[162],[15,27178],{"className":27179,"style":158},[157],[15,27181,27183,27186,27239],{"className":27182},[59],[15,27184],{"className":27185},[146,441],[15,27187,27189],{"className":27188},[238],[15,27190,27192,27231],{"className":27191},[285,286],[15,27193,27195,27228],{"className":27194},[290],[15,27196,27198,27209,27217],{"className":27197,"style":22915},[294],[15,27199,27200,27203],{"style":457},[15,27201],{"className":27202,"style":461},[302],[15,27204,27206],{"className":27205},[59],[15,27207,2465],{"className":27208},[59],[15,27210,27211,27214],{"style":473},[15,27212],{"className":27213,"style":461},[302],[15,27215],{"className":27216,"style":481},[480],[15,27218,27219,27222],{"style":484},[15,27220],{"className":27221,"style":461},[302],[15,27223,27225],{"className":27224},[59],[15,27226,117],{"className":27227},[59],[15,27229,318],{"className":27230},[317],[15,27232,27234],{"className":27233},[290],[15,27235,27237],{"className":27236,"style":3780},[294],[15,27238],{},[15,27240],{"className":27241},[153,441],[15,27243],{"className":27244,"style":406},[157],[15,27246,1289],{"className":27247,"style":1489},[1486,1487,1488],[15,27249],{"className":27250,"style":406},[157],[15,27252,11],{"className":27253},[59,60],[15,27255],{"className":27256,"style":406},[157],[15,27258,4156,27260],{"className":27259},[1486],[15,27261,4160],{"style":4159},[15,27263],{"className":27264,"style":406},[157],[15,27266,27268,27271,27324],{"className":27267},[59],[15,27269],{"className":27270},[146,441],[15,27272,27274],{"className":27273},[238],[15,27275,27277,27316],{"className":27276},[285,286],[15,27278,27280,27313],{"className":27279},[290],[15,27281,27283,27294,27302],{"className":27282,"style":3556},[294],[15,27284,27285,27288],{"style":457},[15,27286],{"className":27287,"style":461},[302],[15,27289,27291],{"className":27290},[59],[15,27292,26349],{"className":27293},[59,60],[15,27295,27296,27299],{"style":473},[15,27297],{"className":27298,"style":461},[302],[15,27300],{"className":27301,"style":481},[480],[15,27303,27304,27307],{"style":484},[15,27305],{"className":27306,"style":461},[302],[15,27308,27310],{"className":27309},[59],[15,27311,11],{"className":27312},[59,60],[15,27314,318],{"className":27315},[317],[15,27317,27319],{"className":27318},[290],[15,27320,27322],{"className":27321,"style":3780},[294],[15,27323],{},[15,27325],{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f_{JS}(1\u002Ft)=f_{JS}(t)",[15,29769,29771,29847],{"className":29770,"ariaHidden":46},[45],[15,29772,29774,29777,29780,29826,29829,29832,29835,29838,29841,29844],{"className":29773},[50],[15,29775],{"className":29776,"style":139},[54],[15,29778,769],{"className":29779},[59,60],[15,29781,29783,29786],{"className":29782},[59],[15,29784,37],{"className":29785,"style":61},[59,60],[15,29787,29789],{"className":29788},[281],[15,29790,29792,29818],{"className":29791},[285,286],[15,29793,29795,29815],{"className":29794},[290],[15,29796,29798],{"className":29797,"style":379},[294],[15,29799,29800,29803],{"style":27535},[15,29801],{"className":29802,"style":303},[302],[15,29804,29806],{"className":29805},[307,308,309,310],[15,29807,29809,29812],{"className":29808},[59,310],[15,29810,23769],{"className":29811,"style":24179},[59,60,310],[15,29813,25763],{"className":29814,"style":25887},[59,60,310],[15,29816,318],{"className":29817},[317],[15,29819,29821],{"className":29820},[290],[15,29822,29824],{"className":29823,"style":7970},[294],[15,29825],{},[15,29827,113],{"className":29828},[146],[15,29830,8591],{"className":29831},[59],[15,29833,769],{"className":29834},[59,60],[15,29836,120],{"className":29837},[153],[15,29839],{"className":29840,"style":158},[157],[15,29842,123],{"className":29843},[162],[15,29845],{"className":29846,"style":158},[157],[15,29848,29850,29853,29899,29902,29905],{"className":29849},[50],[15,29851],{"className":29852,"style":139},[54],[15,29854,29856,29859],{"className":29855},[59],[15,29857,37],{"className":29858,"style":61},[59,60],[15,29860,29862],{"className":29861},[281],[15,29863,29865,29891],{"className":29864},[285,286],[15,29866,29868,29888],{"className":29867},[290],[15,29869,29871],{"className":29870,"style":379},[294],[15,29872,29873,29876],{"style":27535},[15,29874],{"className":29875,"style":303},[302],[15,29877,29879],{"className":29878},[307,308,309,310],[15,29880,29882,29885],{"className":29881},[59,310],[15,29883,23769],{"className":29884,"style":24179},[59,60,310],[15,29886,25763],{"className":29887,"style":25887},[59,60,310],[15,29889,318],{"className":29890},[317],[15,29892,29894],{"className":29893},[290],[15,29895,29897],{"className":29896,"style":7970},[294],[15,29898],{},[15,29900,113],{"className":29901},[146],[15,29903,769],{"className":29904},[59,60],[15,29906,120],{"className":29907},[153],"。因此当 ",[15,29910,29912,29929],{"className":29911,"translate":19},[18],[15,29913,29915],{"className":29914},[23],[25,29916,29917],{"xmlns":27},[29,29918,29919,29927],{},[32,29920,29921,29923,29925],{},[35,29922,206],{},[110,29924,714],{},[35,29926,213],{},[39,29928,719],{"encoding":41},[15,29930,29932,29950],{"className":29931,"ariaHidden":46},[45],[15,29933,29935,29938,29941,29944,29947],{"className":29934},[50],[15,29936],{"className":29937,"style":729},[54],[15,29939,206],{"className":29940,"style":334},[59,60],[15,29942],{"className":29943,"style":158},[157],[15,29945,714],{"className":29946},[162],[15,29948],{"className":29949,"style":158},[157],[15,29951,29953,29956],{"className":29952},[50],[15,29954],{"className":29955,"style":566},[54],[15,29957,213],{"className":29958},[59,60]," 时",[15,29961,29963],{"className":29962,"translate":19},[180],[15,29964,29966,30094],{"className":29965,"translate":19},[18],[15,29967,29969],{"className":29968},[23],[25,29970,29971],{"xmlns":27,"display":189},[29,29972,29973,30091],{},[32,29974,29975,29985,29987,29989,29991,29993,29995,29997,30011,30013,30015,30017,30019,30021,30023,30029,30035,30037,30039,30041,30043,30045,30047,30053,30059,30061,30063,30065,30067,30069,30071,30073,30075,30077,30079,30081,30083,30085,30087,30089],{},[195,29976,29977,29979],{},[35,29978,199],{},[32,29980,29981,29983],{},[35,29982,23769],{},[35,29984,25763],{},[110,29986,113],{"stretchy":112},[35,29988,206],{},[35,29990,210],{"mathvariant":209},[35,29992,213],{},[110,29994,120],{"stretchy":112},[110,29996,123],{},[195,29998,29999,30001],{},[35,30000,199],{},[195,30002,30003,30005],{},[35,30004,37],{},[32,30006,30007,30009],{},[35,30008,23769],{},[35,30010,25763],{},[110,30012,113],{"stretchy":112},[35,30014,206],{},[35,30016,210],{"mathvariant":209},[35,30018,213],{},[110,30020,120],{"stretchy":112},[110,30022,123],{},[238,30024,30025,30027],{},[115,30026,117],{},[115,30028,2465],{},[195,30030,30031,30033],{},[35,30032,199],{},[35,30034,37],{},[110,30036,113],{"stretchy":112},[35,30038,206],{},[35,30040,210],{"mathvariant":209},[35,30042,25593],{},[110,30044,120],{"stretchy":112},[110,30046,2472],{},[238,30048,30049,30051],{},[115,30050,117],{},[115,30052,2465],{},[195,30054,30055,30057],{},[35,30056,199],{},[35,30058,37],{},[110,30060,113],{"stretchy":112},[35,30062,213],{},[35,30064,210],{"mathvariant":209},[35,30066,25593],{},[110,30068,120],{"stretchy":112},[110,30070,551],{"separator":46},[157,30072],{"width":3264},[110,30074,113],{"stretchy":112},[35,30076,37],{},[110,30078,123],{},[35,30080,769],{},[35,30082,4102],{},[110,30084,4105],{},[35,30086,769],{},[110,30088,120],{"stretchy":112},[110,30090,551],{"separator":46},[39,30092,30093],{"encoding":41},"D_{JS}(P\\|Q)=D_{f_{JS}}(P\\|Q)\n=\\frac12D_f(P\\|M)\n+\\frac12D_f(Q\\|M),\\qquad(f=t\\log t),",[15,30095,30097,30173,30293,30425,30572],{"className":30096,"ariaHidden":46},[45],[15,30098,30100,30103,30149,30152,30155,30158,30161,30164,30167,30170],{"className":30099},[50],[15,30101],{"className":30102,"style":139},[54],[15,30104,30106,30109],{"className":30105},[59],[15,30107,199],{"className":30108,"style":277},[59,60],[15,30110,30112],{"className":30111},[281],[15,30113,30115,30141],{"className":30114},[285,286],[15,30116,30118,30138],{"className":30117},[290],[15,30119,30121],{"className":30120,"style":379},[294],[15,30122,30123,30126],{"style":298},[15,30124],{"className":30125,"style":303},[302],[15,30127,30129],{"className":30128},[307,308,309,310],[15,30130,30132,30135],{"className":30131},[59,310],[15,30133,23769],{"className":30134,"style":24179},[59,60,310],[15,30136,25763],{"className":30137,"style":25887},[59,60,310],[15,30139,318],{"className":30140},[317],[15,30142,30144],{"className":30143},[290],[15,30145,30147],{"className":30146,"style":7970},[294],[15,30148],{},[15,30150,113],{"className":30151},[146],[15,30153,206],{"className":30154,"style":334},[59,60],[15,30156,210],{"className":30157},[59],[15,30159,213],{"className":30160},[59,60],[15,30162,120],{"className":30163},[153],[15,30165],{"className":30166,"style":158},[157],[15,30168,123],{"className":30169},[162],[15,30171],{"className":30172,"style":158},[157],[15,30174,30176,30179,30269,30272,30275,30278,30281,30284,30287,30290],{"className":30175},[50],[15,30177],{"className":30178,"style":270},[54],[15,30180,30182,30185],{"className":30181},[59],[15,30183,199],{"className":30184,"style":277},[59,60],[15,30186,30188],{"className":30187},[281],[15,30189,30191,30261],{"className":30190},[285,286],[15,30192,30194,30258],{"className":30193},[290],[15,30195,30197],{"className":30196,"style":295},[294],[15,30198,30199,30202],{"style":298},[15,30200],{"className":30201,"style":303},[302],[15,30203,30205],{"className":30204},[307,308,309,310],[15,30206,30208],{"className":30207},[59,310],[15,30209,30211,30214],{"className":30210},[59,310],[15,30212,37],{"className":30213,"style":61},[59,60,310],[15,30215,30217],{"className":30216},[281],[15,30218,30220,30249],{"className":30219},[285,286],[15,30221,30223,30246],{"className":30222},[290],[15,30224,30226],{"className":30225,"style":6100},[294],[15,30227,30229,30233],{"style":30228},"top:-2.3567em;margin-left:-0.1076em;margin-right:0.0714em;",[15,30230],{"className":30231,"style":30232},[302],"height:2.5em;",[15,30234,30237],{"className":30235},[307,30236,9676,310],"reset-size3",[15,30238,30240,30243],{"className":30239},[59,310],[15,30241,23769],{"className":30242,"style":24179},[59,60,310],[15,30244,25763],{"className":30245,"style":25887},[59,60,310],[15,30247,318],{"className":30248},[317],[15,30250,30252],{"className":30251},[290],[15,30253,30256],{"className":30254,"style":30255},[294],"height:0.1433em;",[15,30257],{},[15,30259,318],{"className":30260},[317],[15,30262,30264],{"className":30263},[290],[15,30265,30267],{"className":30266,"style":325},[294],[15,30268],{},[15,30270,113],{"className":30271},[146],[15,30273,206],{"className":30274,"style":334},[59,60],[15,30276,210],{"className":30277},[59],[15,30279,213],{"className":30280},[59,60],[15,30282,120],{"className":30283},[153],[15,30285],{"className":30286,"style":158},[157],[15,30288,123],{"className":30289},[162],[15,30291],{"className":30292,"style":158},[157],[15,30294,30296,30299,30361,30401,30404,30407,30410,30413,30416,30419,30422],{"className":30295},[50],[15,30297],{"className":30298,"style":25929},[54],[15,30300,30302,30305,30358],{"className":30301},[59],[15,30303],{"className":30304},[146,441],[15,30306,30308],{"className":30307},[238],[15,30309,30311,30350],{"className":30310},[285,286],[15,30312,30314,30347],{"className":30313},[290],[15,30315,30317,30328,30336],{"className":30316,"style":22915},[294],[15,30318,30319,30322],{"style":457},[15,30320],{"className":30321,"style":461},[302],[15,30323,30325],{"className":30324},[59],[15,30326,2465],{"className":30327},[59],[15,30329,30330,30333],{"style":473},[15,30331],{"className":30332,"style":461},[302],[15,30334],{"className":30335,"style":481},[480],[15,30337,30338,30341],{"style":484},[15,30339],{"className":30340,"style":461},[302],[15,30342,30344],{"className":30343},[59],[15,30345,117],{"className":30346},[59],[15,30348,318],{"className":30349},[317],[15,30351,30353],{"className":30352},[290],[15,30354,30356],{"className":30355,"style":3780},[294],[15,30357],{},[15,30359],{"className":30360},[153,441],[15,30362,30364,30367],{"className":30363},[59],[15,30365,199],{"className":30366,"style":277},[59,60],[15,30368,30370],{"className":30369},[281],[15,30371,30373,30393],{"className":30372},[285,286],[15,30374,30376,30390],{"className":30375},[290],[15,30377,30379],{"className":30378,"style":295},[294],[15,30380,30381,30384],{"style":298},[15,30382],{"className":30383,"style":303},[302],[15,30385,30387],{"className":30386},[307,308,309,310],[15,30388,37],{"className":30389,"style":61},[59,60,310],[15,30391,318],{"className":30392},[317],[15,30394,30396],{"className":30395},[290],[15,30397,30399],{"className":30398,"style":325},[294],[15,30400],{},[15,30402,113],{"className":30403},[146],[15,30405,206],{"className":30406,"style":334},[59,60],[15,30408,210],{"className":30409},[59],[15,30411,25593],{"className":30412,"style":25627},[59,60],[15,30414,120],{"className":30415},[153],[15,30417],{"className":30418,"style":2589},[157],[15,30420,2472],{"className":30421},[2593],[15,30423],{"className":30424,"style":2589},[157],[15,30426,30428,30431,30493,30533,30536,30539,30542,30545,30548,30551,30554,30557,30560,30563,30566,30569],{"className":30427},[50],[15,30429],{"className":30430,"style":25929},[54],[15,30432,30434,30437,30490],{"className":30433},[59],[15,30435],{"className":30436},[146,441],[15,30438,30440],{"className":30439},[238],[15,30441,30443,30482],{"className":30442},[285,286],[15,30444,30446,30479],{"className":30445},[290],[15,30447,30449,30460,30468],{"className":30448,"style":22915},[294],[15,30450,30451,30454],{"style":457},[15,30452],{"className":30453,"style":461},[302],[15,30455,30457],{"className":30456},[59],[15,30458,2465],{"className":30459},[59],[15,30461,30462,30465],{"style":473},[15,30463],{"className":30464,"style":461},[302],[15,30466],{"className":30467,"style":481},[480],[15,30469,30470,30473],{"style":484},[15,30471],{"className":30472,"style":461},[302],[15,30474,30476],{"className":30475},[59],[15,30477,117],{"className":30478},[59],[15,30480,318],{"className":30481},[317],[15,30483,30485],{"className":30484},[290],[15,30486,30488],{"className":30487,"style":3780},[294],[15,30489],{},[15,30491],{"className":30492},[153,441],[15,30494,30496,30499],{"className":30495},[59],[15,30497,199],{"className":30498,"style":277},[59,60],[15,30500,30502],{"className":30501},[281],[15,30503,30505,30525],{"className":30504},[285,286],[15,30506,30508,30522],{"className":30507},[290],[15,30509,30511],{"className":30510,"style":295},[294],[15,30512,30513,30516],{"style":298},[15,30514],{"className":30515,"style":303},[302],[15,30517,30519],{"className":30518},[307,308,309,310],[15,30520,37],{"className":30521,"style":61},[59,60,310],[15,30523,318],{"className":30524},[317],[15,30526,30528],{"className":30527},[290],[15,30529,30531],{"className":30530,"style":325},[294],[15,30532],{},[15,30534,113],{"className":30535},[146],[15,30537,213],{"className":30538},[59,60],[15,30540,210],{"className":30541},[59],[15,30543,25593],{"className":30544,"style":25627},[59,60],[15,30546,120],{"className":30547},[153],[15,30549,551],{"className":30550},[573],[15,30552],{"className":30553,"style":3636},[157],[15,30555],{"className":30556,"style":406},[157],[15,30558,113],{"className":30559},[146],[15,30561,37],{"className":30562,"style":61},[59,60],[15,30564],{"className":30565,"style":158},[157],[15,30567,123],{"className":30568},[162],[15,30570],{"className":30571,"style":158},[157],[15,30573,30575,30578,30581,30584,30589,30592,30595,30598],{"className":30574},[50],[15,30576],{"className":30577,"style":139},[54],[15,30579,769],{"className":30580},[59,60],[15,30582],{"className":30583,"style":406},[157],[15,30585,4156,30587],{"className":30586},[1486],[15,30588,4160],{"style":4159},[15,30590],{"className":30591,"style":406},[157],[15,30593,769],{"className":30594},[59,60],[15,30596,120],{"className":30597},[153],[15,30599,551],{"className":30600},[573],[11,30602,30603,30604,30632,30633,30702,30703,30838],{},"两种写法完全一致：JS 既是「两个 f-散度（相对 ",[15,30605,30607,30620],{"className":30606,"translate":19},[18],[15,30608,30610],{"className":30609},[23],[25,30611,30612],{"xmlns":27},[29,30613,30614,30618],{},[32,30615,30616],{},[35,30617,25593],{},[39,30619,25593],{"encoding":41},[15,30621,30623],{"className":30622,"ariaHidden":46},[45],[15,30624,30626,30629],{"className":30625},[50],[15,30627],{"className":30628,"style":664},[54],[15,30630,25593],{"className":30631,"style":25627},[59,60],"）的平均」，也是「一个对称 f-散度」。原文「不是单一 ",[15,30634,30636,30653],{"className":30635,"translate":19},[18],[15,30637,30639],{"className":30638},[23],[25,30640,30641],{"xmlns":27},[29,30642,30643,30651],{},[32,30644,30645],{},[195,30646,30647,30649],{},[35,30648,199],{},[35,30650,37],{},[39,30652,4024],{"encoding":41},[15,30654,30656],{"className":30655,"ariaHidden":46},[45],[15,30657,30659,30662],{"className":30658},[50],[15,30660],{"className":30661,"style":4034},[54],[15,30663,30665,30668],{"className":30664},[59],[15,30666,199],{"className":30667,"style":277},[59,60],[15,30669,30671],{"className":30670},[281],[15,30672,30674,30694],{"className":30673},[285,286],[15,30675,30677,30691],{"className":30676},[290],[15,30678,30680],{"className":30679,"style":295},[294],[15,30681,30682,30685],{"style":298},[15,30683],{"className":30684,"style":303},[302],[15,30686,30688],{"className":30687},[307,308,309,310],[15,30689,37],{"className":30690,"style":61},[59,60,310],[15,30692,318],{"className":30693},[317],[15,30695,30697],{"className":30696},[290],[15,30698,30700],{"className":30699,"style":325},[294],[15,30701],{},"」的说法可修正为：",[998,30704,30705,30706,30756,30757,30837],{},"在 ",[15,30707,30709,30726],{"className":30708,"translate":19},[18],[15,30710,30712],{"className":30711},[23],[25,30713,30714],{"xmlns":27},[29,30715,30716,30724],{},[32,30717,30718,30720,30722],{},[35,30719,206],{},[110,30721,714],{},[35,30723,213],{},[39,30725,719],{"encoding":41},[15,30727,30729,30747],{"className":30728,"ariaHidden":46},[45],[15,30730,30732,30735,30738,30741,30744],{"className":30731},[50],[15,30733],{"className":30734,"style":729},[54],[15,30736,206],{"className":30737,"style":334},[59,60],[15,30739],{"className":30740,"style":158},[157],[15,30742,714],{"className":30743},[162],[15,30745],{"className":30746,"style":158},[157],[15,30748,30750,30753],{"className":30749},[50],[15,30751],{"className":30752,"style":566},[54],[15,30754,213],{"className":30755},[59,60]," 时它是单一 ",[15,30758,30760,30782],{"className":30759,"translate":19},[18],[15,30761,30763],{"className":30762},[23],[25,30764,30765],{"xmlns":27},[29,30766,30767,30779],{},[32,30768,30769],{},[195,30770,30771,30773],{},[35,30772,37],{},[32,30774,30775,30777],{},[35,30776,23769],{},[35,30778,25763],{},[39,30780,30781],{"encoding":41},"f_{JS}",[15,30783,30785],{"className":30784,"ariaHidden":46},[45],[15,30786,30788,30791],{"className":30787},[50],[15,30789],{"className":30790,"style":55},[54],[15,30792,30794,30797],{"className":30793},[59],[15,30795,37],{"className":30796,"style":61},[59,60],[15,30798,30800],{"className":30799},[281],[15,30801,30803,30829],{"className":30802},[285,286],[15,30804,30806,30826],{"className":30805},[290],[15,30807,30809],{"className":30808,"style":379},[294],[15,30810,30811,30814],{"style":27535},[15,30812],{"className":30813,"style":303},[302],[15,30815,30817],{"className":30816},[307,308,309,310],[15,30818,30820,30823],{"className":30819},[59,310],[15,30821,23769],{"className":30822,"style":24179},[59,60,310],[15,30824,25763],{"className":30825,"style":25887},[59,60,310],[15,30827,318],{"className":30828},[317],[15,30830,30832],{"className":30831},[290],[15,30833,30835],{"className":30834,"style":7970},[294],[15,30836],{}," 的 f-散度","；一般情形则用下面的双变量形式。",[11,30840,30841,30844,30845,30886,30887,31002,31003,31006],{},[998,30842,30843],{},"一般情形的双变量形式."," 当 ",[15,30846,30848,30865],{"className":30847,"translate":19},[18],[15,30849,30851],{"className":30850},[23],[25,30852,30853],{"xmlns":27},[29,30854,30855,30863],{},[32,30856,30857,30859,30861],{},[35,30858,206],{},[110,30860,551],{"separator":46},[35,30862,213],{},[39,30864,556],{"encoding":41},[15,30866,30868],{"className":30867,"ariaHidden":46},[45],[15,30869,30871,30874,30877,30880,30883],{"className":30870},[50],[15,30872],{"className":30873,"style":566},[54],[15,30875,206],{"className":30876,"style":334},[59,60],[15,30878,551],{"className":30879},[573],[15,30881],{"className":30882,"style":406},[157],[15,30884,213],{"className":30885},[59,60]," 支撑集可能分离时，",[15,30888,30890,30926],{"className":30889,"translate":19},[18],[15,30891,30893],{"className":30892},[23],[25,30894,30895],{"xmlns":27},[29,30896,30897,30923],{},[32,30898,30899,30901,30903,30905,30915,30917,30919,30921],{},[110,30900,113],{"stretchy":112},[35,30902,769],{},[110,30904,551],{"separator":46},[195,30906,30907,30909],{},[35,30908,37],{},[32,30910,30911,30913],{},[35,30912,23769],{},[35,30914,25763],{},[110,30916,113],{"stretchy":112},[35,30918,769],{},[110,30920,120],{"stretchy":112},[110,30922,120],{"stretchy":112},[39,30924,30925],{"encoding":41},"(t,f_{JS}(t))",[15,30927,30929],{"className":30928,"ariaHidden":46},[45],[15,30930,30932,30935,30938,30941,30944,30947,30993,30996,30999],{"className":30931},[50],[15,30933],{"className":30934,"style":139},[54],[15,30936,113],{"className":30937},[146],[15,30939,769],{"className":30940},[59,60],[15,30942,551],{"className":30943},[573],[15,30945],{"className":30946,"style":406},[157],[15,30948,30950,30953],{"className":30949},[59],[15,30951,37],{"className":30952,"style":61},[59,60],[15,30954,30956],{"className":30955},[281],[15,30957,30959,30985],{"className":30958},[285,286],[15,30960,30962,30982],{"className":30961},[290],[15,30963,30965],{"className":30964,"style":379},[294],[15,30966,30967,30970],{"style":27535},[15,30968],{"className":30969,"style":303},[302],[15,30971,30973],{"className":30972},[307,308,309,310],[15,30974,30976,30979],{"className":30975},[59,310],[15,30977,23769],{"className":30978,"style":24179},[59,60,310],[15,30980,25763],{"className":30981,"style":25887},[59,60,310],[15,30983,318],{"className":30984},[317],[15,30986,30988],{"className":30987},[290],[15,30989,30991],{"className":30990,"style":7970},[294],[15,30992],{},[15,30994,113],{"className":30995},[146],[15,30997,769],{"className":30998},[59,60],[15,31000,19073],{"className":31001},[153]," 的写法会漏掉奇异项。此时把 JS 写成 Csiszár 的",[998,31004,31005],{},"双变量（齐次、联合凸）泛函","最稳妥：",[15,31008,31010],{"className":31009,"translate":19},[180],[15,31011,31013,31070],{"className":31012,"translate":19},[18],[15,31014,31016],{"className":31015},[23],[25,31017,31018],{"xmlns":27,"display":189},[29,31019,31020,31067],{},[32,31021,31022,31032,31034,31036,31038,31040,31042,31044,31046,31049,31051,31053,31055,31057,31059,31061,31063,31065],{},[195,31023,31024,31026],{},[35,31025,199],{},[32,31027,31028,31030],{},[35,31029,23769],{},[35,31031,25763],{},[110,31033,113],{"stretchy":112},[35,31035,206],{},[35,31037,210],{"mathvariant":209},[35,31039,213],{},[110,31041,120],{"stretchy":112},[110,31043,123],{},[110,31045,1289],{},[35,31047,31048],{},"φ",[110,31050,113],{"stretchy":112},[35,31052,11],{},[110,31054,551],{"separator":46},[35,31056,944],{},[110,31058,120],{"stretchy":112},[1299,31060,1301],{},[35,31062,244],{},[35,31064,2366],{},[110,31066,551],{"separator":46},[39,31068,31069],{"encoding":41},"D_{JS}(P\\|Q)=\\int \\varphi(p,q)\\,d\\mu,",[15,31071,31073,31149],{"className":31072,"ariaHidden":46},[45],[15,31074,31076,31079,31125,31128,31131,31134,31137,31140,31143,31146],{"className":31075},[50],[15,31077],{"className":31078,"style":139},[54],[15,31080,31082,31085],{"className":31081},[59],[15,31083,199],{"className":31084,"style":277},[59,60],[15,31086,31088],{"className":31087},[281],[15,31089,31091,31117],{"className":31090},[285,286],[15,31092,31094,31114],{"className":31093},[290],[15,31095,31097],{"className":31096,"style":379},[294],[15,31098,31099,31102],{"style":298},[15,31100],{"className":31101,"style":303},[302],[15,31103,31105],{"className":31104},[307,308,309,310],[15,31106,31108,31111],{"className":31107},[59,310],[15,31109,23769],{"className":31110,"style":24179},[59,60,310],[15,31112,25763],{"className":31113,"style":25887},[59,60,310],[15,31115,318],{"className":31116},[317],[15,31118,31120],{"className":31119},[290],[15,31121,31123],{"className":31122,"style":7970},[294],[15,31124],{},[15,31126,113],{"className":31127},[146],[15,31129,206],{"className":31130,"style":334},[59,60],[15,31132,210],{"className":31133},[59],[15,31135,213],{"className":31136},[59,60],[15,31138,120],{"className":31139},[153],[15,31141],{"className":31142,"style":158},[157],[15,31144,123],{"className":31145},[162],[15,31147],{"className":31148,"style":158},[157],[15,31150,31152,31155,31158,31161,31164,31167,31170,31173,31176,31179,31182,31185,31188,31191],{"className":31151},[50],[15,31153],{"className":31154,"style":5779},[54],[15,31156,1289],{"className":31157,"style":1489},[1486,1487,1488],[15,31159],{"className":31160,"style":406},[157],[15,31162,31048],{"className":31163},[59,60],[15,31165,113],{"className":31166},[146],[15,31168,11],{"className":31169},[59,60],[15,31171,551],{"className":31172},[573],[15,31174],{"className":31175,"style":406},[157],[15,31177,944],{"className":31178,"style":981},[59,60],[15,31180,120],{"className":31181},[153],[15,31183],{"className":31184,"style":406},[157],[15,31186,244],{"className":31187},[59,60],[15,31189,2366],{"className":31190},[59,60],[15,31192,551],{"className":31193},[573],[15,31195,31197],{"className":31196,"translate":19},[180],[15,31198,31200,31284],{"className":31199,"translate":19},[18],[15,31201,31203],{"className":31202},[23],[25,31204,31205],{"xmlns":27,"display":189},[29,31206,31207,31281],{},[32,31208,31209,31211,31213,31215,31217,31219,31221,31223,31229,31231,31233,31235,31237,31253,31255,31257,31259,31261,31277,31279],{},[35,31210,31048],{},[110,31212,113],{"stretchy":112},[35,31214,11],{},[110,31216,551],{"separator":46},[35,31218,944],{},[110,31220,120],{"stretchy":112},[110,31222,123],{},[238,31224,31225,31227],{},[115,31226,117],{},[115,31228,2465],{},[110,31230,230],{"fence":112,"stretchy":46,"minsize":17443,"maxsize":17443},[35,31232,11],{},[35,31234,4102],{},[110,31236,4105],{},[238,31238,31239,31245],{},[32,31240,31241,31243],{},[115,31242,2465],{},[35,31244,11],{},[32,31246,31247,31249,31251],{},[35,31248,11],{},[110,31250,2472],{},[35,31252,944],{},[110,31254,2472],{},[35,31256,944],{},[35,31258,4102],{},[110,31260,4105],{},[238,31262,31263,31269],{},[32,31264,31265,31267],{},[115,31266,2465],{},[35,31268,944],{},[32,31270,31271,31273,31275],{},[35,31272,11],{},[110,31274,2472],{},[35,31276,944],{},[110,31278,257],{"fence":112,"stretchy":46,"minsize":17443,"maxsize":17443},[110,31280,551],{"separator":46},[39,31282,31283],{"encoding":41},"\\varphi(p,q)=\\frac12\\Big[\np\\log\\frac{2p}{p+q}+q\\log\\frac{2q}{p+q}\\Big],",[15,31285,31287,31323,31498],{"className":31286,"ariaHidden":46},[45],[15,31288,31290,31293,31296,31299,31302,31305,31308,31311,31314,31317,31320],{"className":31289},[50],[15,31291],{"className":31292,"style":139},[54],[15,31294,31048],{"className":31295},[59,60],[15,31297,113],{"className":31298},[146],[15,31300,11],{"className":31301},[59,60],[15,31303,551],{"className":31304},[573],[15,31306],{"className":31307,"style":406},[157],[15,31309,944],{"className":31310,"style":981},[59,60],[15,31312,120],{"className":31313},[153],[15,31315],{"className":31316,"style":158},[157],[15,31318,123],{"className":31319},[162],[15,31321],{"className":31322,"style":158},[157],[15,31324,31326,31330,31392,31398,31401,31404,31409,31412,31489,31492,31495],{"className":31325},[50],[15,31327],{"className":31328,"style":31329},[54],"height:2.2019em;vertical-align:-0.8804em;",[15,31331,31333,31336,31389],{"className":31332},[59],[15,31334],{"className":31335},[146,441],[15,31337,31339],{"className":31338},[238],[15,31340,31342,31381],{"className":31341},[285,286],[15,31343,31345,31378],{"className":31344},[290],[15,31346,31348,31359,31367],{"className":31347,"style":22915},[294],[15,31349,31350,31353],{"style":457},[15,31351],{"className":31352,"style":461},[302],[15,31354,31356],{"className":31355},[59],[15,31357,2465],{"className":31358},[59],[15,31360,31361,31364],{"style":473},[15,31362],{"className":31363,"style":461},[302],[15,31365],{"className":31366,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联合凸且一次齐次。在 ",[15,31689,31691,31709],{"className":31690,"translate":19},[18],[15,31692,31694],{"className":31693},[23],[25,31695,31696],{"xmlns":27},[29,31697,31698,31706],{},[32,31699,31700,31702,31704],{},[35,31701,944],{},[110,31703,2905],{},[115,31705,126],{},[39,31707,31708],{"encoding":41},"q>0",[15,31710,31712,31731],{"className":31711,"ariaHidden":46},[45],[15,31713,31715,31719,31722,31725,31728],{"className":31714},[50],[15,31716],{"className":31717,"style":31718},[54],"height:0.7335em;vertical-align:-0.1944em;",[15,31720,944],{"className":31721,"style":981},[59,60],[15,31723],{"className":31724,"style":158},[157],[15,31726,2905],{"className":31727},[162],[15,31729],{"className":31730,"style":158},[157],[15,31732,31734,31737],{"className":31733},[50],[15,31735],{"className":31736,"style":172},[54],[15,31738,126],{"className":31739},[59]," 处 ",[15,31742,31744,31792],{"className":31743,"translate":19},[18],[15,31745,31747],{"className":31746},[23],[25,31748,31749],{"xmlns":27},[29,31750,31751,31789],{},[32,31752,31753,31755,31757,31759,31761,31763,31765,31767,31769,31771,31773,31783,31785,31787],{},[35,31754,31048],{},[110,31756,113],{"stretchy":112},[35,31758,769],{},[35,31760,944],{},[110,31762,551],{"separator":46},[35,31764,944],{},[110,31766,120],{"stretchy":112},[110,31768,123],{},[35,31770,944],{},[1299,31772,1301],{},[195,31774,31775,31777],{},[35,31776,37],{},[32,31778,31779,31781],{},[35,31780,23769],{},[35,31782,25763],{},[110,31784,113],{"stretchy":112},[35,31786,769],{},[110,31788,120],{"stretchy":112},[39,31790,31791],{"encoding":41},"\\varphi(tq,q)=q\\,f_{JS}(t)",[15,31793,31795,31831],{"className":31794,"ariaHidden":46},[45],[15,31796,31798,31801,31804,31807,31810,31813,31816,31819,31822,31825,31828],{"className":31797},[50],[15,31799],{"className":31800,"style":139},[54],[15,31802,31048],{"className":31803},[59,60],[15,31805,113],{"className":31806},[146],[15,31808,26304],{"className":31809,"style":981},[59,60],[15,31811,551],{"className":31812},[573],[15,31814],{"className":31815,"style":406},[157],[15,31817,944],{"className":31818,"style":981},[59,60],[15,31820,120],{"className":31821},[153],[15,31823],{"className":31824,"style":158},[157],[15,31826,123],{"className":31827},[162],[15,31829],{"className":31830,"style":158},[157],[15,31832,31834,31837,31840,31843,31889,31892,31895],{"className":31833},[50],[15,31835],{"className":31836,"style":139},[54],[15,31838,944],{"className":31839,"style":981},[59,60],[15,31841],{"className":31842,"style":406},[157],[15,31844,31846,31849],{"className":31845},[59],[15,31847,37],{"className":31848,"style":61},[59,60],[15,31850,31852],{"className":31851},[281],[15,31853,31855,31881],{"className":31854},[285,286],[15,31856,31858,31878],{"className":31857},[290],[15,31859,31861],{"className":31860,"style":379},[294],[15,31862,31863,31866],{"style":27535},[15,31864],{"className":31865,"style":303},[302],[15,31867,31869],{"className":31868},[307,308,309,310],[15,31870,31872,31875],{"className":31871},[59,310],[15,31873,23769],{"className":31874,"style":24179},[59,60,310],[15,31876,25763],{"className":31877,"style":25887},[59,60,310],[15,31879,318],{"className":31880},[317],[15,31882,31884],{"className":31883},[290],[15,31885,31887],{"className":31886,"style":7970},[294],[15,31888],{},[15,31890,113],{"className":31891},[146],[15,31893,769],{"className":31894},[59,60],[15,31896,120],{"className":31897},[153],"，两者一致；在 ",[15,31900,31902,31928],{"className":31901,"translate":19},[18],[15,31903,31905],{"className":31904},[23],[25,31906,31907],{"xmlns":27},[29,31908,31909,31925],{},[32,31910,31911,31913,31915,31917,31919,31921,31923],{},[35,31912,944],{},[110,31914,123],{},[115,31916,126],{},[110,31918,551],{"separator":46},[35,31920,11],{},[110,31922,2905],{},[115,31924,126],{},[39,31926,31927],{"encoding":41},"q=0,p>0",[15,31929,31931,31949,31976],{"className":31930,"ariaHidden":46},[45],[15,31932,31934,31937,31940,31943,31946],{"className":31933},[50],[15,31935],{"className":31936,"style":2278},[54],[15,31938,944],{"className":31939,"style":981},[59,60],[15,31941],{"className":31942,"style":158},[157],[15,31944,123],{"className":31945},[162],[15,31947],{"className":31948,"style":158},[157],[15,31950,31952,31955,31958,31961,31964,31967,31970,31973],{"className":31951},[50],[15,31953],{"className":31954,"style":9360},[54],[15,31956,126],{"className":31957},[59],[15,31959,551],{"className":31960},[573],[15,31962],{"className":31963,"style":406},[157],[15,31965,11],{"className":31966},[59,60],[15,31968],{"className":31969,"style":158},[157],[15,31971,2905],{"className":31972},[162],[15,31974],{"className":31975,"style":158},[157],[15,31977,31979,31982],{"className":31978},[50],[15,31980],{"className":31981,"style":172},[54],[15,31983,126],{"className":31984},[59]," 处奇异项 ",[15,31987,31989,32015],{"className":31988,"translate":19},[18],[15,31990,31992],{"className":31991},[23],[25,31993,31994],{"xmlns":27},[29,31995,31996,32012],{},[32,31997,31998,32004,32006,32008,32010],{},[238,31999,32000,32002],{},[115,32001,117],{},[115,32003,2465],{},[35,32005,11],{},[35,32007,4102],{},[110,32009,4105],{},[115,32011,2465],{},[39,32013,32014],{"encoding":41},"\\frac12 p\\log 2",[15,32016,32018],{"className":32017,"ariaHidden":46},[45],[15,32019,32021,32024,32092,32095,32098,32103,32106],{"className":32020},[50],[15,32022],{"className":32023,"style":2507},[54],[15,32025,32027,32030,32089],{"className":32026},[59],[15,32028],{"className":32029},[146,441],[15,32031,32033],{"className":32032},[238],[15,32034,32036,32081],{"className":32035},[285,286],[15,32037,32039,32078],{"className":32038},[290],[15,32040,32042,32056,32064],{"className":32041,"style":2526},[294],[15,32043,32044,32047],{"style":2529},[15,32045],{"className":32046,"style":461},[302],[15,32048,32050],{"className":32049},[307,308,309,310],[15,32051,32053],{"className":32052},[59,310],[15,32054,2465],{"className":32055},[59,310],[15,32057,32058,32061],{"style":473},[15,32059],{"className":32060,"style":461},[302],[15,32062],{"className":32063,"style":481},[480],[15,32065,32066,32069],{"style":2552},[15,32067],{"className":32068,"style":461},[302],[15,32070,32072],{"className":32071},[307,308,309,310],[15,32073,32075],{"className":32074},[59,310],[15,32076,117],{"className":32077},[59,310],[15,32079,318],{"className":32080},[317],[15,32082,32084],{"className":32083},[290],[15,32085,32087],{"className":32086,"style":2574},[294],[15,32088],{},[15,32090],{"className":32091},[153,441],[15,32093,11],{"className":32094},[59,60],[15,32096],{"className":32097,"style":406},[157],[15,32099,4156,32101],{"className":32100},[1486],[15,32102,4160],{"style":4159},[15,32104],{"className":32105,"style":406},[157],[15,32107,2465],{"className":32108},[59]," 会被保留下来（这正是 1.1 节讨论的「支撑不重叠」约定）。",[11,32111,32112,32115,32116,32195,32196,15799,32199,32250,32251,32484,32485,32513,32514,32542,32543,32731,32732,32827],{},[998,32113,32114],{},"有界性。"," 注意 ",[15,32117,32119,32140],{"className":32118,"translate":19},[18],[15,32120,32122],{"className":32121},[23],[25,32123,32124],{"xmlns":27},[29,32125,32126,32138],{},[32,32127,32128],{},[195,32129,32130,32132],{},[35,32131,37],{},[32,32133,32134,32136],{},[35,32135,23769],{},[35,32137,25763],{},[39,32139,30781],{"encoding":41},[15,32141,32143],{"className":32142,"ariaHidden":46},[45],[15,32144,32146,32149],{"className":32145},[50],[15,32147],{"className":32148,"style":55},[54],[15,32150,32152,32155],{"className":32151},[59],[15,32153,37],{"className":32154,"style":61},[59,60],[15,32156,32158],{"className":32157},[281],[15,32159,32161,32187],{"className":32160},[285,286],[15,32162,32164,32184],{"className":32163},[290],[15,32165,32167],{"className":32166,"style":379},[294],[15,32168,32169,32172],{"style":27535},[15,32170],{"className":32171,"style":303},[302],[15,32173,32175],{"className":32174},[307,308,309,310],[15,32176,32178,32181],{"className":32177},[59,310],[15,32179,23769],{"className":32180,"style":24179},[59,60,310],[15,32182,25763],{"className":32183,"style":25887},[59,60,310],[15,32185,318],{"className":32186},[317],[15,32188,32190],{"className":32189},[290],[15,32191,32193],{"className":32192,"style":7970},[294],[15,32194],{}," 逐点",[998,32197,32198],{},"无界",[15,32200,32202,32220],{"className":32201,"translate":19},[18],[15,32203,32205],{"className":32204},[23],[25,32206,32207],{"xmlns":27},[29,32208,32209,32217],{},[32,32210,32211,32213,32215],{},[35,32212,769],{},[110,32214,1032],{},[35,32216,1027],{"mathvariant":209},[39,32218,32219],{"encoding":41},"t\\to\\infty",[15,32221,32223,32241],{"className":32222,"ariaHidden":46},[45],[15,32224,32226,32229,32232,32235,32238],{"className":32225},[50],[15,32227],{"className":32228,"style":5345},[54],[15,32230,769],{"className":32231},[59,60],[15,32233],{"className":32234,"style":158},[157],[15,32236,1032],{"className":32237},[162],[15,32239],{"className":32240,"style":158},[157],[15,32242,32244,32247],{"className":32243},[50],[15,32245],{"className":32246,"style":2347},[54],[15,32248,1027],{"className":32249},[59]," 时 ",[15,32252,32254,32302],{"className":32253,"translate":19},[18],[15,32255,32257],{"className":32256},[23],[25,32258,32259],{"xmlns":27},[29,32260,32261,32299],{},[32,32262,32263,32273,32275,32277,32279,32281,32287,32289,32291,32293,32295,32297],{},[195,32264,32265,32267],{},[35,32266,37],{},[32,32268,32269,32271],{},[35,32270,23769],{},[35,32272,25763],{},[110,32274,113],{"stretchy":112},[35,32276,769],{},[110,32278,120],{"stretchy":112},[110,32280,8871],{},[238,32282,32283,32285],{},[115,32284,117],{},[115,32286,2465],{},[35,32288,769],{},[35,32290,4102],{},[110,32292,4105],{},[115,32294,2465],{},[110,32296,1032],{},[35,32298,1027],{"mathvariant":209},[39,32300,32301],{"encoding":41},"f_{JS}(t)\\sim\\frac12 t\\log 2\\to\\infty",[15,32303,32305,32375,32475],{"className":32304,"ariaHidden":46},[45],[15,32306,32308,32311,32357,32360,32363,32366,32369,32372],{"className":32307},[50],[15,32309],{"className":32310,"style":139},[54],[15,32312,32314,32317],{"className":32313},[59],[15,32315,37],{"className":32316,"style":61},[59,60],[15,32318,32320],{"className":32319},[281],[15,32321,32323,32349],{"className":32322},[285,286],[15,32324,32326,32346],{"className":32325},[290],[15,32327,32329],{"className":32328,"style":379},[294],[15,32330,32331,32334],{"style":27535},[15,32332],{"className":32333,"style":303},[302],[15,32335,32337],{"className":32336},[307,308,309,310],[15,32338,32340,32343],{"className":32339},[59,310],[15,32341,23769],{"className":32342,"style":24179},[59,60,310],[15,32344,25763],{"className":32345,"style":25887},[59,60,310],[15,32347,318],{"className":32348},[317],[15,32350,32352],{"className":32351},[290],[15,32353,32355],{"className":32354,"style":7970},[294],[15,32356],{},[15,32358,113],{"className":32359},[146],[15,32361,769],{"className":32362},[59,60],[15,32364,120],{"className":32365},[153],[15,32367],{"className":32368,"style":158},[157],[15,32370,8871],{"className":32371},[162],[15,32373],{"className":32374,"style":158},[157],[15,32376,32378,32381,32449,32452,32455,32460,32463,32466,32469,32472],{"className":32377},[50],[15,32379],{"className":32380,"style":2507},[54],[15,32382,32384,32387,32446],{"className":32383},[59],[15,32385],{"className":32386},[146,441],[15,32388,32390],{"className":32389},[238],[15,32391,32393,32438],{"className":32392},[285,286],[15,32394,32396,32435],{"className":32395},[290],[15,32397,32399,32413,32421],{"className":32398,"style":2526},[294],[15,32400,32401,32404],{"style":2529},[15,32402],{"className":32403,"style":461},[302],[15,32405,32407],{"className":32406},[307,308,309,310],[15,32408,32410],{"className":32409},[59,310],[15,32411,2465],{"className":32412},[59,310],[15,32414,32415,32418],{"style":473},[15,32416],{"className":32417,"style":461},[302],[15,32419],{"className":32420,"style":481},[480],[15,32422,32423,32426],{"style":2552},[15,32424],{"className":32425,"style":461},[302],[15,32427,32429],{"className":32428},[307,308,309,310],[15,32430,32432],{"className":32431},[59,310],[15,32433,117],{"className":32434},[59,310],[15,32436,318],{"className":32437},[317],[15,32439,32441],{"className":32440},[290],[15,32442,32444],{"className":32443,"style":2574},[294],[15,32445],{},[15,32447],{"className":32448},[153,441],[15,32450,769],{"className":32451},[59,60],[15,32453],{"className":32454,"style":406},[157],[15,32456,4156,32458],{"className":32457},[1486],[15,32459,4160],{"style":4159},[15,32461],{"className":32462,"style":406},[157],[15,32464,2465],{"className":32465},[59],[15,32467],{"className":32468,"style":158},[157],[15,32470,1032],{"className":32471},[162],[15,32473],{"className":32474,"style":158},[157],[15,32476,32478,32481],{"className":32477},[50],[15,32479],{"className":32480,"style":2347},[54],[15,32482,1027],{"className":32483},[59],"），但积分有界，因为 ",[15,32486,32488,32501],{"className":32487,"translate":19},[18],[15,32489,32491],{"className":32490},[23],[25,32492,32493],{"xmlns":27},[29,32494,32495,32499],{},[32,32496,32497],{},[35,32498,769],{},[39,32500,769],{"encoding":41},[15,32502,32504],{"className":32503,"ariaHidden":46},[45],[15,32505,32507,32510],{"className":32506},[50],[15,32508],{"className":32509,"style":5345},[54],[15,32511,769],{"className":32512},[59,60]," 大的地方 ",[15,32515,32517,32530],{"className":32516,"translate":19},[18],[15,32518,32520],{"className":32519},[23],[25,32521,32522],{"xmlns":27},[29,32523,32524,32528],{},[32,32525,32526],{},[35,32527,944],{},[39,32529,944],{"encoding":41},[15,32531,32533],{"className":32532,"ariaHidden":46},[45],[15,32534,32536,32539],{"className":32535},[50],[15,32537],{"className":32538,"style":2278},[54],[15,32540,944],{"className":32541,"style":981},[59,60]," 小、权重趋于 0。严格地用两 KL 形式：",[15,32544,32546,32604],{"className":32545,"translate":19},[18],[15,32547,32549],{"className":32548},[23],[25,32550,32551],{"xmlns":27},[29,32552,32553,32601],{},[32,32554,32555,32557,32559,32561,32563,32565,32567,32569,32571,32573,32575,32577,32583,32585,32587,32589,32591,32593,32595,32597,32599],{},[115,32556,2465],{},[35,32558,11],{},[35,32560,941],{"mathvariant":209},[110,32562,113],{"stretchy":112},[35,32564,11],{},[110,32566,2472],{},[35,32568,944],{},[110,32570,120],{"stretchy":112},[110,32572,9954],{},[115,32574,2465],{},[110,32576,10994],{},[32,32578,32579,32581],{},[35,32580,10698],{"mathvariant":209},[35,32582,21919],{"mathvariant":209},[110,32584,113],{"stretchy":112},[35,32586,206],{},[35,32588,210],{"mathvariant":209},[35,32590,25593],{},[110,32592,120],{"stretchy":112},[110,32594,9954],{},[35,32596,4102],{},[110,32598,4105],{},[115,32600,2465],{},[39,32602,32603],{"encoding":41},"2p\u002F(p+q)\\le 2\\Rightarrow \\mathrm{KL}(P\\|M)\\le\\log2",[15,32605,32607,32637,32658,32676,32714],{"className":32606,"ariaHidden":46},[45],[15,32608,32610,32613,32616,32619,32622,32625,32628,32631,32634],{"className":32609},[50],[15,32611],{"className":32612,"style":139},[54],[15,32614,2465],{"className":32615},[59],[15,32617,11],{"className":32618},[59,60],[15,32620,941],{"className":32621},[59],[15,32623,113],{"className":32624},[146],[15,32626,11],{"className":32627},[59,60],[15,32629],{"className":32630,"style":2589},[157],[15,32632,2472],{"className":32633},[2593],[15,32635],{"className":32636,"style":2589},[157],[15,32638,32640,32643,32646,32649,32652,32655],{"className":32639},[50],[15,32641],{"className":32642,"style":139},[54],[15,32644,944],{"className":32645,"style":981},[59,60],[15,32647,120],{"className":32648},[153],[15,32650],{"className":32651,"style":158},[157],[15,32653,9954],{"className":32654},[162],[15,32656],{"className":32657,"style":158},[157],[15,32659,32661,32664,32667,32670,32673],{"className":32660},[50],[15,32662],{"className":32663,"style":172},[54],[15,32665,2465],{"className":32666},[59],[15,32668],{"className":32669,"style":158},[157],[15,32671,10994],{"className":32672},[162],[15,32674],{"className":32675,"style":158},[157],[15,32677,32679,32682,32690,32693,32696,32699,32702,32705,32708,32711],{"className":32678},[50],[15,32680],{"className":32681,"style":139},[54],[15,32683,32685],{"className":32684},[59],[15,32686,32689],{"className":32687},[59,32688],"mathrm","KL",[15,32691,113],{"className":32692},[146],[15,32694,206],{"className":32695,"style":334},[59,60],[15,32697,210],{"className":32698},[59],[15,32700,25593],{"className":32701,"style":25627},[59,60],[15,32703,120],{"className":32704},[153],[15,32706],{"className":32707,"style":158},[157],[15,32709,9954],{"className":32710},[162],[15,32712],{"className":32713,"style":158},[157],[15,32715,32717,32720,32725,32728],{"className":32716},[50],[15,32718],{"className":32719,"style":55},[54],[15,32721,4156,32723],{"className":32722},[1486],[15,32724,4160],{"style":4159},[15,32726],{"className":32727,"style":406},[157],[15,32729,2465],{"className":32730},[59],"，同理 ",[15,32733,32735,32771],{"className":32734,"translate":19},[18],[15,32736,32738],{"className":32737},[23],[25,32739,32740],{"xmlns":27},[29,32741,32742,32768],{},[32,32743,32744,32750,32752,32754,32756,32758,32760,32762,32764,32766],{},[32,32745,32746,32748],{},[35,32747,10698],{"mathvariant":209},[35,32749,21919],{"mathvariant":209},[110,32751,113],{"stretchy":112},[35,32753,213],{},[35,32755,210],{"mathvariant":209},[35,32757,25593],{},[110,32759,120],{"stretchy":112},[110,32761,9954],{},[35,32763,4102],{},[110,32765,4105],{},[115,32767,2465],{},[39,32769,32770],{"encoding":41},"\\mathrm{KL}(Q\\|M)\\le\\log2",[15,32772,32774,32810],{"className":32773,"ariaHidden":46},[45],[15,32775,32777,32780,32786,32789,32792,32795,32798,32801,32804,32807],{"className":32776},[50],[15,32778],{"className":32779,"style":139},[54],[15,32781,32783],{"className":32782},[59],[15,32784,32689],{"className":32785},[59,32688],[15,32787,113],{"className":32788},[146],[15,32790,213],{"className":32791},[59,60],[15,32793,210],{"className":32794},[59],[15,32796,25593],{"className":32797,"style":25627},[59,60],[15,32799,120],{"className":32800},[153],[15,32802],{"className":32803,"style":158},[157],[15,32805,9954],{"className":32806},[162],[15,32808],{"className":32809,"style":158},[157],[15,32811,32813,32816,32821,32824],{"className":32812},[50],[15,32814],{"className":32815,"style":55},[54],[15,32817,4156,32819],{"className":32818},[1486],[15,32820,4160],{"style":4159},[15,32822],{"className":32823,"style":406},[157],[15,32825,2465],{"className":32826},[59],"，故",[15,32829,32831],{"className":32830,"translate":19},[180],[15,32832,32834,32992],{"className":32833,"translate":19},[18],[15,32835,32837],{"className":32836},[23],[25,32838,32839],{"xmlns":27,"display":189},[29,32840,32841,32989],{},[1245,32842,32843,32931],{"rowspacing":1247,"columnalign":1248,"columnspacing":1249},[1251,32844,32845,32875],{},[1254,32846,32847],{},[1257,32848,32849],{"scriptlevel":126,"displaystyle":46},[32,32850,32851,32853,32855,32865,32867,32869,32871,32873],{},[115,32852,126],{},[110,32854,9954],{},[195,32856,32857,32859],{},[35,32858,199],{},[32,32860,32861,32863],{},[35,32862,23769],{},[35,32864,25763],{},[110,32866,113],{"stretchy":112},[35,32868,206],{},[35,32870,210],{"mathvariant":209},[35,32872,213],{},[110,32874,120],{"stretchy":112},[1254,32876,32877],{},[1257,32878,32879],{"scriptlevel":126,"displaystyle":46},[32,32880,32881,32883,32885,32891,32897,32899,32901,32903,32905,32907,32909,32915,32921,32923,32925,32927,32929],{},[32,32882],{},[110,32884,123],{},[238,32886,32887,32889],{},[115,32888,117],{},[115,32890,2465],{},[32,32892,32893,32895],{},[35,32894,10698],{"mathvariant":209},[35,32896,21919],{"mathvariant":209},[110,32898,113],{"stretchy":112},[35,32900,206],{},[35,32902,210],{"mathvariant":209},[35,32904,25593],{},[110,32906,120],{"stretchy":112},[110,32908,2472],{},[238,32910,32911,32913],{},[115,32912,117],{},[115,32914,2465],{},[32,32916,32917,32919],{},[35,32918,10698],{"mathvariant":209},[35,32920,21919],{"mathvariant":209},[110,32922,113],{"stretchy":112},[35,32924,213],{},[35,32926,210],{"mathvariant":209},[35,32928,25593],{},[110,32930,120],{"stretchy":112},[1251,32932,32933,32939],{},[1254,32934,32935],{},[1257,32936,32937],{"scriptlevel":126,"displaystyle":46},[32,32938],{},[1254,32940,32941],{},[1257,32942,32943],{"scriptlevel":126,"displaystyle":46},[32,32944,32945,32947,32949,32955,32957,32959,32961,32963,32969,32971,32973,32975,32977,32979,32981,32983,32986],{},[32,32946],{},[110,32948,9954],{},[238,32950,32951,32953],{},[115,32952,117],{},[115,32954,2465],{},[35,32956,4102],{},[110,32958,4105],{},[115,32960,2465],{},[110,32962,2472],{},[238,32964,32965,32967],{},[115,32966,117],{},[115,32968,2465],{},[35,32970,4102],{},[110,32972,4105],{},[115,32974,2465],{},[110,32976,123],{},[35,32978,4102],{},[110,32980,4105],{},[115,32982,2465],{},[110,32984,32985],{},"≈",[115,32987,32988],{},"0.693.",[39,32990,32991],{"encoding":41},"\\begin{aligned}\n0\\le D_{JS}(P\\|Q)\n&=\\frac12\\mathrm{KL}(P\\|M)\n+\\frac12\\mathrm{KL}(Q\\|M)\\\\[2pt]\n&\\le \\frac12\\log 2+\\frac12\\log 2\n=\\log 2\\approx 0.693 .\n\\end{aligned}",[15,32993,32995],{"className":32994,"ariaHidden":46},[45],[15,32996,32998,33002],{"className":32997},[50],[15,32999],{"className":33000,"style":33001},[54],"height:4.3149em;vertical-align:-1.9074em;",[15,33003,33005],{"className":33004},[59],[15,33006,33008,33123],{"className":33007},[1245],[15,33009,33011],{"className":33010},[1362],[15,33012,33014,33114],{"className":33013},[285,286],[15,33015,33017,33111],{"className":33016},[290],[15,33018,33021,33103],{"className":33019,"style":33020},[294],"height:2.4074em;",[15,33022,33024,33027],{"style":33023},"top:-4.4074em;",[15,33025],{"className":33026,"style":22692},[302],[15,33028,33030,33033,33036,33039,33042,33088,33091,33094,33097,33100],{"className":33029},[59],[15,33031,126],{"className":33032},[59],[15,33034],{"className":33035,"style":158},[157],[15,33037,9954],{"className":33038},[162],[15,33040],{"className":33041,"style":158},[157],[15,33043,33045,33048],{"className":33044},[59],[15,33046,199],{"className":33047,"style":277},[59,60],[15,33049,33051],{"className":33050},[281],[15,33052,33054,33080],{"className":33053},[285,286],[15,33055,33057,33077],{"className":33056},[290],[15,33058,33060],{"className":33059,"style":379},[294],[15,33061,33062,33065],{"style":298},[15,33063],{"className":33064,"style":303},[302],[15,33066,33068],{"className":33067},[307,308,309,310],[15,33069,33071,33074],{"className":33070},[59,310],[15,33072,23769],{"className":33073,"style":24179},[59,60,310],[15,33075,25763],{"className":33076,"style":25887},[59,60,310],[15,33078,318],{"className":33079},[317],[15,33081,33083],{"className":33082},[290],[15,33084,33086],{"className":33085,"style":7970},[294],[15,33087],{},[15,33089,113],{"className":33090},[146],[15,33092,206],{"className":33093,"style":334},[59,60],[15,33095,210],{"className":33096},[59],[15,33098,213],{"className":33099},[59,60],[15,33101,120],{"className":33102},[153],[15,33104,33105,33108],{"style":1905},[15,33106],{"className":33107,"style":22692},[302],[15,33109],{"className":33110},[59],[15,33112,318],{"className":33113},[317],[15,33115,33117],{"className":33116},[290],[15,33118,33121],{"className":33119,"style":33120},[294],"height:1.9074em;",[15,33122],{},[15,33124,33126],{"className":33125},[1453],[15,33127,33129,33546],{"className":33128},[285,286],[15,33130,33132,33543],{"className":33131},[290],[15,33133,33135,33330],{"className":33134,"style":33020},[294],[15,33136,33137,33140],{"style":33023},[15,33138],{"className":33139,"style":22692},[302],[15,33141,33143,33146,33149,33152,33155,33217,33223,33226,33229,33232,33235,33238,33241,33244,33247,33309,33315,33318,33321,33324,33327],{"className":33142},[59],[15,33144],{"className":33145},[59],[15,33147],{"className":33148,"style":158},[157],[15,33150,123],{"className":33151},[162],[15,33153],{"className":33154,"style":158},[157],[15,33156,33158,33161,33214],{"className":33157},[59],[15,33159],{"className":33160},[146,441],[15,33162,33164],{"className":33163},[238],[15,33165,33167,33206],{"className":33166},[285,286],[15,33168,33170,33203],{"className":33169},[290],[15,33171,33173,33184,33192],{"className":33172,"style":22915},[294],[15,33174,33175,33178],{"style":457},[15,33176],{"className":33177,"style":461},[302],[15,33179,33181],{"className":33180},[59],[15,33182,2465],{"className":33183},[59],[15,33185,33186,33189],{"style":473},[15,33187],{"className":33188,"style":461},[302],[15,33190],{"className":33191,"style":481},[480],[15,33193,33194,33197],{"style":484},[15,33195],{"className":33196,"style":461},[302],[15,33198,33200],{"className":33199},[59],[15,33201,117],{"className":33202},[59],[15,33204,318],{"className":33205},[317],[15,33207,33209],{"className":33208},[290],[15,33210,33212],{"className":33211,"style":3780},[294],[15,33213],{},[15,33215],{"className":33216},[153,441],[15,33218,33220],{"className":33219},[59],[15,33221,32689],{"className":33222},[59,32688],[15,33224,113],{"className":33225},[146],[15,33227,206],{"className":33228,"style":334},[59,60],[15,33230,210],{"className":33231},[59],[15,33233,25593],{"className":33234,"style":25627},[59,60],[15,33236,120],{"className":33237},[153],[15,33239],{"className":33240,"style":2589},[157],[15,33242,2472],{"className":33243},[2593],[15,33245],{"className":33246,"style":2589},[157],[15,33248,33250,33253,33306],{"className":33249},[59],[15,33251],{"className":33252},[146,441],[15,33254,33256],{"className":33255},[238],[15,33257,33259,33298],{"className":33258},[285,286],[15,33260,33262,33295],{"className":33261},[290],[15,33263,33265,33276,33284],{"className":33264,"style":22915},[294],[15,33266,33267,33270],{"style":457},[15,33268],{"className":33269,"style":461},[302],[15,33271,33273],{"className":33272},[59],[15,33274,2465],{"className":33275},[59],[15,33277,33278,33281],{"style":473},[15,33279],{"className":33280,"style":461},[302],[15,33282],{"className":33283,"style":481},[480],[15,33285,33286,33289],{"style":484},[15,33287],{"className":33288,"style":461},[302],[15,33290,33292],{"className":33291},[59],[15,33293,117],{"className":33294},[59],[15,33296,318],{"className":33297},[317],[15,33299,33301],{"className":33300},[290],[15,33302,33304],{"className":33303,"style":3780},[294],[15,33305],{},[15,33307],{"className":33308},[153,441],[15,33310,33312],{"className":33311},[59],[15,33313,32689],{"className":33314},[59,32688],[15,33316,113],{"className":33317},[146],[15,33319,213],{"className":33320},[59,60],[15,33322,210],{"className":33323},[59],[15,33325,25593],{"className":33326,"style":25627},[59,60],[15,33328,120],{"className":33329},[153],[15,33331,33332,33335],{"style":1905},[15,33333],{"className":33334,"style":22692},[302],[15,33336,33338,33341,33344,33347,33350,33412,33415,33420,33423,33426,33429,33432,33435,33497,33500,33505,33508,33511,33514,33517,33520,33525,33528,33531,33534,33537,33540],{"className":33337},[59],[15,33339],{"className":33340},[59],[15,33342],{"className":33343,"style":158},[157],[15,33345,9954],{"className":33346},[162],[15,33348],{"className":33349,"style":158},[157],[15,33351,33353,33356,33409],{"className":33352},[59],[15,33354],{"className":33355},[146,441],[15,33357,33359],{"className":33358},[238],[15,33360,33362,33401],{"className":33361},[285,286],[15,33363,33365,33398],{"className":33364},[290],[15,33366,33368,33379,33387],{"className":33367,"style":22915},[294],[15,33369,33370,33373],{"style":457},[15,33371],{"className":33372,"style":461},[302],[15,33374,33376],{"className":33375},[59],[15,33377,2465],{"className":33378},[59],[15,33380,33381,33384],{"style":473},[15,33382],{"className":33383,"style":461},[302],[15,33385],{"className":33386,"style":481},[480],[15,33388,33389,33392],{"style":484},[15,33390],{"className":33391,"style":461},[302],[15,33393,33395],{"className":33394},[59],[15,33396,117],{"className":33397},[59],[15,33399,318],{"className":33400},[317],[15,33402,33404],{"className":33403},[290],[15,33405,33407],{"className":33406,"style":3780},[294],[15,33408],{},[15,33410],{"className":33411},[153,441],[15,33413],{"className":33414,"style":406},[157],[15,33416,4156,33418],{"className":33417},[1486],[15,33419,4160],{"style":4159},[15,33421],{"className":33422,"style":406},[157],[15,33424,2465],{"className":33425},[59],[15,33427],{"className":33428,"style":2589},[157],[15,33430,2472],{"className":33431},[2593],[15,33433],{"className":33434,"style":2589},[157],[15,33436,33438,33441,33494],{"className":33437},[59],[15,33439],{"className":33440},[146,441],[15,33442,33444],{"className":33443},[238],[15,33445,33447,33486],{"className":33446},[285,286],[15,33448,33450,33483],{"className":33449},[290],[15,33451,33453,33464,33472],{"className":33452,"style":22915},[294],[15,33454,33455,33458],{"style":457},[15,33456],{"className":33457,"style":461},[302],[15,33459,33461],{"className":33460},[59],[15,33462,2465],{"className":33463},[59],[15,33465,33466,33469],{"style":473},[15,33467],{"className":33468,"style":461},[302],[15,33470],{"className":33471,"style":481},[480],[15,33473,33474,33477],{"style":484},[15,33475],{"className":33476,"style":461},[302],[15,33478,33480],{"className":33479},[59],[15,33481,117],{"className":33482},[59],[15,33484,318],{"className":33485},[317],[15,33487,33489],{"className":33488},[290],[15,33490,33492],{"className":33491,"style":3780},[294],[15,33493],{},[15,33495],{"className":33496},[153,441],[15,33498],{"className":33499,"style":406},[157],[15,33501,4156,33503],{"className":33502},[1486],[15,33504,4160],{"style":4159},[15,33506],{"className":33507,"style":406},[157],[15,33509,2465],{"className":33510},[59],[15,33512],{"className":33513,"style":158},[157],[15,33515,123],{"className":33516},[162],[15,33518],{"className":33519,"style":158},[157],[15,33521,4156,33523],{"className":33522},[1486],[15,33524,4160],{"style":4159},[15,33526],{"className":33527,"style":406},[157],[15,33529,2465],{"className":33530},[59],[15,33532],{"className":33533,"style":158},[157],[15,33535,32985],{"className":33536},[162],[15,33538],{"className":33539,"style":158},[157],[15,33541,32988],{"className":33542},[59],[15,33544,318],{"className":33545},[317],[15,33547,33549],{"className":33548},[290],[15,33550,33552],{"className":33551,"style":33120},[294],[15,33553],{},[11,33555,33556,33557,33609],{},"等号成立当且仅当 ",[15,33558,33560,33579],{"className":33559,"translate":19},[18],[15,33561,33563],{"className":33562},[23],[25,33564,33565],{"xmlns":27},[29,33566,33567,33576],{},[32,33568,33569,33571,33574],{},[35,33570,206],{},[110,33572,33573],{},"⊥",[35,33575,213],{},[39,33577,33578],{"encoding":41},"P\\perp Q",[15,33580,33582,33600],{"className":33581,"ariaHidden":46},[45],[15,33583,33585,33588,33591,33594,33597],{"className":33584},[50],[15,33586],{"className":33587,"style":12601},[54],[15,33589,206],{"className":33590,"style":334},[59,60],[15,33592],{"className":33593,"style":158},[157],[15,33595,33573],{"className":33596},[162],[15,33598],{"className":33599,"style":158},[157],[15,33601,33603,33606],{"className":33602},[50],[15,33604],{"className":33605,"style":566},[54],[15,33607,213],{"className":33608},[59,60],"（支撑集不重叠）。有界是 JS 相对 KL 的最大优点（梯度消失则是它最大的缺点，二者互为因果）。",[11,33611,33612,33614,33615,33644,33645,33678,33679,33707,33708,33736,33737,33765,33766,33794,33795,2411,33823,33851,33852,9402],{},[998,33613,24219],{}," JS 有一个很直观的信息论解释：先掷一枚公平硬币 ",[15,33616,33618,33632],{"className":33617,"translate":19},[18],[15,33619,33621],{"className":33620},[23],[25,33622,33623],{"xmlns":27},[29,33624,33625,33630],{},[32,33626,33627],{},[35,33628,33629],{},"Z",[39,33631,33629],{"encoding":41},[15,33633,33635],{"className":33634,"ariaHidden":46},[45],[15,33636,33638,33641],{"className":33637},[50],[15,33639],{"className":33640,"style":664},[54],[15,33642,33629],{"className":33643,"style":10786},[59,60],"（正反面各 ",[15,33646,33648,33666],{"className":33647,"translate":19},[18],[15,33649,33651],{"className":33650},[23],[25,33652,33653],{"xmlns":27},[29,33654,33655,33663],{},[32,33656,33657,33659,33661],{},[115,33658,117],{},[35,33660,941],{"mathvariant":209},[115,33662,2465],{},[39,33664,33665],{"encoding":41},"1\u002F2",[15,33667,33669],{"className":33668,"ariaHidden":46},[45],[15,33670,33672,33675],{"className":33671},[50],[15,33673],{"className":33674,"style":139},[54],[15,33676,33665],{"className":33677},[59],"），正面时从 ",[15,33680,33682,33695],{"className":33681,"translate":19},[18],[15,33683,33685],{"className":33684},[23],[25,33686,33687],{"xmlns":27},[29,33688,33689,33693],{},[32,33690,33691],{},[35,33692,206],{},[39,33694,206],{"encoding":41},[15,33696,33698],{"className":33697,"ariaHidden":46},[45],[15,33699,33701,33704],{"className":33700},[50],[15,33702],{"className":33703,"style":664},[54],[15,33705,206],{"className":33706,"style":334},[59,60]," 取样、反面时从 ",[15,33709,33711,33724],{"className":33710,"translate":19},[18],[15,33712,33714],{"className":33713},[23],[25,33715,33716],{"xmlns":27},[29,33717,33718,33722],{},[32,33719,33720],{},[35,33721,213],{},[39,33723,213],{"encoding":41},[15,33725,33727],{"className":33726,"ariaHidden":46},[45],[15,33728,33730,33733],{"className":33729},[50],[15,33731],{"className":33732,"style":566},[54],[15,33734,213],{"className":33735},[59,60]," 取样，得到观测 ",[15,33738,33740,33753],{"className":33739,"translate":19},[18],[15,33741,33743],{"className":33742},[23],[25,33744,33745],{"xmlns":27},[29,33746,33747,33751],{},[32,33748,33749],{},[35,33750,598],{},[39,33752,598],{"encoding":41},[15,33754,33756],{"className":33755,"ariaHidden":46},[45],[15,33757,33759,33762],{"className":33758},[50],[15,33760],{"className":33761,"style":664},[54],[15,33763,598],{"className":33764,"style":8889},[59,60],"；则 JS 正是「看一眼 ",[15,33767,33769,33782],{"className":33768,"translate":19},[18],[15,33770,33772],{"className":33771},[23],[25,33773,33774],{"xmlns":27},[29,33775,33776,33780],{},[32,33777,33778],{},[35,33779,598],{},[39,33781,598],{"encoding":41},[15,33783,33785],{"className":33784,"ariaHidden":46},[45],[15,33786,33788,33791],{"className":33787},[50],[15,33789],{"className":33790,"style":664},[54],[15,33792,598],{"className":33793,"style":8889},[59,60]," 能否判断硬币正反」所蕴含的信息量，即 ",[15,33796,33798,33811],{"className":33797,"translate":19},[18],[15,33799,33801],{"className":33800},[23],[25,33802,33803],{"xmlns":27},[29,33804,33805,33809],{},[32,33806,33807],{},[35,33808,33629],{},[39,33810,33629],{"encoding":41},[15,33812,33814],{"className":33813,"ariaHidden":46},[45],[15,33815,33817,33820],{"className":33816},[50],[15,33818],{"className":33819,"style":664},[54],[15,33821,33629],{"className":33822,"style":10786},[59,60],[15,33824,33826,33839],{"className":33825,"translate":19},[18],[15,33827,33829],{"className":33828},[23],[25,33830,33831],{"xmlns":27},[29,33832,33833,33837],{},[32,33834,33835],{},[35,33836,598],{},[39,33838,598],{"encoding":41},[15,33840,33842],{"className":33841,"ariaHidden":46},[45],[15,33843,33845,33848],{"className":33844},[50],[15,33846],{"className":33847,"style":664},[54],[15,33849,598],{"className":33850,"style":8889},[59,60]," 的",[998,33853,33854],{},"互信息",[15,33856,33858],{"className":33857,"translate":19},[180],[15,33859,33861,33940],{"className":33860,"translate":19},[18],[15,33862,33864],{"className":33863},[23],[25,33865,33866],{"xmlns":27,"display":189},[29,33867,33868,33937],{},[32,33869,33870,33880,33882,33884,33886,33888,33890,33892,33895,33897,33899,33902,33904,33906,33908,33911,33913,33915,33917,33919,33921,33923,33925,33927,33929,33931,33933,33935],{},[195,33871,33872,33874],{},[35,33873,199],{},[32,33875,33876,33878],{},[35,33877,23769],{},[35,33879,25763],{},[110,33881,113],{"stretchy":112},[35,33883,206],{},[35,33885,210],{"mathvariant":209},[35,33887,213],{},[110,33889,120],{"stretchy":112},[110,33891,123],{},[35,33893,33894],{},"I",[110,33896,113],{"stretchy":112},[35,33898,33629],{},[110,33900,33901],{"separator":46},";",[35,33903,598],{},[110,33905,120],{"stretchy":112},[110,33907,123],{},[35,33909,33910],{},"H",[110,33912,113],{"stretchy":112},[35,33914,598],{},[110,33916,120],{"stretchy":112},[110,33918,4815],{},[35,33920,33910],{},[110,33922,113],{"stretchy":112},[35,33924,598],{},[1299,33926,1301],{},[35,33928,4810],{"mathvariant":209},[1299,33930,1301],{},[35,33932,33629],{},[110,33934,120],{"stretchy":112},[35,33936,4377],{"mathvariant":209},[39,33938,33939],{"encoding":41},"D_{JS}(P\\|Q)=I(Z;X)=H(X)-H(X\\,|\\,Z).",[15,33941,33943,34019,34055,34083],{"className":33942,"ariaHidden":46},[45],[15,33944,33946,33949,33995,33998,34001,34004,34007,34010,34013,34016],{"className":33945},[50],[15,33947],{"className":33948,"style":139},[54],[15,33950,33952,33955],{"className":33951},[59],[15,33953,199],{"className":33954,"style":277},[59,60],[15,33956,33958],{"className":33957},[281],[15,33959,33961,33987],{"className":33960},[285,286],[15,33962,33964,33984],{"className":33963},[290],[15,33965,33967],{"className":33966,"style":379},[294],[15,33968,33969,33972],{"style":298},[15,33970],{"className":33971,"style":303},[302],[15,33973,33975],{"className":33974},[307,308,309,310],[15,33976,33978,33981],{"className":33977},[59,310],[15,33979,23769],{"className":33980,"style":24179},[59,60,310],[15,33982,25763],{"className":33983,"style":25887},[59,60,310],[15,33985,318],{"className":33986},[317],[15,33988,33990],{"className":33989},[290],[15,33991,33993],{"className":33992,"style":7970},[294],[15,33994],{},[15,33996,113],{"className":33997},[146],[15,33999,206],{"className":34000,"style":334},[59,60],[15,34002,210],{"className":34003},[59],[15,34005,213],{"className":34006},[59,60],[15,34008,120],{"className":34009},[153],[15,34011],{"className":34012,"style":158},[157],[15,34014,123],{"className":34015},[162],[15,34017],{"className":34018,"style":158},[157],[15,34020,34022,34025,34028,34031,34034,34037,34040,34043,34046,34049,34052],{"className":34021},[50],[15,34023],{"className":34024,"style":139},[54],[15,34026,33894],{"className":34027,"style":8889},[59,60],[15,34029,113],{"className":34030},[146],[15,34032,33629],{"className":34033,"style":10786},[59,60],[15,34035,33901],{"className":34036},[573],[15,34038],{"className":34039,"style":406},[157],[15,34041,598],{"className":34042,"style":8889},[59,60],[15,34044,120],{"className":34045},[153],[15,34047],{"className":34048,"style":158},[157],[15,34050,123],{"className":34051},[162],[15,34053],{"className":34054,"style":158},[157],[15,34056,34058,34061,34065,34068,34071,34074,34077,34080],{"className":34057},[50],[15,34059],{"className":34060,"style":139},[54],[15,34062,33910],{"className":34063,"style":34064},[59,60],"margin-right:0.0813em;",[15,34066,113],{"className":34067},[146],[15,34069,598],{"className":34070,"style":8889},[59,60],[15,34072,120],{"className":34073},[153],[15,34075],{"className":34076,"style":2589},[157],[15,34078,4815],{"className":34079},[2593],[15,34081],{"className":34082,"style":2589},[157],[15,34084,34086,34089,34092,34095,34098,34101,34104,34107,34110,34113],{"className":34085},[50],[15,34087],{"className":34088,"style":139},[54],[15,34090,33910],{"className":34091,"style":34064},[59,60],[15,34093,113],{"className":34094},[146],[15,34096,598],{"className":34097,"style":8889},[59,60],[15,34099],{"className":34100,"style":406},[157],[15,34102,4810],{"className":34103},[59],[15,34105],{"className":34106,"style":406},[157],[15,34108,33629],{"className":34109,"style":10786},[59,60],[15,34111,120],{"className":34112},[153],[15,34114,4377],{"className":34115},[59],[11,34117,34118,34119,34147,34148,34180,34181,34309,34310,34351,34352,34490,34491,34532],{},"由于 ",[15,34120,34122,34135],{"className":34121,"translate":19},[18],[15,34123,34125],{"className":34124},[23],[25,34126,34127],{"xmlns":27},[29,34128,34129,34133],{},[32,34130,34131],{},[35,34132,33629],{},[39,34134,33629],{"encoding":41},[15,34136,34138],{"className":34137,"ariaHidden":46},[45],[15,34139,34141,34144],{"className":34140},[50],[15,34142],{"className":34143,"style":664},[54],[15,34145,33629],{"className":34146,"style":10786},[59,60]," 是概率 ",[15,34149,34151,34168],{"className":34150,"translate":19},[18],[15,34152,34154],{"className":34153},[23],[25,34155,34156],{"xmlns":27},[29,34157,34158,34166],{},[32,34159,34160,34162,34164],{},[115,34161,117],{},[35,34163,941],{"mathvariant":209},[115,34165,2465],{},[39,34167,33665],{"encoding":41},[15,34169,34171],{"className":34170,"ariaHidden":46},[45],[15,34172,34174,34177],{"className":34173},[50],[15,34175],{"className":34176,"style":139},[54],[15,34178,33665],{"className":34179},[59]," 的伯努利变量，",[15,34182,34184,34226],{"className":34183,"translate":19},[18],[15,34185,34187],{"className":34186},[23],[25,34188,34189],{"xmlns":27},[29,34190,34191,34223],{},[32,34192,34193,34195,34197,34199,34201,34203,34205,34207,34209,34211,34213,34215,34217,34219,34221],{},[35,34194,33894],{},[110,34196,113],{"stretchy":112},[35,34198,33629],{},[110,34200,33901],{"separator":46},[35,34202,598],{},[110,34204,120],{"stretchy":112},[110,34206,9954],{},[35,34208,33910],{},[110,34210,113],{"stretchy":112},[35,34212,33629],{},[110,34214,120],{"stretchy":112},[110,34216,123],{},[35,34218,4102],{},[110,34220,4105],{},[115,34222,2465],{},[39,34224,34225],{"encoding":41},"I(Z;X)\\le H(Z)=\\log 2",[15,34227,34229,34265,34292],{"className":34228,"ariaHidden":46},[45],[15,34230,34232,34235,34238,34241,34244,34247,34250,34253,34256,34259,34262],{"className":34231},[50],[15,34233],{"className":34234,"style":139},[54],[15,34236,33894],{"className":34237,"style":8889},[59,60],[15,34239,113],{"className":34240},[146],[15,34242,33629],{"className":34243,"style":10786},[59,60],[15,34245,33901],{"className":34246},[573],[15,34248],{"className":34249,"style":406},[157],[15,34251,598],{"className":34252,"style":8889},[59,60],[15,34254,120],{"className":34255},[153],[15,34257],{"className":34258,"style":158},[157],[15,34260,9954],{"className":34261},[162],[15,34263],{"className":34264,"style":158},[157],[15,34266,34268,34271,34274,34277,34280,34283,34286,34289],{"className":34267},[50],[15,34269],{"className":34270,"style":139},[54],[15,34272,33910],{"className":34273,"style":34064},[59,60],[15,34275,113],{"className":34276},[146],[15,34278,33629],{"className":34279,"style":10786},[59,60],[15,34281,120],{"className":34282},[153],[15,34284],{"className":34285,"style":158},[157],[15,34287,123],{"className":34288},[162],[15,34290],{"className":34291,"style":158},[157],[15,34293,34295,34298,34303,34306],{"className":34294},[50],[15,34296],{"className":34297,"style":55},[54],[15,34299,4156,34301],{"className":34300},[1486],[15,34302,4160],{"style":4159},[15,34304],{"className":34305,"style":406},[157],[15,34307,2465],{"className":34308},[59],"，有界性由此而来，",[15,34311,34313,34331],{"className":34312,"translate":19},[18],[15,34314,34316],{"className":34315},[23],[25,34317,34318],{"xmlns":27},[29,34319,34320,34328],{},[32,34321,34322,34324,34326],{},[35,34323,4102],{},[110,34325,4105],{},[115,34327,2465],{},[39,34329,34330],{"encoding":41},"\\log 2",[15,34332,34334],{"className":34333,"ariaHidden":46},[45],[15,34335,34337,34340,34345,34348],{"className":34336},[50],[15,34338],{"className":34339,"style":55},[54],[15,34341,4156,34343],{"className":34342},[1486],[15,34344,4160],{"style":4159},[15,34346],{"className":34347,"style":406},[157],[15,34349,2465],{"className":34350},[59]," 恰好对应「最多只能获得 1 比特」这一直觉。JS 对称、有界且 ",[15,34353,34355,34380],{"className":34354,"translate":19},[18],[15,34356,34358],{"className":34357},[23],[25,34359,34360],{"xmlns":27},[29,34361,34362,34377],{},[32,34363,34364],{},[34365,34366,34367],"msqrt",{},[195,34368,34369,34371],{},[35,34370,199],{},[32,34372,34373,34375],{},[35,34374,23769],{},[35,34376,25763],{},[39,34378,34379],{"encoding":41},"\\sqrt{D_{JS}}",[15,34381,34383],{"className":34382,"ariaHidden":46},[45],[15,34384,34386,34390],{"className":34385},[50],[15,34387],{"className":34388,"style":34389},[54],"height:1.04em;vertical-align:-0.1883em;",[15,34391,34394],{"className":34392},[59,34393],"sqrt",[15,34395,34397,34481],{"className":34396},[285,286],[15,34398,34400,34478],{"className":34399},[290],[15,34401,34404,34460],{"className":34402,"style":34403},[294],"height:0.8517em;",[15,34405,34407,34410],{"className":34406,"style":11867},[6309],[15,34408],{"className":34409,"style":461},[302],[15,34411,34414],{"className":34412,"style":34413},[59],"padding-left:0.833em;",[15,34415,34417,34420],{"className":34416},[59],[15,34418,199],{"className":34419,"style":277},[59,60],[15,34421,34423],{"className":34422},[281],[15,34424,34426,34452],{"className":34425},[285,286],[15,34427,34429,34449],{"className":34428},[290],[15,34430,34432],{"className":34431,"style":379},[294],[15,34433,34434,34437],{"style":298},[15,34435],{"className":34436,"style":303},[302],[15,34438,34440],{"className":34439},[307,308,309,310],[15,34441,34443,34446],{"className":34442},[59,310],[15,34444,23769],{"className":34445,"style":24179},[59,60,310],[15,34447,25763],{"className":34448,"style":25887},[59,60,310],[15,34450,318],{"className":34451},[317],[15,34453,34455],{"className":34454},[290],[15,34456,34458],{"className":34457,"style":7970},[294],[15,34459],{},[15,34461,34463,34466],{"style":34462},"top:-2.8117em;",[15,34464],{"className":34465,"style":461},[302],[15,34467,34471],{"className":34468,"style":34470},[34469],"hide-tail","min-width:0.853em;height:1.08em;",[6325,34472,34475],{"xmlns":6327,"width":6328,"height":34473,"viewBox":34474,"preserveAspectRatio":6331},"1.08em","0 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,310],[15,35387,318],{"className":35388},[317],[15,35390,35392],{"className":35391},[290],[15,35393,35395],{"className":35394,"style":7970},[294],[15,35396],{},[15,35398,113],{"className":35399},[146],[15,35401,206],{"className":35402,"style":334},[59,60],[15,35404,210],{"className":35405},[59],[15,35407,213],{"className":35408},[59,60],[15,35410,120],{"className":35411},[153],[15,35413,4377],{"className":35414},[59],[11,35416,35417,35418,35683,35684,35809],{},"一般测度上 ",[15,35419,35421,35494],{"className":35420,"translate":19},[18],[15,35422,35424],{"className":35423},[23],[25,35425,35426],{"xmlns":27},[29,35427,35428,35491],{},[32,35429,35430,35440,35442,35444,35446,35448,35450,35452,35469,35471,35473,35475,35477,35479,35481,35483,35485,35487,35489],{},[195,35431,35432,35434],{},[35,35433,199],{},[32,35435,35436,35438],{},[35,35437,5394],{},[35,35439,34876],{},[110,35441,113],{"stretchy":112},[35,35443,206],{},[110,35445,551],{"separator":46},[35,35447,213],{},[110,35449,120],{"stretchy":112},[110,35451,123],{},[195,35453,35454,35461],{},[32,35455,35456,35459],{},[35,35457,35458],{},"sup",[110,35460,4105],{},[32,35462,35463,35465,35467],{},[35,35464,603],{},[110,35466,5911],{},[35,35468,603],{"mathvariant":597},[35,35470,4810],{"mathvariant":209},[35,35472,206],{},[110,35474,113],{"stretchy":112},[35,35476,603],{},[110,35478,120],{"stretchy":112},[110,35480,4815],{},[35,35482,213],{},[110,35484,113],{"stretchy":112},[35,35486,603],{},[110,35488,120],{"stretchy":112},[35,35490,4810],{"mathvariant":209},[39,35492,35493],{"encoding":41},"D_{TV}(P,Q)=\\sup_{A\\in\\mathcal A}|P(A)-Q(A)|",[15,35495,35497,35576,35662],{"className":35496,"ariaHidden":46},[45],[15,35498,35500,35503,35549,35552,35555,35558,35561,35564,35567,35570,35573],{"className":35499},[50],[15,35501],{"className":35502,"style":139},[54],[15,35504,35506,35509],{"className":35505},[59],[15,35507,199],{"className":35508,"style":277},[59,60],[15,35510,35512],{"className":35511},[281],[15,35513,35515,35541],{"className":35514},[285,286],[15,35516,35518,35538],{"className":35517},[290],[15,35519,35521],{"className":35520,"style":379},[294],[15,35522,35523,35526],{"style":298},[15,35524],{"className":35525,"style":303},[302],[15,35527,35529],{"className":35528},[307,308,309,310],[15,35530,35532,35535],{"className":35531},[59,310],[15,35533,5394],{"className":35534,"style":334},[59,60,310],[15,35536,34876],{"className":35537,"style":2589},[59,60,310],[15,35539,318],{"className":35540},[317],[15,35542,35544],{"className":35543},[290],[15,35545,35547],{"className":35546,"style":7970},[294],[15,35548],{},[15,35550,113],{"className":35551},[146],[15,35553,206],{"className":35554,"style":334},[59,60],[15,35556,551],{"className":35557},[573],[15,35559],{"className":35560,"style":406},[157],[15,35562,213],{"className":35563},[59,60],[15,35565,120],{"className":35566},[153],[15,35568],{"className":35569,"style":158},[157],[15,35571,123],{"className":35572},[162],[15,35574],{"className":35575,"style":158},[157],[15,35577,35579,35583,35635,35638,35641,35644,35647,35650,35653,35656,35659],{"className":35578},[50],[15,35580],{"className":35581,"style":35582},[54],"height:1.0215em;vertical-align:-0.2715em;",[15,35584,35586,35589],{"className":35585},[1486],[15,35587,35458],{"className":35588},[1486],[15,35590,35592],{"className":35591},[281],[15,35593,35595,35626],{"className":35594},[285,286],[15,35596,35598,35623],{"className":35597},[290],[15,35599,35602],{"className":35600,"style":35601},[294],"height:0.2342em;",[15,35603,35605,35608],{"style":35604},"top:-2.4559em;margin-right:0.05em;",[15,35606],{"className":35607,"style":303},[302],[15,35609,35611],{"className":35610},[307,308,309,310],[15,35612,35614,35617,35620],{"className":35613},[59,310],[15,35615,603],{"className":35616},[59,60,310],[15,35618,5911],{"className":35619},[162,310],[15,35621,603],{"className":35622},[59,624,310],[15,35624,318],{"className":35625},[317],[15,35627,35629],{"className":35628},[290],[15,35630,35633],{"className":35631,"style":35632},[294],"height:0.2715em;",[15,35634],{},[15,35636],{"className":35637,"style":406},[157],[15,35639,4810],{"className":35640},[59],[15,35642,206],{"className":35643,"style":334},[59,60],[15,35645,113],{"className":35646},[146],[15,35648,603],{"className":35649},[59,60],[15,35651,120],{"className":35652},[153],[15,35654],{"className":35655,"style":2589},[157],[15,35657,4815],{"className":35658},[2593],[15,35660],{"className":35661,"style":2589},[157],[15,35663,35665,35668,35671,35674,35677,35680],{"className":35664},[50],[15,35666],{"className":35667,"style":139},[54],[15,35669,213],{"className":35670},[59,60],[15,35672,113],{"className":35673},[146],[15,35675,603],{"className":35676},[59,60],[15,35678,120],{"className":35679},[153],[15,35681,4810],{"className":35682},[59],"；两个表达式相等（这是全变差的对偶刻画）。注意 ",[15,35685,35687,35721],{"className":35686,"translate":19},[18],[15,35688,35690],{"className":35689},[23],[25,35691,35692],{"xmlns":27},[29,35693,35694,35718],{},[32,35695,35696,35706,35708,35710,35712,35714,35716],{},[195,35697,35698,35700],{},[35,35699,199],{},[32,35701,35702,35704],{},[35,35703,5394],{},[35,35705,34876],{},[110,35707,5911],{},[110,35709,230],{"stretchy":112},[115,35711,126],{},[110,35713,551],{"separator":46},[115,35715,117],{},[110,35717,257],{"stretchy":112},[39,35719,35720],{"encoding":41},"D_{TV}\\in[0,1]",[15,35722,35724,35785],{"className":35723,"ariaHidden":46},[45],[15,35725,35727,35730,35776,35779,35782],{"className":35726},[50],[15,35728],{"className":35729,"style":15965},[54],[15,35731,35733,35736],{"className":35732},[59],[15,35734,199],{"className":35735,"style":277},[59,60],[15,35737,35739],{"className":35738},[281],[15,35740,35742,35768],{"className":35741},[285,286],[15,35743,35745,35765],{"className":35744},[290],[15,35746,35748],{"className":35747,"style":379},[294],[15,35749,35750,35753],{"style":298},[15,35751],{"className":35752,"style":303},[302],[15,35754,35756],{"className":35755},[307,308,309,310],[15,35757,35759,35762],{"className":35758},[59,310],[15,35760,5394],{"className":35761,"style":334},[59,60,310],[15,35763,34876],{"className":35764,"style":2589},[59,60,310],[15,35766,318],{"className":35767},[317],[15,35769,35771],{"className":35770},[290],[15,35772,35774],{"className":35773,"style":7970},[294],[15,35775],{},[15,35777],{"className":35778,"style":158},[157],[15,35780,5911],{"className":35781},[162],[15,35783],{"className":35784,"style":158},[157],[15,35786,35788,35791,35794,35797,35800,35803,35806],{"className":35787},[50],[15,35789],{"className":35790,"style":139},[54],[15,35792,230],{"className":35793},[146],[15,35795,126],{"className":35796},[59],[15,35798,551],{"className":35799},[573],[15,35801],{"className":35802,"style":406},[157],[15,35804,117],{"className":35805},[59],[15,35807,257],{"className":35808},[153],"，总有界。",[11,35811,35812,9402,35815,36201],{},[998,35813,35814],{},"斜共轭",[15,35816,35818,35882],{"className":35817,"translate":19},[18],[15,35819,35821],{"className":35820},[23],[25,35822,35823],{"xmlns":27},[29,35824,35825,35879],{},[32,35826,35827,35829,35831,35837,35839,35845,35847,35849,35851,35853,35859,35861,35863,35865,35867,35869,35871,35873,35875,35877],{},[35,35828,769],{},[110,35830,3301],{},[238,35832,35833,35835],{},[115,35834,117],{},[115,35836,2465],{},[35,35838,4810],{"mathvariant":209},[238,35840,35841,35843],{},[115,35842,117],{},[35,35844,769],{},[110,35846,4815],{},[115,35848,117],{},[35,35850,4810],{"mathvariant":209},[110,35852,123],{},[238,35854,35855,35857],{},[115,35856,117],{},[115,35858,2465],{},[35,35860,4810],{"mathvariant":209},[115,35862,117],{},[110,35864,4815],{},[35,35866,769],{},[35,35868,4810],{"mathvariant":209},[110,35870,123],{},[35,35872,37],{},[110,35874,113],{"stretchy":112},[35,35876,769],{},[110,35878,120],{"stretchy":112},[39,35880,35881],{"encoding":41},"t\\cdot\\frac12|\\frac1t-1|=\\frac12|1-t|=f(t)",[15,35883,35885,35903,36057,36075,36162,36183],{"className":35884,"ariaHidden":46},[45],[15,35886,35888,35891,35894,35897,35900],{"className":35887},[50],[15,35889],{"className":35890,"style":5345},[54],[15,35892,769],{"className":35893},[59,60],[15,35895],{"className":35896,"style":2589},[157],[15,35898,3301],{"className":35899},[2593],[15,35901],{"className":35902,"style":2589},[157],[15,35904,35906,35909,35977,35980,36048,36051,36054],{"className":35905},[50],[15,35907],{"className":35908,"style":2507},[54],[15,35910,35912,35915,35974],{"className":35911},[59],[15,35913],{"className":35914},[146,441],[15,35916,35918],{"className":35917},[238],[15,35919,35921,35966],{"className":35920},[285,286],[15,35922,35924,35963],{"className":35923},[290],[15,35925,35927,35941,35949],{"className":35926,"style":2526},[294],[15,35928,35929,35932],{"style":2529},[15,35930],{"className":35931,"style":461},[302],[15,35933,35935],{"className":35934},[307,308,309,310],[15,35936,35938],{"className":35937},[59,310],[15,35939,2465],{"className":35940},[59,310],[15,35942,35943,35946],{"style":473},[15,35944],{"className":35945,"style":461},[302],[15,35947],{"className":35948,"style":481},[480],[15,35950,35951,35954],{"style":2552},[15,35952],{"className":35953,"style":461},[302],[15,35955,35957],{"className":35956},[307,308,309,310],[15,35958,35960],{"className":35959},[59,310],[15,35961,117],{"className":35962},[59,310],[15,35964,318],{"className":35965},[317],[15,35967,35969],{"className":35968},[290],[15,35970,35972],{"className":35971,"style":2574},[294],[15,35973],{},[15,35975],{"className":35976},[153,441],[15,35978,4810],{"className":35979},[59],[15,35981,35983,35986,36045],{"className":35982},[59],[15,35984],{"className":35985},[146,441],[15,35987,35989],{"className":35988},[238],[15,35990,35992,36037],{"className":35991},[285,286],[15,35993,35995,36034],{"className":35994},[290],[15,35996,35998,36012,36020],{"className":35997,"style":2526},[294],[15,35999,36000,36003],{"style":2529},[15,36001],{"className":36002,"style":461},[302],[15,36004,36006],{"className":36005},[307,308,309,310],[15,36007,36009],{"className":36008},[59,310],[15,36010,769],{"className":36011},[59,60,310],[15,36013,36014,36017],{"style":473},[15,36015],{"className":36016,"style":461},[302],[15,36018],{"className":36019,"style":481},[480],[15,36021,36022,36025],{"style":2552},[15,36023],{"className":36024,"style":461},[302],[15,36026,36028],{"className":36027},[307,308,309,310],[15,36029,36031],{"className":36030},[59,310],[15,36032,117],{"className":36033},[59,310],[15,36035,318],{"className":36036},[317],[15,36038,36040],{"className":36039},[290],[15,36041,36043],{"className":36042,"style":2574},[294],[15,36044],{},[15,36046],{"className":36047},[153,441],[15,36049],{"className":36050,"style":2589},[157],[15,36052,4815],{"className":36053},[2593],[15,36055],{"className":36056,"style":2589},[157],[15,36058,36060,36063,36066,36069,36072],{"className":36059},[50],[15,36061],{"className":36062,"style":139},[54],[15,36064,34705],{"className":36065},[59],[15,36067],{"className":36068,"style":158},[157],[15,36070,123],{"className":36071},[162],[15,36073],{"className":36074,"style":158},[157],[15,36076,36078,36081,36149,36153,36156,36159],{"className":36077},[50],[15,36079],{"className":36080,"style":2507},[54],[15,36082,36084,36087,36146],{"className":36083},[59],[15,36085],{"className":36086},[146,441],[15,36088,36090],{"className":36089},[238],[15,36091,36093,36138],{"className":36092},[285,286],[15,36094,36096,36135],{"className":36095},[290],[15,36097,36099,36113,36121],{"className":36098,"style":2526},[294],[15,36100,36101,36104],{"style":2529},[15,36102],{"className":36103,"style":461},[302],[15,36105,36107],{"className":36106},[307,308,309,310],[15,36108,36110],{"className":36109},[59,310],[15,36111,2465],{"className":36112},[59,310],[15,36114,36115,36118],{"style":473},[15,36116],{"className":36117,"style":461},[302],[15,36119],{"className":36120,"style":481},[480],[15,36122,36123,36126],{"style":2552},[15,36124],{"className":36125,"style":461},[302],[15,36127,36129],{"className":36128},[307,308,309,310],[15,36130,36132],{"className":36131},[59,310],[15,36133,117],{"className":36134},[59,310],[15,36136,318],{"className":36137},[317],[15,36139,36141],{"className":36140},[290],[15,36142,36144],{"className":36143,"style":2574},[294],[15,36145],{},[15,36147],{"className":36148},[153,441],[15,36150,36152],{"className":36151},[59],"∣1",[15,36154],{"className":36155,"style":2589},[157],[15,36157,4815],{"className":36158},[2593],[15,36160],{"className":36161,"style":2589},[157],[15,36163,36165,36168,36171,36174,36177,36180],{"className":36164},[50],[15,36166],{"className":36167,"style":139},[54],[15,36169,769],{"className":36170},[59,60],[15,36172,4810],{"className":36173},[59],[15,36175],{"className":36176,"style":158},[157],[15,36178,123],{"className":36179},[162],[15,36181],{"className":36182,"style":158},[157],[15,36184,36186,36189,36192,36195,36198],{"className":36185},[50],[15,36187],{"className":36188,"style":139},[54],[15,36190,37],{"className":36191,"style":61},[59,60],[15,36193,113],{"className":36194},[146],[15,36196,769],{"className":36197},[59,60],[15,36199,120],{"className":36200},[153],"，所以 TV 是对称的。",[11,36203,36204,36206,36207,36210],{},[998,36205,24219],{}," 全变差是概率测度空间上真正的",[998,36208,36209],{},"度量","（对称、满足三角不等式），它等于两个分布在一件事上能给出的最大概率差：",[15,36212,36214],{"className":36213,"translate":19},[180],[15,36215,36217,36291],{"className":36216,"translate":19},[18],[15,36218,36220],{"className":36219},[23],[25,36221,36222],{"xmlns":27,"display":189},[29,36223,36224,36288],{},[32,36225,36226,36236,36238,36240,36242,36244,36246,36248,36264,36266,36268,36270,36272,36274,36276,36278,36280,36282,36284,36286],{},[195,36227,36228,36230],{},[35,36229,199],{},[32,36231,36232,36234],{},[35,36233,5394],{},[35,36235,34876],{},[110,36237,113],{"stretchy":112},[35,36239,206],{},[110,36241,551],{"separator":46},[35,36243,213],{},[110,36245,120],{"stretchy":112},[110,36247,123],{},[6018,36249,36250,36256],{},[32,36251,36252,36254],{},[35,36253,35458],{},[110,36255,4105],{},[32,36257,36258,36260,36262],{},[35,36259,603],{},[110,36261,5911],{},[35,36263,603],{"mathvariant":597},[110,36265,4810],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[35,36267,206],{},[110,36269,113],{"stretchy":112},[35,36271,603],{},[110,36273,120],{"stretchy":112},[110,36275,4815],{},[35,36277,213],{},[110,36279,113],{"stretchy":112},[35,36281,603],{},[110,36283,120],{"stretchy":112},[110,36285,4810],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[35,36287,4377],{"mathvariant":209},[39,36289,36290],{"encoding":41},"D_{TV}(P,Q)\n=\\sup_{A\\in\\mathcal A}\\big|P(A)-Q(A)\\big| .",[15,36292,36294,36373,36499],{"className":36293,"ariaHidden":46},[45],[15,36295,36297,36300,36346,36349,36352,36355,36358,36361,36364,36367,36370],{"className":36296},[50],[15,36298],{"className":36299,"style":139},[54],[15,36301,36303,36306],{"className":36302},[59],[15,36304,199],{"className":36305,"style":277},[59,60],[15,36307,36309],{"className":36308},[281],[15,36310,36312,36338],{"className":36311},[285,286],[15,36313,36315,36335],{"className":36314},[290],[15,36316,36318],{"className":36317,"style":379},[294],[15,36319,36320,36323],{"style":298},[15,36321],{"className":36322,"style":303},[302],[15,36324,36326],{"className":36325},[307,308,309,310],[15,36327,36329,36332],{"className":36328},[59,310],[15,36330,5394],{"className":36331,"style":334},[59,60,310],[15,36333,34876],{"className":36334,"style":2589},[59,60,310],[15,36336,318],{"className":36337},[317],[15,36339,36341],{"className":36340},[290],[15,36342,36344],{"className":36343,"style":7970},[294],[15,36345],{},[15,36347,113],{"className":36348},[146],[15,36350,206],{"className":36351,"style":334},[59,60],[15,36353,551],{"className":36354},[573],[15,36356],{"className":36357,"style":406},[157],[15,36359,213],{"className":36360},[59,60],[15,36362,120],{"className":36363},[153],[15,36365],{"className":36366,"style":158},[157],[15,36368,123],{"className":36369},[162],[15,36371],{"className":36372,"style":158},[157],[15,36374,36376,36380,36435,36438,36478,36481,36484,36487,36490,36493,36496],{"className":36375},[50],[15,36377],{"className":36378,"style":36379},[54],"height:1.8161em;vertical-align:-0.9661em;",[15,36381,36383],{"className":36382},[1486,13720],[15,36384,36386,36426],{"className":36385},[285,286],[15,36387,36389,36423],{"className":36388},[290],[15,36390,36392,36413],{"className":36391,"style":2347},[294],[15,36393,36395,36398],{"style":36394},"top:-2.1612em;margin-left:0em;",[15,36396],{"className":36397,"style":461},[302],[15,36399,36401],{"className":36400},[307,308,309,310],[15,36402,36404,36407,36410],{"className":36403},[59,310],[15,36405,603],{"className":36406},[59,60,310],[15,36408,5911],{"className":36409},[162,310],[15,36411,603],{"className":36412},[59,624,310],[15,36414,36415,36418],{"style":11867},[15,36416],{"className":36417,"style":461},[302],[15,36419,36420],{},[15,36421,35458],{"className":36422},[1486],[15,36424,318],{"className":36425},[317],[15,36427,36429],{"className":36428},[290],[15,36430,36433],{"className":36431,"style":36432},[294],"height:0.9661em;",[15,36434],{},[15,36436],{"className":36437,"style":406},[157],[15,36439,36441],{"className":36440},[59],[15,36442,36444],{"className":36443},[419,14833],[15,36445,36447,36470],{"className":36446},[285,286],[15,36448,36450,36467],{"className":36449},[290],[15,36451,36453],{"className":36452,"style":14843},[294],[15,36454,36455,36458],{"style":14846},[15,36456],{"className":36457,"style":14850},[302],[15,36459,36461],{"style":36460},"width:0.333em;height:1.2em;",[6325,36462,36464],{"xmlns":6327,"width":35077,"height":9452,"viewBox":36463},"0 0 333 1200",[6333,36465],{"d":36466},"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",[15,36468,318],{"className":36469},[317],[15,36471,36473],{"className":36472},[290],[15,36474,36476],{"className":36475,"style":14870},[294],[15,36477],{},[15,36479,206],{"className":36480,"style":334},[59,60],[15,36482,113],{"className":36483},[146],[15,36485,603],{"className":36486},[59,60],[15,36488,120],{"className":36489},[153],[15,36491],{"className":36492,"style":2589},[157],[15,36494,4815],{"className":36495},[2593],[15,36497],{"className":36498,"style":2589},[157],[15,36500,36502,36505,36508,36511,36514,36517,36554],{"className":36501},[50],[15,36503],{"className":36504,"style":20589},[54],[15,36506,213],{"className":36507},[59,60],[15,36509,113],{"className":36510},[146],[15,36512,603],{"className":36513},[59,60],[15,36515,120],{"className":36516},[153],[15,36518,36520],{"className":36519},[59],[15,36521,36523],{"className":36522},[419,14833],[15,36524,36526,36546],{"className":36525},[285,286],[15,36527,36529,36543],{"className":36528},[290],[15,36530,36532],{"className":36531,"style":14843},[294],[15,36533,36534,36537],{"style":14846},[15,36535],{"className":36536,"style":14850},[302],[15,36538,36539],{"style":36460},[6325,36540,36541],{"xmlns":6327,"width":35077,"height":9452,"viewBox":36463},[6333,36542],{"d":36466},[15,36544,318],{"className":36545},[317],[15,36547,36549],{"className":36548},[290],[15,36550,36552],{"className":36551,"style":14870},[294],[15,36553],{},[15,36555,4377],{"className":36556},[59],[11,36558,36559,36560,2411,36588,36616,36617,36811,36812,36815,36816,36868,36869,37171],{},"它还有一个判决论解释：在等先验下用最优检验区分 ",[15,36561,36563,36576],{"className":36562,"translate":19},[18],[15,36564,36566],{"className":36565},[23],[25,36567,36568],{"xmlns":27},[29,36569,36570,36574],{},[32,36571,36572],{},[35,36573,206],{},[39,36575,206],{"encoding":41},[15,36577,36579],{"className":36578,"ariaHidden":46},[45],[15,36580,36582,36585],{"className":36581},[50],[15,36583],{"className":36584,"style":664},[54],[15,36586,206],{"className":36587,"style":334},[59,60],[15,36589,36591,36604],{"className":36590,"translate":19},[18],[15,36592,36594],{"className":36593},[23],[25,36595,36596],{"xmlns":27},[29,36597,36598,36602],{},[32,36599,36600],{},[35,36601,213],{},[39,36603,213],{"encoding":41},[15,36605,36607],{"className":36606,"ariaHidden":46},[45],[15,36608,36610,36613],{"className":36609},[50],[15,36611],{"className":36612,"style":566},[54],[15,36614,213],{"className":36615},[59,60]," 时，最小平均错误率为 ",[15,36618,36620,36658],{"className":36619,"translate":19},[18],[15,36621,36623],{"className":36622},[23],[25,36624,36625],{"xmlns":27},[29,36626,36627,36655],{},[32,36628,36629,36637,36639,36641,36643,36653],{},[1257,36630,36631],{"scriptlevel":126,"displaystyle":112},[238,36632,36633,36635],{},[115,36634,117],{},[115,36636,2465],{},[110,36638,113],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[115,36640,117],{},[110,36642,4815],{},[195,36644,36645,36647],{},[35,36646,199],{},[32,36648,36649,36651],{},[35,36650,5394],{},[35,36652,34876],{},[110,36654,120],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[39,36656,36657],{"encoding":41},"\\tfrac12\\big(1-D_{TV}\\big)",[15,36659,36661,36753],{"className":36660,"ariaHidden":46},[45],[15,36662,36664,36667,36735,36741,36744,36747,36750],{"className":36663},[50],[15,36665],{"className":36666,"style":20589},[54],[15,36668,36670,36673,36732],{"className":36669},[59],[15,36671],{"className":36672},[146,441],[15,36674,36676],{"className":36675},[238],[15,36677,36679,36724],{"className":36678},[285,286],[15,36680,36682,36721],{"className":36681},[290],[15,36683,36685,36699,36707],{"className":36684,"style":2526},[294],[15,36686,36687,36690],{"style":2529},[15,36688],{"className":36689,"style":461},[302],[15,36691,36693],{"className":36692},[307,308,309,310],[15,36694,36696],{"className":36695},[59,310],[15,36697,2465],{"className":36698},[59,310],[15,36700,36701,36704],{"style":473},[15,36702],{"className":36703,"style":461},[302],[15,36705],{"className":36706,"style":481},[480],[15,36708,36709,36712],{"style":2552},[15,36710],{"className":36711,"style":461},[302],[15,36713,36715],{"className":36714},[307,308,309,310],[15,36716,36718],{"className":36717},[59,310],[15,36719,117],{"className":36720},[59,310],[15,36722,318],{"className":36723},[317],[15,36725,36727],{"className":36726},[290],[15,36728,36730],{"className":36729,"style":2574},[294],[15,36731],{},[15,36733],{"className":36734},[153,441],[15,36736,36738],{"className":36737},[59],[15,36739,113],{"className":36740},[419,9676],[15,36742,117],{"className":36743},[59],[15,36745],{"className":36746,"style":2589},[157],[15,36748,4815],{"className":36749},[2593],[15,36751],{"className":36752,"style":2589},[157],[15,36754,36756,36759,36805],{"className":36755},[50],[15,36757],{"className":36758,"style":20589},[54],[15,36760,36762,36765],{"className":36761},[59],[15,36763,199],{"className":36764,"style":277},[59,60],[15,36766,36768],{"className":36767},[281],[15,36769,36771,36797],{"className":36770},[285,286],[15,36772,36774,36794],{"className":36773},[290],[15,36775,36777],{"className":36776,"style":379},[294],[15,36778,36779,36782],{"style":298},[15,36780],{"className":36781,"style":303},[302],[15,36783,36785],{"className":36784},[307,308,309,310],[15,36786,36788,36791],{"className":36787},[59,310],[15,36789,5394],{"className":36790,"style":334},[59,60,310],[15,36792,34876],{"className":36793,"style":2589},[59,60,310],[15,36795,318],{"className":36796},[317],[15,36798,36800],{"className":36799},[290],[15,36801,36803],{"className":36802,"style":7970},[294],[15,36804],{},[15,36806,36808],{"className":36807},[59],[15,36809,120],{"className":36810},[419,9676],"，因此 TV 直接度量了",[998,36813,36814],{},"可分性","，「TV 小」就等于「几乎分不开」。TV 是这里所有 f-散度中最「弱」的标尺（总在 ",[15,36817,36819,36841],{"className":36818,"translate":19},[18],[15,36820,36822],{"className":36821},[23],[25,36823,36824],{"xmlns":27},[29,36825,36826,36838],{},[32,36827,36828,36830,36832,36834,36836],{},[110,36829,230],{"stretchy":112},[115,36831,126],{},[110,36833,551],{"separator":46},[115,36835,117],{},[110,36837,257],{"stretchy":112},[39,36839,36840],{"encoding":41},"[0,1]",[15,36842,36844],{"className":36843,"ariaHidden":46},[45],[15,36845,36847,36850,36853,36856,36859,36862,36865],{"className":36846},[50],[15,36848],{"className":36849,"style":139},[54],[15,36851,230],{"className":36852},[146],[15,36854,126],{"className":36855},[59],[15,36857,551],{"className":36858},[573],[15,36860],{"className":36861,"style":406},[157],[15,36863,117],{"className":36864},[59],[15,36866,257],{"className":36867},[153]," 内），但按 Csiszár 定理它与 KL 诱导同一个拓扑，所以常常已经够用；在一维实直线上它退化为 Kolmogorov 距离 ",[15,36870,36872,36937],{"className":36871,"translate":19},[18],[15,36873,36875],{"className":36874},[23],[25,36876,36877],{"xmlns":27},[29,36878,36879,36934],{},[32,36880,36881,36891,36893,36903,36905,36912,36914,36916,36918,36920,36926,36928,36930,36932],{},[195,36882,36883,36885],{},[35,36884,199],{},[32,36886,36887,36889],{},[35,36888,5394],{},[35,36890,34876],{},[110,36892,123],{},[195,36894,36895,36901],{},[32,36896,36897,36899],{},[35,36898,35458],{},[110,36900,4105],{},[35,36902,774],{},[35,36904,4810],{"mathvariant":209},[195,36906,36907,36910],{},[35,36908,36909],{},"F",[35,36911,206],{},[110,36913,113],{"stretchy":112},[35,36915,774],{},[110,36917,120],{"stretchy":112},[110,36919,4815],{},[195,36921,36922,36924],{},[35,36923,36909],{},[35,36925,213],{},[110,36927,113],{"stretchy":112},[35,36929,774],{},[110,36931,120],{"stretchy":112},[35,36933,4810],{"mathvariant":209},[39,36935,36936],{"encoding":41},"D_{TV}=\\sup_x|F_P(x)-F_Q(x)|",[15,36938,36940,37001,37113],{"className":36939,"ariaHidden":46},[45],[15,36941,36943,36946,36992,36995,36998],{"className":36942},[50],[15,36944],{"className":36945,"style":15965},[54],[15,36947,36949,36952],{"className":36948},[59],[15,36950,199],{"className":36951,"style":277},[59,60],[15,36953,36955],{"className":36954},[281],[15,36956,36958,36984],{"className":36957},[285,286],[15,36959,36961,36981],{"className":36960},[290],[15,36962,36964],{"className":36963,"style":379},[294],[15,36965,36966,36969],{"style":298},[15,36967],{"className":36968,"style":303},[302],[15,36970,36972],{"className":36971},[307,308,309,310],[15,36973,36975,36978],{"className":36974},[59,310],[15,36976,5394],{"className":36977,"style":334},[59,60,310],[15,36979,34876],{"className":36980,"style":2589},[59,60,310],[15,36982,318],{"className":36983},[317],[15,36985,36987],{"className":36986},[290],[15,36988,36990],{"className":36989,"style":7970},[294],[15,36991],{},[15,36993],{"className":36994,"style":158},[157],[15,36996,123],{"className":36997},[162],[15,36999],{"className":37000,"style":158},[157],[15,37002,37004,37007,37049,37052,37055,37095,37098,37101,37104,37107,37110],{"className":37003},[50],[15,37005],{"className":37006,"style":139},[54],[15,37008,37010,37013],{"className":37009},[1486],[15,37011,35458],{"className":37012},[1486],[15,37014,37016],{"className":37015},[281],[15,37017,37019,37040],{"className":37018},[285,286],[15,37020,37022,37037],{"className":37021},[290],[15,37023,37026],{"className":37024,"style":37025},[294],"height:0.0573em;",[15,37027,37028,37031],{"style":35604},[15,37029],{"className":37030,"style":303},[302],[15,37032,37034],{"className":37033},[307,308,309,310],[15,37035,774],{"className":37036},[59,60,310],[15,37038,318],{"className":37039},[317],[15,37041,37043],{"className":37042},[290],[15,37044,37047],{"className":37045,"style":37046},[294],"height:0.2441em;",[15,37048],{},[15,37050],{"className":37051,"style":406},[157],[15,37053,4810],{"className":37054},[59],[15,37056,37058,37061],{"className":37057},[59],[15,37059,36909],{"className":37060,"style":334},[59,60],[15,37062,37064],{"className":37063},[281],[15,37065,37067,37087],{"className":37066},[285,286],[15,37068,37070,37084],{"className":37069},[290],[15,37071,37073],{"className":37072,"style":379},[294],[15,37074,37075,37078],{"style":11140},[15,37076],{"className":37077,"style":303},[302],[15,37079,37081],{"className":37080},[307,308,309,310],[15,37082,206],{"className":37083,"style":334},[59,60,310],[15,37085,318],{"className":37086},[317],[15,37088,37090],{"className":37089},[290],[15,37091,37093],{"className":37092,"style":7970},[294],[15,37094],{},[15,37096,113],{"className":37097},[146],[15,37099,774],{"className":37100},[59,60],[15,37102,120],{"className":37103},[153],[15,37105],{"className":37106,"style":2589},[157],[15,37108,4815],{"className":37109},[2593],[15,37111],{"className":37112,"style":2589},[157],[15,37114,37116,37119,37159,37162,37165,37168],{"className":37115},[50],[15,37117],{"className":37118,"style":270},[54],[15,37120,37122,37125],{"className":37121},[59],[15,37123,36909],{"className":37124,"style":334},[59,60],[15,37126,37128],{"className":37127},[281],[15,37129,37131,37151],{"className":37130},[285,286],[15,37132,37134,37148],{"className":37133},[290],[15,37135,37137],{"className":37136,"style":379},[294],[15,37138,37139,37142],{"style":11140},[15,37140],{"className":37141,"style":303},[302],[15,37143,37145],{"className":37144},[307,308,309,310],[15,37146,213],{"className":37147},[59,60,310],[15,37149,318],{"className":37150},[317],[15,37152,37154],{"className":37153},[290],[15,37155,37157],{"className":37156,"style":325},[294],[15,37158],{},[15,37160,113],{"className":37161},[146],[15,37163,774],{"className":37164},[59,60],[15,37166,120],{"className":37167},[153],[15,37169,4810],{"className":37170},[59],"。马尔可夫链的混合时间、耦合构造与统计距离（total variation distance）都以它为核心。",[2313,37173,37175],{"id":37174},"_34-卡方散度","3.4 卡方散度",[11,37177,21499],{},[15,37179,37181],{"className":37180,"translate":19},[180],[15,37182,37184,37220],{"className":37183,"translate":19},[18],[15,37185,37187],{"className":37186},[23],[25,37188,37189],{"xmlns":27,"display":189},[29,37190,37191,37217],{},[32,37192,37193,37195,37197,37199,37201,37203,37205,37207,37209,37211],{},[35,37194,37],{},[110,37196,113],{"stretchy":112},[35,37198,769],{},[110,37200,120],{"stretchy":112},[110,37202,123],{},[110,37204,113],{"stretchy":112},[35,37206,769],{},[110,37208,4815],{},[115,37210,117],{},[6816,37212,37213,37215],{},[110,37214,120],{"stretchy":112},[115,37216,2465],{},[39,37218,37219],{"encoding":41},"f(t)=(t-1)^2",[15,37221,37223,37250,37271],{"className":37222,"ariaHidden":46},[45],[15,37224,37226,37229,37232,37235,37238,37241,37244,37247],{"className":37225},[50],[15,37227],{"className":37228,"style":139},[54],[15,37230,37],{"className":37231,"style":61},[59,60],[15,37233,113],{"className":37234},[146],[15,37236,769],{"className":37237},[59,60],[15,37239,120],{"className":37240},[153],[15,37242],{"className":37243,"style":158},[157],[15,37245,123],{"className":37246},[162],[15,37248],{"className":37249,"style":158},[157],[15,37251,37253,37256,37259,37262,37265,37268],{"className":37252},[50],[15,37254],{"className":37255,"style":139},[54],[15,37257,113],{"className":37258},[146],[15,37260,769],{"className":37261},[59,60],[15,37263],{"className":37264,"style":2589},[157],[15,37266,4815],{"className":37267},[2593],[15,37269],{"className":37270,"style":2589},[157],[15,37272,37274,37278,37281],{"className":37273},[50],[15,37275],{"className":37276,"style":37277},[54],"height:1.1141em;vertical-align:-0.25em;",[15,37279,117],{"className":37280},[59],[15,37282,37284,37287],{"className":37283},[153],[15,37285,120],{"className":37286},[153],[15,37288,37290],{"className":37289},[281],[15,37291,37293],{"className":37292},[285],[15,37294,37296],{"className":37295},[290],[15,37297,37299],{"className":37298,"style":19761},[294],[15,37300,37301,37304],{"style":8279},[15,37302],{"className":37303,"style":303},[302],[15,37305,37307],{"className":37306},[307,308,309,310],[15,37308,2465],{"className":37309},[59,310],[11,37311,34708,37312,37377],{},[15,37313,37315,37338],{"className":37314,"translate":19},[18],[15,37316,37318],{"className":37317},[23],[25,37319,37320],{"xmlns":27},[29,37321,37322,37336],{},[32,37323,37324,37326,37328,37330,37332,37334],{},[35,37325,37],{},[110,37327,113],{"stretchy":112},[115,37329,117],{},[110,37331,120],{"stretchy":112},[110,37333,123],{},[115,37335,126],{},[39,37337,129],{"encoding":41},[15,37339,37341,37368],{"className":37340,"ariaHidden":46},[45],[15,37342,37344,37347,37350,37353,37356,37359,37362,37365],{"className":37343},[50],[15,37345],{"className":37346,"style":139},[54],[15,37348,37],{"className":37349,"style":61},[59,60],[15,37351,113],{"className":37352},[146],[15,37354,117],{"className":37355},[59],[15,37357,120],{"className":37358},[153],[15,37360],{"className":37361,"style":158},[157],[15,37363,123],{"className":37364},[162],[15,37366],{"className":37367,"style":158},[157],[15,37369,37371,37374],{"className":37370},[50],[15,37372],{"className":37373,"style":172},[54],[15,37375,126],{"className":37376},[59],"、凸，代入：",[15,37379,37381],{"className":37380,"translate":19},[180],[15,37382,37384,37491],{"className":37383,"translate":19},[18],[15,37385,37387],{"className":37386},[23],[25,37388,37389],{"xmlns":27,"display":189},[29,37390,37391,37488],{},[32,37392,37393,37399,37401,37403,37405,37407,37409,37411,37413,37415,37435,37437,37439,37441,37443,37463,37465,37467,37469,37471,37478,37480,37482,37484,37486],{},[195,37394,37395,37397],{},[35,37396,199],{},[35,37398,37],{},[110,37400,113],{"stretchy":112},[35,37402,206],{},[35,37404,210],{"mathvariant":209},[35,37406,213],{},[110,37408,120],{"stretchy":112},[110,37410,123],{},[110,37412,1289],{},[35,37414,944],{},[6816,37416,37417,37433],{},[32,37418,37419,37421,37427,37429,37431],{},[110,37420,113],{"fence":46},[238,37422,37423,37425],{},[35,37424,11],{},[35,37426,944],{},[110,37428,4815],{},[115,37430,117],{},[110,37432,120],{"fence":46},[115,37434,2465],{},[35,37436,244],{},[35,37438,774],{},[110,37440,123],{},[110,37442,1289],{},[238,37444,37445,37461],{},[32,37446,37447,37449,37451,37453,37455],{},[110,37448,113],{"stretchy":112},[35,37450,11],{},[110,37452,4815],{},[35,37454,944],{},[6816,37456,37457,37459],{},[110,37458,120],{"stretchy":112},[115,37460,2465],{},[35,37462,944],{},[1299,37464,1301],{},[35,37466,244],{},[35,37468,774],{},[110,37470,123],{},[6816,37472,37473,37476],{},[35,37474,37475],{},"χ",[115,37477,2465],{},[110,37479,113],{"stretchy":112},[35,37481,206],{},[35,37483,210],{"mathvariant":209},[35,37485,213],{},[110,37487,120],{"stretchy":112},[39,37489,37490],{"encoding":41},"D_f(P\\|Q)=\\int q\\left(\\frac{p}{q}-1\\right)^2dx\n=\\int 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卡方散度",[15,38293,38295,38356],{"className":38294,"translate":19},[18],[15,38296,38298],{"className":38297},[23],[25,38299,38300],{"xmlns":27},[29,38301,38302,38353],{},[32,38303,38304,38313,38315,38317,38319,38321,38323,38325,38327,38347,38349,38351],{},[7555,38305,38306,38308,38311],{},[35,38307,37475],{},[35,38309,38310],{},"N",[115,38312,2465],{},[110,38314,113],{"stretchy":112},[35,38316,206],{},[35,38318,210],{"mathvariant":209},[35,38320,213],{},[110,38322,120],{"stretchy":112},[110,38324,123],{},[110,38326,1289],{},[238,38328,38329,38345],{},[32,38330,38331,38333,38335,38337,38339],{},[110,38332,113],{"stretchy":112},[35,38334,944],{},[110,38336,4815],{},[35,38338,11],{},[6816,38340,38341,38343],{},[110,38342,120],{"stretchy":112},[115,38344,2465],{},[35,38346,11],{},[1299,38348,1301],{},[35,38350,244],{},[35,38352,774],{},[39,38354,38355],{"encoding":41},"\\chi_N^2(P\\|Q)=\\int\\frac{(q-p)^2}{p}\\,dx",[15,38357,38359,38443],{"className":38358,"ariaHidden":46},[45],[15,38360,38362,38366,38419,38422,38425,38428,38431,38434,38437,38440],{"className":38361},[50],[15,38363],{"className":38364,"style":38365},[54],"height:1.0894em;vertical-align:-0.2753em;",[15,38367,38369,38372],{"className":38368},[59],[15,38370,37475],{"className":38371},[59,60],[15,38373,38375],{"className":38374},[281],[15,38376,38378,38410],{"className":38377},[285,286],[15,38379,38381,38407],{"className":38380},[290],[15,38382,38384,38396],{"className":38383,"style":20054},[294],[15,38385,38387,38390],{"style":38386},"top:-2.4247em;margin-left:0em;margin-right:0.05em;",[15,38388],{"className":38389,"style":303},[302],[15,38391,38393],{"className":38392},[307,308,309,310],[15,38394,38310],{"className":38395,"style":25627},[59,60,310],[15,38397,38398,38401],{"style":6867},[15,38399],{"className":38400,"style":303},[302],[15,38402,38404],{"className":38403},[307,308,309,310],[15,38405,2465],{"className":38406},[59,310],[15,38408,318],{"className":38409},[317],[15,38411,38413],{"className":38412},[290],[15,38414,38417],{"className":38415,"style":38416},[294],"height:0.2753em;",[15,38418],{},[15,38420,113],{"className":38421},[146],[15,38423,206],{"className":38424,"style":334},[59,60],[15,38426,210],{"className":38427},[59],[15,38429,213],{"className":38430},[59,60],[15,38432,120],{"className":38433},[153],[15,38435],{"className":38436,"style":158},[157],[15,38438,123],{"className":38439},[162],[15,38441],{"className":38442,"style":158},[157],[15,38444,38446,38450,38453,38456,38565,38568,38571],{"className":38445},[50],[15,38447],{"className":38448,"style":38449},[54],"height:1.59em;vertical-align:-0.4811em;",[15,38451,1289],{"className":38452,"style":5619},[1486,1487,5618],[15,38454],{"className":38455,"style":406},[157],[15,38457,38459,38462,38562],{"className":38458},[59],[15,38460],{"className":38461},[146,441],[15,38463,38465],{"className":38464},[238],[15,38466,38468,38554],{"className":38467},[285,286],[15,38469,38471,38551],{"className":38470},[290],[15,38472,38475,38489,38497],{"className":38473,"style":38474},[294],"height:1.1089em;",[15,38476,38477,38480],{"style":2529},[15,38478],{"className":38479,"style":461},[302],[15,38481,38483],{"className":38482},[307,308,309,310],[15,38484,38486],{"className":38485},[59,310],[15,38487,11],{"className":38488},[59,60,310],[15,38490,38491,38494],{"style":473},[15,38492],{"className":38493,"style":461},[302],[15,38495],{"className":38496,"style":481},[480],[15,38498,38499,38502],{"style":26444},[15,38500],{"className":38501,"style":461},[302],[15,38503,38505],{"className":38504},[307,308,309,310],[15,38506,38508,38511,38514,38517,38520],{"className":38507},[59,310],[15,38509,113],{"className":38510},[146,310],[15,38512,944],{"className":38513,"style":981},[59,60,310],[15,38515,4815],{"className":38516},[2593,310],[15,38518,11],{"className":38519},[59,60,310],[15,38521,38523,38526],{"className":38522},[153,310],[15,38524,120],{"className":38525},[153,310],[15,38527,38529],{"className":38528},[281],[15,38530,38532],{"className":38531},[285],[15,38533,38535],{"className":38534},[290],[15,38536,38539],{"className":38537,"style":38538},[294],"height:0.8913em;",[15,38540,38542,38545],{"style":38541},"top:-2.931em;margin-right:0.0714em;",[15,38543],{"className":38544,"style":30232},[302],[15,38546,38548],{"className":38547},[307,30236,9676,310],[15,38549,2465],{"className":38550},[59,310],[15,38552,318],{"className":38553},[317],[15,38555,38557],{"className":38556},[290],[15,38558,38560],{"className":38559,"style":2741},[294],[15,38561],{},[15,38563],{"className":38564},[153,441],[15,38566],{"className":38567,"style":406},[157],[15,38569,244],{"className":38570},[59,60],[15,38572,774],{"className":38573},[59,60],[15,38575,38577,38605],{"className":38576,"translate":19},[18],[15,38578,38580],{"className":38579},[23],[25,38581,38582],{"xmlns":27},[29,38583,38584,38602],{},[32,38585,38586,38592,38594,38596,38598,38600],{},[6816,38587,38588,38590],{},[35,38589,37475],{},[115,38591,2465],{},[110,3859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h",[15,38696,38698,38781,38799],{"className":38697,"ariaHidden":46},[45],[15,38699,38701,38704,38772,38775,38778],{"className":38700},[50],[15,38702],{"className":38703,"style":3086},[54],[15,38705,38707,38710,38769],{"className":38706},[59],[15,38708],{"className":38709},[146,441],[15,38711,38713],{"className":38712},[238],[15,38714,38716,38761],{"className":38715},[285,286],[15,38717,38719,38758],{"className":38718},[290],[15,38720,38722,38736,38744],{"className":38721,"style":3105},[294],[15,38723,38724,38727],{"style":2529},[15,38725],{"className":38726,"style":461},[302],[15,38728,38730],{"className":38729},[307,308,309,310],[15,38731,38733],{"className":38732},[59,310],[15,38734,944],{"className":38735,"style":981},[59,60,310],[15,38737,38738,38741],{"style":473},[15,38739],{"className":38740,"style":461},[302],[15,38742],{"className":38743,"style":481},[480],[15,38745,38746,38749],{"style":2855},[15,38747],{"className":38748,"style":461},[302],[15,38750,38752],{"className":38751},[307,308,309,310],[15,38753,38755],{"className":38754},[59,310],[15,38756,11],{"className":38757},[59,60,310],[15,38759,318],{"className":38760},[317],[15,38762,38764],{"className":38763},[290],[15,38765,38767],{"className":38766,"style":2741},[294],[15,38768],{},[15,38770],{"className":38771},[153,441],[15,38773],{"className":38774,"style":158},[157],[15,38776,123],{"className":38777},[162],[15,38779],{"className":38780,"style":158},[157],[15,38782,38784,38787,38790,38793,38796],{"className":38783},[50],[15,38785],{"className":38786,"style":12582},[54],[15,38788,117],{"className":38789},[59],[15,38791],{"className":38792,"style":2589},[157],[15,38794,2472],{"className":38795},[2593],[15,38797],{"className":38798,"style":2589},[157],[15,38800,38802,38805,38808],{"className":38801},[50],[15,38803],{"className":38804,"style":12601},[54],[15,38806,38688],{"className":38807},[59,60],[15,38809,38691],{"className":38810},[59,60]," 展开）有 ",[15,38813,38815,38847],{"className":38814,"translate":19},[18],[15,38816,38818],{"className":38817},[23],[25,38819,38820],{"xmlns":27},[29,38821,38822,38844],{},[32,38823,38824,38830,38832,38838],{},[32,38825,38826,38828],{},[35,38827,10698],{"mathvariant":209},[35,38829,21919],{"mathvariant":209},[110,38831,32985],{},[238,38833,38834,38836],{},[115,38835,117],{},[115,38837,2465],{},[6816,38839,38840,38842],{},[35,38841,37475],{},[115,38843,2465],{},[39,38845,38846],{"encoding":41},"\\mathrm{KL}\\approx\\frac12\\chi^2",[15,38848,38850,38871],{"className":38849,"ariaHidden":46},[45],[15,38851,38853,38856,38862,38865,38868],{"className":38852},[50],[15,38854],{"className":38855,"style":664},[54],[15,38857,38859],{"className":38858},[59],[15,38860,32689],{"className":38861},[59,32688],[15,38863],{"className":38864,"style":158},[157],[15,38866,32985],{"className":38867},[162],[15,38869],{"className":38870,"style":158},[157],[15,38872,38874,38877,38945],{"className":38873},[50],[15,38875],{"className":38876,"style":2507},[54],[15,38878,38880,38883,38942],{"className":38879},[59],[15,38881],{"className":38882},[146,441],[15,38884,38886],{"className":38885},[238],[15,38887,38889,38934],{"className":38888},[285,286],[15,38890,38892,38931],{"className":38891},[290],[15,38893,38895,38909,38917],{"className":38894,"style":2526},[294],[15,38896,38897,38900],{"style":2529},[15,38898],{"className":38899,"style":461},[302],[15,38901,38903],{"className":38902},[307,308,309,310],[15,38904,38906],{"className":38905},[59,310],[15,38907,2465],{"className":38908},[59,310],[15,38910,38911,38914],{"style":473},[15,38912],{"className":38913,"style":461},[302],[15,38915],{"className":38916,"style":481},[480],[15,38918,38919,38922],{"style":2552},[15,38920],{"className":38921,"style":461},[302],[15,38923,38925],{"className":38924},[307,308,309,310],[15,38926,38928],{"className":38927},[59,310],[15,38929,117],{"className":38930},[59,310],[15,38932,318],{"className":38933},[317],[15,38935,38937],{"className":38936},[290],[15,38938,38940],{"className":38939,"style":2574},[294],[15,38941],{},[15,38943],{"className":38944},[153,441],[15,38946,38948,38951],{"className":38947},[59],[15,38949,37475],{"className":38950},[59,60],[15,38952,38954],{"className":38953},[281],[15,38955,38957],{"className":38956},[285],[15,38958,38960],{"className":38959},[290],[15,38961,38963],{"className":38962,"style":20054},[294],[15,38964,38965,38968],{"style":6867},[15,38966],{"className":38967,"style":303},[302],[15,38969,38971],{"className":38970},[307,308,309,310],[15,38972,2465],{"className":38973},[59,310],"、本文约定的 ",[15,38976,38978,39025],{"className":38977,"translate":19},[18],[15,38979,38981],{"className":38980},[23],[25,38982,38983],{"xmlns":27},[29,38984,38985,39022],{},[32,38986,38987,38989,38991,38995,38997,39001,39007,39009,39016],{},[110,38988,1289],{},[110,38990,113],{"stretchy":112},[34365,38992,38993],{},[35,38994,11],{},[110,38996,4815],{},[34365,38998,38999],{},[35,39000,944],{},[6816,39002,39003,39005],{},[110,39004,120],{"stretchy":112},[115,39006,2465],{},[110,39008,32985],{},[238,39010,39011,39013],{},[115,39012,117],{},[115,39014,39015],{},"4",[6816,39017,39018,39020],{},[35,39019,37475],{},[115,39021,2465],{},[39,39023,39024],{"encoding":41},"\\int(\\sqrt p-\\sqrt q)^2\\approx\\frac14\\chi^2",[15,39026,39028,39097,39187],{"className":39027,"ariaHidden":46},[45],[15,39029,39031,39035,39038,39041,39088,39091,39094],{"className":39030},[50],[15,39032],{"className":39033,"style":39034},[54],"height:1.1419em;vertical-align:-0.3369em;",[15,39036,1289],{"className":39037,"style":5619},[1486,1487,5618],[15,39039,113],{"className":39040},[146],[15,39042,39044],{"className":39043},[59,34393],[15,39045,39047,39079],{"className":39046},[285,286],[15,39048,39050,39076],{"className":39049},[290],[15,39051,39054,39063],{"className":39052,"style":39053},[294],"height:0.7031em;",[15,39055,39057,39060],{"className":39056,"style":11867},[6309],[15,39058],{"className":39059,"style":461},[302],[15,39061,11],{"className":39062,"style":34413},[59,60],[15,39064,39066,39069],{"style":39065},"top:-2.6631em;",[15,39067],{"className":39068,"style":461},[302],[15,39070,39072],{"className":39071,"style":34470},[34469],[6325,39073,39074],{"xmlns":6327,"width":6328,"height":34473,"viewBox":34474,"preserveAspectRatio":6331},[6333,39075],{"d":34477},[15,39077,318],{"className":39078},[317],[15,39080,39082],{"className":39081},[290],[15,39083,39086],{"className":39084,"style":39085},[294],"height:0.3369em;",[15,39087],{},[15,39089],{"className":39090,"style":2589},[157],[15,39092,4815],{"className":39093},[2593],[15,39095],{"className":39096,"style":2589},[157],[15,39098,39100,39104,39149,39178,39181,39184],{"className":39099},[50],[15,39101],{"className":39102,"style":39103},[54],"height:1.151em;vertical-align:-0.3369em;",[15,39105,39107],{"className":39106},[59,34393],[15,39108,39110,39141],{"className":39109},[285,286],[15,39111,39113,39138],{"className":39112},[290],[15,39114,39116,39126],{"className":39115,"style":39053},[294],[15,39117,39119,39122],{"className":39118,"style":11867},[6309],[15,39120],{"className":39121,"style":461},[302],[15,39123,944],{"className":39124,"style":39125},[59,60],"margin-right:0.0359em;padding-left:0.833em;",[15,39127,39128,39131],{"style":39065},[15,39129],{"className":39130,"style":461},[302],[15,39132,39134],{"className":39133,"style":34470},[34469],[6325,39135,39136],{"xmlns":6327,"width":6328,"height":34473,"viewBox":34474,"preserveAspectRatio":6331},[6333,39137],{"d":34477},[15,39139,318],{"className":39140},[317],[15,39142,39144],{"className":39143},[290],[15,39145,39147],{"className":39146,"style":39085},[294],[15,39148],{},[15,39150,39152,39155],{"className":39151},[153],[15,39153,120],{"className":39154},[153],[15,39156,39158],{"className":39157},[281],[15,39159,39161],{"className":39160},[285],[15,39162,39164],{"className":39163},[290],[15,39165,39167],{"className":39166,"style":20054},[294],[15,39168,39169,39172],{"style":6867},[15,39170],{"className":39171,"style":303},[302],[15,39173,39175],{"className":39174},[307,308,309,310],[15,39176,2465],{"className":39177},[59,310],[15,39179],{"className":39180,"style":158},[157],[15,39182,32985],{"className":39183},[162],[15,39185],{"className":39186,"style":158},[157],[15,39188,39190,39193,39261],{"className":39189},[50],[15,39191],{"className":39192,"style":2507},[54],[15,39194,39196,39199,39258],{"className":39195},[59],[15,39197],{"className":39198},[146,441],[15,39200,39202],{"className":39201},[238],[15,39203,39205,39250],{"className":39204},[285,286],[15,39206,39208,39247],{"className":39207},[290],[15,39209,39211,39225,39233],{"className":39210,"style":2526},[294],[15,39212,39213,39216],{"style":2529},[15,39214],{"className":39215,"style":461},[302],[15,39217,39219],{"className":39218},[307,308,309,310],[15,39220,39222],{"className":39221},[59,310],[15,39223,39015],{"className":39224},[59,310],[15,39226,39227,39230],{"style":473},[15,39228],{"className":39229,"style":461},[302],[15,39231],{"className":39232,"style":481},[480],[15,39234,39235,39238],{"style":2552},[15,39236],{"className":39237,"style":461},[302],[15,39239,39241],{"className":39240},[307,308,309,310],[15,39242,39244],{"className":39243},[59,310],[15,39245,117],{"className":39246},[59,310],[15,39248,318],{"className":39249},[317],[15,39251,39253],{"className":39252},[290],[15,39254,39256],{"className":39255,"style":2574},[294],[15,39257],{},[15,39259],{"className":39260},[153,441],[15,39262,39264,39267],{"className":39263},[59],[15,39265,37475],{"className":39266},[59,60],[15,39268,39270],{"className":39269},[281],[15,39271,39273],{"className":39272},[285],[15,39274,39276],{"className":39275},[290],[15,39277,39279],{"className":39278,"style":20054},[294],[15,39280,39281,39284],{"style":6867},[15,39282],{"className":39283,"style":303},[302],[15,39285,39287],{"className":39286},[307,308,309,310],[15,39288,2465],{"className":39289},[59,310],"、Bhattacharyya 约定的 ",[15,39292,39294,39339],{"className":39293,"translate":19},[18],[15,39295,39297],{"className":39296},[23],[25,39298,39299],{"xmlns":27},[29,39300,39301,39336],{},[32,39302,39303,39309,39311,39313,39315,39318,39321,39323,39330],{},[6816,39304,39305,39307],{},[35,39306,33910],{},[115,39308,2465],{},[110,39310,123],{},[115,39312,117],{},[110,39314,4815],{},[35,39316,39317],{},"B",[35,39319,39320],{},"C",[110,39322,32985],{},[238,39324,39325,39327],{},[115,39326,117],{},[115,39328,39329],{},"8",[6816,39331,39332,39334],{},[35,39333,37475],{},[115,39335,2465],{},[39,39337,39338],{"encoding":41},"H^2=1-BC\\approx\\frac18\\chi^2",[15,39340,39342,39386,39404,39426],{"className":39341,"ariaHidden":46},[45],[15,39343,39345,39348,39377,39380,39383],{"className":39344},[50],[15,39346],{"className":39347,"style":20054},[54],[15,39349,39351,39354],{"className":39350},[59],[15,39352,33910],{"className":39353,"style":34064},[59,60],[15,39355,39357],{"className":39356},[281],[15,39358,39360],{"className":39359},[285],[15,39361,39363],{"className":39362},[290],[15,39364,39366],{"className":39365,"style":20054},[294],[15,39367,39368,39371],{"style":6867},[15,39369],{"className":39370,"style":303},[302],[15,39372,39374],{"className":39373},[307,308,309,310],[15,39375,2465],{"className":39376},[59,310],[15,39378],{"className":39379,"style":158},[157],[15,39381,123],{"className":39382},[162],[15,39384],{"className":39385,"style":158},[157],[15,39387,39389,39392,39395,39398,39401],{"className":39388},[50],[15,39390],{"className":39391,"style":12582},[54],[15,39393,117],{"className":39394},[59],[15,39396],{"className":39397,"style":2589},[157],[15,39399,4815],{"className":39400},[2593],[15,39402],{"className":39403,"style":2589},[157],[15,39405,39407,39410,39414,39417,39420,39423],{"className":39406},[50],[15,39408],{"className":39409,"style":664},[54],[15,39411,39317],{"className":39412,"style":39413},[59,60],"margin-right:0.0502em;",[15,39415,39320],{"className":39416,"style":10786},[59,60],[15,39418],{"className":39419,"style":158},[157],[15,39421,32985],{"className":39422},[162],[15,39424],{"className":39425,"style":158},[157],[15,39427,39429,39432,39500],{"className":39428},[50],[15,39430],{"className":39431,"style":2507},[54],[15,39433,39435,39438,39497],{"className":39434},[59],[15,39436],{"className":39437},[146,441],[15,39439,39441],{"className":39440},[238],[15,39442,39444,39489],{"className":39443},[285,286],[15,39445,39447,39486],{"className":39446},[290],[15,39448,39450,39464,39472],{"className":39449,"style":2526},[294],[15,39451,39452,39455],{"style":2529},[15,39453],{"className":39454,"style":461},[302],[15,39456,39458],{"className":39457},[307,308,309,310],[15,39459,39461],{"className":39460},[59,310],[15,39462,39329],{"className":39463},[59,310],[15,39465,39466,39469],{"style":473},[15,39467],{"className":39468,"style":461},[302],[15,39470],{"className":39471,"style":481},[480],[15,39473,39474,39477],{"style":2552},[15,39475],{"className":39476,"style":461},[302],[15,39478,39480],{"className":39479},[307,308,309,310],[15,39481,39483],{"className":39482},[59,310],[15,39484,117],{"className":39485},[59,310],[15,39487,318],{"className":39488},[317],[15,39490,39492],{"className":39491},[290],[15,39493,39495],{"className":39494,"style":2574},[294],[15,39496],{},[15,39498],{"className":39499},[153,441],[15,39501,39503,39506],{"className":39502},[59],[15,39504,37475],{"className":39505},[59,60],[15,39507,39509],{"className":39508},[281],[15,39510,39512],{"className":39511},[285],[15,39513,39515],{"className":39514},[290],[15,39516,39518],{"className":39517,"style":20054},[294],[15,39519,39520,39523],{"style":6867},[15,39521],{"className":39522,"style":303},[302],[15,39524,39526],{"className":39525},[307,308,309,310],[15,39527,2465],{"className":39528},[59,310],"（见 4.6 节的局部展开），因此它是所有 f-散度在局部的「公因子」——只是在 ",[15,39531,39533,39561],{"className":39532,"translate":19},[18],[15,39534,39536],{"className":39535},[23],[25,39537,39538],{"xmlns":27},[29,39539,39540,39558],{},[32,39541,39542,39552,39554,39556],{},[6816,39543,39544,39546],{},[35,39545,37],{},[32,39547,39548,39550],{},[110,39549,6822],{"mathvariant":209},[110,39551,6822],{"mathvariant":209},[110,39553,113],{"stretchy":112},[115,39555,117],{},[110,39557,120],{"stretchy":112},[39,39559,39560],{"encoding":41},"f''(1)",[15,39562,39564],{"className":39563,"ariaHidden":46},[45],[15,39565,39567,39570,39602,39605,39608],{"className":39566},[50],[15,39568],{"className":39569,"style":6845},[54],[15,39571,39573,39576],{"className":39572},[59],[15,39574,37],{"className":39575,"style":61},[59,60],[15,39577,39579],{"className":39578},[281],[15,39580,39582],{"className":39581},[285],[15,39583,39585],{"className":39584},[290],[15,39586,39588],{"className":39587,"style":6864},[294],[15,39589,39590,39593],{"style":6867},[15,39591],{"className":39592,"style":303},[302],[15,39594,39596],{"className":39595},[307,308,309,310],[15,39597,39599],{"className":39598},[59,310],[15,39600,21774],{"className":39601},[59,310],[15,39603,113],{"className":39604},[146],[15,39606,117],{"className":39607},[59],[15,39609,120],{"className":39610},[153]," 意义下的「单位」不同。",[11,39613,39614,39616,39617,39949,39950,39978,39979,2747,40007,40091,40092,40120,40121,40149,40150,40178,40179,40207,40208,40236,40237,40240,40241,40399,40400,40460],{},[998,39615,24219],{}," 卡方散度是 Pearson 拟合优度统计量 ",[15,39618,39620,39670],{"className":39619,"translate":19},[18],[15,39621,39623],{"className":39622},[23],[25,39624,39625],{"xmlns":27},[29,39626,39627,39667],{},[32,39628,39629,39635],{},[195,39630,39631,39633],{},[110,39632,13465],{},[35,39634,11967],{},[238,39636,39637,39661],{},[32,39638,39639,39641,39647,39649,39655],{},[110,39640,113],{"stretchy":112},[195,39642,39643,39645],{},[35,39644,19247],{},[35,39646,11967],{},[110,39648,4815],{},[195,39650,39651,39653],{},[35,39652,223],{},[35,39654,11967],{},[6816,39656,39657,39659],{},[110,39658,120],{"stretchy":112},[115,39660,2465],{},[195,39662,39663,39665],{},[35,39664,223],{},[35,39666,11967],{},[39,39668,39669],{"encoding":41},"\\sum_i\\frac{(O_i-E_i)^2}{E_i}",[15,39671,39673],{"className":39672,"ariaHidden":46},[45],[15,39674,39676,39680,39724,39727],{"className":39675},[50],[15,39677],{"className":39678,"style":39679},[54],"height:1.554em;vertical-align:-0.4451em;",[15,39681,39683,39687],{"className":39682},[1486],[15,39684,13465],{"className":39685,"style":39686},[1486,1487,5618],"position:relative;top:0em;",[15,39688,39690],{"className":39689},[281],[15,39691,39693,39715],{"className":39692},[285,286],[15,39694,39696,39712],{"className":39695},[290],[15,39697,39700],{"className":39698,"style":39699},[294],"height:0.162em;",[15,39701,39703,39706],{"style":39702},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[15,39704],{"className":39705,"style":303},[302],[15,39707,39709],{"className":39708},[307,308,309,310],[15,39710,11967],{"className":39711},[59,60,310],[15,39713,318],{"className":39714},[317],[15,39716,39718],{"className":39717},[290],[15,39719,39722],{"className":39720,"style":39721},[294],"height:0.2997em;",[15,39723],{},[15,39725],{"className":39726,"style":406},[157],[15,39728,39730,39733,39946],{"className":39729},[59],[15,39731],{"className":39732},[146,441],[15,39734,39736],{"className":39735},[238],[15,39737,39739,39937],{"className":39738},[285,286],[15,39740,39742,39934],{"className":39741},[290],[15,39743,39745,39799,39807],{"className":39744,"style":38474},[294],[15,39746,39747,39750],{"style":2529},[15,39748],{"className":39749,"style":461},[302],[15,39751,39753],{"className":39752},[307,308,309,310],[15,39754,39756],{"className":39755},[59,310],[15,39757,39759,39762],{"className":39758},[59,310],[15,39760,223],{"className":39761,"style":25887},[59,60,310],[15,39763,39765],{"className":39764},[281],[15,39766,39768,39790],{"className":39767},[285,286],[15,39769,39771,39787],{"className":39770},[290],[15,39772,39775],{"className":39773,"style":39774},[294],"height:0.3281em;",[15,39776,39778,39781],{"style":39777},"top:-2.357em;margin-left:-0.0576em;margin-right:0.0714em;",[15,39779],{"className":39780,"style":30232},[302],[15,39782,39784],{"className":39783},[307,30236,9676,310],[15,39785,11967],{"className":39786},[59,60,310],[15,39788,318],{"className":39789},[317],[15,39791,39793],{"className":39792},[290],[15,39794,39797],{"className":39795,"style":39796},[294],"height:0.143em;",[15,39798],{},[15,39800,39801,39804],{"style":473},[15,39802],{"className":39803,"style":461},[302],[15,39805],{"className":39806,"style":481},[480],[15,39808,39809,39812],{"style":26444},[15,39810],{"className":39811,"style":461},[302],[15,39813,39815],{"className":39814},[307,308,309,310],[15,39816,39818,39821,39862,39865,39905],{"className":39817},[59,310],[15,39819,113],{"className":39820},[146,310],[15,39822,39824,39827],{"className":39823},[59,310],[15,39825,19247],{"className":39826,"style":277},[59,60,310],[15,39828,39830],{"className":39829},[281],[15,39831,39833,39854],{"className":39832},[285,286],[15,39834,39836,39851],{"className":39835},[290],[15,39837,39839],{"className":39838,"style":39774},[294],[15,39840,39842,39845],{"style":39841},"top:-2.357em;margin-left:-0.0278em;margin-right:0.0714em;",[15,39843],{"className":39844,"style":30232},[302],[15,39846,39848],{"className":39847},[307,30236,9676,310],[15,39849,11967],{"className":39850},[59,60,310],[15,39852,318],{"className":39853},[317],[15,39855,39857],{"className":39856},[290],[15,39858,39860],{"className":39859,"style":39796},[294],[15,39861],{},[15,39863,4815],{"className":39864},[2593,310],[15,39866,39868,39871],{"className":39867},[59,310],[15,39869,223],{"className":39870,"style":25887},[59,60,310],[15,39872,39874],{"className":39873},[281],[15,39875,39877,39897],{"className":39876},[285,286],[15,39878,39880,39894],{"className":39879},[290],[15,39881,39883],{"className":39882,"style":39774},[294],[15,39884,39885,39888],{"style":39777},[15,39886],{"className":39887,"style":30232},[302],[15,39889,39891],{"className":39890},[307,30236,9676,310],[15,39892,11967],{"className":39893},[59,60,310],[15,39895,318],{"className":39896},[317],[15,39898,39900],{"className":39899},[290],[15,39901,39903],{"className":39902,"style":39796},[294],[15,39904],{},[15,39906,39908,39911],{"className":39907},[153,310],[15,39909,120],{"className":39910},[153,310],[15,39912,39914],{"className":39913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的总体（无限样本）版本：把观测频率看作 ",[15,39951,39953,39966],{"className":39952,"translate":19},[18],[15,39954,39956],{"className":39955},[23],[25,39957,39958],{"xmlns":27},[29,39959,39960,39964],{},[32,39961,39962],{},[35,39963,206],{},[39,39965,206],{"encoding":41},[15,39967,39969],{"className":39968,"ariaHidden":46},[45],[15,39970,39972,39975],{"className":39971},[50],[15,39973],{"className":39974,"style":664},[54],[15,39976,206],{"className":39977,"style":334},[59,60],"、期望频率看作 ",[15,39980,39982,39995],{"className":39981,"translate":19},[18],[15,39983,39985],{"className":39984},[23],[25,39986,39987],{"xmlns":27},[29,39988,39989,39993],{},[32,39990,39991],{},[35,39992,213],{},[39,39994,213],{"encoding":41},[15,39996,39998],{"className":39997,"ariaHidden":46},[45],[15,39999,40001,40004],{"className":40000},[50],[15,40002],{"className":40003,"style":566},[54],[15,40005,213],{"className":40006},[59,60],[15,40008,40010,40038],{"className":40009,"translate":19},[18],[15,40011,40013],{"className":40012},[23],[25,40014,40015],{"xmlns":27},[29,40016,40017,40035],{},[32,40018,40019,40025,40027,40029,40031,40033],{},[6816,40020,40021,40023],{},[35,40022,37475],{},[115,40024,2465],{},[110,40026,113],{"stretchy":112},[35,40028,206],{},[35,40030,210],{"mathvariant":209},[35,40032,213],{},[110,40034,120],{"stretchy":112},[39,40036,40037],{"encoding":41},"\\chi^2(P\\|Q)",[15,40039,40041],{"className":40040,"ariaHidden":46},[45],[15,40042,40044,40047,40076,40079,40082,40085,40088],{"className":40043},[50],[15,40045],{"className":40046,"style":20029},[54],[15,40048,40050,40053],{"className":40049},[59],[15,40051,37475],{"className":40052},[59,60],[15,40054,40056],{"className":40055},[281],[15,40057,40059],{"className":40058},[285],[15,40060,40062],{"className":40061},[290],[15,40063,40065],{"className":40064,"style":20054},[294],[15,40066,40067,40070],{"style":6867},[15,40068],{"className":40069,"style":303},[302],[15,40071,40073],{"className":40072},[307,308,309,310],[15,40074,2465],{"className":40075},[59,310],[15,40077,113],{"className":40078},[146],[15,40080,206],{"className":40081,"style":334},[59,60],[15,40083,210],{"className":40084},[59],[15,40086,213],{"className":40087},[59,60],[15,40089,120],{"className":40090},[153]," 就是「",[15,40093,40095,40108],{"className":40094,"translate":19},[18],[15,40096,40098],{"className":40097},[23],[25,40099,40100],{"xmlns":27},[29,40101,40102,40106],{},[32,40103,40104],{},[35,40105,206],{},[39,40107,206],{"encoding":41},[15,40109,40111],{"className":40110,"ariaHidden":46},[45],[15,40112,40114,40117],{"className":40113},[50],[15,40115],{"className":40116,"style":664},[54],[15,40118,206],{"className":40119,"style":334},[59,60]," 是否来自 ",[15,40122,40124,40137],{"className":40123,"translate":19},[18],[15,40125,40127],{"className":40126},[23],[25,40128,40129],{"xmlns":27},[29,40130,40131,40135],{},[32,40132,40133],{},[35,40134,213],{},[39,40136,213],{"encoding":41},[15,40138,40140],{"className":40139,"ariaHidden":46},[45],[15,40141,40143,40146],{"className":40142},[50],[15,40144],{"className":40145,"style":566},[54],[15,40147,213],{"className":40148},[59,60],"」的标准化偏差，因此它是独立性检验、拟合优度检验与特征选择的自然尺度。它的分母 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",[15,40209,40211,40224],{"className":40210,"translate":19},[18],[15,40212,40214],{"className":40213},[23],[25,40215,40216],{"xmlns":27},[29,40217,40218,40222],{},[32,40219,40220],{},[35,40221,206],{},[39,40223,206],{"encoding":41},[15,40225,40227],{"className":40226,"ariaHidden":46},[45],[15,40228,40230,40233],{"className":40229},[50],[15,40231],{"className":40232,"style":664},[54],[15,40234,206],{"className":40235,"style":334},[59,60]," 不小」的稀有区域极度敏感（",[998,40238,40239],{},"尾部敏感","）：这既是它能捕捉稀有事件的优点，也是它容易被少量异常值带偏的缺点。在 f-GAN 家族里它对应最小二乘 GAN（LSGAN）：因为 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是二次的，判别器梯度不会像 JS 那样饱和；在鲁棒统计与核方法中，",[15,40401,40403,40421],{"className":40402,"translate":19},[18],[15,40404,40406],{"className":40405},[23],[25,40407,40408],{"xmlns":27},[29,40409,40410,40418],{},[32,40411,40412],{},[6816,40413,40414,40416],{},[35,40415,37475],{},[115,40417,2465],{},[39,40419,40420],{"encoding":41},"\\chi^2",[15,40422,40424],{"className":40423,"ariaHidden":46},[45],[15,40425,40427,40431],{"className":40426},[50],[15,40428],{"className":40429,"style":40430},[54],"height:1.0085em;vertical-align:-0.1944em;",[15,40432,40434,40437],{"className":40433},[59],[15,40435,37475],{"className":40436},[59,60],[15,40438,40440],{"className":40439},[281],[15,40441,40443],{"className":40442},[285],[15,40444,40446],{"className":40445},[290],[15,40447,40449],{"className":40448,"style":20054},[294],[15,40450,40451,40454],{"style":6867},[15,40452],{"className":40453,"style":303},[302],[15,40455,40457],{"className":40456},[307,308,309,310],[15,40458,2465],{"className":40459},[59,310]," 距离还给出了带权核的另一种「局部」相似度。",[2313,40462,40464],{"id":40463},"_35-hellinger-平方","3.5 Hellinger 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凸，代入：",[15,40965,40967],{"className":40966,"translate":19},[180],[15,40968,40970,41060],{"className":40969,"translate":19},[18],[15,40971,40973],{"className":40972},[23],[25,40974,40975],{"xmlns":27,"display":189},[29,40976,40977,41057],{},[32,40978,40979,40985,40987,40989,40991,40993,40995,40997,40999,41001,41023,41025,41027,41029,41031,41033,41037,41039,41043,41049,41051,41053,41055],{},[195,40980,40981,40983],{},[35,40982,199],{},[35,40984,37],{},[110,40986,113],{"stretchy":112},[35,40988,206],{},[35,40990,210],{"mathvariant":209},[35,40992,213],{},[110,40994,120],{"stretchy":112},[110,40996,123],{},[110,40998,1289],{},[35,41000,944],{},[6816,41002,41003,41021],{},[32,41004,41005,41007,41015,41017,41019],{},[110,41006,113],{"fence":46},[34365,41008,41009],{},[238,41010,41011,41013],{},[35,41012,11],{},[35,41014,944],{},[110,41016,4815],{},[115,41018,117],{},[110,41020,120],{"fence":46},[115,41022,2465],{},[35,41024,244],{},[35,41026,774],{},[110,41028,123],{},[110,41030,1289],{},[110,41032,113],{"stretchy":112},[34365,41034,41035],{},[35,41036,11],{},[110,41038,4815],{},[34365,41040,41041],{},[35,41042,944],{},[6816,41044,41045,41047],{},[110,41046,120],{"stretchy":112},[115,41048,2465],{},[1299,41050,1301],{},[35,41052,244],{},[35,41054,774],{},[35,41056,4377],{"mathvariant":209},[39,41058,41059],{"encoding":41},"D_f(P\\|Q)\n=\\int q\\left(\\sqrt{\\frac{p}{q}}-1\\right)^2dx\n=\\int (\\sqrt{p}-\\sqrt{q})^2\\,dx .",[15,41061,41063,41133,41338,41408],{"className":41062,"ariaHidden":46},[45],[15,41064,41066,41069,41109,41112,41115,41118,41121,41124,41127,41130],{"className":41065},[50],[15,41067],{"className":41068,"style":270},[54],[15,41070,41072,41075],{"className":41071},[59],[15,41073,199],{"className":41074,"style":277},[59,60],[15,41076,41078],{"className":41077},[281],[15,41079,41081,41101],{"className":41080},[285,286],[15,41082,41084,41098],{"className":41083},[290],[15,41085,41087],{"className":41086,"style":295},[294],[15,41088,41089,41092],{"style":298},[15,41090],{"className":41091,"style":303},[302],[15,41093,41095],{"className":41094},[307,308,309,310],[15,41096,37],{"className":41097,"style":61},[59,60,310],[15,41099,318],{"className":41100},[317],[15,41102,41104],{"className":41103},[290],[15,41105,41107],{"className":41106,"style":325},[294],[15,41108],{},[15,41110,113],{"className":41111},[146],[15,41113,206],{"className":41114,"style":334},[59,60],[15,41116,210],{"className":41117},[59],[15,41119,213],{"className":41120},[59,60],[15,41122,120],{"className":41123},[153],[15,41125],{"className":41126,"style":158},[157],[15,41128,123],{"className":41129},[162],[15,41131],{"className":41132,"style":158},[157],[15,41134,41136,41140,41143,41146,41149,41152,41320,41323,41326,41329,41332,41335],{"className":41135},[50],[15,41137],{"className":41138,"style":41139},[54],"height:2.6466em;vertical-align:-0.9926em;",[15,41141,1289],{"className":41142,"style":1489},[1486,1487,1488],[15,41144],{"className":41145,"style":406},[157],[15,41147,944],{"className":41148,"style":981},[59,60],[15,41150],{"className":41151,"style":406},[157],[15,41153,41155,41297],{"className":41154},[410],[15,41156,41158,41164,41279,41282,41285,41288,41291],{"className":41157},[410],[15,41159,41161],{"className":41160,"style":415},[146,414],[15,41162,113],{"className":41163},[419,309],[15,41165,41167],{"className":41166},[59,34393],[15,41168,41170,41270],{"className":41169},[285,286],[15,41171,41173,41267],{"className":41172},[290],[15,41174,41177,41250],{"className":41175,"style":41176},[294],"height:1.4474em;",[15,41178,41181,41184],{"className":41179,"style":41180},[6309],"top:-4.4em;",[15,41182],{"className":41183,"style":35071},[302],[15,41185,41188],{"className":41186,"style":41187},[59],"padding-left:1em;",[15,41189,41191,41194,41247],{"className":41190},[59],[15,41192],{"className":41193},[146,441],[15,41195,41197],{"className":41196},[238],[15,41198,41200,41239],{"className":41199},[285,286],[15,41201,41203,41236],{"className":41202},[290],[15,41204,41206,41217,41225],{"className":41205,"style":3556},[294],[15,41207,41208,41211],{"style":457},[15,41209],{"className":41210,"style":461},[302],[15,41212,41214],{"className":41213},[59],[15,41215,944],{"className":41216,"style":981},[59,60],[15,41218,41219,41222],{"style":473},[15,41220],{"className":41221,"style":461},[302],[15,41223],{"className":41224,"style":481},[480],[15,41226,41227,41230],{"style":484},[15,41228],{"className":41229,"style":461},[302],[15,41231,41233],{"className":41232},[59],[15,41234,11],{"className":41235},[59,60],[15,41237,318],{"className":41238},[317],[15,41240,41242],{"className":41241},[290],[15,41243,41245],{"className":41244,"style":506},[294],[15,41246],{},[15,41248],{"className":41249},[153,441],[15,41251,41253,41256],{"style":41252},"top:-3.4074em;",[15,41254],{"className":41255,"style":35071},[302],[15,41257,41260],{"className":41258,"style":41259},[34469],"min-width:1.02em;height:2.48em;",[6325,41261,41264],{"xmlns":6327,"width":6328,"height":41262,"viewBox":41263,"preserveAspectRatio":6331},"2.48em","0 0 400000 2592",[6333,41265],{"d":41266},"M424,2478\nc-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514\nc0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20\ns-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121\ns209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081\nl0 -0c4,-6.7,10,-10,18,-10 H400000\nv40H1014.6\ns-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185\nc-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2z M1001 80\nh400000v40h-400000z",[15,41268,318],{"className":41269},[317],[15,41271,41273],{"className":41272},[290],[15,41274,41277],{"className":41275,"style":41276},[294],"height:0.9926em;",[15,41278],{},[15,41280],{"className":41281,"style":2589},[157],[15,41283,4815],{"className":41284},[2593],[15,41286],{"className":41287,"style":2589},[157],[15,41289,117],{"className":41290},[59],[15,41292,41294],{"className":41293,"style":415},[153,414],[15,41295,120],{"className":41296},[419,309],[15,41298,41300],{"className":41299},[281],[15,41301,41303],{"className":41302},[285],[15,41304,41306],{"className":41305},[290],[15,41307,41309],{"className":41308,"style":37687},[294],[15,41310,41311,41314],{"style":37690},[15,41312],{"className":41313,"style":303},[302],[15,41315,41317],{"className":41316},[307,308,309,310],[15,41318,2465],{"className":41319},[59,310],[15,41321],{"className":41322,"style":406},[157],[15,41324,244],{"className":41325},[59,60],[15,41327,774],{"className":41328},[59,60],[15,41330],{"className":41331,"style":158},[157],[15,41333,123],{"className":41334},[162],[15,41336],{"className":41337,"style":158},[157],[15,41339,41341,41344,41347,41350,41399,41402,41405],{"className":41340},[50],[15,41342],{"className":41343,"style":5779},[54],[15,41345,1289],{"className":41346,"style":1489},[1486,1487,1488],[15,41348,113],{"className":41349},[146],[15,41351,41353],{"className":41352},[59,34393],[15,41354,41356,41390],{"className":41355},[285,286],[15,41357,41359,41387],{"className":41358},[290],[15,41360,41362,41374],{"className":41361,"style":6864},[294],[15,41363,41365,41368],{"className":41364,"style":11867},[6309],[15,41366],{"className":41367,"style":461},[302],[15,41369,41371],{"className":41370,"style":34413},[59],[15,41372,11],{"className":41373},[59,60],[15,41375,41377,41380],{"style":41376},"top:-2.7119em;",[15,41378],{"className":41379,"style":461},[302],[15,41381,41383],{"className":41382,"style":34470},[34469],[6325,41384,41385],{"xmlns":6327,"width":6328,"height":34473,"viewBox":34474,"preserveAspectRatio":6331},[6333,41386],{"d":34477},[15,41388,318],{"className":41389},[317],[15,41391,41393],{"className":41392},[290],[15,41394,41397],{"className":41395,"style":41396},[294],"height:0.2881em;",[15,41398],{},[15,41400],{"className":41401,"style":2589},[157],[15,41403,4815],{"className":41404},[2593],[15,41406],{"className":41407,"style":2589},[157],[15,41409,41411,41415,41462,41491,41494,41497,41500],{"className":41410},[50],[15,41412],{"className":41413,"style":41414},[54],"height:1.1522em;vertical-align:-0.2881em;",[15,41416,41418],{"className":41417},[59,34393],[15,41419,41421,41454],{"className":41420},[285,286],[15,41422,41424,41451],{"className":41423},[290],[15,41425,41427,41439],{"className":41426,"style":6864},[294],[15,41428,41430,41433],{"className":41429,"style":11867},[6309],[15,41431],{"className":41432,"style":461},[302],[15,41434,41436],{"className":41435,"style":34413},[59],[15,41437,944],{"className":41438,"style":981},[59,60],[15,41440,41441,41444],{"style":41376},[15,41442],{"className":41443,"style":461},[302],[15,41445,41447],{"className":41446,"style":34470},[34469],[6325,41448,41449],{"xmlns":6327,"width":6328,"height":34473,"viewBox":34474,"preserveAspectRatio":6331},[6333,41450],{"d":34477},[15,41452,318],{"className":41453},[317],[15,41455,41457],{"className":41456},[290],[15,41458,41460],{"className":41459,"style":41396},[294],[15,41461],{},[15,41463,41465,41468],{"className":41464},[153],[15,41466,120],{"className":41467},[153],[15,41469,41471],{"className":41470},[281],[15,41472,41474],{"className":41473},[285],[15,41475,41477],{"className":41476},[290],[15,41478,41480],{"className":41479,"style":19761},[294],[15,41481,41482,41485],{"style":8279},[15,41483],{"className":41484,"style":303},[302],[15,41486,41488],{"className":41487},[307,308,309,310],[15,41489,2465],{"className":41490},[59,310],[15,41492],{"className":41493,"style":406},[157],[15,41495,244],{"className":41496},[59,60],[15,41498,774],{"className":41499},[59,60],[15,41501,4377],{"className":41502},[59],[11,41504,41505,41506,41664],{},"记 Bhattacharyya 系数 ",[15,41507,41509,41553],{"className":41508,"translate":19},[18],[15,41510,41512],{"className":41511},[23],[25,41513,41514],{"xmlns":27},[29,41515,41516,41550],{},[32,41517,41518,41520,41522,41524,41526,41528,41530,41532,41534,41536,41544,41546,41548],{},[35,41519,39317],{},[35,41521,39320],{},[110,41523,113],{"stretchy":112},[35,41525,206],{},[110,41527,551],{"separator":46},[35,41529,213],{},[110,41531,120],{"stretchy":112},[110,41533,123],{},[110,41535,1289],{},[34365,41537,41538],{},[32,41539,41540,41542],{},[35,41541,11],{},[35,41543,944],{},[1299,41545,1301],{},[35,41547,244],{},[35,41549,774],{},[39,41551,41552],{"encoding":41},"BC(P,Q)=\\int\\sqrt{pq}\\,dx",[15,41554,41556,41595],{"className":41555,"ariaHidden":46},[45],[15,41557,41559,41562,41565,41568,41571,41574,41577,41580,41583,41586,41589,41592],{"className":41558},[50],[15,41560],{"className":41561,"style":139},[54],[15,41563,39317],{"className":41564,"style":39413},[59,60],[15,41566,39320],{"className":41567,"style":10786},[59,60],[15,41569,113],{"className":41570},[146],[15,41572,206],{"className":41573,"style":334},[59,60],[15,41575,551],{"className":41576},[573],[15,41578],{"className":41579,"style":406},[157],[15,41581,213],{"className":41582},[59,60],[15,41584,120],{"className":41585},[153],[15,41587],{"className":41588,"style":158},[157],[15,41590,123],{"className":41591},[162],[15,41593],{"className":41594,"style":158},[157],[15,41596,41598,41601,41604,41607,41655,41658,41661],{"className":41597},[50],[15,41599],{"className":41600,"style":39034},[54],[15,41602,1289],{"className":41603,"style":5619},[1486,1487,5618],[15,41605],{"className":41606,"style":406},[157],[15,41608,41610],{"className":41609},[59,34393],[15,41611,41613,41647],{"className":41612},[285,286],[15,41614,41616,41644],{"className":41615},[290],[15,41617,41619,41632],{"className":41618,"style":39053},[294],[15,41620,41622,41625],{"className":41621,"style":11867},[6309],[15,41623],{"className":41624,"style":461},[302],[15,41626,41628],{"className":41627,"style":34413},[59],[15,41629,41631],{"className":41630,"style":981},[59,60],"pq",[15,41633,41634,41637],{"style":39065},[15,41635],{"className":41636,"style":461},[302],[15,41638,41640],{"className":41639,"style":34470},[34469],[6325,41641,41642],{"xmlns":6327,"width":6328,"height":34473,"viewBox":34474,"preserveAspectRatio":6331},[6333,41643],{"d":34477},[15,41645,318],{"className":41646},[317],[15,41648,41650],{"className":41649},[290],[15,41651,41653],{"className":41652,"style":39085},[294],[15,41654],{},[15,41656],{"className":41657,"style":406},[157],[15,41659,244],{"className":41660},[59,60],[15,41662,774],{"className":41663},[59,60],"，则",[15,41666,41668],{"className":41667,"translate":19},[180],[15,41669,41671,41747],{"className":41670,"translate":19},[18],[15,41672,41674],{"className":41673},[23],[25,41675,41676],{"xmlns":27,"display":189},[29,41677,41678,41744],{},[32,41679,41680,41682,41684,41688,41690,41694,41700,41702,41704,41706,41708,41710,41712,41714,41716,41718,41726,41728,41730,41732,41734,41736,41738,41740,41742],{},[110,41681,1289],{},[110,41683,113],{"stretchy":112},[34365,41685,41686],{},[35,41687,11],{},[110,41689,4815],{},[34365,41691,41692],{},[35,41693,944],{},[6816,41695,41696,41698],{},[110,41697,120],{"stretchy":112},[115,41699,2465],{},[110,41701,123],{},[110,41703,1289],{},[35,41705,11],{},[110,41707,2472],{},[110,41709,1289],{},[35,41711,944],{},[110,41713,4815],{},[115,41715,2465],{},[110,41717,1289],{},[34365,41719,41720],{},[32,41721,41722,41724],{},[35,41723,11],{},[35,41725,944],{},[110,41727,123],{},[115,41729,2465],{},[110,41731,113],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[115,41733,117],{},[110,41735,4815],{},[35,41737,39317],{},[35,41739,39320],{},[110,41741,120],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[35,41743,4377],{"mathvariant":209},[39,41745,41746],{"encoding":41},"\\int(\\sqrt p-\\sqrt q)^2\n=\\int p+\\int q-2\\int\\sqrt{pq}\n=2\\big(1-BC\\big).",[15,41748,41750,41815,41903,41927,41951,42025,42052],{"className":41749,"ariaHidden":46},[45],[15,41751,41753,41756,41759,41762,41806,41809,41812],{"className":41752},[50],[15,41754],{"className":41755,"style":5779},[54],[15,41757,1289],{"className":41758,"style":1489},[1486,1487,1488],[15,41760,113],{"className":41761},[146],[15,41763,41765],{"className":41764},[59,34393],[15,41766,41768,41798],{"className":41767},[285,286],[15,41769,41771,41795],{"className":41770},[290],[15,41772,41774,41783],{"className":41773,"style":6864},[294],[15,41775,41777,41780],{"className":41776,"style":11867},[6309],[15,41778],{"className":41779,"style":461},[302],[15,41781,11],{"className":41782,"style":34413},[59,60],[15,41784,41785,41788],{"style":41376},[15,41786],{"className":41787,"style":461},[302],[15,41789,41791],{"className":41790,"style":34470},[34469],[6325,41792,41793],{"xmlns":6327,"width":6328,"height":34473,"viewBox":34474,"preserveAspectRatio":6331},[6333,41794],{"d":34477},[15,41796,318],{"className":41797},[317],[15,41799,41801],{"className":41800},[290],[15,41802,41804],{"className":41803,"style":41396},[294],[15,41805],{},[15,41807],{"className":41808,"style":2589},[157],[15,41810,4815],{"className":41811},[2593],[15,41813],{"className":41814,"style":2589},[157],[15,41816,41818,41821,41865,41894,41897,41900],{"className":41817},[50],[15,41819],{"className":41820,"style":41414},[54],[15,41822,41824],{"className":41823},[59,34393],[15,41825,41827,41857],{"className":41826},[285,286],[15,41828,41830,41854],{"className":41829},[290],[15,41831,41833,41842],{"className":41832,"style":6864},[294],[15,41834,41836,41839],{"className":41835,"style":11867},[6309],[15,41837],{"className":41838,"style":461},[302],[15,41840,944],{"className":41841,"style":39125},[59,60],[15,41843,41844,41847],{"style":41376},[15,41845],{"className":41846,"style":461},[302],[15,41848,41850],{"className":41849,"style":34470},[34469],[6325,41851,41852],{"xmlns":6327,"width":6328,"height":34473,"viewBox":34474,"preserveAspectRatio":6331},[6333,41853],{"d":34477},[15,41855,318],{"className":41856},[317],[15,41858,41860],{"className":41859},[290],[15,41861,41863],{"className":41862,"style":41396},[294],[15,41864],{},[15,41866,41868,41871],{"className":41867},[153],[15,41869,120],{"className":41870},[153],[15,41872,41874],{"className":41873},[281],[15,41875,41877],{"className":41876},[285],[15,41878,41880],{"className":41879},[290],[15,41881,41883],{"className":41882,"style":19761},[294],[15,41884,41885,41888],{"style":8279},[15,41886],{"className":41887,"style":303},[302],[15,41889,41891],{"className":41890},[307,308,309,310],[15,41892,2465],{"className":41893},[59,310],[15,41895],{"className":41896,"style":158},[157],[15,41898,123],{"className":41899},[162],[15,41901],{"className":41902,"style":158},[157],[15,41904,41906,41909,41912,41915,41918,41921,41924],{"className":41905},[50],[15,41907],{"className":41908,"style":5779},[54],[15,41910,1289],{"className":41911,"style":1489},[1486,1487,1488],[15,41913],{"className":41914,"style":406},[157],[15,41916,11],{"className":41917},[59,60],[15,41919],{"className":41920,"style":2589},[157],[15,41922,2472],{"className":41923},[2593],[15,41925],{"className":41926,"style":2589},[157],[15,41928,41930,41933,41936,41939,41942,41945,41948],{"className":41929},[50],[15,41931],{"className":41932,"style":5779},[54],[15,41934,1289],{"className":41935,"style":1489},[1486,1487,1488],[15,41937],{"className":41938,"style":406},[157],[15,41940,944],{"className":41941,"style":981},[59,60],[15,41943],{"className":41944,"style":2589},[157],[15,41946,4815],{"className":41947},[2593],[15,41949],{"className":41950,"style":2589},[157],[15,41952,41954,41957,41960,41963,41966,41969,42016,42019,42022],{"className":41953},[50],[15,41955],{"className":41956,"style":5779},[54],[15,41958,2465],{"className":41959},[59],[15,41961],{"className":41962,"style":406},[157],[15,41964,1289],{"className":41965,"style":1489},[1486,1487,1488],[15,41967],{"className":41968,"style":406},[157],[15,41970,41972],{"className":41971},[59,34393],[15,41973,41975,42008],{"className":41974},[285,286],[15,41976,41978,42005],{"className":41977},[290],[15,41979,41981,41993],{"className":41980,"style":6864},[294],[15,41982,41984,41987],{"className":41983,"style":11867},[6309],[15,41985],{"className":41986,"style":461},[302],[15,41988,41990],{"className":41989,"style":34413},[59],[15,41991,41631],{"className":41992,"style":981},[59,60],[15,41994,41995,41998],{"style":41376},[15,41996],{"className":41997,"style":461},[302],[15,41999,42001],{"className":42000,"style":34470},[34469],[6325,42002,42003],{"xmlns":6327,"width":6328,"height":34473,"viewBox":34474,"preserveAspectRatio":6331},[6333,42004],{"d":34477},[15,42006,318],{"className":42007},[317],[15,42009,42011],{"className":42010},[290],[15,42012,42014],{"className":42013,"style":41396},[294],[15,42015],{},[15,42017],{"className":42018,"style":158},[157],[15,420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",[15,42077,42079,42097],{"className":42078,"translate":19},[18],[15,42080,42082],{"className":42081},[23],[25,42083,42084],{"xmlns":27},[29,42085,42086,42094],{},[32,42087,42088],{},[6816,42089,42090,42092],{},[35,42091,33910],{},[115,42093,2465],{},[39,42095,42096],{"encoding":41},"H^2",[15,42098,42100],{"className":42099,"ariaHidden":46},[45],[15,42101,42103,42106],{"className":42102},[50],[15,42104],{"className":42105,"style":20054},[54],[15,42107,42109,42112],{"className":42108},[59],[15,42110,33910],{"className":42111,"style":34064},[59,60],[15,42113,42115],{"className":42114},[281],[15,42116,42118],{"className":42117},[285],[15,42119,42121],{"className":42120},[290],[15,42122,42124],{"className":42123,"style":20054},[294],[15,42125,42126,42129],{"style":6867},[15,42127],{"className":42128,"style":303},[302],[15,42130,42132],{"className":42131},[307,308,309,310],[15,42133,2465],{"className":42134},[59,310]," 有两种约定：",[15,42137,42139],{"className":42138,"translate":19},[180],[15,42140,42142,42248],{"className":42141,"translate":19},[18],[15,42143,42145],{"className":42144},[23],[25,42146,42147],{"xmlns":27,"display":189},[29,42148,42149,42245],{},[1245,42150,42151,42186],{"rowspacing":1247,"columnalign":1248,"columnspacing":1249},[1251,42152,42153,42166],{},[1254,42154,42155],{},[1257,42156,42157],{"scriptlevel":126,"displaystyle":46},[7555,42158,42159,42161,42164],{},[35,42160,33910],{},[1299,42162,42163],{},"Bhattacharyya",[115,42165,2465],{},[1254,42167,42168],{},[1257,42169,42170],{"scriptlevel":126,"displaystyle":46},[32,42171,42172,42174,42176,42178,42180,42182,42184],{},[32,42173],{},[110,42175,123],{},[115,42177,117],{},[110,42179,4815],{},[35,42181,39317],{},[35,42183,39320],{},[110,42185,551],{"separator":46},[1251,42187,42188,42207],{},[1254,42189,42190],{},[1257,42191,42192],{"scriptlevel":126,"displaystyle":46},[7555,42193,42194,42196,42205],{},[35,42195,33910],{},[32,42197,42198,42201,42203],{},[1299,42199,42200],{},"this",[1299,42202,1301],{},[35,42204,37],{},[115,42206,2465],{},[1254,42208,42209],{},[1257,42210,42211],{"scriptlevel":126,"displaystyle":46},[32,42212,42213,42215,42217,42219,42221,42223,42225,42227,42229,42231,42233,42235,42243],{},[32,42214],{},[110,42216,123],{},[115,42218,2465],{},[110,42220,113],{"stretchy":112},[115,42222,117],{},[110,42224,4815],{},[35,42226,39317],{},[35,42228,39320],{},[110,42230,120],{"stretchy":112},[110,42232,123],{},[115,42234,2465],{},[7555,42236,42237,42239,42241],{},[35,42238,33910],{},[1299,42240,42163],{},[115,42242,2465],{},[35,42244,4377],{"mathvariant":209},[39,42246,42247],{"encoding":41},"\\begin{aligned}\nH^2_{\\text{Bhattacharyya}}&=1-BC,\\\\[2pt]\nH^2_{\\text{this}\\,f}&=2(1-BC)\n=2H^2_{\\text{Bhattacharyya}} .\n\\end{aligned}",[15,42249,42251],{"className":42250,"ariaHidden":46},[45],[15,42252,42254,42258],{"className":42253},[50],[15,42255],{"className":42256,"style":42257},[54],"height:2.9713em;vertical-align:-1.2357em;",[15,42259,42261],{"className":42260},[59],[15,42262,42264,42430],{"className":42263},[1245],[15,42265,42267],{"className":42266},[1362],[15,42268,42270,42421],{"className":42269},[285,286],[15,42271,42273,42418],{"className":42272},[290],[15,42274,42277,42345],{"className":42275,"style":42276},[294],"height:1.7357em;",[15,42278,42280,42283],{"style":42279},"top:-3.8716em;",[15,42281],{"className":42282,"style":461},[302],[15,42284,42286],{"className":42285},[59],[15,42287,42289,42292],{"className":42288},[59],[15,42290,33910],{"className":42291,"style":34064},[59,60],[15,42293,42295],{"className":42294},[281],[15,42296,42298,42336],{"className":42297},[285,286],[15,42299,42301,42333],{"className":42300},[290],[15,42302,42304,42322],{"className":42303,"style":19761},[294],[15,42305,42307,42310],{"style":42306},"top:-2.453em;margin-left:-0.0813em;margin-right:0.05em;",[15,42308],{"className":42309,"style":303},[302],[15,42311,42313],{"className":42312},[307,308,309,310],[15,42314,42316],{"className":42315},[59,310],[15,42317,42319],{"className":42318},[59,4600,310],[15,42320,42163],{"className":42321},[59,310],[15,42323,42324,42327],{"style":8279},[15,42325],{"className":42326,"style":303},[302],[15,42328,42330],{"className":42329},[307,308,309,310],[15,42331,2465],{"className":42332},[59,310],[15,42334,318],{"className":42335},[317],[15,42337,42339],{"className":42338},[290],[15,42340,42343],{"className":42341,"style":42342},[294],"height:0.3831em;",[15,42344],{},[15,42346,42348,42351],{"style":42347},"top:-2.1474em;",[15,42349],{"className":42350,"style":461},[302],[15,42352,42354],{"className":42353},[59],[15,42355,42357,42360],{"className":42356},[59],[15,42358,33910],{"className":42359,"style":34064},[59,60],[15,42361,42363],{"className":42362},[281],[15,42364,42366,42410],{"className":42365},[285,286],[15,42367,42369,42407],{"className":42368},[290],[15,42370,42372,42396],{"className":42371,"style":19761},[294],[15,42373,42374,42377],{"style":42306},[15,42375],{"className":42376,"style":303},[302],[15,42378,42380],{"className":42379},[307,308,309,310],[15,42381,42383,42389,42393],{"className":42382},[59,310],[15,42384,42386],{"className":42385},[59,4600,310],[15,42387,42200],{"className":42388},[59,310],[15,42390],{"className":42391,"style":42392},[157,310],"margin-right:0.1952em;",[15,42394,37],{"className":42395,"style":61},[59,60,310],[15,42397,42398,42401],{"style":8279},[15,42399],{"className":42400,"style":303},[302],[15,42402,42404],{"className":42403},[307,308,309,310],[15,42405,2465],{"className":42406},[59,310],[15,42408,318],{"className":42409},[317],[15,42411,42413],{"className":42412},[290],[15,42414,42416],{"className":42415,"style":42342},[294],[15,42417],{},[15,42419,318],{"className":42420},[317],[15,42422,42424],{"className":42423},[290],[15,42425,42428],{"className":42426,"style":42427},[294],"height:1.2357em;",[15,42429],{},[15,42431,42433],{"className":42432},[1453],[15,42434,42436,42605],{"className":42435},[285,286],[15,42437,42439,42602],{"className":42438},[290],[15,42440,42442,42483],{"className":42441,"style":42276},[294],[15,42443,42444,42447],{"style":42279},[15,42445],{"className":42446,"style":461},[302],[15,42448,42450,42453,42456,42459,42462,42465,42468,42471,42474,42477,42480],{"className":42449},[59],[15,42451],{"className":42452},[59],[15,42454],{"className":42455,"style":158},[157],[15,42457,123],{"className":42458},[162],[15,42460],{"className":42461,"style":158},[157],[15,42463,117],{"className":42464},[59],[15,42466],{"className":42467,"style":2589},[157],[15,42469,4815],{"className":42470},[2593],[15,42472],{"className":42473,"style":2589},[157],[15,42475,39317],{"className":42476,"style":39413},[59,60],[15,42478,39320],{"className":42479,"style":10786},[59,60],[15,42481,551],{"className":42482},[573],[15,42484,42485,42488],{"style":42347},[15,42486],{"className":42487,"style":461},[302],[15,42489,42491,42494,42497,42500,42503,42506,42509,42512,42515,42518,42521,42524,42527,42530,42533,42536,42539,42542,42599],{"className":42490},[59],[15,42492],{"className":42493},[59],[15,42495],{"className":42496,"style":158},[157],[15,42498,123],{"className":42499},[162],[15,42501],{"className":42502,"style":158},[157],[15,42504,2465],{"className":42505},[59],[15,42507,113],{"className":42508},[146],[15,42510,117],{"className":42511},[59],[15,42513],{"className":42514,"style":2589},[157],[15,42516,4815],{"className":42517},[2593],[15,42519],{"className":42520,"style":2589},[157],[15,42522,39317],{"className":42523,"style":39413},[59,60],[15,42525,39320],{"className":42526,"style":10786},[59,60],[15,42528,120],{"className":42529},[153],[15,42531],{"className":42532,"style":158},[157],[15,42534,123],{"className":42535},[162],[15,42537],{"className":42538,"style":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",[15,42617,42619,42657],{"className":42618,"translate":19},[18],[15,42620,42622],{"className":42621},[23],[25,42623,42624],{"xmlns":27},[29,42625,42626,42654],{},[32,42627,42628,42630,42632,42634,42636,42638,42640,42644,42646,42648],{},[35,42629,37],{},[110,42631,113],{"stretchy":112},[35,42633,769],{},[110,42635,120],{"stretchy":112},[110,42637,123],{},[110,42639,113],{"stretchy":112},[34365,42641,42642],{},[35,42643,769],{},[110,42645,4815],{},[115,42647,117],{},[6816,42649,42650,42652],{},[110,42651,120],{"stretchy":112},[115,42653,2465],{},[39,42655,42656],{"encoding":41},"f(t)=(\\sqrt t-1)^2",[15,42658,42660,42687,42753],{"className":42659,"ariaHidden":46},[45],[15,42661,42663,42666,42669,42672,42675,42678,42681,42684],{"className":42662},[50],[15,42664],{"className":42665,"style":139},[54],[15,42667,37],{"className":42668,"style":61},[59,60],[15,42670,113],{"className":42671},[146],[15,42673,769],{"className":42674},[59,60],[15,42676,120],{"className":42677},[153],[15,42679],{"className":42680,"style":158},[157],[15,42682,123],{"className":42683},[162],[15,42685],{"className":42686,"style":158},[157],[15,42688,42690,42694,42697,42744,42747,42750],{"className":42689},[50],[15,42691],{"className":42692,"style":42693},[54],"height:1.1425em;vertical-align:-0.25em;",[15,42695,113],{"className":42696},[146],[15,42698,42700],{"className":42699},[59,34393],[15,42701,42703,42735],{"className":42702},[285,286],[15,42704,42706,42732],{"className":42705},[290],[15,42707,42710,42719],{"className":42708,"style":42709},[294],"height:0.8925em;",[15,42711,42713,42716],{"className":42712,"style":11867},[6309],[15,42714],{"className":42715,"style":461},[302],[15,42717,769],{"className":42718,"style":34413},[59,60],[15,42720,42722,42725],{"style":42721},"top:-2.8525em;",[15,42723],{"className":42724,"style":461},[302],[15,42726,42728],{"className":42727,"style":34470},[34469],[6325,42729,42730],{"xmlns":6327,"width":6328,"height":34473,"viewBox":34474,"preserveAspectRatio":6331},[6333,42731],{"d":34477},[15,42733,318],{"className":42734},[317],[15,42736,42738],{"className":42737},[290],[15,42739,42742],{"className":42740,"style":42741},[294],"height:0.1475em;",[15,42743],{},[15,42745],{"className":42746,"style":2589},[157],[15,42748,4815],{"className":42749},[2593],[15,42751],{"className":42752,"style":2589},[157],[15,42754,42756,42759,42762],{"className":42755},[50],[15,42757],{"className":42758,"style":20029},[54],[15,42760,117],{"className":42761},[59],[15,42763,42765,42768],{"className":42764},[153],[15,42766,120],{"className":42767},[153],[15,42769,42771],{"className":42770},[281],[15,42772,42774],{"className":42773},[285],[15,42775,42777],{"className":42776},[290],[15,42778,42780],{"className":42779,"style":20054},[294],[15,42781,42782,42785],{"style":6867},[15,42783],{"className":42784,"style":303},[302],[15,42786,42788],{"className":42787},[307,308,309,310],[15,42789,2465],{"className":42790},[59,310],"，故 ",[15,42793,42795,42847],{"className":42794,"translate":19},[18],[15,42796,42798],{"className":42797},[23],[25,42799,42800],{"xmlns":27},[29,42801,42802,42844],{},[32,42803,42804,42810,42812,42814,42822,42824,42826,42828,42832,42834,42838],{},[195,42805,42806,42808],{},[35,42807,199],{},[35,42809,37],{},[110,42811,123],{},[115,42813,2465],{},[7555,42815,42816,42818,42820],{},[35,42817,33910],{},[1299,42819,42163],{},[115,42821,2465],{},[110,42823,123],{},[110,42825,1289],{},[110,42827,113],{"stretchy":112},[34365,42829,42830],{},[35,42831,11],{},[110,42833,4815],{},[34365,42835,42836],{},[35,42837,944],{},[6816,42839,42840,42842],{},[110,42841,120],{"stretchy":112},[115,42843,2465],{},[39,42845,42846],{"encoding":41},"D_f=2H^2_{\\text{Bhattacharyya}}=\\int(\\sqrt p-\\sqrt q)^2",[15,42848,42850,42905,42983,43048],{"className":42849,"ariaHidden":46},[45],[15,42851,42853,42856,42896,42899,42902],{"className":42852},[50],[15,42854],{"className":42855,"style":4034},[54],[15,42857,42859,42862],{"className":42858},[59],[15,42860,199],{"className":42861,"style":277},[59,60],[15,42863,42865],{"className":42864},[281],[15,42866,42868,42888],{"className":42867},[285,286],[15,42869,42871,42885],{"className":42870},[290],[15,42872,42874],{"className":42873,"style":295},[294],[15,42875,42876,42879],{"style":298},[15,42877],{"className":42878,"style":303},[302],[15,42880,42882],{"className":42881},[307,308,309,310],[15,42883,37],{"className":42884,"style":61},[59,60,310],[15,42886,318],{"className":42887},[317],[15,42889,42891],{"className":42890},[290],[15,42892,42894],{"className":42893,"style":325},[294],[15,42895],{},[15,42897],{"className":42898,"style":158},[157],[15,42900,123],{"className":42901},[162],[15,42903],{"className":42904,"style":158},[157],[15,42906,42908,42912,42915,42974,42977,42980],{"className":42907},[50],[15,42909],{"className":42910,"style":42911},[54],"height:1.2333em;vertical-align:-0.4192em;",[15,42913,2465],{"className":42914},[59],[15,42916,42918,42921],{"className":42917},[59],[15,42919,33910],{"className":42920,"style":34064},[59,60],[15,42922,42924],{"className":42923},[281],[15,42925,42927,42965],{"className":42926},[285,286],[15,42928,42930,42962],{"className":42929},[290],[15,42931,42933,42951],{"className":42932,"style":20054},[294],[15,42934,42936,42939],{"style":42935},"top:-2.4169em;margin-left:-0.0813em;margin-right:0.05em;",[15,42937],{"className":42938,"style":303},[302],[15,42940,42942],{"className":42941},[307,308,309,310],[15,42943,42945],{"className":42944},[59,310],[15,42946,42948],{"className":42947},[59,4600,310],[15,42949,42163],{"className":42950},[59,310],[15,42952,42953,42956],{"style":6867},[15,42954],{"className":42955,"style":303},[302],[15,42957,42959],{"className":42958},[307,308,309,310],[15,42960,2465],{"className":42961},[59,310],[15,42963,318],{"className":42964},[317],[15,42966,42968],{"className":42967},[290],[15,42969,42972],{"className":42970,"style":42971},[294],"height:0.4192em;",[15,42973],{},[15,42975],{"className":42976,"style":158},[157],[15,42978,123],{"className":42979},[162],[15,42981],{"className":42982,"style":158},[157],[15,42984,42986,42989,42992,42995,43039,43042,43045],{"className":42985},[50],[15,42987],{"className":42988,"style":39034},[54],[15,42990,1289],{"className":42991,"style":5619},[1486,1487,5618],[15,42993,113],{"className":42994},[146],[15,42996,42998],{"className":42997},[59,34393],[15,42999,43001,43031],{"className":43000},[285,286],[15,43002,43004,43028],{"className":43003},[290],[15,43005,43007,43016],{"className":43006,"style":39053},[294],[15,43008,43010,43013],{"className":43009,"style":11867},[6309],[15,43011],{"className":43012,"style":461},[302],[15,43014,11],{"className":43015,"style":34413},[59,60],[15,43017,43018,43021],{"style":39065},[15,43019],{"className":43020,"style":461},[302],[15,43022,43024],{"className":43023,"style":34470},[34469],[6325,43025,43026],{"xmlns":6327,"width":6328,"height":34473,"viewBox":34474,"preserveAspectRatio":6331},[6333,43027],{"d":34477},[15,43029,318],{"className":43030},[317],[15,43032,43034],{"className":43033},[290],[15,43035,43037],{"className":43036,"style":39085},[294],[15,43038],{},[15,43040],{"className":43041,"style":2589},[157],[15,43043,4815],{"className":43044},[2593],[15,43046],{"className":43047,"style":2589},[157],[15,43049,43051,43054,43098],{"className":43050},[50],[15,43052],{"className":43053,"style":39103},[54],[15,43055,43057],{"className":43056},[59,34393],[15,43058,43060,43090],{"className":43059},[285,286],[15,43061,43063,43087],{"className":43062},[290],[15,43064,43066,43075],{"className":43065,"style":39053},[294],[15,43067,43069,43072],{"className":43068,"style":11867},[6309],[15,43070],{"className":43071,"style":461},[302],[15,43073,944],{"className":43074,"style":39125},[59,60],[15,43076,43077,43080],{"style":39065},[15,43078],{"className":43079,"style":461},[302],[15,43081,43083],{"className":43082,"style":34470},[34469],[6325,43084,43085],{"xmlns":6327,"width":6328,"height":34473,"viewBox":34474,"preserveAspectRatio":6331},[6333,43086],{"d":34477},[15,43088,318],{"className":43089},[317],[15,43091,43093],{"className":43092},[290],[15,43094,43096],{"className":43095,"style":39085},[294],[15,43097],{},[15,43099,43101,43104],{"className":43100},[153],[15,43102,120],{"className":43103},[153],[15,43105,43107],{"className":43106},[281],[15,43108,43110],{"className":43109},[285],[15,43111,43113],{"className":43112},[290],[15,43114,43116],{"className":43115,"style":20054},[294],[15,43117,43118,43121],{"style":6867},[15,43119],{"className":43120,"style":303},[302],[15,43122,43124],{"className":43123},[307,308,309,310],[15,43125,2465],{"className":43126},[59,310],[15,43128,43130,43147],{"className":43129,"translate":19},[18],[15,43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有界（",[15,43187,43189,43205],{"className":43188,"translate":19},[18],[15,43190,43192],{"className":43191},[23],[25,43193,43194],{"xmlns":27},[29,43195,43196,43202],{},[32,43197,43198,43200],{},[110,43199,9954],{},[115,43201,2465],{},[39,43203,43204],{"encoding":41},"\\le 2",[15,43206,43208,43221],{"className":43207,"ariaHidden":46},[45],[15,43209,43211,43215,43218],{"className":43210},[50],[15,43212],{"className":43213,"style":43214},[54],"height:0.7719em;vertical-align:-0.136em;",[15,43216,9954],{"className":43217},[162],[15,43219],{"className":43220,"style":158},[157],[15,43222,43224,43227],{"className":43223},[50],[15,43225],{"className":43226,"style":172},[54],[15,43228,2465],{"className":43229},[59],"）。对称性可直接验算：",[15,43232,43234],{"className":43233,"translate":19},[180],[15,43235,43237,43413],{"className":43236,"translate":19},[18],[15,43238,43240],{"className":43239},[23],[25,43241,43242],{"xmlns":27,"display":189},[29,43243,43244,43410],{},[1245,43245,43246,43304,43348],{"rowspacing":1247,"columnalign":1248,"columnspacing":1249},[1251,43247,43248,43270],{},[1254,43249,43250],{},[1257,43251,43252],{"scriptlevel":126,"displaystyle":46},[32,43253,43254,43256,43258,43260,43262,43264,43266,43268],{},[35,43255,769],{},[1299,43257,1301],{},[35,43259,37],{},[110,43261,113],{"stretchy":112},[115,43263,117],{},[35,43265,941],{"mathvariant":209},[35,43267,769],{},[110,43269,120],{"stretchy":112},[1254,43271,43272],{},[1257,43273,43274],{"scriptlevel":126,"displaystyle":46},[32,43275,43276,43278,43280,43282],{},[32,43277],{},[110,43279,123],{},[35,43281,769],{},[6816,43283,43284,43302],{},[32,43285,43286,43288,43296,43298,43300],{},[110,43287,113],{"fence":46},[238,43289,43290,43292],{},[115,43291,117],{},[34365,43293,43294],{},[35,43295,769],{},[110,43297,4815],{},[115,43299,117],{},[110,43301,120],{"fence":46},[115,43303,2465],{},[1251,43305,43306,43312],{},[1254,43307,43308],{},[1257,43309,43310],{"scriptlevel":126,"displaystyle":46},[32,43311],{},[1254,43313,43314],{},[1257,43315,43316],{"scriptlevel":126,"displaystyle":46},[32,43317,43318,43320,43322,43324,43326,43332,43334,43342,43344,43346],{},[32,43319],{},[110,43321,123],{},[35,43323,769],{},[110,43325,113],{"fence":112,"stretchy":46,"minsize":17443,"maxsize":17443},[238,43327,43328,43330],{},[115,43329,117],{},[35,43331,769],{},[110,43333,4815],{},[238,43335,43336,43338],{},[115,43337,2465],{},[34365,43339,43340],{},[35,43341,769],{},[110,43343,2472],{},[115,43345,117],{},[110,43347,120],{"fence":112,"stretchy":46,"minsize":17443,"maxsize":17443},[1251,43349,43350,43356],{},[1254,43351,43352],{},[1257,43353,43354],{"scriptlevel":126,"displaystyle":46},[32,43355],{},[1254,43357,43358],{},[1257,43359,43360],{"scriptlevel":126,"displaystyle":46},[32,43361,43362,43364,43366,43368,43370,43372,43376,43378,43380,43382,43384,43388,43390,43392,43398,43400,43402,43404,43406,43408],{},[32,43363],{},[110,43365,123],{},[115,43367,117],{},[110,43369,4815],{},[115,43371,2465],{},[34365,43373,43374],{},[35,43375,769],{},[110,43377,2472],{},[35,43379,769],{},[110,43381,123],{},[110,43383,113],{"stretchy":112},[34365,43385,43386],{},[35,43387,769],{},[110,43389,4815],{},[115,43391,117],{},[6816,43393,43394,43396],{},[110,43395,120],{"stretchy":112},[115,43397,2465],{},[110,43399,123],{},[35,43401,37],{},[110,43403,113],{"stretchy":112},[35,43405,769],{},[110,43407,120],{"stretchy":112},[110,43409,551],{"separator":46},[39,43411,43412],{"encoding":41},"\\begin{aligned}\nt\\,f(1\u002Ft)\n&=t\\left(\\frac{1}{\\sqrt t}-1\\right)^2\\\\\n&=t\\Big(\\frac1t-\\frac{2}{\\sqrt t}+1\\Big)\\\\[2pt]\n&=1-2\\sqrt t+t=(\\sqrt t-1)^2=f(t),\n\\end{aligned}",[15,43414,43416],{"className":43415,"ariaHidden":46},[45],[15,43417,43419,43423],{"className":43418},[50],[15,43420],{"className":43421,"style":43422},[54],"height:6.7569em;vertical-align:-3.1284em;",[15,43424,43426],{"className":43425},[59],[15,43427,43429,43503],{"className":43428},[1245],[15,43430,43432],{"className":43431},[1362],[15,43433,43435,43494],{"className":43434},[285,286],[15,43436,43438,43491],{"className":43437},[290],[15,43439,43442,43473,43482],{"className":43440,"style":43441},[294],"height:3.6284em;",[15,43443,43445,43449],{"style":43444},"top:-5.6284em;",[15,43446],{"className":43447,"style":43448},[302],"height:3.654em;",[15,43450,43452,43455,43458,43461,43464,43467,43470],{"className":43451},[59],[15,43453,769],{"className":43454},[59,60],[15,43456],{"className":43457,"style":406},[157],[15,43459,37],{"className":43460,"style":61},[59,60],[15,43462,113],{"className":43463},[146],[15,43465,8591],{"className":43466},[59],[15,43468,769],{"className":43469},[59,60],[15,43471,120],{"className":43472},[153],[15,43474,43476,43479],{"style":43475},"top:-3.057em;",[15,43477],{"className":43478,"style":43448},[302],[15,43480],{"className":43481},[59],[15,43483,43485,43488],{"style":43484},"top:-0.8856em;",[15,43486],{"className":43487,"style":43448},[302],[15,43489],{"className":43490},[59],[15,43492,318],{"className":43493},[317],[15,43495,43497],{"className":43496},[290],[15,43498,43501],{"className":43499,"style":43500},[294],"height:3.1284em;",[15,43502],{},[15,43504,43506],{"className":43505},[1453],[15,43507,43509,44135],{"className":43508},[285,286],[15,43510,43512,44132],{"className":43511},[290],[15,43513,43515,43699,43920],{"className":43514,"style":43441},[294],[15,43516,43517,43520],{"style":43444},[15,43518],{"className":43519,"style":43448},[302],[15,43521,43523,43526,43529,43532,43535,43538,43541],{"className":43522},[59],[15,43524],{"className":43525},[59],[15,43527],{"className":43528,"style":158},[157],[15,43530,123],{"className":43531},[162],[15,43533],{"className":43534,"style":158},[157],[15,43536,769],{"className":43537},[59,60],[15,43539],{"className":43540,"style":406},[157],[15,43542,43544,43676],{"className":43543},[410],[15,43545,43547,43553,43658,43661,43664,43667,43670],{"className":43546},[410],[15,43548,43550],{"className":43549,"style":415},[146,414],[15,43551,113],{"className":43552},[419,309],[15,43554,43556,43559,43655],{"className":43555},[59],[15,43557],{"className":43558},[146,441],[15,43560,43562],{"className":43561},[238],[15,43563,43565,43646],{"className":43564},[285,286],[15,43566,43568,43643],{"className":43567},[290],[15,43569,43571,43624,43632],{"className":43570,"style":22915},[294],[15,43572,43574,43577],{"style":43573},"top:-2.2175em;",[15,43575],{"className":43576,"style":461},[302],[15,43578,43580],{"className":43579},[59],[15,43581,43583],{"className":43582},[59,34393],[15,43584,43586,43616],{"className":43585},[285,286],[15,43587,43589,43613],{"className":43588},[290],[15,43590,43592,43601],{"className":43591,"style":42709},[294],[15,43593,43595,43598],{"className":43594,"style":11867},[6309],[15,43596],{"className":43597,"style":461},[302],[15,43599,769],{"className":43600,"style":34413},[59,60],[15,43602,43603,43606],{"style":42721},[15,43604],{"className":43605,"style":461},[302],[15,43607,43609],{"className":43608,"style":34470},[34469],[6325,43610,43611],{"xmlns":6327,"width":6328,"height":34473,"viewBox":34474,"preserveAspectRatio":6331},[6333,43612],{"d":34477},[15,43614,318],{"className":43615},[317],[15,43617,43619],{"className":43618},[290],[15,43620,43622],{"className":43621,"style":42741},[294],[15,43623],{},[15,43625,43626,43629],{"style":473},[15,43627],{"className":43628,"style":461},[302],[15,43630],{"className":43631,"style":481},[480],[15,43633,43634,43637],{"style":484},[15,43635],{"className":43636,"style":461},[302],[15,43638,43640],{"className":43639},[59],[15,43641,117],{"className":43642},[59],[15,43644,318],{"className":43645},[317],[15,43647,43649],{"className":43648},[290],[15,43650,43653],{"className":43651,"style":43652},[294],"height:0.93em;",[15,43654],{},[15,43656],{"className":43657},[153,441],[15,43659],{"className":43660,"style":2589},[157],[15,43662,4815],{"className":43663},[2593],[15,43665],{"className":43666,"style":2589},[157],[15,43668,117],{"className":43669},[59],[15,43671,43673],{"className":43672,"style":415},[153,414],[15,43674,120],{"className":43675},[419,309],[15,43677,43679],{"className":43678},[281],[15,43680,43682],{"className":43681},[285],[15,43683,43685],{"className":43684},[290],[15,43686,43688],{"className":43687,"style":37687},[294],[15,43689,43690,43693],{"style":37690},[15,43691],{"className":43692,"style":303},[302],[15,43694,43696],{"className":43695},[307,308,309,310],[15,43697,2465],{"className":43698},[59,310],[15,43700,43701,43704],{"style":43475},[15,43702],{"className":43703,"style":43448},[302],[15,43705,43707,43710,43713,43716,43719,43722,43728,43790,43793,43796,43799,43902,43905,43908,43911,43914],{"className":43706},[59],[15,43708],{"className":43709},[59],[15,43711],{"className":43712,"style":158},[157],[15,43714,123],{"className":43715},[162],[15,43717],{"className":43718,"style":158},[157],[15,43720,769],{"className":43721},[59,60],[15,43723,43725],{"className":43724},[59],[15,43726,113],{"className":43727},[419,3722],[15,43729,43731,43734,43787],{"className":43730},[59],[15,43732],{"className":43733},[146,441],[15,43735,43737],{"className":43736},[238],[15,43738,43740,43779],{"className":43739},[285,286],[15,43741,43743,43776],{"className":43742},[290],[15,43744,43746,43757,43765],{"className":43745,"style":22915},[294],[15,43747,43748,43751],{"style":457},[15,43749],{"className":43750,"style":461},[302],[15,43752,43754],{"className":43753},[59],[15,43755,769],{"className":43756},[59,60],[15,43758,43759,43762],{"style":473},[15,43760],{"className":43761,"style":461},[302],[15,43763],{"className":43764,"style":481},[480],[15,43766,43767,43770],{"style":484},[15,43768],{"className":43769,"style":461},[302],[15,43771,43773],{"className":43772},[59],[15,43774,117],{"className":43775},[59],[15,43777,318],{"className":43778},[317],[15,43780,43782],{"className":43781},[290],[15,43783,43785],{"className":43784,"style":3780},[294],[15,43786],{},[15,43788],{"className":43789},[153,441],[15,43791],{"className":43792,"style":2589},[157],[15,43794,4815],{"className":43795},[2593],[15,43797],{"className":43798,"style":2589},[157],[15,43800,43802,43805,43899],{"className":43801},[59],[15,43803],{"className":43804},[146,441],[15,43806,43808],{"className":43807},[238],[15,43809,43811,43891],{"className":43810},[285,286],[15,43812,43814,43888],{"className":43813},[290],[15,43815,43817,43869,43877],{"className":43816,"style":22915},[294],[15,43818,43819,43822],{"style":43573},[15,43820],{"className":43821,"style":461},[302],[15,43823,43825],{"className":43824},[59],[15,43826,43828],{"className":43827},[59,34393],[15,43829,43831,43861],{"className":43830},[285,286],[15,43832,43834,43858],{"className":43833},[290],[15,43835,43837,43846],{"className":43836,"style":42709},[294],[15,43838,43840,43843],{"className":43839,"style":11867},[6309],[15,43841],{"className":43842,"style":461},[302],[15,43844,769],{"className":43845,"style":34413},[59,60],[15,43847,43848,43851],{"style":42721},[15,43849],{"className":43850,"style":461},[302],[15,43852,43854],{"className":43853,"style":34470},[34469],[6325,43855,43856],{"xmlns":6327,"width":6328,"height":34473,"viewBox":34474,"preserveAspectRatio":6331},[6333,43857],{"d":34477},[15,43859,318],{"className":43860},[317],[15,43862,43864],{"className":43863},[290],[15,43865,43867],{"className":43866,"style":42741},[294],[15,43868],{},[15,43870,43871,43874],{"style":473},[15,43872],{"className":43873,"style":461},[302],[15,43875],{"className":43876,"style":481},[480],[15,43878,43879,43882],{"style":484},[15,43880],{"className":43881,"style":461},[302],[15,43883,43885],{"className":43884},[59],[15,43886,2465],{"className":43887},[59],[15,43889,318],{"className":43890},[317],[15,43892,43894],{"className":43893},[290],[15,43895,43897],{"className":43896,"style":43652},[294],[15,43898],{},[15,43900],{"className":43901},[153,441],[15,43903],{"className":43904,"style":2589},[157],[15,43906,2472],{"className":43907},[2593],[15,43909],{"className":43910,"style":2589},[157],[15,43912,117],{"className":43913},[59],[15,43915,43917],{"className":43916},[59],[15,43918,120],{"className":43919},[419,3722],[15,43921,43922,43925],{"style":43484},[15,43923],{"className":43924,"style":43448},[302],[15,43926,43928,43931,43934,43937,43940,43943,43946,43949,43952,43955,43999,44002,44005,44008,44011,44014,44017,44020,44023,44067,44070,44073,44076,44079,44108,44111,44114,44117,44120,44123,44126,44129],{"className":43927},[59],[15,43929],{"className":43930},[59],[15,43932],{"className":43933,"style":158},[157],[15,43935,123],{"className":43936},[162],[15,43938],{"className":43939,"style":158},[157],[15,43941,117],{"className":43942},[59],[15,43944],{"className":43945,"style":2589},[157],[15,43947,4815],{"className":43948},[2593],[15,43950],{"className":43951,"style":2589},[157],[15,43953,2465],{"className":43954},[59],[15,43956,43958],{"className":43957},[59,34393],[15,43959,43961,43991],{"className":43960},[285,286],[15,43962,43964,43988],{"className":43963},[290],[15,43965,43967,43976],{"className":43966,"style":40563},[294],[15,43968,43970,43973],{"className":43969,"style":11867},[6309],[15,43971],{"className":43972,"style":461},[302],[15,43974,769],{"className":43975,"style":34413},[59,60],[15,43977,43978,43981],{"style":40578},[15,43979],{"className":43980,"style":461},[302],[15,43982,43984],{"className":43983,"style":34470},[34469],[6325,43985,43986],{"xmlns":6327,"width":6328,"height":34473,"viewBox":34474,"preserveAspectRatio":6331},[6333,43987],{"d":34477},[15,43989,318],{"className":43990},[317],[15,43992,43994],{"className":43993},[290],[15,43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Hellinger",[15,44276,44278,44296],{"className":44277,"translate":19},[18],[15,44279,44281],{"className":44280},[23],[25,44282,44283],{"xmlns":27},[29,44284,44285,44293],{},[32,44286,44287],{},[6816,44288,44289,44291],{},[32,44290],{},[115,44292,2465],{},[39,44294,44295],{"encoding":41},"^2",[15,44297,44299],{"className":44298,"ariaHidden":46},[45],[15,44300,44302,44305],{"className":44301},[50],[15,44303],{"className":44304,"style":20054},[54],[15,44306,44308,44310],{"className":44307},[59],[15,44309],{},[15,44311,44313],{"className":44312},[281],[15,44314,44316],{"className":44315},[285],[15,44317,44319],{"className":44318},[290],[15,44320,44322],{"className":44321,"style":20054},[294],[15,44323,44324,44327],{"style":6867},[15,44325],{"className":44326,"style":303},[302],[15,44328,44330],{"className":44329},[307,308,309,310],[15,44331,2465],{"className":44332},[59,310]," 也满足对称判别准则（4.3 节）。",[11,44335,44336,44338,44339,44453,44454,44457,44458,44514,44515,44571,44572,44730,44731,44759],{},[998,44337,24219],{}," Hellinger 与 Bhattacharyya 系数 ",[15,44340,44342,44370],{"className":44341,"translate":19},[18],[15,44343,44345],{"className":44344},[23],[25,44346,44347],{"xmlns":27},[29,44348,44349,44367],{},[32,44350,44351,44353,44355,44357,44359],{},[35,44352,39317],{},[35,44354,39320],{},[110,44356,123],{},[110,44358,1289],{},[34365,44360,44361],{},[32,44362,44363,44365],{},[35,44364,11],{},[35,44366,944],{},[39,44368,44369],{"encoding":41},"BC=\\int\\sqrt{pq}",[15,44371,44373,44394],{"className":44372,"ariaHidden":46},[45],[15,44374,44376,44379,44382,44385,44388,44391],{"className":44375},[50],[15,44377],{"className":44378,"style":664},[54],[15,44380,39317],{"className":44381,"style":39413},[59,60],[15,44383,39320],{"className":44384,"style":10786},[59,60],[15,44386],{"className":44387,"style":158},[157],[15,44389,123],{"className":44390},[162],[15,44392],{"className":44393,"style":158},[157],[15,44395,44397,44400,44403,44406],{"className":44396},[50],[15,44398],{"className":44399,"style":39034},[54],[15,44401,1289],{"className":44402,"style":5619},[1486,1487,5618],[15,44404],{"className":44405,"style":406},[157],[15,44407,44409],{"className":44408},[59,34393],[15,44410,44412,44445],{"className":44411},[285,286],[15,44413,44415,44442],{"className":44414},[290],[15,44416,44418,44430],{"className":44417,"style":39053},[294],[15,44419,44421,44424],{"className":44420,"style":11867},[6309],[15,44422],{"className":44423,"style":461},[302],[15,44425,44427],{"className":44426,"style":34413},[59],[15,44428,41631],{"className":44429,"style":981},[59,60],[15,44431,44432,44435],{"style":39065},[15,44433],{"className":44434,"style":461},[302],[15,44436,44438],{"className":44437,"style":34470},[34469],[6325,44439,44440],{"xmlns":6327,"width":6328,"height":34473,"viewBox":34474,"preserveAspectRatio":6331},[6333,44441],{"d":34477},[15,44443,318],{"className":44444},[317],[15,44446,44448],{"className":44447},[290],[15,44449,44451],{"className":44450,"style":39085},[294],[15,44452],{}," 把「两个分布的",[998,44455,44456],{},"重叠程度","」量化：",[15,44459,44461,44481],{"className":44460,"translate":19},[18],[15,44462,44464],{"className":44463},[23],[25,44465,44466],{"xmlns":27},[29,44467,44468,44478],{},[32,44469,44470,44472,44474,44476],{},[35,44471,39317],{},[35,44473,39320],{},[110,44475,123],{},[115,44477,117],{},[39,44479,44480],{"encoding":41},"BC=1",[15,44482,44484,44505],{"className":44483,"ariaHidden":46},[45],[15,44485,44487,44490,44493,44496,44499,44502],{"className":44486},[50],[15,44488],{"className":44489,"style":664},[54],[15,44491,39317],{"className":44492,"style":39413},[59,60],[15,44494,39320],{"className":44495,"style":10786},[59,60],[15,44497],{"className":44498,"style":158},[157],[15,44500,123],{"className":44501},[162],[15,44503],{"className":44504,"style":158},[157],[15,44506,44508,44511],{"className":44507},[50],[15,44509],{"className":44510,"style":172},[54],[15,44512,117],{"className":44513},[59]," 表示完全重合，",[15,44516,44518,44538],{"className":44517,"translate":19},[18],[15,44519,44521],{"className":44520},[23],[25,44522,44523],{"xmlns":27},[29,44524,44525,44535],{},[32,44526,44527,44529,44531,44533],{},[35,44528,39317],{},[35,44530,39320],{},[110,44532,123],{},[115,44534,126],{},[39,44536,44537],{"encoding":41},"BC=0",[15,44539,44541,44562],{"className":44540,"ariaHidden":46},[45],[15,44542,44544,44547,44550,44553,44556,44559],{"className":44543},[50],[15,44545],{"className":44546,"style":664},[54],[15,44548,39317],{"className":44549,"style":39413},[59,60],[15,44551,39320],{"className":44552,"style":10786},[59,60],[15,44554],{"className":44555,"style":158},[157],[15,44557,123],{"className":44558},[162],[15,44560],{"className":44561,"style":158},[157],[15,44563,44565,44568],{"className":44564},[50],[15,44566],{"className":44567,"style":172},[54],[15,44569,126],{"className":44570},[59]," 表示支撑分离。由它可造出真正的距离 ",[15,44573,44575,44616],{"className":44574,"translate":19},[18],[15,44576,44578],{"className":44577},[23],[25,44579,44580],{"xmlns":27},[29,44581,44582,44613],{},[32,44583,44584,44587,44589,44591,44593,44595,44597,44599],{},[35,44585,44586],{},"ρ",[110,44588,113],{"stretchy":112},[35,44590,206],{},[110,44592,551],{"separator":46},[35,44594,213],{},[110,44596,120],{"stretchy":112},[110,44598,123],{},[34365,44600,44601],{},[32,44602,44603,44605,44607,44609,44611],{},[115,44604,2465],{},[110,44606,4815],{},[115,44608,2465],{},[35,44610,39317],{},[35,44612,39320],{},[39,44614,44615],{"encoding":41},"\\rho(P,Q)=\\sqrt{2-2BC}",[15,44617,44619,44655],{"className":44618,"ariaHidden":46},[45],[15,44620,44622,44625,44628,44631,44634,44637,44640,44643,44646,44649,44652],{"className":44621},[50],[15,44623],{"className":44624,"style":139},[54],[15,44626,44586],{"className":44627},[59,60],[15,44629,113],{"className":44630},[146],[15,44632,206],{"className":44633,"style":334},[59,60],[15,44635,551],{"className":44636},[573],[15,44638],{"className":44639,"style":406},[157],[15,44641,213],{"className":44642},[59,60],[15,44644,120],{"className":44645},[153],[15,44647],{"className":44648,"style":158},[157],[15,44650,123],{"className":44651},[162],[15,44653],{"className":44654,"style":158},[157],[15,44656,44658,44662],{"className":44657},[50],[15,44659],{"className":44660,"style":44661},[54],"height:1.04em;vertical-align:-0.155em;",[15,44663,44665],{"className":44664},[59,34393],[15,44666,44668,44721],{"className":44667},[285,286],[15,44669,44671,44718],{"className":44670},[290],[15,44672,44675,44705],{"className":44673,"style":44674},[294],"height:0.885em;",[15,44676,44678,44681],{"className":44677,"style":11867},[6309],[15,44679],{"className":44680,"style":461},[302],[15,44682,44684,44687,44690,44693,44696,44699,44702],{"className":44683,"style":34413},[59],[15,44685,2465],{"className":44686},[59],[15,44688],{"className":44689,"style":2589},[157],[15,44691,4815],{"className":44692},[2593],[15,44694],{"className":44695,"style":2589},[157],[15,44697,2465],{"className":44698},[59],[15,44700,39317],{"className":44701,"style":39413},[59,60],[15,44703,39320],{"className":44704,"style":10786},[59,60],[15,44706,44708,44711],{"style":44707},"top:-2.845em;",[15,44709],{"className":44710,"style":461},[302],[15,44712,44714],{"className":44713,"style":34470},[34469],[6325,44715,44716],{"xmlns":6327,"width":6328,"height":34473,"viewBox":34474,"preserveAspectRatio":6331},[6333,44717],{"d":34477},[15,44719,318],{"className":44720},[317],[15,44722,44724],{"className":44723},[290],[15,44725,44728],{"className":44726,"style":44727},[294],"height:0.155em;",[15,44729],{},"，而且它对绝对连续性不作要求，支撑分离时也只取到有限的最大值 ",[15,44732,44734,44747],{"className":44733,"translate":19},[18],[15,44735,44737],{"className":44736},[23],[25,44738,44739],{"xmlns":27},[29,44740,44741,44745],{},[32,44742,44743],{},[115,44744,2465],{},[39,44746,2465],{"encoding":41},[15,44748,44750],{"className":44749,"ariaHidden":46},[45],[15,44751,44753,44756],{"className":44752},[50],[15,44754],{"className":44755,"style":172},[54],[15,44757,2465],{"className":44758},[59],"，不会像 KL 那样发散——这让它在数值与统计上格外稳健。Hellinger 距离与 Fisher 信息度量在无穷小尺度上相容（4.6 节），并在非参数估计的 Le Cam 三点法、量子信息中的保真度（fidelity）、以及 Hellinger–Fisher–Rao 几何里扮演中心角色。",[2313,44761,44763],{"id":44762},"_36-α-散度","3.6 α-散度",[11,44765,14265,44766,44847],{},[15,44767,44769,44798],{"className":44768,"translate":19},[18],[15,44770,44772],{"className":44771},[23],[25,44773,44774],{"xmlns":27},[29,44775,44776,44795],{},[32,44777,44778,44781,44783,44785,44787,44789,44791,44793],{},[35,44779,44780],{},"α",[110,44782,5911],{},[110,44784,113],{"stretchy":112},[110,44786,4815],{},[115,44788,117],{},[110,44790,551],{"separator":46},[115,44792,117],{},[110,44794,120],{"stretchy":112},[39,44796,44797],{"encoding":41},"\\alpha\\in(-1,1)",[15,44799,44801,44820],{"className":44800,"ariaHidden":46},[45],[15,44802,44804,44807,44811,44814,44817],{"className":44803},[50],[15,44805],{"className":44806,"style":5926},[54],[15,44808,44780],{"className":44809,"style":44810},[59,60],"margin-right:0.0037em;",[15,44812],{"className":44813,"style":158},[157],[15,44815,5911],{"className":44816},[162],[15,44818],{"className":44819,"style":158},[157],[15,44821,44823,44826,44829,44832,44835,44838,44841,44844],{"className":44822},[50],[15,44824],{"className":44825,"style":139},[54],[15,44827,113],{"className":44828},[146],[15,44830,4815],{"className":44831},[59],[15,44833,117],{"className":44834},[59],[15,44836,551],{"className":44837},[573],[15,44839],{"className":44840,"style":406},[157],[15,44842,117],{"className":44843},[59],[15,44845,120],{"className":44846},[153],"，取",[15,44849,44851],{"className":44850,"translate":19},[180],[15,44852,44854,44928],{"className":44853,"translate":19},[18],[15,44855,44857],{"className":44856},[23],[25,44858,44859],{"xmlns":27,"display":189},[29,44860,44861,44925],{},[32,44862,44863,44869,44871,44873,44875,44877,44893,44923],{},[195,44864,44865,44867],{},[35,44866,37],{},[35,44868,44780],{},[110,44870,113],{"stretchy":112},[35,44872,769],{},[110,44874,120],{"stretchy":112},[110,44876,123],{},[238,44878,44879,44881],{},[115,44880,39015],{},[32,44882,44883,44885,44887],{},[115,44884,117],{},[110,44886,4815],{},[6816,44888,44889,44891],{},[35,44890,44780],{},[115,44892,2465],{},[32,44894,44895,44897,44899,44901,44921],{},[110,44896,113],{"fence":46},[115,44898,117],{},[110,44900,4815],{},[6816,44902,44903,44905],{},[35,44904,769],{},[32,44906,44907,44909,44911,44913,44915,44917,44919],{},[110,44908,113],{"stretchy":112},[115,44910,117],{},[110,44912,2472],{},[35,44914,44780],{},[110,44916,120],{"stretchy":112},[35,44918,941],{"mathvariant":209},[115,44920,2465],{},[110,44922,120],{"fence":46},[35,44924,4377],{"mathvariant":209},[39,44926,44927],{"encoding":41},"f_\\alpha(t)=\\frac{4}{1-\\alpha^2}\n\\left(1-t^{(1+\\alpha)\u002F2}\\right).",[15,44929,44931,44996],{"className":44930,"ariaHidden":46},[45],[15,44932,44934,44937,44978,44981,44984,44987,44990,44993],{"className":44933},[50],[15,44935],{"className":44936,"style":139},[54],[15,44938,44940,44943],{"className":44939},[59],[15,44941,37],{"className":44942,"style":61},[59,60],[15,44944,44946],{"className":44945},[281],[15,44947,44949,44970],{"className":44948},[285,286],[15,44950,44952,44967],{"className":44951},[290],[15,44953,44956],{"className":44954,"style":44955},[294],"height:0.1514em;",[15,44957,44958,44961],{"style":27535},[15,44959],{"className":44960,"style":303},[302],[15,44962,44964],{"className":44963},[307,308,309,310],[15,44965,44780],{"className":44966,"style":44810},[59,60,310],[15,44968,318],{"className":44969},[317],[15,44971,44973],{"className":44972},[290],[15,44974,44976],{"className":44975,"style":7970},[294],[15,44977],{},[15,44979,113],{"className":44980},[146],[15,44982,769],{"className":44983},[59,60],[15,44985,120],{"className":44986},[153],[15,44988],{"className":44989,"style":158},[157],[15,44991,123],{"className":44992},[162],[15,44994],{"className":44995,"style":158},[157],[15,44997,44999,45002,45104,45107,45183,45186],{"className":44998},[50],[15,45000],{"className":45001,"style":26642},[54],[15,45003,45005,45008,45101],{"className":45004},[59],[15,45006],{"className":45007},[146,441],[15,45009,45011],{"className":45010},[238],[15,45012,45014,45093],{"className":45013},[285,286],[15,45015,45017,45090],{"className":45016},[290],[15,45018,45020,45071,45079],{"className":45019,"style":22915},[294],[15,45021,45022,45025],{"style":457},[15,45023],{"className":45024,"style":461},[302],[15,45026,45028,45031,45034,45037,45040],{"className":45027},[59],[15,45029,117],{"className":45030},[59],[15,45032],{"className":45033,"style":2589},[157],[15,45035,4815],{"className":45036},[2593],[15,45038],{"className":45039,"style":2589},[157],[15,45041,45043,45046],{"className":45042},[59],[15,45044,44780],{"className":45045,"style":44810},[59,60],[15,45047,45049],{"className":45048},[281],[15,45050,45052],{"className":45051},[285],[15,45053,45055],{"className":45054},[290],[15,45056,45059],{"className":45057,"style":45058},[294],"height:0.7401em;",[15,45060,45062,45065],{"style":45061},"top:-2.989em;margin-right:0.05em;",[15,45063],{"className":45064,"style":303},[302],[15,45066,45068],{"className":45067},[307,308,309,310],[15,45069,2465],{"className":450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",[15,45193,45195,45236],{"className":45194,"translate":19},[18],[15,45196,45198],{"className":45197},[23],[25,45199,45200],{"xmlns":27},[29,45201,45202,45233],{},[32,45203,45204,45207,45209,45221,45223,45225,45227,45229,45231],{},[35,45205,45206],{},"a",[110,45208,123],{},[238,45210,45211,45219],{},[32,45212,45213,45215,45217],{},[115,45214,117],{},[110,45216,2472],{},[35,45218,44780],{},[115,45220,2465],{},[110,45222,5911],{},[110,45224,113],{"stretchy":112},[115,45226,126],{},[110,45228,551],{"separator":46},[115,45230,117],{},[110,45232,120],{"stretchy":112},[39,45234,45235],{"encoding":41},"a=\\frac{1+\\alpha}{2}\\in(0,1)",[15,45237,45239,45257,45346],{"className":45238,"ariaHidden":46},[45],[15,45240,45242,45245,45248,45251,45254],{"className":45241},[50],[15,45243],{"className":45244,"style":2347},[54],[15,45246,45206],{"className":45247},[59,60],[15,45249],{"className":45250,"style":158},[157],[15,45252,123],{"className":45253},[162],[15,45255],{"className":45256,"style":158},[157],[15,45258,45260,45263,45337,45340,45343],{"className":45259},[50],[15,45261],{"className":45262,"style":2507},[54],[15,45264,45266,45269,45334],{"className":45265},[59],[15,45267],{"className":45268},[146,441],[15,45270,45272],{"className":45271},[238],[15,45273,45275,45326],{"className":45274},[285,286],[15,45276,45278,45323],{"className":45277},[290],[15,45279,45281,45295,45303],{"className":45280,"style":2526},[294],[15,45282,45283,45286],{"style":2529},[15,45284],{"className":45285,"style":461},[302],[15,45287,45289],{"className":45288},[307,308,309,310],[15,45290,45292],{"className":45291},[59,310],[15,45293,2465],{"className":45294},[59,310],[15,45296,45297,45300],{"style":473},[15,45298],{"className":45299,"style":461},[302],[15,45301],{"className":45302,"style":481},[480],[15,45304,45305,45308],{"style":2552},[15,45306],{"className":45307,"style":461},[302],[15,45309,45311],{"className":45310},[307,308,309,310],[15,45312,45314,45317,45320],{"className":45313},[59,310],[15,45315,117],{"className":45316},[59,310],[15,45318,2472],{"className":45319},[2593,310],[15,45321,44780],{"className":45322,"style":44810},[59,60,310],[15,45324,318],{"className":45325},[317],[15,45327,45329],{"className":45328},[290],[15,45330,45332],{"className":45331,"style":2574},[294],[15,45333],{},[15,45335],{"className":45336},[153,441],[15,45338],{"className":45339,"style":158},[157],[15,45341,5911],{"className":45342},[162],[15,45344],{"className":45345,"style":158},[157],[15,45347,45349,45352,45355,45358,45361,45364,45367],{"className":45348},[50],[15,45350],{"className":45351,"style":139},[54],[15,45353,113],{"className":45354},[146],[15,45356,126],{"className":45357},[59],[15,45359,551],{"className":45360},[573],[15,45362],{"className":45363,"style":406},[157],[15,45365,117],{"className":45366},[59],[15,45368,120],{"className":45369},[153],"（于是 ",[15,45372,45374,45422],{"className":45373,"translate":19},[18],[15,45375,45377],{"className":45376},[23],[25,45378,45379],{"xmlns":27},[29,45380,45381,45419],{},[32,45382,45383,45399,45401],{},[238,45384,45385,45387],{},[115,45386,39015],{},[32,45388,45389,45391,45393],{},[115,45390,117],{},[110,45392,4815],{},[6816,45394,45395,45397],{},[35,45396,44780],{},[115,45398,2465],{},[110,45400,123],{},[238,45402,45403,45405],{},[115,45404,117],{},[32,45406,45407,45409,45411,45413,45415,45417],{},[35,45408,45206],{},[110,45410,113],{"stretchy":112},[115,45412,117],{},[110,45414,4815],{},[35,45416,45206],{},[110,45418,120],{"stretchy":112},[39,45420,45421],{"encoding":41},"\\frac{4}{1-\\alpha^2}=\\frac{1}{a(1-a)}",[15,45423,45425,45543],{"className":45424,"ariaHidden":46},[45],[15,45426,45428,45432,45534,45537,45540],{"className":45427},[50],[15,45429],{"className":45430,"style":45431},[54],"height:1.2484em;vertical-align:-0.4033em;",[15,45433,45435,45438,45531],{"className":45434},[59],[15,45436],{"className":45437},[146,441],[15,45439,45441],{"className":45440},[238],[15,45442,45444,45523],{"className":45443},[285,286],[15,45445,45447,45520],{"className":45446},[290],[15,45448,45450,45498,45506],{"className":45449,"style":2526},[294],[15,45451,45452,45455],{"style":2529},[15,45453],{"className":45454,"style":461},[302],[15,45456,45458],{"className":45457},[307,308,309,310],[15,45459,45461,45464,45467],{"className":45460},[59,310],[15,45462,117],{"className":45463},[59,310],[15,45465,4815],{"className":45466},[2593,310],[15,45468,45470,45473],{"className":45469},[59,310],[15,45471,44780],{"className":45472,"style":44810},[59,60,310],[15,45474,45476],{"className":45475},[281],[15,45477,45479],{"className":45478},[285],[15,45480,45482],{"className":45481},[290],[15,45483,45486],{"className":45484,"style":45485},[294],"height:0.7463em;",[15,45487,45489,45492],{"style":45488},"top:-2.786em;margin-right:0.0714em;",[15,45490],{"className":45491,"style":30232},[302],[15,45493,45495],{"className":45494},[307,30236,9676,310],[15,45496,2465],{"className":45497},[59,310],[15,45499,45500,45503],{"style":473},[15,45501],{"className":45502,"style":461},[302],[15,45504],{"className":45505,"style":481},[480],[15,45507,45508,45511],{"style":2552},[15,45509],{"className":45510,"style":461},[302],[15,45512,45514],{"className":45513},[307,308,309,310],[15,45515,45517],{"className":45516},[59,310],[15,45518,39015],{"className":45519},[59,310],[15,45521,318],{"className":45522},[317],[15,45524,45526],{"className":45525},[290],[15,45527,45529],{"className":45528,"style":16433},[294],[15,45530],{},[15,45532],{"className":45533},[153,441],[15,45535],{"className":45536,"style":158},[157],[15,45538,123],{"className":45539},[162],[15,45541],{"className":45542,"style":158},[157],[15,45544,45546,45550],{"className":45545},[50],[15,45547],{"className":45548,"style":45549},[54],"height:1.3651em;vertical-align:-0.52em;",[15,45551,45553,45556,45631],{"className":45552},[59],[15,45554],{"className":45555},[146,441],[15,45557,45559],{"className":45558},[238],[15,45560,45562,45622],{"className":45561},[285,286],[15,45563,45565,45619],{"className":45564},[290],[15,45566,45568,45597,45605],{"className":45567,"style":2526},[294],[15,45569,45570,45573],{"style":2529},[15,45571],{"className":45572,"style":461},[302],[15,45574,45576],{"className":45575},[307,308,309,310],[15,45577,45579,45582,45585,45588,45591,45594],{"className":45578},[59,310],[15,45580,45206],{"className":45581},[59,60,310],[15,45583,113],{"className":45584},[146,310],[15,45586,117],{"className":45587},[59,310],[15,45589,4815],{"className":45590},[2593,310],[15,45592,45206],{"className":45593},[59,60,310],[15,45595,120],{"className":45596},[153,310],[15,45598,45599,45602],{"style":473},[15,45600],{"className":45601,"style":461},[302],[15,45603],{"className":45604,"style":481},[480],[15,45606,45607,45610],{"style":2552},[15,45608],{"className":45609,"style":461},[302],[15,45611,45613],{"className":45612},[307,308,309,310],[15,45614,45616],{"className":45615},[59,310],[15,45617,117],{"className":45618},[59,310],[15,45620,318],{"className":45621},[317],[15,45623,45625],{"className":45624},[290],[15,45626,45629],{"className":45627,"style":45628},[294],"height:0.52em;",[15,45630],{},[15,45632],{"className":45633},[153,441],"）：",[15,45636,45638],{"className":45637,"translate":19},[180],[15,45639,45641,45727],{"className":45640,"translate":19},[18],[15,45642,45644],{"className":45643},[23],[25,45645,45646],{"xmlns":27,"display":189},[29,45647,45648,45724],{},[32,45649,45650,45656,45658,45660,45662,45664,45666,45668,45686,45722],{},[195,45651,45652,45654],{},[35,45653,199],{},[35,45655,44780],{},[110,45657,113],{"stretchy":112},[35,45659,206],{},[35,45661,210],{"mathvariant":209},[35,45663,213],{},[110,45665,120],{"stretchy":112},[110,45667,123],{},[238,45669,45670,45672],{},[115,45671,117],{},[32,45673,45674,45676,45678,45680,45682,45684],{},[35,45675,45206],{},[110,45677,113],{"stretchy":112},[115,45679,117],{},[110,45681,4815],{},[35,45683,45206],{},[110,45685,120],{"stretchy":112},[32,45687,45688,45690,45692,45694,45696,45702,45714,45716,45718,45720],{},[110,45689,113],{"fence":46},[115,45691,117],{},[110,45693,4815],{},[110,45695,1289],{},[6816,45697,45698,45700],{},[35,45699,11],{},[35,45701,45206],{},[6816,45703,45704,45706],{},[35,45705,944],{},[32,45707,45708,45710,45712],{},[115,45709,117],{},[110,45711,4815],{},[35,45713,45206],{},[1299,45715,1301],{},[35,45717,244],{},[35,45719,774],{},[110,45721,120],{"fence":46},[35,45723,4377],{"mathvariant":209},[39,45725,45726],{"encoding":41},"D_\\alpha(P\\|Q)=\\frac{1}{a(1-a)}\n\\left(1-\\int p^{a}q^{1-a}\\,dx\\right).",[15,45728,45730,45800],{"className":45729,"ariaHidden":46},[45],[15,45731,45733,45736,45776,45779,45782,45785,45788,45791,45794,45797],{"className":45732},[50],[15,45734],{"className":45735,"style":139},[54],[15,45737,45739,45742],{"className":45738},[59],[15,45740,199],{"className":45741,"style":277},[59,60],[15,45743,45745],{"className":45744},[281],[15,45746,45748,45768],{"className":45747},[285,286],[15,45749,45751,45765],{"className":45750},[290],[15,45752,45754],{"className":45753,"style":44955},[294],[15,45755,45756,45759],{"style":298},[15,45757],{"className":45758,"style":303},[302],[15,45760,45762],{"className":45761},[307,308,309,310],[15,45763,44780],{"className":45764,"style":44810},[59,60,310],[15,45766,318],{"className":45767},[317],[15,45769,45771],{"className":45770},[290],[15,45772,45774],{"className":45773,"style":7970},[294],[15,45775],{},[15,45777,113],{"className":45778},[146],[15,45780,206],{"className":45781,"style":334},[59,60],[15,45783,210],{"className":45784},[59],[15,45786,213],{"className":45787},[59,60],[15,45789,120],{"className":45790},[153],[15,45792],{"className":45793,"style":158},[157],[15,45795,123],{"className":45796},[162],[15,45798],{"className":45799,"style":158},[157],[15,45801,45803,45806,45889,45892,46005,46008],{"className":45802},[50],[15,45804],{"className":45805,"style":359},[54],[15,45807,45809,45812,45886],{"className":45808},[59],[15,45810],{"className":45811},[146,441],[15,45813,45815],{"className":45814},[238],[15,45816,45818,45878],{"className":45817},[285,286],[15,45819,45821,45875],{"className":45820},[290],[15,45822,45824,45856,45864],{"className":45823,"style":22915},[294],[15,45825,45826,45829],{"style":457},[15,45827],{"className":45828,"style":461},[302],[15,45830,45832,45835,45838,45841,45844,45847,45850,45853],{"className":45831},[59],[15,45833,45206],{"className":45834},[59,60],[15,45836,113],{"className":45837},[146],[15,45839,117],{"className":45840},[59],[15,45842],{"className":45843,"style":2589},[157],[15,45845,4815],{"className":45846},[2593],[15,45848],{"className":45849,"style":2589},[157],[15,45851,45206],{"className":45852},[59,60],[15,45854,120],{"className":45855},[153],[15,45857,45858,45861],{"style":473},[15,45859],{"className":45860,"style":461},[302],[15,45862],{"className":45863,"style":481},[480],[15,45865,45866,45869],{"style":484},[15,45867],{"className":45868,"style":461},[302],[15,45870,45872],{"className":45871},[59],[15,45873,117],{"className":45874},[59],[15,45876,318],{"className":45877},[317],[15,45879,45881],{"className":45880},[290],[15,45882,45884],{"className":45883,"style":1603},[294],[15,45885],{},[15,45887],{"className":45888},[153,441],[15,45890],{"className":45891,"style":406},[157],[15,45893,45895,45901,45904,45907,45910,45913,45916,45919,45952,45990,45993,45996,45999],{"className":45894},[410],[15,45896,45898],{"className":45897,"style":415},[146,414],[15,45899,113],{"className":45900},[419,309],[15,45902,117],{"className":45903},[59],[15,45905],{"className":45906,"style":2589},[157],[15,45908,4815],{"className":45909},[2593],[15,45911],{"className":45912,"style":2589},[157],[15,45914,1289],{"className":45915,"style":1489},[1486,1487,1488],[15,45917],{"className":45918,"style":406},[157],[15,45920,45922,45925],{"className":45921},[59],[15,45923,11],{"className":45924},[59,60],[15,45926,45928],{"className":45927},[281],[15,45929,45931],{"className":45930},[285],[15,45932,45934],{"className":45933},[290],[15,45935,45938],{"className":45936,"style":45937},[294],"height:0.7144em;",[15,45939,45940,45943],{"style":8279},[15,45941],{"className":45942,"style":303},[302],[15,45944,45946],{"className":45945},[307,308,309,310],[15,45947,45949],{"className":45948},[59,310],[15,45950,45206],{"className":45951},[59,60,310],[15,45953,45955,45958],{"className":45954},[59],[15,45956,944],{"className":45957,"style":981},[59,60],[15,45959,45961],{"className":45960},[281],[15,45962,45964],{"className":45963},[285],[15,45965,45967],{"className":45966},[290],[15,45968,45970],{"className":45969,"style":19761},[294],[15,45971,45972,45975],{"style":8279},[15,45973],{"className":45974,"style":303},[302],[15,45976,45978],{"className":45977},[307,308,309,310],[15,45979,45981,45984,45987],{"className":45980},[59,310],[15,45982,117],{"className":45983},[59,310],[15,45985,4815],{"className":45986},[2593,310],[15,45988,45206],{"className":45989},[59,60,310],[15,45991],{"className":45992,"style":406},[157],[15,45994,244],{"className":45995},[59,60],[15,45997,774],{"className":45998},[59,60],[15,46000,46002],{"className":46001,"style":415},[153,414],[15,46003,120],{"className":46004},[419,309],[15,46006],{"className":46007,"style":406},[157],[15,46009,4377],{"className":46010},[59],[11,46012,46013,46014,46017,46018,32250,46073,9402],{},"这正是 Csiszár \u002F Amari 的 ",[998,46015,46016],{},"α-散度","（差一个参数重命名）。",[15,46019,46021,46043],{"className":46020,"translate":19},[18],[15,46022,46024],{"className":46023},[23],[25,46025,46026],{"xmlns":27},[29,46027,46028,46040],{},[32,46029,46030,46032,46034,46036,46038],{},[35,46031,45206],{},[110,46033,123],{},[115,46035,117],{},[35,46037,941],{"mathvariant":209},[115,46039,2465],{},[39,46041,46042],{"encoding":41},"a=1\u002F2",[15,46044,46046,46064],{"className":46045,"ariaHidden":46},[45],[15,46047,46049,46052,46055,46058,46061],{"className":46048},[50],[15,46050],{"className":46051,"style":2347},[54],[15,46053,45206],{"className":46054},[59,60],[15,46056],{"className":46057,"style":158},[157],[15,46059,123],{"className":46060},[162],[15,46062],{"className":46063,"style":158},[157],[15,46065,46067,46070],{"className":46066},[50],[15,46068],{"className":46069,"style":139},[54],[15,46071,33665],{"className":46072},[59],[15,46074,46076,46110],{"className":46075,"translate":19},[18],[15,46077,46079],{"className":46078},[23],[25,46080,46081],{"xmlns":27},[29,46082,46083,46107],{},[32,46084,46085,46103,46105],{},[238,46086,46087,46089],{},[115,46088,117],{},[32,46090,46091,46093,46095,46097,46099,46101],{},[35,46092,45206],{},[110,46094,113],{"stretchy":112},[115,46096,117],{},[110,46098,4815],{},[35,46100,45206],{},[110,46102,120],{"stretchy":112},[110,46104,123],{},[115,46106,39015],{},[39,46108,46109],{"encoding":41},"\\frac{1}{a(1-a)}=4",[15,46111,46113,46211],{"className":46112,"ariaHidden":46},[45],[15,46114,46116,46119,46202,46205,46208],{"className":46115},[50],[15,46117],{"className":46118,"style":45549},[54],[15,46120,46122,46125,46199],{"className":46121},[59],[15,46123],{"className":46124},[146,441],[15,46126,46128],{"className":46127},[238],[15,46129,46131,46191],{"className":46130},[285,286],[15,46132,46134,46188],{"className":46133},[290],[15,46135,46137,46166,46174],{"className":46136,"style":2526},[294],[15,46138,46139,46142],{"style":2529},[15,46140],{"className":46141,"style":461},[302],[15,46143,46145],{"className":46144},[307,308,309,310],[15,46146,46148,46151,46154,46157,46160,46163],{"className":46147},[59,310],[15,46149,45206],{"className":46150},[59,60,310],[15,46152,113],{"className":46153},[146,310],[15,46155,117],{"className":46156},[59,310],[15,46158,4815],{"className":46159},[2593,310],[15,46161,45206],{"className":46162},[59,60,310],[15,46164,120],{"className":46165},[153,310],[15,46167,46168,46171],{"style":473},[15,46169],{"className":46170,"style":461},[302],[15,46172],{"className":46173,"style":481},[480],[15,46175,46176,46179],{"style":2552},[15,46177],{"className":46178,"style":461},[302],[15,46180,46182],{"className":46181},[307,308,309,310],[15,46183,46185],{"className":46184},[59,310],[15,46186,117],{"className":46187},[59,310],[15,46189,318],{"className":46190},[317],[15,46192,46194],{"className":46193},[290],[15,46195,46197],{"className":46196,"style":45628},[294],[15,46198],{},[15,46200],{"className":46201},[153,441],[15,46203],{"className":46204,"style":158},[157],[15,46206,123],{"className":46207},[162],[15,46209],{"className":46210,"style":158},[157],[15,46212,46214,46217],{"className":46213},[50],[15,46215],{"className":46216,"style":172},[54],[15,46218,39015],{"className":46219},[59],[15,46221,46223],{"className":46222,"translate":19},[180],[15,46224,46226,46292],{"className":46225,"translate":19},[18],[15,46227,46229],{"className":46228},[23],[25,46230,46231],{"xmlns":27,"display":189},[29,46232,46233,46289],{},[32,46234,46235,46247,46249,46251,46253,46255,46257,46259,46261,46275,46277,46279,46287],{},[195,46236,46237,46239],{},[35,46238,199],{},[32,46240,46241,46243,46245],{},[35,46242,44780],{},[110,46244,123],{},[115,46246,126],{},[110,46248,113],{"stretchy":112},[35,46250,206],{},[35,46252,210],{"mathvariant":209},[35,46254,213],{},[110,46256,120],{"stretchy":112},[110,46258,123],{},[115,46260,39015],{},[32,46262,46263,46265,46267,46269,46271,46273],{},[110,46264,113],{"fence":46},[115,46266,117],{},[110,46268,4815],{},[35,46270,39317],{},[35,46272,39320],{},[110,46274,120],{"fence":46},[110,46276,123],{},[115,46278,39015],{},[7555,46280,46281,46283,46285],{},[35,46282,33910],{},[1299,46284,42163],{},[115,46286,2465],{},[110,46288,551],{"separator":46},[39,46290,46291],{"encoding":41},"D_{\\alpha=0}(P\\|Q)=4\\left(1-BC\\right)\n=4H^2_{\\text{Bhattacharyya}},",[15,46293,46295,46374,46422],{"className":46294,"ariaHidden":46},[45],[15,46296,46298,46301,46350,46353,46356,46359,46362,46365,46368,46371],{"className":46297},[50],[15,46299],{"className":46300,"style":139},[54],[15,46302,46304,46307],{"className":46303},[59],[15,46305,199],{"className":46306,"style":277},[59,60],[15,46308,46310],{"className":46309},[281],[15,46311,46313,46342],{"className":46312},[285,286],[15,46314,46316,46339],{"className":46315},[290],[15,46317,46319],{"className":46318,"style":11137},[294],[15,46320,46321,46324],{"style":298},[15,46322],{"className":46323,"style":303},[302],[15,46325,46327],{"className":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是 Hellinger 型散度。两个端点极限要小心：",[15,46547,46549,46567],{"className":46548,"translate":19},[18],[15,46550,46552],{"className":46551},[23],[25,46553,46554],{"xmlns":27},[29,46555,46556,46564],{},[32,46557,46558],{},[195,46559,46560,46562],{},[35,46561,37],{},[35,46563,44780],{},[39,46565,46566],{"encoding":41},"f_\\alpha",[15,46568,46570],{"className":46569,"ariaHidden":46},[45],[15,46571,46573,46576],{"className":46572},[50],[15,46574],{"className":46575,"style":55},[54],[15,46577,46579,46582],{"className":46578},[59],[15,46580,37],{"className":46581,"style":61},[59,60],[15,46583,46585],{"className":46584},[281],[15,46586,46588,46608],{"className":46587},[285,286],[15,46589,46591,46605],{"className":46590},[290],[15,46592,46594],{"className":46593,"style":44955},[294],[15,46595,46596,46599],{"style":27535},[15,46597],{"className":46598,"style":303},[302],[15,46600,46602],{"className":46601},[307,308,309,310],[15,46603,44780],{"className":46604,"style":44810},[59,60,310],[15,46606,318],{"className":46607},[317],[15,46609,46611],{"className":46610},[290],[15,46612,46614],{"className":46613,"style":7970},[294],[15,46615],{},[15,46617,46619,46640],{"className":46618,"translate":19},[18],[15,46620,46622],{"className":46621},[23],[25,46623,46624],{"xmlns":27},[29,46625,46626,46637],{},[32,46627,46628,46630,46632,46635],{},[35,46629,44780],{},[110,46631,1032],{},[110,46633,46634],{},"±",[115,46636,117],{},[39,46638,46639],{"encoding":41},"\\alpha\\to\\pm1",[15,46641,46643,46661],{"className":46642,"ariaHidden":46},[45],[15,46644,46646,46649,46652,46655,46658],{"className":46645},[50],[15,46647],{"className":46648,"style":2347},[54],[15,46650,44780],{"className":46651,"style":44810},[59,60],[15,46653],{"className":46654,"style":158},[157],[15,46656,1032],{"className":46657},[162],[15,46659],{"className":46660,"style":158},[157],[15,46662,46664,46667,46670],{"className":46663},[50],[15,46665],{"className":46666,"style":12582},[54],[15,46668,46634],{"className":46669},[59],[15,46671,117],{"className":46672},[59]," 时逐点发散，但",[998,46675,46676],{},"散度值","收敛到 KL：",[15,46679,46681],{"className":46680,"translate":19},[180],[15,46682,46684,46813],{"className":46683,"translate":19},[18],[15,46685,46687],{"className":46686},[23],[25,46688,46689],{"xmlns":27,"display":189},[29,46690,46691,46810],{},[32,46692,46693,46710,46716,46718,46720,46722,46724,46726,46728,46738,46740,46742,46744,46746,46748,46750,46752,46770,46776,46778,46780,46782,46784,46786,46788,46798,46800,46802,46804,46806,46808],{},[6018,46694,46695,46702],{},[32,46696,46697,46700],{},[35,46698,46699],{},"lim",[110,46701,4105],{},[32,46703,46704,46706,46708],{},[35,46705,44780],{},[110,46707,1032],{},[115,46709,117],{},[195,46711,46712,46714],{},[35,46713,199],{},[35,46715,44780],{},[110,46717,113],{"stretchy":112},[35,46719,206],{},[35,46721,210],{"mathvariant":209},[35,46723,213],{},[110,46725,120],{"stretchy":112},[110,46727,123],{},[195,46729,46730,46732],{},[35,46731,199],{},[32,46733,46734,46736],{},[35,46735,10698],{},[35,46737,21919],{},[110,46739,113],{"stretchy":112},[35,46741,206],{},[35,46743,210],{"mathvariant":209},[35,46745,213],{},[110,46747,120],{"stretchy":112},[110,46749,551],{"separator":46},[157,46751],{"width":3264},[6018,46753,46754,46760],{},[32,46755,46756,46758],{},[35,46757,46699],{},[110,46759,4105],{},[32,46761,46762,46764,46766,46768],{},[35,46763,44780],{},[110,46765,1032],{},[110,46767,4815],{},[115,46769,117],{},[195,46771,46772,46774],{},[35,46773,199],{},[35,46775,44780],{},[110,46777,113],{"stretchy":112},[35,46779,206],{},[35,46781,210],{"mathvariant":209},[35,46783,213],{},[110,46785,120],{"stretchy":112},[110,46787,123],{},[195,46789,46790,46792],{},[35,46791,199],{},[32,46793,46794,46796],{},[35,46795,10698],{},[35,46797,21919],{},[110,46799,113],{"stretchy":112},[35,46801,213],{},[35,46803,210],{"mathvariant":209},[35,46805,206],{},[110,46807,120],{"stretchy":112},[35,46809,4377],{"mathvariant":209},[39,46811,46812],{"encoding":41},"\\lim_{\\alpha\\to1}D_\\alpha(P\\|Q)=D_{KL}(P\\|Q),\n\\qquad\n\\lim_{\\alpha\\to-1}D_\\alpha(P\\|Q)=D_{KL}(Q\\|P).",[15,46814,46816,46945,47146],{"className":46815,"ariaHidden":46},[45],[15,46817,46819,46823,46878,46881,46921,46924,46927,46930,46933,46936,46939,46942],{"className":46818},[50],[15,46820],{"className":46821,"style":46822},[54],"height:1.4671em;vertical-align:-0.7171em;",[15,46824,46826],{"className":46825},[1486,13720],[15,46827,46829,46869],{"className":46828},[285,286],[15,46830,46832,46866],{"className":46831},[290],[15,46833,46835,46856],{"className":46834,"style":12601},[294],[15,46836,46838,46841],{"style":46837},"top:-2.3829em;margin-left:0em;",[15,46839],{"className":46840,"style":461},[302],[15,46842,46844],{"className":46843},[307,308,309,310],[15,46845,46847,46850,46853],{"className":46846},[59,310],[15,46848,44780],{"className":46849,"style":44810},[59,60,310],[15,46851,1032],{"className":46852},[162,310],[15,46854,117],{"className":46855},[59,310],[15,46857,46858,46861],{"style":11867},[15,46859],{"className":46860,"style":461},[302],[15,46862,46863],{},[15,46864,46699],{"className":46865},[1486],[15,46867,318],{"className":46868},[317],[15,46870,46872],{"className":46871},[290],[15,46873,46876],{"className":46874,"style":46875},[294],"height:0.7171em;",[15,46877],{},[15,46879],{"className":46880,"style":406},[157],[15,46882,46884,46887],{"className":46883},[59],[15,46885,199],{"className":46886,"style":277},[59,60],[15,46888,46890],{"className":46889},[281],[15,46891,46893,46913],{"className":46892},[285,286],[15,46894,46896,46910],{"className":46895},[290],[15,46897,46899],{"className":46898,"style":44955},[294],[15,46900,46901,46904],{"style":298},[15,46902],{"className":46903,"style":303},[302],[15,46905,46907],{"className":46906},[307,308,309,310],[15,46908,44780],{"className":46909,"style":44810},[59,60,310],[15,46911,318],{"className":46912},[317],[15,46914,46916],{"className":46915},[290],[15,46917,46919],{"className":46918,"style":7970},[294],[15,46920],{},[15,46922,113],{"className":46923},[146],[15,46925,206],{"className":46926,"style":334},[59,60],[15,46928,210],{"className":46929},[59],[15,46931,213],{"className":46932},[59,60],[15,46934,120],{"className":46935},[153],[15,46937],{"className":46938,"style":158},[157],[15,46940,123],{"className":46941},[162],[15,46943],{"className":46944,"style":158},[157],[15,46946,46948,46952,46998,47001,47004,47007,47010,47013,47016,47019,47022,47079,47082,47122,47125,47128,47131,47134,47137,47140,47143],{"className":46947},[50],[15,46949],{"className":46950,"style":46951},[54],"height:1.5254em;vertical-align:-0.7754em;",[15,46953,46955,46958],{"className":46954},[59],[15,46956,199],{"className":46957,"style":277},[59,60],[15,46959,46961],{"className":46960},[281],[15,46962,46964,46990],{"className":46963},[285,286],[15,46965,46967,46987],{"className":46966},[290],[15,46968,46970],{"className":46969,"style":379},[294],[15,46971,46972,46975],{"style":298},[15,46973],{"className":46974,"style":303},[302],[15,46976,46978],{"className":46977},[307,308,309,310],[15,4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",[15,47222,47224,47242],{"className":47223,"translate":19},[18],[15,47225,47227],{"className":47226},[23],[25,47228,47229],{"xmlns":27},[29,47230,47231,47239],{},[32,47232,47233,47235,47237],{},[35,47234,44780],{},[110,47236,1032],{},[115,47238,117],{},[39,47240,47241],{"encoding":41},"\\alpha\\to1",[15,47243,47245,47263],{"className":47244,"ariaHidden":46},[45],[15,47246,47248,47251,47254,47257,47260],{"className":47247},[50],[15,47249],{"className":47250,"style":2347},[54],[15,47252,44780],{"className":47253,"style":44810},[59,60],[15,47255],{"className":47256,"style":158},[157],[15,47258,1032],{"className":47259},[162],[15,47261],{"className":47262,"style":158},[157],[15,47264,47266,47269],{"className":47265},[50],[15,47267],{"className":47268,"style":172},[54],[15,47270,117],{"className":47271},[59]," 的极限。"," 令 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0v40h399900v-40z",[15,48931,48933,48936],{"style":48932},"top:-2.0919em;",[15,48934],{"className":48935,"style":303},[302],[15,48937,48939],{"className":48938},[307,308,309,310,48908],[15,48940,48942,48945,48948],{"className":48941},[59,310],[15,48943,15716],{"className":48944,"style":15765},[59,60,310],[15,48946,1032],{"className":48947},[162,310],[15,48949,126],{"className":48950},[59,310],[15,48952,318],{"className":48953},[317],[15,48955,48957],{"className":48956},[290],[15,48958,48961],{"className":48959,"style":48960},[294],"height:0.6081em;",[15,48962],{},[15,48964],{"className":48965,"style":158},[157],[15,48967,48969,48972],{"className":48968},[59],[15,48970,199],{"className":48971,"style":277},[59,60],[15,48973,48975],{"className":48974},[281],[15,48976,48978,49004],{"className":48977},[285,286],[15,48979,48981,49001],{"className":48980},[290],[15,48982,48984],{"className":48983,"style":379},[294],[15,48985,48986,48989],{"style":298},[15,48987],{"className":48988,"style":303},[302],[15,48990,48992],{"className":48991},[307,308,309,310],[15,48993,48995,48998],{"className":48994},[59,310],[15,48996,10698],{"className":48997,"style":10786},[59,60,310],[15,48999,21919],{"className":49000},[59,60,310],[15,49002,318],{"className":49003},[317],[15,49005,49007],{"className":49006},[290],[15,49008,49010],{"className":49009,"style":7970},[294],[15,49011],{},[15,49013,113],{"className":49014},[146],[15,49016,206],{"className":49017,"style":334},[59,60],[15,49019,210],{"className":49020},[59],[15,49022,213],{"className":49023},[59,60],[15,49025,120],{"className":49026},[153],[15,49028,4377],{"className":49029},[59],[15,49031,318],{"className":49032},[317],[15,49034,49036],{"className":49035},[290],[15,49037,49039],{"className":49038,"style":48498},[294],[15,49040],{},[11,49042,49043,49099,49100,49129],{},[15,49044,49046,49066],{"className":49045,"translate":19},[18],[15,49047,49049],{"className":49048},[23],[25,49050,49051],{"xmlns":27},[29,49052,49053,49063],{},[32,49054,49055,49057,49059,49061],{},[35,49056,44780],{},[110,49058,1032],{},[110,49060,4815],{},[115,49062,117],{},[39,49064,49065],{"encoding":41},"\\alpha\\to-1",[15,49067,49069,49087],{"className":49068,"ariaHidden":46},[45],[15,49070,49072,49075,49078,49081,49084],{"className":49071},[50],[15,49073],{"className":49074,"style":2347},[54],[15,49076,44780],{"className":49077,"style":44810},[59,60],[15,49079],{"className":49080,"style":158},[157],[15,49082,1032],{"className":49083},[162],[15,49085],{"className":49086,"style":158},[157],[15,49088,49090,49093,49096],{"className":49089},[50],[15,49091],{"className":49092,"style":12582},[54],[15,49094,4815],{"className":49095},[59],[15,49097,117],{"className":49098},[59]," 对称地给出反向 KL。所以 α-散度把 KL（两个方向）与 Hellinger 连成一条曲线，",[15,49101,49103,49117],{"className":49102,"translate":19},[18],[15,49104,49106],{"className":49105},[23],[25,49107,49108],{"xmlns":27},[29,49109,49110,49114],{},[32,49111,49112],{},[35,49113,44780],{},[39,49115,49116],{"encoding":41},"\\alpha",[15,49118,49120],{"className":49119,"ariaHidden":46},[45],[15,49121,49123,49126],{"className":49122},[50],[15,49124],{"className":49125,"style":2347},[54],[15,49127,44780],{"className":49128,"style":44810},[59,60]," 是「在对称与不对称之间连续插值」的旋钮。",[11,49131,49132,49134,49135,49163,49164,49198,49199,49227,49228,49278,49279,49282,49283,49311,49312,49340,49341,49369,49370,49398],{},[998,49133,24219],{}," α-散度的价值在于它把「方向」与「强度」一并参数化：",[15,49136,49138,49151],{"className":49137,"translate":19},[18],[15,49139,49141],{"className":49140},[23],[25,49142,49143],{"xmlns":27},[29,49144,49145,49149],{},[32,49146,49147],{},[35,49148,44780],{},[39,49150,49116],{"encoding":41},[15,49152,49154],{"className":49153,"ariaHidden":46},[45],[15,49155,49157,49160],{"className":49156},[50],[15,49158],{"className":49159,"style":2347},[54],[15,49161,44780],{"className":49162,"style":44810},[59,60]," 从 ",[15,49165,49167,49183],{"className":49166,"translate":19},[18],[15,49168,49170],{"className":49169},[23],[25,49171,49172],{"xmlns":27},[29,49173,49174,49180],{},[32,49175,49176,49178],{},[110,49177,4815],{},[115,49179,117],{},[39,49181,49182],{"encoding":41},"-1",[15,49184,49186],{"className":49185,"ariaHidden":46},[45],[15,49187,49189,49192,49195],{"className":49188},[50],[15,49190],{"className":49191,"style":12582},[54],[15,49193,4815],{"className":49194},[59],[15,49196,117],{"className":49197},[59]," 扫到 ",[15,49200,49202,49215],{"className":49201,"translate":19},[18],[15,49203,49205],{"className":49204},[23],[25,49206,49207],{"xmlns":27},[29,49208,49209,49213],{},[32,49210,49211],{},[115,49212,117],{},[39,49214,117],{"encoding":41},[15,49216,49218],{"className":49217,"ariaHidden":46},[45],[15,49219,49221,49224],{"className":49220},[50],[15,49222],{"className":49223,"style":172},[54],[15,49225,117],{"className":49226},[59]," 时，散度由反向 KL（mode-seeking）连续过渡到正向 KL（mass-covering），中间在 ",[15,49229,49231,49248],{"className":49230,"translate":19},[18],[15,49232,49234],{"className":49233},[23],[25,49235,49236],{"xmlns":27},[29,49237,49238,49246],{},[32,49239,49240,49242,49244],{},[35,49241,44780],{},[110,49243,123],{},[115,49245,126],{},[39,49247,46514],{"encoding":41},[15,49249,49251,49269],{"className":49250,"ariaHidden":46},[45],[15,49252,49254,49257,49260,49263,49266],{"className":49253},[50],[15,49255],{"className":49256,"style":2347},[54],[15,49258,44780],{"className":49259,"style":44810},[59,60],[15,49261],{"className":49262,"style":158},[157],[15,49264,123],{"className":49265},[162],[15,49267],{"className":49268,"style":158},[157],[15,49270,49272,49275],{"className":49271},[50],[15,49273],{"className":49274,"style":172},[54],[15,49276,126],{"className":49277},[59]," 处恰好经过 Hellinger 这个对称点。在 Amari 的",[998,49280,49281],{},"信息几何","里，α-族是唯一同时拥有 ",[15,49284,49286,49299],{"className":49285,"translate":19},[18],[15,49287,49289],{"className":49288},[23],[25,49290,49291],{"xmlns":27},[29,49292,49293,49297],{},[32,49294,49295],{},[35,49296,44780],{},[39,49298,49116],{"encoding":41},[15,49300,49302],{"className":49301,"ariaHidden":46},[45],[15,49303,49305,49308],{"className":49304},[50],[15,49306],{"className":49307,"style":2347},[54],[15,49309,44780],{"className":49310,"style":44810},[59,60],"-对偶平坦结构的单参数散度族：它诱导的度量是 Fisher 度量，诱导的仿射联络是 ",[15,49313,49315,49328],{"className":49314,"translate":19},[18],[15,49316,49318],{"className":49317},[23],[25,49319,49320],{"xmlns":27},[29,49321,49322,49326],{},[32,49323,49324],{},[35,49325,44780],{},[39,49327,49116],{"encoding":41},[15,49329,49331],{"className":49330,"ariaHidden":46},[45],[15,49332,49334,49337],{"className":49333},[50],[15,49335],{"className":49336,"style":2347},[54],[15,49338,44780],{"className":49339,"style":44810},[59,60],"-联络，这正是它被称为「散度范本」的原因。在机器学习中，",[15,49342,49344,49357],{"className":49343,"translate":19},[18],[15,49345,49347],{"className":49346},[23],[25,49348,49349],{"xmlns":27},[29,49350,49351,49355],{},[32,49352,49353],{},[35,49354,44780],{},[39,49356,49116],{"encoding":41},[15,49358,49360],{"className":49359,"ariaHidden":46},[45],[15,49361,49363,49366],{"className":49362},[50],[15,49364],{"className":49365,"style":2347},[54],[15,49367,44780],{"className":49368,"style":44810},[59,60],"-GAN 与 ",[15,49371,49373,49386],{"className":49372,"translate":19},[18],[15,49374,49376],{"className":49375},[23],[25,49377,49378],{"xmlns":27},[29,49379,49380,49384],{},[32,49381,49382],{},[35,49383,44780],{},[39,49385,49116],{"encoding":41},[15,49387,49389],{"className":49388,"ariaHidden":46},[45],[15,49390,49392,49395],{"className":49391},[50],[15,49393],{"className":49394,"style":2347},[54],[15,49396,44780],{"className":49397,"style":44810},[59,60],"-VI 用一个参数调节「对异常值的鲁棒性」与「模式覆盖」之间的取舍；在信号处理中它用于独立成分分析（ICA）与鲁棒估计。",[2313,49400,49402],{"id":49401},"_37-与-rényi-散度的关系","3.7 与 Rényi 散度的关系",[11,49404,49405],{},"Rényi 散度定义为",[15,49407,49409],{"className":49408,"translate":19},[180],[15,49410,49412,49540],{"className":49411,"translate":19},[18],[15,49413,49415],{"className":49414},[23],[25,49416,49417],{"xmlns":27,"display":189},[29,49418,49419,49537],{},[1245,49420,49421,49501],{"rowspacing":1247,"columnalign":1248,"columnspacing":1249},[1251,49422,49423,49447],{},[1254,49424,49425],{},[1257,49426,49427],{"scriptlevel":126,"displaystyle":46},[32,49428,49429,49437,49439,49441,49443,49445],{},[7555,49430,49431,49433,49435],{},[35,49432,199],{},[35,49434,44780],{},[35,49436,1035],{},[110,49438,113],{"stretchy":112},[35,49440,206],{},[35,49442,210],{"mathvariant":209},[35,49444,213],{},[110,49446,120],{"stretchy":112},[1254,49448,49449],{},[1257,49450,49451],{"scriptlevel":126,"displaystyle":46},[32,49452,49453,49455,49457,49469,49471,49473,49475,49481,49493,49495,49497,49499],{},[32,49454],{},[110,49456,123],{},[238,49458,49459,49461],{},[115,49460,117],{},[32,49462,49463,49465,49467],{},[35,49464,44780],{},[110,49466,4815],{},[115,49468,117],{},[35,49470,4102],{},[110,49472,4105],{},[110,49474,1289],{},[6816,49476,49477,49479],{},[35,49478,11],{},[35,49480,44780],{},[6816,49482,49483,49485],{},[35,49484,944],{},[32,49486,49487,49489,49491],{},[115,49488,117],{},[110,49490,4815],{},[35,49492,44780],{},[1299,49494,1301],{},[35,49496,244],{},[35,49498,774],{},[110,49500,551],{"separator":46},[1251,49502,49503,49509],{},[1254,49504,49505],{},[1257,49506,49507],{"scriptlevel":126,"displaystyle":46},[32,49508],{},[1254,49510,49511],{},[1257,49512,49513],{"scriptlevel":126,"displaystyle":46},[32,49514,49515,49517,49519,49521,49523,49525,49527,49529,49531,49534],{},[32,49516],{},[157,49518],{"width":3264},[35,49520,44780],{},[110,49522,2905],{},[115,49524,126],{},[110,49526,551],{"separator":46},[1299,49528,4285],{},[35,49530,44780],{},[110,49532,49533],{"mathvariant":209},"≠",[115,49535,49536],{},"1.",[39,49538,49539],{"encoding":41},"\\begin{aligned}\nD^{R}_{\\alpha}(P\\|Q)\n&=\\frac{1}{\\alpha-1}\n\\log\\int p^{\\alpha}q^{1-\\alpha}\\,dx,\\\\[2pt]\n&\\qquad \\alpha>0,\\ \\alpha\\ne1 .\n\\end{aligned}",[15,49541,49543],{"className":49542,"ariaHidden":46},[45],[15,49544,49546,49550],{"className":49545},[50],[15,49547],{"className":49548,"style":49549},[54],"height:3.7223em;vertical-align:-1.6111em;",[15,49551,49553],{"className":49552},[59],[15,49554,49556,49673],{"className":49555},[1245],[15,49557,49559],{"className":49558},[1362],[15,49560,49562,49664],{"className":49561},[285,286],[15,49563,49565,49661],{"className":49564},[290],[15,49566,49569,49652],{"className":49567,"style":49568},[294],"height:2.1111em;",[15,49570,49572,49575],{"style":49571},"top:-4.1111em;",[15,49573],{"className":49574,"style":6275},[302],[15,49576,49578,49637,49640,49643,49646,49649],{"className":49577},[59],[15,49579,49581,49584],{"className":49580},[59],[15,49582,199],{"className":49583,"style":277},[59,60],[15,49585,49587],{"className":49586},[281],[15,49588,49590,49629],{"className":49589},[285,286],[15,49591,49593,49626],{"className":49592},[290],[15,49594,49596,49611],{"className":49595,"style":38538},[294],[15,49597,49599,49602],{"style":49598},"top:-2.453em;margin-left:-0.0278em;margin-right:0.05em;",[15,49600],{"className":49601,"style":303},[302],[15,49603,49605],{"className":49604},[307,308,309,310],[15,49606,49608],{"className":49607},[59,310],[15,49609,44780],{"className":49610,"style":44810},[59,60,310],[15,49612,49613,49616],{"style":8279},[15,49614],{"className":49615,"style":303},[302],[15,49617,49619],{"className":49618},[307,308,309,310],[15,49620,49622],{"className":49621},[59,310],[15,49623,1035],{"className":49624,"style":49625},[59,60,310],"margin-right:0.0077em;",[15,49627,318],{"className":49628},[317],[15,49630,49632],{"className":49631},[290],[15,49633,49635],{"className":49634,"style":8301},[294],[15,49636],{},[15,49638,113],{"className":49639},[146],[15,49641,206],{"className":49642,"style":334},[59,60],[15,49644,210],{"className":49645},[59],[15,49647,213],{"className":49648},[59,60],[15,49650,120],{"className":49651},[153],[15,49653,49655,49658],{"style":49654},"top:-2.1089em;",[15,49656],{"className":49657,"style":6275},[302],[15,49659],{"className":49660},[59],[15,49662,318],{"className":49663},[317],[15,49665,49667],{"className":49666},[290],[15,49668,49671],{"className":49669,"style":49670},[294],"height:1.6111em;",[15,49672],{},[15,49674,49676],{"className":49675},[1453],[15,49677,49679,49967],{"className":49678},[285,286],[15,49680,49682,49964],{"className":49681},[290],[15,49683,49685,49878],{"className":49684,"style":49568},[294],[15,49686,49687,49690],{"style":49571},[15,49688],{"className":49689,"style":6275},[302],[15,49691,49693,49696,49699,49702,49705,49779,49782,49787,49790,49793,49796,49828,49866,49869,49872,49875],{"className":49692},[59],[15,49694],{"className":49695},[59],[15,49697],{"className":49698,"style":158},[157],[15,49700,123],{"className":49701},[162],[15,49703],{"className":49704,"style":158},[157],[15,49706,49708,49711,49776],{"className":49707},[59],[15,49709],{"className":49710},[146,441],[15,49712,49714],{"className":49713},[238],[15,49715,49717,49768],{"className":49716},[285,286],[15,49718,49720,49765],{"className":49719},[290],[15,49721,49723,49746,49754],{"className":49722,"style":22915},[294],[15,49724,49725,49728],{"style":457},[15,49726],{"className":49727,"style":461},[302],[15,49729,49731,49734,49737,49740,49743],{"className":49730},[59],[15,49732,44780],{"className":49733,"style":44810},[59,60],[15,49735],{"className":49736,"style":2589},[157],[15,49738,4815],{"className":49739},[2593],[15,49741],{"className":49742,"style":2589},[157],[15,49744,117],{"className":49745},[59],[15,49747,49748,49751],{"style":473},[15,49749],{"className":49750,"style":461},[302],[15,49752],{"className":49753,"style":481},[480],[15,49755,49756,49759],{"style":484},[15,49757],{"className":49758,"style":461},[302],[15,49760,49762],{"className":49761},[59],[15,49763,117],{"className":49764},[59],[15,49766,318],{"className":49767},[317],[15,49769,49771],{"className":49770},[290],[15,49772,49774],{"className":49773,"style":17019},[294],[15,49775],{},[15,49777],{"className":49778},[153,441],[15,49780],{"className":49781,"style":406},[157],[15,49783,4156,49785],{"className":49784},[1486],[15,49786,4160],{"style":4159},[15,49788],{"className":49789,"style":406},[157],[15,49791,1289],{"className":49792,"style":1489},[1486,1487,1488],[15,49794],{"className":49795,"style":406},[157],[15,49797,49799,49802],{"className":49798},[59],[15,49800,11],{"className":49801},[59,60],[15,49803,49805],{"className":49804},[281],[15,49806,49808],{"className":49807},[285],[15,49809,49811],{"className":49810},[290],[15,49812,49814],{"className":49813,"style":45937},[294],[15,49815,49816,49819],{"style":8279},[15,49817],{"className":49818,"style":303},[302],[15,49820,49822],{"className":49821},[307,308,309,310],[15,49823,49825],{"className":49824},[59,310],[15,49826,44780],{"className":49827,"style":44810},[59,60,310],[15,49829,49831,49834],{"className":49830},[59],[15,49832,944],{"className":49833,"style":981},[59,60],[15,49835,49837],{"className":49836},[281],[15,49838,49840],{"className":49839},[285],[15,49841,49843],{"className":49842},[290],[15,49844,49846],{"className":49845,"style":19761},[294],[15,49847,49848,49851],{"style":8279},[15,49849],{"className":49850,"style":303},[302],[15,49852,49854],{"className":49853},[307,308,309,310],[15,49855,49857,49860,49863],{"className":49856},[59,310],[15,49858,117],{"className":49859},[59,310],[15,49861,4815],{"className":49862},[2593,310],[15,49864,44780],{"className":49865,"style":44810},[59,60,310],[15,49867],{"className":49868,"style":406},[157],[15,49870,244],{"className":49871},[59,60],[15,49873,774],{"className":49874},[59,60],[15,49876,551],{"className":49877},[573],[15,49879,49880,49883],{"style":49654},[15,49881],{"className":49882,"style":6275},[302],[15,49884,49886,49889,49892,49895,49898,49901,49904,49907,49910,49913,49916,49919,49922,49958,49961],{"className":49885},[59],[15,49887],{"className":49888},[59],[15,49890],{"className":49891,"style":3636},[157],[15,49893,44780],{"className":49894,"style":44810},[59,60],[15,49896],{"className":49897,"style":158},[157],[15,49899,2905],{"className":49900},[162],[15,49902],{"className":49903,"style":158},[157],[15,49905,126],{"className":49906},[59],[15,49908,551],{"className":49909},[573],[15,49911,4285],{"className":49912},[157],[15,49914],{"className":49915,"style":406},[157],[15,49917,44780],{"className":49918,"style":44810},[59,60],[15,49920],{"className":49921,"style":158},[157],[15,49923,49925,49952,49955],{"className":49924},[162],[15,49926,49928],{"className":49927},[162],[15,49929,49931],{"className":49930},[59,4402],[15,49932,49934],{"className":49933},[4406],[15,49935,49937,49940,49949],{"className":49936},[4410],[15,49938],{"className":49939,"style":55},[54],[15,49941,49943],{"className":49942},[4417],[15,49944,49946],{"className":49945},[59],[15,49947,4424],{"className":49948},[162],[15,49950],{"className":49951},[4428],[15,49953],{"className":49954},[157,4432],[15,49956,123],{"className":49957},[162],[15,49959],{"className":49960,"style":158},[157],[15,49962,49536],{"className":49963},[59],[15,49965,318],{"className":49966},[317],[15,49968,49970],{"className":49969},[290],[15,49971,49973],{"className":49972,"style":49670},[294],[15,49974],{},[11,49976,49977,49978,50010,50011,50062,50063,50239,50240,50441],{},"它不是（直接的）f-散度，因为 ",[15,49979,49981,49996],{"className":49980,"translate":19},[18],[15,49982,49984],{"className":49983},[23],[25,49985,49986],{"xmlns":27},[29,49987,49988,49994],{},[32,49989,49990,49992],{},[35,49991,4102],{},[110,49993,4105],{},[39,49995,21609],{"encoding":41},[15,49997,49999],{"className":49998,"ariaHidden":46},[45],[15,50000,50002,50005],{"className":50001},[50],[15,50003],{"className":50004,"style":55},[54],[15,50006,4156,50008],{"className":50007},[1486],[15,50009,4160],{"style":4159}," 包在积分外面（这正是它满足可乘可加性、但一般不满足数据处理不等式的根源：",[15,50012,50014,50032],{"className":50013,"translate":19},[18],[15,50015,50017],{"className":50016},[23],[25,50018,50019],{"xmlns":27},[29,50020,50021,50029],{},[32,50022,50023,50025,50027],{},[35,50024,44780],{},[110,50026,2905],{},[115,50028,117],{},[39,50030,50031],{"encoding":41},"\\alpha>1",[15,50033,50035,50053],{"className":50034,"ariaHidden":46},[45],[15,50036,50038,50041,50044,50047,50050],{"className":50037},[50],[15,50039],{"className":50040,"style":5926},[54],[15,50042,44780],{"className":50043,"style":44810},[59,60],[15,50045],{"className":50046,"style":158},[157],[15,50048,2905],{"className":50049},[162],[15,50051],{"className":50052,"style":158},[157],[15,50054,50056,50059],{"className":50055},[50],[15,50057],{"className":50058,"style":172},[54],[15,50060,117],{"className":50061},[59]," 的 Rényi 散度可以在随机变换下增大）。它与 α-散度是\"同一积分的两种包装\"：沿用 3.6 节的指数记号 ",[15,50064,50066,50105],{"className":50065,"translate":19},[18],[15,50067,50069],{"className":50068},[23],[25,50070,50071],{"xmlns":27},[29,50072,50073,50103],{},[32,50074,50075,50077,50079,50091,50093,50095,50097,50099,50101],{},[35,50076,45206],{},[110,50078,123],{},[238,50080,50081,50089],{},[32,50082,50083,50085,50087],{},[115,50084,117],{},[110,50086,2472],{},[35,50088,44780],{},[115,50090,2465],{},[110,50092,5911],{},[110,50094,113],{"stretchy":112},[115,50096,126],{},[110,50098,551],{"separator":46},[115,50100,117],{},[110,50102,120],{"stretchy":112},[39,50104,45235],{"encoding":41},[15,50106,50108,50126,50215],{"className":50107,"ariaHidden":46},[45],[15,50109,50111,50114,50117,50120,50123],{"className":50110},[50],[15,50112],{"className":50113,"style":2347},[54],[15,50115,45206],{"className":50116},[59,60],[15,50118],{"className":50119,"style":158},[157],[15,50121,123],{"className":50122},[162],[15,50124],{"className":50125,"style":158},[157],[15,50127,50129,50132,50206,50209,50212],{"className":50128},[50],[15,50130],{"className":50131,"style":2507},[54],[15,50133,50135,50138,50203],{"className":50134},[59],[15,50136],{"className":50137},[146,441],[15,50139,50141],{"className":50140},[238],[15,50142,50144,50195],{"className":50143},[285,286],[15,50145,50147,50192],{"className":50146},[290],[15,50148,50150,50164,50172],{"className":50149,"style":2526},[294],[15,50151,50152,50155],{"style":2529},[15,50153],{"className":50154,"style":461},[302],[15,50156,50158],{"className":50157},[307,308,309,310],[15,50159,50161],{"className":50160},[59,310],[15,50162,2465],{"className":50163},[59,310],[15,50165,50166,50169],{"style":473},[15,50167],{"className":50168,"style":461},[302],[15,50170],{"className":50171,"style":481},[480],[15,50173,50174,50177],{"style":2552},[15,50175],{"className":50176,"style":461},[302],[15,50178,50180],{"className":50179},[307,308,309,310],[15,50181,50183,50186,50189],{"className":50182},[59,310],[15,50184,117],{"className":50185},[59,310],[15,50187,2472],{"className":50188},[2593,310],[15,50190,44780],{"className":50191,"style":44810},[59,60,310],[15,50193,318],{"className":50194},[317],[15,50196,50198],{"className":50197},[290],[15,50199,50201],{"className":50200,"style":2574},[294],[15,50202],{},[15,50204],{"className":50205},[153,441],[15,50207],{"className":50208,"style":158},[157],[15,50210,5911],{"className":50211},[162],[15,50213],{"className":50214,"style":158},[157],[15,50216,50218,50221,50224,50227,50230,50233,50236],{"className":50217},[50],[15,50219],{"className":50220,"style":139},[54],[15,50222,113],{"className":50223},[146],[15,50225,126],{"className":50226},[59],[15,50228,551],{"className":50229},[573],[15,50231],{"className":50232,"style":406},[157],[15,50234,117],{"className":50235},[59],[15,50237,120],{"className":50238},[153],"， ",[15,50241,50243,50289],{"className":50242,"translate":19},[18],[15,50244,50246],{"className":50245},[23],[25,50247,50248],{"xmlns":27},[29,50249,50250,50286],{},[32,50251,50252,50258,50260,50262,50268,50280,50282,50284],{},[195,50253,50254,50256],{},[35,50255,33629],{},[35,50257,45206],{},[110,50259,123],{},[110,50261,1289],{},[6816,50263,50264,50266],{},[35,50265,11],{},[35,50267,45206],{},[6816,50269,50270,50272],{},[35,50271,944],{},[32,50273,50274,50276,50278],{},[115,50275,117],{},[110,50277,4815],{},[35,50279,45206],{},[1299,50281,1301],{},[35,50283,244],{},[35,50285,774],{},[39,50287,50288],{"encoding":41},"Z_a=\\int p^{a}q^{1-a}\\,dx",[15,50290,50292,50348],{"className":50291,"ariaHidden":46},[45],[15,50293,50295,50298,50339,50342,50345],{"className":50294},[50],[15,50296],{"className":50297,"style":15965},[54],[15,50299,50301,50304],{"className":50300},[59],[15,50302,33629],{"className":50303,"style":10786},[59,60],[15,50305,50307],{"className":50306},[281],[15,50308,50310,50331],{"className":50309},[285,286],[15,50311,50313,50328],{"className":50312},[290],[15,50314,50316],{"className":50315,"style":44955},[294],[15,50317,50319,50322],{"style":50318},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[15,50320],{"className":50321,"style":303},[302],[15,50323,50325],{"className":50324},[307,308,309,310],[15,50326,45206],{"className":50327},[59,60,310],[15,50329,318],{"className":50330},[317],[15,50332,50334],{"className":50333},[290],[15,50335,50337],{"className":50336,"style":7970},[294],[15,50338],{},[15,50340],{"className":50341,"style":158},[157],[15,50343,123],{"className":50344},[162],[15,50346],{"className":50347,"style":158},[157],[15,50349,50351,50355,50358,50361,50394,50432,50435,50438],{"className":50350},[50],[15,50352],{"className":50353,"style":50354},[54],"height:1.1202em;vertical-align:-0.3061em;",[15,50356,1289],{"className":50357,"style":5619},[1486,1487,5618],[15,50359],{"className":50360,"style":406},[157],[15,50362,50364,50367],{"className":50363},[59],[15,50365,11],{"className":50366},[59,60],[15,50368,50370],{"className":50369},[281],[15,50371,50373],{"className":50372},[285],[15,50374,50376],{"className":50375},[290],[15,50377,50380],{"className":50378,"style":50379},[294],"height:0.6644em;",[15,50381,50382,50385],{"style":6867},[15,50383],{"className":50384,"style":303},[302],[15,50386,50388],{"className":50387},[307,308,309,310],[15,50389,50391],{"className":50390},[59,310],[15,50392,45206],{"className":50393},[59,60,310],[15,50395,50397,50400],{"className":50396},[59],[15,50398,944],{"className":50399,"style":981},[59,60],[15,50401,50403],{"className":50402},[281],[15,50404,50406],{"className":50405},[285],[15,50407,50409],{"className":50408},[290],[15,50410,50412],{"className":50411,"style":20054},[294],[15,50413,50414,50417],{"style":6867},[15,50415],{"className":50416,"style":303},[302],[15,50418,50420],{"className":50419},[307,308,309,310],[15,50421,50423,50426,50429],{"className":50422},[59,310],[15,50424,117],{"className":50425},[59,310],[15,50427,4815],{"className":50428},[2593,310],[15,50430,45206],{"className":50431},[59,60,310],[15,50433],{"className":50434,"style":406},[157],[15,50436,244],{"className":50437},[59,60],[15,50439,774],{"className":50440},[59,60],"，则 3.6 节给出",[15,50443,50445],{"className":50444,"translate":19},[180],[15,50446,50448,50494],{"className":50447,"translate":19},[18],[15,50449,50451],{"className":50450},[23],[25,50452,50453],{"xmlns":27,"display":189},[29,50454,50455,50491],{},[32,50456,50457,50463,50465,50467,50469,50471,50473,50475,50477,50479,50481,50483,50489],{},[195,50458,50459,50461],{},[35,50460,33629],{},[35,50462,45206],{},[110,50464,123],{},[115,50466,117],{},[110,50468,4815],{},[35,50470,45206],{},[110,50472,113],{"stretchy":112},[115,50474,117],{},[110,50476,4815],{},[35,50478,45206],{},[110,50480,120],{"stretchy":112},[1299,50482,1301],{},[195,50484,50485,50487],{},[35,50486,199],{},[35,50488,44780],{},[110,50490,551],{"separator":46},[39,50492,50493],{"encoding":41},"Z_a=1-a(1-a)\\,D_\\alpha ,",[15,50495,50497,50552,50570,50594],{"className":50496,"ariaHidden":46},[45],[15,50498,50500,50503,50543,50546,50549],{"className":50499},[50],[15,50501],{"className":50502,"style":15965},[54],[15,50504,50506,50509],{"className":50505},[59],[15,50507,33629],{"className":50508,"style":10786},[59,60],[15,50510,50512],{"className":50511},[281],[15,50513,50515,50535],{"className":50514},[285,286],[15,50516,50518,50532],{"className":50517},[290],[15,50519,50521],{"className":50520,"style":44955},[294],[15,50522,50523,50526],{"style":50318},[15,50524],{"className":50525,"style":303},[302],[15,50527,50529],{"className":50528},[307,308,309,310],[15,50530,45206],{"className":50531},[59,60,310],[15,50533,318],{"className":50534},[317],[15,50536,50538],{"className":50537},[290],[15,50539,50541],{"className":50540,"style":7970},[294],[15,50542],{},[15,50544],{"className":50545,"style":158},[157],[15,50547,123],{"className":50548},[162],[15,50550],{"className":50551,"style":158},[157],[15,50553,50555,50558,50561,50564,50567],{"className":50554},[50],[15,50556],{"className":50557,"style":12582},[54],[15,50559,117],{"className":50560},[59],[15,50562],{"className":50563,"style":2589},[157],[15,50565,4815],{"className":50566},[2593],[15,50568],{"className":50569,"style":2589},[157],[15,50571,50573,50576,50579,50582,50585,50588,50591],{"className":50572},[50],[15,50574],{"className":50575,"style":139},[54],[15,50577,45206],{"className":50578},[59,60],[15,50580,113],{"className":50581},[146],[15,50583,117],{"className":50584},[59],[15,50586],{"className":50587,"style":2589},[157],[15,50589,4815],{"className":50590},[2593],[15,50592],{"className":50593,"style":2589},[157],[15,50595,50597,50600,50603,50606,50609,50649],{"className":50596},[50],[15,50598],{"className":50599,"style":139},[54],[15,50601,45206],{"className":50602},[59,60],[15,50604,120],{"className":50605},[153],[15,50607],{"className":50608,"style":406},[157],[15,50610,50612,50615],{"className":50611},[59],[15,50613,199],{"className":50614,"style":277},[59,60],[15,50616,50618],{"className":50617},[281],[15,50619,50621,50641],{"className":50620},[285,286],[15,50622,50624,50638],{"className":50623},[290],[15,50625,50627],{"className":50626,"style":44955},[294],[15,50628,50629,50632],{"style":298},[15,50630],{"className":50631,"style":303},[302],[15,50633,50635],{"className":50634},[307,308,309,310],[15,50636,44780],{"className":50637,"style":44810},[59,60,310],[15,50639,318],{"className":50640},[317],[15,50642,50644],{"className":50643},[290],[15,50645,50647],{"className":50646,"style":7970},[294],[15,50648],{},[15,50650,551],{"className":50651},[573],[11,50653,50654,50655,50658,50659,50687,50688,50750,50751,50779],{},"而把 Rényi 的",[998,50656,50657],{},"阶数","取为 ",[15,50660,50662,50675],{"className":50661,"translate":19},[18],[15,50663,50665],{"className":50664},[23],[25,50666,50667],{"xmlns":27},[29,50668,50669,50673],{},[32,50670,50671],{},[35,50672,45206],{},[39,50674,45206],{"encoding":41},[15,50676,50678],{"className":50677,"ariaHidden":46},[45],[15,50679,50681,50684],{"className":50680},[50],[15,50682],{"className":50683,"style":2347},[54],[15,50685,45206],{"className":50686},[59,60],"（注意 order ",[15,50689,50691,50705],{"className":50690,"translate":19},[18],[15,50692,50694],{"className":50693},[23],[25,50695,50696],{"xmlns":27},[29,50697,50698,50702],{},[32,50699,50700],{},[110,50701,49533],{"mathvariant":209},[39,50703,50704],{"encoding":41},"\\neq",[15,50706,50708],{"className":50707,"ariaHidden":46},[45],[15,50709,50711,50714],{"className":50710},[50],[15,50712],{"className":50713,"style":55},[54],[15,50715,50717,50744,50747],{"className":50716},[162],[15,50718,50720],{"className":50719},[162],[15,50721,50723],{"className":50722},[59,4402],[15,50724,50726],{"className":50725},[4406],[15,50727,50729,50732,50741],{"className":50728},[4410],[15,50730],{"className":50731,"style":55},[54],[15,50733,50735],{"className":50734},[4417],[15,50736,50738],{"className":50737},[59],[15,50739,4424],{"className":50740},[162],[15,50742],{"className":50743},[4428],[15,50745],{"className":50746},[157,4432],[15,50748,123],{"className":50749},[162]," 3.6 的参数 ",[15,50752,50754,50767],{"className":50753,"translate":19},[18],[15,50755,50757],{"className":50756},[23],[25,50758,50759],{"xmlns":27},[29,50760,50761,50765],{},[32,50762,50763],{},[35,50764,44780],{},[39,50766,49116],{"encoding":41},[15,50768,50770],{"className":50769,"ariaHidden":46},[45],[15,50771,50773,50776],{"className":50772},[50],[15,50774],{"className":50775,"style":2347},[54],[15,50777,44780],{"className":50778,"style":44810},[59,60],"，只是经过线性换算）：",[15,50781,50783],{"className":50782,"translate":19},[180],[15,50784,50786,50890],{"className":50785,"translate":19},[18],[15,50787,50789],{"className":50788},[23],[25,50790,50791],{"xmlns":27,"display":189},[29,50792,50793,50887],{},[32,50794,50795,50803,50805,50807,50809,50811,50813,50815,50827,50829,50831,50837,50839,50841,50853,50855,50857,50859,50861,50863,50865,50867,50869,50871,50873,50875,50877,50883,50885],{},[7555,50796,50797,50799,50801],{},[35,50798,199],{},[35,50800,45206],{},[35,50802,1035],{},[110,50804,113],{"stretchy":112},[35,50806,206],{},[35,50808,210],{"mathvariant":209},[35,50810,213],{},[110,50812,120],{"stretchy":112},[110,50814,123],{},[238,50816,50817,50819],{},[115,50818,117],{},[32,50820,50821,50823,50825],{},[35,50822,45206],{},[110,50824,4815],{},[115,50826,117],{},[35,50828,4102],{},[110,50830,4105],{},[195,50832,50833,50835],{},[35,50834,33629],{},[35,50836,45206],{},[110,50838,123],{},[110,50840,4815],{},[238,50842,50843,50845],{},[115,50844,117],{},[32,50846,50847,50849,50851],{},[115,50848,117],{},[110,50850,4815],{},[35,50852,45206],{},[1299,50854,1301],{},[35,50856,4102],{},[110,50858,4105],{},[110,50860,113],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[115,50862,117],{},[110,50864,4815],{},[35,50866,45206],{},[110,50868,113],{"stretchy":112},[115,50870,117],{},[110,50872,4815],{},[35,50874,45206],{},[110,50876,120],{"stretchy":112},[195,50878,50879,50881],{},[35,50880,199],{},[35,50882,44780],{},[110,50884,120],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[35,50886,4377],{"mathvariant":209},[39,50888,50889],{"encoding":41},"D^{R}_{a}(P\\|Q)=\\frac{1}{a-1}\\log Z_a\n=-\\frac{1}{1-a}\\,\\log\\big(1-a(1-a)D_\\alpha\\big).",[15,50891,50893,50981,51121,51236,51260],{"className":50892,"ariaHidden":46},[45],[15,50894,50896,50900,50957,50960,50963,50966,50969,50972,50975,50978],{"className":50895},[50],[15,50897],{"className":50898,"style":50899},[54],"height:1.1413em;vertical-align:-0.25em;",[15,50901,50903,50906],{"className":50902},[59],[15,50904,199],{"className":50905,"style":277},[59,60],[15,50907,50909],{"className":50908},[281],[15,50910,50912,50949],{"className":50911},[285,286],[15,50913,50915,50946],{"className":50914},[290],[15,50916,50918,50932],{"className":50917,"style":38538},[294],[15,50919,50920,50923],{"style":49598},[15,50921],{"className":50922,"style":303},[302],[15,50924,50926],{"className":50925},[307,308,309,310],[15,50927,50929],{"className":50928},[59,310],[15,50930,45206],{"className":50931},[59,60,310],[15,50933,50934,50937],{"style":8279},[15,50935],{"className":50936,"style":303},[302],[15,50938,50940],{"className":50939},[307,308,309,310],[15,50941,50943],{"className":50942},[59,310],[15,50944,1035],{"className":50945,"style":49625},[59,60,310],[15,50947,318],{"className":50948},[317],[15,50950,50952],{"className":50951},[290],[15,50953,50955],{"className":50954,"style":8301},[294],[15,50956],{},[15,50958,113],{"className":50959},[146],[15,50961,206],{"className":50962,"style":334},[59,60],[15,50964,210],{"className":50965},[59],[15,50967,213],{"className":50968},[59,60],[15,50970,120],{"className":50971},[153],[15,50973],{"className":50974,"style":158},[157],[15,50976,123],{"className":50977},[162],[15,50979],{"className":50980,"style":158},[157],[15,50982,50984,50987,51061,51064,51069,51072,51112,51115,51118],{"className":50983},[50],[15,50985],{"className":50986,"style":26642},[54],[15,50988,50990,50993,51058],{"className":50989},[59],[15,50991],{"className":50992},[146,441],[15,50994,50996],{"className":50995},[238],[15,50997,50999,51050],{"className":50998},[285,286],[15,51000,51002,51047],{"className":51001},[290],[15,51003,51005,51028,51036],{"className":51004,"style":22915},[294],[15,51006,51007,51010],{"style":457},[15,51008],{"className":51009,"style":461},[302],[15,51011,51013,51016,51019,51022,51025],{"className":51012},[59],[15,51014,45206],{"className":51015},[59,60],[15,51017],{"className":51018,"style":2589},[157],[15,51020,4815],{"className":51021},[2593],[15,51023],{"className":51024,"style":2589},[157],[15,51026,117],{"className":51027},[59],[15,51029,51030,51033],{"style":473},[15,51031],{"className":51032,"style":461},[302],[15,51034],{"className":51035,"style":481},[480],[15,51037,51038,51041],{"style":484},[15,51039],{"className":51040,"style":461},[302],[15,51042,51044],{"className":51043},[59],[15,51045,117],{"className":51046},[59],[15,51048,318],{"className":51049},[317],[15,51051,51053],{"className":51052},[290],[15,51054,51056],{"className":51055,"style":17019},[294],[15,51057],{},[15,51059],{"className":51060},[153,441],[15,51062],{"className":51063,"style":406},[157],[15,51065,4156,51067],{"className":51066},[1486],[15,51068,4160],{"style":4159},[15,51070],{"className":51071,"style":406},[157],[15,51073,51075,51078],{"className":51074},[59],[15,51076,33629],{"className":51077,"style":10786},[59,60],[15,51079,51081],{"className":51080},[281],[15,51082,51084,51104],{"className":51083},[285,286],[15,51085,51087,51101],{"className":51086},[290],[15,51088,51090],{"className":51089,"style":44955},[294],[15,51091,51092,51095],{"style":50318},[15,51093],{"className":51094,"style":303},[302],[15,51096,51098],{"className":51097},[307,308,309,310],[15,51099,45206],{"className":51100},[59,60,310],[15,51102,318],{"className":51103},[317],[15,51105,51107],{"className":51106},[290],[15,51108,51110],{"className":51109,"style":7970},[294],[15,51111],{},[15,51113],{"className":51114,"style":158},[157],[15,51116,123],{"className":51117},[162],[15,51119],{"className":51120,"style":158},[157],[15,51122,51124,51127,51130,51204,51207,51210,51215,51218,51224,51227,51230,51233],{"className":51123},[50],[15,51125],{"className":51126,"style":26642},[54],[15,51128,4815],{"className":51129},[59],[15,51131,51133,51136,51201],{"className":51132},[59],[15,51134],{"className":51135},[146,441],[15,51137,51139],{"className":51138},[238],[15,51140,51142,51193],{"className":51141},[285,286],[15,51143,51145,51190],{"className":51144},[290],[15,51146,51148,51171,51179],{"className":51147,"style":22915},[294],[15,51149,51150,51153],{"style":457},[15,51151],{"className":51152,"style":461},[302],[15,51154,51156,51159,51162,51165,51168],{"className":51155},[59],[15,51157,117],{"className":51158},[59],[15,51160],{"className":51161,"style":2589},[157],[15,51163,4815],{"className":51164},[2593],[15,51166],{"className":51167,"style":2589},[157],[15,51169,45206],{"className":51170},[59,60],[15,51172,51173,51176],{"style":473},[15,51174],{"className":51175,"style":461},[302],[15,51177],{"className":51178,"style":481},[480],[15,51180,51181,51184],{"style":484},[15,51182],{"className":51183,"style":461},[302],[15,51185,51187],{"className":51186},[59],[15,51188,117],{"className":51189},[59],[15,51191,318],{"className":51192},[317],[15,51194,51196],{"className":51195},[290],[15,51197,51199],{"className":51198,"style":17019},[294],[15,51200],{},[15,51202],{"className":51203},[153,441],[15,51205],{"className":51206,"style":406},[157],[15,51208],{"className":51209,"style":406},[157],[15,51211,4156,51213],{"className":51212},[1486],[15,51214,4160],{"style":4159},[15,51216],{"className":51217,"style":406},[157],[15,51219,51221],{"className":51220},[59],[15,51222,113],{"className":51223},[419,9676],[15,51225,117],{"className":51226},[59],[15,51228],{"className":51229,"style":2589},[157],[15,51231,4815],{"className":51232},[2593],[15,51234],{"className":51235,"style":2589},[157],[15,51237,51239,51242,51245,51248,51251,51254,51257],{"className":51238},[50],[15,51240],{"className":51241,"style":139},[54],[15,51243,45206],{"className":51244},[59,60],[15,51246,113],{"className":51247},[146],[15,51249,117],{"className":51250},[59],[15,51252],{"className":51253,"style":2589},[157],[15,51255,4815],{"className":51256},[2593],[15,51258],{"className":51259,"style":2589},[157],[15,51261,51263,51266,51269,51272,51312,51318],{"className":51262},[50],[15,51264],{"className":51265,"style":20589},[54],[15,51267,45206],{"className":51268},[59,60],[15,51270,120],{"className":51271},[153],[15,51273,51275,51278],{"className":51274},[59],[15,51276,199],{"className":51277,"style":277},[59,60],[15,51279,51281],{"className":51280},[281],[15,51282,51284,51304],{"className":51283},[285,286],[15,51285,51287,51301],{"className":51286},[290],[15,51288,51290],{"className":51289,"style":44955},[294],[15,51291,51292,51295],{"style":298},[15,51293],{"className":51294,"style":303},[302],[15,51296,51298],{"className":51297},[307,308,309,310],[15,51299,44780],{"className":51300,"style":44810},[59,60,310],[15,51302,318],{"className":51303},[317],[15,51305,51307],{"className":51306},[290],[15,51308,51310],{"className":51309,"style":7970},[294],[15,51311],{},[15,51313,51315],{"className":51314},[59],[15,51316,120],{"className":51317},[419,9676],[15,51319,4377],{"className":51320},[59],[11,51322,51323,51324,51471,51472,51500,51501,51533],{},"这说明两者是一条\"幂—对数\"的两级包装链：f-散度是对似然比幂 ",[15,51325,51327,51353],{"className":51326,"translate":19},[18],[15,51328,51330],{"className":51329},[23],[25,51331,51332],{"xmlns":27},[29,51333,51334,51350],{},[32,51335,51336,51338,51344],{},[110,51337,113],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[238,51339,51340,51342],{},[35,51341,11],{},[35,51343,944],{},[6816,51345,51346,51348],{},[110,51347,120],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[35,51349,45206],{},[39,51351,51352],{"encoding":41},"\\big(\\frac pq\\big)^a",[15,51354,51356],{"className":51355,"ariaHidden":46},[45],[15,51357,51359,51363,51369,51437],{"className":51358},[50],[15,51360],{"className":51361,"style":51362},[54],"height:1.3854em;vertical-align:-0.4811em;",[15,51364,51366],{"className":51365},[59],[15,51367,113],{"className":51368},[419,9676],[15,51370,51372,51375,51434],{"className":51371},[59],[15,51373],{"className":51374},[146,441],[15,51376,51378],{"className":51377},[238],[15,51379,51381,51426],{"className":51380},[285,286],[15,51382,51384,51423],{"className":51383},[290],[15,51385,51387,51401,51409],{"className":51386,"style":3105},[294],[15,51388,51389,51392],{"style":2529},[15,51390],{"className":51391,"style":461},[302],[15,51393,51395],{"className":51394},[307,308,309,310],[15,51396,51398],{"className":51397},[59,310],[15,51399,944],{"className":51400,"style":981},[59,60,310],[15,51402,51403,51406],{"style":473},[15,51404],{"className":51405,"style":461},[302],[15,51407],{"className":51408,"style":481},[480],[15,51410,51411,51414],{"style":2855},[15,51412],{"className":51413,"style":461},[302],[15,51415,51417],{"className":51416},[307,308,309,310],[15,51418,51420],{"className":51419},[59,310],[15,51421,11],{"className":51422},[59,60,310],[15,51424,318],{"className":51425},[317],[15,51427,51429],{"className":51428},[290],[15,51430,51432],{"className":51431,"style":2741},[294],[15,51433],{},[15,51435],{"className":51436},[153,441],[15,51438,51440,51446],{"className":51439},[59],[15,51441,51443],{"className":51442},[59],[15,51444,120],{"className":51445},[419,9676],[15,51447,51449],{"className":51448},[281],[15,51450,51452],{"className":51451},[285],[15,51453,51455],{"className":51454},[290],[15,51456,51459],{"className":51457,"style":51458},[294],"height:0.9043em;",[15,51460,51462,51465],{"style":51461},"top:-3.3029em;margin-right:0.05em;",[15,51463],{"className":51464,"style":303},[302],[15,51466,51468],{"className":51467},[307,308,309,310],[15,51469,45206],{"className":51470},[59,60,310]," 做 ",[15,51473,51475,51488],{"className":51474,"translate":19},[18],[15,51476,51478],{"className":51477},[23],[25,51479,51480],{"xmlns":27},[29,51481,51482,51486],{},[32,51483,51484],{},[35,51485,213],{},[39,51487,213],{"encoding":41},[15,51489,51491],{"className":51490,"ariaHidden":46},[45],[15,51492,51494,51497],{"className":51493},[50],[15,51495],{"className":51496,"style":566},[54],[15,51498,213],{"className":51499},[59,60]," 下的线性平均，Rényi 在外面再套一层对数。所以 f-散度族与 Rényi 族\"同源\"：后者是前者经 ",[15,51502,51504,51519],{"className":51503,"translate":19},[18],[15,51505,51507],{"className":51506},[23],[25,51508,51509],{"xmlns":27},[29,51510,51511,51517],{},[32,51512,51513,51515],{},[35,51514,4102],{},[110,51516,4105],{},[39,51518,21609],{"encoding":41},[15,51520,51522],{"className":51521,"ariaHidden":46},[45],[15,51523,51525,51528],{"className":51524},[50],[15,51526],{"className":51527,"style":55},[54],[15,51529,4156,51531],{"className":51530},[1486],[15,51532,4160],{"style":4159}," 变换后的\"信息量\"版本。",[11,51535,51536,51538,51539,51542],{},[998,51537,24219],{}," Rényi 散度在",[998,51540,51541],{},"假设检验","中直接就是误差指数：Chernoff 信息",[15,51544,51546],{"className":51545,"translate":19},[180],[15,51547,51549,51672],{"className":51548,"translate":19},[18],[15,51550,51552],{"className":51551},[23],[25,51553,51554],{"xmlns":27,"display":189},[29,51555,51556,51669],{},[32,51557,51558,51560,51562,51564,51566,51568,51570,51572,51592,51594,51596,51598,51600,51602,51610,51612,51614,51616,51618,51620,51622,51624,51626,51628,51649,51651,51657],{},[35,51559,39320],{},[110,51561,113],{"stretchy":112},[35,51563,206],{},[110,51565,551],{"separator":46},[35,51567,213],{},[110,51569,120],{"stretchy":112},[110,51571,123],{},[6018,51573,51574,51580],{},[32,51575,51576,51578],{},[35,51577,35458],{},[110,51579,4105],{},[32,51581,51582,51584,51586,51588,51590],{},[115,51583,126],{},[110,51585,7253],{},[35,51587,11343],{},[110,51589,7253],{},[115,51591,117],{},[110,51593,113],{"stretchy":112},[115,51595,117],{},[110,51597,4815],{},[35,51599,11343],{},[110,51601,120],{"stretchy":112},[7555,51603,51604,51606,51608],{},[35,51605,199],{},[35,51607,11343],{},[35,51609,1035],{},[110,51611,113],{"stretchy":112},[35,51613,206],{},[35,51615,210],{"mathvariant":209},[35,51617,213],{},[110,51619,120],{"stretchy":112},[110,51621,123],{},[110,51623,4815],{},[35,51625,4102],{},[110,51627,4105],{},[6018,51629,51630,51637],{},[32,51631,51632,51635],{},[35,51633,51634],{},"min",[110,51636,4105],{},[32,51638,51639,51641,51643,51645,51647],{},[115,51640,126],{},[110,51642,9954],{},[35,51644,11343],{},[110,51646,9954],{},[115,51648,117],{},[110,51650,1289],{},[6816,51652,51653,51655],{},[35,51654,11],{},[35,51656,11343],{},[6816,51658,51659,51661],{},[35,51660,944],{},[32,51662,51663,51665,51667],{},[115,51664,117],{},[110,51666,4815],{},[35,51668,11343],{},[39,51670,51671],{"encoding":41},"C(P,Q)=\\sup_{0\u003C\\lambda\u003C1}\n(1-\\lambda)D^{R}_{\\lambda}(P\\|Q)\n=-\\log\\min_{0\\le\\lambda\\le1}\n\\int p^\\lambda 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引理给出的单侧指数则是 ",[15,52049,52051,52083],{"className":52050,"translate":19},[18],[15,52052,52054],{"className":52053},[23],[25,52055,52056],{"xmlns":27},[29,52057,52058,52080],{},[32,52059,52060,52068,52070],{},[7555,52061,52062,52064,52066],{},[35,52063,199],{},[115,52065,117],{},[35,52067,1035],{},[110,52069,123],{},[195,52071,52072,52074],{},[35,52073,199],{},[32,52075,52076,52078],{},[35,52077,10698],{},[35,52079,21919],{},[39,52081,52082],{"encoding":41},"D^{R}_{1}=D_{KL}",[15,52084,52086,52162],{"className":52085,"ariaHidden":46},[45],[15,52087,52089,52093,52153,52156,52159],{"className":52088},[50],[15,52090],{"className":52091,"style":52092},[54],"height:1.0894em;vertical-align:-0.2481em;",[15,52094,52096,52099],{"className":52095},[59],[15,52097,199],{"className":52098,"style":277},[59,60],[15,52100,52102],{"className":52101},[281],[15,52103,52105,52144],{"className":52104},[285,286],[15,52106,52108,52141],{"className":52107},[290],[15,52109,52112,52127],{"className":52110,"style":52111},[294],"height:0.8413em;",[15,52113,52115,52118],{"style":52114},"top:-2.4519em;margin-left:-0.0278em;margin-right:0.05em;",[15,52116],{"className":52117,"style":303},[302],[15,52119,52121],{"className":52120},[307,308,309,310],[15,52122,52124],{"className":52123},[59,310],[15,52125,117],{"className":52126},[59,310],[15,52128,52129,52132],{"style":6867},[15,52130],{"className":52131,"style":303},[302],[15,52133,52135],{"className":52134},[307,308,309,310],[15,52136,52138],{"className":52137},[59,310],[15,52139,1035],{"className":52140,"style":49625},[59,60,310],[15,52142,318],{"className":52143},[317],[15,52145,52147],{"className":52146},[290],[15,52148,52151],{"className":52149,"style":52150},[294],"height:0.2481em;",[15,52152],{},[15,52154],{"className":52155,"style":158},[157],[15,52157,123],{"className":52158},[162],[15,52160],{"className":52161,"style":158},[157],[15,52163,52165,52168],{"className":52164},[50],[15,52166],{"className":52167,"style":15965},[54],[15,52169,52171,52174],{"className":52170},[59],[15,52172,199],{"className":52173,"style":277},[59,60],[15,52175,52177],{"className":52176},[281],[15,52178,52180,52206],{"className":52179},[285,286],[15,52181,52183,52203],{"className":52182},[290],[15,52184,52186],{"className":52185,"style":379},[294],[15,52187,52188,52191],{"style":298},[15,52189],{"className":52190,"style":303},[302],[15,52192,52194],{"className":52193},[307,308,309,310],[15,52195,52197,52200],{"className":52196},[59,310],[15,52198,10698],{"className":52199,"style":10786},[59,60,310],[15,52201,21919],{"className":52202},[59,60,310],[15,52204,318],{"className":52205},[317],[15,52207,52209],{"className":52208},[290],[15,52210,52212],{"className":52211,"style":7970},[294],[15,52213],{},"。对应的 Rényi 熵 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时退化为 Shannon 熵、在 ",[15,52338,52340,52358],{"className":52339,"translate":19},[18],[15,52341,52343],{"className":52342},[23],[25,52344,52345],{"xmlns":27},[29,52346,52347,52355],{},[32,52348,52349,52351,52353],{},[35,52350,44780],{},[110,52352,1032],{},[35,52354,1027],{"mathvariant":209},[39,52356,52357],{"encoding":41},"\\alpha\\to\\infty",[15,52359,52361,52379],{"className":52360,"ariaHidden":46},[45],[15,52362,52364,52367,52370,52373,52376],{"className":52363},[50],[15,52365],{"className":52366,"style":2347},[54],[15,52368,44780],{"className":52369,"style":44810},[59,60],[15,52371],{"className":52372,"style":158},[157],[15,52374,1032],{"className":52375},[162],[15,52377],{"className":52378,"style":158},[157],[15,52380,52382,52385],{"className":52381},[50],[15,52383],{"className":52384,"style":2347},[54],[15,52386,1027],{"className":52387},[59]," 时退化为 min-entropy 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",[998,52510,52511],{},"Rényi 差分隐私（RDP）"," 中被用来对隐私预算做更紧致的记账。它与 f-散度族的\"同源\"关系也给出一条实用经验：讨论可加性、信息几何与变分表示时用 f-散度更顺手，讨论误差指数与熵时用它的对数版本更自然。",[2313,52514,52516],{"id":52515},"_38-汇总表","3.8 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",[15,54520,54522,54549],{"className":54521,"translate":19},[18],[15,54523,54525],{"className":54524},[23],[25,54526,54527],{"xmlns":27},[29,54528,54529,54547],{},[32,54530,54531,54533,54535,54537,54539,54541,54543,54545],{},[35,54532,44780],{},[110,54534,5911],{},[110,54536,113],{"stretchy":112},[110,54538,4815],{},[115,54540,117],{},[110,54542,551],{"separator":46},[115,54544,117],{},[110,54546,120],{"stretchy":112},[39,54548,44797],{"encoding":41},[15,54550,54552,54570],{"className":54551,"ariaHidden":46},[45],[15,54553,54555,54558,54561,54564,54567],{"className":54554},[50],[15,54556],{"className":54557,"style":5926},[54],[15,54559,44780],{"className":54560,"style":44810},[59,60],[15,54562],{"className":54563,"style":158},[157],[15,54565,5911],{"className":54566},[162],[15,54568],{"className":54569,"style":158},[157],[15,54571,54573,54576,54579,54582,54585,54588,54591,54594],{"className":54572},[50],[15,54574],{"className":54575,"style":139},[54],[15,54577,113],{"className":54578},[146],[15,54580,4815],{"className":54581},[59],[15,54583,117],{"className":54584},[59],[15,54586,551],{"className":54587},[573],[15,54589],{"className":54590,"style":406},[157],[15,54592,117],{"className":54593},[59],[15,54595,120],{"className":54596},[153],[52621,54598,54599],{},[15,54600,54602,54658],{"className":54601,"translate":19},[18],[15,54603,54605],{"className":54604},[23],[25,54606,54607],{"xmlns":27},[29,54608,54609,54655],{},[32,54610,54611,54627,54629,54631,54633,54653],{},[238,54612,54613,54615],{},[115,54614,39015],{},[32,54616,54617,54619,54621],{},[115,54618,117],{},[110,54620,4815],{},[6816,54622,54623,54625],{},[35,54624,44780],{},[115,54626,2465],{},[110,54628,113],{"stretchy":112},[115,54630,117],{},[110,54632,4815],{},[6816,54634,54635,54637],{},[35,54636,769],{},[32,54638,54639,54641,54643,54645,54647,54649,54651],{},[110,54640,113],{"stretchy":112},[115,54642,117],{},[110,54644,2472],{},[35,54646,44780],{},[110,54648,120],{"stretchy":112},[35,54650,941],{"mathvariant":209},[115,54652,2465],{},[110,54654,120],{"stretchy":112},[39,54656,54657],{"encoding":41},"\\frac{4}{1-\\alpha^2}(1-t^{(1+\\alpha)\u002F2})",[15,54659,54661,54782],{"className":54660,"ariaHidden":46},[45],[15,54662,54664,54667,54767,54770,54773,54776,54779],{"className":54663},[50],[15,54665],{"className":54666,"style":45431},[54],[15,54668,54670,54673,54764],{"className":54669},[59],[15,54671],{"className":54672},[146,441],[15,54674,54676],{"className":54675},[238],[15,54677,54679,54756],{"className":54678},[285,286],[15,54680,54682,54753],{"className":54681},[290],[15,54683,54685,54731,54739],{"className":54684,"style":2526},[294],[15,54686,54687,54690],{"style":2529},[15,54688],{"className":54689,"style":461},[302],[15,54691,54693],{"className":54692},[307,308,309,310],[15,54694,54696,54699,54702],{"className":54695},[59,310],[15,54697,117],{"className":54698},[59,310],[15,54700,4815],{"className":54701},[2593,310],[15,54703,54705,54708],{"className":54704},[59,310],[15,54706,44780],{"className":54707,"style":44810},[59,60,310],[15,54709,54711],{"className":54710},[281],[15,54712,54714],{"className":54713},[285],[15,54715,54717],{"className":54716},[290],[15,54718,54720],{"className":54719,"style":45485},[294],[15,54721,54722,54725],{"style":45488},[15,54723],{"className":54724,"style":30232},[302],[15,54726,54728],{"className":54727},[307,30236,9676,310],[15,54729,2465],{"className":54730},[59,310],[15,54732,54733,54736],{"style":473},[15,54734],{"className":54735,"style":461},[302],[15,54737],{"className":54738,"style":481},[480],[15,54740,54741,54744],{"style":2552},[15,54742],{"className":54743,"style":461},[302],[15,54745,54747],{"className":54746},[307,308,309,310],[15,54748,54750],{"className":54749},[59,310],[15,54751,39015],{"className":54752},[59,310],[15,54754,318],{"className":54755},[317],[15,54757,54759],{"className":54758},[290],[15,54760,54762],{"className":54761,"style":16433},[294],[15,54763],{},[15,54765],{"className":54766},[153,441],[15,54768,113],{"className":54769},[146],[15,54771,117],{"className":54772},[59],[15,54774],{"className":54775,"style":2589},[157],[15,54777,4815],{"className":54778},[2593],[15,54780],{"className":54781,"style":2589},[157],[15,54783,54785,54789,54837],{"className":54784},[50],[15,54786],{"className":54787,"style":54788},[54],"height:1.138em;vertical-align:-0.25em;",[15,54790,54792,54795],{"className":54791},[59],[15,54793,769],{"className":54794},[59,60],[15,54796,54798],{"className":54797},[281],[15,54799,54801],{"className":54800},[285],[15,54802,54804],{"className":54803},[290],[15,54805,54808],{"className":54806,"style":54807},[294],"height:0.888em;",[15,54809,54810,54813],{"style":6867},[15,54811],{"className":54812,"style":303},[302],[15,54814,54816],{"className":54815},[307,308,309,310],[15,54817,54819,54822,54825,54828,54831,54834],{"className":54818},[59,310],[15,54820,113],{"className":54821},[146,310],[15,54823,117],{"className":54824},[59,310],[15,54826,2472],{"className":54827},[2593,310],[15,54829,44780],{"className":54830,"style":44810},[59,60,310],[15,54832,120],{"className":54833},[153,310],[15,54835,45176],{"className":54836},[59,310],[15,54838,120],{"className":54839},[153],[52621,54841,52781],{},[52621,54843,52781],{},[52621,54845,54846,54847,53501],{},"有限（",[15,54848,54850,54870],{"className":54849,"translate":19},[18],[15,54851,54853],{"className":54852},[23],[25,54854,54855],{"xmlns":27},[29,54856,54857,54867],{},[32,54858,54859,54861,54863,54865],{},[35,54860,44780],{},[110,54862,49533],{"mathvariant":209},[110,54864,46634],{},[115,54866,117],{},[39,54868,54869],{"encoding":41},"\\alpha\\ne\\pm1",[15,54871,54873,54924],{"className":54872,"ariaHidden":46},[45],[15,54874,54876,54879,54882,54885,54921],{"className":54875},[50],[15,54877],{"className":54878,"style":55},[54],[15,54880,44780],{"className":54881,"style":44810},[59,60],[15,54883],{"className":54884,"style":158},[157],[15,54886,54888,54915,54918],{"className":54887},[162],[15,54889,54891],{"className":54890},[162],[15,54892,54894],{"className":54893},[59,4402],[15,54895,54897],{"className":54896},[4406],[15,54898,54900,54903,54912],{"className":54899},[4410],[15,54901],{"className":54902,"style":55},[54],[15,54904,54906],{"className":54905},[4417],[15,54907,54909],{"className":54908},[59],[15,54910,4424],{"className":54911},[162],[15,54913],{"className":54914},[4428],[15,54916],{"className":54917},[157,4432],[15,54919,123],{"className":54920},[162],[15,54922],{"className":54923,"style":158},[157],[15,54925,54927,54930,54933],{"className":54926},[50],[15,54928],{"className":54929,"style":12582},[54],[15,54931,46634],{"className":54932},[59],[15,54934,117],{"className":54935},[59],[528,54937,54939],{"id":54938},"_4-基本性质","4. 基本性质",[11,54941,54942,54943,15799,55007,55164,55165,10182],{},"下列性质逐个给出推导。约定 ",[15,54944,54946,54968],{"className":54945,"translate":19},[18],[15,54947,54949],{"className":54948},[23],[25,54950,54951],{"xmlns":27},[29,54952,54953,54965],{},[32,54954,54955,54957,54959,54961,54963],{},[35,54956,206],{},[110,54958,551],{"separator":46},[35,54960,213],{},[110,54962,714],{},[35,54964,2366],{},[39,54966,54967],{"encoding":41},"P,Q\\ll\\mu",[15,54969,54971,54998],{"className":54970,"ariaHidden":46},[45],[15,54972,54974,54977,54980,54983,54986,54989,54992,54995],{"className":54973},[50],[15,54975],{"className":54976,"style":566},[54],[15,54978,206],{"className":54979,"style":334},[59,60],[15,54981,551],{"className":54982},[573],[15,54984],{"className":54985,"style":406},[157],[15,54987,213],{"className":54988},[59,60],[15,54990],{"className":54991,"style":158},[157],[15,54993,714],{"className":54994},[162],[15,54996],{"className":54997,"style":158},[157],[15,54999,55001,55004],{"className":55000},[50],[15,55002],{"className":55003,"style":2278},[54],[15,55005,2366],{"className":55006},[59,60],[15,55008,55010,55042],{"className":55009,"translate":19},[18],[15,55011,55013],{"className":55012},[23],[25,55014,55015],{"xmlns":27},[29,55016,55017,55039],{},[32,55018,55019,55021,55023,55029,55031,55033,55035,55037],{},[35,55020,2366],{},[110,55022,123],{},[238,55024,55025,55027],{},[115,55026,117],{},[115,55028,2465],{},[110,55030,113],{"stretchy":112},[35,55032,206],{},[110,55034,2472],{},[35,55036,213],{},[110,55038,120],{"stretchy":112},[39,55040,55041],{"encoding":41},"\\mu=\\frac12(P+Q)",[15,55043,55045,55063,55152],{"className":55044,"ariaHidden":46},[45],[15,55046,55048,55051,55054,55057,55060],{"className":55047},[50],[15,55049],{"className":55050,"style":2278},[54],[15,55052,2366],{"className":55053},[59,60],[15,55055],{"className":55056,"style":158},[157],[15,55058,123],{"className":55059},[162],[15,55061],{"className":55062,"style":158},[157],[15,55064,55066,55069,55137,55140,55143,55146,55149],{"className":55065},[50],[15,55067],{"className":55068,"style":2507},[54],[15,55070,55072,55075,55134],{"className":55071},[59],[15,55073],{"className":55074},[146,441],[15,55076,55078],{"className":55077},[238],[15,55079,55081,55126],{"className":55080},[285,286],[15,55082,55084,55123],{"className":55083},[290],[15,55085,55087,55101,55109],{"className":55086,"style":2526},[294],[15,55088,55089,55092],{"style":2529},[15,55090],{"className":55091,"style":461},[302],[15,55093,55095],{"className":55094},[307,308,309,310],[15,55096,55098],{"className":55097},[59,310],[15,55099,2465],{"className":55100},[59,310],[15,55102,55103,55106],{"style":473},[15,55104],{"className":55105,"style":461},[302],[15,55107],{"className":55108,"style":481},[480],[15,55110,55111,55114],{"style":2552},[15,55112],{"className":55113,"style":461},[302],[15,55115,55117],{"className":55116},[307,308,309,310],[15,55118,55120],{"className":55119},[59,310],[15,55121,117],{"className":55122},[59,310],[15,55124,318],{"className":55125},[317],[15,55127,55129],{"className":55128},[290],[15,55130,55132],{"className":55131,"style":2574},[294],[15,55133],{},[15,55135],{"className":55136},[153,441],[15,55138,113],{"className":55139},[146],[15,55141,206],{"className":55142,"style":334},[59,60],[15,55144],{"className":55145,"style":2589},[157],[15,55147,2472],{"className":55148},[2593],[15,55150],{"className":55151,"style":2589},[157],[15,55153,55155,55158,55161],{"className":55154},[50],[15,55156],{"className":55157,"style":139},[54],[15,55159,213],{"className":55160},[59,60],[15,55162,120],{"className":55163},[153]," 或任一控制测度），",[15,55166,55168,55193],{"className":55167,"translate":19},[18],[15,55169,55171],{"className":55170},[23],[25,55172,55173],{"xmlns":27},[29,55174,55175,55191],{},[32,55176,55177,55179,55181,55183,55185,55187,55189],{},[35,55178,769],{},[110,55180,123],{},[35,55182,244],{},[35,55184,206],{},[35,55186,941],{"mathvariant":209},[35,55188,244],{},[35,55190,213],{},[39,55192,5335],{"encoding":41},[15,55194,55196,55214],{"className":55195,"ariaHidden":46},[45],[15,55197,55199,55202,55205,55208,55211],{"className":55198},[50],[15,55200],{"className":55201,"style":5345},[54],[15,55203,769],{"className":55204},[59,60],[15,55206],{"className":55207,"style":158},[157],[15,55209,123],{"className":55210},[162],[15,55212],{"className":55213,"style":158},[157],[15,55215,55217,55220,55223,55226,55229,55232],{"className":55216},[50],[15,55218],{"className":55219,"style":139},[54],[15,55221,244],{"className":55222},[59,60],[15,55224,206],{"className":55225,"style":334},[59,60],[15,55227,941],{"className":55228},[59],[15,55230,244],{"className":55231},[59,60],[15,55233,213],{"className":55234},[59,60],[2313,55236,55238],{"id":55237},"_41-非负性","4.1 非负性",[993,55240,55241,55288,55413],{},[11,55242,55243,55246,55247,2747],{},[998,55244,55245],{},"命题 4.1（非负性）"," 对一切概率分布 ",[15,55248,55250,55267],{"className":55249,"translate":19},[18],[15,55251,55253],{"className":55252},[23],[25,55254,55255],{"xmlns":27},[29,55256,55257,55265],{},[32,55258,55259,55261,55263],{},[35,55260,206],{},[110,55262,551],{"separator":46},[35,55264,213],{},[39,55266,556],{"encoding":41},[15,55268,55270],{"className":55269,"ariaHidden":46},[45],[15,55271,55273,55276,55279,55282,55285],{"className":55272},[50],[15,55274],{"className":55275,"style":566},[54],[15,55277,206],{"className":55278,"style":334},[59,60],[15,55280,551],{"className":55281},[573],[15,55283],{"className":55284,"style":406},[157],[15,55286,213],{"className":55287},[59,60],[15,55289,55291],{"className":55290,"translate":19},[180],[15,55292,55294,55328],{"className":55293,"translate":19},[18],[15,55295,55297],{"className":55296},[23],[25,55298,55299],{"xmlns":27,"display":189},[29,55300,55301,55325],{},[32,55302,55303,55309,55311,55313,55315,55317,55319,55321,55323],{},[195,55304,55305,55307],{},[35,55306,199],{},[35,55308,37],{},[110,55310,113],{"stretchy":112},[35,55312,206],{},[35,55314,210],{"mathvariant":209},[35,55316,213],{},[110,55318,120],{"stretchy":112},[110,55320,6930],{},[115,55322,126],{},[110,55324,551],{"separator":46},[39,55326,55327],{"encoding":41},"D_f(P\\|Q)\\ge 0,",[15,55329,55331,55401],{"className":55330,"ariaHidden":46},[45],[15,55332,55334,55337,55377,55380,55383,55386,55389,55392,55395,55398],{"className":55333},[50],[15,55335],{"className":55336,"style":270},[54],[15,55338,55340,55343],{"className":55339},[59],[15,55341,199],{"className":55342,"style":277},[59,60],[15,55344,55346],{"className":55345},[281],[15,55347,55349,55369],{"className":55348},[285,286],[15,55350,55352,55366],{"className":55351},[290],[15,55353,55355],{"className":55354,"style":295},[294],[15,55356,55357,55360],{"style":298},[15,55358],{"className":55359,"style":303},[302],[15,55361,55363],{"className":55362},[307,308,309,310],[15,55364,37],{"className":55365,"style":61},[59,60,310],[15,55367,318],{"className":55368},[317],[15,55370,55372],{"className":55371},[290],[15,55373,55375],{"className":55374,"style":325},[294],[15,55376],{},[15,55378,113],{"className":55379},[146],[15,55381,206],{"className":55382,"style":334},[59,60],[15,55384,210],{"className":55385},[59],[15,55387,213],{"className":55388},[59,60],[15,55390,120],{"className":55391},[153],[15,55393],{"className":55394,"style":158},[157],[15,55396,6930],{"className":55397},[162],[15,55399],{"className":55400,"style":158},[157],[15,55402,55404,55407,55410],{"className":55403},[50],[15,55405],{"className":55406,"style":9360},[54],[15,55408,126],{"className":55409},[59],[15,55411,551],{"className":55412},[573],[11,55414,55415,55416,55444,55445,10182],{},"且当 ",[15,55417,55419,55432],{"className":55418,"translate":19},[18],[15,55420,55422],{"className":55421},[23],[25,55423,55424],{"xmlns":27},[29,55425,55426,55430],{},[32,55427,55428],{},[35,55429,37],{},[39,55431,37],{"encoding":41},[15,55433,55435],{"className":55434,"ariaHidden":46},[45],[15,55436,55438,55441],{"className":55437},[50],[15,55439],{"className":55440,"style":55},[54],[15,55442,37],{"className":55443,"style":61},[59,60]," 严格凸时，等号成立当且仅当 ",[15,55446,55448,55465],{"className":55447,"translate":19},[18],[15,55449,55451],{"className":55450},[23],[25,55452,55453],{"xmlns":27},[29,55454,55455,55463],{},[32,55456,55457,55459,55461],{},[35,55458,206],{},[110,55460,123],{},[35,55462,213],{},[39,55464,10273],{"encoding":41},[15,55466,55468,55486],{"className":55467,"ariaHidden":46},[45],[15,55469,55471,55474,55477,55480,55483],{"className":55470},[50],[15,55472],{"className":55473,"style":664},[54],[15,55475,206],{"className":55476,"style":334},[59,60],[15,55478],{"className":55479,"style":158},[157],[15,55481,123],{"className":55482},[162],[15,55484],{"className":55485,"style":158},[157],[15,55487,55489,55492],{"className":55488},[50],[15,55490],{"className":55491,"style":566},[54],[15,55493,213],{"className":55494},[59,60],[11,55496,55497,55500,55501,55530,55531,55559,55560,3855,55689,55717,55718,55768,55769,55797,55798,55966,55967,56059,56060,56110,56111,56161],{},[998,55498,55499],{},"证明。"," Jensen 已给出 ",[15,55502,55504,55518],{"className":55503,"translate":19},[18],[15,55505,55507],{"className":55506},[23],[25,55508,55509],{"xmlns":27},[29,55510,55511,55515],{},[32,55512,55513],{},[110,55514,6930],{},[39,55516,55517],{"encoding":41},"\\ge",[15,55519,55521],{"className":55520,"ariaHidden":46},[45],[15,55522,55524,55527],{"className":55523},[50],[15,55525],{"className":55526,"style":43214},[54],[15,55528,6930],{"className":55529},[162],"（见 2.1）。等号条件：若 ",[15,55532,55534,55547],{"className":55533,"translate":19},[18],[15,55535,55537],{"className":55536},[23],[25,55538,55539],{"xmlns":27},[29,55540,55541,55545],{},[32,55542,55543],{},[35,55544,37],{},[39,55546,37],{"encoding":41},[15,55548,55550],{"className":55549,"ariaHidden":46},[45],[15,55551,55553,55556],{"className":55552},[50],[15,55554],{"className":55555,"style":55},[54],[15,55557,37],{"className":55558,"style":61},[59,60]," 严格凸，Jensen 取等号当且仅当 ",[15,55561,55563,55595],{"className":55562,"translate":19},[18],[15,55564,55566],{"className":55565},[23],[25,55567,55568],{"xmlns":27},[29,55569,55570,55592],{},[32,55571,55572,55574,55576,55582,55584,55586,55588,55590],{},[35,55573,769],{},[110,55575,123],{},[195,55577,55578,55580],{},[35,55579,223],{"mathvariant":222},[35,55581,213],{},[110,55583,230],{"stretchy":112},[35,55585,769],{},[110,55587,257],{"stretchy":112},[110,55589,123],{},[115,55591,117],{},[39,55593,55594],{"encoding":41},"t=\\mathbb E_Q[t]=1",[15,55596,55598,55616,55680],{"className":55597,"ariaHidden":46},[45],[15,55599,55601,55604,55607,55610,55613],{"className":55600},[50],[15,55602],{"className":55603,"style":5345},[54],[15,55605,769],{"className":55606},[59,60],[15,55608],{"className":55609,"style":158},[157],[15,55611,123],{"className":55612},[162],[15,55614],{"className":55615,"style":158},[157],[15,55617,55619,55622,55662,55665,55668,55671,55674,55677],{"className":55618},[50],[15,55620],{"className":55621,"style":270},[54],[15,55623,55625,55628],{"className":55624},[59],[15,55626,223],{"className":55627},[59,366],[15,55629,55631],{"className":55630},[281],[15,55632,55634,55654],{"className":55633},[285,286],[15,55635,55637,55651],{"className":55636},[290],[15,55638,55640],{"className":55639,"style":379},[294],[15,55641,55642,55645],{"style":382},[15,55643],{"className":55644,"style":303},[302],[15,55646,55648],{"className":55647},[307,308,309,310],[15,55649,213],{"className":55650},[59,60,310],[15,55652,318],{"className":55653},[317],[15,55655,55657],{"className":55656},[290],[15,55658,55660],{"className":55659,"style":325},[294],[15,55661],{},[15,55663,230],{"className":55664},[146],[15,55666,769],{"className":55667},[59,60],[15,55669,257],{"className":55670},[153],[15,55672],{"className":55673,"style":158},[157],[15,55675,123],{"className":55676},[162],[15,55678],{"className":55679,"style":158},[157],[15,55681,55683,55686],{"className":55682},[50],[15,55684],{"className":55685,"style":172},[54],[15,55687,117],{"className":55688},[59],[15,55690,55692,55705],{"className":55691,"translate":19},[18],[15,55693,55695],{"className":55694},[23],[25,55696,55697],{"xmlns":27},[29,55698,55699,55703],{},[32,55700,55701],{},[35,55702,213],{},[39,55704,213],{"encoding":41},[15,55706,55708],{"className":55707,"ariaHidden":46},[45],[15,55709,55711,55714],{"className":55710},[50],[15,55712],{"className":55713,"style":566},[54],[15,55715,213],{"className":55716},[59,60]," 下几乎必然成立，即 ",[15,55719,55721,55738],{"className":55720,"translate":19},[18],[15,55722,55724],{"className":55723},[23],[25,55725,55726],{"xmlns":27},[29,55727,55728,55736],{},[32,55729,55730,55732,55734],{},[35,55731,206],{},[110,55733,123],{},[35,55735,213],{},[39,55737,10273],{"encoding":41},[15,55739,55741,55759],{"className":55740,"ariaHidden":46},[45],[15,55742,55744,55747,55750,55753,55756],{"className":55743},[50],[15,55745],{"className":55746,"style":664},[54],[15,55748,206],{"className":55749,"style":334},[59,60],[15,55751],{"className":55752,"style":158},[157],[15,55754,123],{"className":55755},[162],[15,55757],{"className":55758,"style":158},[157],[15,55760,55762,55765],{"className":55761},[50],[15,55763],{"className":55764,"style":566},[54],[15,55766,213],{"className":55767},[59,60],"。若 ",[15,55770,55772,55785],{"className":55771,"translate":19},[18],[15,55773,55775],{"className":55774},[23],[25,55776,55777],{"xmlns":27},[29,55778,55779,55783],{},[32,55780,55781],{},[35,55782,37],{},[39,55784,37],{"encoding":41},[15,55786,55788],{"className":55787,"ariaHidden":46},[45],[15,55789,55791,55794],{"className":55790},[50],[15,55792],{"className":55793,"style":55},[54],[15,55795,37],{"className":55796,"style":61},[59,60]," 只是凸而非严格凸（如 ",[15,55799,55801,55838],{"className":55800,"translate":19},[18],[15,55802,55804],{"className":55803},[23],[25,55805,55806],{"xmlns":27},[29,55807,55808,55836],{},[32,55809,55810,55812,55814,55816,55818,55820,55826,55828,55830,55832,55834],{},[35,55811,37],{},[110,55813,113],{"stretchy":112},[35,55815,769],{},[110,55817,120],{"stretchy":112},[110,55819,123],{},[238,55821,55822,55824],{},[115,55823,117],{},[115,55825,2465],{},[35,55827,4810],{"mathvariant":209},[35,55829,769],{},[110,55831,4815],{},[115,55833,117],{},[35,55835,4810],{"mathvariant":209},[39,55837,34582],{"encoding":41},[15,55839,55841,55868,55957],{"className":55840,"ariaHidden":46},[45],[15,55842,55844,55847,55850,55853,55856,55859,55862,55865],{"className":55843},[50],[15,55845],{"className":55846,"style":139},[54],[15,55848,37],{"className":55849,"style":61},[59,60],[15,55851,113],{"className":55852},[146],[15,55854,769],{"className":55855},[59,60],[15,55857,120],{"className":55858},[153],[15,55860],{"className":55861,"style":158},[157],[15,55863,123],{"className":55864},[162],[15,55866],{"className":55867,"style":158},[157],[15,55869,55871,55874,55942,55945,55948,55951,55954],{"className":55870},[50],[15,55872],{"className":55873,"style":2507},[54],[15,55875,55877,55880,55939],{"className":55876},[59],[15,55878],{"className":55879},[146,441],[15,55881,55883],{"className":55882},[238],[15,55884,55886,55931],{"className":55885},[285,286],[15,55887,55889,55928],{"className":55888},[290],[15,55890,55892,55906,55914],{"className":55891,"style":2526},[294],[15,55893,55894,55897],{"style":2529},[15,55895],{"className":55896,"style":461},[302],[15,55898,55900],{"className":55899},[307,308,309,310],[15,55901,55903],{"className":55902},[59,310],[15,55904,2465],{"className":55905},[59,310],[15,55907,55908,55911],{"style":473},[15,55909],{"className":55910,"style":461},[302],[15,55912],{"className":55913,"style":481},[480],[15,55915,55916,55919],{"style":2552},[15,55917],{"className":55918,"style":461},[302],[15,55920,55922],{"className":55921},[307,308,309,310],[15,55923,55925],{"className":55924},[59,310],[15,55926,117],{"className":55927},[59,310],[15,55929,318],{"className":55930},[317],[15,55932,55934],{"className":55933},[290],[15,55935,55937],{"className":55936,"style":2574},[294],[15,55938],{},[15,55940],{"className":55941},[153,441],[15,55943,4810],{"className":55944},[59],[15,55946,769],{"className":55947},[59,60],[15,55949],{"className":55950,"style":2589},[157],[15,55952,4815],{"className":55953},[2593],[15,55955],{"className":55956,"style":2589},[157],[15,55958,55960,55963],{"className":55959},[50],[15,55961],{"className":55962,"style":139},[54],[15,55964,34705],{"className":55965},[59],"），则只能保证 ",[15,55968,55970,55992],{"className":55969,"translate":19},[18],[15,55971,55973],{"className":55972},[23],[25,55974,55975],{"xmlns":27},[29,55976,55977,55989],{},[32,55978,55979,55985,55987],{},[195,55980,55981,55983],{},[35,55982,199],{},[35,55984,37],{},[110,55986,123],{},[115,55988,126],{},[39,55990,55991],{"encoding":41},"D_f=0",[15,55993,55995,56050],{"className":55994,"ariaHidden":46},[45],[15,55996,55998,56001,56041,56044,56047],{"className":55997},[50],[15,55999],{"className":56000,"style":4034},[54],[15,56002,56004,56007],{"className":56003},[59],[15,56005,199],{"className":56006,"style":277},[59,60],[15,56008,56010],{"className":56009},[281],[15,56011,56013,56033],{"className":56012},[285,286],[15,56014,56016,56030],{"className":56015},[290],[15,56017,56019],{"className":56018,"style":295},[294],[15,56020,56021,56024],{"style":298},[15,56022],{"className":56023,"style":303},[302],[15,56025,56027],{"className":56026},[307,308,309,310],[15,56028,37],{"className":56029,"style":61},[59,60,310],[15,56031,318],{"className":56032},[317],[15,56034,56036],{"className":56035},[290],[15,56037,56039],{"className":56038,"style":325},[294],[15,56040],{},[15,56042],{"className":56043,"style":158},[157],[15,56045,123],{"className":56046},[162],[15,56048],{"className":56049,"style":158},[157],[15,56051,56053,56056],{"className":56052},[50],[15,56054],{"className":56055,"style":172},[54],[15,56057,126],{"className":56058},[59]," 时不必然 ",[15,56061,56063,56080],{"className":56062,"translate":19},[18],[15,56064,56066],{"className":56065},[23],[25,56067,56068],{"xmlns":27},[29,56069,56070,56078],{},[32,56071,56072,56074,56076],{},[35,56073,206],{},[110,56075,123],{},[35,56077,213],{},[39,56079,10273],{"encoding":41},[15,56081,56083,56101],{"className":56082,"ariaHidden":46},[45],[15,56084,56086,56089,56092,56095,56098],{"className":56085},[50],[15,56087],{"className":56088,"style":664},[54],[15,56090,206],{"className":56091,"style":334},[59,60],[15,56093],{"className":56094,"style":158},[157],[15,56096,123],{"className":56097},[162],[15,56099],{"className":56100,"style":158},[157],[15,56102,56104,56107],{"className":56103},[50],[15,56105],{"className":56106,"style":566},[54],[15,56108,213],{"className":56109},[59,60],"：TV 是反例，",[15,56112,56114,56131],{"className":56113,"translate":19},[18],[15,56115,56117],{"className":56116},[23],[25,56118,56119],{"xmlns":27},[29,56120,56121,56129],{},[32,56122,56123,56125,56127],{},[35,56124,206],{},[110,56126,123],{},[35,56128,213],{},[39,56130,10273],{"encoding":41},[15,56132,56134,56152],{"className":56133,"ariaHidden":46},[45],[15,56135,56137,56140,56143,56146,56149],{"className":56136},[50],[15,56138],{"className":56139,"style":664},[54],[15,56141,206],{"className":56142,"style":334},[59,60],[15,56144],{"className":56145,"style":158},[157],[15,56147,123],{"className":56148},[162],[15,56150],{"className":56151,"style":158},[157],[15,56153,56155,56158],{"className":56154},[50],[15,56156],{"className":56157,"style":566},[54],[15,56159,213],{"className":56160},[59,60]," 时才是唯一使 TV 为 0 的一对（TV 也对）。",[2313,56163,56165],{"id":56164},"_42-联合凸性","4.2 联合凸性",[993,56167,56168,56640],{},[11,56169,56170,56173,56174,2747,56370,2747,56566,41664],{},[998,56171,56172],{},"命题 4.2（联合凸性）"," 若 ",[15,56175,56177,56218],{"className":56176,"translate":19},[18],[15,56178,56180],{"className":56179},[23],[25,56181,56182],{"xmlns":27},[29,56183,56184,56216],{},[32,56185,56186,56188,56190,56192,56198,56200,56202,56204,56206,56208,56210],{},[35,56187,206],{},[110,56189,123],{},[35,56191,11343],{},[195,56193,56194,56196],{},[35,56195,206],{},[115,56197,117],{},[110,56199,2472],{},[110,56201,113],{"stretchy":112},[115,56203,117],{},[110,56205,4815],{},[35,56207,11343],{},[110,56209,120],{"stretchy":112},[195,56211,56212,56214],{},[35,56213,206],{},[115,56215,2465],{},[39,56217,11450],{"encoding":41},[15,56219,56221,56239,56297,56318],{"className":56220,"ariaHidden":46},[45],[15,56222,56224,56227,56230,56233,56236],{"className":56223},[50],[15,56225],{"className":56226,"style":664},[54],[15,56228,206],{"className":56229,"style":334},[59,60],[15,56231],{"className":56232,"style":158},[157],[15,56234,123],{"className":56235},[162],[15,56237],{"className":56238,"style":158},[157],[15,56240,56242,56245,56248,56288,56291,56294],{"className":56241},[50],[15,56243],{"className":56244,"style":11478},[54],[15,56246,11343],{"className":56247},[59,60],[15,56249,56251,56254],{"className":56250},[59],[15,56252,206],{"className":56253,"style":334},[59,60],[15,56255,56257],{"className":56256},[281],[15,56258,56260,56280],{"className":56259},[285,286],[15,56261,56263,56277],{"className":56262},[290],[15,56264,56266],{"className":56265,"style":11137},[294],[15,56267,56268,56271],{"style":11140},[15,56269],{"className":56270,"style":303},[302],[15,56272,56274],{"className":56273},[307,308,309,310],[15,56275,117],{"className":56276},[59,310],[15,56278,318],{"className":56279},[317],[15,56281,56283],{"className":56282},[290],[15,56284,56286],{"className":56285,"style":7970},[294],[15,56287],{},[15,56289],{"className":56290,"style":2589},[157],[15,56292,2472],{"className":56293},[2593],[15,56295],{"className":56296,"style":2589},[157],[15,56298,56300,56303,56306,56309,56312,56315],{"className":56299},[50],[15,56301],{"className":56302,"style":139},[54],[15,56304,113],{"className":56305},[146],[15,56307,117],{"className":56308},[59],[15,56310],{"className":56311,"style":2589},[157],[15,56313,4815],{"className":56314},[2593],[15,56316],{"className":56317,"style":2589},[157],[15,56319,56321,56324,56327,56330],{"className":56320},[50],[15,56322],{"className":56323,"style":139},[54],[15,56325,11343],{"className":56326},[59,60],[15,56328,120],{"className":56329},[153],[15,56331,56333,56336],{"className":56332},[59],[15,56334,206],{"className":56335,"style":334},[59,60],[15,56337,56339],{"className":56338},[281],[15,56340,56342,56362],{"className":56341},[285,286],[15,56343,56345,56359],{"className":56344},[290],[15,56346,56348],{"className":56347,"style":11137},[294],[15,56349,56350,56353],{"style":11140},[15,56351],{"className":56352,"style":303},[302],[15,56354,56356],{"className":56355},[307,308,309,310],[15,56357,2465],{"className":56358},[59,310],[15,56360,318],{"className":56361},[317],[15,56363,56365],{"className":56364},[290],[15,56366,56368],{"className":56367,"style":7970},[294],[15,56369],{},[15,56371,56373,56414],{"className":56372,"translate":19},[18],[15,56374,56376],{"className":56375},[23],[25,56377,56378],{"xmlns":27},[29,56379,56380,56412],{},[32,56381,56382,56384,56386,56388,56394,56396,56398,56400,56402,56404,56406],{},[35,56383,213],{},[110,56385,123],{},[35,56387,11343],{},[195,56389,56390,56392],{},[35,56391,213],{},[115,56393,117],{},[110,56395,2472],{},[110,56397,113],{"stretchy":112},[115,56399,117],{},[110,56401,4815],{},[35,56403,11343],{},[110,56405,120],{"stretchy":112},[195,56407,56408,56410],{},[35,56409,213],{},[115,56411,2465],{},[39,56413,11648],{"encoding":41},[15,56415,56417,56435,56493,56514],{"className":56416,"ariaHidden":46},[45],[15,56418,56420,56423,56426,56429,56432],{"className":56419},[50],[15,56421],{"className":56422,"style":566},[54],[15,56424,213],{"className":56425},[59,60],[15,56427],{"className":56428,"style":158},[157],[15,56430,123],{"className":56431},[162],[15,56433],{"className":56434,"style":158},[157],[15,56436,56438,56441,56444,56484,56487,56490],{"className":56437},[50],[15,56439],{"className":56440,"style":55},[54],[15,56442,11343],{"className":56443},[59,60],[15,56445,56447,56450],{"className":56446},[59],[15,56448,213],{"className":56449},[59,60],[15,56451,56453],{"className":56452},[281],[15,56454,56456,56476],{"className":56455},[285,286],[15,56457,56459,56473],{"className":56458},[290],[15,56460,56462],{"className":56461,"style":11137},[294],[15,56463,56464,56467],{"style":382},[15,56465],{"className":56466,"style":303},[302],[15,56468,56470],{"className":56469},[307,308,309,310],[15,56471,117],{"className":56472},[59,310],[15,56474,318],{"className":56475},[317],[15,56477,56479],{"className":56478},[290],[15,56480,56482],{"className":56481,"style":7970},[294],[15,56483],{},[15,56485],{"className":56486,"style":2589},[157],[15,56488,2472],{"className":56489},[2593],[1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\\lambda D_f(P_1\\|Q_1)\n+(1-\\lambda)D_f(P_2\\|Q_2).",[15,56741,56743,56813,56960,56981],{"className":56742,"ariaHidden":46},[45],[15,56744,56746,56749,56789,56792,56795,56798,56801,56804,56807,56810],{"className":56745},[50],[15,56747],{"className":56748,"style":270},[54],[15,56750,56752,56755],{"className":56751},[59],[15,56753,199],{"className":56754,"style":277},[59,60],[15,56756,56758],{"className":56757},[281],[15,56759,56761,56781],{"className":56760},[285,286],[15,56762,56764,56778],{"className":56763},[290],[15,56765,56767],{"className":56766,"style":295},[294],[15,56768,56769,56772],{"style":298},[15,56770],{"className":56771,"style":303},[302],[15,56773,56775],{"className":56774},[307,308,309,310],[15,56776,37],{"className":56777,"style":61},[59,60,310],[15,56779,318],{"className":56780},[317],[15,56782,56784],{"className":56783},[290],[15,56785,56787],{"className":56786,"style":325},[294],[15,56788],{},[15,56790,113],{"className":56791},[146],[15,56793,206],{"className":56794,"style":334},[59,60],[15,56796,210],{"className":56797},[59],[15,56799,213],{"className":56800},[59,60],[15,56802,120],{"className":56803},[153],[15,56805],{"className":56806,"style":158},[157],[15,56808,9954],{"className":56809},[162],[15,56811],{"className":56812,"style":158},[157],[15,56814,56816,56819,56822,56862,56865,56905,56908,56948,56951,56954,56957],{"className":56815},[50],[15,56817],{"className":56818,"style":270},[54],[15,56820,11343],{"className":56821},[59,60],[15,56823,56825,56828],{"className":56824},[59],[15,56826,199],{"className":56827,"style":277},[59,60],[15,56829,56831],{"className":56830},[281],[15,56832,56834,56854],{"className":56833},[285,286],[15,56835,56837,56851],{"className":56836},[290],[15,56838,56840],{"className":56839,"style":295},[294],[15,56841,56842,56845],{"style":298},[15,56843],{"className":56844,"style":303},[302],[15,56846,56848],{"className":56847},[307,308,309,310],[15,56849,37],{"className":56850,"style":61},[59,60,310],[15,56852,318],{"className":56853},[317],[15,56855,56857],{"className":56856},[290],[15,56858,56860],{"className":56859,"style":325},[294],[15,56861],{},[15,56863,113],{"className":56864},[146],[15,56866,56868,56871],{"className":56867},[59],[15,56869,206],{"className":56870,"style":334},[59,60],[15,56872,56874],{"className":56873},[281],[15,56875,56877,56897],{"className":56876},[285,286],[15,56878,56880,56894],{"className":56879},[290],[15,56881,56883],{"className":56882,"style":11137},[294],[15,56884,56885,56888],{"style":11140},[15,56886],{"className":56887,"style":303},[302],[15,56889,56891],{"className":56890},[307,308,309,310],[15,56892,117],{"className":56893},[59,310],[15,56895,318],{"className":56896},[317],[15,56898,56900],{"className":56899},[290],[15,56901,56903],{"className":56902,"style":7970},[294],[15,56904],{},[15,56906,210],{"className":56907},[59],[15,56909,56911,56914],{"className":56910},[59],[15,56912,213],{"className":56913},[59,60],[15,56915,56917],{"className":56916},[281],[15,56918,56920,56940],{"className":56919},[285,286],[15,56921,56923,56937],{"className":56922},[290],[15,56924,56926],{"className":56925,"style":11137},[294],[15,56927,56928,56931],{"style":382},[15,56929],{"className":56930,"style":303},[302],[15,56932,56934],{"className":56933},[307,308,309,310],[15,56935,117],{"className":56936},[59,310],[15,56938,318],{"className":56939},[317],[15,56941,56943],{"className":56942},[290],[15,56944,56946],{"className":56945,"style":7970},[294],[15,56947],{},[15,56949,120],{"className":56950},[153],[15,56952],{"className":56953,"style":2589},[157],[15,56955,2472],{"className":56956},[2593],[15,56958],{"className":56959,"style":2589},[157],[15,56961,56963,56966,56969,56972,56975,56978],{"className":56962},[50],[15,56964],{"className":56965,"style":139},[54],[15,56967,113],{"className":56968},[146],[15,56970,117],{"className":56971},[59],[15,56973],{"className":56974,"style":2589},[157],[15,56976,4815],{"className":56977},[2593],[15,56979],{"className":56980,"style":2589},[157],[15,56982,56984,56987,56990,56993,57033,57036,57076,57079,57119,57122],{"className":56983},[50],[15,56985],{"className":56986,"style":270},[54],[15,56988,11343],{"className":56989},[59,60],[15,56991,120],{"className":56992},[153],[15,56994,56996,56999],{"className":56995},[59],[15,56997,199],{"className":56998,"style":277},[59,60],[15,57000,57002],{"className":57001},[281],[15,57003,57005,57025],{"className":57004},[285,286],[15,57006,57008,57022],{"className":57007},[290],[15,57009,57011],{"className":57010,"style":295},[294],[15,57012,57013,57016],{"style":298},[15,57014],{"className":57015,"style":303},[302],[15,57017,57019],{"className":57018},[307,308,309,310],[15,57020,37],{"className":57021,"style":61},[59,60,310],[15,57023,318],{"className":57024},[317],[15,57026,57028],{"className":57027},[290],[15,57029,57031],{"className":57030,"style":325},[294],[15,57032],{},[15,57034,113],{"className":57035},[146],[15,57037,57039,57042],{"className":57038},[59],[15,57040,2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这是 ",[15,57131,57133,57146],{"className":57132,"translate":19},[18],[15,57134,57136],{"className":57135},[23],[25,57137,57138],{"xmlns":27},[29,57139,57140,57144],{},[32,57141,57142],{},[35,57143,37],{},[39,57145,37],{"encoding":41},[15,57147,57149],{"className":57148,"ariaHidden":46},[45],[15,57150,57152,57155],{"className":57151},[50],[15,57153],{"className":57154,"style":55},[54],[15,57156,37],{"className":57157,"style":61},[59,60]," 凸性加上「",[15,57160,57162,57178],{"className":57161,"translate":19},[18],[15,57163,57165],{"className":57164},[23],[25,57166,57167],{"xmlns":27},[29,57168,57169,57175],{},[32,57170,57171,57173],{},[35,57172,35458],{},[110,57174,4105],{},[39,57176,57177],{"encoding":41},"\\sup",[15,57179,57181],{"className":57180,"ariaHidden":46},[45],[15,57182,57184,57187],{"className":57183},[50],[15,57185],{"className":57186,"style":2278},[54],[15,57188,35458],{"className":57189},[1486]," 与线性运算」的标准结果，也可用 f-散度的变分表示（4.4 节）直接看：",[15,57192,57194,57274],{"className":57193,"translate":19},[18],[15,57195,57197],{"className":57196},[23],[25,57198,57199],{"xmlns":27},[29,57200,57201,57271],{},[32,57202,57203,57209,57211,57213,57215,57217,57219,57221,57231,57233,57239,57241,57243,57245,57247,57253,57255,57261,57263,57265,57267,57269],{},[195,57204,57205,57207],{},[35,57206,199],{},[35,57208,37],{},[110,57210,113],{"stretchy":112},[35,57212,206],{},[35,57214,210],{"mathvariant":209},[35,57216,213],{},[110,57218,120],{"stretchy":112},[110,57220,123],{},[195,57222,57223,57229],{},[32,57224,57225,57227],{},[35,57226,35458],{},[110,57228,4105],{},[35,57230,4160],{},[110,57232,2900],{"stretchy":112},[195,57234,57235,57237],{},[35,57236,223],{"mathvariant":222},[35,57238,206],{},[110,57240,230],{"stretchy":112},[35,57242,4160],{},[110,57244,257],{"stretchy":112},[110,57246,4815],{},[195,57248,57249,57251],{},[35,57250,223],{"mathvariant":222},[35,57252,213],{},[110,57254,230],{"stretchy":112},[6816,57256,57257,57259],{},[35,57258,37],{},[110,57260,7575],{},[110,57262,113],{"stretchy":112},[35,57264,4160],{},[110,57266,120],{"stretchy":112},[110,57268,257],{"stretchy":112},[110,57270,2910],{"stretchy":112},[39,57272,57273],{"encoding":41},"D_f(P\\|Q)=\\sup_g\\{\\mathbb E_P[g]-\\mathbb E_Q[f^*(g)]\\}",[15,57275,57277,57347,57456],{"className":57276,"ariaHidden":46},[45],[15,57278,57280,57283,57323,57326,57329,57332,57335,57338,57341,57344],{"className":57279},[50],[15,57281],{"className":57282,"style":270},[54],[15,57284,57286,57289],{"className":57285},[59],[15,57287,199],{"className":57288,"style":277},[59,60],[15,57290,57292],{"className":57291},[281],[15,57293,57295,57315],{"className":57294},[285,286],[15,57296,57298,57312],{"className":57297},[290],[15,57299,57301],{"className":57300,"style":295},[294],[15,57302,57303,57306],{"style":298},[15,57304],{"className":57305,"style":303},[302],[15,57307,57309],{"className":57308},[307,308,309,310],[15,57310,37],{"className":57311,"style":61},[59,60,310],[15,57313,318],{"className":57314},[317],[15,57316,57318],{"className":57317},[290],[15,57319,57321],{"className":57320,"style":325},[294],[15,57322],{},[15,57324,113],{"className":57325},[146],[15,57327,206],{"className":57328,"style":334},[59,60],[15,57330,210],{"className":57331},[59],[15,57333,213],{"className":57334},[59,60],[15,57336,120],{"className":57337},[153],[15,57339],{"className":57340,"style":158},[157],[15,57342,123],{"className":57343},[162],[15,57345],{"className":57346,"style":158},[157],[15,57348,57350,57354,57395,57398,57438,57441,57444,57447,57450,57453],{"className":57349},[50],[15,57351],{"className":57352,"style":57353},[54],"height:1.1302em;vertical-align:-0.3802em;",[15,57355,57357,57360],{"className":57356},[1486],[15,57358,35458],{"className":57359},[1486],[15,57361,57363],{"className":57362},[281],[15,57364,57366,57386],{"className":57365},[285,286],[15,57367,57369,57383],{"className":57368},[290],[15,57370,57372],{"className":57371,"style":37025},[294],[15,57373,57374,57377],{"style":35604},[15,57375],{"className":57376,"style":303},[302],[15,57378,57380],{"className":57379},[307,308,309,310],[15,57381,4160],{"className":57382,"style":981},[59,60,310],[15,57384,318],{"className":57385},[317],[15,57387,57389],{"className":57388},[290],[15,57390,57393],{"className":57391,"style":57392},[294],"height:0.3802em;",[15,57394],{},[15,57396,2900],{"className":57397},[146],[15,57399,57401,57404],{"className":57400},[59],[15,57402,223],{"className":57403},[59,366],[15,57405,57407],{"className":57406},[281],[15,57408,57410,57430],{"className":57409},[285,286],[15,57411,57413,57427],{"className":57412},[290],[15,57414,57416],{"className":57415,"style":379},[294],[15,57417,57418,57421],{"style":382},[15,57419],{"className":57420,"style":303},[302],[15,57422,57424],{"className":57423},[307,308,309,310],[15,57425,206],{"className":57426,"style":334},[59,60,310],[15,57428,318],{"className":57429},[317],[15,57431,57433],{"className":57432},[290],[15,57434,57436],{"className":57435,"style":7970},[294],[15,57437],{},[15,57439,230],{"className":57440},[146],[15,57442,4160],{"className":57443,"style":981},[59,60],[15,57445,257],{"className":57446},[153],[15,57448],{"className":57449,"style":2589},[157],[15,57451,4815],{"className":57452},[2593],[15,57454],{"className":57455,"style":2589},[157],[15,57457,57459,57462,57502,57505,57534,57537,57540],{"className":57458},[50],[15,57460],{"className":57461,"style":270},[54],[15,57463,57465,57468],{"className":57464},[59],[15,57466,223],{"className":57467},[59,366],[15,57469,57471],{"className":57470},[281],[15,57472,57474,57494],{"className":57473},[285,286],[15,57475,57477,57491],{"className":57476},[290],[15,57478,57480],{"className":57479,"style":379},[294],[15,57481,57482,57485],{"style":382},[15,57483],{"className":57484,"style":303},[302],[15,57486,57488],{"className":57487},[307,308,309,310],[15,57489,213],{"className":57490},[59,60,310],[15,57492,318],{"className":57493},[317],[15,57495,57497],{"className":57496},[290],[15,57498,57500],{"className":57499,"style":325},[294],[15,57501],{},[15,57503,230],{"className":57504},[146],[15,57506,57508,57511],{"className":57507},[59],[15,57509,37],{"className":57510,"style":61},[59,60],[15,57512,57514],{"className":57513},[281],[15,57515,57517],{"className":57516},[285],[15,57518,57520],{"className":57519},[290],[15,57521,57523],{"className":57522,"style":8501},[294],[15,57524,57525,57528],{"style":6867},[15,57526],{"className":57527,"style":303},[302],[15,57529,57531],{"className":57530},[307,308,309,310],[15,57532,7575],{"className":57533},[2593,310],[15,57535,113],{"className":57536},[146],[15,57538,4160],{"className":57539,"style":981},[59,60],[15,57541,57543],{"className":57542},[153],")]}","，右端关于 ",[15,57546,57548,57569],{"className":57547,"translate":19},[18],[15,57549,57551],{"className":57550},[23],[25,57552,57553],{"xmlns":27},[29,57554,57555,57567],{},[32,57556,57557,57559,57561,57563,57565],{},[110,57558,113],{"stretchy":112},[35,57560,206],{},[110,57562,551],{"separator":46},[35,57564,213],{},[110,57566,120],{"stretchy":112},[39,57568,11035],{"encoding":41},[15,57570,57572],{"className":57571,"ariaHidden":46},[45],[15,57573,57575,57578,57581,57584,57587,57590,57593],{"className":57574},[50],[15,57576],{"className":57577,"style":139},[54],[15,57579,113],{"className":57580},[146],[15,57582,206],{"className":57583,"style":334},[59,60],[15,57585,551],{"className":57586},[573],[15,57588],{"className":57589,"style":406},[157],[15,57591,213],{"className":57592},[59,60],[15,57594,120],{"className":57595},[153]," 是凸的（",[15,57598,57600,57624],{"className":57599,"translate":19},[18],[15,57601,57603],{"className":57602},[23],[25,57604,57605],{"xmlns":27},[29,57606,57607,57621],{},[32,57608,57609,57615,57617,57619],{},[195,57610,57611,57613],{},[35,57612,223],{"mathvariant":222},[35,57614,206],{},[110,57616,230],{"stretchy":112},[35,57618,4160],{},[110,57620,257],{"stretchy":112},[39,57622,57623],{"encoding":41},"\\mathbb E_P[g]",[15,57625,57627],{"className":57626,"ariaHidden":46},[45],[15,57628,57630,57633,57673,57676,57679],{"className":57629},[50],[15,57631],{"className":57632,"style":139},[54],[15,57634,57636,57639],{"className":57635},[59],[15,57637,223],{"className":57638},[59,366],[15,57640,57642],{"className":57641},[281],[15,57643,57645,57665],{"className":57644},[285,286],[15,57646,57648,57662],{"className":57647},[290],[15,57649,57651],{"className":57650,"style":379},[294],[15,57652,57653,57656],{"style":382},[15,57654],{"className":57655,"style":303},[302],[15,57657,57659],{"className":57658},[307,308,309,310],[15,57660,206],{"className":57661,"style":334},[59,60,310],[15,57663,318],{"className":57664},[317],[15,57666,57668],{"className":57667},[290],[15,57669,57671],{"className":57670,"style":7970},[294],[15,57672],{},[15,57674,230],{"className":57675},[146],[15,57677,4160],{"className":57678,"style":981},[59,60],[15,57680,257],{"className":57681},[153]," 线性、",[15,57684,57686,57722],{"className":57685,"translate":19},[18],[15,57687,57689],{"className":57688},[23],[25,57690,57691],{"xmlns":27},[29,57692,57693,57719],{},[32,57694,57695,57697,57703,57705,57711,57713,57715,57717],{},[110,57696,4815],{},[195,57698,57699,57701],{},[35,57700,223],{"mathvariant":222},[35,57702,213],{},[110,57704,230],{"stretchy":112},[6816,57706,57707,57709],{},[35,57708,37],{},[110,57710,7575],{},[110,57712,113],{"stretchy":112},[35,57714,4160],{},[110,57716,120],{"stretchy":112},[110,57718,257],{"stretchy":112},[39,57720,57721],{"encoding":41},"-\\mathbb E_Q[f^*(g)]",[15,57723,57725],{"className":57724,"ariaHidden":46},[45],[15,57726,57728,57731,57734,57774,57777,57806,57809,57812],{"className":57727},[50],[15,57729],{"className":57730,"style":270},[54],[15,57732,4815],{"className":57733},[59],[15,57735,57737,57740],{"className":57736},[59],[15,57738,223],{"className":57739},[59,366],[15,57741,57743],{"className":57742},[281],[15,57744,57746,57766],{"className":57745},[285,286],[15,57747,57749,57763],{"className":57748},[290],[15,57750,57752],{"className":57751,"style":379},[294],[15,57753,57754,57757],{"style":382},[15,57755],{"className":57756,"style":303},[302],[15,57758,57760],{"className":57759},[307,308,309,310],[15,57761,213],{"className":57762},[59,60,310],[15,57764,318],{"className":57765},[317],[15,57767,57769],{"className":57768},[290],[15,57770,57772],{"className":57771,"style":325},[294],[15,57773],{},[15,57775,230],{"className":57776},[146],[15,57778,57780,57783],{"className":57779},[59],[15,57781,37],{"className":57782,"style":61},[59,60],[15,57784,57786],{"className":57785},[281],[15,57787,57789],{"className":57788},[285],[15,57790,57792],{"className":57791},[290],[15,57793,57795],{"className":57794,"style":8501},[294],[15,57796,57797,57800],{"style":6867},[15,57798],{"className":57799,"style":303},[302],[15,57801,57803],{"className":57802},[307,308,309,310],[15,57804,7575],{"className":57805},[2593,310],[15,57807,113],{"className":57808},[146],[15,57810,4160],{"className":57811,"style":981},[59,60],[15,57813,20416],{"className":57814},[153]," 线性），取上确界后仍凸。",[11,57817,57818,57819,2747,57847,57965,57966,2747,57994,10969,58111,9402,58114,58230,58231,58259,58260,58288,58289,58317],{},"等价地，固定 ",[15,57820,57822,57835],{"className":57821,"translate":19},[18],[15,57823,57825],{"className":57824},[23],[25,57826,57827],{"xmlns":27},[29,57828,57829,57833],{},[32,57830,57831],{},[35,57832,213],{},[39,57834,213],{"encoding":41},[15,57836,57838],{"className":57837,"ariaHidden":46},[45],[15,57839,57841,57844],{"className":57840},[50],[15,57842],{"className":57843,"style":566},[54],[15,57845,213],{"className":57846},[59,60],[15,57848,57850,57882],{"className":57849,"translate":19},[18],[15,57851,57853],{"className":57852},[23],[25,57854,57855],{"xmlns":27},[29,57856,57857,57879],{},[32,57858,57859,57861,57863,57869,57871,57873,57875,57877],{},[35,57860,206],{},[110,57862,8686],{},[195,57864,57865,57867],{},[35,57866,199],{},[35,57868,37],{},[110,57870,113],{"stretchy":112},[35,57872,206],{},[35,57874,210],{"mathvariant":209},[35,57876,213],{},[110,57878,120],{"stretchy":112},[39,57880,57881],{"encoding":41},"P\\mapsto D_f(P\\|Q)",[15,57883,57885,57904],{"className":57884,"ariaHidden":46},[45],[15,57886,57888,57892,57895,57898,57901],{"className":57887},[50],[15,57889],{"className":57890,"style":57891},[54],"height:0.6943em;vertical-align:-0.011em;",[15,57893,206],{"className":57894,"style":334},[59,60],[15,57896],{"className":57897,"style":158},[157],[15,57899,8686],{"className":57900},[162],[15,57902],{"className":57903,"style":158},[157],[15,57905,57907,57910,57950,57953,57956,57959,57962],{"className":57906},[50],[15,57908],{"className":57909,"style":270},[54],[15,57911,57913,57916],{"className":57912},[59],[15,57914,199],{"className":57915,"style":277},[59,60],[15,57917,57919],{"className":57918},[281],[15,57920,57922,57942],{"className":57921},[285,286],[15,57923,57925,57939],{"className":57924},[290],[15,57926,57928],{"className":57927,"style":295},[294],[15,57929,57930,57933],{"style":298},[15,57931],{"className":57932,"style":303},[302],[15,57934,57936],{"className":57935},[307,308,309,310],[15,57937,37],{"className":57938,"style":61},[59,60,310],[15,57940,318],{"className":57941},[317],[15,57943,57945],{"className":57944},[290],[15,57946,57948],{"className":57947,"style":325},[294],[15,57949],{},[15,57951,113],{"className":57952},[146],[15,57954,206],{"className":57955,"style":334},[59,60],[15,57957,210],{"className":57958},[59],[15,57960,213],{"className":57961},[59,60],[15,57963,120],{"className":57964},[153]," 凸；固定 ",[15,57967,57969,57982],{"className":57968,"translate":19},[18],[15,57970,57972],{"className":57971},[23],[25,57973,57974],{"xmlns":27},[29,57975,57976,57980],{},[32,57977,57978],{},[35,57979,206],{},[39,57981,206],{"encoding":41},[15,57983,57985],{"className":57984,"ariaHidden":46},[45],[15,57986,57988,57991],{"className":57987},[50],[15,57989],{"className":57990,"style":664},[54],[15,57992,206],{"className":57993,"style":334},[59,60],[15,57995,57997,58029],{"className":57996,"translate":19},[18],[15,57998,58000],{"className":57999},[23],[25,58001,58002],{"xmlns":27},[29,58003,58004,58026],{},[32,58005,58006,58008,58010,58016,58018,58020,58022,58024],{},[35,58007,213],{},[110,58009,8686],{},[195,58011,58012,58014],{},[35,58013,199],{},[35,58015,37],{},[110,58017,113],{"stretchy":112},[35,58019,206],{},[35,58021,210],{"mathvariant":209},[35,58023,213],{},[110,58025,120],{"stretchy":112},[39,58027,58028],{"encoding":41},"Q\\mapsto D_f(P\\|Q)",[15,58030,58032,58050],{"className":58031,"ariaHidden":46},[45],[15,58033,58035,58038,58041,58044,58047],{"className":58034},[50],[15,58036],{"className":58037,"style":566},[54],[15,58039,213],{"className":58040},[59,60],[15,58042],{"className":58043,"style":158},[157],[15,58045,8686],{"className":58046},[162],[15,58048],{"className":58049,"style":158},[157],[15,58051,58053,58056,58096,58099,58102,58105,58108],{"className":58052},[50],[15,58054],{"className":58055,"style":270},[54],[15,58057,58059,58062],{"className":58058},[59],[15,58060,199],{"className":58061,"style":277},[59,60],[15,58063,58065],{"className":58064},[281],[15,58066,58068,58088],{"className":58067},[285,286],[15,58069,58071,58085],{"className":58070},[290],[15,58072,58074],{"className":58073,"style":295},[294],[15,58075,58076,58079],{"style":298},[15,58077],{"className":58078,"style":303},[302],[15,58080,58082],{"className":58081},[307,308,309,310],[15,58083,37],{"className":58084,"style":61},[59,60,310],[15,58086,318],{"className":58087},[317],[15,58089,58091],{"className":58090},[290],[15,58092,58094],{"className":58093,"style":325},[294],[15,58095],{},[15,58097,113],{"className":58098},[146],[15,58100,206],{"className":58101,"style":334},[59,60],[15,58103,210],{"className":58104},[59],[15,58106,213],{"className":58107},[59,60],[15,58109,120],{"className":58110},[153],[998,58112,58113],{},"注意",[15,58115,58117,58148],{"className":58116,"translate":19},[18],[15,58118,58120],{"className":58119},[23],[25,58121,58122],{"xmlns":27},[29,58123,58124,58146],{},[32,58125,58126,58128,58130,58136,58138,58140,58142,58144],{},[35,58127,213],{},[110,58129,8686],{},[195,58131,58132,58134],{},[35,58133,199],{},[35,58135,37],{},[110,58137,113],{"stretchy":112},[35,58139,206],{},[35,58141,210],{"mathvariant":209},[35,58143,213],{},[110,58145,120],{"stretchy":112},[39,58147,58028],{"encoding":41},[15,58149,58151,58169],{"className":58150,"ariaHidden":46},[45],[15,58152,58154,58157,58160,58163,58166],{"className":58153},[50],[15,58155],{"className":58156,"style":566},[54],[15,58158,213],{"className":58159},[59,60],[15,58161],{"className":58162,"style":158},[157],[15,58164,8686],{"className":58165},[162],[15,58167],{"className":58168,"style":158},[157],[15,58170,58172,58175,58215,58218,58221,58224,58227],{"className":58171},[50],[15,58173],{"className":58174,"style":270},[54],[15,58176,58178,58181],{"className":58177},[59],[15,58179,199],{"className":58180,"style":277},[59,60],[15,58182,58184],{"className":58183},[281],[15,58185,58187,58207],{"className":58186},[285,286],[15,58188,58190,58204],{"className":58189},[290],[15,58191,58193],{"className":58192,"style":295},[294],[15,58194,58195,58198],{"style":298},[15,58196],{"className":58197,"style":303},[302],[15,58199,58201],{"className":58200},[307,308,309,310],[15,58202,37],{"className":58203,"style":61},[59,60,310],[15,58205,318],{"className":58206},[317],[15,58208,58210],{"className":58209},[290],[15,58211,58213],{"className":58212,"style":325},[294],[15,58214],{},[15,58216,113],{"className":58217},[146],[15,58219,206],{"className":58220,"style":334},[59,60],[15,58222,210],{"className":58223},[59],[15,58225,213],{"className":58226},[59,60],[15,58228,120],{"className":58229},[153]," 凸是「以 ",[15,58232,58234,58247],{"className":58233,"translate":19},[18],[15,58235,58237],{"className":58236},[23],[25,58238,58239],{"xmlns":27},[29,58240,58241,58245],{},[32,58242,58243],{},[35,58244,213],{},[39,58246,213],{"encoding":41},[15,58248,58250],{"className":58249,"ariaHidden":46},[45],[15,58251,58253,58256],{"className":58252},[50],[15,58254],{"className":58255,"style":566},[54],[15,58257,213],{"className":58258},[59,60]," 为参照」的凸性，它与「",[15,58261,58263,58276],{"className":58262,"translate":19},[18],[15,58264,58266],{"className":58265},[23],[25,58267,58268],{"xmlns":27},[29,58269,58270,58274],{},[32,58271,58272],{},[35,58273,213],{},[39,58275,213],{"encoding":41},[15,58277,58279],{"className":58278,"ariaHidden":46},[45],[15,58280,58282,58285],{"className":58281},[50],[15,58283],{"className":58284,"style":566},[54],[15,58286,213],{"className":58287},[59,60]," 出现在分母上」的直觉相反，这个凸性正是变分推断里「对 ",[15,58290,58292,58305],{"className":58291,"translate":19},[18],[15,58293,58295],{"className":58294},[23],[25,58296,58297],{"xmlns":27},[29,58298,58299,58303],{},[32,58300,58301],{},[35,58302,944],{},[39,58304,944],{"encoding":41},[15,58306,58308],{"className":58307,"ariaHidden":46},[45],[15,58309,58311,58314],{"className":58310},[50],[15,58312],{"className":58313,"style":2278},[54],[15,58315,944],{"className":58316,"style":981},[59,60]," 做平均场优化收敛到全局最优」的根据之一。",[2313,58319,58321],{"id":58320},"_43-斜对称关系","4.3 斜对称关系",[993,58323,58324,58426,58644],{},[11,58325,58326,58329,58330,58425],{},[998,58327,58328],{},"定义 4.1（斜共轭）"," 对 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t>0",[15,58495,58497,58587,58635],{"className":58496,"ariaHidden":46},[45],[15,58498,58500,58503,58569,58572,58575,58578,58581,58584],{"className":58499},[50],[15,58501],{"className":58502,"style":139},[54],[15,58504,58506,58509],{"className":58505},[59],[15,58507,37],{"className":58508,"style":61},[59,60],[15,58510,58512],{"className":58511},[281],[15,58513,58515,58561],{"className":58514},[285,286],[15,58516,58518,58558],{"className":58517},[290],[15,58519,58521,58544],{"className":58520,"style":8250},[294],[15,58522,58523,58526],{"style":8253},[15,58524],{"className":58525,"style":303},[302],[15,58527,58529],{"className":58528},[307,308,309,310],[15,58530,58532,58535,58538,58541],{"className":58531},[59,310],[15,58533,7563],{"className":58534},[59,60,310],[15,58536,7566],{"className":58537,"style":8269},[59,60,310],[15,58539,7569],{"className":58540},[59,60,310],[15,58542,7572],{"className":58543,"style":8276},[59,60,310],[15,58545,58546,58549],{"style":8279},[15,58547],{"className":58548,"style":303},[302],[15,58550,58552],{"className":58551},[307,308,309,310],[15,58553,58555],{"className":58554},[59,310],[15,58556,7575],{"className":58557},[59,310],[15,58559,318],{"className":58560},[317],[15,58562,58564],{"className":58563},[290],[15,58565,58567],{"className":58566,"style":8301},[294],[15,58568],{},[15,58570,113],{"className":58571},[146],[15,58573,769],{"className":58574},[59,60],[15,58576,120],{"className":58577},[153],[15,58579],{"className":58580,"style":158},[157],[15,58582,123],{"className":58583},[162],[15,58585],{"className":58586,"style":158},[157],[15,58588,58590,58593,58596,58599,58602,58605,58608,58611,58614,58617,58620,58623,58626,58629,58632],{"className":58589},[50],[15,58591],{"className":58592,"style":139},[54],[15,58594,769],{"className":58595},[59,60],[15,58597],{"className":58598,"style":406},[157],[15,58600,37],{"className":58601,"style":61},[59,60],[15,58603,113],{"className":58604},[146],[15,58606,8591],{"className":58607},[59],[15,58609,769],{"className":58610},[59,60],[15,58612,120],{"className":58613},[153],[15,58615,551],{"className":58616},[573],[15,58618],{"className":58619,"style":3636},[157],[15,58621],{"className":58622,"style":406},[157],[15,58624,769],{"className":58625},[59,60],[15,58627],{"className":58628,"style":158},[157],[15,58630,2905],{"className":58631},[162],[15,58633],{"className":58634,"style":158},[157],[15,58636,58638,58641],{"className":58637},[50],[15,58639],{"className":58640,"style":172},[54],[15,58642,126],{"className":58643},[59],[11,58645,58646,58647,33851,58675,10182],{},"为 ",[15,58648,58650,58663],{"className":58649,"translate":19},[18],[15,58651,58653],{"className":58652},[23],[25,58654,58655],{"xmlns":27},[29,58656,58657,58661],{},[32,58658,58659],{},[35,58660,37],{},[39,58662,37],{"encoding":41},[15,58664,58666],{"className":58665,"ariaHidden":46},[45],[15,58667,58669,58672],{"className":58668},[50],[15,58670],{"className":58671,"style":55},[54],[15,58673,37],{"className":58674,"style":61},[59,60],[998,58676,35814],{},[993,58678,58679,58726],{},[11,58680,58681,58684,58685,2747],{},[998,58682,58683],{},"命题 4.3（斜对称）"," 对任意概率分布 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f(t)\\,dQ\n=\\int f\\!\\left(\\frac1s\\right) s\\,dP\\\\\n&=\\int 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对称（",[15,59917,59919,59965],{"className":59918,"translate":19},[18],[15,59920,59922],{"className":59921},[23],[25,59923,59924],{"xmlns":27},[29,59925,59926,59962],{},[32,59927,59928,59934,59936,59938,59940,59942,59944,59946,59952,59954,59956,59958,59960],{},[195,59929,59930,59932],{},[35,59931,199],{},[35,59933,37],{},[110,59935,113],{"stretchy":112},[35,59937,206],{},[35,59939,210],{"mathvariant":209},[35,59941,213],{},[110,59943,120],{"stretchy":112},[110,59945,123],{},[195,59947,59948,59950],{},[35,59949,199],{},[35,59951,37],{},[110,59953,113],{"stretchy":112},[35,59955,213],{},[35,59957,210],{"mathvariant":209},[35,59959,206],{},[110,59961,120],{"stretchy":112},[39,59963,59964],{"encoding":41},"D_f(P\\|Q)=D_f(Q\\|P)",[15,59966,59968,60038],{"className":59967,"ariaHidden":46},[45],[15,59969,59971,59974,60014,60017,60020,60023,60026,60029,60032,60035],{"className":59970},[50],[15,59972],{"className":59973,"style":270},[54],[15,59975,59977,59980],{"className":59976},[59],[15,59978,199],{"className":59979,"style":277},[59,60],[15,59981,59983],{"className":59982},[281],[15,59984,59986,60006],{"className":59985},[285,286],[15,59987,59989,60003],{"className":59988},[290],[15,59990,59992],{"className":59991,"style":295},[294],[15,59993,59994,59997],{"style":298},[15,59995],{"className":59996,"style":303},[302],[15,59998,60000],{"className":59999},[307,308,309,310],[15,60001,37],{"className":60002,"style":61},[59,60,310],[15,60004,318],{"className":60005},[317],[15,60007,60009],{"className":60008},[290],[15,60010,60012],{"className":60011,"style":325},[294],[15,60013],{},[15,60015,113],{"className":60016},[146],[15,60018,206],{"className":60019,"style":334},[59,60],[15,60021,210],{"className":60022},[59],[15,60024,213],{"className":60025},[59,60],[15,60027,120],{"className":60028},[153],[15,60030],{"className":60031,"style":158},[157],[15,60033,123],{"className":60034},[162],[15,60036],{"className":60037,"style":158},[157],[15,60039,60041,60044,60084,60087,60090,60093,60096],{"className":60040},[50],[15,60042],{"className":60043,"style":270},[54],[15,60045,60047,60050],{"className":60046},[59],[15,60048,199],{"className":60049,"style":277},[59,60],[15,60051,60053],{"className":60052},[281],[15,60054,60056,60076],{"className":60055},[285,286],[15,60057,60059,60073],{"className":60058},[290],[15,60060,60062],{"className":60061,"style":295},[294],[15,60063,60064,60067],{"style":298},[15,60065],{"className":60066,"style":303},[302],[15,60068,60070],{"className":60069},[307,308,309,310],[15,60071,37],{"className":60072,"style":61},[59,60,310],[15,60074,318],{"className":60075},[317],[15,60077,60079],{"className":60078},[290],[15,60080,60082],{"className":60081,"style":325},[294],[15,60083],{},[15,60085,113],{"className":60086},[146],[15,60088,213],{"className":60089},[59,60],[15,60091,210],{"className":60092},[59],[15,60094,206],{"className":60095,"style":334},[59,60],[15,60097,120],{"className":60098},[153]," 对所有 ",[15,60101,60103,60120],{"className":60102,"translate":19},[18],[15,60104,60106],{"className":60105},[23],[25,60107,60108],{"xmlns":27},[29,60109,60110,60118],{},[32,60111,60112,60114,60116],{},[35,60113,206],{},[110,60115,551],{"separator":46},[35,60117,213],{},[39,60119,556],{"encoding":41},[15,60121,60123],{"className":60122,"ariaHidden":46},[45],[15,60124,60126,60129,60132,60135,60138],{"className":60125},[50],[15,60127],{"className":60128,"style":566},[54],[15,60130,206],{"className":60131,"style":334},[59,60],[15,60133,551],{"className":60134},[573],[15,60136],{"className":60137,"style":406},[157],[15,60139,213],{"className":60140},[59,60],"）当且仅当 ",[15,60143,60145,60176],{"className":60144,"translate":19},[18],[15,60146,60148],{"className":60147},[23],[25,60149,60150],{"xmlns":27},[29,60151,60152,60174],{},[32,60153,60154,60170,60172],{},[7555,60155,60156,60158,60168],{},[35,60157,37],{},[32,60159,60160,60162,60164,60166],{},[35,60161,7563],{},[35,60163,7566],{},[35,60165,7569],{},[35,60167,7572],{},[110,60169,7575],{"lspace":1249,"rspace":1249},[110,60171,123],{},[35,60173,37],{},[39,60175,44179],{"encoding":41},[15,60177,60179,60260],{"className":60178,"ariaHidden":46},[45],[15,60180,60182,60185,60251,60254,60257],{"className":60181},[50],[15,60183],{"className":60184,"style":44189},[54],[15,60186,60188,60191],{"className":60187},[59],[15,60189,37],{"className":60190,"style":61},[59,60],[15,60192,60194],{"className":60193},[281],[15,60195,60197,60243],{"className":60196},[285,286],[15,60198,60200,60240],{"className":60199},[290],[15,60201,60203,60226],{"className":60202,"style":8501},[294],[15,60204,60205,60208],{"style":8504},[15,60206],{"className":60207,"style":303},[302],[15,60209,60211],{"className":60210},[307,308,309,310],[15,60212,60214,60217,60220,60223],{"className":60213},[59,310],[15,60215,7563],{"className":60216},[59,60,310],[15,60218,7566],{"className":60219,"style":8269},[59,60,310],[15,60221,7569],{"className":60222},[59,60,310],[15,60224,7572],{"className":60225,"style":8276},[59,60,310],[15,60227,60228,60231],{"style":6867},[15,60229],{"className":60230,"style":303},[302],[15,60232,60234],{"className":60233},[307,308,309,310],[15,60235,60237],{"className":60236},[59,310],[15,60238,7575],{"className":60239},[59,310],[15,60241,318],{"className":60242},[317],[15,60244,60246],{"className":60245},[290],[15,60247,60249],{"className":60248,"style":8549},[294],[15,60250],{},[15,60252],{"className":60253,"style":158},[157],[15,60255,123],{"className":60256},[162],[15,60258],{"className":60259,"style":158},[157],[15,60261,60263,60266],{"className":60262},[50],[15,60264],{"className":60265,"style":55},[54],[15,60267,37],{"className":60268,"style":61},[59,60],"（模去 ",[15,60271,60273,60296],{"className":60272,"translate":19},[18],[15,60274,60276],{"className":60275},[23],[25,60277,60278],{"xmlns":27},[29,60279,60280,60294],{},[32,60281,60282,60284,60286,60288,60290,60292],{},[35,60283,5908],{},[110,60285,113],{"stretchy":112},[35,60287,769],{},[110,60289,4815],{},[115,60291,117],{},[110,60293,120],{"stretchy":112},[39,60295,6622],{"encoding":41},[15,60297,60299,60323],{"className":60298,"ariaHidden":46},[45],[15,60300,60302,60305,60308,60311,60314,60317,60320],{"className":60301},[50],[15,60303],{"className":60304,"style":139},[54],[15,60306,5908],{"className":60307},[59,60],[15,60309,113],{"className":60310},[146],[15,60312,769],{"className":60313},[59,60],[15,60315],{"className":60316,"style":2589},[157],[15,60318,4815],{"className":60319},[2593],[15,60321],{"className":60322,"style":2589},[157],[15,60324,60326,60329,60332],{"className":60325},[50],[15,60327],{"className":60328,"style":139},[54],[15,60330,117],{"className":60331},[59],[15,60333,120],{"className":60334},[153],"）。验证 TV：",[15,60337,60339,60393],{"className":60338,"translate":19},[18],[15,60340,60342],{"className":60341},[23],[25,60343,60344],{"xmlns":27},[29,60345,60346,60390],{},[32,60347,60348,60350,60352,60358,60360,60362,60364,60366,60368,60370,60372,60374,60380,60382,60384,60386,60388],{},[35,60349,769],{},[110,60351,3301],{},[238,60353,60354,60356],{},[115,60355,117],{},[115,60357,2465],{},[35,60359,4810],{"mathvariant":209},[115,60361,117],{},[35,60363,941],{"mathvariant":209},[35,60365,769],{},[110,60367,4815],{},[115,60369,117],{},[35,60371,4810],{"mathvariant":209},[110,60373,123],{},[238,60375,60376,60378],{},[115,60377,117],{},[115,60379,2465],{},[35,60381,4810],{"mathvariant":209},[35,60383,769],{},[110,60385,4815],{},[115,60387,117],{},[35,60389,4810],{"mathvariant":209},[39,60391,60392],{"encoding":41},"t\\cdot\\frac12|1\u002Ft-1|=\\frac12|t-1|",[15,60394,60396,60414,60504,60522,60611],{"className":60395,"ariaHidden":46},[45],[15,60397,60399,60402,60405,60408,60411],{"className":60398},[50],[15,60400],{"className":60401,"style":5345},[54],[15,60403,769],{"className":60404},[59,60],[15,60406],{"className":60407,"style":2589},[157],[15,60409,3301],{"className":60410},[2593],[15,60412],{"className":60413,"style":2589},[157],[15,60415,60417,60420,60488,60492,60495,60498,60501],{"className":60416},[50],[15,60418],{"className":60419,"style":2507},[54],[15,60421,60423,60426,60485],{"className":60422},[59],[15,60424],{"className":60425},[146,441],[15,60427,60429],{"className":60428},[238],[15,60430,60432,60477],{"className":60431},[285,286],[15,60433,60435,60474],{"className":60434},[290],[15,60436,60438,60452,60460],{"className":60437,"style":2526},[294],[15,60439,60440,60443],{"style":2529},[15,60441],{"className":60442,"style":461},[302],[15,60444,60446],{"className":60445},[307,308,309,310],[15,60447,60449],{"className":60448},[59,310],[15,60450,2465],{"className":60451},[59,310],[15,60453,60454,60457],{"style":473},[15,60455],{"className":60456,"style":461},[302],[15,60458],{"className":60459,"style":481},[480],[15,60461,60462,60465],{"style":2552},[15,60463],{"className":60464,"style":461},[302],[15,60466,60468],{"className":60467},[307,308,309,310],[15,60469,60471],{"className":60470},[59,310],[15,60472,117],{"className":60473},[59,310],[15,60475,318],{"className":60476},[317],[15,60478,60480],{"className":60479},[290],[15,60481,60483],{"className":60482,"style":2574},[294],[15,60484],{},[15,60486],{"className":60487},[153,441],[15,60489,60491],{"className":60490},[59],"∣1\u002F",[15,60493,769],{"className":60494},[59,60],[15,60496],{"className":60497,"style":2589},[157],[15,60499,4815],{"className":60500},[2593],[15,60502],{"className":60503,"style":2589},[157],[15,60505,60507,60510,60513,60516,60519],{"className":60506},[50],[15,60508],{"className":60509,"style":139},[54],[15,60511,34705],{"className":60512},[59],[15,60514],{"className":60515,"style":158},[157],[15,60517,123],{"className":60518},[162],[15,60520],{"className":60521,"style":158},[157],[15,60523,60525,60528,60596,60599,60602,60605,60608],{"className":60524},[50],[15,60526],{"className":60527,"style":2507},[54],[15,60529,60531,60534,60593],{"className":60530},[59],[15,60532],{"className":60533},[146,441],[15,60535,60537],{"className":60536},[238],[15,60538,60540,60585],{"className":60539},[285,286],[15,60541,60543,60582],{"className":60542},[290],[15,60544,60546,60560,60568],{"className":60545,"style":2526},[294],[15,60547,60548,60551],{"style":2529},[15,60549],{"className":60550,"style":461},[302],[15,60552,60554],{"className":60553},[307,308,309,310],[15,60555,60557],{"className":60556},[59,310],[15,60558,2465],{"className":60559},[59,310],[15,60561,60562,60565],{"style":473},[15,60563],{"className":60564,"style":461},[302],[15,60566],{"className":60567,"style":481},[480],[15,60569,60570,60573],{"style":2552},[15,60571],{"className":60572,"style":461},[302],[15,60574,60576],{"className":60575},[307,308,309,310],[15,60577,60579],{"className":60578},[59,310],[15,60580,117],{"className":60581},[59,310],[15,60583,318],{"className":60584},[317],[15,60586,60588],{"className":60587},[290],[15,60589,60591],{"className":60590,"style":2574},[294],[15,60592],{},[15,60594],{"className":60595},[153,441],[15,60597,4810],{"className":60598},[59],[15,60600,769],{"className":60601},[59,60],[15,60603],{"className":60604,"style":2589},[157],[15,60606,4815],{"className":60607},[2593],[15,60609],{"className":60610,"style":2589},[157],[15,60612,60614,60617],{"className":60613},[50],[15,60615],{"className":60616,"style":139},[54],[15,60618,34705],{"className":60619},[59],"；JS 的 ",[15,60622,60624,60659],{"className":60623,"translate":19},[18],[15,60625,60627],{"className":60626},[23],[25,60628,60629],{"xmlns":27},[29,60630,60631,60657],{},[32,60632,60633,60635,60637,60639,60641,60643,60645,60647,60649,60651,60653,60655],{},[35,60634,769],{},[35,60636,37],{},[110,60638,113],{"stretchy":112},[115,60640,117],{},[35,60642,941],{"mathvariant":209},[35,60644,769],{},[110,60646,120],{"stretchy":112},[110,60648,123],{},[35,60650,37],{},[110,60652,113],{"stretchy":112},[35,60654,769],{},[110,60656,120],{"stretchy":112},[39,60658,23595],{"encoding":41},[15,60660,60662,60695],{"className":60661,"ariaHidden":46},[45],[15,60663,60665,60668,60671,60674,60677,60680,60683,60686,60689,60692],{"className":60664},[50],[15,60666],{"className":60667,"style":139},[54],[15,60669,769],{"className":60670},[59,60],[15,60672,37],{"className":60673,"style":61},[59,60],[15,60675,113],{"className":60676},[146],[15,60678,8591],{"className":60679},[59],[15,60681,769],{"className":60682},[59,60],[15,60684,120],{"className":60685},[153],[15,60687],{"className":60688,"style":158},[157],[15,60690,123],{"className":60691},[162],[15,60693],{"className":60694,"style":158},[157],[15,60696,60698,60701,60704,60707,60710],{"className":60697},[50],[15,60699],{"className":60700,"style":139},[54],[15,60702,37],{"className":60703,"style":61},[59,60],[15,60705,113],{"className":60706},[146],[15,60708,769],{"className":60709},[59,60],[15,60711,120],{"className":60712},[153]," 也已验算；KL 显然不满足，故不对称。",[2313,60715,60717],{"id":60716},"_44-变分表示","4.4 变分表示",[993,60719,60720,60814,61040,61043,61419],{},[11,60721,60722,11064,60725,60784,60785,60813],{},[998,60723,60724],{},"定理 4.1（变分表示 \u002F Fenchel 对偶）",[15,60726,60728,60746],{"className":60727,"translate":19},[18],[15,60729,60731],{"className":60730},[23],[25,60732,60733],{"xmlns":27},[29,60734,60735,60743],{},[32,60736,60737],{},[6816,60738,60739,60741],{},[35,60740,37],{},[110,60742,7575],{},[39,60744,60745],{"encoding":41},"f^*",[15,60747,60749],{"className":60748,"ariaHidden":46},[45],[15,60750,60752,60755],{"className":60751},[50],[15,60753],{"className":60754,"style":55},[54],[15,60756,60758,60761],{"className":60757},[59],[15,60759,37],{"className":60760,"style":61},[59,60],[15,60762,60764],{"className":60763},[281],[15,60765,60767],{"className":60766},[285],[15,60768,60770],{"className":60769},[290],[15,60771,60773],{"className":60772,"style":8501},[294],[15,60774,60775,60778],{"style":6867},[15,60776],{"className":60777,"style":303},[302],[15,60779,60781],{"className":60780},[307,308,309,310],[15,60782,7575],{"className":60783},[2593,310]," 为 ",[15,60786,60788,60801],{"className":60787,"translate":19},[18],[15,60789,60791],{"className":60790},[23],[25,60792,60793],{"xmlns":27},[29,60794,60795,60799],{},[32,60796,60797],{},[35,60798,37],{},[39,60800,37],{"encoding":41},[15,60802,60804],{"className":60803,"ariaHidden":46},[45],[15,60805,60807,60810],{"className":60806},[50],[15,60808],{"className":60809,"style":55},[54],[15,60811,37],{"className":60812,"style":61},[59,60]," 的凸共轭（Fenchel 共轭）",[15,60815,60817],{"className":60816,"translate":19},[180],[15,60818,60820,60882],{"className":60819,"translate":19},[18],[15,60821,60823],{"className":60822},[23],[25,60824,60825],{"xmlns":27,"display":189},[29,60826,60827,60879],{},[32,60828,60829,60835,60837,60839,60841,60843,60859,60861,60863,60865,60867,60869,60871,60873,60875,60877],{},[6816,60830,60831,60833],{},[35,60832,37],{},[110,60834,7575],{},[110,60836,113],{"stretchy":112},[35,60838,40263],{},[110,60840,120],{"stretchy":112},[110,60842,123],{},[6018,60844,60845,60851],{},[32,60846,60847,60849],{},[35,60848,35458],{},[110,60850,4105],{},[32,60852,60853,60855,60857],{},[35,60854,769],{},[110,60856,6930],{},[115,60858,126],{},[110,60860,2900],{"stretchy":112},[35,60862,40263],{},[35,60864,769],{},[110,60866,4815],{},[35,60868,37],{},[110,60870,113],{"stretchy":112},[35,60872,769],{},[110,60874,120],{"stretchy":112},[110,60876,2910],{"stretchy":112},[110,60878,551],{"separator":46},[39,60880,60881],{"encoding":41},"f^*(y)=\\sup_{t\\ge0}\\{yt-f(t)\\},",[15,60883,60885,60938,61018],{"className":60884,"ariaHidden":46},[45],[15,60886,60888,60891,60920,60923,60926,60929,60932,60935],{"className":60887},[50],[15,60889],{"className":60890,"style":139},[54],[15,60892,60894,60897],{"className":60893},[59],[15,60895,37],{"className":60896,"style":61},[59,60],[15,60898,60900],{"className":60899},[281],[15,60901,60903],{"className":60902},[285],[15,60904,60906],{"className":60905},[290],[15,60907,60909],{"className":60908,"style":8250},[294],[15,60910,60911,60914],{"style":8279},[15,60912],{"className":60913,"style":303},[302],[15,60915,60917],{"className":60916},[307,308,309,310],[15,60918,7575],{"className":60919},[2593,310],[15,60921,113],{"className":60922},[146],[15,60924,40263],{"className":60925,"style":981},[59,60],[15,60927,120],{"className":60928},[153],[15,60930],{"className":60931,"style":158},[157],[15,60933,123],{"className":60934},[162],[15,60936],{"className":60937,"style":158},[157],[15,60939,60941,60945,61000,61003,61006,61009,61012,61015],{"className":60940},[50],[15,60942],{"className":60943,"style":60944},[54],"height:1.7567em;vertical-align:-1.0067em;",[15,60946,60948],{"className":60947},[1486,13720],[15,60949,60951,60991],{"className":60950},[285,286],[15,60952,60954,60988],{"className":60953},[290],[15,60955,60957,60978],{"className":60956,"style":2347},[294],[15,60958,60960,60963],{"style":60959},"top:-2.1885em;margin-left:0em;",[15,60961],{"className":60962,"style":461},[302],[15,60964,60966],{"className":60965},[307,308,309,310],[15,60967,60969,60972,60975],{"className":60968},[59,310],[15,60970,769],{"className":60971},[59,60,310],[15,60973,6930],{"className":60974},[162,310],[15,60976,126],{"className":60977},[59,310],[15,60979,60980,60983],{"style":11867},[15,60981],{"className":60982,"style":461},[302],[15,60984,60985],{},[15,60986,35458],{"className":60987},[1486],[15,60989,318],{"className":60990},[317],[15,60992,60994],{"className":60993},[290],[15,60995,60998],{"className":60996,"style":60997},[294],"height:1.0067em;",[15,60999],{},[15,61001,2900],{"className":61002},[146],[15,61004,40263],{"className":61005,"style":981},[59,60],[15,61007,769],{"className":61008},[59,60],[15,61010],{"className":61011,"style":2589},[157],[15,61013,4815],{"className":61014},[2593],[15,61016],{"className":61017,"style":2589},[157],[15,61019,61021,61024,61027,61030,61033,61037],{"className":61020},[50],[15,61022],{"className":61023,"style":139},[54],[15,61025,37],{"className":61026,"style":61},[59,60],[15,61028,113],{"className":61029},[146],[15,61031,769],{"className":61032},[59,60],[15,61034,61036],{"className":61035},[153],")}",[15,61038,551],{"className":61039},[573],[11,61041,61042],{},"则",[15,61044,61046],{"className":61045,"translate":19},[180],[15,61047,61049,61133],{"className":61048,"translate":19},[18],[15,61050,61052],{"className":61051},[23],[25,61053,61054],{"xmlns":27,"display":189},[29,61055,61056,61130],{},[32,61057,61058,61064,61066,61068,61070,61072,61074,61076,61086,61128],{},[195,61059,61060,61062],{},[35,61061,199],{},[35,61063,37],{},[110,61065,113],{"stretchy":112},[35,61067,206],{},[35,61069,210],{"mathvariant":209},[35,61071,213],{},[110,61073,120],{"stretchy":112},[110,61075,123],{},[6018,61077,61078,61084],{},[32,61079,61080,61082],{},[35,61081,35458],{},[110,61083,4105],{},[35,61085,4160],{},[32,61087,61088,61090,61096,61098,61100,61102,61104,61110,61112,61118,61120,61122,61124,61126],{},[110,61089,2900],{"fence":46},[195,61091,61092,61094],{},[35,61093,223],{"mathvariant":222},[35,61095,206],{},[110,61097,230],{"stretchy":112},[35,61099,4160],{},[110,61101,257],{"stretchy":112},[110,61103,4815],{},[195,61105,61106,61108],{},[35,61107,223],{"mathvariant":222},[35,61109,213],{},[110,61111,230],{"stretchy":112},[6816,61113,61114,61116],{},[35,61115,37],{},[110,61117,7575],{},[110,61119,113],{"stretchy":112},[35,61121,4160],{},[110,61123,120],{"stretchy":112},[110,61125,257],{"stretchy":112},[110,61127,2910],{"fence":46},[110,61129,551],{"separator":46},[39,61131,61132],{"encoding":41},"D_f(P\\|Q)=\\sup_{g}\n\\left\\{\\mathbb E_P[g]\n-\\mathbb E_Q[f^*(g)]\\right\\},",[15,61134,61136,61206],{"className":61135,"ariaHidden":46},[45],[15,61137,61139,61142,61182,61185,61188,61191,61194,61197,61200,61203],{"className":61138},[50],[15,61140],{"className":61141,"style":270},[54],[15,61143,61145,61148],{"className":61144},[59],[15,61146,199],{"className":61147,"style":277},[59,60],[15,61149,61151],{"className":61150},[281],[15,61152,61154,61174],{"className":61153},[285,286],[15,61155,61157,61171],{"className":61156},[290],[15,61158,61160],{"className":61159,"style":295},[294],[15,61161,61162,61165],{"style":298},[15,61163],{"className":61164,"style":303},[302],[15,61166,61168],{"className":61167},[307,308,309,310],[15,61169,37],{"className":61170,"style":61},[59,60,310],[15,61172,318],{"className":61173},[317],[15,61175,61177],{"className":61176},[290],[15,61178,61180],{"className":61179,"style":325},[294],[15,61181],{},[15,61183,113],{"className":61184},[146],[15,61186,206],{"className":61187,"style":334},[59,60],[15,61189,210],{"className":61190},[59],[15,61192,213],{"className":61193},[59,60],[15,61195,120],{"className":61196},[153],[15,61198],{"className":61199,"style":158},[157],[15,61201,123],{"className":61202},[162],[15,61204],{"className":61205,"style":158},[157],[15,61207,61209,61213,61262,61265,61413,61416],{"className":61208},[50],[15,61210],{"className":61211,"style":61212},[54],"height:1.7805em;vertical-align:-1.0305em;",[15,61214,61216],{"className":61215},[1486,13720],[15,61217,61219,61253],{"className":61218},[285,286],[15,61220,61222,61250],{"className":61221},[290],[15,61223,61225,61240],{"className":61224,"style":2347},[294],[15,61226,61228,61231],{"style":61227},"top:-2.2056em;margin-left:0em;",[15,61229],{"className":61230,"style":461},[302],[15,61232,61234],{"className":61233},[307,308,309,310],[15,61235,61237],{"className":61236},[59,310],[15,61238,4160],{"className":61239,"style":981},[59,60,310],[15,61241,61242,61245],{"style":11867},[15,61243],{"className":61244,"style":461},[302],[15,61246,61247],{},[15,61248,35458],{"className":61249},[1486],[15,61251,318],{"className":61252},[317],[15,61254,61256],{"className":61255},[290],[15,61257,61260],{"className":61258,"style":61259},[294],"height:1.0305em;",[15,61261],{},[15,61263],{"className":61264,"style":406},[157],[15,61266,61268,61271,61311,61314,61317,61320,61323,61326,61329,61369,61372,61401,61404,61407,61410],{"className":61267},[410],[15,61269,2900],{"className":61270,"style":415},[146,414],[15,61272,61274,61277],{"className":61273},[59],[15,61275,223],{"className":61276},[59,366],[15,61278,61280],{"className":61279},[281],[15,61281,61283,61303],{"className":61282},[285,286],[15,61284,61286,61300],{"className":61285},[290],[15,61287,61289],{"className":61288,"style":379},[294],[15,61290,61291,61294],{"style":382},[15,61292],{"className":61293,"style":303},[302],[15,61295,61297],{"className":61296},[307,308,309,310],[15,61298,206],{"className":61299,"style":334},[59,60,310],[15,61301,318],{"className":61302},[317],[15,61304,61306],{"className":61305},[290],[15,61307,61309],{"className":61308,"style":7970},[294],[15,61310],{},[15,61312,230],{"className":61313},[146],[15,61315,4160],{"className":61316,"style":981},[59,60],[15,61318,257],{"className":61319},[153],[15,61321],{"className":61322,"style":2589},[157],[15,61324,4815],{"className":61325},[2593],[15,61327],{"className":61328,"style":2589},[157],[15,61330,61332,61335],{"className":61331},[59],[15,61333,223],{"className":61334},[59,366],[15,61336,61338],{"className":61337},[281],[15,61339,61341,61361],{"className":61340},[285,286],[15,61342,61344,61358],{"className":61343},[290],[15,61345,61347],{"className":61346,"style":379},[294],[15,61348,61349,61352],{"style":382},[15,61350],{"className":61351,"style":303},[302],[15,61353,61355],{"className":61354},[307,308,309,310],[15,61356,213],{"className":61357},[59,60,310],[15,61359,318],{"className":61360},[317],[15,61362,61364],{"className":61363},[290],[15,61365,61367],{"className":61366,"style":325},[294],[15,61368],{},[15,61370,230],{"className":61371},[146],[15,61373,61375,61378],{"className":61374},[59],[15,61376,37],{"className":61377,"style":61},[59,60],[15,61379,61381],{"className":61380},[281],[15,61382,61384],{"className":61383},[285],[15,61385,61387],{"className":61386},[290],[15,61388,61390],{"className":61389,"style":8250},[294],[15,61391,61392,61395],{"style":8279},[15,61393],{"className":61394,"style":303},[302],[15,61396,61398],{"className":61397},[307,308,309,310],[15,61399,7575],{"className":61400},[2593,310],[15,61402,113],{"className":61403},[146],[15,61405,4160],{"className":61406,"style":981},[59,60],[15,61408,20416],{"className":61409},[153],[15,61411,2910],{"className":61412,"style":415},[153,414],[15,61414],{"className":61415,"style":406},[157],[15,61417,551],{"className":61418},[573],[11,61420,61421,61422,10182],{},"其中上确界取遍所有使两个期望都有定义的可测函数 ",[15,61423,61425,61456],{"className":61424,"translate":19},[18],[15,61426,61428],{"className":61427},[23],[25,61429,61430],{"xmlns":27},[29,61431,61432,61453],{},[32,61433,61434,61436,61438,61440,61442,61445,61447],{},[35,61435,4160],{},[110,61437,1018],{},[35,61439,598],{"mathvariant":597},[110,61441,1032],{},[35,61443,61444],{"mathvariant":209},"dom",[110,61446,4105],{},[6816,61448,61449,61451],{},[35,61450,37],{},[110,61452,7575],{},[39,61454,61455],{"encoding":41},"g:\\mathcal X\\to\\operatorname{dom} f^*",[15,61457,61459,61477,61495],{"className":61458,"ariaHidden":46},[45],[15,61460,61462,61465,61468,61471,61474],{"className":61461},[50],[15,61463],{"className":61464,"style":2278},[54],[15,61466,4160],{"className":61467,"style":981},[59,60],[15,61469],{"className":61470,"style":158},[157],[15,61472,1018],{"className":61473},[162],[15,61475],{"className":61476,"style":158},[157],[15,61478,61480,61483,61486,61489,61492],{"className":61479},[50],[15,61481],{"className":61482,"style":664},[54],[15,61484,598],{"className":61485,"style":625},[59,624],[15,61487],{"className":61488,"style":158},[157],[15,61490,1032],{"className":61491},[162],[15,61493],{"className":61494,"style":158},[157],[15,61496,61498,61501,61507,61510],{"className":61497},[50],[15,61499],{"className":61500,"style":55},[54],[15,61502,61504],{"className":61503},[1486],[15,61505,61444],{"className":61506},[59,32688],[15,61508],{"className":61509,"style":406},[157],[15,61511,61513,61516],{"className":61512},[59],[15,61514,37],{"className":61515,"style":61},[59,60],[15,61517,61519],{"className":61518},[281],[15,61520,61522],{"className":61521},[285],[15,61523,61525],{"className":61524},[290],[15,61526,61528],{"className":61527,"style":8501},[294],[15,61529,61530,61533],{"style":6867},[15,61531],{"className":61532,"style":303},[302],[15,61534,61536],{"className":61535},[307,308,309,310],[15,61537,7575],{"className":61538},[2593,310],[11,61540,61541,61544,61545,61573],{},[998,61542,61543],{},"证明（Fenchel 对偶）。"," 由共轭函数的性质（Fenchel–Moreau，",[15,61546,61548,61561],{"className":61547,"translate":19},[18],[15,61549,61551],{"className":61550},[23],[25,61552,61553],{"xmlns":27},[29,61554,61555,61559],{},[32,61556,61557],{},[35,61558,37],{},[39,61560,37],{"encoding":41},[15,61562,61564],{"className":61563,"ariaHidden":46},[45],[15,61565,61567,61570],{"className":61566},[50],[15,61568],{"className":61569,"style":55},[54],[15,61571,37],{"className":61572,"style":61},[59,60]," 凸闭），",[15,61575,61577],{"className":61576,"translate":19},[180],[15,61578,61580,61647],{"className":61579,"translate":19},[18],[15,61581,61583],{"className":61582},[23],[25,61584,61585],{"xmlns":27,"display":189},[29,61586,61587,61644],{},[32,61588,61589,61591,61593,61595,61597,61599,61609,61611,61613,61615,61617,61623,61625,61627,61629,61631,61633,61635,61638,61640,61642],{},[35,61590,37],{},[110,61592,113],{"stretchy":112},[35,61594,769],{},[110,61596,120],{"stretchy":112},[110,61598,123],{},[6018,61600,61601,61607],{},[32,61602,61603,61605],{},[35,61604,35458],{},[110,61606,4105],{},[35,61608,40263],{},[110,61610,2900],{"stretchy":112},[35,61612,40263],{},[35,61614,769],{},[110,61616,4815],{},[6816,61618,61619,61621],{},[35,61620,37],{},[110,61622,7575],{},[110,61624,113],{"stretchy":112},[35,61626,40263],{},[110,61628,120],{"stretchy":112},[110,61630,2910],{"stretchy":112},[110,61632,551],{"separator":46},[157,61634],{"width":3264},[35,61636,61637],{"mathvariant":209},"∀",[35,61639,769],{},[110,61641,6930],{},[115,61643,5766],{},[39,61645,61646],{"encoding":41},"f(t)=\\sup_{y}\\{yt-f^*(y)\\},\n\\qquad \\forall t\\ge0 .",[15,61648,61650,61677,61748,61816],{"className":61649,"ariaHidden":46},[45],[15,61651,61653,61656,61659,61662,61665,61668,61671,61674],{"className":61652},[50],[15,61654],{"className":61655,"style":139},[54],[15,61657,37],{"className":61658,"style":61},[59,60],[15,61660,113],{"className":61661},[146],[15,61663,769],{"className":61664},[59,60],[15,61666,120],{"className":61667},[153],[15,61669],{"className":61670,"style":158},[157],[15,61672,123],{"className":61673},[162],[15,61675],{"className":61676,"style":158},[157],[15,61678,61680,61683,61730,61733,61736,61739,61742,61745],{"className":61679},[50],[15,61681],{"className":61682,"style":61212},[54],[15,61684,61686],{"className":61685},[1486,13720],[15,61687,61689,61722],{"className":61688},[285,286],[15,61690,61692,61719],{"className":61691},[290],[15,61693,61695,61709],{"className":61694,"style":2347},[294],[15,61696,61697,61700],{"style":61227},[15,61698],{"className":61699,"style":461},[302],[15,61701,61703],{"className":61702},[307,308,309,310],[15,61704,61706],{"className":61705},[59,310],[15,61707,40263],{"className":61708,"style":981},[59,60,310],[15,61710,61711,61714],{"style":11867},[15,61712],{"className":61713,"style":461},[302],[15,61715,61716],{},[15,61717,35458],{"className":61718},[1486],[15,61720,318],{"className":61721},[317],[15,61723,61725],{"className":61724},[290],[15,61726,61728],{"className":61727,"style":61259},[294],[15,61729],{},[15,61731,2900],{"className":61732},[146],[15,61734,40263],{"className":61735,"style":981},[59,60],[15,61737,769],{"className":61738},[59,60],[15,61740],{"className":61741,"style":2589},[157],[15,61743,4815],{"className":61744},[2593],[15,61746],{"className":61747,"style":2589},[157],[15,61749,61751,61754,61783,61786,61789,61792,61795,61798,61801,61804,61807,61810,61813],{"className":61750},[50],[15,61752],{"className":61753,"style":139},[54],[15,61755,61757,61760],{"className":61756},[59],[15,61758,37],{"className":61759,"style":61},[59,60],[15,61761,61763],{"className":61762},[281],[15,61764,61766],{"className":61765},[285],[15,61767,61769],{"className":61768},[290],[15,61770,61772],{"className":61771,"style":8250},[294],[15,61773,61774,61777],{"style":8279},[15,61775],{"className":61776,"style":303},[302],[15,61778,61780],{"className":61779},[307,308,309,310],[15,61781,7575],{"className":61782},[2593,310],[15,61784,113],{"className":61785},[146],[15,61787,40263],{"className":61788,"style":981},[59,60],[15,61790,61036],{"className":61791},[153],[15,61793,551],{"className":61794},[573],[15,61796],{"className":61797,"style":3636},[157],[15,61799],{"className":61800,"style":406},[157],[15,61802,61637],{"className":61803},[59],[15,61805,769],{"className":61806},[59,60],[15,61808],{"className":61809,"style":158},[157],[15,61811,6930],{"className":61812},[162],[15,61814],{"className":61815,"style":158},[157],[15,61817,61819,61822],{"className":61818},[50],[15,61820],{"className":61821,"style":172},[54],[15,61823,5766],{"className":61824},[59],[11,61826,17050,61827,61961,61962,61990],{},[15,61828,61830,61860],{"className":61829,"translate":19},[18],[15,61831,61833],{"className":61832},[23],[25,61834,61835],{"xmlns":27},[29,61836,61837,61857],{},[32,61838,61839,61841,61843],{},[35,61840,769],{},[110,61842,123],{},[238,61844,61845,61851],{},[32,61846,61847,61849],{},[35,61848,244],{},[35,61850,206],{},[32,61852,61853,61855],{},[35,61854,244],{},[35,61856,213],{},[39,61858,61859],{"encoding":41},"t=\\frac{dP}{dQ}",[15,61861,61863,61881],{"className":61862,"ariaHidden":46},[45],[15,61864,61866,61869,61872,61875,61878],{"className":61865},[50],[15,61867],{"className":61868,"style":5345},[54],[15,61870,769],{"className":61871},[59,60],[15,61873],{"className":61874,"style":158},[157],[15,61876,123],{"className":61877},[162],[15,61879],{"className":61880,"style":158},[157],[15,61882,61884,61887],{"className":61883},[50],[15,61885],{"className":61886,"style":2670},[54],[15,61888,61890,61893,61958],{"className":61889},[59],[15,61891],{"className":61892},[146,441],[15,61894,61896],{"className":61895},[238],[15,61897,61899,61950],{"className":61898},[285,286],[15,61900,61902,61947],{"className":61901},[290],[15,61903,61905,61922,61930],{"className":61904,"style":2689},[294],[15,61906,61907,61910],{"style":2529},[15,61908],{"className":61909,"style":461},[302],[15,61911,61913],{"className":61912},[307,308,309,310],[15,61914,61916,61919],{"className":61915},[59,310],[15,61917,244],{"className":61918},[59,60,310],[15,61920,213],{"className":61921},[59,60,310],[15,61923,61924,61927],{"style":473},[15,61925],{"className":61926,"style":461},[302],[15,61928],{"className":61929,"style":481},[480],[15,61931,61932,61935],{"style":2552},[15,61933],{"className":61934,"style":461},[302],[15,61936,61938],{"className":61937},[307,308,309,310],[15,61939,61941,61944],{"className":61940},[59,310],[15,61942,244],{"className":61943},[59,60,310],[15,61945,206],{"className":61946,"style":334},[59,60,310],[15,61948,318],{"className":61949},[317],[15,61951,61953],{"className":61952},[290],[15,61954,61956],{"className":61955,"style":2741},[294],[15,61957],{},[15,61959],{"className":61960},[153,441]," 代入并对 ",[15,61963,61965,61978],{"className":61964,"translate":19},[18],[15,61966,61968],{"className":61967},[23],[25,61969,61970],{"xmlns":27},[29,61971,61972,61976],{},[32,61973,61974],{},[35,61975,213],{},[39,61977,213],{"encoding":41},[15,61979,61981],{"className":61980,"ariaHidden":46},[45],[15,61982,61984,61987],{"className":61983},[50],[15,61985],{"className":61986,"style":566},[54],[15,61988,213],{"className":61989},[59,60]," 积分：",[15,61992,61994],{"className":61993,"translate":19},[180],[15,61995,61997,62239],{"className":61996,"translate":19},[18],[15,61998,62000],{"className":61999},[23],[25,62001,62002],{"xmlns":27,"display":189},[29,62003,62004,62236],{},[1245,62005,62006,62092,62166],{"rowspacing":1247,"columnalign":1248,"columnspacing":1249},[1251,62007,62008,62030],{},[1254,62009,62010],{},[1257,62011,62012],{"scriptlevel":126,"displaystyle":46},[32,62013,62014,62020,62022,62024,62026,62028],{},[195,62015,62016,62018],{},[35,62017,199],{},[35,62019,37],{},[110,62021,113],{"stretchy":112},[35,62023,206],{},[35,62025,210],{"mathvariant":209},[35,62027,213],{},[110,62029,120],{"stretchy":112},[1254,62031,62032],{},[1257,62033,62034],{"scriptlevel":126,"displaystyle":46},[32,62035,62036,62038,62040,62042,62052,62054,62056,62058,62072,62074,62080,62082,62084,62086,62088,62090],{},[32,62037],{},[110,62039,123],{},[110,62041,1289],{},[6018,62043,62044,62050],{},[32,62045,62046,62048],{},[35,62047,35458],{},[110,62049,4105],{},[35,62051,40263],{},[110,62053,2900],{"fence":112,"stretchy":46,"minsize":17443,"maxsize":17443},[35,62055,40263],{},[1299,62057,1301],{},[238,62059,62060,62066],{},[32,62061,62062,62064],{},[35,62063,244],{},[35,62065,206],{},[32,62067,62068,62070],{},[35,62069,244],{},[35,62071,213],{},[110,62073,4815],{},[6816,62075,62076,62078],{},[35,62077,37],{},[110,62079,7575],{},[110,62081,113],{"stretchy":112},[35,62083,40263],{},[110,62085,120],{"stretchy":112},[110,62087,2910],{"fence":112,"stretchy":46,"minsize":17443,"maxsize":17443},[35,62089,244],{},[35,62091,213],{},[1251,62093,62094,62100],{},[1254,62095,62096],{},[1257,62097,62098],{"scriptlevel":126,"displaystyle":46},[32,62099],{},[1254,62101,62102],{},[1257,62103,62104],{"scriptlevel":126,"displaystyle":46},[32,62105,62106,62108,62110,62112,62114,62124,62126,62128,62130,62132,62146,62148,62154,62156,62158,62160,62162,62164],{},[32,62107],{},[1299,62109,4285],{},[110,62111,6930],{},[1299,62113,4285],{},[6018,62115,62116,62122],{},[32,62117,62118,62120],{},[35,62119,35458],{},[110,62121,4105],{},[35,62123,40263],{},[110,62125,1289],{},[110,62127,2900],{"fence":112,"stretchy":46,"minsize":17443,"maxsize":17443},[35,62129,40263],{},[1299,62131,1301],{},[238,62133,62134,62140],{},[32,62135,62136,62138],{},[35,62137,244],{},[35,62139,206],{},[32,62141,62142,62144],{},[35,62143,244],{},[35,62145,213],{},[110,62147,4815],{},[6816,62149,62150,62152],{},[35,62151,37],{},[110,62153,7575],{},[110,62155,113],{"stretchy":112},[35,62157,40263],{},[110,62159,120],{"stretchy":112},[110,62161,2910],{"fence":112,"stretchy":46,"minsize":17443,"maxsize":17443},[35,62163,244],{},[35,62165,213],{},[1251,62167,62168,62174],{},[1254,62169,62170],{},[1257,62171,62172],{"scriptlevel":126,"displaystyle":46},[32,62173],{},[1254,62175,62176],{},[1257,62177,62178],{"scriptlevel":126,"displaystyle":46},[32,62179,62180,62182,62184,62194,62196,62202,62204,62206,62208,62210,62216,62218,62224,62226,62228,62230,62232,62234],{},[32,62181],{},[110,62183,123],{},[6018,62185,62186,62192],{},[32,62187,62188,62190],{},[35,62189,35458],{},[110,62191,4105],{},[35,62193,40263],{},[110,62195,2900],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[195,62197,62198,62200],{},[35,62199,223],{"mathvariant":222},[35,62201,206],{},[110,62203,230],{"stretchy":112},[35,62205,40263],{},[110,62207,257],{"stretchy":112},[110,62209,4815],{},[195,62211,62212,62214],{},[35,62213,223],{"mathvariant":222},[35,62215,213],{},[110,62217,230],{"stretchy":112},[6816,62219,62220,62222],{},[35,62221,37],{},[110,62223,7575],{},[110,62225,113],{"stretchy":112},[35,62227,40263],{},[110,62229,120],{"stretchy":112},[110,62231,257],{"stretchy":112},[110,62233,2910],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[35,62235,4377],{"mathvariant":209},[39,62237,62238],{"encoding":41},"\\begin{aligned}\nD_f(P\\|Q)\n&=\\int \\sup_y\\Big\\{y\\,\\frac{dP}{dQ}\n-f^*(y)\\Big\\}dQ\\\\\n&\\ \\ge\\ \\sup_y\\int\\Big\\{y\\,\\frac{dP}{dQ}\n-f^*(y)\\Big\\}dQ\\\\\n&=\\sup_y\\big\\{\\mathbb E_P[y]\n-\\mathbb E_Q[f^*(y)]\\big\\}.\n\\end{aligned}",[15,62240,62242],{"className":62241,"ariaHidden":46},[45],[15,62243,62245,62249],{"className":62244},[50],[15,62246],{"className":62247,"style":62248},[54],"height:7.2845em;vertical-align:-3.3923em;",[15,62250,62252],{"className":62251},[59],[15,62253,62255,62363],{"className":62254},[1245],[15,62256,62258],{"className":62257},[1362],[15,62259,62261,62354],{"className":62260},[285,286],[15,62262,62264,62351],{"className":62263},[290],[15,62265,62268,62333,62342],{"className":62266,"style":62267},[294],"height:3.8923em;",[15,62269,62271,62275],{"style":62270},"top:-5.8923em;",[15,62272],{"className":62273,"style":62274},[302],"height:3.3714em;",[15,62276,62278,62318,62321,62324,62327,62330],{"className":62277},[59],[15,62279,62281,62284],{"className":62280},[59],[15,62282,199],{"className":62283,"style":277},[59,60],[15,62285,62287],{"className":62286},[281],[15,62288,62290,62310],{"className":62289},[285,286],[15,62291,62293,62307],{"className":62292},[290],[15,62294,62296],{"className":62295,"style":295},[294],[15,62297,62298,62301],{"style":298},[15,62299],{"className":62300,"style":303},[302],[15,62302,62304],{"className":62303},[307,308,309,310],[15,62305,37],{"className":62306,"style":61},[59,60,310],[15,62308,318],{"className":62309},[317],[15,62311,62313],{"className":62312},[290],[15,62314,62316],{"className":62315,"style":325},[294],[15,62317],{},[15,62319,113],{"className":62320},[146],[15,62322,206],{"className":62323,"style":334},[59,60],[15,62325,210],{"className":62326},[59],[15,62328,213],{"className":62329},[59,60],[15,62331,120],{"className":62332},[153],[15,62334,62336,62339],{"style":62335},"top:-3.1903em;",[15,62337],{"className":62338,"style":62274},[302],[15,62340],{"className":62341},[59],[15,62343,62345,62348],{"style":62344},"top:-1.0097em;",[15,62346],{"className":62347,"style":62274},[302],[15,62349],{"className":62350},[59],[15,62352,318],{"className":62353},[317],[15,62355,62357],{"className":62356},[290],[15,62358,62361],{"className":62359,"style":62360},[294],"height:3.3923em;",[15,62362],{},[15,62364,62366],{"className":62365},[1453],[15,62367,62369,63029],{"className":62368},[285,286],[15,62370,62372,63026],{"className":62371},[290],[15,62373,62375,62587,62805],{"className":62374,"style":62267},[294],[15,62376,62377,62380],{"style":62270},[15,62378],{"className":62379,"style":62274},[302],[15,62381,62383,62386,62389,62392,62395,62398,62401,62445,62448,62454,62457,62460,62528,62531,62534,62537,62566,62569,62572,62575,62581,62584],{"className":62382},[59],[15,62384],{"className":62385},[59],[15,62387],{"className":62388,"style":158},[157],[15,62390,123],{"className":62391},[162],[15,62393],{"className":62394,"style":158},[157],[15,62396,1289],{"className":62397,"style":1489},[1486,1487,1488],[15,62399],{"className":62400,"style":406},[157],[15,62402,62404],{"className":62403},[1486,13720],[15,62405,62407,62437],{"className":62406},[285,286],[15,62408,62410,62434],{"className":62409},[290],[15,62411,62413,62424],{"className":62412,"style":2347},[294],[15,62414,62415,62418],{"style":61227},[15,62416],{"className":62417,"style":461},[302],[15,62419,62421],{"className":62420},[307,308,309,310],[15,62422,40263],{"className":62423,"style":981},[59,60,310],[15,62425,62426,62429],{"style":11867},[15,62427],{"className":62428,"style":461},[302],[15,62430,62431],{},[15,62432,35458],{"className":62433},[1486],[15,62435,318],{"className":62436},[317],[15,62438,62440],{"className":62439},[290],[15,62441,62443],{"className":62442,"style":61259},[294],[15,62444],{},[15,62446],{"className":62447,"style":406},[157],[15,62449,62451],{"className":62450},[59],[15,62452,2900],{"className":62453},[419,3722],[15,62455,40263],{"className":62456,"style":981},[59,60],[15,62458],{"className":62459,"style":406},[157],[15,62461,62463,62466,62525],{"className":62462},[59],[15,62464],{"className":62465},[146,441],[15,62467,62469],{"className":62468},[238],[15,62470,62472,62517],{"className":62471},[285,286],[15,62473,62475,62514],{"className":62474},[290],[15,62476,62478,62492,62500],{"className":62477,"style":454},[294],[15,62479,62480,62483],{"style":457},[15,62481],{"className":62482,"style":461},[302],[15,62484,62486,62489],{"className":62485},[59],[15,62487,244],{"className":62488},[59,60],[15,62490,213],{"className":62491},[59,60],[15,62493,62494,62497],{"style":473},[15,62495],{"className":62496,"style":461},[302],[15,62498],{"className":62499,"style":481},[480],[15,62501,62502,62505],{"style":484},[15,62503],{"className":62504,"style":461},[302],[15,62506,62508,62511],{"className":62507},[59],[15,62509,244],{"className":62510},[59,60],[15,62512,206],{"className":62513,"style":334},[59,60],[15,62515,318],{"className":62516},[317],[15,62518,62520],{"className":62519},[290],[15,62521,62523],{"className":62522,"style":506},[294],[15,62524],{},[15,62526],{"className":62527},[153,441],[15,62529],{"className":62530,"style":2589},[157],[15,62532,4815],{"className":62533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",[15,63041,63043,63056],{"className":63042,"translate":19},[18],[15,63044,63046],{"className":63045},[23],[25,63047,63048],{"xmlns":27},[29,63049,63050,63054],{},[32,63051,63052],{},[110,63053,6930],{},[39,63055,55517],{"encoding":41},[15,63057,63059],{"className":63058,"ariaHidden":46},[45],[15,63060,63062,63065],{"className":63061},[50],[15,63063],{"className":63064,"style":43214},[54],[15,63066,6930],{"className":63067},[162],"，因为「积分的上确界 ≥ 上确界的积分」。反向不等式需要一个可测选择\u002F极大值论证：由于 ",[15,63070,63072,63085],{"className":63071,"translate":19},[18],[15,63073,63075],{"className":63074},[23],[25,63076,63077],{"xmlns":27},[29,63078,63079,63083],{},[32,63080,63081],{},[35,63082,37],{},[39,63084,37],{"encoding":41},[15,63086,63088],{"className":63087,"ariaHidden":46},[45],[15,63089,63091,63094],{"className":63090},[50],[15,63092],{"className":63093,"style":55},[54],[15,63095,37],{"className":63096,"style":61},[59,60]," 凸，",[15,63099,63101,63155],{"className":63100,"translate":19},[18],[15,63102,63104],{"className":63103},[23],[25,63105,63106],{"xmlns":27},[29,63107,63108,63152],{},[32,63109,63110,63112,63114,63116,63118,63120,63130,63132,63134,63136,63138,63144,63146,63148,63150],{},[35,63111,37],{},[110,63113,113],{"stretchy":112},[35,63115,769],{},[110,63117,120],{"stretchy":112},[110,63119,123],{},[195,63121,63122,63128],{},[32,63123,63124,63126],{},[35,63125,35458],{},[110,63127,4105],{},[35,63129,40263],{},[110,63131,2900],{"stretchy":112},[35,63133,40263],{},[35,63135,769],{},[110,63137,4815],{},[6816,63139,63140,63142],{},[35,63141,37],{},[110,63143,7575],{},[110,63145,113],{"stretchy":112},[35,63147,40263],{},[110,63149,120],{"stretchy":112},[110,63151,2910],{"stretchy":112},[39,63153,63154],{"encoding":41},"f(t)=\\sup_y\\{yt-f^*(y)\\}",[15,63156,63158,63185,63249],{"className":63157,"ariaHidden":46},[45],[15,63159,63161,63164,63167,63170,63173,63176,63179,63182],{"className":63160},[50],[15,63162],{"className":63163,"style":139},[54],[15,63165,37],{"className":63166,"style":61},[59,60],[15,63168,113],{"className":63169},[146],[15,63171,769],{"className":63172},[59,60],[15,63174,120],{"className":63175},[153],[15,63177],{"className":63178,"style":158},[157],[15,63180,123],{"className":63181},[162],[15,63183],{"className":63184,"style":158},[157],[15,63186,63188,63191,63231,63234,63237,63240,63243,63246],{"className":63187},[50],[15,63189],{"className":63190,"style":57353},[54],[15,63192,63194,63197],{"className":63193},[1486],[15,63195,35458],{"className":63196},[1486],[15,63198,63200],{"className":63199},[281],[15,63201,63203,63223],{"className":63202},[285,286],[15,63204,63206,63220],{"className":63205},[290],[15,63207,63209],{"className":63208,"style":37025},[294],[15,63210,63211,63214],{"style":35604},[15,63212],{"className":63213,"style":303},[302],[15,63215,63217],{"className":63216},[307,308,309,310],[15,63218,40263],{"className":63219,"style":981},[59,60,310],[15,63221,318],{"className":63222},[317],[15,63224,63226],{"className":63225},[290],[15,63227,63229],{"className":63228,"style":57392},[294],[15,63230],{},[15,63232,2900],{"className":63233},[146],[15,63235,40263],{"className":63236,"style":981},[59,60],[15,63238,769],{"className":63239},[59,60],[15,63241],{"className":63242,"style":2589},[157],[15,63244,4815],{"className":63245},[2593],[15,63247],{"className":63248,"style":2589},[157],[15,63250,63252,63255,63284,63287,63290],{"className":63251},[50],[15,63253],{"className":63254,"style":139},[54],[15,63256,63258,63261],{"className":63257},[59],[15,63259,37],{"className":63260,"style":61},[59,60],[15,63262,63264],{"className":63263},[281],[15,63265,63267],{"className":63266},[285],[15,63268,63270],{"className":63269},[290],[15,63271,63273],{"className":63272,"style":8501},[294],[15,63274,63275,63278],{"style":6867},[15,63276],{"className":63277,"style":303},[302],[15,63279,63281],{"className":63280},[307,308,309,310],[15,63282,7575],{"className":63283},[2593,310],[15,63285,113],{"className":63286},[146],[15,63288,40263],{"className":63289,"style":981},[59,60],[15,63291,61036],{"className":63292},[153]," 中的 ",[15,63295,63297,63310],{"className":63296,"translate":19},[18],[15,63298,63300],{"className":63299},[23],[25,63301,63302],{"xmlns":27},[29,63303,63304,63308],{},[32,63305,63306],{},[35,63307,40263],{},[39,63309,40263],{"encoding":41},[15,63311,63313],{"className":63312,"ariaHidden":46},[45],[15,63314,63316,63319],{"className":63315},[50],[15,63317],{"className":63318,"style":2278},[54],[15,63320,40263],{"className":63321,"style":981},[59,60]," 可取为 ",[15,63324,63326,63350],{"className":63325,"translate":19},[18],[15,63327,63329],{"className":63328},[23],[25,63330,63331],{"xmlns":27},[29,63332,63333,63347],{},[32,63334,63335,63341,63343,63345],{},[6816,63336,63337,63339],{},[35,63338,37],{},[110,63340,6822],{"mathvariant":209,"lspace":1249,"rspace":1249},[110,63342,113],{"stretchy":112},[35,63344,769],{},[110,63346,120],{"stretchy":112},[39,63348,63349],{"encoding":41},"f'(t)",[15,63351,63353],{"className":63352,"ariaHidden":46},[45],[15,63354,63356,63359,63391,63394,63397],{"className":63355},[50],[15,63357],{"className":63358,"style":6845},[54],[15,63360,63362,63365],{"className":63361},[59],[15,63363,37],{"className":63364,"style":61},[59,60],[15,63366,63368],{"className":63367},[281],[15,63369,63371],{"className":63370},[285],[15,63372,63374],{"className":63373},[290],[15,63375,63377],{"className":63376,"style":6864},[294],[15,63378,63379,63382],{"style":6867},[15,63380],{"className":63381,"style":303},[302],[15,63383,63385],{"className":63384},[307,308,309,310],[15,63386,63388],{"className":63387},[59,310],[15,63389,6822],{"className":63390},[59,310],[15,63392,113],{"className":63393},[146],[15,63395,769],{"className":63396},[59,60],[15,63398,120],{"className":63399},[153],"（在可微点），而 ",[15,63402,63404,63432],{"className":63403,"translate":19},[18],[15,63405,63407],{"className":63406},[23],[25,63408,63409],{"xmlns":27},[29,63410,63411,63429],{},[32,63412,63413,63415,63417,63423,63425,63427],{},[35,63414,769],{},[110,63416,8686],{},[6816,63418,63419,63421],{},[35,63420,37],{},[110,63422,6822],{"mathvariant":209,"lspace":1249,"rspace":1249},[110,63424,113],{"stretchy":112},[35,63426,769],{},[110,63428,120],{"stretchy":112},[39,63430,63431],{"encoding":41},"t\\mapsto f'(t)",[15,63433,63435,63453],{"className":63434,"ariaHidden":46},[45],[15,63436,63438,63441,63444,63447,63450],{"className":63437},[50],[15,63439],{"className":63440,"style":8713},[54],[15,63442,769],{"className":63443},[59,60],[15,63445],{"className":63446,"style":158},[157],[15,63448,8686],{"className":63449},[162],[15,63451],{"className":63452,"style":158},[157],[15,63454,63456,63459,63491,63494,63497],{"className":63455},[50],[15,63457],{"className":63458,"style":6845},[54],[15,63460,63462,63465],{"className":63461},[59],[15,63463,37],{"className":63464,"style":61},[59,60],[15,63466,63468],{"className":63467},[281],[15,63469,63471],{"className":63470},[285],[15,63472,63474],{"className":63473},[290],[15,63475,63477],{"className":63476,"style":6864},[294],[15,63478,63479,63482],{"style":6867},[15,63480],{"className":63481,"style":303},[302],[15,63483,63485],{"className":63484},[307,308,309,310],[15,63486,63488],{"className":63487},[59,310],[15,63489,6822],{"className":63490},[59,310],[15,63492,113],{"className":63493},[146],[15,63495,769],{"className":63496},[59,60],[15,63498,120],{"className":63499},[153]," 单调，于是 ",[15,63502,63504,63550],{"className":63503,"translate":19},[18],[15,63505,63507],{"className":63506},[23],[25,63508,63509],{"xmlns":27},[29,63510,63511,63547],{},[32,63512,63513,63519,63521,63523,63525,63527,63529,63535,63537,63539,63541,63543,63545],{},[6816,63514,63515,63517],{},[35,63516,4160],{},[110,63518,7575],{},[110,63520,113],{"stretchy":112},[35,63522,774],{},[110,63524,120],{"stretchy":112},[110,63526,1018],{},[110,63528,123],{},[6816,63530,63531,63533],{},[35,63532,37],{},[110,63534,6822],{"mathvariant":209,"lspace":1249,"rspace":1249},[110,63536,113],{"stretchy":112},[35,63538,769],{},[110,63540,113],{"stretchy":112},[35,63542,774],{},[110,63544,120],{"stretchy":112},[110,63546,120],{"stretchy":112},[39,63548,63549],{"encoding":41},"g^*(x):=f'(t(x))",[15,63551,63553,63607],{"className":63552,"ariaHidden":46},[45],[15,63554,63556,63559,63588,63591,63594,63597,63600,63604],{"className":63555},[50],[15,63557],{"className":63558,"style":139},[54],[15,63560,63562,63565],{"className":63561},[59],[15,63563,4160],{"className":63564,"style":981},[59,60],[15,63566,63568],{"className":63567},[281],[15,63569,63571],{"className":63570},[285],[15,63572,63574],{"className":63573},[290],[15,63575,63577],{"className":63576,"style":8501},[294],[15,63578,63579,63582],{"style":6867},[15,63580],{"className":63581,"style":303},[302],[15,63583,63585],{"className":63584},[307,308,309,310],[15,63586,7575],{"className":63587},[2593,310],[15,63589,113],{"className":63590},[146],[15,63592,774],{"className":63593},[59,60],[15,63595,120],{"className":63596},[153],[15,63598],{"className":63599,"style":158},[157],[15,63601,63603],{"className":63602},[162],":=",[15,63605],{"className":63606,"style":158},[157],[15,63608,63610,63613,63645,63648,63651,63654,63657],{"className":63609},[50],[15,63611],{"className":63612,"style":6845},[54],[15,63614,63616,63619],{"className":63615},[59],[15,63617,37],{"className":63618,"style":61},[59,60],[15,63620,63622],{"className":63621},[281],[15,63623,63625],{"className":63624},[285],[15,63626,63628],{"className":63627},[290],[15,63629,63631],{"className":63630,"style":6864},[294],[15,63632,63633,63636],{"style":6867},[15,63634],{"className":63635,"style":303},[302],[15,63637,63639],{"className":63638},[307,308,309,310],[15,63640,63642],{"className":63641},[59,310],[15,63643,6822],{"className":63644},[59,310],[15,63646,113],{"className":63647},[146],[15,63649,769],{"className":63650},[59,60],[15,63652,113],{"className":63653},[146],[15,63655,774],{"className":63656},[59,60],[15,63658,19073],{"className":63659},[153]," 是可测的，并且对每个 ",[15,63662,63664,63677],{"className":63663,"translate":19},[18],[15,63665,63667],{"className":63666},[23],[25,63668,63669],{"xmlns":27},[29,63670,63671,63675],{},[32,63672,63673],{},[35,63674,774],{},[39,63676,774],{"encoding":41},[15,63678,63680],{"className":63679,"ariaHidden":46},[45],[15,63681,63683,63686],{"className":63682},[50],[15,63684],{"className":63685,"style":2347},[54],[15,63687,774],{"className":63688},[59,60]," 达到上确界。于是不等式取等号。",[11,63691,63692,63695],{},[998,63693,63694],{},"最优解。"," 达到上确界的函数为",[15,63697,63699],{"className":63698,"translate":19},[180],[15,63700,63702,63764],{"className":63701,"translate":19},[18],[15,63703,63705],{"className":63704},[23],[25,63706,63707],{"xmlns":27,"display":189},[29,63708,63709,63761],{},[32,63710,63711,63717,63719,63721,63723,63725,63731,63733,63759],{},[6816,63712,63713,63715],{},[35,63714,4160],{},[110,63716,7575],{},[110,63718,113],{"stretchy":112},[35,63720,774],{},[110,63722,120],{"stretchy":112},[110,63724,123],{},[6816,63726,63727,63729],{},[35,63728,37],{},[110,63730,6822],{"mathvariant":209,"lspace":1249,"rspace":1249},[1299,63732,1306],{},[32,63734,63735,63737,63751,63753,63755,63757],{},[110,63736,113],{"fence":46},[238,63738,63739,63745],{},[32,63740,63741,63743],{},[35,63742,244],{},[35,63744,206],{},[32,63746,63747,63749],{},[35,63748,244],{},[35,63750,213],{},[110,63752,113],{"stretchy":112},[35,63754,774],{},[110,63756,120],{"stretchy":112},[110,63758,120],{"fence":46},[110,63760,551],{"separator":46},[39,63762,63763],{"encoding":41},"g^*(x)=f'\\!\\left(\\frac{dP}{dQ}(x)\\right),",[15,63765,63767,63820],{"className":63766,"ariaHidden":46},[45],[15,63768,63770,63773,63802,63805,63808,63811,63814,63817],{"className":63769},[50],[15,63771],{"className":63772,"style":139},[54],[15,63774,63776,63779],{"className":63775},[59],[15,63777,4160],{"className":63778,"style":981},[59,60],[15,63780,63782],{"className":63781},[281],[15,63783,63785],{"className":63784},[285],[15,63786,63788],{"className":63787},[290],[15,63789,63791],{"className":63790,"style":8250},[294],[15,63792,63793,63796],{"style":8279},[15,63794],{"className":63795,"style":303},[302],[15,63797,63799],{"className":63798},[307,308,309,310],[15,63800,7575],{"className":63801},[2593,310],[15,63803,113],{"className":63804},[146],[15,63806,774],{"className":63807},[59,60],[15,63809,120],{"className":63810},[153],[15,63812],{"className":63813,"style":158},[157],[15,63815,123],{"className":63816},[162],[15,63818],{"className":63819,"style":158},[157],[15,63821,63823,63826,63858,63861,63864,63956,63959],{"className":63822},[50],[15,63824],{"className":63825,"style":359},[54],[15,63827,63829,63832],{"className":63828},[59],[15,63830,37],{"className":63831,"style":61},[59,60],[15,63833,63835],{"className":63834},[281],[15,63836,63838],{"className":63837},[285],[15,63839,63841],{"className":63840},[290],[15,63842,63844],{"className":63843,"style":28997},[294],[15,63845,63846,63849],{"style":8279},[15,63847],{"className":63848,"style":303},[302],[15,63850,63852],{"className":63851},[307,308,309,310],[15,63853,63855],{"className":63854},[59,310],[15,63856,6822],{"className":63857},[59,310],[15,63859],{"className":63860,"style":1514},[157],[15,63862],{"className":63863,"style":406},[157],[15,63865,63867,63873,63941,63944,63947,63950],{"className":63866},[410],[15,63868,63870],{"className":63869,"style":415},[146,414],[15,63871,113],{"className":63872},[419,309],[15,63874,63876,63879,63938],{"className":63875},[59],[15,63877],{"className":63878},[146,441],[15,63880,63882],{"className":63881},[238],[15,63883,63885,63930],{"className":63884},[285,286],[15,63886,63888,63927],{"className":63887},[290],[15,63889,63891,63905,63913],{"className":63890,"style":454},[294],[15,63892,63893,63896],{"style":457},[15,63894],{"className":63895,"style":461},[302],[15,63897,63899,63902],{"className":63898},[59],[15,63900,244],{"className":63901},[59,60],[15,63903,213],{"className":63904},[59,60],[15,63906,63907,63910],{"style":473},[15,63908],{"className":63909,"style":461},[302],[15,63911],{"className":63912,"style":481},[480],[15,63914,63915,63918],{"style":484},[15,63916],{"className":63917,"style":461},[302],[15,63919,63921,63924],{"className":63920},[59],[15,63922,244],{"className":63923},[59,60],[15,63925,206],{"className":63926,"style":334},[59,60],[15,63928,318],{"className":63929},[317],[15,63931,63933],{"className":63932},[290],[15,63934,63936],{"className":63935,"style":506},[294],[15,63937],{},[15,63939],{"className":63940},[153,441],[15,63942,113],{"className":63943},[146],[15,63945,774],{"className":63946},[59,60],[15,63948,120],{"className":63949},[153],[15,63951,63953],{"className":63952,"style":415},[153,414],[15,63954,120],{"className":63955},[419,309],[15,63957],{"className":63958,"style":406},[157],[15,63960,551],{"className":63961},[573],[11,63963,63964,63965,3855,63993,64021,64022,64082,64083,64155,64156,45634,64275,64459],{},"条件是 ",[15,63966,63968,63981],{"className":63967,"translate":19},[18],[15,63969,63971],{"className":63970},[23],[25,63972,63973],{"xmlns":27},[29,63974,63975,63979],{},[32,63976,63977],{},[35,63978,37],{},[39,63980,37],{"encoding":41},[15,63982,63984],{"className":63983,"ariaHidden":46},[45],[15,63985,63987,63990],{"className":63986},[50],[15,63988],{"className":63989,"style":55},[54],[15,63991,37],{"className":63992,"style":61},[59,60],[15,63994,63996,64009],{"className":63995,"translate":19},[18],[15,63997,63999],{"className":63998},[23],[25,64000,64001],{"xmlns":27},[29,64002,64003,64007],{},[32,64004,64005],{},[35,64006,769],{},[39,64008,769],{"encoding":41},[15,64010,64012],{"className":64011,"ariaHidden":46},[45],[15,64013,64015,64018],{"className":64014},[50],[15,64016],{"className":64017,"style":5345},[54],[15,64019,769],{"className":64020},[59,60]," 处可微、且 ",[15,64023,64025,64043],{"className":64024,"translate":19},[18],[15,64026,64028],{"className":64027},[23],[25,64029,64030],{"xmlns":27},[29,64031,64032,64040],{},[32,64033,64034],{},[6816,64035,64036,64038],{},[35,64037,4160],{},[110,64039,7575],{},[39,64041,64042],{"encoding":41},"g^*",[15,64044,64046],{"className":64045,"ariaHidden":46},[45],[15,64047,64049,64053],{"className":64048},[50],[15,64050],{"className":64051,"style":64052},[54],"height:0.8831em;vertical-align:-0.1944em;",[15,64054,64056,64059],{"className":64055},[59],[15,64057,4160],{"className":64058,"style":981},[59,60],[15,64060,64062],{"className":64061},[281],[15,64063,64065],{"className":64064},[285],[15,64066,64068],{"className":64067},[290],[15,64069,64071],{"className":64070,"style":8501},[294],[15,64072,64073,64076],{"style":6867},[15,64074],{"className":64075,"style":303},[302],[15,64077,64079],{"className":64078},[307,308,309,310],[15,64080,7575],{"className":64081},[2593,310]," 落在 ",[15,64084,64086,64108],{"className":64085,"translate":19},[18],[15,64087,64089],{"className":64088},[23],[25,64090,64091],{"xmlns":27},[29,64092,64093,64105],{},[32,64094,64095,64097,64099],{},[35,64096,61444],{"mathvariant":209},[110,64098,4105],{},[6816,64100,64101,64103],{},[35,64102,37],{},[110,64104,7575],{},[39,64106,64107],{"encoding":41},"\\operatorname{dom} f^*",[15,64109,64111],{"className":64110,"ariaHidden":46},[45],[15,64112,64114,64117,64123,64126],{"className":64113},[50],[15,64115],{"className":64116,"style":55},[54],[15,64118,64120],{"className":64119},[1486],[15,64121,61444],{"className":64122},[59,32688],[15,64124],{"className":64125,"style":406},[157],[15,64127,64129,64132],{"className":64128},[59],[15,64130,37],{"className":64131,"style":61},[59,60],[15,64133,64135],{"className":64134},[281],[15,64136,64138],{"className":64137},[285],[15,64139,64141],{"className":64140},[290],[15,64142,64144],{"className":64143,"style":8501},[294],[15,64145,64146,64149],{"style":6867},[15,64147],{"className":64148,"style":303},[302],[15,64150,64152],{"className":64151},[307,308,309,310],[15,64153,7575],{"className":64154},[2593,310]," 内。例如 KL（",[15,64157,64159,64189],{"className":64158,"translate":19},[18],[15,64160,64162],{"className":64161},[23],[25,64163,64164],{"xmlns":27},[29,64165,64166,64186],{},[32,64167,64168,64174,64176,64178,64180,64182,64184],{},[6816,64169,64170,64172],{},[35,64171,37],{},[110,64173,6822],{"mathvariant":209,"lspace":1249,"rspace":1249},[110,64175,123],{},[35,64177,4102],{},[110,64179,4105],{},[35,64181,769],{},[110,64183,2472],{},[115,64185,117],{},[39,64187,64188],{"encoding":41},"f'=\\log t+1",[15,64190,64192,64240,64266],{"className":64191,"ariaHidden":46},[45],[15,64193,64195,64199,64231,64234,64237],{"className":64194},[50],[15,64196],{"className":64197,"style":64198},[54],"height:0.9463em;vertical-align:-0.1944em;",[15,64200,64202,64205],{"className":64201},[59],[15,64203,37],{"className":64204,"style":61},[59,60],[15,64206,64208],{"className":64207},[281],[15,64209,64211],{"className":64210},[285],[15,64212,64214],{"className":64213},[290],[15,64215,64217],{"className":64216,"style":6864},[294],[15,64218,64219,64222],{"style":6867},[15,64220],{"className":64221,"style":303},[302],[15,64223,64225],{"className":64224},[307,308,309,310],[15,64226,64228],{"className":64227},[59,310],[15,64229,6822],{"className":64230},[59,310],[15,64232],{"className":64233,"style":158},[157],[15,64235,123],{"className":64236},[162],[15,64238],{"className":64239,"style":158},[157],[15,64241,64243,64246,64251,64254,64257,64260,64263],{"className":64242},[50],[15,64244],{"className":64245,"style":55},[54],[15,64247,4156,64249],{"className":64248},[1486],[15,64250,4160],{"style":4159},[15,64252],{"className":64253,"style":406},[157],[15,64255,769],{"className":64256},[59,60],[15,64258],{"className":64259,"style":2589},[157],[15,64261,2472],{"className":64262},[2593],[15,64264],{"className":64265,"style":2589},[157],[15,64267,64269,64272],{"className":64268},[50],[15,64270],{"className":64271,"style":172},[54],[15,64273,117],{"className":64274},[59],[15,64276,64278,64312],{"className":64277,"translate":19},[18],[15,64279,64281],{"className":64280},[23],[25,64282,64283],{"xmlns":27},[29,64284,64285,64309],{},[32,64286,64287,64293,64295,64297,64299,64305,64307],{},[6816,64288,64289,64291],{},[35,64290,4160],{},[110,64292,7575],{},[110,64294,123],{},[35,64296,4102],{},[110,64298,4105],{},[238,64300,64301,64303],{},[35,64302,11],{},[35,64304,944],{},[110,64306,2472],{},[115,64308,117],{},[39,64310,64311],{"encoding":41},"g^*=\\log\\frac{p}{q}+1",[15,64313,64315,64359,64450],{"className":64314,"ariaHidden":46},[45],[15,64316,64318,64321,64350,64353,64356],{"className":64317},[50],[15,64319],{"className":64320,"style":64052},[54],[15,64322,64324,64327],{"className":64323},[59],[15,64325,4160],{"className":64326,"style":981},[59,60],[15,64328,64330],{"className":64329},[281],[15,64331,64333],{"className":64332},[285],[15,64334,64336],{"className":64335},[290],[15,64337,64339],{"className":64338,"style":8501},[294],[15,64340,64341,64344],{"style":6867},[15,64342],{"className":64343,"style":303},[302],[15,64345,64347],{"className":64346},[307,308,309,310],[15,64348,7575],{"className":64349},[2593,310],[15,64351],{"className":64352,"style":158},[157],[15,64354,123],{"className":64355},[162],[15,64357],{"className":64358,"style":158},[157],[15,64360,64362,64365,64370,64373,64441,64444,64447],{"className":64361},[50],[15,64363],{"className":64364,"style":3086},[54],[15,64366,4156,64368],{"className":64367},[1486],[15,64369,4160],{"style":4159},[15,64371],{"className":64372,"style":406},[157],[15,64374,64376,64379,64438],{"className":64375},[59],[15,64377],{"className":64378},[146,441],[15,64380,64382],{"className":64381},[238],[15,64383,64385,64430],{"className":64384},[285,286],[15,64386,64388,64427],{"className":64387},[290],[15,64389,64391,64405,64413],{"className":64390,"style":3105},[294],[15,64392,64393,64396],{"style":2529},[15,64394],{"className":64395,"style":461},[302],[15,64397,64399],{"className":64398},[307,308,309,310],[15,64400,64402],{"className":64401},[59,310],[15,64403,944],{"className":64404,"style":981},[59,60,310],[15,64406,64407,64410],{"style":473},[15,64408],{"className":64409,"style":461},[302],[15,64411],{"className":64412,"style":481},[480],[15,64414,64415,64418],{"style":2855},[15,64416],{"className":64417,"style":461},[302],[15,64419,64421],{"className":64420},[307,308,309,310],[15,64422,64424],{"className":64423},[59,310],[15,64425,11],{"className":64426},[59,60,310],[15,64428,318],{"className":64429},[317],[15,64431,64433],{"className":64432},[290],[15,64434,64436],{"className":64435,"style":2741},[294],[15,64437],{},[15,64439],{"className":64440},[153,441],[15,64442],{"className":64443,"style":2589},[157],[15,64445,2472],{"className":64446},[2593],[15,64448],{"className":64449,"style":2589},[157],[15,64451,64453,64456],{"className":64452},[50],[15,64454],{"className":64455,"style":172},[54],[15,64457,117],{"className":64458},[59],"；这解释了为什么「对判别器做 log 密度比」是 f-GAN 的最佳策略。",[11,64461,64462,64524,64525,64624,64625,45634],{},[998,64463,64464,64465,64523],{},"常用 ",[15,64466,64468,64485],{"className":64467,"translate":19},[18],[15,64469,64471],{"className":64470},[23],[25,64472,64473],{"xmlns":27},[29,64474,64475,64483],{},[32,64476,64477],{},[6816,64478,64479,64481],{},[35,64480,37],{},[110,64482,7575],{},[39,64484,60745],{"encoding":41},[15,64486,64488],{"className":64487,"ariaHidden":46},[45],[15,64489,64491,64494],{"className":64490},[50],[15,64492],{"className":64493,"style":55},[54],[15,64495,64497,64500],{"className":64496},[59],[15,64498,37],{"className":64499,"style":61},[59,60],[15,64501,64503],{"className":64502},[281],[15,64504,64506],{"className":64505},[285],[15,64507,64509],{"className":64508},[290],[15,64510,64512],{"className":64511,"style":8501},[294],[15,64513,64514,64517],{"style":6867},[15,64515],{"className":64516,"style":303},[302],[15,64518,64520],{"className":64519},[307,308,309,310],[15,64521,7575],{"className":64522},[2593,310]," 表","（结合 ",[15,64526,64528,64558],{"className":64527,"translate":19},[18],[15,64529,64531],{"className":64530},[23],[25,64532,64533],{"xmlns":27},[29,64534,64535,64555],{},[32,64536,64537,64539,64541,64543,64545,64547,64549,64551,64553],{},[110,64538,1289],{},[35,64540,244],{},[35,64542,206],{},[110,64544,123],{},[110,64546,1289],{},[35,64548,244],{},[35,64550,213],{},[110,64552,123],{},[115,64554,117],{},[39,64556,64557],{"encoding":41},"\\int dP=\\int dQ=1",[15,64559,64561,64588,64615],{"className":64560,"ariaHidden":46},[45],[15,64562,64564,64567,64570,64573,64576,64579,64582,64585],{"className":64563},[50],[15,64565],{"className":64566,"style":5614},[54],[15,64568,1289],{"className":64569,"style":5619},[1486,1487,5618],[15,64571],{"className":64572,"style":406},[157],[15,64574,244],{"className":64575},[59,60],[15,64577,206],{"className":64578,"style":334},[59,60],[15,64580],{"className":64581,"style":158},[157],[15,64583,123],{"className":64584},[162],[15,64586],{"className":64587,"style":158},[157],[15,64589,64591,64594,64597,64600,64603,64606,64609,64612],{"className":64590},[50],[15,64592],{"className":64593,"style":5614},[54],[15,64595,1289],{"className":64596,"style":5619},[1486,1487,5618],[15,64598],{"className":64599,"style":406},[157],[15,64601,244],{"className":64602},[59,60],[15,64604,213],{"className":64605},[59,60],[15,64607],{"className":64608,"style":158},[157],[15,64610,123],{"className":64611},[162],[15,64613],{"className":64614,"style":158},[157],[15,64616,64618,64621],{"className":64617},[50],[15,64619],{"className":64620,"style":172},[54],[15,64622,117],{"className":64623},[59]," 可加任意常数 ",[15,64626,64628,64641],{"className":64627,"translate":19},[18],[15,64629,64631],{"className":64630},[23],[25,64632,64633],{"xmlns":27},[29,64634,64635,64639],{},[32,64636,64637],{},[35,64638,5908],{},[39,64640,5908],{"encoding":41},[15,64642,64644],{"className":64643,"ariaHidden":46},[45],[15,64645,64647,64650],{"className":64646},[50],[15,64648],{"className":64649,"style":2347},[54],[15,64651,5908],{"className":64652},[59,60],[52518,64654,64655,64856],{},[52521,64656,64657],{},[52524,64658,64659,64704,64780,64853],{},[52527,64660,64661],{},[15,64662,64664,64683],{"className":64663,"translate":19},[18],[15,64665,64667],{"className":64666},[23],[25,64668,64669],{"xmlns":27},[29,64670,64671,64681],{},[32,64672,64673,64675,64677,64679],{},[35,64674,37],{},[110,64676,113],{"stretchy":112},[35,64678,769],{},[110,64680,120],{"stretchy":112},[39,64682,52554],{"encoding":41},[15,64684,64686],{"className":64685,"ariaHidden":46},[45],[15,64687,64689,64692,64695,64698,64701],{"className":64688},[50],[15,64690],{"className":64691,"style":139},[54],[15,64693,37],{"className":64694,"style":61},[59,60],[15,64696,113],{"className":64697},[146],[15,64699,769],{"className":64700},[59,60],[15,64702,120],{"className":64703},[153],[52527,64705,64706],{},[15,64707,64709,64733],{"className":64708,"translate":19},[18],[15,64710,64712],{"className":64711},[23],[25,64713,64714],{"xmlns":27},[29,64715,64716,64730],{},[32,64717,64718,64724,64726,64728],{},[6816,64719,64720,64722],{},[35,64721,37],{},[110,64723,7575],{},[110,64725,113],{"stretchy":112},[35,64727,40263],{},[110,64729,120],{"stretchy":112},[39,64731,64732],{"encoding":41},"f^*(y)",[15,64734,64736],{"className":64735,"ariaHidden":46},[45],[15,64737,64739,64742,64771,64774,64777],{"className":64738},[50],[15,64740],{"className":64741,"style":139},[54],[15,64743,64745,64748],{"className":64744},[59],[15,64746,37],{"className":64747,"style":61},[59,60],[15,64749,64751],{"className":64750},[281],[15,64752,64754],{"className":64753},[285],[15,64755,64757],{"className":64756},[290],[15,64758,64760],{"className":64759,"style":8501},[294],[15,64761,64762,64765],{"style":6867},[15,64763],{"className":64764,"style":303},[302],[15,64766,64768],{"className":64767},[307,308,309,310],[15,64769,7575],{"className":64770},[2593,310],[15,64772,113],{"className":64773},[146],[15,64775,40263],{"className":64776,"style":981},[59,60],[15,64778,120],{"className":64779},[153],[52527,64781,64782],{},[15,64783,64785,64806],{"className":64784,"translate":19},[18],[15,64786,64788],{"className":64787},[23],[25,64789,64790],{"xmlns":27},[29,64791,64792,64804],{},[32,64793,64794,64796,64798],{},[35,64795,61444],{"mathvariant":209},[110,64797,4105],{},[6816,64799,64800,64802],{},[35,64801,37],{},[110,64803,7575],{},[39,64805,64107],{"encoding":41},[15,64807,64809],{"className":64808,"ariaHidden":46},[45],[15,64810,64812,64815,64821,64824],{"className":64811},[50],[15,64813],{"className":64814,"style":55},[54],[15,64816,64818],{"className":64817},[1486],[15,64819,61444],{"className":64820},[59,32688],[15,64822],{"className":64823,"style":406},[157],[15,64825,64827,64830],{"className":64826},[59],[15,64828,37],{"className":64829,"style":61},[59,60],[15,64831,64833],{"className":64832},[281],[15,64834,64836],{"className":64835},[285],[15,64837,64839],{"className":64838},[290],[15,64840,64842],{"className":64841,"style":8501},[294],[15,64843,64844,64847],{"style":6867},[15,64845],{"className":64846,"style":303},[302],[15,64848,64850],{"className":64849},[307,308,309,310],[15,64851,7575],{"className":64852},[2593,310],[52527,64854,64855],{},"对应 GAN 损失",[52616,64857,64858,65021,65262,65895,66199,66538],{},[52524,64859,64860,64911,64987,65018],{},[52621,64861,64862,64910],{},[15,64863,64865,64884],{"className":64864,"translate":19},[18],[15,64866,64868],{"className":64867},[23],[25,64869,64870],{"xmlns":27},[29,64871,64872,64882],{},[32,64873,64874,64876,64878,64880],{},[35,64875,769],{},[35,64877,4102],{},[110,64879,4105],{},[35,64881,769],{},[39,64883,52752],{"encoding":41},[15,64885,64887],{"className":64886,"ariaHidden":46},[45],[15,64888,64890,64893,64896,64899,64904,64907],{"className":64889},[50],[15,64891],{"className":64892,"style":55},[54],[15,64894,769],{"className":64895},[59,60],[15,64897],{"className":64898,"style":406},[157],[15,64900,4156,64902],{"className":64901},[1486],[15,64903,4160],{"style":4159},[15,64905],{"className":64906,"style":406},[157],[15,64908,769],{"className":64909},[59,60],"（KL）",[52621,64912,64913],{},[15,64914,64916,64940],{"className":64915,"translate":19},[18],[15,64917,64919],{"className":64918},[23],[25,64920,64921],{"xmlns":27},[29,64922,64923,64937],{},[32,64924,64925],{},[6816,64926,64927,64929],{},[35,64928,7569],{},[32,64930,64931,64933,64935],{},[35,64932,40263],{},[110,64934,4815],{},[115,64936,117],{},[39,64938,64939],{"encoding":41},"e^{y-1}",[15,64941,64943],{"className":64942,"ariaHidden":46},[45],[15,64944,64946,64949],{"className":64945},[50],[15,64947],{"className":64948,"style":20054},[54],[15,64950,64952,64955],{"className":64951},[59],[15,64953,7569],{"className":64954},[59,60],[15,64956,64958],{"className":64957},[281],[15,64959,64961],{"className":64960},[285],[15,64962,64964],{"className":64963},[290],[15,64965,64967],{"className":64966,"style":20054},[294],[15,64968,64969,64972],{"style":6867},[15,64970],{"className":64971,"style":303},[302],[15,64973,64975],{"className":64974},[307,308,309,310],[15,64976,64978,64981,64984],{"className":64977},[59,310],[15,64979,40263],{"className":64980,"style":981},[59,60,310],[15,64982,4815],{"className":64983},[2593,310],[15,64985,117],{"className":64986},[59,310],[52621,64988,64989],{},[15,64990,64992,65006],{"className":64991,"translate":19},[18],[15,64993,64995],{"className":64994},[23],[25,64996,64997],{"xmlns":27},[29,64998,64999,65003],{},[32,65000,65001],{},[35,65002,1035],{"mathvariant":222},[39,65004,65005],{"encoding":41},"\\mathbb R",[15,65007,65009],{"className":65008,"ariaHidden":46},[45],[15,65010,65012,65015],{"className":65011},[50],[15,65013],{"className":65014,"style":1099},[54],[15,65016,1035],{"className":65017},[59,366],[52621,65019,65020],{},"MINE-f \u002F 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assName":66408,"style":66409},[54],"height:1.415em;vertical-align:-0.345em;",[15,66411,66413,66416,66502],{"className":66412},[59],[15,66414],{"className":66415},[146,441],[15,66417,66419],{"className":66418},[238],[15,66420,66422,66494],{"className":66421},[285,286],[15,66423,66425,66491],{"className":66424},[290],[15,66426,66429,66443,66451],{"className":66427,"style":66428},[294],"height:1.07em;",[15,66430,66431,66434],{"style":2529},[15,66432],{"className":66433,"style":461},[302],[15,66435,66437],{"className":66436},[307,308,309,310],[15,66438,66440],{"className":66439},[59,310],[15,66441,39015],{"className":66442},[59,310],[15,66444,66445,66448],{"style":473},[15,66446],{"className":66447,"style":461},[302],[15,66449],{"className":66450,"style":481},[480],[15,66452,66453,66456],{"style":2855},[15,66454],{"className":66455,"style":461},[302],[15,66457,66459],{"className":66458},[307,308,309,310],[15,66460,66462],{"className":66461},[59,310],[15,66463,66465,66468],{"className":66464},[59,310],[15,66466,40263],{"className":66467,"style":981},[59,60,310],[15,66469,66471],{"className":66470},[281],[15,66472,66474],{"className":66473},[285],[15,66475,66477],{"className":66476},[290],[15,66478,66480],{"className":66479,"style":38538},[294],[15,66481,66482,66485],{"style":38541},[15,66483],{"className":66484,"style":30232},[302],[15,66486,66488],{"className":66487},[307,30236,9676,310],[15,66489,2465],{"className":66490},[59,310],[15,66492,318],{"className":66493},[317],[15,66495,66497],{"className":66496},[290],[15,66498,66500],{"className":66499,"style":2574},[294],[15,66501],{},[15,66503],{"className":66504},[153,441],[52621,66506,66507],{},[15,66508,66510,66523],{"className":66509,"translate":19},[18],[15,66511,66513],{"className":66512},[23],[25,66514,66515],{"xmlns":27},[29,66516,66517,66521],{},[32,66518,66519],{},[35,66520,1035],{"mathvariant":222},[39,66522,65005],{"encoding":41},[15,66524,66526],{"className":66525,"ariaHidden":46},[45],[15,66527,66529,66532],{"className":66528},[50],[15,66530],{"className":66531,"style":1099},[54],[15,66533,1035],{"className":66534},[59,366],[52621,66536,66537],{},"LSGAN",[52524,66539,66540,66676,66788,66841],{},[52621,66541,66542,66675],{},[15,66543,66545,66572],{"className":66544,"translate":19},[18],[15,66546,66548],{"className":66547},[23],[25,66549,66550],{"xmlns":27},[29,66551,66552,66570],{},[32,66553,66554,66556,66560,66562,66564],{},[110,66555,113],{"stretchy":112},[34365,66557,66558],{},[35,66559,769],{},[110,66561,4815],{},[115,66563,117],{},[6816,66565,66566,66568],{},[110,66567,120],{"stretchy":112},[115,66569,2465],{},[39,66571,54333],{"encoding":41},[15,66573,66575,66637],{"className":66574,"ariaHidden":46},[45],[15,66576,66578,66581,66584,66628,66631,66634],{"className":66577},[50],[15,66579],{"className":66580,"style":42693},[54],[15,66582,113],{"className":66583},[146],[15,66585,66587],{"className":66586},[59,34393],[15,66588,66590,66620],{"className":66589},[285,286],[15,66591,66593,66617],{"className":66592},[290],[15,66594,66596,66605],{"className":66595,"style":42709},[294],[15,66597,66599,66602],{"className":66598,"style":11867},[6309],[15,66600],{"className":66601,"style":461},[302],[15,66603,769],{"className":66604,"style":34413},[59,60],[15,66606,66607,66610],{"style":42721},[15,66608],{"className":66609,"style":461},[302],[15,66611,66613],{"className":66612,"style":34470},[34469],[6325,66614,66615],{"xmlns":6327,"width":6328,"height":34473,"viewBox":34474,"preserveAspectRatio":6331},[6333,66616],{"d":34477},[15,66618,318],{"className":66619},[317],[15,66621,66623],{"className":66622},[290],[15,66624,66626],{"className":66625,"style":42741},[294],[15,66627],{},[15,66629],{"className":66630,"style":2589},[157],[15,66632,4815],{"className":66633},[2593],[15,66635],{"className":66636,"style":2589},[157],[15,66638,66640,66643,66646],{"className":66639},[50],[15,66641],{"className":66642,"style":20029},[54],[15,66644,117],{"className":66645},[59],[15,66647,66649,66652],{"className":66648},[153],[15,66650,120],{"className":66651},[153],[15,66653,66655],{"className":66654},[281],[15,66656,66658],{"className":66657},[285],[15,66659,66661],{"className":66660},[290],[15,66662,66664],{"className":66663,"style":20054},[294],[15,66665,66666,66669],{"style":6867},[15,66667],{"className":66668,"style":303},[302],[15,66670,66672],{"className":66671},[307,308,309,310],[15,66673,2465],{"className":66674},[59,310],"（Hellinger²）",[52621,66677,66678],{},[15,66679,66681,66705],{"className":66680,"translate":19},[18],[15,66682,66684],{"className":66683},[23],[25,66685,66686],{"xmlns":27},[29,66687,66688,66702],{},[32,66689,66690],{},[238,66691,66692,66694],{},[35,66693,40263],{},[32,66695,66696,66698,66700],{},[115,66697,117],{},[110,66699,4815],{},[35,66701,40263],{},[39,66703,66704],{"encoding":41},"\\frac{y}{1-y}",[15,66706,66708],{"className":66707,"ariaHidden":46},[45],[15,66709,66711,66714],{"className":66710},[50],[15,66712],{"className":66713,"style":3086},[54],[15,66715,66717,66720,66785],{"className":66716},[59],[15,66718],{"className":66719},[146,441],[15,66721,66723],{"className":66722},[238],[15,66724,66726,66777],{"className":66725},[285,286],[15,66727,66729,66774],{"className":66728},[290],[15,66730,66732,66752,66760],{"className":66731,"style":3105},[294],[15,66733,66734,66737],{"style":2529},[15,66735],{"className":66736,"style":461},[302],[15,66738,66740],{"className":66739},[307,308,309,310],[15,66741,66743,66746,66749],{"className":66742},[59,310],[15,66744,117],{"className":66745},[59,310],[15,66747,4815],{"className":66748},[2593,310],[15,66750,40263],{"className":66751,"style":981},[59,60,310],[15,66753,66754,66757],{"style":473},[15,66755],{"className":66756,"style":461},[302],[15,66758],{"className":66759,"style":481},[480],[15,66761,66762,66765],{"style":2855},[15,66763],{"className":66764,"style":461},[302],[15,66766,66768],{"className":66767},[307,308,309,310],[15,66769,66771],{"className":66770},[59,310],[15,66772,40263],{"className":66773,"style":981},[59,60,310],[15,66775,318],{"className":66776},[317],[15,66778,66780],{"className":66779},[290],[15,66781,66783],{"className":66782,"style":2741},[294],[15,66784],{},[15,66786],{"className":66787},[153,441],[52621,66789,66790],{},[15,66791,66793,66811],{"className":66792,"translate":19},[18],[15,66794,66796],{"className":66795},[23],[25,66797,66798],{"xmlns":27},[29,66799,66800,66808],{},[32,66801,66802,66804,66806],{},[35,66803,40263],{},[110,66805,7253],{},[115,66807,117],{},[39,66809,66810],{"encoding":41},"y\u003C1",[15,66812,66814,66832],{"className":66813,"ariaHidden":46},[45],[15,66815,66817,66820,66823,66826,66829],{"className":66816},[50],[15,66818],{"className":66819,"style":31718},[54],[15,66821,40263],{"className":66822,"style":981},[59,60],[15,66824],{"className":66825,"style":158},[157],[15,66827,7253],{"className":66828},[162],[15,66830],{"className":66831,"style":158},[157],[15,66833,66835,66838],{"className":66834},[50],[15,66836],{"className":66837,"style":172},[54],[15,66839,117],{"className":66840},[59],[52621,66842,66198],{},[11,66844,66845,66846,66904,66905,67151,67152,67313,67314,67579,67580,20982,67740,67856],{},"以 ",[15,66847,66849,66866],{"className":66848,"translate":19},[18],[15,66850,66852],{"className":66851},[23],[25,66853,66854],{"xmlns":27},[29,66855,66856,66864],{},[32,66857,66858],{},[6816,66859,66860,66862],{},[35,66861,37475],{},[115,66863,2465],{},[39,66865,40420],{"encoding":41},[15,66867,66869],{"className":66868,"ariaHidden":46},[45],[15,66870,66872,66875],{"className":66871},[50],[15,66873],{"className":66874,"style":40430},[54],[15,66876,66878,66881],{"className":66877},[59],[15,66879,37475],{"className":66880},[59,60],[15,66882,66884],{"className":66883},[281],[15,66885,66887],{"className":66886},[285],[15,66888,66890],{"className":66889},[290],[15,66891,66893],{"className":66892,"style":20054},[294],[15,66894,66895,66898],{"style":6867},[15,66896],{"className":66897,"style":303},[302],[15,66899,66901],{"className":66900},[307,308,309,310],[15,66902,2465],{"className":66903},[59,310]," 为例验证：",[15,66906,66908,66968],{"className":66907,"translate":19},[18],[15,66909,66911],{"className":66910},[23],[25,66912,66913],{"xmlns":27},[29,66914,66915,66965],{},[32,66916,66917,66923,66925,66927,66929,66931,66941,66943,66945,66947,66949,66951,66953,66955,66957,66963],{},[6816,66918,66919,66921],{},[35,66920,37],{},[110,66922,7575],{},[110,66924,113],{"stretchy":112},[35,66926,40263],{},[110,66928,120],{"stretchy":112},[110,66930,123],{},[195,66932,66933,66939],{},[32,66934,66935,66937],{},[35,66936,35458],{},[110,66938,4105],{},[35,66940,769],{},[110,66942,2900],{"stretchy":112},[35,66944,769],{},[35,66946,40263],{},[110,66948,4815],{},[110,66950,113],{"stretchy":112},[35,66952,769],{},[110,66954,4815],{},[115,66956,117],{},[6816,66958,66959,66961],{},[110,66960,120],{"stretchy":112},[115,66962,2465],{},[110,66964,2910],{"stretchy":112},[39,66966,66967],{"encoding":41},"f^*(y)=\\sup_t\\{ty-(t-1)^2\\}",[15,66969,66971,67024,67089,67110],{"className":66970,"ariaHidden":46},[45],[15,66972,66974,66977,67006,67009,67012,67015,67018,67021],{"className":66973},[50],[15,66975],{"className":66976,"style":139},[54],[15,66978,66980,66983],{"className":66979},[59],[15,66981,37],{"className":66982,"style":61},[59,60],[15,66984,66986],{"className":66985},[281],[15,66987,66989],{"className":66988},[285],[15,66990,66992],{"className":66991},[290],[15,66993,66995],{"className":66994,"style":8501},[294],[15,66996,66997,67000],{"style":6867},[15,66998],{"className":66999,"style":303},[302],[15,67001,67003],{"className":67002},[307,308,309,310],[15,67004,7575],{"className":67005},[2593,310],[15,67007,113],{"className":67008},[146],[15,67010,40263],{"className":67011,"style":981},[59,60],[15,67013,120],{"className":67014},[153],[15,67016],{"className":67017,"style":158},[157],[15,67019,123],{"className":67020},[162],[15,67022],{"className":67023,"style":158},[157],[15,67025,67027,67030,67071,67074,67077,67080,67083,67086],{"className":67026},[50],[15,67028],{"className":67029,"style":139},[54],[15,67031,67033,67036],{"className":67032},[1486],[15,67034,35458],{"className":67035},[1486],[15,67037,67039],{"className":67038},[281],[15,67040,67042,67063],{"className":67041},[285,286],[15,67043,67045,67060],{"className":67044},[290],[15,67046,67049],{"className":67047,"style":67048},[294],"height:0.1864em;",[15,67050,67051,67054],{"style":35604},[15,67052],{"className":67053,"style":303},[302],[15,67055,67057],{"className":67056},[307,308,309,310],[15,67058,769],{"className":67059},[59,60,310],[15,67061,318],{"className":67062},[317],[15,67064,67066],{"className":67065},[290],[15,67067,67069],{"className":67068,"style":37046},[294],[15,67070],{},[15,67072,2900],{"className":67073},[146],[15,67075,769],{"className":67076},[59,60],[15,67078,40263],{"className":67079,"style":981},[59,60],[15,67081],{"className":67082,"style":2589},[157],[15,67084,4815],{"className":67085},[2593],[15,67087],{"className":67088,"style":2589},[157],[15,67090,67092,67095,67098,67101,67104,67107],{"className":67091},[50],[15,67093],{"className":67094,"style":139},[54],[15,67096,113],{"className":67097},[146],[15,67099,769],{"className":67100},[59,60],[15,67102],{"className":67103,"style":2589},[157],[15,67105,4815],{"className":67106},[2593],[15,67108],{"className":67109,"style":2589},[157],[15,67111,67113,67116,67119,67148],{"className":67112},[50],[15,67114],{"className":67115,"style":20029},[54],[15,67117,117],{"className":67118},[59],[15,67120,67122,67125],{"className":67121},[153],[15,67123,120],{"className":67124},[153],[15,67126,67128],{"className":67127},[281],[15,67129,67131],{"className":67130},[285],[15,67132,67134],{"className":67133},[290],[15,67135,67137],{"className":67136,"style":20054},[294],[15,67138,67139,67142],{"style":6867},[15,67140],{"className":67141,"style":303},[302],[15,67143,67145],{"className":67144},[307,308,309,310],[15,67146,2465],{"className":67147},[59,310],[15,67149,2910],{"className":67150},[153],"，驻点 ",[15,67153,67155,67199],{"className":67154,"translate":19},[18],[15,67156,67158],{"className":67157},[23],[25,67159,67160],{"xmlns":27},[29,67161,67162,67196],{},[32,67163,67164,67166,67168,67170,67172,67174,67176,67178,67180,67182,67184,67186,67188,67190,67192,67194],{},[35,67165,40263],{},[110,67167,123],{},[115,67169,2465],{},[110,67171,113],{"stretchy":112},[35,67173,769],{},[110,67175,4815],{},[115,67177,117],{},[110,67179,120],{"stretchy":112},[110,67181,10994],{},[35,67183,769],{},[110,67185,123],{},[115,67187,117],{},[110,67189,2472],{},[35,67191,40263],{},[35,67193,941],{"mathvariant":209},[115,67195,2465],{},[39,67197,67198],{"encoding":41},"y=2(t-1)\\Rightarrow t=1+y\u002F2",[15,67200,67202,67220,67244,67265,67283,67301],{"className":67201,"ariaHidden":46},[45],[15,67203,67205,67208,67211,67214,67217],{"className":67204},[50],[15,67206],{"className":67207,"style":2278},[54],[15,67209,40263],{"className":67210,"style":981},[59,60],[15,67212],{"className":67213,"style":158},[157],[15,67215,123],{"className":67216},[162],[15,67218],{"className":67219,"style":158},[157],[15,67221,67223,67226,67229,67232,67235,67238,67241],{"className":67222},[50],[15,67224],{"className":67225,"style":139},[54],[15,67227,2465],{"className":67228},[59],[15,67230,113],{"className":67231},[146],[15,67233,769],{"className":67234},[59,60],[15,67236],{"className":67237,"style":2589},[157],[15,67239,4815],{"className":67240},[2593],[15,67242],{"className":67243,"style":2589},[157],[15,67245,67247,67250,67253,67256,67259,67262],{"className":67246},[50],[15,67248],{"className":67249,"style":139},[54],[15,67251,117],{"className":67252},[59],[15,67254,120],{"className":67255},[153],[15,67257],{"className":67258,"style":158},[157],[15,67260,10994],{"className":67261},[162],[15,67263],{"className":67264,"style":158},[157],[15,67266,67268,67271,67274,67277,67280],{"className":67267},[50],[15,67269],{"className":67270,"style":5345},[54],[15,67272,769],{"className":67273},[59,60],[15,67275],{"className":67276,"style":158},[157],[15,67278,123],{"className":67279},[162],[15,67281],{"className":67282,"style":158},[157],[15,67284,67286,67289,67292,67295,67298],{"className":67285},[50],[15,67287],{"className":67288,"style":12582},[54],[15,67290,117],{"className":67291},[59],[15,67293],{"className":67294,"style":2589},[157],[15,67296,2472],{"className":67297},[2593],[15,67299],{"className":67300,"style":2589},[157],[15,67302,67304,67307,67310],{"className":67303},[50],[15,67305],{"className":67306,"style":139},[54],[15,67308,40263],{"className":67309,"style":981},[59,60],[15,67311,45176],{"className":67312},[59],"，得 ",[15,67315,67317,67381],{"className":67316,"translate":19},[18],[15,67318,67320],{"className":67319},[23],[25,67321,67322],{"xmlns":27},[29,67323,67324,67378],{},[32,67325,67326,67332,67334,67336,67338,67340,67342,67344,67346,67348,67350,67352,67358,67360,67362,67364,67366,67368,67374,67376],{},[6816,67327,67328,67330],{},[35,67329,37],{},[110,67331,7575],{},[110,67333,123],{},[35,67335,40263],{},[110,67337,113],{"stretchy":112},[115,67339,117],{},[110,67341,2472],{},[35,67343,40263],{},[35,67345,941],{"mathvariant":209},[115,67347,2465],{},[110,67349,120],{"stretchy":112},[110,67351,4815],{},[6816,67353,67354,67356],{},[35,67355,40263],{},[115,67357,2465],{},[35,67359,941],{"mathvariant":209},[115,67361,39015],{},[110,67363,123],{},[35,67365,40263],{},[110,67367,2472],{},[6816,67369,67370,67372],{},[35,67371,40263],{},[115,67373,2465],{},[35,67375,941],{"mathvariant":209},[115,67377,39015],{},[39,67379,67380],{"encoding":41},"f^*=y(1+y\u002F2)-y^2\u002F4=y+y^2\u002F4",[15,67382,67384,67428,67452,67476,67523,67541],{"className":67383,"ariaHidden":46},[45],[15,67385,67387,67390,67419,67422,67425],{"className":67386},[50],[15,67388],{"className":67389,"style":55},[54],[15,67391,67393,67396],{"className":67392},[59],[15,67394,37],{"className":67395,"style":61},[59,60],[15,67397,67399],{"className":67398},[281],[15,67400,67402],{"className":67401},[285],[15,67403,67405],{"className":67404},[290],[15,67406,67408],{"className":67407,"style":8501},[294],[15,67409,67410,67413],{"style":6867},[15,67411],{"className":67412,"style":303},[302],[15,67414,67416],{"className":67415},[307,308,309,310],[15,67417,7575],{"className":67418},[2593,310],[15,67420],{"className":67421,"style":158},[157],[15,67423,123],{"className":67424},[162],[15,67426],{"className":67427,"style":158},[157],[15,67429,67431,67434,67437,67440,67443,67446,67449],{"className":67430},[50],[15,67432],{"className":67433,"style":139},[54],[15,67435,40263],{"className":67436,"style":981},[59,60],[15,67438,113],{"className":67439},[146],[15,67441,117],{"className":67442},[59],[15,67444],{"className":67445,"style":2589},[157],[15,67447,2472],{"className":67448},[2593],[15,67450],{"className":67451,"style":2589},[157],[15,67453,67455,67458,67461,67464,67467,67470,67473],{"className":67454},[50],[15,67456],{"className":67457,"style":139},[54],[15,67459,40263],{"className":67460,"style":981},[59,60],[15,67462,45176],{"className":67463},[59],[15,67465,120],{"className":67466},[153],[15,67468],{"className":67469,"style":2589},[157],[15,67471,4815],{"className":67472},[2593],[15,67474],{"className":67475,"style":2589},[157],[15,67477,67479,67482,67511,67514,67517,67520],{"className":67478},[50],[15,67480],{"className":67481,"style":20029},[54],[15,67483,67485,67488],{"className":67484},[59],[15,67486,40263],{"className":67487,"style":981},[59,60],[15,67489,67491],{"className":67490},[281],[15,67492,67494],{"className":67493},[285],[15,67495,67497],{"className":67496},[290],[15,67498,67500],{"className":67499,"style":20054},[294],[15,67501,67502,67505],{"style":6867},[15,67503],{"className":67504,"style":303},[302],[15,67506,67508],{"className":67507},[307,308,309,310],[15,67509,2465],{"className":67510},[59,310],[15,67512,40398],{"className":67513},[59],[15,67515],{"className":67516,"style":158},[157],[15,67518,123],{"className":67519},[162],[15,67521],{"className":67522,"style":158},[157],[15,67524,67526,67529,67532,67535,67538],{"className":67525},[50],[15,67527],{"className":67528,"style":40347},[54],[15,67530,40263],{"className":67531,"style":981},[59,60],[15,67533],{"className":67534,"style":2589},[157],[15,67536,2472],{"className":67537},[2593],[15,67539],{"className":67540,"style":2589},[157],[15,67542,67544,67547,67576],{"className":67543},[50],[15,67545],{"className":67546,"style":20029},[54],[15,67548,67550,67553],{"className":67549},[59],[15,67551,40263],{"className":67552,"style":981},[59,60],[15,67554,67556],{"className":67555},[281],[15,67557,67559],{"className":67558},[285],[15,67560,67562],{"className":67561},[290],[15,67563,67565],{"className":67564,"style":20054},[294],[15,67566,67567,67570],{"style":6867},[15,67568],{"className":67569,"style":303},[302],[15,67571,67573],{"className":67572},[307,308,309,310],[15,67574,2465],{"className":67575},[59,310],[15,67577,40398],{"className":67578},[59],"。以 Hellinger² 为例：",[15,67581,67583,67619],{"className":67582,"translate":19},[18],[15,67584,67586],{"className":67585},[23],[25,67587,67588],{"xmlns":27},[29,67589,67590,67616],{},[32,67591,67592,67594,67596,67598,67600,67602,67604,67606,67608,67612,67614],{},[35,67593,37],{},[110,67595,113],{"stretchy":112},[35,67597,769],{},[110,67599,120],{"stretchy":112},[110,67601,123],{},[35,67603,769],{},[110,67605,4815],{},[115,67607,2465],{},[34365,67609,67610],{},[35,67611,769],{},[110,67613,2472],{},[115,67615,117],{},[39,67617,67618],{"encoding":41},"f(t)=t-2\\sqrt t+1",[15,67620,67622,67649,67668,67731],{"className":67621,"ariaHidden":46},[45],[15,67623,67625,67628,67631,67634,67637,67640,67643,67646],{"className":67624},[50],[15,67626],{"className":67627,"style":139},[54],[15,67629,37],{"className":67630,"style":61},[59,60],[15,67632,113],{"className":67633},[146],[15,67635,769],{"className":67636},[59,60],[15,67638,120],{"className":67639},[153],[15,67641],{"className":67642,"style":158},[157],[15,67644,123],{"className":67645},[162],[15,67647],{"className":67648,"style":158},[157],[15,67650,67652,67656,67659,67662,67665],{"className":67651},[50],[15,67653],{"className":67654,"style":67655},[54],"height:0.6984em;vertical-align:-0.0833em;",[15,67657,769],{"className":67658},[59,60],[15,67660],{"className":67661,"style":2589},[157],[15,67663,4815],{"className":67664},[2593],[15,67666],{"className":67667,"style":2589},[157],[15,67669,67671,67675,67678,67722,67725,67728],{"className":67670},[50],[15,67672],{"className":67673,"style":67674},[54],"height:1.04em;vertical-align:-0.1475em;",[15,67676,2465],{"className":67677},[59],[15,67679,67681],{"className":67680},[59,34393],[15,67682,67684,67714],{"className":67683},[285,286],[15,67685,67687,67711],{"className":67686},[290],[15,67688,67690,67699],{"className":67689,"style":42709},[294],[15,67691,67693,67696],{"className":67692,"style":11867},[6309],[15,67694],{"className":67695,"style":461},[302],[15,67697,769],{"className":67698,"style":34413},[59,60],[15,67700,67701,67704],{"style":42721},[15,67702],{"className":67703,"style":461},[302],[15,67705,67707],{"className":67706,"style":34470},[34469],[6325,67708,67709],{"xmlns":6327,"width":6328,"height":34473,"viewBox":34474,"preserveAspectRatio":6331},[6333,67710],{"d":34477},[15,67712,318],{"className":67713},[317],[15,67715,67717],{"className":67716},[290],[15,67718,67720],{"className":67719,"style":42741},[294],[15,67721],{},[15,67723],{"className":67724,"style":2589},[157],[15,67726,2472],{"className":67727},[2593],[15,67729],{"className":67730,"style":2589},[157],[15,67732,67734,67737],{"className":67733},[50],[15,67735],{"className":67736,"style":172},[54],[15,67738,117],{"className":67739},[59],[15,67741,67743,67767],{"className":67742,"translate":19},[18],[15,67744,67746],{"className":67745},[23],[25,67747,67748],{"xmlns":27},[29,67749,67750,67764],{},[32,67751,67752,67754,67756,67760,67762],{},[35,67753,15404],{},[110,67755,123],{},[34365,67757,67758],{},[35,67759,769],{},[110,67761,6930],{},[115,67763,126],{},[39,67765,67766],{"encoding":41},"u=\\sqrt t\\ge0",[15,67768,67770,67788,67847],{"className":67769,"ariaHidden":46},[45],[15,67771,67773,67776,67779,67782,67785],{"className":67772},[50],[15,67774],{"className":67775,"style":2347},[54],[15,67777,15404],{"className":67778},[59,60],[15,67780],{"className":67781,"style":158},[157],[15,67783,123],{"className":67784},[162],[15,67786],{"className":67787,"style":158},[157],[15,67789,67791,67794,67838,67841,67844],{"className":67790},[50],[15,67792],{"className":67793,"style":67674},[54],[15,67795,67797],{"className":67796},[59,34393],[15,67798,67800,67830],{"className":67799},[285,286],[15,67801,67803,67827],{"className":67802},[290],[15,67804,67806,67815],{"className":67805,"style":42709},[294],[15,67807,67809,67812],{"className":67808,"style":11867},[6309],[15,67810],{"className":67811,"style":461},[302],[15,67813,769],{"className":67814,"style":34413},[59,60],[15,67816,67817,67820],{"style":42721},[15,67818],{"className":67819,"style":461},[302],[15,67821,67823],{"className":67822,"style":34470},[34469],[6325,67824,67825],{"xmlns":6327,"width":6328,"height":34473,"viewBox":34474,"preserveAspectRatio":6331},[6333,67826],{"d":34477},[15,67828,318],{"className":67829},[317],[15,67831,67833],{"className":67832},[290],[15,67834,67836],{"className":67835,"style":42741},[294],[15,67837],{},[15,67839],{"className":67840,"style":158},[157],[15,67842,6930],{"className":67843},[162],[15,67845],{"className":67846,"style":158},[157],[15,67848,67850,67853],{"className":67849},[50],[15,67851],{"className":67852,"style":172},[54],[15,67854,126],{"className":67855},[59],"，把共轭写成有限上确界",[15,67858,67860],{"className":67859,"translate":19},[180],[15,67861,67863,67937],{"className":67862,"translate":19},[18],[15,67864,67866],{"className":67865},[23],[25,67867,67868],{"xmlns":27,"display":189},[29,67869,67870,67934],{},[32,67871,67872,67878,67880,67882,67884,67886,67902,67904,67906,67908,67910,67912,67914,67920,67922,67924,67926,67928,67930,67932],{},[6816,67873,67874,67876],{},[35,67875,37],{},[110,67877,7575],{},[110,67879,113],{"stretchy":112},[35,67881,40263],{},[110,67883,120],{"stretchy":112},[110,67885,123],{},[6018,67887,67888,67894],{},[32,67889,67890,67892],{},[35,67891,35458],{},[110,67893,4105],{},[32,67895,67896,67898,67900],{},[35,67897,15404],{},[110,67899,6930],{},[115,67901,126],{},[110,67903,2900],{"fence":112,"stretchy":46,"minsize":17443,"maxsize":17443},[110,67905,113],{"stretchy":112},[35,67907,40263],{},[110,67909,4815],{},[115,67911,117],{},[110,67913,120],{"stretchy":112},[6816,67915,67916,67918],{},[35,67917,15404],{},[115,67919,2465],{},[110,67921,2472],{},[115,67923,2465],{},[35,67925,15404],{},[110,67927,4815],{},[115,67929,117],{},[110,67931,2910],{"fence":112,"stretchy":46,"minsize":17443,"maxsize":17443},[110,67933,551],{"separator":46},[39,67935,67936],{"encoding":41},"f^*(y)=\\sup_{u\\ge0}\\Big\\{(y-1)u^2+2u-1\\Big\\},",[15,67938,67940,67993,68077,68127,68148],{"className":67939,"ariaHidden":46},[45],[15,67941,67943,67946,67975,67978,67981,67984,67987,67990],{"className":67942},[50],[15,67944],{"className":67945,"style":139},[54],[15,67947,67949,67952],{"className":67948},[59],[15,67950,37],{"className":67951,"style":61},[59,60],[15,67953,67955],{"className":67954},[281],[15,67956,67958],{"className":67957},[285],[15,67959,67961],{"className":67960},[290],[15,67962,67964],{"className":67963,"style":8250},[294],[15,67965,67966,67969],{"style":8279},[15,67967],{"className":67968,"style":303},[302],[15,67970,67972],{"className":67971},[307,308,309,310],[15,67973,7575],{"className":67974},[2593,310],[15,67976,113],{"className":67977},[146],[15,67979,40263],{"className":67980,"style":981},[59,60],[15,67982,120],{"className":67983},[153],[15,67985],{"className":67986,"style":158},[157],[15,67988,123],{"className":67989},[162],[15,67991],{"className":67992,"style":158},[157],[15,67994,67996,68000,68053,68056,68062,68065,68068,68071,68074],{"className":67995},[50],[15,67997],{"className":67998,"style":67999},[54],"height:2.1567em;vertical-align:-1.0067em;",[15,68001,68003],{"className":68002},[1486,13720],[15,68004,68006,68045],{"className":68005},[285,286],[15,68007,68009,68042],{"className":68008},[290],[15,68010,68012,68032],{"className":68011,"style":2347},[294],[15,68013,68014,68017],{"style":60959},[15,68015],{"className":68016,"style":461},[302],[15,68018,68020],{"className":68019},[307,308,309,310],[15,68021,68023,68026,68029],{"className":68022},[59,310],[15,68024,15404],{"className":68025},[59,60,310],[15,68027,6930],{"className":68028},[162,310],[15,68030,126],{"className":68031},[59,310],[15,68033,68034,68037],{"style":11867},[15,68035],{"className":68036,"style":461},[302],[15,68038,68039],{},[15,68040,35458],{"className":68041},[1486],[15,68043,318],{"className":68044},[317],[15,68046,68048],{"className":68047},[290],[15,68049,68051],{"className":68050,"style":60997},[294],[15,68052],{},[15,68054],{"className":68055,"style":406},[157],[15,68057,68059],{"className":68058},[59],[15,68060,2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",[15,68171,68173,68191],{"className":68172,"translate":19},[18],[15,68174,68176],{"className":68175},[23],[25,68177,68178],{"xmlns":27},[29,68179,68180,68188],{},[32,68181,68182,68184,68186],{},[35,68183,40263],{},[110,68185,6930],{},[115,68187,117],{},[39,68189,68190],{"encoding":41},"y\\ge1",[15,68192,68194,68212],{"className":68193,"ariaHidden":46},[45],[15,68195,68197,68200,68203,68206,68209],{"className":68196},[50],[15,68198],{"className":68199,"style":15421},[54],[15,68201,40263],{"className":68202,"style":981},[59,60],[15,68204],{"className":68205,"style":158},[157],[15,68207,6930],{"className":68208},[162],[15,68210],{"className":68211,"style":158},[157],[15,68213,68215,68218],{"className":68214},[50],[15,68216],{"className":68217,"style":172},[54],[15,68219,117],{"className":68220},[59]," 时上确界为 ",[15,68223,68225,68240],{"className":68224,"translate":19},[18],[15,68226,68228],{"className":68227},[23],[25,68229,68230],{"xmlns":27},[29,68231,68232,68238],{},[32,68233,68234,68236],{},[110,68235,2472],{},[35,68237,1027],{"mathvariant":209},[39,68239,4243],{"encoding":41},[15,68241,68243],{"className":68242,"ariaHidden":46},[45],[15,68244,68246,68249,68252],{"className":68245},[50],[15,68247],{"className":68248,"style":4253},[54],[15,68250,2472],{"className":68251},[59],[15,68253,1027],{"className":68254},[59],"；当 ",[15,68257,68259,68276],{"className":68258,"translate":19},[18],[15,68260,68262],{"className":68261},[23],[25,68263,68264],{"xmlns":27},[29,68265,68266,68274],{},[32,68267,68268,68270,68272],{},[35,68269,40263],{},[110,68271,7253],{},[115,68273,117],{},[39,68275,66810],{"encoding":41},[15,68277,68279,68297],{"className":68278,"ariaHidden":46},[45],[15,68280,68282,68285,68288,68291,68294],{"className":68281},[50],[15,68283],{"className":68284,"style":31718},[54],[15,68286,40263],{"className":68287,"style":981},[59,60],[15,68289],{"className":68290,"style":158},[157],[15,68292,7253],{"className":68293},[162],[15,68295],{"className":68296,"style":158},[157],[15,68298,68300,68303],{"className":68299},[50],[15,68301],{"className":68302,"style":172},[54],[15,68304,117],{"className":68305},[59]," 时上式是开口向下的二次函数，最大点 ",[15,68308,68310,68340],{"className":68309,"translate":19},[18],[15,68311,68313],{"className":68312},[23],[25,68314,68315],{"xmlns":27},[29,68316,68317,68337],{},[32,68318,68319,68321,68323],{},[35,68320,15404],{},[110,68322,123],{},[1257,68324,68325],{"scriptlevel":126,"displaystyle":46},[238,68326,68327,68329],{},[115,68328,117],{},[32,68330,68331,68333,68335],{},[115,68332,117],{},[110,68334,4815],{},[35,68336,40263],{},[39,68338,68339],{"encoding":41},"u=\\dfrac{1}{1-y}",[15,68341,68343,68361],{"className":68342,"ariaHidden":46},[45],[15,68344,68346,68349,68352,68355,68358],{"className":68345},[50],[15,68347],{"className":68348,"style":2347},[54],[15,68350,15404],{"className":68351},[59,60],[15,68353],{"className":68354,"style":158},[157],[15,68356,123],{"className":68357},[162],[15,68359],{"className":68360,"style":158},[157],[15,68362,68364,68367],{"className":68363},[50],[15,68365],{"className":68366,"style":31329},[54],[15,68368,68370,68373,68438],{"className":68369},[59],[15,68371],{"className":68372},[146,441],[15,68374,68376],{"className":68375},[238],[15,68377,68379,68430],{"className":68378},[285,286],[15,68380,68382,68427],{"className":68381},[290],[15,68383,68385,68408,68416],{"className":68384,"style":22915},[294],[15,68386,68387,68390],{"style":457},[15,68388],{"className":68389,"style":461},[302],[15,68391,68393,68396,68399,68402,68405],{"className":68392},[59],[15,68394,117],{"className":68395},[59],[15,68397],{"className":68398,"style":2589},[157],[15,68400,4815],{"className":68401},[2593],[15,68403],{"className":68404,"style":2589},[157],[15,68406,40263],{"className":68407,"style":981},[59,60],[15,68409,68410,68413],{"style":473},[15,68411],{"className":68412,"style":461},[302],[15,68414],{"className":68415,"style":481},[480],[15,68417,68418,68421],{"style":484},[15,68419],{"className":68420,"style":461},[302],[15,68422,68424],{"className":68423},[59],[15,68425,117],{"className":68426},[59],[15,68428,318],{"className":68429},[317],[15,68431,68433],{"className":68432},[290],[15,68434,68436],{"className":68435,"style":506},[294],[15,68437],{},[15,68439],{"className":68440},[153,441],"，代回得",[15,68443,68445],{"className":68444,"translate":19},[180],[15,68446,68448,68544],{"className":68447,"translate":19},[18],[15,68449,68451],{"className":68450},[23],[25,68452,68453],{"xmlns":27,"display":189},[29,68454,68455,68541],{},[32,68456,68457,68463,68465,68467,68469,68471,68497,68499,68501,68513,68515,68517,68519,68531,68533,68535,68537,68539],{},[6816,68458,68459,68461],{},[35,68460,37],{},[110,68462,7575],{},[110,68464,113],{"stretchy":112},[35,68466,40263],{},[110,68468,120],{"stretchy":112},[110,68470,123],{},[238,68472,68473,68481],{},[32,68474,68475,68477,68479],{},[35,68476,40263],{},[110,68478,4815],{},[115,68480,117],{},[32,68482,68483,68485,68487,68489,68491],{},[110,68484,113],{"stretchy":112},[115,68486,117],{},[110,68488,4815],{},[35,68490,40263],{},[6816,68492,68493,68495],{},[110,68494,120],{"stretchy":112},[115,68496,2465],{},[110,68498,2472],{},[115,68500,2465],{},[238,68502,68503,68505],{},[115,68504,117],{},[32,68506,68507,68509,68511],{},[115,68508,117],{},[110,68510,4815],{},[35,68512,40263],{},[110,68514,4815],{},[115,68516,117],{},[110,68518,123],{},[238,68520,68521,68523],{},[35,68522,40263],{},[32,68524,68525,68527,68529],{},[115,68526,117],{},[110,68528,4815],{},[35,68530,40263],{},[110,68532,551],{"separator":46},[157,68534],{"width":3264},[35,68536,40263],{},[110,68538,7253],{},[115,68540,49536],{},[39,68542,68543],{"encoding":41},"f^*(y)=\\frac{y-1}{(1-y)^2}+2\\frac{1}{1-y}-1\n=\\frac{y}{1-y},\\qquad y\u003C1 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ssName":69639,"style":16433},[294],[15,69641],{},[15,69643],{"className":69644},[153,441],"。令 ",[15,69647,69649,69683],{"className":69648,"translate":19},[18],[15,69650,69652],{"className":69651},[23],[25,69653,69654],{"xmlns":27},[29,69655,69656,69680],{},[32,69657,69658,69660,69662,69674,69676,69678],{},[35,69659,40263],{},[110,69661,123],{},[7555,69663,69664,69666,69672],{},[35,69665,37],{},[32,69667,69668,69670],{},[35,69669,23769],{},[35,69671,25763],{},[110,69673,6822],{"mathvariant":209,"lspace":1249,"rspace":1249},[110,69675,113],{"stretchy":112},[35,69677,769],{},[110,69679,120],{"stretchy":112},[39,69681,69682],{"encoding":41},"y=f_{JS}'(t)",[15,69684,69686,69704],{"className":69685,"ariaHidden":46},[45],[15,69687,69689,69692,69695,69698,69701],{"className":69688},[50],[15,69690],{"className":69691,"style":2278},[54],[15,69693,40263],{"className":69694,"style":981},[59,60],[15,69696],{"className":69697,"style":158},[157],[15,69699,123],{"className":69700},[162],[15,69702],{"className":69703,"style":158},[157],[15,69705,69707,69710,69770,69773,69776],{"className":69706},[50],[15,69708],{"className":69709,"style":69403},[54],[15,69711,69713,69716],{"className":69712},[59],[15,69714,37],{"className":69715,"style":61},[59,60],[15,69717,69719],{"className":69718},[281],[15,69720,69722,69762],{"className":69721},[285,286],[15,69723,69725,69759],{"className":69724},[290],[15,69726,69728,69745],{"className":69727,"style":6864},[294],[15,69729,69730,69733],{"style":69424},[15,69731],{"className":69732,"style":303},[302],[15,69734,69736],{"className":69735},[307,308,309,310],[15,69737,69739,69742],{"className":69738},[59,310],[15,69740,23769],{"className":69741,"style":24179},[59,60,310],[15,69743,25763],{"className":69744,"style":25887},[59,60,310],[15,69746,69747,69750],{"style":6867},[15,69748],{"className":69749,"style":303},[302],[15,69751,69753],{"className":69752},[307,308,309,310],[15,69754,69756],{"className":69755},[59,310],[15,69757,6822],{"className":69758},[59,310],[15,69760,318],{"className":69761},[317],[15,69763,69765],{"className":69764},[290],[15,69766,69768],{"className":69767,"style":38416},[294],[15,69769],{},[15,69771,113],{"className":69772},[146],[15,69774,769],{"className":69775},[59,60],[15,69777,120],{"className":69778},[153],[15,69780,69782,69822],{"className":69781,"translate":19},[18],[15,69783,69785],{"className":69784},[23],[25,69786,69787],{"xmlns":27},[29,69788,69789,69819],{},[32,69790,69791,69801,69803],{},[6816,69792,69793,69795],{},[35,69794,7569],{},[32,69796,69797,69799],{},[115,69798,2465],{},[35,69800,40263],{},[110,69802,123],{},[238,69804,69805,69811],{},[32,69806,69807,69809],{},[115,69808,2465],{},[35,69810,769],{},[32,69812,69813,69815,69817],{},[35,69814,769],{},[110,69816,2472],{},[115,69818,117],{},[39,69820,69821],{"encoding":41},"e^{2y}=\\frac{2t}{t+1}",[15,69823,69825,69875],{"className":69824,"ariaHidden":46},[45],[15,69826,69828,69831,69866,69869,69872],{"className":69827},[50],[15,69829],{"className":69830,"style":20054},[54],[15,69832,69834,69837],{"className":69833},[59],[15,69835,7569],{"className":69836},[59,60],[15,69838,69840],{"className":69839},[281],[15,69841,69843],{"className":69842},[285],[15,69844,69846],{"className":69845},[290],[15,69847,69849],{"className":69848,"style":20054},[294],[15,69850,69851,69854],{"style":6867},[15,69852],{"className":69853,"style":303},[302],[15,69855,69857],{"className":69856},[307,308,309,310],[15,69858,69860,69863],{"className":69859},[59,310],[15,69861,2465],{"className":69862},[59,310],[15,69864,40263],{"className":69865,"style":981},[59,60,310],[15,69867],{"className":69868,"style":158},[157],[15,69870,123],{"className":69871},[162],[15,69873],{"className":69874,"style":158},[157],[15,69876,69878,69881],{"className":69877},[50],[15,69879],{"className":69880,"style":45431},[54],[15,69882,69884,69887,69955],{"className":69883},[59],[15,69885],{"className":69886},[146,441],[15,69888,69890],{"className":69889},[238],[15,69891,69893,69947],{"className":69892},[285,286],[15,69894,69896,69944],{"className":69895},[290],[15,69897,69899,69919,69927],{"className":69898,"style":2526},[294],[15,69900,69901,69904],{"style":2529},[15,69902],{"className":69903,"style":461},[302],[15,69905,69907],{"className":69906},[307,308,309,310],[15,69908,69910,69913,69916],{"className":69909},[59,310],[15,69911,769],{"className":69912},[59,60,310],[15,69914,2472],{"className":69915},[2593,310],[15,69917,117],{"className":69918},[59,310],[15,69920,69921,69924],{"style":473},[15,69922],{"className":69923,"style":461},[302],[15,69925],{"className":69926,"style":481},[480],[15,69928,69929,69932],{"style":2552},[15,69930],{"className":69931,"style":461},[302],[15,69933,69935],{"className":69934},[307,308,309,310],[15,69936,69938,69941],{"className":69937},[59,310],[15,69939,2465],{"className":69940},[59,310],[15,69942,769],{"className":69943},[59,60,310],[15,69945,318],{"className":69946},[317],[15,69948,69950],{"className":69949},[290],[15,69951,69953],{"className":69952,"style":16433},[294],[15,69954],{},[15,69956],{"className":69957},[153,441],"，解得 ",[15,69960,69962,70008],{"className":69961,"translate":19},[18],[15,69963,69965],{"className":69964},[23],[25,69966,69967],{"xmlns":27},[29,69968,69969,70005],{},[32,69970,69971,69973,69975],{},[35,69972,769],{},[110,69974,123],{},[1257,69976,69977],{"scriptlevel":126,"displaystyle":46},[238,69978,69979,69989],{},[6816,69980,69981,69983],{},[35,69982,7569],{},[32,69984,69985,69987],{},[115,69986,2465],{},[35,69988,40263],{},[32,69990,69991,69993,69995],{},[115,69992,2465],{},[110,69994,4815],{},[6816,69996,69997,69999],{},[35,69998,7569],{},[32,70000,70001,70003],{},[115,70002,2465],{},[35,70004,40263],{},[39,70006,70007],{"encoding":41},"t=\\dfrac{e^{2y}}{2-e^{2y}}",[15,70009,70011,70029],{"className":70010,"ariaHidden":46},[45],[15,70012,70014,70017,70020,70023,70026],{"className":70013},[50],[15,70015],{"className":70016,"style":5345},[54],[15,70018,769],{"className":70019},[59,60],[15,70021],{"className":70022,"style":158},[157],[15,70024,123],{"className":70025},[162],[15,70027],{"className":70028,"style":158},[157],[15,70030,70032,70036],{"className":70031},[50],[15,70033],{"className":70034,"style":70035},[54],"height:2.2604em;vertical-align:-0.7693em;",[15,70037,70039,70042,70171],{"className":70038},[59],[15,70040],{"className":70041},[146,441],[15,70043,70045],{"className":70044},[238],[15,70046,70048,70163],{"className":70047},[285,286],[15,70049,70051,70160],{"className":70050},[290],[15,70052,70054,70109,70117],{"className":70053,"style":37749},[294],[15,70055,70056,70059],{"style":457},[15,70057],{"className":70058,"style":461},[302],[15,70060,70062,70065,70068,70071,70074],{"className":70061},[59],[15,70063,2465],{"className":70064},[59],[15,70066],{"className":70067,"style":2589},[157],[15,70069,4815],{"className":70070},[2593],[15,70072],{"className":70073,"style":2589},[157],[15,70075,70077,70080],{"className":70076},[59],[15,70078,7569],{"className":70079},[59,60],[15,70081,70083],{"className":70082},[281],[15,70084,70086],{"className":70085},[285],[15,70087,70089],{"className":70088},[290],[15,70090,70092],{"className":70091,"style":45058},[294],[15,70093,70094,70097],{"style":45061},[15,70095],{"className":70096,"style":303},[302],[15,70098,70100],{"className":70099},[307,308,309,310],[15,70101,70103,70106],{"className":70102},[59,310],[15,70104,2465],{"className":70105},[59,310],[15,70107,40263],{"className":70108,"style":981},[59,60,310],[15,70110,70111,70114],{"style":473},[15,70112],{"className":70113,"style":461},[302],[15,70115],{"className":70116,"style":481},[480],[15,70118,70119,70122],{"style":484},[15,70120],{"className":70121,"style":461},[302],[15,70123,70125],{"className":70124},[59],[15,70126,70128,70131],{"className":70127},[59],[15,70129,7569],{"className":70130},[59,60],[15,70132,70134],{"className":70133},[281],[15,70135,70137],{"className":70136},[285],[15,70138,70140],{"className":70139},[290],[15,70141,70143],{"className":70142,"style":20054},[294],[15,70144,70145,70148],{"style":6867},[15,70146],{"className":70147,"style":303},[302],[15,70149,70151],{"className":70150},[307,308,309,310],[15,70152,70154,70157],{"className":70153},[59,310],[15,70155,2465],{"className":70156},[59,310],[15,70158,40263],{"className":70159,"style":981},[59,60,310],[15,70161,318],{"className":70162},[317],[15,70164,70166],{"className":70165},[290],[15,70167,70169],{"className":70168,"style":17019},[294],[15,70170],{},[15,70172],{"className":70173},[153,441],"（这要求 ",[15,70176,70178,70204],{"className":70177,"translate":19},[18],[15,70179,70181],{"className":70180},[23],[25,70182,70183],{"xmlns":27},[29,70184,70185,70201],{},[32,70186,70187,70197,70199],{},[6816,70188,70189,70191],{},[35,70190,7569],{},[32,70192,70193,70195],{},[115,70194,2465],{},[35,70196,40263],{},[110,70198,7253],{},[115,70200,2465],{},[39,70202,70203],{"encoding":41},"e^{2y}\u003C2",[15,70205,70207,70258],{"className":70206,"ariaHidden":46},[45],[15,70208,70210,70214,70249,70252,70255],{"className":70209},[50],[15,70211],{"className":70212,"style":70213},[54],"height:0.8532em;vertical-align:-0.0391em;",[15,70215,70217,70220],{"className":70216},[59],[15,70218,7569],{"className":70219},[59,60],[15,70221,70223],{"className":70222},[281],[15,70224,70226],{"className":70225},[285],[15,70227,70229],{"className":70228},[290],[15,70230,70232],{"className":70231,"style":20054},[294],[15,70233,70234,70237],{"style":6867},[15,70235],{"className":70236,"style":303},[302],[15,70238,70240],{"className":70239},[307,308,309,310],[15,70241,70243,70246],{"className":70242},[59,310],[15,70244,2465],{"className":70245},[59,310],[15,70247,40263],{"className":70248,"style":981},[59,60,310],[15,70250],{"className":70251,"style":158},[157],[15,70253,7253],{"className":70254},[162],[15,70256],{"className":70257,"style":158},[157],[15,70259,70261,70264],{"className":70260},[50],[15,70262],{"className":70263,"style":172},[54],[15,70265,2465],{"className":70266},[59],[15,70268,70270,70297],{"className":70269,"translate":19},[18],[15,70271,70273],{"className":70272},[23],[25,70274,70275],{"xmlns":27},[29,70276,70277,70295],{},[32,70278,70279,70281,70283],{},[35,70280,40263],{},[110,70282,7253],{},[238,70284,70285,70293],{},[32,70286,70287,70289,70291],{},[35,70288,4102],{},[110,70290,4105],{},[115,70292,2465],{},[115,70294,2465],{},[39,70296,65780],{"encoding":41},[15,70298,70300,70318],{"className":70299,"ariaHidden":46},[45],[15,70301,70303,70306,70309,70312,70315],{"className":70302},[50],[15,70304],{"className":70305,"style":31718},[54],[15,70307,40263],{"className":70308,"style":981},[59,60],[15,70310],{"className":70311,"style":158},[157],[15,70313,7253],{"className":70314},[162],[15,70316],{"className":70317,"style":158},[157],[15,70319,70321,70324],{"className":70320},[50],[15,70322],{"className":70323,"style":65808},[54],[15,70325,70327,70330,70404],{"className":70326},[59],[15,70328],{"className":70329},[146,441],[15,70331,70333],{"className":70332},[238],[15,70334,70336,70396],{"className":70335},[285,286],[15,70337,70339,70393],{"className":70338},[290],[15,70340,70342,70356,70364],{"className":70341,"style":2827},[294],[15,70343,70344,70347],{"style":2529},[15,70345],{"className":70346,"style":461},[302],[15,70348,70350],{"className":70349},[307,308,309,310],[15,70351,70353],{"className":70352},[59,310],[15,70354,2465],{"className":70355},[59,310],[15,70357,70358,70361],{"style":473},[15,70359],{"className":70360,"style":461},[302],[15,70362],{"className":70363,"style":481},[480],[15,70365,70366,70369],{"style":2855},[15,70367],{"className":70368,"style":461},[302],[15,70370,70372],{"className":70371},[307,308,309,310],[15,70373,70375,70387,70390],{"className":70374},[59,310],[15,70376,70378,70381,70384],{"className":70377},[1486,310],[15,70379,47684],{"className":70380},[310],[15,70382,47688],{"className":70383},[310],[15,70385,4160],{"className":70386,"style":4159},[310],[15,70388],{"className":70389,"style":42392},[157,310],[15,70391,2465],{"className":70392},[59,310],[15,70394,318],{"className":70395},[317],[15,70397,70399],{"className":70398},[290],[15,70400,70402],{"className":70401,"style":2574},[294],[15,70403],{},[15,70405],{"className":70406},[153,441],"）。于是 ",[15,70409,70411,70465],{"className":70410,"translate":19},[18],[15,70412,70414],{"className":70413},[23],[25,70415,70416],{"xmlns":27},[29,70417,70418,70462],{},[32,70419,70420,70422,70424,70436,70438,70440,70442,70444,70446,70448,70450,70460],{},[35,70421,4102],{},[110,70423,4105],{},[238,70425,70426,70434],{},[32,70427,70428,70430,70432],{},[35,70429,769],{},[110,70431,2472],{},[115,70433,117],{},[115,70435,2465],{},[110,70437,123],{},[110,70439,4815],{},[35,70441,4102],{},[110,70443,4105],{},[110,70445,113],{"stretchy":112},[115,70447,2465],{},[110,70449,4815],{},[6816,70451,70452,70454],{},[35,70453,7569],{},[32,70455,70456,70458],{},[115,70457,2465],{},[35,70459,40263],{},[110,70461,120],{"stretchy":112},[39,70463,70464],{"encoding":41},"\\log\\frac{t+1}{2}=-\\log(2-e^{2y})",[15,70466,70468,70565,70597],{"className":70467,"ariaHidden":46},[45],[15,70469,70471,70474,70479,70482,70556,70559,70562],{"className":70470},[50],[15,70472],{"className":70473,"style":2507},[54],[15,70475,4156,70477],{"className":70476},[1486],[15,70478,4160],{"style":4159},[15,70480],{"className":70481,"style":406},[157],[15,70483,70485,70488,70553],{"className":70484},[59],[15,70486],{"className":70487},[146,441],[15,70489,70491],{"className":70490},[238],[15,70492,70494,70545],{"className":70493},[285,286],[15,70495,70497,70542],{"className":70496},[290],[15,70498,70500,70514,70522],{"className":70499,"style":2526},[294],[15,70501,70502,70505],{"style":2529},[15,70503],{"className":70504,"style":461},[302],[15,70506,70508],{"className":70507},[307,308,309,310],[15,70509,70511],{"className":70510},[59,310],[15,70512,2465],{"className":70513},[59,310],[15,70515,70516,70519],{"style":473},[15,70517],{"className":70518,"style":461},[302],[15,70520],{"className":70521,"style":481},[480],[15,70523,70524,70527],{"style":2552},[15,70525],{"className":70526,"style":461},[302],[15,70528,70530],{"className":70529},[307,308,309,310],[15,70531,70533,70536,70539],{"className":70532},[59,310],[15,70534,769],{"className":70535},[59,60,310],[15,70537,2472],{"className":70538},[2593,310],[15,70540,117],{"className":70541},[59,310],[15,70543,318],{"className":70544},[317],[15,70546,70548],{"className":70547},[290],[15,70549,70551],{"className":70550,"style":2574},[294],[15,70552],{},[15,70554],{"className":70555},[153,441],[15,70557],{"className":70558,"style":158},[157],[15,70560,123],{"className":70561},[162],[15,70563],{"className":70564,"style":158},[157],[15,70566,70568,70571,70574,70577,70582,70585,70588,70591,70594],{"className":70567},[50],[15,70569],{"className":70570,"style":139},[54],[15,70572,4815],{"className":70573},[59],[15,70575],{"className":70576,"style":406},[157],[15,70578,4156,70580],{"className":70579},[1486],[15,70581,4160],{"style":4159},[15,70583,113],{"className":70584},[146],[15,70586,2465],{"className":70587},[59],[15,70589],{"className":70590,"style":2589},[157],[15,70592,4815],{"className":70593},[2593],[15,70595],{"className":70596,"style":2589},[157],[15,70598,70600,70603,70638],{"className":70599},[50],[15,70601],{"className":70602,"style":20029},[54],[15,70604,70606,70609],{"className":70605},[59],[15,70607,7569],{"className":70608},[59,60],[15,70610,70612],{"className":70611},[281],[15,70613,70615],{"className":70614},[285],[15,70616,70618],{"className":70617},[290],[15,70619,70621],{"className":70620,"style":20054},[294],[15,70622,70623,70626],{"style":6867},[15,70624],{"className":70625,"style":303},[302],[15,70627,70629],{"className":70628},[307,308,309,310],[15,70630,70632,70635],{"className":70631},[59,310],[15,70633,2465],{"className":70634},[59,310],[15,70636,40263],{"className":70637,"style":981},[59,60,310],[15,70639,120],{"className":70640},[153],"，代入",[15,70643,70645],{"className":70644,"translate":19},[180],[15,70646,70648,70846],{"className":70647,"translate":19},[18],[15,70649,70651],{"className":70650},[23],[25,70652,70653],{"xmlns":27,"display":189},[29,70654,70655,70843],{},[1245,70656,70657,70709,70791],{"rowspacing":1247,"columnalign":1248,"columnspacing":1249},[1251,70658,70659,70677],{},[1254,70660,70661],{},[1257,70662,70663],{"scriptlevel":126,"displaystyle":46},[32,70664,70665,70671,70673,70675],{},[6816,70666,70667,70669],{},[35,70668,37],{},[110,70670,7575],{},[110,70672,113],{"stretchy":112},[35,70674,40263],{},[110,70676,120],{"stretchy":112},[1254,70678,70679],{},[1257,70680,70681],{"scriptlevel":126,"displaystyle":46},[32,70682,70683,70685,70687,70689,70691,70693,70703,70705,70707],{},[32,70684],{},[110,70686,123],{},[35,70688,40263],{},[35,70690,769],{},[110,70692,4815],{},[195,70694,70695,70697],{},[35,70696,37],{},[32,70698,70699,70701],{},[35,70700,23769],{},[35,70702,25763],{},[110,70704,113],{"stretchy":112},[35,70706,769],{},[110,70708,120],{"stretchy":112},[1251,70710,70711,70717],{},[1254,70712,70713],{},[1257,70714,70715],{"scriptlevel":126,"displaystyle":46},[32,70716],{},[1254,70718,70719],{},[1257,70720,70721],{"scriptlevel":126,"displaystyle":46},[32,70722,70723,707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",[15,71841,71843,71861],{"className":71842,"translate":19},[18],[15,71844,71846],{"className":71845},[23],[25,71847,71848],{"xmlns":27},[29,71849,71850,71858],{},[32,71851,71852,71854,71856],{},[35,71853,769],{},[110,71855,9954],{},[115,71857,117],{},[39,71859,71860],{"encoding":41},"t\\le1",[15,71862,71864,71882],{"className":71863,"ariaHidden":46},[45],[15,71865,71867,71870,71873,71876,71879],{"className":71866},[50],[15,71868],{"className":71869,"style":43214},[54],[15,71871,769],{"className":71872},[59,60],[15,71874],{"className":71875,"style":158},[157],[15,71877,9954],{"className":71878},[162],[15,71880],{"className":71881,"style":158},[157],[15,71883,71885,71888],{"className":71884},[50],[15,71886],{"className":71887,"style":172},[54],[15,71889,117],{"className":71890},[59]," 一侧，",[15,71893,71895,71951],{"className":71894,"translate":19},[18],[15,71896,71898],{"className":71897},[23],[25,71899,71900],{"xmlns":27},[29,71901,71902,71948],{},[32,71903,71904,71906,71908,71910,71922,71924,71926,71928,71930,71932,71938,71940,71942],{},[35,71905,40263],{},[35,71907,769],{},[110,71909,4815],{},[238,71911,71912,71920],{},[32,71913,71914,71916,71918],{},[115,71915,117],{},[110,71917,4815],{},[35,71919,769],{},[115,71921,2465],{},[110,71923,123],{},[35,71925,769],{},[110,71927,113],{"stretchy":112},[35,71929,40263],{},[110,71931,2472],{},[238,71933,71934,71936],{},[115,71935,117],{},[115,71937,2465],{},[110,71939,120],{"stretchy":112},[110,71941,4815],{},[238,71943,71944,71946],{},[115,71945,117],{},[115,71947,2465],{},[39,71949,71950],{"encoding":41},"yt-\\frac{1-t}{2}=t(y+\\frac12)-\\frac12",[15,71952,71954,71975,72064,72088,72174],{"className":71953,"ariaHidden":46},[45],[15,71955,71957,71960,71963,71966,71969,71972],{"className":71956},[50],[15,71958],{"className":71959,"style":26300},[54],[15,71961,40263],{"className":71962,"style":981},[59,60],[15,71964,769],{"className":71965},[59,60],[15,71967],{"className":71968,"style":2589},[157],[15,71970,4815],{"className":71971},[2593],[15,71973],{"className":71974,"style":2589},[157],[15,71976,71978,71981,72055,72058,72061],{"className":71977},[50],[15,71979],{"className":71980,"style":2507},[54],[15,71982,71984,71987,72052],{"className":71983},[59],[15,71985],{"className":71986},[146,441],[15,71988,71990],{"className":71989},[238],[15,71991,71993,72044],{"className":71992},[285,286],[15,71994,71996,72041],{"className":71995},[290],[15,71997,71999,72013,72021],{"className":71998,"style":2526},[294],[15,72000,72001,72004],{"style":2529},[15,72002],{"className":72003,"style":461},[302],[15,72005,72007],{"className":72006},[307,308,309,310],[15,72008,72010],{"className":72009},[59,310],[15,72011,2465],{"className":72012},[59,310],[15,72014,72015,72018],{"style":473},[15,72016],{"className":72017,"style":461},[302],[15,72019],{"className":72020,"style":481},[480],[15,72022,72023,72026],{"style":2552},[15,72024],{"className":72025,"style":461},[302],[15,72027,72029],{"className":72028},[307,308,309,310],[15,72030,72032,72035,72038],{"className":72031},[59,310],[15,72033,117],{"className":72034},[59,310],[15,72036,4815],{"className":72037},[2593,310],[15,72039,769],{"className":72040},[59,60,310],[15,72042,318],{"className":72043},[317],[15,72045,72047],{"className":72046},[290],[15,72048,72050],{"className":72049,"style":2574},[294],[15,72051],{},[15,72053],{"className":72054},[153,441],[15,72056],{"className":72057,"style":158},[157],[15,72059,123],{"className":72060},[162],[15,72062],{"className":72063,"style":158},[157],[15,72065,72067,72070,72073,72076,72079,72082,72085],{"className":72066},[50],[15,72068],{"className":72069,"style":139},[54],[15,72071,769],{"className":72072},[59,60],[15,72074,113],{"className":72075},[146],[15,72077,40263],{"className":72078,"style":981},[59,60],[15,72080],{"className":72081,"style":2589},[157],[15,72083,2472],{"className":72084},[2593],[15,72086],{"className":72087,"style":2589},[157],[15,72089,72091,72094,72162,72165,72168,72171],{"className":72090},[50],[15,72092],{"className":72093,"style":2507},[54],[15,72095,72097,72100,72159],{"className":72096},[59],[15,72098],{"className":72099},[146,441],[15,72101,72103],{"className":72102},[238],[15,72104,72106,72151],{"className":72105},[285,286],[15,72107,72109,72148],{"className":72108},[290],[15,72110,72112,72126,72134],{"className":72111,"style":2526},[294],[15,72113,72114,72117],{"style":2529},[15,72115],{"className":72116,"style":461},[302],[15,72118,72120],{"className":72119},[307,308,309,310],[15,72121,72123],{"className":72122},[59,310],[15,72124,2465],{"className":72125},[59,310],[15,72127,72128,72131],{"style":473},[15,72129],{"className":72130,"style":461},[302],[15,72132],{"className":72133,"style":481},[480],[15,72135,72136,72139],{"style":2552},[15,72137],{"className":72138,"style":461},[302],[15,72140,72142],{"className":72141},[307,308,309,310],[15,72143,72145],{"className":72144},[59,310],[15,72146,117],{"className":72147},[59,310],[15,72149,318],{"className":72150},[317],[15,72152,72154],{"className":72153},[290],[15,72155,72157],{"className":72156,"style":2574},[294],[15,72158],{},[15,72160],{"className":72161},[153,441],[15,72163,120],{"className":72164},[153],[15,72166],{"className":72167,"style":2589},[157],[15,72169,4815],{"className":72170},[2593],[15,72172],{"className":72173,"style":2589},[157],[15,72175,72177,72180],{"className":72176},[50],[15,72178],{"className":72179,"style":2507},[54],[15,72181,72183,72186,72245],{"className":72182},[59],[15,72184],{"className":72185},[146,441],[15,72187,72189],{"className":72188},[238],[15,72190,72192,72237],{"className":72191},[285,286],[15,72193,72195,72234],{"className":72194},[290],[15,72196,72198,72212,72220],{"className":72197,"style":2526},[294],[15,72199,72200,72203],{"style":2529},[15,72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",[15,72250,72252,72270],{"className":72251,"translate":19},[18],[15,72253,72255],{"className":72254},[23],[25,72256,72257],{"xmlns":27},[29,72258,72259,72267],{},[32,72260,72261,72263,72265],{},[35,72262,769],{},[110,72264,6930],{},[115,72266,117],{},[39,72268,72269],{"encoding":41},"t\\ge1",[15,72271,72273,72291],{"className":72272,"ariaHidden":46},[45],[15,72274,72276,72279,72282,72285,72288],{"className":72275},[50],[15,72277],{"className":72278,"style":43214},[54],[15,72280,769],{"className":72281},[59,60],[15,72283],{"className":72284,"style":158},[157],[15,72286,6930],{"className":72287},[162],[15,72289],{"className":72290,"style":158},[157],[15,72292,72294,72297],{"className":72293},[50],[15,72295],{"className":72296,"style":172},[54],[15,72298,117],{"className":72299},[59],[15,72301,72303,72359],{"className":72302,"translate":19},[18],[15,72304,72306],{"className":72305},[23],[25,72307,72308],{"xmlns":27},[29,72309,72310,72356],{},[32,72311,72312,72314,72316,72318,72330,72332,72334,72336,72338,72340,72346,72348,72350],{},[35,72313,40263],{},[35,72315,769],{},[110,72317,4815],{},[238,72319,72320,72328],{},[32,72321,72322,72324,72326],{},[35,72323,769],{},[110,72325,4815],{},[115,72327,117],{},[115,72329,2465],{},[110,72331,123],{},[35,72333,769],{},[110,72335,113],{"stretchy":112},[35,72337,40263],{},[110,72339,4815],{},[238,72341,72342,72344],{},[115,72343,117],{},[115,72345,2465],{},[110,72347,120],{"stretchy":112},[110,72349,2472],{},[238,72351,72352,72354],{},[115,72353,117],{},[115,72355,2465],{},[39,72357,72358],{"encoding":41},"yt-\\frac{t-1}{2}=t(y-\\frac12)+\\frac12",[15,72360,72362,72383,72472,72496,72582],{"className":72361,"ariaHidden":46},[45],[15,72363,72365,72368,72371,72374,72377,72380],{"className":72364},[50],[15,72366],{"className":72367,"style":26300},[54],[15,72369,40263],{"className":72370,"style":981},[59,60],[15,72372,769],{"className":72373},[59,60],[15,72375],{"className":72376,"style":2589},[157],[15,72378,4815],{"className":72379},[2593],[15,72381],{"className":72382,"style":2589},[157],[15,72384,72386,72389,72463,72466,72469],{"className":72385},[50],[15,72387],{"className":72388,"style":2507},[54],[15,72390,72392,72395,72460],{"className":72391},[59],[15,72393],{"className":72394},[146,441],[15,72396,72398],{"className":72397},[238],[15,72399,72401,72452],{"className":72400},[285,286],[15,72402,72404,72449],{"className":72403},[290],[15,72405,72407,72421,72429],{"className":72406,"style":2526},[294],[15,72408,72409,72412],{"style":2529},[15,72410],{"className":72411,"style":461},[302],[15,72413,72415],{"className":72414},[307,308,309,310],[15,72416,72418],{"className":72417},[59,310],[15,72419,2465],{"className":72420},[59,310],[15,72422,72423,72426],{"style":473},[15,72424],{"className":72425,"style":461},[302],[15,72427],{"className":72428,"style":481},[480],[15,72430,72431,72434],{"style":2552},[15,72432],{"className":72433,"style":461},[302],[15,72435,72437],{"className":72436},[307,308,309,310],[15,72438,72440,72443,72446],{"className":72439},[59,310],[15,72441,769],{"className":72442},[59,60,310],[15,72444,4815],{"className":72445},[2593,310],[15,72447,117],{"className":72448},[59,310],[15,72450,318],{"className":72451},[317],[15,72453,72455],{"className":72454},[290],[15,72456,72458],{"className":72457,"style":2574},[294],[15,72459],{},[15,72461],{"className":72462},[153,441],[15,72464],{"className":72465,"style":158},[157],[15,72467,123],{"className":72468},[162],[15,72470],{"className":72471,"style":158},[157],[15,72473,72475,72478,72481,72484,72487,72490,72493],{"className":72474},[50],[15,72476],{"className":72477,"style":139},[54],[15,72479,769],{"className":72480},[59,60],[15,72482,113],{"className":72483},[146],[15,72485,40263],{"className":72486,"style":981},[59,60],[15,72488],{"className":72489,"style":2589},[157],[15,72491,4815],{"className":72492},[2593],[15,72494],{"className":72495,"style":2589},[157],[15,72497,72499,72502,72570,72573,72576,72579],{"className":72498},[50],[15,72500],{"className":72501,"style":2507},[54],[15,72503,72505,72508,72567],{"className":72504},[59],[15,72506],{"className":72507},[146,441],[15,72509,72511],{"className":72510},[238],[15,72512,72514,72559],{"className":72513},[285,286],[15,72515,72517,72556],{"className":72516},[290],[15,72518,72520,72534,72542],{"className":72519,"style":2526},[294],[15,72521,72522,72525],{"style":2529},[15,72523],{"className":72524,"style":461},[302],[15,72526,72528],{"className":72527},[307,308,309,310],[15,72529,72531],{"className":72530},[59,310],[15,72532,2465],{"className":72533},[59,310],[15,72535,72536,72539],{"style":473},[15,72537],{"className":72538,"style":461},[302],[15,72540],{"className":72541,"style":481},[480],[15,72543,72544,72547],{"style":2552},[15,72545],{"className":72546,"style":461},[302],[15,72548,72550],{"className":72549},[307,308,309,310],[15,72551,72553],{"className":72552},[59,310],[15,72554,117],{"className":72555},[59,310],[15,72557,318],{"className":72558},[317],[15,72560,72562],{"className":72561},[290],[15,72563,72565],{"className":72564,"style":2574},[294],[15,72566],{},[15,72568],{"className":72569},[153,441],[15,72571,120],{"className":72572},[153],[15,72574],{"className":72575,"style":2589},[157],[15,72577,2472],{"className":72578},[2593],[15,72580],{"className":72581,"style":2589},[157],[15,72583,72585,72588],{"className":72584},[50],[15,72586],{"className":72587,"style":2507},[54],[15,72589,72591,72594,72653],{"className":72590},[59],[15,72592],{"className":72593},[146,441],[15,72595,72597],{"className":72596},[238],[15,72598,72600,72645],{"className":72599},[285,286],[15,72601,72603,72642],{"className":72602},[290],[15,72604,72606,72620,72628],{"className":72605,"style":2526},[294],[15,72607,72608,72611],{"style":2529},[15,72609],{"className":72610,"style":461},[302],[15,72612,72614],{"className":72613},[307,308,309,310],[15,72615,72617],{"className":72616},[59,310],[15,72618,2465],{"className":72619},[59,310],[15,72621,72622,72625],{"style":473},[15,72623],{"className":72624,"style":461},[302],[15,72626],{"className":72627,"style":481},[480],[15,72629,72630,72633],{"style":2552},[15,72631],{"className":72632,"style":461},[302],[15,72634,72636],{"className":72635},[307,308,309,310],[15,72637,72639],{"className":72638},[59,310],[15,72640,117],{"className":72641},[59,310],[15,72643,318],{"className":72644},[317],[15,72646,72648],{"className":72647},[290],[15,72649,72651],{"className":72650,"style":2574},[294],[15,72652],{},[15,72654],{"className":72655},[153,441],"。两段都是线性的：当 ",[15,72658,72660,72685],{"className":72659,"translate":19},[18],[15,72661,72663],{"className":72662},[23],[25,72664,72665],{"xmlns":27},[29,72666,72667,72683],{},[32,72668,72669,72671,72673,72675,72677],{},[110,72670,4810],{"stretchy":112},[35,72672,40263],{},[110,72674,4810],{"stretchy":112},[110,72676,9954],{},[238,72678,72679,72681],{},[115,72680,117],{},[115,72682,2465],{},[39,72684,66094],{"encoding":41},[15,72686,72688,72712],{"className":72687,"ariaHidden":46},[45],[15,72689,72691,72694,72697,72700,72703,72706,72709],{"className":72690},[50],[15,72692],{"className":72693,"style":139},[54],[15,72695,4810],{"className":72696},[146],[15,72698,40263],{"className":72699,"style":981},[59,60],[15,72701,4810],{"className":72702},[153],[15,72704],{"className":72705,"style":158},[157],[15,72707,9954],{"className":72708},[162],[15,72710],{"className":72711,"style":158},[157],[15,72713,72715,72718],{"className":72714},[50],[15,72716],{"className":72717,"style":2507},[54],[15,72719,72721,72724,72783],{"className":72720},[59],[15,72722],{"className":72723},[146,441],[15,72725,72727],{"className":72726},[238],[15,72728,72730,72775],{"className":72729},[285,286],[15,72731,72733,72772],{"className":72732},[290],[15,72734,72736,72750,72758],{"className":72735,"style":2526},[294],[15,72737,72738,72741],{"style":2529},[15,72739],{"className":72740,"style":461},[302],[15,72742,72744],{"className":72743},[307,308,309,310],[15,72745,72747],{"className":72746},[59,310],[15,72748,2465],{"className":72749},[59,310],[15,72751,72752,72755],{"style":473},[15,72753],{"className":72754,"style":461},[302],[15,72756],{"className":72757,"style":481},[480],[15,72759,72760,72763],{"style":2552},[15,72761],{"className":72762,"style":461},[302],[15,72764,72766],{"className":72765},[307,308,309,310],[15,72767,72769],{"className":72768},[59,310],[15,72770,117],{"className":72771},[59,310],[15,72773,318],{"className":72774},[317],[15,72776,72778],{"className":72777},[290],[15,72779,72781],{"className":72780,"style":2574},[294],[15,72782],{},[15,72784],{"className":72785},[153,441]," 时两段的最大值都在 ",[15,72788,72790,72808],{"className":72789,"translate":19},[18],[15,72791,72793],{"className":72792},[23],[25,72794,72795],{"xmlns":27},[29,72796,72797,72805],{},[32,72798,72799,72801,72803],{},[35,72800,769],{},[110,72802,123],{},[115,72804,117],{},[39,72806,72807],{"encoding":41},"t=1",[15,72809,72811,72829],{"className":72810,"ariaHidden":46},[45],[15,72812,72814,72817,72820,72823,72826],{"className":72813},[50],[15,72815],{"className":72816,"style":5345},[54],[15,72818,769],{"className":72819},[59,60],[15,72821],{"className":72822,"style":158},[157],[15,72824,123],{"className":72825},[162],[15,72827],{"className":72828,"style":158},[157],[15,72830,72832,72835],{"className":72831},[50],[15,72833],{"className":72834,"style":172},[54],[15,72836,117],{"className":72837},[59]," 取到，为 ",[15,72840,72842,72855],{"className":72841,"translate":19},[18],[15,72843,72845],{"className":72844},[23],[25,72846,72847],{"xmlns":27},[29,72848,72849,72853],{},[32,72850,72851],{},[35,72852,40263],{},[39,72854,40263],{"encoding":41},[15,72856,72858],{"className":72857,"ariaHidden":46},[45],[15,72859,72861,72864],{"className":72860},[50],[15,72862],{"className":72863,"style":2278},[54],[15,72865,40263],{"className":72866,"style":981},[59,60],[15,72868,72870,72892],{"className":72869,"translate":19},[18],[15,72871,72873],{"className":72872},[23],[25,72874,72875],{"xmlns":27},[29,72876,72877,72889],{},[32,72878,72879,72881,72883],{},[35,72880,40263],{},[110,72882,2905],{},[238,72884,72885,72887],{},[115,72886,117],{},[115,72888,2465],{},[39,72890,72891],{"encoding":41},"y>\\frac12",[15,72893,72895,72913],{"className":72894,"ariaHidden":46},[45],[15,72896,72898,72901,72904,72907,72910],{"className":72897},[50],[15,72899],{"className":72900,"style":31718},[54],[15,72902,40263],{"className":72903,"style":981},[59,60],[15,72905],{"className":72906,"style":158},[157],[15,72908,2905],{"className":72909},[162],[15,72911],{"className":72912,"style":158},[157],[15,72914,72916,72919],{"className":72915},[50],[15,72917],{"className":72918,"style":2507},[54],[15,72920,72922,72925,72984],{"className":72921},[59],[15,72923],{"className":72924},[146,441],[15,72926,72928],{"className":72927},[238],[15,72929,72931,72976],{"className":72930},[285,286],[15,72932,72934,72973],{"className":72933},[290],[15,72935,72937,72951,72959],{"className":72936,"style":2526},[294],[15,72938,72939,72942],{"style":2529},[15,72940],{"className":72941,"style":461},[302],[15,72943,72945],{"className":72944},[307,308,309,310],[15,72946,72948],{"className":72947},[59,310],[15,72949,2465],{"className":72950},[59,310],[15,72952,72953,72956],{"style":473},[15,72954],{"className":72955,"style":461},[302],[15,72957],{"className":72958,"style":481},[480],[15,72960,72961,72964],{"style":2552},[15,72962],{"className":72963,"style":461},[302],[15,72965,72967],{"className":72966},[307,308,309,310],[15,72968,72970],{"className":72969},[59,310],[15,72971,117],{"className":72972},[59,310],[15,72974,318],{"className":72975},[317],[15,72977,72979],{"className":72978},[290],[15,72980,72982],{"className":72981,"style":2574},[294],[15,72983],{},[15,72985],{"className":72986},[153,441]," 时第二段严格递增，上确界为 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时第一段严格递减，上确界在 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Kantorovich 型对偶。",[11,74483,74484],{},"这个对偶表示在机器学习中非常有用，例如 f-GAN 就是基于此构造的（第 5 节详述）。",[2313,74486,74488],{"id":74487},"_45-数据处理不等式","4.5 数据处理不等式",[993,74490,74491,74594],{},[11,74492,74493,11064,74496,74524,74525,5647,74559,74593],{},[998,74494,74495],{},"命题 4.4（数据处理不等式）",[15,74497,74499,74512],{"className":74498,"translate":19},[18],[15,74500,74502],{"className":74501},[23],[25,74503,74504],{"xmlns":27},[29,74505,74506,74510],{},[32,74507,74508],{},[35,74509,10698],{},[39,74511,10698],{"encoding":41},[15,74513,74515],{"className":74514,"ariaHidden":46},[45],[15,74516,74518,74521],{"className":74517},[50],[15,74519],{"className":74520,"style":664},[54],[15,74522,10698],{"className":74523,"style":10786},[59,60]," 是一个马尔可夫核（随机变换），记 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为输出分布，则",[15,74595,74597],{"className":74596,"translate":19},[180],[15,74598,74600,74652],{"className":74599,"translate":19},[18],[15,74601,74603],{"className":74602},[23],[25,74604,74605],{"xmlns":27,"display":189},[29,74606,74607,74649],{},[32,74608,74609,74615,74617,74619,74621,74623,74625,74627,74629,74631,74637,74639,74641,74643,74645,74647],{},[195,74610,74611,74613],{},[35,74612,199],{},[35,74614,37],{},[110,74616,113],{"stretchy":112},[35,74618,206],{},[35,74620,10698],{},[35,74622,210],{"mathvariant":209},[35,74624,213],{},[35,74626,10698],{},[110,74628,120],{"stretchy":112},[110,74630,9954],{},[195,74632,74633,74635],{},[35,74634,199],{},[35,74636,37],{},[110,74638,113],{"stretchy":112},[35,74640,206],{},[35,74642,210],{"mathvariant":209},[35,74644,213],{},[110,74646,120],{"stretchy":112},[35,74648,4377],{"mathvariant":209},[39,74650,74651],{"encoding":41},"D_f(PK\\|QK)\\le 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则 ",[15,75044,75046,75068],{"className":75045,"translate":19},[18],[15,75047,75049],{"className":75048},[23],[25,75050,75051],{"xmlns":27},[29,75052,75053,75065],{},[32,75054,75055,75057,75059,75061,75063],{},[35,75056,206],{},[35,75058,10698],{},[110,75060,714],{},[35,75062,213],{},[35,75064,10698],{},[39,75066,75067],{"encoding":41},"PK\\ll QK",[15,75069,75071,75092],{"className":75070,"ariaHidden":46},[45],[15,75072,75074,75077,75080,75083,75086,75089],{"className":75073},[50],[15,75075],{"className":75076,"style":729},[54],[15,75078,206],{"className":75079,"style":334},[59,60],[15,75081,10698],{"className":75082,"style":10786},[59,60],[15,75084],{"className":75085,"style":158},[157],[15,75087,714],{"className":75088},[162],[15,75090],{"className":75091,"style":158},[157],[15,75093,75095,75098,75101],{"className":75094},[50],[15,75096],{"className":75097,"style":566},[54],[15,75099,213],{"className":75100},[59,60],[15,75102,10698],{"className":75103,"style":10786},[59,60],"，且对任意可测集 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E_{QK}\\Big[\nf\\big(\\mathbb E_Q[t|Y]\\big)\\Big]\\\\\n&\\le \\mathbb E_{QK}\\Big[\n\\mathbb E_Q[f(t)|Y]\\Big]\\\\\n&=\\mathbb E_Q[f(t)]=D_f(P\\|Q).\\qquad\\square\n\\end{aligned}",[15,77054,77056],{"className":77055,"ariaHidden":46},[45],[15,77057,77059,77063],{"className":77058},[50],[15,77060],{"className":77061,"style":77062},[54],"height:5.4em;vertical-align:-2.45em;",[15,77064,77066],{"className":77065},[59],[15,77067,77069,77181],{"className":77068},[1245],[15,77070,77072],{"className":77071},[1362],[15,77073,77075,77172],{"className":77074},[285,286],[15,77076,77078,77169],{"className":77077},[290],[15,77079,77082,77152,77160],{"className":77080,"style":77081},[294],"height:2.95em;",[15,77083,77085,77088],{"style":77084},"top:-4.95em;",[15,77086],{"className":77087,"style":17592},[302],[15,77089,77091,77131,77134,77137,77140,77143,77146,77149],{"className":77090},[59],[15,77092,77094,77097],{"className":77093},[59],[15,77095,199],{"className":77096,"style":277},[59,60],[15,77098,77100],{"className":77099},[281],[15,77101,77103,77123],{"className":77102},[285,286],[15,77104,77106,77120],{"className":77105},[290],[15,77107,77109],{"className":77108,"style":295},[294],[15,77110,77111,77114],{"style":298},[15,77112],{"className":77113,"style":303},[302],[15,77115,77117],{"className":77116},[307,308,309,310],[15,77118,37],{"className":77119,"style":61},[59,60,310],[15,77121,318],{"className":77122},[317],[15,77124,77126],{"className":77125},[290],[15,77127,77129],{"className":77128,"style":325},[294],[15,77130],{},[15,77132,113],{"className":77133},[146],[15,77135,206],{"className":77136,"style":334},[59,60],[15,77138,10698],{"className":77139,"style":10786},[59,60],[15,77141,210],{"className":77142},[59],[15,77144,213],{"className":77145},[59,60],[15,77147,10698],{"className":77148,"style":10786},[59,60],[15,77150,120],{"className":77151},[153],[15,77153,77154,77157],{"style":14846},[15,77155],{"className":77156,"style":17592},[302],[15,77158],{"className":77159},[59],[15,77161,77163,77166],{"style":77162},"top:-1.06em;",[15,77164],{"className":77165,"style":17592},[302],[15,77167],{"className":77168},[59],[15,77170,318],{"className":77171},[317],[15,77173,77175],{"className":77174},[290],[15,77176,77179],{"className":77177,"style":77178},[294],"height:2.45em;",[15,77180],{},[15,77182,77184],{"className":77183},[1453],[15,77185,77187,77634],{"className":77186},[285,286],[15,77188,77190,77631],{"className":77189},[290],[15,77191,77193,77341,77483],{"className":77192,"style":77081},[294],[15,77194,77195,77198],{"style":77084},[15,77196],{"className":77197,"style":17592},[302],[15,77199,77201,77204,77207,77210,77213,77259,77265,77268,77274,77314,77317,77320,77323,77326,77329,77335],{"className":77200},[59],[15,77202],{"className":77203},[59],[15,77205],{"className":77206,"style":158},[157],[15,77208,123],{"className":77209},[162],[15,77211],{"className":77212,"style":158},[157],[15,77214,77216,77219],{"className":77215},[59],[15,77217,223],{"className":77218},[59,366],[15,77220,77222],{"className":77221},[281],[15,77223,77225,77251],{"className":77224},[285,286],[15,77226,77228,77248],{"className":77227},[290],[15,77229,77231],{"className":77230,"style":379},[294],[15,77232,77233,77236],{"style":382},[15,77234],{"className":77235,"style":303},[302],[15,77237,77239],{"className":77238},[307,308,309,310],[15,77240,77242,77245],{"className":77241},[59,310],[15,77243,213],{"className":77244},[59,60,310],[15,77246,10698],{"className":77247,"style":10786},[59,60,310],[15,77249,318],{"className":77250},[317],[15,77252,77254],{"className":77253},[290],[15,77255,77257],{"className":77256,"style":325},[294],[15,77258],{},[15,77260,77262],{"className":77261},[59],[15,77263,230],{"className":77264},[419,3722],[15,77266,37],{"className":77267,"style":61},[59,60],[15,77269,77271],{"className":77270},[59],[15,77272,113],{"className":77273},[419,9676],[15,77275,77277,77280],{"className":77276},[59],[15,77278,223],{"className":77279},[59,366],[15,77281,77283],{"className":77282},[281],[15,77284,77286,77306],{"className":77285},[285,286],[15,77287,77289,77303],{"className":77288},[290],[15,77290,77292],{"className":77291,"style":379},[294],[15,77293,77294,77297],{"style":382},[15,77295],{"className":77296,"style":303},[302],[15,77298,77300],{"className":77299},[307,308,309,310],[15,77301,213],{"className":77302},[59,60,310],[15,77304,318],{"className":77305},[317],[15,77307,77309],{"className":77308},[290],[15,77310,77312],{"className":77311,"style":325},[294],[15,77313],{},[15,77315,230],{"className":77316},[146],[15,77318,769],{"className":77319},[59,60],[15,77321,4810],{"className":77322},[59],[15,77324,5403],{"className":77325,"style":2589},[59,60],[15,77327,257],{"className":77328},[153],[15,77330,77332],{"className":77331},[59],[15,77333,120],{"className":77334},[419,9676],[15,77336,77338],{"className":77337},[59],[15,77339,257],{"className":77340},[419,3722],[15,77342,77343,77346],{"style":14846},[15,77344],{"className":77345,"style":17592},[302],[15,77347,77349,77352,77355,77358,77361,77407,77413,77453,77456,77459,77462,77465,77468,77471,77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严格凸，等号成立当且仅当 ",[15,77677,77679,77707],{"className":77678,"translate":19},[18],[15,77680,77682],{"className":77681},[23],[25,77683,77684],{"xmlns":27},[29,77685,77686,77704],{},[32,77687,77688,77694,77696,77698,77700,77702],{},[195,77689,77690,77692],{},[35,77691,223],{"mathvariant":222},[35,77693,213],{},[110,77695,230],{"stretchy":112},[35,77697,769],{},[35,77699,4810],{"mathvariant":209},[35,77701,5403],{},[110,77703,257],{"stretchy":112},[39,77705,77706],{"encoding":41},"\\mathbb E_Q[t|Y]",[15,77708,77710],{"className":77709,"ariaHidden":46},[45],[15,77711,77713,77716,77756,77759,77762,77765,77768],{"className":77712},[50],[15,77714],{"className":77715,"style":270},[54],[15,77717,77719,77722],{"className":77718},[59],[15,77720,223],{"className":77721},[59,366],[15,77723,77725],{"className":77724},[281],[15,77726,77728,77748],{"className":77727},[285,286],[15,77729,77731,77745],{"className":77730},[290],[15,77732,77734],{"className":77733,"style":379},[294],[15,77735,77736,77739],{"style":382},[15,77737],{"className":77738,"style":303},[302],[15,77740,77742],{"className":77741},[307,308,309,310],[15,77743,213],{"className":77744},[59,60,310],[15,77746,318],{"className":77747},[317],[15,77749,77751],{"className":77750},[290],[15,77752,77754],{"className":77753,"style":325},[294],[15,77755],{},[15,77757,230],{"className":77758},[146],[15,77760,769],{"className":77761},[59,60],[15,77763,4810],{"className":77764},[59],[15,77766,5403],{"className":77767,"style":2589},[59,60],[15,77769,257],{"className":77770},[153]," 本身是 ",[15,77773,77775,77795],{"className":77774,"translate":19},[18],[15,77776,77778],{"className":77777},[23],[25,77779,77780],{"xmlns":27},[29,77781,77782,77792],{},[32,77783,77784,77786,77788,77790],{},[35,77785,2334],{},[110,77787,113],{"stretchy":112},[35,77789,5403],{},[110,77791,120],{"stretchy":112},[39,77793,77794],{"encoding":41},"\\sigma(Y)",[15,77796,77798],{"className":77797,"ariaHidden":46},[45],[15,77799,77801,77804,77807,77810,77813],{"className":77800},[50],[15,77802],{"className":77803,"style":139},[54],[15,77805,2334],{"className":77806,"style":981},[59,60],[15,77808,113],{"className":77809},[146],[15,77811,5403],{"className":77812,"style":2589},[59,60],[15,77814,120],{"className":77815},[153],"-可测的（对每个 ",[15,77818,77820,77833],{"className":77819,"translate":19},[18],[15,77821,77823],{"className":77822},[23],[25,77824,77825],{"xmlns":27},[29,77826,77827,77831],{},[32,77828,77829],{},[35,77830,40263],{},[39,77832,40263],{"encoding":41},[15,77834,77836],{"className":77835,"ariaHidden":46},[45],[15,77837,77839,77842],{"className":77838},[50],[15,77840],{"className":77841,"style":2278},[54],[15,77843,40263],{"className":77844,"style":981},[59,60]," 是常数），即 ",[15,77847,77849,77874],{"className":77848,"translate":19},[18],[15,77850,77852],{"className":77851},[23],[25,77853,77854],{"xmlns":27},[29,77855,77856,77872],{},[32,77857,77858,77860,77862,77864,77866,77868,77870],{},[35,77859,769],{},[110,77861,123],{},[35,77863,244],{},[35,77865,206],{},[35,77867,941],{"mathvariant":209},[35,77869,244],{},[35,77871,213],{},[39,77873,5335],{"encoding":41},[15,77875,77877,77895],{"className":77876,"ariaHidden":46},[45],[15,77878,77880,77883,77886,77889,77892],{"className":77879},[50],[15,77881],{"className":77882,"style":5345},[54],[15,77884,769],{"className":77885},[59,60],[15,77887],{"className":77888,"style":158},[157],[15,77890,123],{"className":77891},[162],[15,77893],{"className":77894,"style":158},[157],[15,77896,77898,77901,77904,77907,77910,77913],{"className":77897},[50],[15,77899],{"className":77900,"style":139},[54],[15,77902,244],{"className":77903},[59,60],[15,77905,206],{"className":77906,"style":334},[59,60],[15,77908,941],{"className":77909},[59],[15,77911,244],{"className":77912},[59,60],[15,77914,213],{"className":77915},[59,60]," 可表为 ",[15,77918,77920,77933],{"className":77919,"translate":19},[18],[15,77921,77923],{"className":77922},[23],[25,77924,77925],{"xmlns":27},[29,77926,77927,77931],{},[32,77928,77929],{},[35,77930,5403],{},[39,77932,5403],{"encoding":41},[15,77934,77936],{"className":77935,"ariaHidden":46},[45],[15,77937,77939,77942],{"className":77938},[50],[15,77940],{"className":77941,"style":664},[54],[15,77943,5403],{"className":77944,"style":2589},[59,60]," 的函数。此时 ",[15,77947,77949,77962],{"className":77948,"translate":19},[18],[15,77950,77952],{"className":77951},[23],[25,77953,77954],{"xmlns":27},[29,77955,77956,77960],{},[32,77957,77958],{},[35,77959,5403],{},[39,77961,5403],{"encoding":41},[15,77963,77965],{"className":77964,"ariaHidden":46},[45],[15,77966,77968,77971],{"className":77967},[50],[15,77969],{"className":77970,"style":664},[54],[15,77972,5403],{"className":77973,"style":2589},[59,60]," 是关于 ",[15,77976,77978,77995],{"className":77977,"translate":19},[18],[15,77979,77981],{"className":77980},[23],[25,77982,77983],{"xmlns":27},[29,77984,77985,77993],{},[32,77986,77987,77989,77991],{},[35,77988,206],{},[110,77990,551],{"separator":46},[35,77992,213],{},[39,77994,556],{"encoding":41},[15,77996,77998],{"className":77997,"ariaHidden":46},[45],[15,77999,78001,78004,78007,78010,78013],{"className":78000},[50],[15,78002],{"className":78003,"style":566},[54],[15,78005,206],{"className":78006,"style":334},[59,60],[15,78008,551],{"className":78009},[573],[15,78011],{"className":78012,"style":406},[157],[15,78014,213],{"className":78015},[59,60],[998,78017,78018],{},"充分统计量","：随机变换没有丢信息。这正是 f-散度能担当「信息量」的原因。",[11,78021,78022,78025,78026,78067],{},[998,78023,78024],{},"信息单调性（Csiszár 的更进一步结果）。"," 对 α-散度族，还有更强的「信息单调性」：把 ",[15,78027,78029,78046],{"className":78028,"translate":19},[18],[15,78030,78032],{"className":78031},[23],[25,78033,78034],{"xmlns":27},[29,78035,78036,78044],{},[32,78037,78038,78040,78042],{},[35,78039,206],{},[110,78041,551],{"separator":46},[35,78043,213],{},[39,78045,556],{"encoding":41},[15,78047,78049],{"className":78048,"ariaHidden":46},[45],[15,78050,78052,78055,78058,78061,78064],{"className":78051},[50],[15,78053],{"className":78054,"style":566},[54],[15,78056,206],{"className":78057,"style":334},[59,60],[15,78059,551],{"className":78060},[573],[15,78062],{"className":78063,"style":406},[157],[15,78065,213],{"className":78066},[59,60]," 粗粒化（把样本空间划分得更粗）时散度不增，且存在严格单调的「信息序」。这构成了 Chentsov–Morozova 定理与信息几何中单调度量的基础。",[2313,78069,78071],{"id":78070},"_46-局部二次展开","4.6 局部二次展开",[993,78073,78074,78487],{},[11,78075,78076,11064,78079,78107,78108,5647,78182,20982,78286,78435,78436,29959],{},[998,78077,78078],{},"命题 4.5（局部二次展开）",[15,78080,78082,78095],{"className":78081,"translate":19},[18],[15,78083,78085],{"className":78084},[23],[25,78086,78087],{"xmlns":27},[29,78088,78089,78093],{},[32,78090,78091],{},[35,78092,38691],{},[39,78094,38691],{"encoding":41},[15,78096,78098],{"className":78097,"ariaHidden":46},[45],[15,78099,78101,78104],{"className":78100},[50],[15,78102],{"className":78103,"style":12601},[54],[15,78105,38691],{"className":78106},[59,60]," 满足 ",[15,78109,78111,78137],{"className":78110,"translate":19},[18],[15,78112,78114],{"className":78113},[23],[25,78115,78116],{"xmlns":27},[29,78117,78118,78134],{},[32,78119,78120,78122,78124,78126,78128,78130,78132],{},[110,78121,1289],{},[35,78123,38691],{},[1299,78125,1301],{},[35,78127,244],{},[35,78129,206],{},[110,78131,123],{},[115,78133,126],{},[39,78135,78136],{"encoding":41},"\\int 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h^2\\,dP\u003C\\infty",[15,78216,78218,78277],{"className":78217,"ariaHidden":46},[45],[15,78219,78221,78224,78227,78230,78259,78262,78265,78268,78271,78274],{"className":78220},[50],[15,78222],{"className":78223,"style":50354},[54],[15,78225,1289],{"className":78226,"style":5619},[1486,1487,5618],[15,78228],{"className":78229,"style":406},[157],[15,78231,78233,78236],{"className":78232},[59],[15,78234,38691],{"className":78235},[59,60],[15,78237,78239],{"className":78238},[281],[15,78240,78242],{"className":78241},[285],[15,78243,78245],{"className":78244},[290],[15,78246,78248],{"className":78247,"style":20054},[294],[15,78249,78250,78253],{"style":6867},[15,78251],{"className":78252,"style":303},[302],[15,78254,78256],{"className":78255},[307,308,309,310],[15,78257,2465],{"className":78258},[59,310],[15,78260],{"className":78261,"style":406},[157],[15,78263,244],{"className":78264},[59,60],[15,78266,206],{"className":78267,"style":334},[59,60],[15,78269],{"className":78270,"style":158},[157],[15,78272,7253],{"className":78273},[162],[15,78275],{"className":78276,"style":158},[157],[15,78278,78280,78283],{"className":78279},[50],[15,78281],{"className":78282,"style":2347},[54],[15,78284,1027],{"className":78285},[59],[15,78287,78289,78329],{"className":78288,"translate":19},[18],[15,78290,78292],{"className":78291},[23],[25,78293,78294],{"xmlns":27},[29,78295,78296,78326],{},[32,78297,78298,78300,78306,78308,78310,78312,78314,78316,78318,78320,78322,78324],{},[35,78299,244],{},[195,78301,78302,78304],{},[35,78303,213],{},[35,78305,38688],{},[110,78307,123],{},[110,78309,113],{"stretchy":112},[115,78311,117],{},[110,78313,2472],{},[35,78315,38688],{},[35,78317,38691],{},[110,78319,120],{"stretchy":112},[1299,78321,1301],{},[35,78323,244],{},[35,78325,206],{},[39,78327,78328],{"encoding":41},"dQ_\\varepsilon=(1+\\varepsilon h)\\,dP",[15,78330,78332,78390,78411],{"className":78331,"ariaHidden":46},[45],[15,78333,78335,78338,78341,78381,78384,78387],{"className":78334},[50],[15,78336],{"className":78337,"style":55},[54],[15,78339,244],{"className":78340},[59,60],[15,78342,78344,78347],{"className":78343},[59],[15,78345,213],{"className":78346},[59,60],[15,78348,78350],{"className":78349},[281],[15,78351,78353,78373],{"className":78352},[285,286],[15,78354,78356,78370],{"className":78355},[290],[15,78357,78359],{"className":78358,"style":44955},[294],[15,78360,78361,78364],{"style":382},[15,78362],{"className":78363,"style":303},[302],[15,78365,78367],{"className":78366},[307,308,309,310],[15,78368,38688],{"className":78369},[59,60,310],[15,78371,318],{"className":78372},[317],[15,78374,78376],{"className":78375},[290],[15,78377,78379],{"className":78378,"style":7970},[294],[15,78380],{},[15,78382],{"className":78383,"style":158},[157],[15,78385,123],{"className":78386},[162],[15,78388],{"className":78389,"style":158},[157],[15,78391,78393,78396,78399,78402,78405,78408],{"className":78392},[50],[15,78394],{"className":78395,"style":139},[54],[15,78397,113],{"className":78398},[146],[15,78400,117],{"className":78401},[59],[15,78403],{"className":78404,"style":2589},[157],[15,78406,2472],{"className":78407},[2593],[15,78409],{"className":78410,"style":2589},[157],[15,78412,78414,78417,78420,78423,78426,78429,78432],{"className":78413},[50],[15,78415],{"className":78416,"style":139},[54],[15,78418,38688],{"className":78419},[59,60],[15,78421,38691],{"className":78422},[59,60],[15,78424,120],{"className":78425},[153],[15,78427],{"className":78428,"style":406},[157],[15,78430,244],{"className":78431},[59,60],[15,78433,206],{"className":78434,"style":334},[59,60],"，则当 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h^{2}\\,dP\n+O(\\varepsilon^{3}).",[15,78590,78592,78699,78904],{"className":78591,"ariaHidden":46},[45],[15,78593,78595,78598,78638,78641,78681,78684,78687,78690,78693,78696],{"className":78594},[50],[15,78596],{"className":78597,"style":270},[54],[15,78599,78601,78604],{"className":78600},[59],[15,78602,199],{"className":78603,"style":277},[59,60],[15,78605,78607],{"className":78606},[281],[15,78608,78610,78630],{"className":78609},[285,286],[15,78611,78613,78627],{"className":78612},[290],[15,78614,78616],{"className":78615,"style":295},[294],[15,78617,78618,78621],{"style":298},[15,78619],{"className":78620,"style":303},[302],[15,78622,78624],{"className":78623},[307,308,309,310],[15,78625,37],{"className":78626,"style":61},[59,60,310],[15,78628,318],{"className":78629},[317],[15,78631,78633],{"className":78632},[290],[15,78634,78636],{"className":78635,"style":325},[294],[15,78637],{},[15,78639,113],{"className":78640},[146],[15,78642,78644,78647],{"className":78643},[59],[15,78645,213],{"className":78646},[59,60],[15,78648,78650],{"className":78649},[281],[15,78651,78653,78673],{"className":78652},[285,286],[15,78654,78656,78670],{"className":78655},[290],[15,78657,78659],{"className":78658,"style":44955},[294],[15,78660,78661,78664],{"style":382},[15,78662],{"className":78663,"style":303},[302],[15,78665,78667],{"className":78666},[307,308,309,310],[15,78668,38688],{"className":78669},[59,60,310],[15,78671,318],{"className":78672},[317],[15,78674,78676],{"className":78675},[290],[15,78677,78679],{"className":78678,"style":7970},[294],[15,78680],{},[15,78682,210],{"className":78683},[59],[15,78685,206],{"className":78686,"style":334},[59,60],[15,78688,120],{"className":78689},[153],[15,78691],{"className":78692,"style":158},[157],[15,78694,123],{"className":78695},[162],[15,78697],{"className":78698,"style":158},[157],[15,78700,78702,78706,78807,78810,78842,78845,78848,78851,78854,78886,78889,78892,78895,78898,78901],{"className":78701},[50],[15,78703],{"className":78704,"style":78705},[54],"height:2.2911em;vertical-align:-0.8622em;",[15,78707,78709,78712,78804],{"className":78708},[59],[15,78710],{"className":78711},[146,441],[15,78713,78715],{"className":78714},[238],[15,78716,78718,78796],{"className":78717},[285,286],[15,78719,78721,78793],{"className":78720},[290],[15,78722,78725,78736,78744],{"className":78723,"style":78724},[294],"height:1.4289em;",[15,78726,78727,78730],{"style":457},[15,78728],{"className":78729,"style":461},[302],[15,78731,78733],{"className":78732},[59],[15,78734,2465],{"className":78735},[59],[15,78737,78738,78741],{"style":473},[15,78739],{"className":78740,"style":461},[302],[15,78742],{"className":78743,"style":481},[480],[15,78745,78746,78749],{"style":484},[15,78747],{"className":78748,"style":461},[302],[15,78750,78752,78784,78787,78790],{"className":78751},[59],[15,78753,78755,78758],{"className":78754},[59],[15,78756,37],{"className":78757,"style":61},[59,60],[15,78759,78761],{"className":78760},[281],[15,78762,78764],{"className":78763},[285],[15,78765,78767],{"className":78766},[290],[15,78768,78770],{"className":78769,"style":6864},[294],[15,78771,78772,78775],{"style":6867},[15,78773],{"className":78774,"style":303},[302],[15,78776,78778],{"className":78777},[307,308,309,310],[15,78779,78781],{"className":78780},[59,310],[15,78782,21774],{"className":78783},[59,310],[15,78785,113],{"className":78786},[146],[15,78788,117],{"className":78789},[59],[15,78791,120],{"className":78792},[153],[15,78794,318],{"className":78795},[317],[15,78797,78799],{"className":78798},[290],[15,78800,78802],{"className":78801,"style":3780},[294],[15,78803],{},[15,78805],{"className":78806},[153,441],[15,78808],{"className":78809,"style":406},[157],[15,78811,78813,78816],{"className":78812},[59],[15,78814,38688],{"className":78815},[59,60],[15,78817,78819],{"className":78818},[281],[15,78820,78822],{"className":78821},[285],[15,78823,78825],{"className":78824},[290],[15,78826,78828],{"className":78827,"style":19761},[294],[15,78829,78830,78833],{"style":8279},[15,78831],{"className":78832,"style":303},[302],[15,78834,78836],{"className":78835},[307,308,309,310],[15,78837,78839],{"className":78838},[59,310],[15,78840,2465],{"className":78841},[59,310],[15,78843],{"className":78844,"style":1514},[157],[15,78846],{"className":78847,"style":406},[157],[15,78849,1289],{"className":78850,"style":1489},[1486,1487,1488],[15,78852],{"className":78853,"style":406},[157],[15,78855,78857,78860],{"className":78856},[59],[15,78858,38691],{"className":78859},[59,60],[15,78861,78863],{"className":78862},[281],[15,78864,78866],{"className":78865},[285],[15,78867,78869],{"className":78868},[290],[15,78870,78872],{"className":78871,"style":19761},[294],[15,78873,78874,78877],{"style":8279},[15,78875],{"className":78876,"style":303},[302],[15,78878,78880],{"className":78879},[307,308,309,310],[15,78881,78883],{"className":78882},[59,310],[15,78884,2465],{"className":78885},[59,310],[15,78887],{"className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记 ",[15,78960,78962,79002],{"className":78961,"translate":19},[18],[15,78963,78965],{"className":78964},[23],[25,78966,78967],{"xmlns":27},[29,78968,78969,78999],{},[32,78970,78971,78973,78975,78977,78983,78985,78987,78989,78991,78993,78995,78997],{},[35,78972,769],{},[110,78974,123],{},[35,78976,244],{},[195,78978,78979,78981],{},[35,78980,213],{},[35,78982,38688],{},[35,78984,941],{"mathvariant":209},[35,78986,244],{},[35,78988,206],{},[110,78990,123],{},[115,78992,117],{},[110,78994,2472],{},[35,78996,38688],{},[35,78998,38691],{},[39,79000,79001],{"encoding":41},"t=dQ_\\varepsilon\u002FdP=1+\\varepsilon h",[15,79003,79005,79023,79090,79108],{"className":79004,"ariaHidden":46},[45],[15,79006,79008,79011,79014,79017,79020],{"className":79007},[50],[15,79009],{"className":79010,"style":5345},[54],[15,79012,769],{"className":79013},[59,60],[15,79015],{"className":79016,"style":158},[157],[15,79018,123],{"className":79019},[162],[15,79021],{"className":79022,"style":158},[157],[15,79024,79026,79029,79032,79072,79075,79078,79081,79084,79087],{"className":79025},[50],[15,79027],{"className":79028,"style":139},[54],[15,79030,244],{"className":79031},[59,60],[15,79033,79035,79038],{"className":79034},[59],[15,79036,213],{"className":79037},[59,60],[15,79039,79041],{"className":79040},[281],[15,79042,79044,79064],{"className":79043},[285,286],[15,79045,79047,79061],{"className":79046},[290],[15,79048,79050],{"className":79049,"style":44955},[294],[15,79051,79052,79055],{"style":382},[15,79053],{"className":79054,"style":303},[302],[15,79056,79058],{"className":79057},[307,308,309,310],[15,79059,38688],{"className":79060},[59,60,310],[15,79062,318],{"className":79063},[317],[15,79065,79067],{"className":79066},[290],[15,79068,79070],{"className":79069,"style":7970},[294],[15,79071],{},[15,79073,941],{"className":79074},[59],[15,79076,244],{"className":79077},[59,60],[15,79079,206],{"className":79080,"style":334},[59,60],[15,79082],{"className":79083,"style":158},[157],[15,79085,123],{"className":79086},[162],[15,79088],{"className":79089,"style":158},[157],[15,79091,79093,79096,79099,79102,79105],{"className":79092},[50],[15,79094],{"className":79095,"style":12582},[54],[15,79097,117],{"className":79098},[59],[15,79100],{"className":79101,"style":2589},[157],[15,79103,2472],{"className":79104},[2593],[15,79106],{"className":79107,"style":2589},[157],[15,79109,79111,79114,79117],{"className":79110},[50],[15,79112],{"className":79113,"style":12601},[54],[15,79115,38688],{"className":79116},[59,60],[15,79118,38691],{"className":79119},[59,60],"，由 ",[15,79122,79124,79147],{"className":79123,"translate":19},[18],[15,79125,79127],{"className":79126},[23],[25,79128,79129],{"xmlns":27},[29,79130,79131,79145],{},[32,79132,79133,79135,79137,79139,79141,79143],{},[35,79134,37],{},[110,79136,113],{"stretchy":112},[115,79138,117],{},[110,79140,120],{"stretchy":112},[110,79142,123],{},[115,79144,126],{},[39,79146,129],{"encoding":41},[15,79148,79150,79177],{"className":79149,"ariaHidden":46},[45],[15,79151,79153,79156,79159,79162,79165,79168,79171,79174],{"className":79152},[50],[15,79154],{"className":79155,"style":139},[54],[15,79157,37],{"className":79158,"style":61},[59,60],[15,79160,113],{"className":79161},[146],[15,79163,117],{"className":79164},[59],[15,79166,120],{"className":79167},[153],[15,79169],{"className":79170,"style":158},[157],[15,79172,123],{"className":79173},[162],[15,79175],{"className":79176,"style":158},[157],[15,79178,79180,79183],{"className":79179},[50],[15,79181],{"className":79182,"style":172},[54],[15,79184,126],{"className":79185},[59],[15,79187,79189,79214],{"className":79188,"translate":19},[18],[15,79190,79192],{"className":79191},[23],[25,79193,79194],{"xmlns":27},[29,79195,79196,79212],{},[32,79197,79198,79200,79202,79204,79206,79208,79210],{},[110,79199,1289],{},[35,79201,38691],{},[1299,79203,1301],{},[35,79205,244],{},[35,79207,206],{},[110,79209,123],{},[115,79211,126],{},[39,79213,78136],{"encoding":41},[15,79215,79217,79250],{"className":79216,"ariaHidden":46},[45],[15,79218,79220,79223,79226,79229,79232,79235,79238,79241,79244,79247],{"className":79219},[50],[15,79221],{"className":79222,"style":5614},[54],[15,79224,1289],{"className":79225,"style":5619},[1486,1487,5618],[15,79227],{"className":79228,"style":406},[157],[15,79230,38691],{"className":79231},[59,60],[15,79233],{"className":79234,"style":406},[157],[15,79236,244],{"className":79237},[59,60],[15,79239,206],{"className":79240,"style":334},[59,60],[15,79242],{"className":79243,"style":158},[157],[15,79245,123],{"className":79246},[162],[15,79248],{"className":79249,"style":158},[157],[15,79251,79253,79256],{"className":79252},[50],[15,79254],{"className":79255,"style":172},[54],[15,79257,126],{"className":79258},[59]," 做 Taylor 展开：",[15,79261,79263],{"className":79262,"translate":19},[180],[15,79264,79266,79494],{"className":79265,"translate":19},[18],[15,79267,79269],{"className":79268},[23],[25,79270,79271],{"xmlns":27,"display":189},[29,79272,79273,79491],{},[1245,79274,79275,79337,79411],{"rowspacing":1247,"columnalign":1248,"columnspacing":1249},[1251,79276,79277,79303],{},[1254,79278,79279],{},[1257,79280,79281],{"scriptlevel":126,"displaystyle":46},[32,79282,79283,79289,79291,79297,79299,79301],{},[195,79284,79285,79287],{},[35,79286,199],{},[35,79288,37],{},[110,79290,113],{"stretchy":112},[195,79292,79293,79295],{},[35,79294,213],{},[35,79296,38688],{},[35,79298,210],{"mathvariant":209},[35,79300,206],{},[110,79302,120],{"stretchy":112},[1254,79304,79305],{},[1257,79306,79307],{"scriptlevel":126,"displaystyle":46},[32,79308,79309,79311,79313,79319,79321,79323,79325,79327,79329,79331,79333,79335],{},[32,79310],{},[110,79312,123],{},[195,79314,79315,79317],{},[35,79316,223],{"mathvariant":222},[35,79318,206],{},[110,79320,230],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[35,79322,37],{},[110,79324,113],{"stretchy":112},[115,79326,117],{},[110,79328,2472],{},[35,79330,38688],{},[35,79332,38691],{},[110,79334,120],{"stretchy":112},[110,79336,257],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[1251,79338,79339,79345],{},[1254,79340,79341],{},[1257,79342,79343],{"scriptlevel":126,"displaystyle":46},[32,79344],{},[1254,79346,79347],{},[1257,79348,79349],{"scriptlevel":126,"displaystyle":46},[32,79350,79351,79353,79355,79373,79375,79377,79383,79385,79387,79389],{},[32,79352],{},[110,79354,123],{},[6018,79356,79357,79371],{},[6018,79358,79359,79369],{},[32,79360,79361,79363,79365,79367],{},[35,79362,37],{},[110,79364,113],{"stretchy":112},[115,79366,117],{},[110,79368,120],{"stretchy":112},[110,79370,6042],{"stretchy":46},[115,79372,126],{},[110,79374,2472],{},[35,79376,38688],{},[6816,79378,79379,79381],{},[35,79380,37],{},[110,79382,6822],{"mathvariant":209,"lspace":1249,"rspace":1249},[110,79384,113],{"stretchy":112},[115,79386,117],{},[110,79388,120],{"stretchy":112},[6018,79390,79391,79409],{},[6018,79392,79393,79407],{},[32,79394,79395,79401,79403,79405],{},[195,79396,79397,79399],{},[35,79398,223],{"mathvariant":222},[35,79400,206],{},[110,79402,230],{"stretchy":112},[35,79404,38691],{},[110,79406,257],{"stretchy":112},[110,79408,6042],{"stretchy":46},[115,79410,126],{},[1251,79412,79413,79419],{},[1254,79414,79415],{},[1257,79416,79417],{"scriptlevel":126,"displaystyle":46},[32,79418],{},[1254,79420,79421],{},[1257,79422,79423],{"scriptlevel":126,"displaystyle":46},[32,79424,79425,79427,79429,79431,79441,79451,79453,79455,79457,79459,79465,79467,79473,79475,79477,79479,79481,79487,79489],{},[32,79426],{},[157,79428],{"width":3264},[110,79430,2472],{},[238,79432,79433,79439],{},[6816,79434,79435,79437],{},[35,79436,38688],{},[115,79438,2465],{},[115,79440,2465],{},[6816,79442,79443,79445],{},[35,79444,37],{},[32,79446,79447,79449],{},[110,79448,6822],{"mathvariant":209},[110,79450,6822],{"mathvariant":209},[110,79452,113],{"stretchy":112},[115,79454,117],{},[110,79456,120],{"stretchy":112},[1299,79458,1301],{},[195,79460,79461,79463],{},[35,79462,223],{"mathvariant":222},[35,79464,206],{},[110,79466,230],{"stretchy":112},[6816,79468,79469,79471],{},[35,79470,38691],{},[115,79472,2465],{},[110,79474,257],{"stretchy":112},[110,79476,2472],{},[35,79478,19247],{},[110,79480,113],{"stretchy":112},[6816,79482,79483,79485],{},[35,79484,38688],{},[115,79486,40761],{},[110,79488,120],{"stretchy":112},[35,79490,4377],{"mathvariant":209},[39,79492,79493],{"encoding":41},"\\begin{aligned}\nD_f(Q_\\varepsilon\\|P)\n&=\\mathbb E_P\\big[f(1+\\varepsilon h)\\big]\\\\[2pt]\n&=\\underbrace{f(1)}_{0}\n+\\varepsilon f'(1)\n\\underbrace{\\mathbb E_P[h]}_{0}\\\\\n&\\qquad+\\frac{\\varepsilon^2}{2}f''(1)\\,\n\\mathbb 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f-散度都只差一个常数 ",[15,80408,80410,80437],{"className":80409,"translate":19},[18],[15,80411,80413],{"className":80412},[23],[25,80414,80415],{"xmlns":27},[29,80416,80417,80435],{},[32,80418,80419,80429,80431,80433],{},[6816,80420,80421,80423],{},[35,80422,37],{},[32,80424,80425,80427],{},[110,80426,6822],{"mathvariant":209},[110,80428,6822],{"mathvariant":209},[110,80430,113],{"stretchy":112},[115,80432,117],{},[110,80434,120],{"stretchy":112},[39,80436,39560],{"encoding":41},[15,80438,80440],{"className":80439,"ariaHidden":46},[45],[15,80441,80443,80446,80478,80481,80484],{"className":80442},[50],[15,80444],{"className":80445,"style":6845},[54],[15,80447,80449,80452],{"className":80448},[59],[15,80450,37],{"className":80451,"style":61},[59,60],[15,80453,80455],{"className":80454},[281],[15,80456,80458],{"className":80457},[285],[15,80459,80461],{"className":80460},[290],[15,80462,80464],{"className":80463,"style":6864},[294],[15,80465,80466,80469],{"style":6867},[15,80467],{"className":80468,"style":303},[302],[15,80470,80472],{"className":80471},[307,308,309,310],[15,80473,80475],{"className":80474},[59,310],[15,80476,21774],{"className":80477},[59,310],[15,80479,113],{"className":80480},[146],[15,80482,117],{"className":80483},[59],[15,80485,120],{"className":80486},[153],"，形状完全一致。",[15,80489,80491,80520],{"className":80490,"translate":19},[18],[15,80492,80494],{"className":80493},[23],[25,80495,80496],{"xmlns":27},[29,80497,80498,80518],{},[32,80499,80500,80502,80504,80506,80508,80510,80512,80514,80516],{},[35,80501,37],{},[110,80503,113],{"stretchy":112},[35,80505,769],{},[110,80507,120],{"stretchy":112},[110,80509,123],{},[35,80511,769],{},[35,80513,4102],{},[110,80515,4105],{},[35,80517,769],{},[39,80519,4110],{"encoding":41},[15,80521,80523,80550],{"className":80522,"ariaHidden":46},[45],[15,80524,80526,80529,80532,80535,80538,80541,80544,80547],{"className":80525},[50],[15,80527],{"className":80528,"style":139},[54],[15,80530,37],{"className":80531,"style":61},[59,60],[15,80533,113],{"className":80534},[146],[15,80536,769],{"className":80537},[59,60],[15,80539,120],{"className":80540},[153],[15,80542],{"className":80543,"style":158},[157],[15,80545,123],{"className":80546},[162],[15,80548],{"className":80549,"style":158},[157],[15,80551,80553,80556,80559,80562,80567,80570],{"className":80552},[50],[15,80554],{"className":80555,"style":55},[54],[15,80557,769],{"className":80558},[59,60],[15,80560],{"className":80561,"style":406},[157],[15,80563,4156,80565],{"className":80564},[1486],[15,80566,4160],{"style":4159},[15,80568],{"className":80569,"style":406},[157],[15,80571,769],{"className":80572},[59,60],[15,80574,80576,80608],{"className":80575,"translate":19},[18],[15,80577,80579],{"className":80578},[23],[25,80580,80581],{"xmlns":27},[29,80582,80583,80605],{},[32,80584,80585,80595,80597,80599,80601,80603],{},[6816,80586,80587,80589],{},[35,80588,37],{},[32,80590,80591,80593],{},[110,80592,6822],{"mathvariant":209},[110,80594,6822],{"mathvariant":209},[110,80596,113],{"stretchy":112},[115,80598,117],{},[110,80600,120],{"stretchy":112},[110,80602,123],{},[115,80604,117],{},[39,80606,80607],{"encoding":41},"f''(1)=1",[15,80609,80611,80667],{"className":80610,"ariaHidden":46},[45],[15,80612,80614,80617,80649,80652,80655,80658,80661,80664],{"className":80613},[50],[15,80615],{"className":80616,"style":6845},[54],[15,80618,80620,80623],{"className":80619},[59],[15,80621,37],{"className":80622,"style":61},[59,60],[15,80624,80626],{"className":80625},[281],[15,80627,80629],{"className":80628},[285],[15,80630,80632],{"className":80631},[290],[15,80633,80635],{"className":80634,"style":6864},[294],[15,80636,80637,80640],{"style":6867},[15,80638],{"className":80639,"style":303},[302],[15,80641,80643],{"className":80642},[307,308,309,310],[15,80644,80646],{"className":80645},[59,310],[15,80647,21774],{"className":80648},[59,310],[15,80650,113],{"className":80651},[146],[15,80653,117],{"className":80654},[59],[15,80656,120],{"className":80657},[153],[15,80659],{"className":80660,"style":158},[157],[15,80662,123],{"className":80663},[162],[15,80665],{"className":80666,"style":158},[157],[15,80668,80670,80673],{"className":80669},[50],[15,80671],{"className":80672,"style":172},[54],[15,80674,117],{"className":80675},[59],"）即得 ",[15,80678,80680,80716],{"className":80679,"translate":19},[18],[15,80681,80683],{"className":80682},[23],[25,80684,80685],{"xmlns":27},[29,80686,80687,80713],{},[32,80688,80689,80699,80701,80707],{},[195,80690,80691,80693],{},[35,80692,199],{},[32,80694,80695,80697],{},[35,80696,10698],{},[35,80698,21919],{},[110,80700,32985],{},[238,80702,80703,80705],{},[115,80704,117],{},[115,80706,2465],{},[6816,80708,80709,80711],{},[35,80710,37475],{},[115,80712,2465],{},[39,80714,80715],{"encoding":41},"D_{KL}\\approx\\frac12\\chi^2",[15,80717,80719,80780],{"className":80718,"ariaHidden":46},[45],[15,80720,80722,80725,80771,80774,80777],{"className":80721},[50],[15,80723],{"className":80724,"style":15965},[54],[15,80726,80728,80731],{"className":80727},[59],[15,80729,199],{"className":80730,"style":277},[59,60],[15,80732,80734],{"className":80733},[281],[15,80735,80737,80763],{"className":80736},[285,286],[15,80738,80740,80760],{"className":80739},[290],[15,80741,80743],{"className":80742,"style":379},[294],[15,80744,80745,80748],{"style":298},[15,80746],{"className":80747,"style":303},[302],[15,80749,80751],{"className":80750},[307,308,309,310],[15,80752,80754,80757],{"className":80753},[59,310],[15,80755,10698],{"className":80756,"style":10786},[59,60,310],[15,80758,21919],{"className":80759},[59,60,310],[15,80761,318],{"className":80762},[317],[15,80764,80766],{"className":80765},[290],[15,80767,80769],{"className":80768,"style":7970},[294],[15,80770],{},[15,80772],{"className":80773,"style":158},[157],[15,80775,32985],{"className":80776},[162],[15,80778],{"className":80779,"style":158},[157],[15,80781,80783,80786,80854],{"className":80782},[50],[15,80784],{"className":80785,"style":2507},[54],[15,80787,80789,80792,80851],{"className":80788},[59],[15,80790],{"className":80791},[146,441],[15,80793,80795],{"className":80794},[238],[15,80796,80798,80843],{"className":80797},[285,286],[15,80799,80801,80840],{"className":80800},[290],[15,80802,80804,80818,80826],{"className":80803,"style":2526},[294],[15,80805,80806,80809],{"style":2529},[15,80807],{"className":80808,"style":461},[302],[15,80810,80812],{"className":80811},[307,308,309,310],[15,80813,80815],{"className":80814},[59,310],[15,80816,2465],{"className":80817},[59,310],[15,80819,80820,80823],{"style":473},[15,80821],{"className":80822,"style":461},[302],[15,80824],{"className":80825,"style":481},[480],[15,80827,80828,80831],{"style":2552},[15,80829],{"className":80830,"style":461},[302],[15,80832,80834],{"className":80833},[307,308,309,310],[15,80835,80837],{"className":80836},[59,310],[15,80838,117],{"className":80839},[59,310],[15,80841,318],{"className":80842},[317],[15,80844,80846],{"className":80845},[290],[15,80847,80849],{"className":80848,"style":2574},[294],[15,80850],{},[15,80852],{"className":80853},[153,441],[15,80855,80857,80860],{"className":80856},[59],[15,80858,37475],{"className":80859},[59,60],[15,80861,80863],{"className":80862},[281],[15,80864,80866],{"className":80865},[285],[15,80867,80869],{"className":80868},[290],[15,80870,80872],{"className":80871,"style":20054},[294],[15,80873,80874,80877],{"style":6867},[15,80875],{"className":80876,"style":303},[302],[15,80878,80880],{"className":80879},[307,308,309,310],[15,80881,2465],{"className":80882},[59,310],"；取 ",[15,80885,80887,80924],{"className":80886,"translate":19},[18],[15,80888,80890],{"className":80889},[23],[25,80891,80892],{"xmlns":27},[29,80893,80894,80922],{},[32,80895,80896,80898,80900,80902,80904,80906,80908,80912,80914,80916],{},[35,80897,37],{},[110,80899,113],{"stretchy":112},[35,80901,769],{},[110,80903,120],{"stretchy":112},[110,80905,123],{},[110,80907,113],{"stretchy":112},[34365,80909,80910],{},[35,80911,769],{},[110,80913,4815],{},[115,80915,117],{},[6816,80917,80918,80920],{},[110,80919,120],{"stretchy":112},[115,80921,2465],{},[39,80923,42656],{"encoding":41},[15,80925,80927,80954,81016],{"className":80926,"ariaHidden":46},[45],[15,80928,80930,80933,80936,80939,80942,80945,80948,80951],{"className":80929},[50],[15,80931],{"className":80932,"style":139},[54],[15,80934,37],{"className":80935,"style":61},[59,60],[15,80937,113],{"className":80938},[146],[15,80940,769],{"className":80941},[59,60],[15,80943,120],{"className":80944},[153],[15,80946],{"className":80947,"style":158},[157],[15,80949,123],{"className":80950},[162],[15,80952],{"className":80953,"style":158},[157],[15,80955,80957,80960,80963,81007,81010,81013],{"className":80956},[50],[15,80958],{"className":80959,"style":42693},[54],[15,80961,113],{"className":80962},[146],[15,80964,80966],{"className":80965},[59,34393],[15,80967,80969,80999],{"className":80968},[285,286],[15,80970,80972,80996],{"className":80971},[290],[15,80973,80975,80984],{"className":80974,"style":42709},[294],[15,80976,80978,80981],{"className":80977,"style":11867},[6309],[15,80979],{"className":80980,"style":461},[302],[15,80982,769],{"className":80983,"style":34413},[59,60],[15,80985,80986,80989],{"style":42721},[15,80987],{"className":80988,"style":461},[302],[15,80990,80992],{"className":80991,"style":34470},[34469],[6325,80993,80994],{"xmlns":6327,"width":6328,"height":34473,"viewBox":34474,"preserveAspectRatio":6331},[6333,80995],{"d":34477},[15,80997,318],{"className":80998},[317],[15,81000,81002],{"className":81001},[290],[15,81003,81005],{"className":81004,"style":42741},[294],[15,81006],{},[15,81008],{"className":81009,"style":2589},[157],[15,81011,4815],{"className":81012},[2593],[15,81014],{"className":81015,"style":2589},[157],[15,81017,81019,81022,81025],{"className":81018},[50],[15,81020],{"className":81021,"style":20029},[54],[15,81023,117],{"className":81024},[59],[15,81026,81028,81031],{"className":81027},[153],[15,81029,120],{"className":81030},[153],[15,81032,81034],{"className":81033},[281],[15,81035,81037],{"className":81036},[285],[15,81038,81040],{"className":81039},[290],[15,81041,81043],{"className":81042,"style":20054},[294],[15,81044,81045,81048],{"style":6867},[15,81046],{"className":81047,"style":303},[302],[15,81049,81051],{"className":81050},[307,308,309,310],[15,81052,2465],{"className":81053},[59,310],[15,81055,81057,81093],{"className":81056,"translate":19},[18],[15,81058,81060],{"className":81059},[23],[25,81061,81062],{"xmlns":27},[29,81063,81064,81090],{},[32,81065,81066,81076,81078,81080,81082,81084],{},[6816,81067,81068,81070],{},[35,81069,37],{},[32,81071,81072,81074],{},[110,81073,6822],{"mathvariant":209},[110,81075,6822],{"mathvariant":209},[110,81077,113],{"stretchy":112},[115,81079,117],{},[110,81081,120],{"stretchy":112},[110,81083,123],{},[238,81085,81086,81088],{},[115,81087,117],{},[115,81089,2465],{},[39,81091,81092],{"encoding":41},"f''(1)=\\frac12",[15,81094,81096,81152],{"className":81095,"ariaHidden":46},[45],[15,81097,81099,81102,81134,81137,81140,81143,81146,81149],{"className":81098},[50],[15,81100],{"className":81101,"style":6845},[54],[15,81103,81105,81108],{"className":81104},[59],[15,81106,37],{"className":81107,"style":61},[59,60],[15,81109,81111],{"className":81110},[281],[15,81112,81114],{"className":81113},[285],[15,81115,81117],{"className":81116},[290],[15,81118,81120],{"className":81119,"style":6864},[294],[15,81121,81122,81125],{"style":6867},[15,81123],{"className":81124,"style":303},[302],[15,81126,81128],{"className":81127},[307,308,309,310],[15,81129,81131],{"className":81130},[59,310],[15,81132,21774],{"className":81133},[59,310],[15,81135,113],{"className":81136},[146],[15,81138,117],{"className":81139},[59],[15,81141,120],{"className":81142},[153],[15,81144],{"className":81145,"style":158},[157],[15,81147,123],{"className":81148},[162],[15,81150],{"className":81151,"style":158},[157],[15,81153,81155,81158],{"className":81154},[50],[15,81156],{"className":81157,"style":2507},[54],[15,81159,81161,81164,81223],{"className":81160},[59],[15,81162],{"className":81163},[146,441],[15,81165,81167],{"className":81166},[238],[15,81168,81170,81215],{"className":81169},[285,286],[15,81171,81173,81212],{"className":81172},[290],[15,81174,81176,81190,81198],{"className":81175,"style":2526},[294],[15,81177,81178,81181],{"style":2529},[15,81179],{"className":81180,"style":461},[302],[15,81182,81184],{"className":81183},[307,308,309,310],[15,81185,81187],{"className":81186},[59,310],[15,81188,2465],{"className":81189},[59,310],[15,81191,81192,81195],{"style":473},[15,81193],{"className":81194,"style":461},[302],[15,81196],{"className":81197,"style":481},[480],[15,81199,81200,81203],{"style":2552},[15,81201],{"className":81202,"style":461},[302],[15,81204,81206],{"className":81205},[307,308,309,310],[15,81207,81209],{"className":81208},[59,310],[15,81210,117],{"className":81211},[59,310],[15,81213,318],{"className":81214},[317],[15,81216,81218],{"className":81217},[290],[15,81219,81221],{"className":81220,"style":2574},[294],[15,81222],{},[15,81224],{"className":81225},[153,441],[15,81227,81229,81274],{"className":81228,"translate":19},[18],[15,81230,81232],{"className":81231},[23],[25,81233,81234],{"xmlns":27},[29,81235,81236,81272],{},[32,81237,81238,81240,81242,81246,81248,81252,81258,81260,81266],{},[110,81239,1289],{},[110,81241,113],{"stretchy":112},[34365,81243,81244],{},[35,81245,11],{},[110,81247,4815],{},[34365,81249,81250],{},[35,81251,944],{},[6816,81253,81254,81256],{},[110,81255,120],{"stretchy":112},[115,81257,2465],{},[110,81259,32985],{},[238,81261,81262,81264],{},[115,81263,117],{},[115,81265,39015],{},[6816,81267,81268,81270],{},[35,81269,37475],{},[115,81271,2465],{},[39,81273,39024],{"encoding":41},[15,81275,81277,81342,81430],{"className":81276,"ariaHidden":46},[45],[15,81278,81280,81283,81286,81289,81333,81336,81339],{"className":81279},[50],[15,81281],{"className":81282,"style":39034},[54],[15,81284,1289],{"className":81285,"style":5619},[1486,1487,5618],[15,81287,113],{"className":81288},[146],[15,81290,81292],{"className":81291},[59,34393],[15,81293,81295,81325],{"className":81294},[285,286],[15,81296,81298,81322],{"className":81297},[290],[15,81299,81301,81310],{"className":81300,"style":39053},[294],[15,81302,81304,81307],{"className":81303,"style":11867},[6309],[15,81305],{"className":81306,"style":461},[302],[15,81308,11],{"className":81309,"style":34413},[59,60],[15,81311,81312,81315],{"style":39065},[15,81313],{"className":81314,"style":461},[302],[15,81316,81318],{"className":81317,"style":34470},[34469],[6325,81319,81320],{"xmlns":6327,"width":6328,"height":34473,"viewBox":34474,"preserveAspectRatio":6331},[6333,81321],{"d":34477},[15,81323,318],{"className":81324},[317],[15,81326,81328],{"className":81327},[290],[15,81329,81331],{"className":81330,"style":39085},[294],[15,81332],{},[15,81334],{"className":81335,"style":2589},[157],[15,81337,4815],{"className":81338},[2593],[15,81340],{"className":81341,"style":2589},[157],[15,81343,81345,81348,81392,81421,81424,81427],{"className":81344},[50],[15,81346],{"className":81347,"style":39103},[54],[15,81349,81351],{"className":81350},[59,34393],[15,81352,81354,81384],{"className":81353},[285,286],[15,81355,81357,81381],{"className":81356},[290],[15,81358,81360,81369],{"className":81359,"style":39053},[294],[15,81361,81363,81366],{"className":81362,"style":11867},[6309],[15,81364],{"className":81365,"style":461},[302],[15,81367,944],{"className":81368,"style":39125},[59,60],[15,81370,81371,81374],{"style":39065},[15,81372],{"className":81373,"style":461},[302],[15,81375,81377],{"className":81376,"style":34470},[34469],[6325,81378,81379],{"xmlns":6327,"width":6328,"height":34473,"viewBox":34474,"preserveAspectRatio":6331},[6333,81380],{"d":34477},[15,81382,318],{"className":81383},[317],[15,81385,81387],{"className":81386},[290],[15,81388,81390],{"className":81389,"style":39085},[294],[15,81391],{},[15,81393,81395,81398],{"className":81394},[153],[15,81396,120],{"className":81397},[153],[15,81399,81401],{"className":81400},[281],[15,81402,81404],{"className":81403},[285],[15,81405,81407],{"className":81406},[290],[15,81408,81410],{"className":81409,"style":20054},[294],[15,81411,81412,81415],{"style":6867},[15,81413],{"className":81414,"style":303},[302],[15,81416,81418],{"className":81417},[307,308,309,310],[15,81419,2465],{"className":81420},[59,310],[15,81422],{"className":81423,"style":158},[157],[15,81425,32985],{"className":81426},[162],[15,81428],{"className":81429,"style":158},[157],[15,81431,81433,81436,81504],{"className":81432},[50],[15,81434],{"className":81435,"style":2507},[54],[15,81437,81439,81442,81501],{"className":81438},[59],[15,81440],{"className":81441},[146,441],[15,81443,81445],{"className":81444},[238],[15,81446,81448,81493],{"className":81447},[285,286],[15,81449,81451,81490],{"className":81450},[290],[15,81452,81454,81468,81476],{"className":81453,"style":2526},[294],[15,81455,81456,81459],{"style":2529},[15,81457],{"className":81458,"style":461},[302],[15,81460,81462],{"className":81461},[307,308,309,310],[15,81463,81465],{"className":81464},[59,310],[15,81466,39015],{"className":81467},[59,310],[15,81469,81470,81473],{"style":473},[15,81471],{"className":81472,"style":461},[302],[15,81474],{"className":81475,"style":481},[480],[15,81477,81478,81481],{"style":2552},[15,81479],{"className":81480,"style":461},[302],[15,81482,81484],{"className":81483},[307,308,309,310],[15,81485,81487],{"className":81486},[59,310],[15,81488,117],{"className":81489},[59,310],[15,81491,318],{"className":81492},[317],[15,81494,81496],{"className":81495},[290],[15,81497,81499],{"className":81498,"style":2574},[294],[15,81500],{},[15,81502],{"className":81503},[153,441],[15,81505,81507,81510],{"className":81506},[59],[15,81508,37475],{"className":81509},[59,60],[15,81511,81513],{"className":81512},[281],[15,81514,81516],{"className":81515},[285],[15,81517,81519],{"className":81518},[290],[15,81520,81522],{"className":81521,"style":20054},[294],[15,81523,81524,81527],{"style":6867},[15,81525],{"className":81526,"style":303},[302],[15,81528,81530],{"className":81529},[307,308,309,310],[15,81531,2465],{"className":81532},[59,310],"，除以 2 或换算成 ",[15,81535,81537,81557],{"className":81536,"translate":19},[18],[15,81538,81540],{"className":81539},[23],[25,81541,81542],{"xmlns":27},[29,81543,81544,81554],{},[32,81545,81546,81548,81550,81552],{},[115,81547,117],{},[110,81549,4815],{},[35,81551,39317],{},[35,81553,39320],{},[39,81555,81556],{"encoding":41},"1-BC",[15,81558,81560,81578],{"className":81559,"ariaHidden":46},[45],[15,81561,81563,81566,81569,81572,81575],{"className":81562},[50],[15,81564],{"className":81565,"style":12582},[54],[15,81567,117],{"className":81568},[59],[15,81570],{"className":81571,"style":2589},[157],[15,81573,4815],{"className":81574},[2593],[15,81576],{"className":81577,"style":2589},[157],[15,81579,81581,81584,81587],{"className":81580},[50],[15,81582],{"className":81583,"style":664},[54],[15,81585,39317],{"className":81586,"style":39413},[59,60],[15,81588,39320],{"className":81589,"style":10786},[59,60]," 就是 3.4 节引用的 Bhattacharyya 约定 ",[15,81592,81594,81626],{"className":81593,"translate":19},[18],[15,81595,81597],{"className":81596},[23],[25,81598,81599],{"xmlns":27},[29,81600,81601,81623],{},[32,81602,81603,81609,81611,81617],{},[6816,81604,81605,81607],{},[35,81606,33910],{},[115,81608,2465],{},[110,81610,32985],{},[238,81612,81613,81615],{},[115,81614,117],{},[115,81616,39329],{},[6816,81618,81619,81621],{},[35,81620,37475],{},[115,81622,2465],{},[39,81624,81625],{"encoding":41},"H^2\\approx\\frac18\\chi^2",[15,81627,81629,81673],{"className":81628,"ariaHidden":46},[45],[15,81630,81632,81635,81664,81667,81670],{"className":81631},[50],[15,81633],{"className":81634,"style":20054},[54],[15,81636,81638,81641],{"className":81637},[59],[15,81639,33910],{"className":81640,"style":34064},[59,60],[15,81642,81644],{"className":81643},[281],[15,81645,81647],{"className":81646},[285],[15,81648,81650],{"className":81649},[290],[15,81651,81653],{"className":81652,"style":20054},[294],[15,81654,81655,81658],{"style":6867},[15,81656],{"className":81657,"style":303},[302],[15,81659,81661],{"className":81660},[307,308,309,310],[15,81662,2465],{"className":81663},[59,310],[15,81665],{"className":81666,"style":158},[157],[15,81668,32985],{"className":81669},[162],[15,81671],{"className":81672,"style":158},[157],[15,81674,81676,81679,81747],{"className":81675},[50],[15,81677],{"className":81678,"style":2507},[54],[15,81680,81682,81685,81744],{"className":81681},[59],[15,81683],{"className":81684},[146,441],[15,81686,81688],{"className":81687},[238],[15,81689,81691,81736],{"className":81690},[285,286],[15,81692,81694,81733],{"className":81693},[290],[15,81695,81697,81711,81719],{"className":81696,"style":2526},[294],[15,81698,81699,81702],{"style":2529},[15,81700],{"className":81701,"style":461},[302],[15,81703,81705],{"className":81704},[307,308,309,310],[15,81706,81708],{"className":81707},[59,310],[15,81709,39329],{"className":81710},[59,310],[15,81712,81713,81716],{"style":473},[15,81714],{"className":81715,"style":461},[302],[15,81717],{"className":81718,"style":481},[480],[15,81720,81721,81724],{"style":2552},[15,81722],{"className":81723,"style":461},[302],[15,81725,81727],{"className":81726},[307,308,309,310],[15,81728,81730],{"className":81729},[59,310],[15,81731,117],{"className":81732},[59,310],[15,81734,318],{"className":81735},[317],[15,81737,81739],{"className":81738},[290],[15,81740,81742],{"className":81741,"style":2574},[294],[15,81743],{},[15,81745],{"className":81746},[153,441],[15,81748,81750,81753],{"className":81749},[59],[15,81751,37475],{"className":81752},[59,60],[15,81754,81756],{"className":81755},[281],[15,81757,81759],{"className":81758},[285],[15,81760,81762],{"className":81761},[290],[15,81763,81765],{"className":81764,"style":20054},[294],[15,81766,81767,81770],{"style":6867},[15,81768],{"className":81769,"style":303},[302],[15,81771,81773],{"className":81772},[307,308,309,310],[15,81774,2465],{"className":81775},[59,310],"。共同因子 ",[15,81778,81780,81832],{"className":81779,"translate":19},[18],[15,81781,81783],{"className":81782},[23],[25,81784,81785],{"xmlns":27},[29,81786,81787,81829],{},[32,81788,81789,81795,81797,81803,81805,81807,81809,81815,81817,81823,81825,81827],{},[6816,81790,81791,81793],{},[35,81792,38688],{},[115,81794,2465],{},[110,81796,1289],{},[6816,81798,81799,81801],{},[35,81800,38691],{},[115,81802,2465],{},[35,81804,244],{},[35,81806,206],{},[110,81808,123],{},[6816,81810,81811,81813],{},[35,81812,37475],{},[115,81814,2465],{},[110,81816,113],{"stretchy":112},[195,81818,81819,81821],{},[35,81820,213],{},[35,81822,38688],{},[35,81824,210],{"mathvariant":209},[35,81826,206],{},[110,81828,120],{"stretchy":112},[39,81830,81831],{"encoding":41},"\\varepsilon^2\\int h^2 dP=\\chi^2(Q_\\varepsilon\\|P)",[15,81833,81835,81923],{"className":81834,"ariaHidden":46},[45],[15,81836,81838,81841,81870,81873,81876,81879,81908,81911,81914,81917,81920],{"className":81837},[50],[15,81839],{"className":81840,"style":50354},[54],[15,81842,81844,81847],{"className":81843},[59],[15,81845,38688],{"className":81846},[59,60],[15,81848,81850],{"className":81849},[281],[15,81851,81853],{"className":81852},[285],[15,81854,81856],{"className":81855},[290],[15,81857,81859],{"className":81858,"style":20054},[294],[15,81860,81861,81864],{"style":6867},[15,81862],{"className":81863,"style":303},[302],[15,81865,81867],{"className":81866},[307,308,309,310],[15,81868,2465],{"className":81869},[59,310],[15,81871],{"className":81872,"style":406},[157],[15,81874,1289],{"className":81875,"style":5619},[1486,1487,5618],[15,81877],{"className":81878,"style":406},[157],[15,81880,81882,81885],{"className":81881},[59],[15,81883,38691],{"className":81884},[59,60],[15,81886,81888],{"className":81887},[281],[15,81889,81891],{"className":81890},[285],[15,81892,81894],{"className":81893},[290],[15,81895,81897],{"className":81896,"style":20054},[294],[15,81898,81899,81902],{"style":6867},[15,81900],{"className":81901,"style":303},[302],[15,81903,81905],{"className":81904},[307,308,309,310],[15,81906,2465],{"className":81907},[59,310],[15,81909,244],{"className":81910},[59,60],[15,81912,206],{"className":81913,"style":334},[59,60],[15,81915],{"className":81916,"style":158},[157],[15,81918,123],{"className":81919},[162],[15,81921],{"className":81922,"style":158},[157],[15,81924,81926,81929,81958,81961,82001,82004,82007],{"className":81925},[50],[15,81927],{"className":81928,"style":20029},[54],[15,81930,81932,81935],{"className":81931},[59],[15,81933,37475],{"className":81934},[59,60],[15,81936,81938],{"className":81937},[281],[15,81939,81941],{"className":81940},[285],[15,81942,81944],{"className":81943},[290],[15,81945,81947],{"className":81946,"style":20054},[294],[15,81948,81949,81952],{"style":6867},[15,81950],{"className":81951,"style":303},[302],[15,81953,81955],{"className":81954},[307,308,309,310],[15,81956,2465],{"className":81957},[59,310],[15,81959,113],{"className":81960},[146],[15,81962,81964,81967],{"className":81963},[59],[15,81965,213],{"className":81966},[59,60],[15,81968,81970],{"className":81969},[281],[15,81971,81973,81993],{"className":81972},[285,286],[15,81974,81976,81990],{"className":81975},[290],[15,81977,81979],{"className":81978,"style":44955},[294],[15,81980,81981,81984],{"style":382},[15,81982],{"className":81983,"style":303},[302],[15,81985,81987],{"className":81986},[307,308,309,310],[15,81988,38688],{"className":81989},[59,60,310],[15,81991,318],{"className":81992},[317],[15,81994,81996],{"className":81995},[290],[15,81997,81999],{"className":81998,"style":7970},[294],[15,82000],{},[15,82002,210],{"className":82003},[59],[15,82005,206],{"className":82006,"style":334},[59,60],[15,82008,120],{"className":82009},[153]," 就是",[998,82012,82013],{},"Fisher 信息度量","：它给出统计流形上的黎曼度量，而 ",[15,82016,82018,82045],{"className":82017,"translate":19},[18],[15,82019,82021],{"className":82020},[23],[25,82022,82023],{"xmlns":27},[29,82024,82025,82043],{},[32,82026,82027,82037,82039,82041],{},[6816,82028,82029,82031],{},[35,82030,37],{},[32,82032,82033,82035],{},[110,82034,6822],{"mathvariant":209},[110,82036,6822],{"mathvariant":209},[110,82038,113],{"stretchy":112},[115,82040,117],{},[110,82042,120],{"stretchy":112},[39,82044,39560],{"encoding":41},[15,82046,82048],{"className":82047,"ariaHidden":46},[45],[15,82049,82051,82054,82086,82089,82092],{"className":82050},[50],[15,82052],{"className":82053,"style":6845},[54],[15,82055,82057,82060],{"className":82056},[59],[15,82058,37],{"className":82059,"style":61},[59,60],[15,82061,82063],{"className":82062},[281],[15,82064,82066],{"className":82065},[285],[15,82067,82069],{"className":82068},[290],[15,82070,82072],{"className":82071,"style":6864},[294],[15,82073,82074,82077],{"style":6867},[15,82075],{"className":82076,"style":303},[302],[15,82078,82080],{"className":82079},[307,308,309,310],[15,82081,82083],{"className":82082},[59,310],[15,82084,21774],{"className":82085},[59,310],[15,82087,113],{"className":82088},[146],[15,82090,117],{"className":82091},[59],[15,82093,120],{"className":82094},[153]," 只是「单位选择」。这也解释了为什么 GAN 中「选 JS 还是选 KL」在小偏差区几乎无差别，差别只在远离平衡或支撑不重叠时才显现。",[2313,82097,82099],{"id":82098},"_47-不等式网络","4.7 不等式网络",[993,82101,82102,82298,82804,83265],{},[11,82103,82104,78958,82107,82297],{},[998,82105,82106],{},"命题 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D_{TV}(P,Q)\\le\n\\sqrt{\\tfrac12 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2 步（二元问题的精确最小值）。"," 固定差 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TV 值），对 ",[15,84574,84576,84589],{"className":84575,"translate":19},[18],[15,84577,84579],{"className":84578},[23],[25,84580,84581],{"xmlns":27},[29,84582,84583,84587],{},[32,84584,84585],{},[35,84586,44780],{},[39,84588,49116],{"encoding":41},[15,84590,84592],{"className":84591,"ariaHidden":46},[45],[15,84593,84595,84598],{"className":84594},[50],[15,84596],{"className":84597,"style":2347},[54],[15,84599,44780],{"className":84600,"style":44810},[59,60]," 最小化二元 KL。取对称中心 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",[15,84869,84871,84893],{"className":84870,"translate":19},[18],[15,84872,84874],{"className":84873},[23],[25,84875,84876],{"xmlns":27},[29,84877,84878,84890],{},[32,84879,84880,84882,84884,84886,84888],{},[115,84881,117],{},[110,84883,4815],{},[35,84885,44780],{},[110,84887,123],{},[35,84889,84158],{},[39,84891,84892],{"encoding":41},"1-\\alpha=\\beta",[15,84894,84896,84914,84932],{"className":84895,"ariaHidden":46},[45],[15,84897,84899,84902,84905,84908,84911],{"className":84898},[50],[15,84900],{"className":84901,"style":12582},[54],[15,84903,117],{"className":84904},[59],[15,84906],{"className":84907,"style":2589},[157],[15,84909,4815],{"className":84910},[2593],[15,84912],{"className":84913,"style":2589},[157],[15,84915,84917,84920,84923,84926,84929],{"className":84916},[50],[15,84918],{"className":84919,"style":2347},[54],[15,84921,44780],{"className":84922,"style":44810},[59,60],[15,84924],{"className":84925,"style":158},[157],[15,84927,123],{"className":84928},[162],[15,84930],{"className":84931,"style":158},[157],[15,84933,84935,84938],{"className":84934},[50],[15,84936],{"className":84937,"style":55},[54],[15,84939,84158],{"className":84940,"style":84380},[59,60],[15,84942,84944,84966],{"className":84943,"translate":19},[18],[15,84945,84947],{"className":84946},[23],[25,84948,84949],{"xmlns":27},[29,84950,84951,84963],{},[32,84952,84953,84955,84957,84959,84961],{},[115,84954,117],{},[110,84956,4815],{},[35,84958,84158],{},[110,84960,123],{},[35,84962,44780],{},[39,84964,84965],{"encoding":41},"1-\\beta=\\alpha",[15,84967,84969,84987,85005],{"className":84968,"ariaHidden":46},[45],[15,84970,84972,84975,84978,84981,84984],{"className":84971},[50],[15,84973],{"className":84974,"style":12582},[54],[15,84976,117],{"className":84977},[59],[15,84979],{"className":84980,"style":2589},[157],[15,84982,4815],{"className":84983},[2593],[15,84985],{"className":84986,"style":2589},[157],[15,84988,84990,84993,84996,84999,85002],{"className":84989},[50],[15,84991],{"className":84992,"style":55},[54],[15,84994,84158],{"className":84995,"style":84380},[59,60],[15,84997],{"className":84998,"style":158},[157],[15,85000,123],{"className":85001},[162],[15,85003],{"className":85004,"style":158},[157],[15,85006,85008,85011],{"className":85007},[50],[15,85009],{"className":85010,"style":2347},[54],[15,85012,44780],{"className":85013,"style":44810},[59,60],"，算得",[15,85016,85018],{"className":85017,"translate":19},[180],[15,85019,85021,85277],{"className":85020,"translate":19},[18],[15,85022,85024],{"className":85023},[23],[25,85025,85026],{"xmlns":27,"display":189},[29,85027,85028,85274],{},[1245,85029,85030,85088,85138,85212],{"rowspacing":1247,"columnalign":1248,"columnspacing":1249},[1251,85031,85032,85038],{},[1254,85033,85034],{},[1257,85035,85036],{"scriptlevel":126,"displaystyle":46},[32,85037],{},[1254,85039,85040],{},[1257,85041,85042],{"scriptlevel":126,"displaystyle":46},[32,85043,85044,85046,85056,85058,85068,85070,85084,85086],{},[32,85045],{},[195,85047,85048,85050],{},[35,85049,199],{},[32,85051,85052,85054],{},[35,85053,10698],{},[35,85055,21919],{},[110,85057,113],{"fence":112,"stretchy":46,"minsize":17443,"maxsize":17443},[32,85059,85060,85062,85064,85066],{},[35,85061,39317],{"mathvariant":209},[35,85063,7569],{"mathvariant":209},[35,85065,16454],{"mathvariant":209},[35,85067,82337],{"mathvariant":209},[110,85069,113],{"fence":112,"stretchy":46,"minsize":17443,"maxsize":17443},[1257,85071,85072],{"scriptlevel":126,"displaystyle":112},[238,85073,85074,85082],{},[32,85075,85076,85078,85080],{},[115,85077,117],{},[110,85079,2472],{},[35,85081,15716],{},[115,85083,2465],{},[110,85085,120],{"fence":112,"stretchy":46,"minsize":17443,"maxsize":17443},[110,85087,120],{"fence":112,"stretchy":46,"minsize":17443,"maxsize":17443},[1251,85089,85090,85096],{},[1254,85091,85092],{},[1257,85093,85094],{"scriptlevel":126,"displaystyle":46},[32,85095],{},[1254,85097,85098],{},[1257,85099,85100],{"scriptlevel":126,"displaystyle":46},[32,85101,85102,85104,85106,85108,85118,85120,85134,85136],{},[32,85103],{},[157,85105],{"width":3264},[110,85107,210],{"fence":112,"stretchy":46,"minsize":17443,"maxsize":17443},[32,85109,85110,85112,85114,85116],{},[35,85111,39317],{"mathvariant":209},[35,85113,7569],{"mathvariant":209},[35,85115,16454],{"mathvariant":209},[35,85117,82337],{"mathvariant":209},[110,85119,113],{"fence":112,"stretchy":46,"minsize":17443,"maxsize":17443},[1257,85121,85122],{"scriptlevel":126,"displaystyle":112},[238,85123,85124,85132],{},[32,85125,85126,85128,85130],{},[115,85127,117],{},[110,85129,4815],{},[35,85131,15716],{},[115,85133,2465],{},[110,85135,120],{"fence":112,"stretchy":46,"minsize":17443,"maxsize":17443},[110,85137,120],{"fence":112,"stretchy":46,"minsize":17443,"maxsize":17443},[1251,85139,85140,85146],{},[1254,85141,85142],{},[1257,85143,85144],{"scriptlevel":126,"displaystyle":46},[32,85145],{},[1254,85147,85148],{},[1257,85149,85150],{"scriptlevel":126,"displaystyle":46},[32,85151,85152,85154,85156,85158,85160,85162,85170,85172,85174,85176,85178,85186,85188,85190,85192,85194,85196,85198,85200,85202,85204],{},[32,85153],{},[110,85155,123],{},[35,85157,44780],{},[35,85159,4102],{},[110,85161,4105],{},[1257,85163,85164],{"scriptlevel":126,"displaystyle":112},[238,85165,85166,85168],{},[35,85167,44780],{},[35,85169,84158],{},[110,85171,2472],{},[35,85173,84158],{},[35,85175,4102],{},[110,85177,4105],{},[1257,85179,85180],{"scriptlevel":126,"displaystyle":112},[238,85181,85182,85184],{},[35,85183,84158],{},[35,85185,44780],{},[110,85187,123],{},[110,85189,113],{"stretchy":112},[35,85191,44780],{},[110,85193,4815],{},[35,85195,84158],{},[110,85197,120],{"stretchy":112},[1299,85199,1301],{},[35,85201,4102],{},[110,85203,4105],{},[1257,85205,85206],{"scriptlevel":126,"displaystyle":112},[238,85207,85208,85210],{},[35,85209,44780],{},[35,85211,84158],{},[1251,85213,85214,85220],{},[1254,85215,85216],{},[1257,85217,85218],{"scriptlevel":126,"displaystyle":46},[32,85219],{},[1254,85221,85222],{},[1257,85223,85224],{"scriptlevel":126,"displaystyle":46},[32,85225,85226,85228,85230,85232,85234,85236,85238,85258,85260,85262,85264,85266,85268,85270,85272],{},[32,85227],{},[110,85229,123],{},[35,85231,15716],{},[1299,85233,1301],{},[35,85235,4102],{},[110,85237,4105],{},[1257,85239,85240],{"scriptlevel":126,"displaystyle":112},[238,85241,85242,85250],{},[32,85243,85244,85246,85248],{},[115,85245,117],{},[110,85247,2472],{},[35,85249,15716],{},[32,85251,85252,85254,85256],{},[115,85253,117],{},[110,85255,4815],{},[35,85257,15716],{},[110,85259,123],{},[110,85261,1018],{},[1299,85263,1301],{},[35,85265,38691],{},[110,85267,113],{"stretchy":112},[35,85269,15716],{},[110,85271,120],{"stretchy":112},[35,85273,4377],{"mathvariant":209},[39,85275,85276],{"encoding":41},"\\begin{aligned}\n&D_{KL}\\Big(\\mathrm{Bern}\n\\Big(\\tfrac{1+\\delta}2\\Big)\\Big)\\\\\n&\\qquad\\Big\\|\\mathrm{Bern}\n\\Big(\\tfrac{1-\\delta}2\\Big)\\Big)\\\\\n&=\\alpha\\log\\tfrac{\\alpha}{\\beta}\n+\\beta\\log\\tfrac{\\beta}{\\alpha}\n=(\\alpha-\\beta)\\,\\log\\tfrac{\\alpha}{\\beta}\\\\\n&=\\delta\\,\\log\\tfrac{1+\\delta}{1-\\delta}\n=:\\,h(\\delta).\n\\end{aligned}",[15,85278,85280],{"className":85279,"ariaHidden":46},[45],[15,85281,85283,85287],{"className":85282},[50],[15,85284],{"className":85285,"style":85286},[54],"height:7.1968em;vertical-align:-3.3484em;",[15,85288,85290],{"className":85289},[59],[15,85291,85293,85354],{"className":85292},[1245],[15,85294,85296],{"className":85295},[1362],[15,85297,85299,85345],{"className":85298},[285,286],[15,85300,85302,85342],{"className":85301},[290],[15,85303,85306,85315,85324,85333],{"className":85304,"style":85305},[294],"height:3.8484em;",[15,85307,85309,85312],{"style":85308},"top:-5.8484em;",[15,85310],{"className":85311,"style":17592},[302],[15,85313],{"className":85314},[59],[15,85316,85318,85321],{"style":85317},"top:-3.7484em;",[15,85319],{"className":85320,"style":17592},[302],[15,85322],{"className":85323},[59],[15,85325,85327,85330],{"style":85326},"top:-1.8661em;",[15,85328],{"className":85329,"style":17592},[302],[15,85331],{"className":85332},[59],[15,85334,85336,85339],{"style":85335},"top:-0.2049em;",[15,85337],{"className":85338,"style":17592},[302],[15,85340],{"className":85341},[59],[15,85343,318],{"className":85344},[317],[15,85346,85348],{"className":85347},[290],[15,85349,85352],{"className":85350,"style":85351},[294],"height:3.3484em;",[15,85353],{},[15,85355,85357],{"className":85356},[1453],[15,85358,85360,86134],{"className":85359},[285,286],[15,85361,85363,86131],{"className":85362},[290],[15,85364,85366,85527,85681,85987],{"className":85365,"style":85305},[294],[15,85367,85368,85371],{"style":85308},[15,85369],{"className":85370,"style":17592},[302],[15,85372,85374,85377,85423,85429,85435,85441,85515,85521],{"className":85373},[59],[15,85375],{"className":85376},[59],[15,85378,85380,85383],{"className":85379},[59],[15,85381,199],{"className":85382,"style":277},[59,60],[15,85384,85386],{"className":85385},[281],[15,85387,85389,85415],{"className":85388},[285,286],[15,85390,85392,85412],{"className":85391},[290],[15,85393,85395],{"className":85394,"style":379},[294],[15,85396,85397,85400],{"style":298},[15,85398],{"className":85399,"style":303},[302],[15,85401,85403],{"className":85402},[307,308,309,310],[15,85404,85406,85409],{"className":85405},[59,310],[15,85407,10698],{"className":85408,"style":10786},[59,60,310],[15,85410,21919],{"className":85411},[59,60,310],[15,85413,318],{"className":85414},[317],[15,85416,85418],{"className":85417},[290],[15,85419,85421],{"className":85420,"style":7970},[294],[15,85422],{},[15,85424,85426],{"className":85425},[59],[15,85427,113],{"className":85428},[419,3722],[15,85430,85432],{"className":85431},[59],[15,85433,84349],{"className":85434},[59,32688],[15,85436,85438],{"className":85437},[59],[15,85439,113],{"className":85440},[419,3722],[15,85442,85444,85447,85512],{"className":85443},[59],[15,85445],{"className":85446},[146,44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时不平衡质量挤出 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时值为 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",[15,86512,86514,86541],{"className":86513,"translate":19},[18],[15,86515,86517],{"className":86516},[23],[25,86518,86519],{"xmlns":27},[29,86520,86521,86539],{},[32,86522,86523,86525,86527],{},[35,86524,44780],{},[110,86526,123],{},[238,86528,86529,86537],{},[32,86530,86531,86533,86535],{},[115,86532,117],{},[110,86534,2472],{},[35,86536,15716],{},[115,86538,2465],{},[39,86540,84632],{"encoding":41},[15,86542,86544,86562],{"className":86543,"ariaHidden":46},[45],[15,86545,86547,86550,86553,86556,86559],{"className":86546},[50],[15,86548],{"className":86549,"style":2347},[54],[15,86551,44780],{"className":86552,"style":44810},[59,60],[15,86554],{"className":86555,"style":158},[157],[15,86557,123],{"className":86558},[162],[15,86560],{"className":86561,"style":158},[157],[15,86563,86565,86568],{"className":86564},[50],[15,86566],{"className":86567,"style":84660},[54],[15,86569,86571,86574,86639],{"className":86570},[59],[15,86572],{"className":86573},[146,441],[15,86575,86577],{"className":86576},[238],[15,86578,86580,86631],{"className":86579},[285,286],[15,86581,86583,86628],{"className":86582},[290],[15,86584,86586,86600,86608],{"className":86585,"style":2689},[294],[15,86587,86588,86591],{"style":2529},[15,86589],{"className":86590,"style":461},[302],[15,86592,86594],{"className":86593},[307,308,309,310],[15,86595,86597],{"className":86596},[59,310],[15,86598,2465],{"className":86599},[59,310],[15,86601,86602,86605],{"style":473},[15,86603],{"className":86604,"style":461},[302],[15,86606],{"className":86607,"style":481},[480],[15,86609,86610,86613],{"style":2552},[15,86611],{"className":86612,"style":461},[302],[15,86614,86616],{"className":86615},[307,308,309,310],[15,86617,86619,86622,86625],{"className":86618},[59,310],[15,86620,117],{"className":86621},[59,310],[15,86623,2472],{"className":86624},[2593,310],[15,86626,15716],{"className":86627,"style":15765},[59,60,310],[15,86629,318],{"className":86630},[317],[15,86632,86634],{"className":86633},[290],[15,86635,86637],{"className":86636,"style":2574},[294],[15,86638],{},[15,86640],{"className":86641},[153,441],"，且是",[998,86644,86645],{},"精确最小值","（Vajda 界）：",[15,86648,86650,86706],{"className":86649,"translate":19},[18],[15,86651,86653],{"className":86652},[23],[25,86654,86655],{"xmlns":27},[29,86656,86657,86703],{},[32,86658,86659,86669,86671,86673,86675,86677,86679,86681,86683,86685,86691,86693,86695,86697,86699,86701],{},[195,86660,86661,86663],{},[35,86662,199],{},[32,86664,86665,86667],{},[35,86666,10698],{},[35,86668,21919],{},[110,86670,113],{"stretchy":112},[35,86672,206],{},[35,86674,210],{"mathvariant":209},[35,86676,213],{},[110,86678,120],{"stretchy":112},[110,86680,6930],{},[35,86682,38691],{},[110,86684,113],{"stretchy":112},[32,86686,86687,86689],{},[35,86688,5394],{"mathvariant":209},[35,86690,34876],{"mathvariant":209},[110,86692,113],{"stretchy":112},[35,86694,206],{},[110,86696,551],{"separator":46},[35,86698,213],{},[110,86700,120],{"stretchy":112},[110,86702,120],{"stretchy":112},[39,86704,86705],{"encoding":41},"D_{KL}(P\\|Q)\\ge h(\\mathrm{TV}(P,Q))",[15,86707,86709,86785],{"className":86708,"ariaHidden":46},[45],[15,86710,86712,86715,86761,86764,86767,86770,86773,86776,86779,86782],{"className":86711},[50],[15,86713],{"className":86714,"style":139},[54],[15,86716,86718,86721],{"className":86717},[59],[15,86719,199],{"className":86720,"style":277},[59,60],[15,86722,86724],{"className":86723},[281],[15,86725,86727,86753],{"className":86726},[285,286],[15,86728,86730,86750],{"className":86729},[290],[15,86731,86733],{"className":86732,"style":379},[294],[15,86734,86735,86738],{"style":298},[15,86736],{"className":86737,"style":303},[302],[15,86739,86741],{"className":86740},[307,308,309,310],[15,86742,86744,86747],{"className":86743},[59,310],[15,86745,10698],{"className":86746,"style":10786},[59,60,310],[15,86748,21919],{"className":86749},[59,60,310],[15,86751,318],{"className":86752},[317],[15,86754,86756],{"className":86755},[290],[15,86757,86759],{"className":86758,"style":7970},[294],[15,86760],{},[15,86762,113],{"className":86763},[146],[15,86765,206],{"className":86766,"style":334},[59,60],[15,86768,210],{"className":86769},[59],[15,86771,213],{"className":86772},[59,60],[15,86774,120],{"className":86775},[153],[15,86777],{"className":86778,"style":158},[157],[15,86780,6930],{"className":86781},[162],[15,86783],{"className":86784,"style":158},[157],[15,86786,86788,86791,86794,86797,86804,86807,86810,86813,86816,86819],{"className":86787},[50],[15,86789],{"className":86790,"style":139},[54],[15,86792,38691],{"className":86793},[59,60],[15,86795,113],{"className":86796},[146],[15,86798,86800],{"className":86799},[59],[15,86801,86803],{"className":86802,"style":4159},[59,32688],"TV",[15,86805,113],{"className":86806},[146],[15,86808,206],{"className":86809,"style":334},[59,60],[15,86811,551],{"className":86812},[573],[15,86814],{"className":86815,"style":406},[157],[15,86817,213],{"className":86818},[59,60],[15,86820,19073],{"className":86821},[153],"，其中 ",[15,86824,86826,86872],{"className":86825,"translate":19},[18],[15,86827,86829],{"className":86828},[23],[25,86830,86831],{"xmlns":27},[29,86832,86833,86869],{},[32,86834,86835,86837,86839,86841,86843,86845,86847,86849,86851],{},[35,86836,38691],{},[110,86838,113],{"stretchy":112},[35,86840,15716],{},[110,86842,120],{"stretchy":112},[110,86844,123],{},[35,86846,15716],{},[35,86848,4102],{},[110,86850,4105],{},[238,86852,86853,86861],{},[32,86854,86855,86857,86859],{},[115,86856,117],{},[110,86858,2472],{},[35,86860,15716],{},[32,86862,86863,86865,86867],{},[115,86864,117],{},[110,86866,4815],{},[35,86868,15716],{},[39,86870,86871],{"encoding":41},"h(\\delta)=\\delta\\log\\frac{1+\\delta}{1-\\delta}",[15,86873,86875,86902],{"className":86874,"ariaHidden":46},[45],[15,86876,86878,86881,86884,86887,86890,86893,86896,86899],{"className":86877},[50],[15,86879],{"className":86880,"style":139},[54],[15,86882,38691],{"className":86883},[59,60],[15,86885,113],{"className":86886},[146],[15,86888,15716],{"className":86889,"style":15765},[59,60],[15,86891,120],{"className":86892},[153],[15,86894],{"className":86895,"style":158},[157],[15,86897,123],{"className":86898},[162],[15,86900],{"className":86901,"style":158},[157],[15,86903,86905,86908,86911,86914,86919,86922],{"className":86904},[50],[15,86906],{"className":86907,"style":16357},[54],[15,86909,15716],{"className":86910,"style":15765},[59,60],[15,86912],{"className":86913,"style":406},[157],[15,86915,4156,86917],{"className":86916},[1486],[15,86918,4160],{"style":4159},[15,86920],{"className":86921,"style":406},[157],[15,86923,86925,86928,86999],{"className":86924},[59],[15,86926],{"className":86927},[146,441],[15,86929,86931],{"className":86930},[238],[15,86932,86934,86991],{"className":86933},[285,286],[15,86935,86937,86988],{"className":86936},[290],[15,86938,86940,86960,86968],{"className":86939,"style":2689},[294],[15,86941,86942,86945],{"style":2529},[15,86943],{"className":86944,"style":461},[302],[15,86946,86948],{"className":86947},[307,308,309,310],[15,86949,86951,86954,86957],{"className":86950},[59,310],[15,86952,117],{"className":86953},[59,310],[15,86955,4815],{"className":86956},[2593,310],[15,86958,15716],{"className":86959,"style":15765},[59,60,310],[15,86961,86962,86965],{"style":473},[15,86963],{"className":86964,"style":461},[302],[15,86966],{"className":86967,"style":481},[480],[15,86969,86970,86973],{"style":2552},[15,86971],{"className":86972,"style":461},[302],[15,86974,86976],{"className":86975},[307,308,309,310],[15,86977,86979,86982,86985],{"className":86978},[59,310],[15,86980,117],{"className":86981},[59,310],[15,86983,2472],{"className":86984},[2593,310],[15,86986,15716],{"className":86987,"style":15765},[59,60,310],[15,86989,318],{"className":86990},[317],[15,86992,86994],{"className":86993},[290],[15,86995,86997],{"className":86996,"style":16433},[294],[15,86998],{},[15,87000],{"className":87001},[153,441],[15,87003,87005,87023],{"className":87004,"translate":19},[18],[15,87006,87008],{"className":87007},[23],[25,87009,87010],{"xmlns":27},[29,87011,87012,87020],{},[32,87013,87014,87016,87018],{},[35,87015,15716],{},[110,87017,1032],{},[115,87019,117],{},[39,87021,87022],{"encoding":41},"\\delta\\to1",[15,87024,87026,87044],{"className":87025,"ariaHidden":46},[45],[15,87027,87029,87032,87035,87038,87041],{"className":87028},[50],[15,87030],{"className":87031,"style":12601},[54],[15,87033,15716],{"className":87034,"style":15765},[59,60],[15,87036],{"className":87037,"style":158},[157],[15,87039,1032],{"className":87040},[162],[15,87042],{"className":87043,"style":158},[157],[15,87045,87047,87050],{"className":87046},[50],[15,87048],{"className":87049,"style":172},[54],[15,87051,117],{"className":87052},[59]," 时发散（与 KL 在支撑分离时为 ",[15,87055,87057,87070],{"className":87056,"translate":19},[18],[15,87058,87060],{"className":87059},[23],[25,87061,87062],{"xmlns":27},[29,87063,87064,87068],{},[32,87065,87066],{},[35,87067,1027],{"mathvariant":209},[39,87069,3872],{"encoding":41},[15,87071,87073],{"className":87072,"ariaHidden":46},[45],[15,87074,87076,87079],{"className":87075},[50],[15,87077],{"className":87078,"style":2347},[54],[15,87080,1027],{"className":87081},[59]," 一致）。",[11,87084,87085,87088],{},[998,87086,87087],{},"第 3 步（初等放缩）。"," 由",[15,87090,87092],{"className":87091,"translate":19},[180],[15,87093,87095,87181],{"className":87094,"translate":19},[18],[15,87096,87098],{"className":87097},[23],[25,87099,87100],{"xmlns":27,"display":189},[29,87101,87102,87178],{},[32,87103,87104,87106,87108,87126,87128,87130,87170,87172,87174,87176],{},[35,87105,4102],{},[110,87107,4105],{},[238,87109,87110,87118],{},[32,87111,87112,87114,87116],{},[115,87113,117],{},[110,87115,2472],{},[35,87117,15716],{},[32,87119,87120,87122,87124],{},[115,87121,117],{},[110,87123,4815],{},[35,87125,15716],{},[110,87127,123],{},[115,87129,2465],{},[32,87131,87132,87134,87136,87138,87148,87150,87161,87163,87166,87168],{},[110,87133,113],{"fence":46},[35,87135,15716],{},[110,87137,2472],{},[238,87139,87140,87146],{},[6816,87141,87142,87144],{},[35,87143,15716],{},[115,87145,40761],{},[115,87147,40761],{},[110,87149,2472],{},[238,87151,87152,87159],{},[6816,87153,87154,87156],{},[35,87155,15716],{},[115,87157,87158],{},"5",[115,87160,87158],{},[110,87162,2472],{},[110,87164,87165],{},"⋯",[1299,87167,1301],{},[110,87169,120],{"fence":46},[110,87171,6930],{},[115,87173,2465],{},[35,87175,15716],{},[110,87177,551],{"separator":46},[39,87179,87180],{"encoding":41},"\\log\\frac{1+\\delta}{1-\\delta}\n=2\\left(\\delta+\\frac{\\delta^3}3\n+\\frac{\\delta^5}5+\\cdots\\right)\n\\ge 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Pinsker 不等式中的 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",[15,88350,88352,88386],{"className":88351,"translate":19},[18],[15,88353,88355],{"className":88354},[23],[25,88356,88357],{"xmlns":27},[29,88358,88359,88383],{},[32,88360,88361,88367,88369,88371,88373],{},[32,88362,88363,88365],{},[35,88364,10698],{"mathvariant":209},[35,88366,21919],{"mathvariant":209},[110,88368,6930],{},[115,88370,2465],{},[1299,88372,1301],{},[6816,88374,88375,88381],{},[32,88376,88377,88379],{},[35,88378,5394],{"mathvariant":209},[35,88380,34876],{"mathvariant":209},[115,88382,2465],{},[39,88384,88385],{"encoding":41},"\\mathrm{KL}\\ge2\\,\\mathrm{TV}^2",[15,88387,88389,88411],{"className":88388,"ariaHidden":46},[45],[15,88390,88392,88396,88402,88405,88408],{"className":88391},[50],[15,88393],{"className":88394,"style":88395},[54],"height:0.8193em;vertical-align:-0.136em;",[15,88397,88399],{"className":88398},[59],[15,88400,32689],{"className":88401},[59,32688],[15,88403],{"className":88404,"style":158},[157],[15,88406,6930],{"className":88407},[162],[15,88409],{"className":88410,"style":158},[157],[15,88412,88414,88418,88421,88424],{"className":88413},[50],[15,88415],{"className":88416,"style":88417},[54],"height:0.8873em;",[15,88419,2465],{"className":88420},[59],[15,88422],{"className":88423,"style":406},[157],[15,88425,88427,88433],{"className":88426},[59],[15,88428,88430],{"className":88429},[59],[15,88431,86803],{"className":88432,"style":4159},[59,32688],[15,88434,88436],{"className":88435},[281],[15,88437,88439],{"className":88438},[285],[15,88440,88442],{"className":88441},[290],[15,88443,88445],{"className":88444,"style":88417},[294],[15,88446,88448,88451],{"style":88447},"top:-3.1362em;margin-right:0.05em;",[15,88449],{"className":88450,"style":303},[302],[15,88452,88454],{"className":88453},[307,308,309,310],[15,88455,2465],{"className":88456},[59,310]," 的系数 2）是最优的：取两分布仅在极小 ",[15,88459,88461,88475],{"className":88460,"translate":19},[18],[15,88462,88464],{"className":88463},[23],[25,88465,88466],{"xmlns":27},[29,88467,88468,88472],{},[32,88469,88470],{},[35,88471,38688],{},[39,88473,88474],{"encoding":41},"\\varepsilon",[15,88476,88478],{"className":88477,"ariaHidden":46},[45],[15,88479,88481,88484],{"className":88480},[50],[15,88482],{"className":88483,"style":2347},[54],[15,88485,38688],{"className":88486},[59,60]," 上偏离最优对称配置，可让 ",[15,88489,88491,88523],{"className":88490,"translate":19},[18],[15,88492,88494],{"className":88493},[23],[25,88495,88496],{"xmlns":27},[29,88497,88498,88520],{},[32,88499,88500,88502,88504,88506,88508,88510,88516,88518],{},[35,88501,38691],{},[110,88503,113],{"stretchy":112},[35,88505,15716],{},[110,88507,120],{"stretchy":112},[110,88509,941],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[6816,88511,88512,88514],{},[35,88513,15716],{},[115,88515,2465],{},[110,88517,1032],{},[115,88519,2465],{},[39,88521,88522],{"encoding":41},"h(\\delta)\\big\u002F\\delta^2\\to 2",[15,88524,88526,88588],{"className":88525,"ariaHidden":46},[45],[15,88527,88529,88532,88535,88538,88541,88544,88550,88579,88582,88585],{"className":88528},[50],[15,88530],{"className":88531,"style":20589},[54],[15,88533,38691],{"className":88534},[59,60],[15,88536,113],{"className":88537},[146],[15,88539,15716],{"className":88540,"style":15765},[59,60],[15,88542,120],{"className":88543},[153],[15,88545,88547],{"className":88546},[59],[15,88548,941],{"className":88549},[419,9676],[15,88551,88553,88556],{"className":88552},[59],[15,88554,15716],{"className":88555,"style":15765},[59,60],[15,88557,88559],{"className":88558},[281],[15,88560,88562],{"className":88561},[285],[15,88563,88565],{"className":88564},[290],[15,88566,88568],{"className":88567,"style":20054},[294],[15,88569,88570,88573],{"style":6867},[15,88571],{"className":88572,"style":303},[302],[15,88574,88576],{"className":88575},[307,308,309,310],[15,88577,2465],{"className":88578},[59,310],[15,88580],{"className":88581,"style":158},[157],[15,88583,1032],{"className":88584},[162],[15,88586],{"className":88587,"style":158},[157],[15,88589,88591,88594],{"className":88590},[50],[15,88592],{"className":88593,"style":172},[54],[15,88595,2465],{"className":88596},[59],[15,88598,88600,88618],{"className":88599,"translate":19},[18],[15,88601,88603],{"className":88602},[23],[25,88604,88605],{"xmlns":27},[29,88606,88607,88615],{},[32,88608,88609,88611,88613],{},[35,88610,15716],{},[110,88612,1032],{},[115,88614,126],{},[39,88616,88617],{"encoding":41},"\\delta\\to0",[15,88619,88621,88639],{"className":88620,"ariaHidden":46},[45],[15,88622,88624,88627,88630,88633,88636],{"className":88623},[50],[15,88625],{"className":88626,"style":12601},[54],[15,88628,15716],{"className":88629,"style":15765},[59,60],[15,88631],{"className":88632,"style":158},[157],[15,88634,1032],{"className":88635},[162],[15,88637],{"className":88638,"style":158},[157],[15,88640,88642,88645],{"className":88641},[50],[15,88643],{"className":88644,"style":172},[54],[15,88646,126],{"className":88647},[59],"），故常数不能改进。",[11,88650,88651,88652,88721],{},"顺便指出：对一般 f-散度也有平行的\"固定 TV 时最小化 ",[15,88653,88655,88672],{"className":88654,"translate":19},[18],[15,88656,88658],{"className":88657},[23],[25,88659,88660],{"xmlns":27},[29,88661,88662,88670],{},[32,88663,88664],{},[195,88665,88666,88668],{},[35,88667,199],{},[35,88669,37],{},[39,88671,4024],{"encoding":41},[15,88673,88675],{"className":88674,"ariaHidden":46},[45],[15,88676,88678,88681],{"className":88677},[50],[15,88679],{"className":88680,"style":4034},[54],[15,88682,88684,88687],{"className":88683},[59],[15,88685,199],{"className":88686,"style":277},[59,60],[15,88688,88690],{"className":88689},[281],[15,88691,88693,88713],{"className":88692},[285,286],[15,88694,88696,88710],{"className":88695},[290],[15,88697,88699],{"className":88698,"style":295},[294],[15,88700,88701,88704],{"style":298},[15,88702],{"className":88703,"style":303},[302],[15,88705,88707],{"className":88706},[307,308,309,310],[15,88708,37],{"className":88709,"style":61},[59,60,310],[15,88711,318],{"className":88712},[317],[15,88714,88716],{"className":88715},[290],[15,88717,88719],{"className":88718,"style":325},[294],[15,88720],{},"\"函数（Fedotov–Harremoës 2003 给出完整刻画），即每个 f-散度都自带一个 TV 的下界函数；这是 4.7 末拓扑结论的定量基础。",[11,88723,88724,88727,88728,88756,88757,88760],{},[998,88725,88726],{},"Bretagnolle–Huber 的推导。"," 由 Rényi 散度在阶数 ",[15,88729,88731,88744],{"className":88730,"translate":19},[18],[15,88732,88734],{"className":88733},[23],[25,88735,88736],{"xmlns":27},[29,88737,88738,88742],{},[32,88739,88740],{},[35,88741,44780],{},[39,88743,49116],{"encoding":41},[15,88745,88747],{"className":88746,"ariaHidden":46},[45],[15,88748,88750,88753],{"className":88749},[50],[15,88751],{"className":88752,"style":2347},[54],[15,88754,44780],{"className":88755,"style":44810},[59,60]," 上",[998,88758,88759],{},"单调不减","（Hölder 不等式的直接推论），",[15,88762,88764],{"className":88763,"translate":19},[180],[15,88765,88767,88865],{"className":88766,"translate":19},[18],[15,88768,88770],{"className":88769},[23],[25,88771,88772],{"xmlns":27,"display":189},[29,88773,88774,88862],{},[32,88775,88776,88786,88788,88790,88792,88794,88796,88798,88806,88808,88810,88812,88814,88816,88818,88820,88822,88836,88838,88840,88842,88844,88846,88848,88850,88852,88854,88856,88858,88860],{},[195,88777,88778,88780],{},[35,88779,199],{},[32,88781,88782,88784],{},[35,88783,10698],{},[35,88785,21919],{},[110,88787,113],{"stretchy":112},[35,88789,206],{},[35,88791,210],{"mathvariant":209},[35,88793,213],{},[110,88795,120],{"stretchy":112},[110,88797,123],{},[7555,88799,88800,88802,88804],{},[35,88801,199],{},[115,88803,117],{},[35,88805,1035],{},[110,88807,113],{"stretchy":112},[35,88809,206],{},[35,88811,210],{"mathvariant":209},[35,88813,213],{},[110,88815,120],{"stretchy":112},[1299,88817,4285],{},[110,88819,6930],{},[1299,88821,4285],{},[7555,88823,88824,88826,88834],{},[35,88825,199],{},[32,88827,88828,88830,88832],{},[115,88829,117],{},[35,88831,941],{"mathvariant":209},[115,88833,2465],{},[35,88835,1035],{},[110,88837,113],{"stretchy":112},[35,88839,206],{},[35,88841,210],{"mathvariant":209},[35,88843,213],{},[110,88845,120],{"stretchy":112},[110,88847,123],{},[110,88849,4815],{},[115,88851,2465],{},[35,88853,4102],{},[110,88855,4105],{},[35,88857,39317],{},[35,88859,39320],{},[35,88861,4377],{"mathvariant":209},[39,88863,88864],{"encoding":41},"D_{KL}(P\\|Q)=D^{R}_{1}(P\\|Q)\n\\ \\ge\\ D^{R}_{1\u002F2}(P\\|Q)=-2\\log BC 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-2\\log BC\\ge 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串起来就有 ",[15,92246,92248,92280],{"className":92247,"translate":19},[18],[15,92249,92251],{"className":92250},[23],[25,92252,92253],{"xmlns":27},[29,92254,92255,92277],{},[32,92256,92257,92263,92265],{},[32,92258,92259,92261],{},[35,92260,5394],{"mathvariant":209},[35,92262,34876],{"mathvariant":209},[110,92264,9954],{},[34365,92266,92267],{},[195,92268,92269,92271],{},[35,92270,199],{},[32,92272,92273,92275],{},[35,92274,10698],{},[35,92276,21919],{},[39,92278,92279],{"encoding":41},"\\mathrm{TV}\\le\\sqrt{D_{KL}}",[15,92281,92283,92304],{"className":92282,"ariaHidden":46},[45],[15,92284,92286,92289,92295,92298,92301],{"className":92285},[50],[15,92287],{"className":92288,"style":88395},[54],[15,92290,92292],{"className":92291},[59],[15,92293,86803],{"className":92294,"style":4159},[59,32688],[15,92296],{"className":92297,"style":158},[157],[15,92299,9954],{"className":92300},[162],[15,92302],{"className":92303,"style":158},[157],[15,92305,92307,92310],{"className":92306},[50],[15,92308],{"className":92309,"style":34389},[54],[15,92311,92313],{"className":92312},[59,34393],[15,92314,92316,92392],{"className":92315},[285,286],[15,92317,92319,92389],{"className":92318},[290],[15,92320,92322,92377],{"className":92321,"style":34403},[294],[15,92323,92325,92328],{"className":92324,"style":11867},[6309],[15,92326],{"className":92327,"style":461},[302],[15,92329,92331],{"className":92330,"style":34413},[59],[15,92332,92334,92337],{"className":92333},[59],[15,92335,199],{"className":92336,"style":277},[59,60],[15,92338,92340],{"className":92339},[281],[15,92341,92343,92369],{"className":92342},[285,286],[15,92344,92346,92366],{"className":92345},[290],[15,92347,92349],{"className":92348,"style":379},[294],[15,92350,92351,92354],{"style":298},[15,92352],{"className":92353,"style":303},[302],[15,92355,92357],{"className":92356},[307,308,309,310],[15,92358,92360,92363],{"className":92359},[59,310],[15,92361,10698],{"className":92362,"style":10786},[59,60,310],[15,92364,21919],{"className":92365},[59,60,310],[15,92367,318],{"className":92368},[317],[15,92370,92372],{"className":92371},[290],[15,92373,92375],{"className":92374,"style":7970},[294],[15,92376],{},[15,92378,92379,92382],{"style":34462},[15,92380],{"className":92381,"style":461},[302],[15,92383,92385],{"className":92384,"style":34470},[34469],[6325,92386,92387],{"xmlns":6327,"width":6328,"height":34473,"viewBox":34474,"preserveAspectRatio":6331},[6333,92388],{"d":34477},[15,92390,318],{"className":92391},[317],[15,92393,92395],{"className":92394},[290],[15,92396,92398],{"className":92397,"style":34487},[294],[15,92399],{},"（比 Pinsker 常数差一个 ",[15,92402,92404,92420],{"className":92403,"translate":19},[18],[15,92405,92407],{"className":92406},[23],[25,92408,92409],{"xmlns":27},[29,92410,92411,92417],{},[32,92412,92413],{},[34365,92414,92415],{},[115,92416,2465],{},[39,92418,92419],{"encoding":41},"\\sqrt2",[15,92421,92423],{"className":92422,"ariaHidden":46},[45],[15,92424,92426,92430],{"className":92425},[50],[15,92427],{"className":92428,"style":92429},[54],"height:1.04em;vertical-align:-0.1328em;",[15,92431,92433],{"className":92432},[59,34393],[15,92434,92436,92468],{"className":92435},[285,286],[15,92437,92439,92465],{"className":92438},[290],[15,92440,92443,92452],{"className":92441,"style":92442},[294],"height:0.9072em;",[15,92444,92446,92449],{"className":92445,"style":11867},[6309],[15,92447],{"className":92448,"style":461},[302],[15,92450,2465],{"className":92451,"style":34413},[59],[15,92453,92455,92458],{"style":92454},"top:-2.8672em;",[15,92456],{"className":92457,"style":461},[302],[15,92459,92461],{"className":92460,"style":34470},[34469],[6325,92462,92463],{"xmlns":6327,"width":6328,"height":34473,"viewBox":34474,"preserveAspectRatio":6331},[6333,92464],{"d":34477},[15,92466,318],{"className":92467},[317],[15,92469,92471],{"className":92470},[290],[15,92472,92475],{"className":92473,"style":92474},[294],"height:0.1328em;",[15,92476],{},[993,92478,92479,92632,94160,95652,96039,96605],{},[11,92480,92481,11064,92484,92512,92513,92578,92579,92631],{},[998,92482,92483],{},"定理 4.2（拓扑关系的准确版）",[15,92485,92487,92500],{"className":92486,"translate":19},[18],[15,92488,92490],{"className":92489},[23],[25,92491,92492],{"xmlns":27},[29,92493,92494,92498],{},[32,92495,92496],{},[35,92497,37],{},[39,92499,37],{"encoding":41},[15,92501,92503],{"className":92502,"ariaHidden":46},[45],[15,92504,92506,92509],{"className":92505},[50],[15,92507],{"className":92508,"style":55},[54],[15,92510,37],{"className":92511,"style":61},[59,60]," 凸、",[15,92514,92516,92539],{"className":92515,"translate":19},[18],[15,92517,92519],{"className":92518},[23],[25,92520,92521],{"xmlns":27},[29,92522,92523,92537],{},[32,92524,92525,92527,92529,92531,92533,92535],{},[35,92526,37],{},[110,92528,113],{"stretchy":112},[115,92530,117],{},[110,92532,120],{"stretchy":112},[110,92534,123],{},[115,92536,126],{},[39,92538,129],{"encoding":41},[15,92540,92542,92569],{"className":92541,"ariaHidden":46},[45],[15,92543,92545,92548,92551,92554,92557,92560,92563,92566],{"className":92544},[50],[15,92546],{"className":92547,"style":139},[54],[15,92549,37],{"className":92550,"style":61},[59,60],[15,92552,113],{"className":92553},[146],[15,92555,117],{"className":92556},[59],[15,92558,120],{"className":92559},[153],[15,92561],{"className":92562,"style":158},[157],[15,92564,123],{"className":92565},[162],[15,92567],{"className":92568,"style":158},[157],[15,92570,92572,92575],{"className":92571},[50],[15,92573],{"className":92574,"style":172},[54],[15,92576,126],{"className":92577},[59]," 且在 ",[15,92580,92582,92604],{"className":92581,"translate":19},[18],[15,92583,92585],{"className":92584},[23],[25,92586,92587],{"xmlns":27},[29,92588,92589,92601],{},[32,92590,92591,92593,92595,92597,92599],{},[110,92592,113],{"stretchy":112},[115,92594,126],{},[110,92596,551],{"separator":46},[35,92598,1027],{"mathvariant":209},[110,92600,120],{"stretchy":112},[39,92602,92603],{"encoding":41},"(0,\\infty)",[15,92605,92607],{"className":92606,"ariaHidden":46},[45],[15,92608,92610,92613,92616,92619,92622,92625,92628],{"className":92609},[50],[15,92611],{"className":92612,"style":139},[54],[15,92614,113],{"className":92615},[146],[15,92617,126],{"className":92618},[59],[15,92620,551],{"className":92621},[573],[15,92623],{"className":92624,"style":406},[157],[15,92626,1027],{"className":92627},[59],[15,92629,120],{"className":92630},[153]," 上连续。",[11,92633,92634,6496,92637,92970,92971,93042,93043,5647,93150,93299,93300,93455,93456,93606,93607,93665,93666,93794,93795,93852,93853,94159],{},[998,92635,92636],{},"(a)（\"下来\"方向，一般成立）",[15,92638,92640,92710],{"className":92639,"translate":19},[18],[15,92641,92643],{"className":92642},[23],[25,92644,92645],{"xmlns":27},[29,92646,92647,92707],{},[32,92648,92649,92655,92657,92663,92665,92667,92669,92671,92673,92675,92677,92679,92689,92691,92697,92699,92701,92703,92705],{},[195,92650,92651,92653],{},[35,92652,199],{},[35,92654,37],{},[110,92656,113],{"stretchy":112},[195,92658,92659,92661],{},[35,92660,206],{},[35,92662,82337],{},[35,92664,210],{"mathvariant":209},[35,92666,213],{},[110,92668,120],{"stretchy":112},[110,92670,1032],{},[115,92672,126],{},[1299,92674,4285],{},[110,92676,4288],{},[1299,92678,4285],{},[195,92680,92681,92683],{},[35,92682,199],{},[32,92684,92685,92687],{},[35,92686,5394],{},[35,92688,34876],{},[110,92690,113],{"stretchy":112},[195,92692,92693,92695],{},[35,92694,206],{},[35,92696,82337],{},[110,92698,551],{"separator":46},[35,92700,213],{},[110,92702,120],{"stretchy":112},[110,92704,1032],{},[115,92706,126],{},[39,92708,92709],{"encoding":41},"D_f(P_n\\|Q)\\to0\\ \\Longrightarrow\\ D_{TV}(P_n,Q)\\to0",[15,92711,92713,92820,92845,92961],{"className":92712,"ariaHidden":46},[45],[15,92714,92716,92719,92759,92762,92802,92805,92808,92811,92814,92817],{"className":92715},[50],[15,92717],{"className":92718,"style":270},[54],[15,92720,92722,92725],{"className":92721},[59],[15,92723,199],{"className":92724,"style":277},[59,60],[15,92726,92728],{"className":92727},[281],[15,92729,92731,92751],{"className":92730},[285,286],[15,92732,92734,92748],{"className":92733},[290],[15,92735,92737],{"className":92736,"style":295},[294],[15,92738,92739,92742],{"style":298},[15,92740],{"className":92741,"style":303},[302],[15,92743,92745],{"className":92744},[307,308,309,310],[15,92746,37],{"className":92747,"style":61},[59,60,310],[15,92749,318],{"className":92750},[317],[15,92752,92754],{"className":92753},[290],[15,92755,92757],{"className":92756,"style":325},[294],[15,92758],{},[15,92760,113],{"className":92761},[146],[15,92763,92765,92768],{"className":92764},[59],[15,92766,206],{"className":92767,"style":334},[59,60],[15,92769,92771],{"className":92770},[281],[15,92772,92774,92794],{"className":92773},[285,286],[15,92775,92777,92791],{"className":92776},[290],[15,92778,92780],{"className":92779,"style":44955},[294],[15,92781,92782,92785],{"style":11140},[15,92783],{"className":92784,"style":303},[302],[15,92786,92788],{"className":92787},[307,308,309,310],[15,92789,82337],{"className":92790},[59,60,310],[15,92792,318],{"className":92793},[317],[15,92795,92797],{"className":92796},[290],[15,92798,92800],{"className":92799,"style":7970},[294],[15,92801],{},[15,92803,210],{"className":92804},[59],[15,92806,213],{"className":92807},[59,60],[15,92809,120],{"className":92810},[153],[15,92812],{"className":92813,"style":158},[157],[15,92815,1032],{"className":92816},[162],[15,92818],{"className":92819,"style":158},[157],[15,92821,92823,92827,92830,92833,92836,92839,92842],{"className":92822},[50],[15,92824],{"className":92825,"style":92826},[54],"height:0.6684em;vertical-align:-0.024em;",[15,92828,126],{"className":92829},[59],[15,92831,4285],{"className":92832},[157],[15,92834],{"className":92835,"style":158},[157],[15,92837,4288],{"className":92838},[162],[15,92840,4285],{"className":92841},[157],[15,92843],{"className":92844,"style":158},[157],[15,92846,92848,92851,92897,92900,92940,92943,92946,92949,92952,92955,92958],{"className":92847},[50],[15,92849],{"className":92850,"style":139},[54],[15,92852,92854,92857],{"className":92853},[59],[15,92855,199],{"className":92856,"style":277},[59,60],[15,92858,92860],{"className":92859},[281],[15,92861,92863,92889],{"className":92862},[285,286],[15,92864,92866,92886],{"className":92865},[290],[15,92867,92869],{"className":92868,"style":379},[294],[15,92870,92871,92874],{"style":298},[15,92872],{"className":92873,"style":303},[302],[15,92875,92877],{"className":92876},[307,308,309,310],[15,92878,92880,92883],{"className":92879},[59,310],[15,92881,5394],{"className":92882,"style":334},[59,60,310],[15,92884,34876],{"className":92885,"style":2589},[59,60,310],[15,92887,318],{"className":92888},[317],[15,92890,92892],{"className":92891},[290],[15,92893,92895],{"className":92894,"style":7970},[294],[15,92896],{},[15,92898,113],{"className":92899},[146],[15,92901,92903,92906],{"className":92902},[59],[15,92904,206],{"className":92905,"style":334},[59,60],[15,92907,92909],{"className":92908},[281],[15,92910,92912,92932],{"className":92911},[285,286],[15,92913,92915,92929],{"className":92914},[290],[15,92916,92918],{"className":92917,"style":44955},[294],[15,92919,92920,92923],{"style":11140},[15,92921],{"className":92922,"style":303},[302],[15,92924,92926],{"className":92925},[307,308,309,310],[15,92927,82337],{"className":92928},[59,60,310],[15,92930,318],{"className":92931},[317],[15,92933,92935],{"className":92934},[290],[15,92936,92938],{"className":92937,"style":7970},[294],[15,92939],{},[15,92941,551],{"className":92942},[573],[15,92944],{"className":92945,"style":406},[157],[15,92947,213],{"className":92948},[59,60],[15,92950,120],{"className":92951},[153],[15,92953],{"className":92954,"style":158},[157],[15,92956,1032],{"className":92957},[162],[15,92959],{"className":92960,"style":158},[157],[15,92962,92964,92967],{"className":92963},[50],[15,92965],{"className":92966,"style":172},[54],[15,92968,126],{"className":92969},[59],"。每个适度正则的 f-散度都自带一个连续的 TV 下界函数 ",[15,92972,92974,92992],{"className":92973,"translate":19},[18],[15,92975,92977],{"className":92976},[23],[25,92978,92979],{"xmlns":27},[29,92980,92981,92989],{},[32,92982,92983],{},[195,92984,92985,92987],{},[35,92986,4160],{},[35,92988,37],{},[39,92990,92991],{"encoding":41},"g_f",[15,92993,92995],{"className":92994,"ariaHidden":46},[45],[15,92996,92998,93002],{"className":92997},[50],[15,92999],{"className":93000,"style":93001},[54],"height:0.7167em;vertical-align:-0.2861em;",[15,93003,93005,93008],{"className":93004},[59],[15,93006,4160],{"className":93007,"style":981},[59,60],[15,93009,93011],{"className":93010},[281],[15,93012,93014,93034],{"className":93013},[285,286],[15,93015,93017,93031],{"className":93016},[290],[15,93018,93020],{"className":93019,"style":295},[294],[15,93021,93022,93025],{"style":18869},[15,93023],{"className":93024,"style":303},[302],[15,93026,93028],{"className":93027},[307,308,309,310],[15,93029,37],{"className":93030,"style":61},[59,60,310],[15,93032,318],{"className":93033},[317],[15,93035,93037],{"className":93036},[290],[15,93038,93040],{"className":93039,"style":325},[294],[15,93041],{},"（Vajda \u002F Fedotov–Harremoës），",[15,93044,93046,93074],{"className":93045,"translate":19},[18],[15,93047,93049],{"className":93048},[23],[25,93050,93051],{"xmlns":27},[29,93052,93053,93071],{},[32,93054,93055,93061,93063,93065,93067,93069],{},[195,93056,93057,93059],{},[35,93058,4160],{},[35,93060,37],{},[110,93062,113],{"stretchy":112},[115,93064,126],{},[110,93066,120],{"stretchy":112},[110,93068,123],{},[115,93070,126],{},[39,93072,93073],{"encoding":41},"g_f(0)=0",[15,93075,93077,93141],{"className":93076,"ariaHidden":46},[45],[15,93078,93080,93083,93123,93126,93129,93132,93135,93138],{"className":93079},[50],[15,93081],{"className":93082,"style":270},[54],[15,93084,93086,93089],{"className":93085},[59],[15,93087,4160],{"className":93088,"style":981},[59,60],[15,93090,93092],{"className":93091},[281],[15,93093,93095,93115],{"className":93094},[285,286],[15,93096,93098,93112],{"className":93097},[290],[15,93099,93101],{"className":93100,"style":295},[294],[15,93102,93103,93106],{"style":18869},[15,93104],{"className":93105,"style":303},[302],[15,93107,93109],{"className":93108},[307,308,309,310],[15,93110,37],{"className":93111,"style":61},[59,60,310],[15,93113,318],{"className":93114},[317],[15,93116,93118],{"className":93117},[290],[15,93119,93121],{"className":93120,"style":325},[294],[15,93122],{},[15,93124,113],{"className":93125},[146],[15,93127,126],{"className":93128},[59],[15,93130,120],{"className":93131},[153],[15,93133],{"className":93134,"style":158},[157],[15,93136,123],{"className":93137},[162],[15,93139],{"className":93140,"style":158},[157],[15,93142,93144,93147],{"className":93143},[50],[15,93145],{"className":93146,"style":172},[54],[15,93148,126],{"className":93149},[59],[15,93151,93153,93193],{"className":93152,"translate":19},[18],[15,93154,93156],{"className":93155},[23],[25,93157,93158],{"xmlns":27},[29,93159,93160,93190],{},[32,93161,93162,93168,93170,93172,93174,93176,93178,93180,93182,93184,93186,93188],{},[195,93163,93164,93166],{},[35,93165,4160],{},[35,93167,37],{},[110,93169,113],{"stretchy":112},[35,93171,769],{},[110,93173,120],{"stretchy":112},[110,93175,2905],{},[115,93177,126],{},[1299,93179,4285],{},[110,93181,113],{"stretchy":112},[35,93183,769],{},[110,93185,2905],{},[115,93187,126],{},[110,93189,120],{"stretchy":112},[39,93191,93192],{"encoding":41},"g_f(t)>0\\ (t>0)",[15,93194,93196,93260,93287],{"className":93195,"ariaHidden":46},[45],[15,93197,93199,93202,93242,93245,93248,93251,93254,93257],{"className":93198},[50],[15,93200],{"className":93201,"style":270},[54],[15,93203,93205,93208],{"className":93204},[59],[15,93206,4160],{"className":93207,"style":981},[59,60],[15,93209,93211],{"className":93210},[281],[15,93212,93214,93234],{"className":93213},[285,286],[15,93215,93217,93231],{"className":93216},[290],[15,93218,93220],{"className":93219,"style":295},[294],[15,93221,93222,93225],{"style":18869},[15,93223],{"className":93224,"style":303},[302],[15,93226,93228],{"className":93227},[307,308,309,310],[15,93229,37],{"className":93230,"style":61},[59,60,310],[15,93232,318],{"className":93233},[317],[15,93235,93237],{"className":93236},[290],[15,93238,93240],{"className":93239,"style":325},[294],[15,93241],{},[15,93243,113],{"className":93244},[146],[15,93246,769],{"className":93247},[59,60],[15,93249,120],{"className":93250},[153],[15,93252],{"className":93253,"style":158},[157],[15,93255,2905],{"className":93256},[162],[15,93258],{"className":93259,"style":158},[157],[15,93261,93263,93266,93269,93272,93275,93278,93281,93284],{"className":93262},[50],[15,93264],{"className":93265,"style":139},[54],[15,93267,126],{"className":93268},[59],[15,93270,4285],{"className":93271},[157],[15,93273,113],{"className":93274},[146],[15,93276,769],{"className":93277},[59,60],[15,93279],{"className":93280,"style":158},[157],[15,93282,2905],{"className":93283},[162],[15,93285],{"className":93286,"style":158},[157],[15,93288,93290,93293,93296],{"className":93289},[50],[15,93291],{"className":93292,"style":139},[54],[15,93294,126],{"className":93295},[59],[15,93297,120],{"className":93298},[153],"，使 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给 ",[15,93457,93459,93497],{"className":93458,"translate":19},[18],[15,93460,93462],{"className":93461},[23],[25,93463,93464],{"xmlns":27},[29,93465,93466,93494],{},[32,93467,93468,93470,93472,93482,93484],{},[115,93469,2465],{},[1299,93471,1301],{},[6816,93473,93474,93480],{},[32,93475,93476,93478],{},[35,93477,5394],{"mathvariant":209},[35,93479,34876],{"mathvariant":209},[115,93481,2465],{},[110,93483,9954],{},[195,93485,93486,93488],{},[35,93487,199],{},[32,93489,93490,93492],{},[35,93491,10698],{},[35,93493,21919],{},[39,93495,93496],{"encoding":41},"2\\,\\mathrm{TV}^2\\le 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给 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经由 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",[15,95251,95253,95297],{"className":95252,"translate":19},[18],[15,95254,95256],{"className":95255},[23],[25,95257,95258],{"xmlns":27},[29,95259,95260,95294],{},[32,95261,95262,95268,95270,95276,95278,95280,95282,95284,95290,95292],{},[32,95263,95264,95266],{},[35,95265,5394],{"mathvariant":209},[35,95267,34876],{"mathvariant":209},[110,95269,113],{"stretchy":112},[195,95271,95272,95274],{},[35,95273,944],{},[35,95275,82337],{},[110,95277,551],{"separator":46},[35,95279,11],{},[110,95281,120],{"stretchy":112},[110,95283,32985],{},[195,95285,95286,95288],{},[35,95287,15716],{},[35,95289,82337],{},[110,95291,1032],{},[115,95293,126],{},[39,95295,95296],{"encoding":41},"\\mathrm{TV}(q_n,p)\\approx\\delta_n\\to0",[15,95298,95300,95376,95431],{"className":95299,"ariaHidden":46},[45],[15,95301,95303,95306,95312,95315,95355,95358,95361,95364,95367,95370,95373],{"className":95302},[50],[15,95304],{"className":95305,"style":139},[54],[15,95307,95309],{"className":95308},[59],[15,95310,86803],{"className":95311,"style":4159},[59,32688],[15,95313,113],{"className":95314},[146],[15,95316,95318,95321],{"className":95317},[59],[15,95319,944],{"className":95320,"style":981},[59,60],[15,95322,95324],{"className":95323},[281],[15,95325,95327,95347],{"className":95326},[285,286],[15,95328,95330,95344],{"className":95329},[290],[15,95331,95333],{"className":95332,"style":44955},[294],[15,95334,95335,95338],{"style":18869},[15,95336],{"className":95337,"style":303},[302],[15,95339,95341],{"className":95340},[307,308,309,310],[15,95342,82337],{"className":95343},[59,60,310],[15,95345,318],{"className":95346},[317],[15,95348,95350],{"className":95349},[290],[15,95351,95353],{"className":95352,"style":7970},[294],[15,95354],{},[15,95356,551],{"className":95357},[573],[15,95359],{"className":95360,"style":406},[157],[15,95362,11],{"className":95363},[59,60],[15,95365,120],{"className":95366},[153],[15,95368],{"className":95369,"style":158},[157],[15,95371,32985],{"className":95372},[162],[15,95374],{"className":95375,"style":158},[157],[15,95377,95379,95382,95422,95425,95428],{"className":95378},[50],[15,95380],{"className":95381,"style":11478},[54],[15,95383,95385,95388],{"className":95384},[59],[15,95386,15716],{"className":95387,"style":15765},[59,60],[15,95389,95391],{"className":95390},[281],[15,95392,95394,95414],{"className":95393},[285,286],[15,95395,95397,95411],{"className":95396},[290],[15,95398,95400],{"className":95399,"style":44955},[294],[15,95401,95402,95405],{"style":94786},[15,95403],{"className":95404,"style":303},[302],[15,95406,95408],{"className":95407},[307,308,309,310],[15,95409,82337],{"className":95410},[59,60,310],[15,95412,318],{"className":95413},[317],[15,95415,95417],{"className":95416},[290],[15,95418,95420],{"className":95419,"style":7970},[294],[15,95421],{},[15,95423],{"className":95424,"style":158},[157],[15,95426,1032],{"className":95427},[162],[15,95429],{"className":95430,"style":158},[157],[15,95432,95434,95437],{"className":95433},[50],[15,95435],{"className":95436,"style":172},[54],[15,95438,126],{"className":95439},[59],"，但 ",[15,95442,95444,95493],{"className":95443,"translate":19},[18],[15,95445,95447],{"className":95446},[23],[25,95448,95449],{"xmlns":27},[29,95450,95451,95490],{},[32,95452,95453,95459,95461,95467,95469,95471,95473,95476,95482,95484,95486,95488],{},[32,95454,95455,95457],{},[35,95456,10698],{"mathvariant":209},[35,95458,21919],{"mathvariant":209},[110,95460,113],{"stretchy":112},[195,95462,95463,95465],{},[35,95464,944],{},[35,95466,82337],{},[35,95468,210],{"mathvariant":209},[35,95470,11],{},[110,95472,120],{"stretchy":112},[110,95474,95475],{},"≳",[195,95477,95478,95480],{},[35,95479,15716],{},[35,95481,82337],{},[110,95483,3301],{},[35,95485,82337],{},[110,95487,1032],{},[35,95489,1027],{"mathvariant":209},[39,95491,95492],{"encoding":41},"\\mathrm{KL}(q_n\\|p)\\gtrsim \\delta_n\\cdot n\\to\\infty",[15,95494,95496,95569,95624,95642],{"className":95495,"ariaHidden":46},[45],[15,95497,95499,95502,95508,95511,95551,95554,95557,95560,95563,95566],{"className":95498},[50],[15,95500],{"className":95501,"style":139},[54],[15,95503,95505],{"className":95504},[59],[15,95506,32689],{"className":95507},[59,32688],[15,95509,113],{"className":95510},[146],[15,95512,95514,95517],{"className":95513},[59],[15,95515,944],{"className":95516,"style":981},[59,60],[15,95518,95520],{"className":95519},[281],[15,95521,95523,95543],{"className":95522},[285,286],[15,95524,95526,95540],{"className":95525},[290],[15,95527,95529],{"className":95528,"style":44955},[294],[15,95530,95531,95534],{"style":18869},[15,95532],{"className":95533,"style":303},[302],[15,95535,95537],{"className":95536},[307,308,309,310],[15,95538,82337],{"className":95539},[59,60,310],[15,95541,318],{"className":95542},[317],[15,95544,95546],{"className":95545},[290],[15,95547,95549],{"className":95548,"style":7970},[294],[15,95550],{},[15,95552,210],{"className":95553},[59],[15,95555,11],{"className":95556},[59,60],[15,95558,120],{"className":95559},[153],[15,95561],{"className":95562,"style":158},[157],[15,95564,95475],{"className":95565},[162,59829],[15,95567],{"className":95568,"style":158},[157],[15,95570,95572,95575,95615,95618,95621],{"className":95571},[50],[15,95573],{"className":95574,"style":11478},[54],[15,95576,95578,95581],{"className":95577},[59],[15,95579,15716],{"className":95580,"style":15765},[59,60],[15,95582,95584],{"className":95583},[281],[15,95585,95587,95607],{"className":95586},[285,286],[15,95588,95590,95604],{"className":95589},[290],[15,95591,95593],{"className":95592,"style":44955},[294],[15,95594,95595,95598],{"style":94786},[15,95596],{"className":95597,"style":303},[302],[15,95599,95601],{"className":95600},[307,308,309,310],[15,95602,82337],{"className":95603},[59,60,310],[15,95605,318],{"className":95606},[317],[15,95608,95610],{"className":95609},[290],[15,95611,95613],{"className":95612,"style":7970},[294],[15,95614],{},[15,95616],{"className":95617,"style":2589},[157],[15,95619,3301],{"className":95620},[2593],[15,95622],{"className":95623,"style":2589},[157],[15,95625,95627,95630,95633,95636,95639],{"className":95626},[50],[15,95628],{"className":95629,"style":2347},[54],[15,95631,82337],{"className":95632},[59,60],[15,95634],{"className":95635,"style":158},[157],[15,95637,1032],{"className":95638},[162],[15,95640],{"className":95641,"style":158},[157],[15,95643,95645,95648],{"className":95644},[50],[15,95646],{"className":95647,"style":2347},[54],[15,95649,1027],{"className":95650},[59],"。反例的机制很典型：TV 只控制\"总质量差\"，控制不了小测度集合上密度的巨大尖峰，而 KL 对小分母极其敏感——这正是 KL\u002Fχ² \"无界\"分歧的拓扑体现。",[11,95653,95654,95657,95658,95686,95687,60099,95753,95818,95819,23650],{},[998,95655,95656],{},"(c)（有限字母表\u002F一致比值界下，全部等价）"," 若似然比一致有界（例如：",[15,95659,95661,95674],{"className":95660,"translate":19},[18],[15,95662,95664],{"className":95663},[23],[25,95665,95666],{"xmlns":27},[29,95667,95668,95672],{},[32,95669,95670],{},[35,95671,213],{},[39,95673,213],{"encoding":41},[15,95675,95677],{"className":95676,"ariaHidden":46},[45],[15,95678,95680,95683],{"className":95679},[50],[15,95681],{"className":95682,"style":566},[54],[15,95684,213],{"className":95685},[59,60]," 在有限字母表上且 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",[15,96786,96788,96832],{"className":96787,"translate":19},[18],[15,96789,96791],{"className":96790},[23],[25,96792,96793],{"xmlns":27},[29,96794,96795,96829],{},[32,96796,96797,96799,96801,96803,96805,96807,96815,96817,96819,96827],{},[35,96798,769],{},[110,96800,5911],{},[110,96802,230],{"stretchy":112},[115,96804,117],{},[35,96806,941],{"mathvariant":209},[195,96808,96809,96811,96813],{},[35,96810,769],{},[35,96812,51634],{},[110,96814,4105],{},[110,96816,551],{"separator":46},[1299,96818,1301],{},[195,96820,96821,96823,96825],{},[35,96822,769],{},[35,96824,52413],{},[110,96826,4105],{},[110,96828,257],{"stretchy":112},[39,96830,96831],{"encoding":41},"t\\in[1\u002Ft_{\\min},\\,t_{\\max}]",[15,96833,96835,96853],{"className":96834,"ariaHidden":46},[45],[15,96836,96838,96841,96844,96847,96850],{"className":96837},[50],[15,96839],{"className":96840,"style":7268},[54],[15,96842,769],{"className":96843},[59,60],[15,96845],{"className":96846,"style":158},[157],[15,96848,5911],{"className":96849},[162],[15,96851],{"className":96852,"style":158},[157],[15,96854,96856,96859,96862,96865,96918,96921,96924,96927,96979],{"className":96855},[50],[15,96857],{"className":96858,"style":139},[54],[15,96860,230],{"className":96861},[146],[15,96863,8591],{"className":96864},[59],[15,96866,96868,96871],{"className":96867},[59],[15,96869,769],{"className":96870},[59,60],[15,96872,96874],{"className":96873},[281],[15,96875,96877,96910],{"className":96876},[285,286],[15,96878,96880,96907],{"className":96879},[290],[15,96881,96884],{"className":96882,"style":96883},[294],"height:0.3175em;",[15,96885,96886,96889],{"style":382},[15,96887],{"className":96888,"style":303},[302],[15,96890,96892],{"className":96891},[307,308,309,310],[15,96893,96895],{"className":96894},[59,310],[15,96896,96898,96901,96904],{"className":96897},[1486,310],[15,96899,26349],{"className":96900},[310],[15,96902,11967],{"className":96903},[310],[15,96905,82337],{"className":96906},[310],[15,96908,318],{"className":96909},[317],[15,96911,96913],{"className":96912},[290],[15,96914,96916],{"className":96915,"style":7970},[294],[15,96917],{},[15,96919,551],{"className":96920},[573],[15,96922],{"className":96923,"style":406},[157],[15,96925],{"className":96926,"style":406},[157],[15,96928,96930,96933],{"className":96929},[59],[15,96931,769],{"className":96932},[59,60],[15,96934,96936],{"className":96935},[281],[15,96937,96939,96971],{"className":96938},[285,286],[15,96940,96942,96968],{"className":96941},[290],[15,96943,96945],{"className":96944,"style":44955},[294],[15,96946,96947,96950],{"style":382},[15,96948],{"className":96949,"style":303},[302],[15,96951,96953],{"className":96952},[307,308,309,310],[15,96954,96956],{"className":96955},[59,310],[15,96957,96959,96962,96965],{"className":96958},[1486,310],[15,96960,26349],{"className":96961},[310],[15,96963,45206],{"className":96964},[310],[15,96966,774],{"className":96967},[310],[15,96969,318],{"className":96970},[317],[15,96972,96974],{"className":96973},[290],[15,96975,96977],{"className":96976,"style":7970},[294],[15,96978],{},[15,96980,257],{"className":96981},[153]," 的一致 Lipschitz 性给出）。",[11,96984,96985,96986,96989],{},"也就是说，",[998,96987,96988],{},"\"所有 f-散度诱导同一拓扑\"必须在有限\u002F紧设定或一致比值界下读","：此时它们都等价于全变差拓扑（也叫强拓扑），只是\"度量标尺\"不同；而在一般测度空间上，KL 型拓扑严格强于 TV 拓扑。这一分层对理解第 6 节 f-散度与 Wasserstein 的真正差异很重要：f-散度族的内部差异是\"强度\"差异，而它们与 Wasserstein 的差异是\"性质\"差异。",[2313,96991,96993],{"id":96992},"_48-一般不是度量","4.8 一般不是度量",[11,96995,96996],{},"f-散度一般不对称，也不满足三角不等式，因此通常不是距离度量。但某些特殊形式可以诱导出度量：",[10029,96998,96999,98511],{},[6667,97000,97001,97004,97005,97677,97680,97681,97709,97710,97783,97784,98044,98045,98450,98451,98510],{},[998,97002,97003],{},"Hellinger 距离","是度量：",[15,97006,97008],{"className":97007,"translate":19},[180],[15,97009,97011,97115],{"className":97010,"translate":19},[18],[15,97012,97014],{"className":97013},[23],[25,97015,97016],{"xmlns":27,"display":189},[29,97017,97018,97112],{},[32,97019,97020,97026,97028,97030,97032,97034,97036,97038,97076,97078,97090,97092,97110],{},[195,97021,97022,97024],{},[35,97023,44586],{},[35,97025,33910],{},[110,97027,113],{"stretchy":112},[35,97029,206],{},[110,97031,551],{"separator":46},[35,97033,213],{},[110,97035,120],{"stretchy":112},[110,97037,123],{},[34365,97039,97040],{},[32,97041,97042,97050,97052,97054,97058,97060,97064,97070,97072,97074],{},[1257,97043,97044],{"scriptlevel":126,"displaystyle":112},[238,97045,97046,97048],{},[115,97047,117],{},[115,97049,2465],{},[110,97051,1289],{},[110,97053,113],{"stretchy":112},[34365,97055,97056],{},[35,97057,11],{},[110,97059,4815],{},[34365,97061,97062],{},[35,97063,944],{},[6816,97065,97066,97068],{},[110,97067,120],{"stretchy":112},[115,97069,2465],{},[1299,97071,1301],{},[35,97073,244],{},[35,97075,774],{},[110,97077,123],{},[34365,97079,97080],{},[32,97081,97082,97084,97086,97088],{},[115,97083,117],{},[110,97085,4815],{},[35,97087,39317],{},[35,97089,39320],{},[110,97091,123],{},[34365,97093,97094],{},[32,97095,97096,97098,97100,97102],{},[115,97097,2465],{},[110,97099,4815],{},[115,97101,2465],{},[7555,97103,97104,97106,97108],{},[35,97105,33910],{},[1299,97107,42163],{},[115,97109,2465],{},[35,97111,4377],{"mathvariant":209},[39,97113,97114],{"encoding":41},"\\rho_H(P,Q)\n=\\sqrt{\\tfrac12\\int(\\sqrt p-\\sqrt q)^2\\,dx}\n=\\sqrt{1-BC}\n=\\sqrt{2-2H^2_{\\text{Bhattacharyya}}}.",[15,97116,97118,97191,97463,97544],{"className":97117,"ariaHidden":46},[45],[15,97119,97121,97124,97164,97167,97170,97173,97176,97179,97182,97185,97188],{"className":97120},[50],[15,97122],{"className":97123,"style":139},[54],[15,97125,97127,97130],{"className":97126},[59],[15,97128,44586],{"className":97129},[59,60],[15,97131,97133],{"className":97132},[281],[15,97134,97136,97156],{"className":97135},[285,286],[15,97137,97139,97153],{"className":97138},[290],[15,97140,97142],{"className":97141,"style":379},[294],[15,97143,97144,97147],{"style":382},[15,97145],{"className":97146,"style":303},[302],[15,97148,97150],{"className":97149},[307,308,309,310],[15,97151,33910],{"className":97152,"style":34064},[59,60,310],[15,97154,318],{"className":97155},[317],[15,97157,97159],{"className":97158},[290],[15,97160,97162],{"className":97161,"style":7970},[294],[15,97163],{},[15,97165,113],{"className":97166},[146],[15,97168,206],{"className":97169,"style":334},[59,60],[15,97171,551],{"className":97172},[573],[15,97174],{"className":97175,"style":406},[157],[15,97177,213],{"className":97178},[59,60],[15,97180,120],{"className":97181},[153],[15,97183],{"className":97184,"style":158},[157],[15,97186,123],{"className":97187},[162],[15,97189],{"className":97190,"style":158},[157],[15,97192,97194,97198,97454,97457,97460],{"className":97193},[50],[15,97195],{"className":97196,"style":97197},[54],"height:3.04em;vertical-align:-1.1572em;",[15,97199,97201],{"className":97200},[59,34393],[15,97202,97204,97446],{"className":97203},[285,286],[15,97205,97207,97443],{"className":97206},[290],[15,97208,97210,97431],{"className":97209,"style":91152},[294],[15,97211,97213,97216],{"className":97212,"style":91156},[6309],[15,97214],{"className":97215,"style":91160},[302],[15,97217,97219,97287,97290,97293,97296,97340,97343,97346,97349,97393,97422,97425,97428],{"className":97218,"style":41187},[59],[15,97220,97222,97225,97284],{"className":97221},[59],[15,97223],{"className":97224},[146,441],[15,97226,97228],{"className":97227},[238],[15,97229,97231,97276],{"className":97230},[285,286],[15,97232,97234,97273],{"className":97233},[290],[15,97235,97237,97251,97259],{"className":97236,"style":2526},[294],[15,97238,97239,97242],{"style":2529},[15,97240],{"className":97241,"style":461},[302],[15,97243,97245],{"className":97244},[307,308,309,310],[15,97246,97248],{"className":97247},[59,310],[15,97249,2465],{"className":97250},[59,310],[15,97252,97253,97256],{"style":473},[15,97254],{"className":97255,"style":461},[302],[15,97257],{"className":97258,"style":481},[480],[15,97260,97261,97264],{"style":2552},[15,97262],{"className":97263,"style":461},[302],[15,97265,97267],{"className":97266},[307,308,309,310],[15,97268,97270],{"className":97269},[59,310],[15,97271,117],{"className":97272},[59,310],[15,97274,318],{"className":97275},[317],[15,97277,97279],{"className":97278},[290],[15,97280,97282],{"className":97281,"style":2574},[294],[15,97283],{},[15,97285],{"className":97286},[153,441],[15,97288],{"className":97289,"style":406},[157],[15,97291,1289],{"className":97292,"style":1489},[1486,1487,1488],[15,97294,113],{"className":97295},[146],[15,97297,97299],{"className":97298},[59,34393],[15,97300,97302,97332],{"className":97301},[285,286],[15,97303,97305,97329],{"className":97304},[290],[15,97306,97308,97317],{"className":97307,"style":6864},[294],[15,97309,97311,97314],{"className":97310,"style":11867},[6309],[15,97312],{"className":97313,"style":461},[302],[15,97315,11],{"className":97316,"style":34413},[59,60],[15,97318,97319,97322],{"style":41376},[15,97320],{"className":97321,"style":461},[302],[15,97323,97325],{"className":97324,"style":34470},[34469],[6325,97326,97327],{"xmlns":6327,"width":6328,"height":34473,"viewBox":34474,"preserveAspectRatio":6331},[6333,97328],{"d":34477},[15,97330,318],{"className":97331},[317],[15,97333,97335],{"className":97334},[290],[15,97336,97338],{"className":97337,"style":41396},[294],[15,97339],{},[15,97341],{"className":97342,"style":2589},[157],[15,97344,4815],{"className":97345},[2593],[15,97347],{"className":97348,"style":2589},[157],[15,97350,97352],{"className":97351},[59,34393],[15,97353,97355,97385],{"className":97354},[285,286],[15,97356,97358,97382],{"className":97357},[290],[15,97359,97361,97370],{"className":97360,"style":6864},[294],[15,97362,97364,97367],{"className":97363,"style":11867},[6309],[15,97365],{"className":97366,"style":461},[302],[15,97368,944],{"className":97369,"style":39125},[59,60],[15,97371,97372,97375],{"style":41376},[15,97373],{"className":97374,"style":461},[302],[15,97376,97378],{"className":97377,"style":34470},[34469],[6325,97379,97380],{"xmlns":6327,"width":6328,"height":34473,"viewBox":34474,"preserveAspectRatio":6331},[6333,97381],{"d":34477},[15,97383,318],{"className":97384},[317],[15,97386,97388],{"className":97387},[290],[15,97389,97391],{"className":97390,"style":41396},[294],[15,97392],{},[1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em;",[15,97673],{},[15,97675,4377],{"className":97676},[59],[97678,97679],"br",{},"它有一个特别漂亮的几何解释：把分布 ",[15,97682,97684,97697],{"className":97683,"translate":19},[18],[15,97685,97687],{"className":97686},[23],[25,97688,97689],{"xmlns":27},[29,97690,97691,97695],{},[32,97692,97693],{},[35,97694,206],{},[39,97696,206],{"encoding":41},[15,97698,97700],{"className":97699,"ariaHidden":46},[45],[15,97701,97703,97706],{"className":97702},[50],[15,97704],{"className":97705,"style":664},[54],[15,97707,206],{"className":97708,"style":334},[59,60]," 映到单位球面上的点 ",[15,97711,97713,97729],{"className":97712,"translate":19},[18],[15,97714,97716],{"className":97715},[23],[25,97717,97718],{"xmlns":27},[29,97719,97720,97726],{},[32,97721,97722],{},[34365,97723,97724],{},[35,97725,11],{},[39,97727,97728],{"encoding":41},"\\sqrt 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就是单位球面上的欧氏距离——度量公理（对称、三角不等式）由 ",[15,98452,98454,98472],{"className":98453,"translate":19},[18],[15,98455,98457],{"className":98456},[23],[25,98458,98459],{"xmlns":27},[29,98460,98461,98469],{},[32,98462,98463],{},[6816,98464,98465,98467],{},[35,98466,21919],{},[115,98468,2465],{},[39,98470,98471],{"encoding":41},"L^2",[15,98473,98475],{"className":98474,"ariaHidden":46},[45],[15,98476,98478,98481],{"className":98477},[50],[15,98479],{"className":98480,"style":20054},[54],[15,98482,98484,98487],{"className":98483},[59],[15,98485,21919],{"className":98486},[59,60],[15,98488,98490],{"className":98489},[281],[15,98491,98493],{"className":98492},[285],[15,98494,98496],{"className":98495},[290],[15,98497,98499],{"className":98498,"style":20054},[294],[15,98500,98501,98504],{"style":6867},[15,98502],{"className":98503,"style":303},[302],[15,98505,98507],{"className":98506},[307,308,309,310],[15,98508,2465],{"className":98509},[59,310]," 距离直接继承，非退化由\"开平方是单射\"保证；",[6667,98512,98513,98829,98830,98909],{},[998,98514,98515,98653,98654,98828],{},[15,98516,98518,98548],{"className":98517,"translate":19},[18],[15,98519,98521],{"className":98520},[23],[25,98522,98523],{"xmlns":27},[29,98524,98525,98545],{},[32,98526,98527],{},[34365,98528,98529],{},[32,98530,98531,98533,98535],{},[115,98532,2465],{},[1299,98534,1301],{},[195,98536,98537,98539],{},[35,98538,199],{},[32,98540,98541,98543],{},[35,98542,23769],{},[35,98544,25763],{},[39,98546,98547],{"encoding":41},"\\sqrt{2\\,D_{JS}}",[15,98549,98551],{"className":98550,"ariaHidden":46},[45],[15,98552,98554,98557],{"className":98553},[50],[15,98555],{"className":98556,"style":34389},[54],[15,98558,98560],{"className":98559},[59,34393],[15,98561,98563,98645],{"className":98562},[285,286],[15,98564,98566,98642],{"className":98565},[290],[15,98567,98569,98630],{"className":98568,"style":34403},[294],[15,98570,98572,98575],{"className":98571,"style":11867},[6309],[15,98573],{"className":98574,"style":461},[302],[15,98576,98578,98581,98584],{"className":98577,"style":34413},[59],[15,98579,2465],{"className":98580},[59],[15,98582],{"className":98583,"style":406},[157],[15,98585,98587,98590],{"className":98586},[59],[15,98588,199],{"className":98589,"style":277},[59,60],[15,98591,98593],{"className":98592},[281],[15,98594,98596,98622],{"className":98595},[285,286],[15,98597,98599,98619],{"className":98598},[290],[15,98600,98602],{"className":98601,"style":379},[294],[15,98603,98604,98607],{"style":298},[15,98605],{"className":98606,"style":303},[302],[15,98608,98610],{"className":98609},[307,308,309,310],[15,98611,98613,98616],{"className":98612},[59,310],[15,98614,23769],{"className":98615,"style":24179},[59,60,310],[15,98617,25763],{"className":98618,"style":25887},[59,60,310],[15,98620,318],{"className":98621},[317],[15,98623,98625],{"className":98624},[290],[15,98626,98628],{"className":98627,"style":7970},[294],[15,98629],{},[15,98631,98632,98635],{"style":34462},[15,98633],{"className":98634,"style":461},[302],[15,98636,98638],{"className":98637,"style":34470},[34469],[6325,98639,98640],{"xmlns":6327,"width":6328,"height":34473,"viewBox":34474,"preserveAspectRatio":6331},[6333,98641],{"d":34477},[15,98643,318],{"className":98644},[317],[15,98646,98648],{"className":98647},[290],[15,98649,98651],{"className":98650,"style":34487},[294],[15,98652],{},"（等价地 ",[15,98655,98657,98685],{"className":98656,"translate":19},[18],[15,98658,98660],{"className":98659},[23],[25,98661,98662],{"xmlns":27},[29,98663,98664,98682],{},[32,98665,98666],{},[34365,98667,98668],{},[195,98669,98670,98672],{},[35,98671,199],{},[195,98673,98674,98676],{},[35,98675,37],{},[32,98677,98678,98680],{},[35,98679,23769],{},[35,98681,25763],{},[39,98683,98684],{"encoding":41},"\\sqrt{D_{f_{JS}}}",[15,98686,98688],{"className":98687,"ariaHidden":46},[45],[15,98689,98691,98695],{"className":98690},[50],[15,98692],{"className":98693,"style":98694},[54],"height:1.24em;vertical-align:-0.3564em;",[15,98696,98698],{"className":98697},[59,34393],[15,98699,98701,98819],{"className":98700},[285,286],[15,98702,98704,98816],{"className":98703},[290],[15,98705,98708,98803],{"className":98706,"style":98707},[294],"height:0.8836em;",[15,98709,98711,98714],{"className":98710,"style":83577},[6309],[15,98712],{"className":98713,"style":14850},[302],[15,98715,98717],{"className":98716,"style":41187},[59],[15,98718,98720,98723],{"className":98719},[59],[15,98721,199],{"className":98722,"style":277},[59,60],[15,98724,98726],{"className":98725},[281],[15,98727,98729,98795],{"className":98728},[285,286],[15,98730,98732,98792],{"className":98731},[290],[15,98733,98735],{"className":98734,"style":295},[294],[15,98736,98737,98740],{"style":298},[15,98738],{"className":98739,"style":303},[302],[15,98741,98743],{"className":98742},[307,308,309,310],[15,98744,98746],{"className":98745},[59,310],[15,98747,98749,98752],{"className":98748},[59,310],[15,98750,37],{"className":98751,"style":61},[59,60,310],[15,98753,98755],{"className":98754},[281],[15,98756,98758,98784],{"className":98757},[285,286],[15,98759,98761,98781],{"className":98760},[290],[15,98762,98764],{"className":98763,"style":6100},[294],[15,98765,98766,98769],{"style":30228},[15,98767],{"className":98768,"style":30232},[302],[15,98770,98772],{"className":98771},[307,30236,9676,310],[15,98773,98775,98778],{"className":98774},[59,310],[15,98776,23769],{"className":98777,"style":24179},[59,60,310],[15,98779,25763],{"className":98780,"style":25887},[59,60,310],[15,98782,318],{"className":98783},[317],[15,98785,98787],{"className":98786},[290],[15,98788,98790],{"className":98789,"style":30255},[294],[15,98791],{},[15,98793,318],{"className":98794},[317],[15,98796,98798],{"className":98797},[290],[15,98799,98801],{"className":98800,"style":325},[294],[15,98802],{},[15,98804,98806,98809],{"style":98805},"top:-2.8436em;",[15,98807],{"className":98808,"style":14850},[302],[15,98810,98812],{"className":98811,"style":83669},[34469],[6325,98813,98814],{"xmlns":6327,"width":6328,"height":83672,"viewBox":83673,"preserveAspectRatio":6331},[6333,98815],{"d":83676},[15,98817,318],{"className":98818},[317],[15,98820,98822],{"className":98821},[290],[15,98823,98826],{"className":98824,"style":98825},[294],"height:0.3564em;",[15,98827],{}," 规范化后）是度量","（Endres–Schindelin 的结论），其证明用到了 ",[15,98831,98833,98854],{"className":98832,"translate":19},[18],[15,98834,98836],{"className":98835},[23],[25,98837,98838],{"xmlns":27},[29,98839,98840,98852],{},[32,98841,98842],{},[195,98843,98844,98846],{},[35,98845,37],{},[32,98847,98848,98850],{},[35,98849,23769],{},[35,98851,25763],{},[39,98853,30781],{"encoding":41},[15,98855,98857],{"className":98856,"ariaHidden":46},[45],[15,98858,98860,98863],{"className":98859},[50],[15,98861],{"className":98862,"style":55},[54],[15,98864,98866,98869],{"className":98865},[59],[15,98867,37],{"className":98868,"style":61},[59,60],[15,98870,98872],{"className":98871},[281],[15,98873,98875,98901],{"className":98874},[285,286],[15,98876,98878,98898],{"className":98877},[290],[15,98879,98881],{"className":98880,"style":379},[294],[15,98882,98883,98886],{"style":27535},[15,98884],{"className":98885,"style":303},[302],[15,98887,98889],{"className":98888},[307,308,309,310],[15,98890,98892,98895],{"className":98891},[59,310],[15,98893,23769],{"className":98894,"style":24179},[59,60,310],[15,98896,25763],{"className":98897,"style":25887},[59,60,310],[15,98899,318],{"className":98900},[317],[15,98902,98904],{"className":98903},[290],[15,98905,98907],{"className":98906,"style":7970},[294],[15,98908],{}," 对应生成元的\"负定型\"性质。",[11,98911,98912,98913,99029,99030,99110,99111,99139,99140,99168],{},"反过来，不对称显然排除度量；即使对称，",[15,98914,98916,98936],{"className":98915,"translate":19},[18],[15,98917,98919],{"className":98918},[23],[25,98920,98921],{"xmlns":27},[29,98922,98923,98933],{},[32,98924,98925],{},[34365,98926,98927],{},[195,98928,98929,98931],{},[35,98930,199],{},[35,98932,37],{},[39,98934,98935],{"encoding":41},"\\sqrt{D_f}",[15,98937,98939],{"className":98938,"ariaHidden":46},[45],[15,98940,98942,98945],{"className":98941},[50],[15,98943],{"className":98944,"style":98694},[54],[15,98946,98948],{"className":98947},[59,34393],[15,98949,98951,99021],{"className":98950},[285,286],[15,98952,98954,99018],{"className":98953},[290],[15,98955,98957,99006],{"className":98956,"style":98707},[294],[15,98958,98960,98963],{"className":98959,"style":83577},[6309],[15,98961],{"className":98962,"style":14850},[302],[15,98964,98966],{"className":98965,"style":41187},[59],[15,98967,98969,98972],{"className":98968},[59],[15,98970,199],{"className":98971,"style":277},[59,60],[15,98973,98975],{"className":98974},[281],[15,98976,98978,98998],{"className":98977},[285,286],[15,98979,98981,98995],{"className":98980},[290],[15,98982,98984],{"className":98983,"style":295},[294],[15,98985,98986,98989],{"style":298},[15,98987],{"className":98988,"style":303},[302],[15,98990,98992],{"className":98991},[307,308,309,310],[15,98993,37],{"className":98994,"style":61},[59,60,310],[15,98996,318],{"className":98997},[317],[15,98999,99001],{"className":99000},[290],[15,99002,99004],{"className":99003,"style":325},[294],[15,99005],{},[15,99007,99008,99011],{"style":98805},[15,99009],{"className":99010,"style":14850},[302],[15,99012,99014],{"className":99013,"style":83669},[34469],[6325,99015,99016],{"xmlns":6327,"width":6328,"height":83672,"viewBox":83673,"preserveAspectRatio":6331},[6333,99017],{"d":83676},[15,99019,318],{"className":99020},[317],[15,99022,99024],{"className":99023},[290],[15,99025,99027],{"className":99026,"style":98825},[294],[15,99028],{}," 也不自动是度量。例如全变差 ",[15,99031,99033,99055],{"className":99032,"translate":19},[18],[15,99034,99036],{"className":99035},[23],[25,99037,99038],{"xmlns":27},[29,99039,99040,99052],{},[32,99041,99042],{},[195,99043,99044,99046],{},[35,99045,199],{},[32,99047,99048,99050],{},[35,99049,5394],{},[35,99051,34876],{},[39,99053,99054],{"encoding":41},"D_{TV}",[15,99056,99058],{"className":99057,"ariaHidden":46},[45],[15,99059,99061,99064],{"className":99060},[50],[15,99062],{"className":99063,"style":15965},[54],[15,99065,99067,99070],{"className":99066},[59],[15,99068,199],{"className":99069,"style":277},[59,60],[15,99071,99073],{"className":99072},[281],[15,99074,99076,99102],{"className":99075},[285,286],[15,99077,99079,99099],{"className":99078},[290],[15,99080,99082],{"className":99081,"style":379},[294],[15,99083,99084,99087],{"style":298},[15,99085],{"className":99086,"style":303},[302],[15,99088,99090],{"className":99089},[307,308,309,310],[15,99091,99093,99096],{"className":99092},[59,310],[15,99094,5394],{"className":99095,"style":334},[59,60,310],[15,99097,34876],{"className":99098,"style":2589},[59,60,310],[15,99100,318],{"className":99101},[317],[15,99103,99105],{"className":99104},[290],[15,99106,99108],{"className":99107,"style":7970},[294],[15,99109],{}," 本身就是度量，但若取某凸积 ",[15,99112,99114,99127],{"className":99113,"translate":19},[18],[15,99115,99117],{"className":99116},[23],[25,99118,99119],{"xmlns":27},[29,99120,99121,99125],{},[32,99122,99123],{},[35,99124,37],{},[39,99126,37],{"encoding":41},[15,99128,99130],{"className":99129,"ariaHidden":46},[45],[15,99131,99133,99136],{"className":99132},[50],[15,99134],{"className":99135,"style":55},[54],[15,99137,37],{"className":99138,"style":61},[59,60]," 的散度再做开方，三角不等式可能失效——这依赖 ",[15,99141,99143,99156],{"className":99142,"translate":19},[18],[15,99144,99146],{"className":99145},[23],[25,99147,99148],{"xmlns":27},[29,99149,99150,99154],{},[32,99151,99152],{},[35,99153,37],{},[39,99155,37],{"encoding":41},[15,99157,99159],{"className":99158,"ariaHidden":46},[45],[15,99160,99162,99165],{"className":99161},[50],[15,99163],{"className":99164,"style":55},[54],[15,99166,37],{"className":99167,"style":61},[59,60]," 是否落在\"负定（negative type）\"函数类中；不是自动成立的。",[11,99170,99171,99174,99175,99177,99178,99228,99229,99257,99258,99337],{},[998,99172,99173],{},"与度量的对照结论："," f-散度的正确身份是",[998,99176,52529],{},"（divergence）：非负、在 ",[15,99179,99181,99198],{"className":99180,"translate":19},[18],[15,99182,99184],{"className":99183},[23],[25,99185,99186],{"xmlns":27},[29,99187,99188,99196],{},[32,99189,99190,99192,99194],{},[35,99191,206],{},[110,99193,123],{},[35,99195,213],{},[39,99197,10273],{"encoding":41},[15,99199,99201,99219],{"className":99200,"ariaHidden":46},[45],[15,99202,99204,99207,99210,99213,99216],{"className":99203},[50],[15,99205],{"className":99206,"style":664},[54],[15,99208,206],{"className":99209,"style":334},[59,60],[15,99211],{"className":99212,"style":158},[157],[15,99214,123],{"className":99215},[162],[15,99217],{"className":99218,"style":158},[157],[15,99220,99222,99225],{"className":99221},[50],[15,99223],{"className":99224,"style":566},[54],[15,99226,213],{"className":99227},[59,60]," 处取零，但不要求对称与三角不等式。当我们需要真距离时（例如做收敛分析、中心化、 ",[15,99230,99232,99245],{"className":99231,"translate":19},[18],[15,99233,99235],{"className":99234},[23],[25,99236,99237],{"xmlns":27},[29,99238,99239,99243],{},[32,99240,99241],{},[35,99242,7566],{},[39,99244,7566],{"encoding":41},[15,99246,99248],{"className":99247,"ariaHidden":46},[45],[15,99249,99251,99254],{"className":99250},[50],[15,99252],{"className":99253,"style":12601},[54],[15,99255,7566],{"className":99256,"style":8269},[59,60],"-中心聚类），就从 f-散度族里挑那些 ",[15,99259,99261,99277],{"className":99260,"translate":19},[18],[15,99262,99264],{"className":99263},[23],[25,99265,99266],{"xmlns":27},[29,99267,99268,99274],{},[32,99269,99270],{},[34365,99271,99272],{},[110,99273,3301],{"lspace":1249,"rspace":1249},[39,99275,99276],{"encoding":41},"\\sqrt{\\cdot}",[15,99278,99280],{"className":99279,"ariaHidden":46},[45],[15,99281,99283,99287],{"className":99282},[50],[15,99284],{"className":99285,"style":99286},[54],"height:1.04em;vertical-align:-0.2328em;",[15,99288,99290],{"className":99289},[59,34393],[15,99291,99293,99328],{"className":99292},[285,286],[15,99294,99296,99325],{"className":99295},[290],[15,99297,99300,99312],{"className":99298,"style":99299},[294],"height:0.8072em;",[15,99301,99303,99306],{"className":99302,"style":11867},[6309],[15,99304],{"className":99305,"style":461},[302],[15,99307,99309],{"className":99308,"style":34413},[59],[15,99310,3301],{"className":99311},[59],[15,99313,99315,99318],{"style":99314},"top:-2.7672em;",[15,99316],{"className":99317,"style":461},[302],[15,99319,99321],{"className":99320,"style":34470},[34469],[6325,99322,99323],{"xmlns":6327,"width":6328,"height":34473,"viewBox":34474,"preserveAspectRatio":6331},[6333,99324],{"d":34477},[15,99326,318],{"className":99327},[317],[15,99329,99331],{"className":99330},[290],[15,99332,99335],{"className":99333,"style":99334},[294],"height:0.2328em;",[15,99336],{}," 后成为度量的成员（Hellinger、JS、TV），或干脆换成真正的度量类（第 6 节的 IPM）。",[528,99339,99341],{"id":99340},"_5-变分表示与-f-gan","5. 变分表示与 f-GAN",[2313,99343,99345],{"id":99344},"_51-可训练目标","5.1 可训练目标",[11,99347,99348],{},"4.4 节给出精确对偶：",[15,99350,99352],{"className":99351,"translate":19},[180],[15,99353,99355,99449],{"className":99354,"translate":19},[18],[15,99356,99358],{"className":99357},[23],[25,99359,99360],{"xmlns":27,"display":189},[29,99361,99362,99446],{},[32,99363,99364,99370,99372,99374,99376,99378,99380,99382,99392,99394,99400,99402,99404,99406,99408,99410,99412,99414,99420,99422,99428,99430,99432,99434,99436,99438,99440,99442,99444],{},[195,99365,99366,99368],{},[35,99367,199],{},[35,99369,37],{},[110,99371,113],{"stretchy":112},[35,99373,206],{},[35,99375,210],{"mathvariant":209},[35,99377,213],{},[110,99379,120],{"stretchy":112},[110,99381,123],{},[6018,99383,99384,99390],{},[32,99385,99386,99388],{},[35,99387,35458],{},[110,99389,4105],{},[35,99391,4160],{},[110,99393,2900],{"fence":112,"stretchy":46,"minsize":17443,"maxsize":17443},[195,99395,99396,99398],{},[35,99397,223],{"mathvariant":222},[35,99399,206],{},[110,99401,230],{"stretchy":112},[35,99403,4160],{},[110,99405,113],{"stretchy":112},[35,99407,598],{},[110,99409,120],{"stretchy":112},[110,99411,257],{"stretchy":112},[110,99413,4815],{},[195,99415,99416,99418],{},[35,99417,223],{"mathvariant":222},[35,99419,213],{},[110,99421,230],{"stretchy":112},[6816,99423,99424,99426],{},[35,99425,37],{},[110,99427,7575],{},[110,99429,113],{"stretchy":112},[35,99431,4160],{},[110,99433,113],{"stretchy":112},[35,99435,598],{},[110,99437,120],{"stretchy":112},[110,99439,120],{"stretchy":112},[110,99441,257],{"stretchy":112},[110,99443,2910],{"fence":112,"stretchy":46,"minsize":17443,"maxsize":17443},[35,99445,4377],{"mathvariant":209},[39,99447,99448],{"encoding":41},"D_f(P\\|Q)\n=\\sup_{g}\\Big\\{\\mathbb E_P[g(X)]\n-\\mathbb E_Q[f^*(g(X))]\\Big\\}.",[15,99450,99452,99522,99649],{"className":99451,"ariaHidden":46},[45],[15,99453,99455,99458,99498,99501,99504,99507,99510,99513,99516,99519],{"className":99454},[50],[15,99456],{"className":99457,"style":270},[54],[15,99459,99461,99464],{"className":99460},[59],[15,99462,199],{"className":99463,"style":277},[59,60],[15,99465,99467],{"className":99466},[281],[15,99468,99470,99490],{"className":99469},[285,286],[15,99471,99473,99487],{"className":99472},[290],[15,99474,99476],{"className":99475,"style":295},[294],[15,99477,99478,99481],{"style":298},[15,99479],{"className":99480,"style":303},[302],[15,99482,99484],{"className":99483},[307,308,309,310],[15,99485,37],{"className":99486,"style":61},[59,60,310],[15,99488,318],{"className":99489},[317],[15,99491,99493],{"className":99492},[290],[15,99494,99496],{"className":99495,"style":325},[294],[15,99497],{},[15,99499,113],{"className":99500},[146],[15,99502,206],{"className":99503,"style":334},[59,60],[15,99505,210],{"className":99506},[59],[15,99508,213],{"className":99509},[59,60],[15,99511,120],{"className":99512},[153],[15,99514],{"className":99515,"style":158},[157],[15,99517,123],{"className":99518},[162],[15,99520],{"className":99521,"style":158},[157],[15,99523,99525,99529,99576,99579,99585,99625,99628,99631,99634,99637,99640,99643,99646],{"className":99524},[50],[15,99526],{"className":99527,"style":99528},[54],"height:2.1805em;vertical-align:-1.0305em;",[15,99530,99532],{"className":99531},[1486,13720],[15,99533,99535,99568],{"className":99534},[285,286],[15,99536,99538,99565],{"className":99537},[290],[15,99539,99541,99555],{"className":99540,"style":2347},[294],[15,99542,99543,99546],{"style":61227},[15,99544],{"className":99545,"style":461},[302],[15,99547,99549],{"className":99548},[307,308,309,310],[15,99550,99552],{"className":99551},[59,310],[15,99553,4160],{"className":99554,"style":981},[59,60,310],[15,99556,99557,99560],{"style":11867},[15,99558],{"className":99559,"style":461},[302],[15,99561,99562],{},[15,99563,35458],{"className":99564},[1486],[15,99566,318],{"className":99567},[317],[15,99569,99571],{"className":99570},[290],[15,99572,99574],{"className":99573,"style":61259},[294],[15,99575],{},[15,99577],{"className":99578,"style":406},[157],[15,99580,99582],{"className":99581},[59],[15,99583,2900],{"className":99584},[419,3722],[15,99586,99588,99591],{"className":99587},[59],[15,99589,223],{"className":99590},[59,366],[15,99592,99594],{"className":99593},[281],[15,99595,99597,99617],{"className":99596},[285,286],[15,99598,99600,99614],{"className":99599},[290],[15,99601,99603],{"className":99602,"style":379},[294],[15,99604,99605,99608],{"style":382},[15,99606],{"className":99607,"style":303},[302],[15,99609,99611],{"className":99610},[307,308,309,310],[15,99612,206],{"className":99613,"style":334},[59,60,310],[15,99615,318],{"className":99616},[317],[15,99618,99620],{"className":99619},[290],[15,99621,99623],{"className":99622,"style":7970},[294],[15,99624],{},[15,99626,230],{"className":99627},[146],[15,99629,4160],{"className":99630,"style":981},[59,60],[15,99632,113],{"className":99633},[146],[15,99635,598],{"className":99636,"style":8889},[59,60],[15,99638,20416],{"className":99639},[153],[15,99641],{"className":99642,"style":2589},[157],[15,99644,4815],{"className":99645},[2593],[15,99647],{"className":99648,"style":2589},[157],[15,99650,99652,99655,99695,99698,99727,99730,99733,99736,99739,99743,99749],{"className":99651},[50],[15,99653],{"className":99654,"style":68154},[54],[15,99656,99658,99661],{"className":99657},[59],[15,99659,223],{"className":99660},[59,366],[15,99662,99664],{"className":99663},[281],[15,99665,99667,99687],{"className":99666},[285,286],[15,99668,99670,99684],{"className":99669},[290],[15,99671,99673],{"className":99672,"style":379},[294],[15,99674,99675,99678],{"style":382},[15,99676],{"className":99677,"style":303},[302],[15,99679,99681],{"className":99680},[307,308,309,310],[15,99682,213],{"className":99683},[59,60,310],[15,99685,318],{"className":99686},[317],[15,99688,99690],{"className":99689},[290],[15,99691,99693],{"className":99692,"style":325},[294],[15,99694],{},[15,99696,230],{"className":99697},[146],[15,99699,99701,99704],{"className":99700},[59],[15,99702,37],{"className":99703,"style":61},[59,60],[15,99705,99707],{"className":99706},[281],[15,99708,99710],{"className":99709},[285],[15,99711,99713],{"className":99712},[290],[15,99714,99716],{"className":99715,"style":8250},[294],[15,99717,99718,99721],{"style":8279},[15,99719],{"className":99720,"style":303},[302],[15,99722,99724],{"className":99723},[307,308,309,310],[15,99725,7575],{"className":99726},[2593,310],[15,99728,113],{"className":99729},[146],[15,99731,4160],{"className":99732,"style":981},[59,60],[15,99734,113],{"className":99735},[146],[15,99737,598],{"className":99738,"style":8889},[59,60],[15,99740,99742],{"className":99741},[153],"))]",[15,99744,99746],{"className":99745},[59],[15,99747,2910],{"className":99748},[419,3722],[15,99750,4377],{"className":99751},[59],[11,99753,68169,99754,99782,99783,99811,99812,99881,99882,99885,99886,99957],{},[15,99755,99757,99770],{"className":99756,"translate":19},[18],[15,99758,99760],{"className":99759},[23],[25,99761,99762],{"xmlns":27},[29,99763,99764,99768],{},[32,99765,99766],{},[35,99767,206],{},[39,99769,206],{"encoding":41},[15,99771,99773],{"className":99772,"ariaHidden":46},[45],[15,99774,99776,99779],{"className":99775},[50],[15,99777],{"className":99778,"style":664},[54],[15,99780,206],{"className":99781,"style":334},[59,60]," 是数据分布、",[15,99784,99786,99799],{"className":99785,"translate":19},[18],[15,99787,99789],{"className":99788},[23],[25,99790,99791],{"xmlns":27},[29,99792,99793,99797],{},[32,99794,99795],{},[35,99796,213],{},[39,99798,213],{"encoding":41},[15,99800,99802],{"className":99801,"ariaHidden":46},[45],[15,99803,99805,99808],{"className":99804},[50],[15,99806],{"className":99807,"style":566},[54],[15,99809,213],{"className":99810},[59,60]," 是生成分布时，右端可直接用两批样本做蒙特卡洛估计，得到 ",[15,99813,99815,99832],{"className":99814,"translate":19},[18],[15,99816,99818],{"className":99817},[23],[25,99819,99820],{"xmlns":27},[29,99821,99822,99830],{},[32,99823,99824],{},[195,99825,99826,99828],{},[35,99827,199],{},[35,99829,37],{},[39,99831,4024],{"encoding":41},[15,99833,99835],{"className":99834,"ariaHidden":46},[45],[15,99836,99838,99841],{"className":99837},[50],[15,99839],{"className":99840,"style":4034},[54],[15,99842,99844,99847],{"className":99843},[59],[15,99845,199],{"className":99846,"style":277},[59,60],[15,99848,99850],{"className":99849},[281],[15,99851,99853,99873],{"className":99852},[285,286],[15,99854,99856,99870],{"className":99855},[290],[15,99857,99859],{"className":99858,"style":295},[294],[15,99860,99861,99864],{"style":298},[15,99862],{"className":99863,"style":303},[302],[15,99865,99867],{"className":99866},[307,308,309,310],[15,99868,37],{"className":99869,"style":61},[59,60,310],[15,99871,318],{"className":99872},[317],[15,99874,99876],{"className":99875},[290],[15,99877,99879],{"className":99878,"style":325},[294],[15,99880],{}," 的一个",[998,99883,99884],{},"下界","，并用一个网络 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逼近最优解。于是",[15,99959,99961],{"className":99960,"translate":19},[180],[15,99962,99964,100094],{"className":99963,"translate":19},[18],[15,99965,99967],{"className":99966},[23],[25,99968,99969],{"xmlns":27,"display":189},[29,99970,99971,100091],{},[32,99972,99973,99984,99986,99988,99990,99992,99994,100000,100002,100004,100010,100024,100030,100032,100038,100040,100042,100048,100063,100069,100071,100077,100079,100085,100087,100089],{},[195,99974,99975,99982],{},[11815,99976,99977,99979],{"accent":46},[35,99978,199],{},[110,99980,99981],{"stretchy":46},"^",[35,99983,37],{},[110,99985,113],{"stretchy":112},[35,99987,206],{},[35,99989,210],{"mathvariant":209},[35,99991,213],{},[110,99993,33901],{"separator":46},[195,99995,99996,99998],{},[35,99997,4160],{},[35,99999,99904],{},[110,100001,120],{"stretchy":112},[110,100003,123],{},[238,100005,100006,100008],{},[115,100007,117],{},[35,100009,82337],{},[48346,100011,100012,100014,100022],{},[110,100013,13465],{},[32,100015,100016,100018,100020],{},[35,100017,11967],{},[110,100019,123],{},[115,100021,117],{},[35,100023,82337],{},[195,100025,100026,100028],{},[35,100027,4160],{},[35,100029,99904],{},[110,100031,113],{"stretchy":112},[195,100033,100034,100036],{},[35,100035,774],{},[35,100037,11967],{},[110,100039,120],{"stretchy":112},[110,100041,4815],{},[238,100043,100044,100046],{},[115,100045,117],{},[35,100047,26349],{},[48346,100049,100050,100052,100061],{},[110,100051,13465],{},[32,100053,100054,100057,100059],{},[35,100055,100056],{},"j",[110,100058,123],{},[115,100060,117],{},[35,100062,26349],{},[6816,100064,100065,100067],{},[35,100066,37],{},[110,100068,7575],{},[110,100070,113],{"stretchy":112},[195,100072,100073,100075],{},[35,100074,4160],{},[35,100076,99904],{},[110,100078,113],{"stretchy":112},[195,100080,100081,100083],{},[35,100082,40263],{},[35,100084,100056],{},[110,100086,120],{"stretchy":112},[110,100088,120],{"stretchy":112},[35,100090,4377],{"mathvariant":209},[39,100092,100093],{"encoding":41},"\\widehat D_f(P\\|Q;g_\\omega)\n=\\frac1n\\sum_{i=1}^n g_\\omega(x_i)\n-\\frac1m\\sum_{j=1}^m f^*(g_\\omega(y_j)).",[15,100095,100097,100250,100486],{"className":100096,"ariaHidden":46},[45],[15,100098,100100,100104,100180,100183,100186,100189,100192,100195,100198,100238,100241,100244,100247],{"className":100099},[50],[15,100101],{"className":100102,"style":100103},[54],"height:1.2094em;vertical-align:-0.2861em;",[15,100105,100107,100146],{"className":100106},[59],[15,100108,100110],{"className":100109},[59,11855],[15,100111,100113],{"className":100112},[285],[15,100114,100116],{"className":100115},[290],[15,100117,100120,100128],{"className":100118,"style":100119},[294],"height:0.9233em;",[15,100121,100122,100125],{"style":11867},[15,100123],{"className":100124,"style":461},[302],[15,100126,199],{"className":100127,"style":277},[59,60],[15,100129,100132,100135],{"className":100130,"style":100131},[6309],"width:calc(100% - 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的最优解为",[15,101470,101472],{"className":101471,"translate":19},[180],[15,101473,101475,101539],{"className":101474,"translate":19},[18],[15,101476,101478],{"className":101477},[23],[25,101479,101480],{"xmlns":27,"display":189},[29,101481,101482,101536],{},[32,101483,101484,101490,101492,101494,101496,101498,101504,101506,101534],{},[6816,101485,101486,101488],{},[35,101487,4160],{},[110,101489,7575],{},[110,101491,113],{"stretchy":112},[35,101493,774],{},[110,101495,120],{"stretchy":112},[110,101497,123],{},[6816,101499,101500,101502],{},[35,101501,37],{},[110,101503,6822],{"mathvariant":209,"lspace":1249,"rspace":1249},[1299,101505,1306],{},[32,101507,101508,101510,101532],{},[110,101509,113],{"fence":46},[238,101511,101512,101522],{},[32,101513,101514,101516,101518,101520],{},[35,101515,11],{},[110,101517,113],{"stretchy":112},[35,101519,774],{},[110,101521,120],{"stretchy":112},[32,101523,101524,101526,101528,101530],{},[35,101525,944],{},[110,101527,113],{"stretchy":112},[35,101529,774],{},[110,101531,120],{"stretchy":112},[110,101533,120],{"fence":46},[110,101535,551],{"separator":46},[39,101537,101538],{"encoding":41},"g^*(x)=f'\\!\\left(\\frac{p(x)}{q(x)}\\right),",[15,101540,101542,101595],{"className":101541,"ariaHidden":46},[45],[15,101543,101545,101548,101577,101580,101583,101586,101589,101592],{"className":101544},[50],[15,101546],{"className":101547,"style":139},[54],[15,101549,101551,101554],{"className":101550},[59],[15,101552,4160],{"className":101553,"style":981},[59,60],[15,101555,101557],{"className":101556},[281],[15,101558,101560],{"className":101559},[285],[15,101561,101563],{"className":101562},[290],[15,101564,101566],{"className":101565,"style":8250},[294],[15,101567,101568,101571],{"style":8279},[15,101569],{"className":101570,"style":303},[302],[15,101572,101574],{"className":101573},[307,308,309,310],[15,101575,7575],{"className":101576},[2593,310],[15,101578,113],{"className":101579},[146],[15,101581,774],{"className":101582},[59,60],[15,101584,120],{"className":101585},[153],[15,101587],{"className":101588,"style":158},[157],[15,101590,123],{"className":101591},[162],[15,101593],{"className":101594,"style":158},[157],[15,101596,101598,101601,101633,101636,101639,101734,101737],{"className":101597},[50],[15,101599],{"className":101600,"style":359},[54],[15,101602,101604,101607],{"className":101603},[59],[15,101605,37],{"className":101606,"style":61},[59,60],[15,101608,101610],{"className":101609},[281],[15,101611,101613],{"className":101612},[285],[15,101614,101616],{"className":101615},[290],[15,101617,101619],{"className":101618,"style":28997},[294],[15,101620,101621,101624],{"style":8279},[15,101622],{"className":101623,"style":303},[302],[15,101625,101627],{"className":101626},[307,308,309,310],[15,101628,101630],{"className":101629},[59,310],[15,101631,6822],{"className":101632},[59,310],[15,101634],{"className":101635,"style":1514},[157],[15,101637],{"className":101638,"style":406},[157],[15,101640,101642,101648,101728],{"className":101641},[410],[15,101643,101645],{"className":101644,"style":415},[146,414],[15,101646,113],{"className":101647},[419,309],[15,101649,101651,101654,101725],{"className":101650},[59],[15,101652],{"className":101653},[146,441],[15,101655,101657],{"className":101656},[238],[15,101658,101660,101717],{"className":101659},[285,286],[15,101661,101663,101714],{"className":101662},[290],[15,101664,101666,101686,101694],{"className":101665,"style":1545},[294],[15,101667,101668,101671],{"style":457},[15,101669],{"className":101670,"style":461},[302],[15,101672,101674,101677,101680,101683],{"className":101673},[59],[15,101675,944],{"className":101676,"style":981},[59,60],[15,101678,113],{"className":101679},[146],[15,101681,774],{"className":101682},[59,60],[15,101684,120],{"className":101685},[153],[15,101687,101688,101691],{"style":473},[15,101689],{"className":101690,"style":461},[302],[15,101692],{"className":101693,"style":481},[480],[15,101695,101696,101699],{"style":484},[15,101697],{"className":101698,"style":461},[302],[15,101700,101702,101705,101708,101711],{"className":101701},[59],[15,101703,11],{"className":101704},[59,60],[15,101706,113],{"className":101707},[146],[15,101709,774],{"className":101710},[59,60],[15,101712,120],{"className":101713},[153],[15,101715,318],{"className":101716},[317],[15,101718,101720],{"className":101719},[290],[15,101721,101723],{"className":101722,"style":1603},[294],[15,101724],{},[15,101726],{"className":101727},[153,441],[15,101729,101731],{"className":101730,"style":415},[153,414],[15,101732,120],{"className":101733},[419,309],[15,101735],{"className":101736,"style":406},[157],[15,101738,551],{"className":101739},[573],[11,101741,101742,101743,101805],{},"即判别器学到的量是似然比经 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之后的取值。",[11,101807,101808,101810,101811,101855,101856,102019,102020,102063],{},[998,101809,55499],{}," 目标对 ",[15,101812,101814,101834],{"className":101813,"translate":19},[18],[15,101815,101817],{"className":101816},[23],[25,101818,101819],{"xmlns":27},[29,101820,101821,101831],{},[32,101822,101823,101825,101827,101829],{},[35,101824,4160],{},[110,101826,113],{"stretchy":112},[35,101828,774],{},[110,101830,120],{"stretchy":112},[39,101832,101833],{"encoding":41},"g(x)",[15,101835,101837],{"className":101836,"ariaHidden":46},[45],[15,101838,101840,101843,101846,101849,101852],{"className":101839},[50],[15,101841],{"className":101842,"style":139},[54],[15,101844,4160],{"className":101845,"style":981},[59,60],[15,101847,113],{"className":101848},[146],[15,101850,774],{"className":101851},[59,60],[15,101853,120],{"className":101854},[153]," 的逐点依赖是 ",[15,101857,101859,101915],{"className":101858,"translate":19},[18],[15,101860,101862],{"className":101861},[23],[25,101863,101864],{"xmlns":27},[29,101865,101866,101912],{},[32,101867,101868,101870,101872,101874,101876,101878,101880,101882,101884,101886,101888,101890,101892,101894,101900,101902,101904,101906,101908,101910],{},[35,101869,11],{},[110,101871,113],{"stretchy":112},[35,101873,774],{},[110,101875,120],{"stretchy":112},[35,101877,4160],{},[110,101879,113],{"stretchy":112},[35,101881,774],{},[110,101883,120],{"stretchy":112},[110,101885,4815],{},[35,101887,944],{},[110,101889,113],{"stretchy":112},[35,101891,774],{},[110,101893,120],{"stretchy":112},[6816,101895,101896,101898],{},[35,101897,37],{},[110,101899,7575],{},[110,101901,113],{"stretchy":112},[35,101903,4160],{},[110,101905,113],{"stretchy":112},[35,101907,774],{},[110,101909,120],{"stretchy":112},[110,101911,120],{"stretchy":112},[39,101913,101914],{"encoding":41},"p(x)g(x)-q(x)f^*(g(x))",[15,101916,101918,101957],{"className":101917,"ariaHidden":46},[45],[15,101919,101921,101924,101927,101930,101933,101936,101939,101942,101945,101948,101951,101954],{"className":101920},[50],[15,101922],{"className":101923,"style":139},[54],[15,101925,11],{"className":101926},[59,60],[15,101928,113],{"className":101929},[146],[15,101931,774],{"className":101932},[59,60],[15,101934,120],{"className":101935},[153],[15,101937,4160],{"className":101938,"style":981},[59,60],[15,101940,113],{"className":101941},[146],[15,101943,774],{"className":101944},[59,60],[15,101946,120],{"className":101947},[153],[15,101949],{"className":101950,"style":2589},[157],[15,101952,4815],{"className":101953},[2593],[15,101955],{"className":101956,"style":2589},[157],[15,101958,101960,101963,101966,101969,101972,101975,102004,102007,102010,102013,102016],{"className":101959},[50],[15,101961],{"className":101962,"style":139},[54],[15,101964,944],{"className":101965,"style":981},[59,60],[15,101967,113],{"className":101968},[146],[15,101970,774],{"className":101971},[59,60],[15,101973,120],{"className":101974},[153],[15,101976,101978,101981],{"className":101977},[59],[15,101979,37],{"className":101980,"style":61},[59,60],[15,101982,101984],{"className":101983},[281],[15,101985,101987],{"className":101986},[285],[15,101988,101990],{"className":101989},[290],[15,101991,101993],{"className":101992,"style":8501},[294],[15,101994,101995,101998],{"style":6867},[15,101996],{"className":101997,"style":303},[302],[15,101999,102001],{"className":102000},[307,308,309,310],[15,102002,7575],{"className":102003},[2593,310],[15,102005,113],{"className":102006},[146],[15,102008,4160],{"className":102009,"style":981},[59,60],[15,102011,113],{"className":102012},[146],[15,102014,774],{"className":102015},[59,60],[15,102017,19073],{"className":102018},[153],"，对 ",[15,102021,102023,102042],{"className":102022,"translate":19},[18],[15,102024,102026],{"className":102025},[23],[25,102027,102028],{"xmlns":27},[29,102029,102030,102040],{},[32,102031,102032,102034,102036,102038],{},[35,102033,4160],{},[110,102035,113],{"stretchy":112},[35,102037,774],{},[110,102039,120],{"stretchy":112},[39,102041,101833],{"encoding":41},[15,102043,102045],{"className":102044,"ariaHidden":46},[45],[15,102046,102048,102051,102054,102057,102060],{"className":102047},[50],[15,102049],{"className":102050,"style":139},[54],[15,102052,4160],{"className":102053,"style":981},[59,60],[15,102055,113],{"className":102056},[146],[15,102058,774],{"className":102059},[59,60],[15,102061,120],{"className":102062},[153]," 求导：",[15,102065,102067],{"className":102066,"translate":19},[180],[15,102068,102070,102190],{"className":102069,"translate":19},[18],[15,102071,102073],{"className":102072},[23],[25,102074,102075],{"xmlns":27,"display":189},[29,102076,102077,102187],{},[32,102078,102079,102081,102083,102085,102087,102089,102091,102093,102095,102097,102099,102109,102111,102113,102115,102117,102119,102121,102123,102125,102127,102129,102131,102141,102143,102149,102151,102153,102155,102157,102159,102181,102183,102185],{},[35,102080,11],{},[110,102082,113],{"stretchy":112},[35,102084,774],{},[110,102086,120],{"stretchy":112},[110,102088,4815],{},[35,102090,944],{},[110,102092,113],{"stretchy":112},[35,102094,774],{},[110,102096,120],{"stretchy":112},[1299,102098,1301],{},[6816,102100,102101,102103],{},[35,102102,37],{},[32,102104,102105,102107],{},[110,102106,7575],{},[110,102108,6822],{"mathvariant":209},[110,102110,113],{"stretchy":112},[35,102112,4160],{},[110,102114,113],{"stretchy":112},[35,102116,774],{},[110,102118,120],{"stretchy":112},[110,102120,120],{"stretchy":112},[110,102122,123],{},[115,102124,126],{},[1299,102126,4285],{},[110,102128,4288],{},[1299,102130,4285],{},[6816,102132,102133,102135],{},[35,102134,37],{},[32,102136,102137,102139],{},[110,102138,7575],{},[110,102140,6822],{"mathvariant":209},[110,102142,113],{"stretchy":112},[6816,102144,102145,102147],{},[35,102146,4160],{},[110,102148,7575],{},[110,102150,113],{"stretchy":112},[35,102152,774],{},[110,102154,120],{"stretchy":112},[110,102156,120],{"stretchy":112},[110,102158,123],{},[238,102160,102161,102171],{},[32,102162,102163,102165,102167,102169],{},[35,102164,11],{},[110,102166,113],{"stretchy":112},[35,102168,774],{},[110,102170,120],{"stretchy":112},[32,102172,102173,102175,102177,102179],{},[35,102174,944],{},[110,102176,113],{"stretchy":112},[35,102178,774],{},[110,102180,120],{"stretchy":112},[110,102182,123],{},[35,102184,769],{},[35,102186,4377],{"mathvariant":209},[39,102188,102189],{"encoding":41},"p(x)-q(x)\\,f^{*\\prime}(g(x))=0\n\\ \\Longrightarrow\\\nf^{*\\prime}(g^*(x))=\\frac{p(x)}{q(x)}=t .",[15,102191,102193,102220,102301,102325,102416,102512],{"className":102192,"ariaHidden":46},[45],[15,102194,102196,102199,102202,102205,102208,102211,102214,102217],{"className":102195},[50],[15,102197],{"className":102198,"style":139},[54],[15,102200,11],{"className":102201},[59,60],[15,102203,113],{"className":102204},[146],[15,102206,774],{"className":102207},[59,60],[15,102209,120],{"className":102210},[153],[15,102212],{"className":102213,"style":2589},[157],[15,102215,4815],{"className":102216},[2593],[15,102218],{"className":102219,"style":2589},[157],[15,102221,102223,102227,102230,102233,102236,102239,102242,102277,102280,102283,102286,102289,102292,102295,102298],{"className":102222},[50],[15,102224],{"className":102225,"style":102226},[54],"height:1.0519em;vertical-align:-0.25em;",[15,102228,944],{"className":102229,"style":981},[59,60],[15,102231,113],{"className":102232},[146],[15,102234,774],{"className":102235},[59,60],[15,102237,120],{"className":102238},[153],[15,102240],{"className":102241,"style":406},[157],[15,102243,102245,102248],{"className":102244},[59],[15,102246,37],{"className":102247,"style":61},[59,60],[15,102249,102251],{"className":102250},[281],[15,102252,102254],{"className":102253},[285],[15,102255,102257],{"className":102256},[290],[15,102258,102260],{"className":102259,"style":28997},[294],[15,102261,102262,102265],{"style":8279},[15,102263],{"className":102264,"style":303},[302],[15,102266,102268],{"className":102267},[307,308,309,310],[15,102269,102271,102274],{"className":102270},[59,310],[15,102272,7575],{"className":102273},[59,310],[15,102275,6822],{"className":102276},[59,310],[15,102278,113],{"className":102279},[146],[15,102281,4160],{"className":102282,"style":981},[59,60],[15,102284,113],{"className":102285},[146],[15,102287,774],{"className":102288},[59,60],[15,102290,19073],{"className":102291},[153],[15,102293],{"className":102294,"style":158},[157],[15,102296,123],{"className":102297},[162],[15,102299],{"className":102300,"style":158},[157],[15,102302,102304,102307,102310,102313,102316,102319,102322],{"className":102303},[50],[15,102305],{"className":102306,"style":92826},[54],[15,102308,126],{"className":102309},[59],[15,102311,4285],{"className":102312},[157],[15,102314],{"className":102315,"style":158},[157],[15,102317,4288],{"className":102318},[162],[15,102320,4285],{"className":102321},[157],[15,102323],{"className":102324,"style":158},[157],[15,102326,102328,102331,102366,102369,102398,102401,102404,102407,102410,102413],{"className":102327},[50],[15,102329],{"className":102330,"style":102226},[54],[15,102332,102334,102337],{"className":102333},[59],[15,102335,37],{"className":102336,"style":61},[59,60],[15,102338,102340],{"className":102339},[281],[15,102341,102343],{"className":102342},[285],[15,102344,102346],{"className":102345},[290],[15,102347,102349],{"className":102348,"style":28997},[294],[15,102350,102351,102354],{"style":8279},[15,102352],{"className":102353,"style":303},[302],[15,102355,102357],{"className":102356},[307,308,309,310],[15,102358,102360,102363],{"className":102359},[59,310],[15,102361,7575],{"className":102362},[59,310],[15,102364,6822],{"className":102365},[59,310],[15,102367,113],{"className":102368},[146],[15,102370,102372,102375],{"className":102371},[59],[15,102373,4160],{"className":102374,"style":981},[59,60],[15,102376,102378],{"className":102377},[281],[15,102379,102381],{"className":102380},[285],[15,102382,102384],{"className":102383},[290],[15,102385,102387],{"className":102386,"style":8250},[294],[15,102388,102389,102392],{"style":8279},[15,102390],{"className":102391,"style":303},[302],[15,102393,102395],{"className":102394},[307,308,309,310],[15,102396,7575],{"className":102397},[2593,310],[15,102399,113],{"className":102400},[146],[15,102402,774],{"className":102403},[59,60],[15,102405,19073],{"className":102406},[153],[15,102408],{"className":102409,"style":158},[157],[15,102411,123],{"className":102412},[162],[15,102414],{"className":102415,"style":158},[157],[15,102417,102419,102423,102503,102506,102509],{"className":102418},[50],[15,102420],{"className":102421,"style":102422},[54],"height:2.363em;vertical-align:-0.936em;",[15,102424,102426,102429,102500],{"className":102425},[59],[15,102427],{"className":102428},[146,441],[15,102430,102432],{"className":102431},[238],[15,102433,102435,102492],{"className":102434},[285,286],[15,102436,102438,102489],{"className":102437},[290],[15,102439,102441,102461,102469],{"className":102440,"style":1545},[294],[15,102442,102443,102446],{"style":457},[15,102444],{"className":102445,"style":461},[302],[15,102447,102449,102452,102455,102458],{"className":102448},[59],[15,102450,944],{"className":102451,"style":981},[59,60],[15,102453,113],{"className":102454},[146],[15,102456,774],{"className":102457},[59,60],[15,102459,120],{"className":102460},[153],[15,102462,102463,102466],{"style":473},[15,102464],{"className":102465,"style":461},[302],[15,102467],{"className":102468,"style":481},[480],[15,102470,102471,102474],{"style":484},[15,102472],{"className":102473,"style":461},[302],[15,102475,102477,102480,102483,102486],{"className":102476},[59],[15,102478,11],{"className":102479},[59,60],[15,102481,113],{"className":102482},[146],[15,102484,774],{"className":102485},[59,60],[15,102487,120],{"className":102488},[153],[15,102490,318],{"className":102491},[317],[15,102493,102495],{"className":102494},[290],[15,102496,102498],{"className":102497,"style":1603},[294],[15,102499],{},[15,102501],{"className":102502},[153,441],[15,102504],{"className":102505,"style":158},[157],[15,102507,123],{"className":102508},[162],[15,102510],{"className":102511,"style":158},[157],[15,102513,102515,102518,102521],{"className":102514},[50],[15,102516],{"className":102517,"style":5345},[54],[15,102519,769],{"className":102520},[59,60],[15,102522,4377],{"className":102523},[59],[11,102525,34118,102526,102733,102734,102890,102891,102949,102950,103044,103045,103048,103049,103118,103119,103200,103201,103229],{},[15,102527,102529,102575],{"className":102528,"translate":19},[18],[15,102530,102532],{"className":102531},[23],[25,102533,102534],{"xmlns":27},[29,102535,102536,102572],{},[32,102537,102538,102540,102546,102552,102554,102556,102562],{},[110,102539,113],{"stretchy":112},[6816,102541,102542,102544],{},[35,102543,37],{},[110,102545,7575],{},[6816,102547,102548,102550],{},[110,102549,120],{"stretchy":112},[110,102551,6822],{"mathvariant":209,"lspace":1249,"rspace":1249},[110,102553,123],{},[110,102555,113],{"stretchy":112},[6816,102557,102558,102560],{},[35,102559,37],{},[110,102561,6822],{"mathvariant":209,"lspace":1249,"rspace":1249},[6816,102563,102564,102566],{},[110,102565,120],{"stretchy":112},[32,102567,102568,102570],{},[110,102569,4815],{},[115,102571,117],{},[39,102573,102574],{"encoding":41},"(f^*)'=(f')^{-1}",[15,102576,102578,102657],{"className":102577,"ariaHidden":46},[45],[15,102579,102581,102584,102587,102616,102648,102651,102654],{"className":102580},[50],[15,102582],{"className":102583,"style":6845},[54],[15,102585,113],{"className":102586},[146],[15,102588,102590,102593],{"className":102589},[59],[15,102591,37],{"className":102592,"style":61},[59,60],[15,102594,102596],{"className":102595},[281],[15,102597,102599],{"className":102598},[285],[15,102600,102602],{"className":102601},[290],[15,102603,102605],{"className":102604,"style":8501},[294],[15,102606,102607,102610],{"style":6867},[15,102608],{"className":102609,"style":303},[302],[15,102611,102613],{"className":102612},[307,308,309,310],[15,102614,7575],{"className":102615},[2593,310],[15,102617,102619,102622],{"className":102618},[153],[15,102620,120],{"className":102621},[153],[15,102623,102625],{"className":102624},[281],[15,102626,102628],{"className":102627},[285],[15,102629,102631],{"className":102630},[290],[15,102632,102634],{"className":102633,"style":6864},[294],[15,102635,102636,102639],{"style":6867},[15,102637],{"className":102638,"style":303},[302],[15,102640,102642],{"className":102641},[307,308,309,310],[15,102643,102645],{"className":102644},[59,310],[15,102646,6822],{"className":102647},[59,310],[15,102649],{"className":102650,"style":158},[157],[15,102652,123],{"className":102653},[162],[15,102655],{"className":102656,"style":158},[157],[15,102658,102660,102663,102666,102698],{"className":102659},[50],[15,102661],{"className":102662,"style":20029},[54],[15,102664,113],{"className":102665},[146],[15,102667,102669,102672],{"className":102668},[59],[15,102670,37],{"className":102671,"style":61},[59,60],[15,102673,102675],{"className":102674},[281],[15,102676,102678],{"className":102677},[285],[15,102679,102681],{"className":102680},[290],[15,102682,102684],{"className":102683,"style":6864},[294],[15,102685,102686,102689],{"style":6867},[15,102687],{"className":102688,"style":303},[302],[15,102690,102692],{"className":102691},[307,308,309,310],[15,102693,102695],{"className":102694},[59,310],[15,102696,6822],{"className":102697},[59,310],[15,102699,102701,102704],{"className":102700},[153],[15,102702,120],{"className":102703},[153],[15,102705,102707],{"className":102706},[281],[15,102708,102710],{"className":102709},[285],[15,102711,102713],{"className":102712},[290],[15,102714,102716],{"className":102715,"style":20054},[294],[15,102717,102718,102721],{"style":6867},[15,102719],{"className":102720,"style":303},[302],[15,102722,102724],{"className":102723},[307,308,309,310],[15,102725,102727,102730],{"className":102726},[59,310],[15,102728,4815],{"className":102729},[59,310],[15,102731,117],{"className":102732},[59,310],"（Fenchel 的互逆关系），即得 ",[15,102735,102737,102781],{"className":102736,"translate":19},[18],[15,102738,102740],{"className":102739},[23],[25,102741,102742],{"xmlns":27},[29,102743,102744,102778],{},[32,102745,102746,102752,102754,102756,102758,102760,102766,102768,102770,102772,102774,102776],{},[6816,102747,102748,102750],{},[35,102749,4160],{},[110,102751,7575],{},[110,102753,113],{"stretchy":112},[35,102755,774],{},[110,102757,120],{"stretchy":112},[110,102759,123],{},[6816,102761,102762,102764],{},[35,102763,37],{},[110,102765,6822],{"mathvariant":209,"lspace":1249,"rspace":1249},[110,102767,113],{"stretchy":112},[35,102769,769],{},[110,102771,113],{"stretchy":112},[35,102773,774],{},[110,102775,120],{"stretchy":112},[110,102777,120],{"stretchy":112},[39,102779,102780],{"encoding":41},"g^*(x)=f'(t(x))",[15,102782,102784,102837],{"className":102783,"ariaHidden":46},[45],[15,102785,102787,102790,102819,102822,102825,102828,102831,102834],{"className":102786},[50],[15,102788],{"className":102789,"style":139},[54],[15,102791,102793,102796],{"className":102792},[59],[15,102794,4160],{"className":102795,"style":981},[59,60],[15,102797,102799],{"className":102798},[281],[15,102800,102802],{"className":102801},[285],[15,102803,102805],{"className":102804},[290],[15,102806,102808],{"className":102807,"style":8501},[294],[15,102809,102810,102813],{"style":6867},[15,102811],{"className":102812,"style":303},[302],[15,102814,102816],{"className":102815},[307,308,309,310],[15,102817,7575],{"className":102818},[2593,310],[15,102820,113],{"className":102821},[146],[15,102823,774],{"className":102824},[59,60],[15,102826,120],{"className":102827},[153],[15,102829],{"className":102830,"style":158},[157],[15,102832,123],{"className":102833},[162],[15,102835],{"className":102836,"style":158},[157],[15,102838,102840,102843,102875,102878,102881,102884,102887],{"className":102839},[50],[15,102841],{"className":102842,"style":6845},[54],[15,102844,102846,102849],{"className":102845},[59],[15,102847,37],{"className":102848,"style":61},[59,60],[15,102850,102852],{"className":102851},[281],[15,102853,102855],{"className":102854},[285],[15,102856,102858],{"className":102857},[290],[15,102859,102861],{"className":102860,"style":6864},[294],[15,102862,102863,102866],{"style":6867},[15,102864],{"className":102865,"style":303},[302],[15,102867,102869],{"className":102868},[307,308,309,310],[15,102870,102872],{"className":102871},[59,310],[15,102873,6822],{"className":102874},[59,310],[15,102876,113],{"className":102877},[146],[15,102879,769],{"className":102880},[59,60],[15,102882,113],{"className":102883},[146],[15,102885,774],{"className":102886},[59,60],[15,102888,19073],{"className":102889},[153],"。把 ",[15,102892,102894,102911],{"className":102893,"translate":19},[18],[15,102895,102897],{"className":102896},[23],[25,102898,102899],{"xmlns":27},[29,102900,102901,102909],{},[32,102902,102903],{},[6816,102904,102905,102907],{},[35,102906,4160],{},[110,102908,7575],{},[39,102910,64042],{"encoding":41},[15,102912,102914],{"className":102913,"ariaHidden":46},[45],[15,102915,102917,102920],{"className":102916},[50],[15,102918],{"className":102919,"style":64052},[54],[15,102921,102923,102926],{"className":102922},[59],[15,102924,4160],{"className":102925,"style":981},[59,60],[15,102927,102929],{"className":102928},[281],[15,102930,102932],{"className":102931},[285],[15,102933,102935],{"className":102934},[290],[15,102936,102938],{"className":102937,"style":8501},[294],[15,102939,102940,102943],{"style":6867},[15,102941],{"className":102942,"style":303},[302],[15,102944,102946],{"className":102945},[307,308,309,310],[15,102947,7575],{"className":102948},[2593,310]," 代回即得 ",[15,102951,102953,102980],{"className":102952,"translate":19},[18],[15,102954,102956],{"className":102955},[23],[25,102957,102958],{"xmlns":27},[29,102959,102960,102978],{},[32,102961,102962,102968,102970,102972,102974,102976],{},[195,102963,102964,102966],{},[35,102965,199],{},[35,102967,37],{},[110,102969,113],{"stretchy":112},[35,102971,206],{},[35,102973,210],{"mathvariant":209},[35,102975,213],{},[110,102977,120],{"stretchy":112},[39,102979,5241],{"encoding":41},[15,102981,102983],{"className":102982,"ariaHidden":46},[45],[15,102984,102986,102989,103029,103032,103035,103038,103041],{"className":102985},[50],[15,102987],{"className":102988,"style":270},[54],[15,102990,102992,102995],{"className":102991},[59],[15,102993,199],{"className":102994,"style":277},[59,60],[15,102996,102998],{"className":102997},[281],[15,102999,103001,103021],{"className":103000},[285,286],[15,103002,103004,103018],{"className":103003},[290],[15,103005,103007],{"className":103006,"style":295},[294],[15,103008,103009,103012],{"style":298},[15,103010],{"className":103011,"style":303},[302],[15,103013,103015],{"className":103014},[307,308,309,310],[15,103016,37],{"className":103017,"style":61},[59,60,310],[15,103019,318],{"className":103020},[317],[15,103022,103024],{"className":103023},[290],[15,103025,103027],{"className":103026,"style":325},[294],[15,103028],{},[15,103030,113],{"className":103031},[146],[15,103033,206],{"className":103034,"style":334},[59,60],[15,103036,210],{"className":103037},[59],[15,103039,213],{"className":103040},[59,60],[15,103042,120],{"className":103043},[153],"（这正是定理 4.1 所给的最优解）。",[998,103046,103047],{},"实践含义："," 判别器网络 ",[15,103050,103052,103069],{"className":103051,"translate":19},[18],[15,103053,103055],{"className":103054},[23],[25,103056,103057],{"xmlns":27},[29,103058,103059,103067],{},[32,103060,103061],{},[195,103062,103063,103065],{},[35,103064,4160],{},[35,103066,99904],{},[39,103068,99907],{"encoding":41},[15,103070,103072],{"className":103071,"ariaHidden":46},[45],[15,103073,103075,103078],{"className":103074},[50],[15,103076],{"className":103077,"style":2278},[54],[15,103079,103081,103084],{"className":103080},[59],[15,103082,4160],{"className":103083,"style":981},[59,60],[15,103085,103087],{"className":103086},[281],[15,103088,103090,103110],{"className":103089},[285,286],[15,103091,103093,103107],{"className":103092},[290],[15,103094,103096],{"className":103095,"style":44955},[294],[15,103097,103098,103101],{"style":18869},[15,103099],{"className":103100,"style":303},[302],[15,103102,103104],{"className":103103},[307,308,309,310],[15,103105,99904],{"className":103106,"style":981},[59,60,310],[15,103108,318],{"className":103109},[317],[15,103111,103113],{"className":103112},[290],[15,103114,103116],{"className":103115,"style":7970},[294],[15,103117],{}," 学的是 ",[15,103120,103122,103147],{"className":103121,"translate":19},[18],[15,103123,103125],{"className":103124},[23],[25,103126,103127],{"xmlns":27},[29,103128,103129,103144],{},[32,103130,103131,103137,103139,103142],{},[6816,103132,103133,103135],{},[35,103134,37],{},[110,103136,6822],{"mathvariant":209,"lspace":1249,"rspace":1249},[110,103138,113],{"stretchy":112},[1299,103140,103141],{},"似然比",[110,103143,120],{"stretchy":112},[39,103145,103146],{"encoding":41},"f'(\\text{似然比})",[15,103148,103150],{"className":103149,"ariaHidden":46},[45],[15,103151,103153,103156,103188,103191,103197],{"className":103152},[50],[15,103154],{"className":103155,"style":6845},[54],[15,103157,103159,103162],{"className":103158},[59],[15,103160,37],{"className":103161,"style":61},[59,60],[15,103163,103165],{"className":103164},[281],[15,103166,103168],{"className":103167},[285],[15,103169,103171],{"className":103170},[290],[15,103172,103174],{"className":103173,"style":6864},[294],[15,103175,103176,103179],{"style":6867},[15,103177],{"className":103178,"style":303},[302],[15,103180,103182],{"className":103181},[307,308,309,310],[15,103183,103185],{"className":103184},[59,310],[15,103186,6822],{"className":103187},[59,310],[15,103189,113],{"className":103190},[146],[15,103192,103194],{"className":103193},[59,4600],[15,103195,103141],{"className":103196},[59,4604],[15,103198,120],{"className":103199},[153],"，而不是似然比本身；不同 ",[15,103202,103204,103217],{"className":103203,"translate":19},[18],[15,103205,103207],{"className":103206},[23],[25,103208,103209],{"xmlns":27},[29,103210,103211,103215],{},[32,103212,103213],{},[35,103214,37],{},[39,103216,37],{"encoding":41},[15,103218,103220],{"className":103219,"ariaHidden":46},[45],[15,103221,103223,103226],{"className":103222},[50],[15,103224],{"className":103225,"style":55},[54],[15,103227,37],{"className":103228,"style":61},[59,60]," 等价于对似然比做了不同的单调「整形」，因此决定了梯度的尺度与饱和方式。",[2313,103231,103233],{"id":103232},"_53-各散度的-gan-目标","5.3 各散度的 GAN 目标",[11,103235,103236,103237,9402],{},"取上表的 ",[15,103238,103240,103257],{"className":103239,"translate":19},[18],[15,103241,103243],{"className":103242},[23],[25,103244,103245],{"xmlns":27},[29,103246,103247,103255],{},[32,103248,103249],{},[6816,103250,103251,103253],{},[35,103252,37],{},[110,103254,7575],{},[39,103256,60745],{"encoding":41},[15,103258,103260],{"className":103259,"ariaHidden":46},[45],[15,103261,103263,103266],{"className":103262},[50],[15,103264],{"className":103265,"style":55},[54],[15,103267,103269,103272],{"className":103268},[59],[15,103270,37],{"className":103271,"style":61},[59,60],[15,103273,103275],{"className":103274},[281],[15,103276,103278],{"className":103277},[285],[15,103279,103281],{"className":103280},[290],[15,103282,103284],{"className":103283,"style":8501},[294],[15,103285,103286,103289],{"style":6867},[15,103287],{"className":103288,"style":303},[302],[15,103290,103292],{"className":103291},[307,308,309,310],[15,103293,7575],{"className":103294},[2593,310],[10029,103296,103297],{},[6667,103298,103299,15799,103301,45634],{},[998,103300,32689],{},[15,103302,103304,103336],{"className":103303,"translate":19},[18],[15,103305,103307],{"className":103306},[23],[25,103308,103309],{"xmlns":27},[29,103310,103311,103333],{},[32,103312,103313,103319,103321],{},[6816,103314,103315,103317],{},[35,103316,37],{},[110,103318,7575],{},[110,103320,123],{},[6816,103322,103323,103325],{},[35,103324,7569],{},[32,103326,103327,103329,103331],{},[35,103328,40263],{},[110,103330,4815],{},[115,103332,117],{},[39,103334,103335],{"encoding":41},"f^*=e^{y-1}",[15,103337,103339,103383],{"className":103338,"ariaHidden":46},[45],[15,103340,103342,103345,103374,103377,103380],{"className":103341},[50],[15,103343],{"className":103344,"style":55},[54],[15,103346,103348,103351],{"className":103347},[59],[15,103349,37],{"className":103350,"style":61},[59,60],[15,103352,103354],{"className":103353},[281],[15,103355,103357],{"className":103356},[285],[15,103358,103360],{"className":103359},[290],[15,103361,103363],{"className":103362,"style":8501},[294],[15,103364,103365,103368],{"style":6867},[15,103366],{"className":103367,"style":303},[302],[15,103369,103371],{"className":103370},[307,308,309,310],[15,103372,7575],{"className":103373},[2593,310],[15,103375],{"className":103376,"style":158},[157],[15,103378,123],{"className":103379},[162],[15,103381],{"className":103382,"style":158},[157],[15,103384,103386,103389],{"className":103385},[50],[15,103387],{"className":103388,"style":20054},[54],[15,103390,103392,103395],{"className":103391},[59],[15,103393,7569],{"className":103394},[59,60],[15,103396,103398],{"className":103397},[281],[15,103399,103401],{"className":103400},[285],[15,103402,103404],{"className":103403},[290],[15,103405,103407],{"className":103406,"style":20054},[294],[15,103408,103409,103412],{"style":6867},[15,103410],{"className":103411,"style":303},[302],[15,103413,103415],{"className":103414},[307,308,309,310],[15,103416,103418,103421,103424],{"className":103417},[59,310],[15,103419,40263],{"className":103420,"style":981},[59,60,310],[15,103422,4815],{"className":103423},[2593,310],[15,103425,117],{"className":103426},[59,310],[15,103428,103430],{"className":103429,"translate":19},[180],[15,103431,103433,103519],{"className":103432,"translate":19},[18],[15,103434,103436],{"className":103435},[23],[25,103437,103438],{"xmlns":27,"display":189},[29,103439,103440,103516],{},[32,103441,103442,103452,103454,103456,103458,103460,103462,103464,103474,103476,103482,103484,103486,103488,103490,103496,103498,103510,103512,103514],{},[195,103443,103444,103446],{},[35,103445,199],{},[32,103447,103448,103450],{},[35,103449,10698],{},[35,103451,21919],{},[110,103453,113],{"stretchy":112},[35,103455,206],{},[35,103457,210],{"mathvariant":209},[35,103459,213],{},[110,103461,120],{"stretchy":112},[110,103463,123],{},[6018,103465,103466,103472],{},[32,103467,103468,103470],{},[35,103469,35458],{},[110,103471,4105],{},[35,103473,4160],{},[110,103475,2900],{"fence":112,"stretchy":46,"minsize":17443,"maxsize":17443},[195,103477,103478,103480],{},[35,103479,223],{"mathvariant":222},[35,103481,206],{},[110,103483,230],{"stretchy":112},[35,103485,4160],{},[110,103487,257],{"stretchy":112},[110,103489,4815],{},[195,103491,103492,103494],{},[35,103493,223],{"mathvariant":222},[35,103495,213],{},[110,103497,230],{"stretchy":112},[6816,103499,103500,103502],{},[35,103501,7569],{},[32,103503,103504,103506,103508],{},[35,103505,4160],{},[110,103507,4815],{},[115,103509,117],{},[110,103511,257],{"stretchy":112},[110,103513,2910],{"fence":112,"stretchy":46,"minsize":17443,"maxsize":17443},[35,103515,4377],{"mathvariant":209},[39,103517,103518],{"encoding":41},"D_{KL}(P\\|Q)\n=\\sup_g\\Big\\{\\mathbb E_P[g]\n-\\mathbb E_Q[e^{g-1}]\\Big\\}.",[15,103520,103522,103598,103715],{"className":103521,"ariaHidden":46},[45],[15,103523,103525,103528,103574,103577,103580,103583,103586,103589,103592,103595],{"className":103524},[50],[15,103526],{"className":103527,"style":139},[54],[15,103529,103531,103534],{"className":103530},[59],[15,103532,199],{"className":103533,"style":277},[59,60],[15,103535,103537],{"className":103536},[281],[15,103538,103540,103566],{"className":103539},[285,286],[15,103541,103543,103563],{"className":103542},[290],[15,103544,103546],{"className":103545,"style":379},[294],[15,103547,103548,103551],{"style":298},[15,103549],{"className":103550,"style":303},[302],[15,103552,103554],{"className":103553},[307,308,309,310],[15,103555,103557,103560],{"className":103556},[59,310],[15,103558,10698],{"className":103559,"style":10786},[59,60,310],[15,103561,21919],{"className":103562},[59,60,310],[15,103564,318],{"className":103565},[317],[15,103567,103569],{"className":103568},[290],[15,103570,103572],{"className":103571,"style":7970},[294],[15,103573],{},[15,103575,113],{"className":103576},[146],[15,103578,206],{"className":103579,"style":334},[59,60],[15,103581,210],{"className":103582},[59],[15,103584,213],{"className":103585},[59,60],[15,103587,120],{"className":103588},[153],[15,103590],{"className":103591,"style":158},[157],[15,103593,123],{"className":103594},[162],[15,103596],{"className":103597,"style":158},[157],[15,103599,103601,103604,103648,103651,103657,103697,103700,103703,103706,103709,103712],{"className":103600},[50],[15,103602],{"className":103603,"style":99528},[54],[15,103605,103607],{"className":103606},[1486,13720],[15,103608,103610,103640],{"className":103609},[285,286],[15,103611,103613,103637],{"className":103612},[290],[15,103614,103616,103627],{"className":103615,"style":2347},[294],[15,103617,103618,103621],{"style":61227},[15,103619],{"className":103620,"style":461},[302],[15,103622,103624],{"className":103623},[307,308,309,310],[15,103625,4160],{"className":103626,"style":981},[59,60,310],[15,103628,103629,103632],{"style":11867},[15,103630],{"className":103631,"style":461},[302],[15,103633,103634],{},[15,103635,35458],{"className":103636},[1486],[15,103638,318],{"className":103639},[317],[15,103641,103643],{"className":103642},[290],[15,103644,103646],{"className":103645,"style":61259},[294],[15,103647],{},[15,103649],{"className":103650,"style":406},[157],[15,103652,103654],{"className":103653},[59],[15,103655,2900],{"className":103656},[419,3722],[15,103658,103660,103663],{"className":103659},[59],[15,103661,223],{"className":103662},[59,366],[15,103664,103666],{"className":103665},[281],[15,103667,103669,103689],{"className":103668},[285,286],[15,103670,103672,103686],{"className":103671},[290],[15,103673,103675],{"className":103674,"style":379},[294],[15,103676,103677,103680],{"style":382},[15,103678],{"className":103679,"style":303},[302],[15,103681,103683],{"className":103682},[307,308,309,310],[15,103684,206],{"className":103685,"style":334},[59,60,310],[15,103687,318],{"className":103688},[317],[15,103690,103692],{"className":103691},[290],[15,103693,103695],{"className":103694,"style":7970},[294],[15,103696],{},[15,103698,230],{"className":103699},[146],[15,103701,4160],{"className":103702,"style":981},[59,60],[15,103704,257],{"className":103705},[153],[15,103707],{"className":103708,"style":2589},[157],[15,103710,4815],{"className":103711},[2593],[15,103713],{"className":103714,"style":2589},[157],[15,103716,103718,103721,103761,103764,103802,103805,103811],{"className":103717},[50],[15,103719],{"className":103720,"style":68154},[54],[15,103722,103724,103727],{"className":103723},[59],[15,103725,223],{"className":103726},[59,366],[15,103728,103730],{"className":103729},[281],[15,103731,103733,103753],{"className":103732},[285,286],[15,103734,103736,103750],{"className":103735},[290],[15,103737,103739],{"className":103738,"style":379},[294],[15,103740,103741,103744],{"style":382},[15,103742],{"className":103743,"style":303},[302],[15,103745,103747],{"className":103746},[307,308,309,310],[15,103748,213],{"className":103749},[59,60,310],[15,103751,318],{"className":103752},[317],[15,103754,103756],{"className":103755},[290],[15,103757,103759],{"className":103758,"style":325},[294],[15,103760],{},[15,103762,230],{"className":103763},[146],[15,103765,103767,103770],{"className":103766},[59],[15,103768,7569],{"className":103769},[59,60],[15,103771,103773],{"className":103772},[281],[15,103774,103776],{"className":103775},[285],[15,103777,103779],{"className":103778},[290],[15,103780,103782],{"className":103781,"style":19761},[294],[15,103783,103784,103787],{"style":8279},[15,103785],{"className":103786,"style":303},[302],[15,103788,103790],{"className":103789},[307,308,309,310],[15,103791,103793,103796,103799],{"className":103792},[59,310],[15,103794,4160],{"className":103795,"style":981},[59,60,310],[15,103797,4815],{"className":103798},[2593,310],[15,103800,117],{"className":103801},[59,310],[15,103803,257],{"className":103804},[153],[15,103806,103808],{"className":103807},[59],[15,103809,2910],{"className":103810},[419,3722],[15,103812,4377],{"className":103813},[59],[11,103815,103816,103817,104025,104026,104099,104100,104132,104133,104161],{},"（这是「MINE-f」式下界。经典 Donsker–Varadhan 下界是 ",[15,103818,103820,103866],{"className":103819,"translate":19},[18],[15,103821,103823],{"className":103822},[23],[25,103824,103825],{"xmlns":27},[29,103826,103827,103863],{},[32,103828,103829,103835,103837,103839,103841,103843,103845,103847,103853,103855,103861],{},[195,103830,103831,103833],{},[35,103832,223],{"mathvariant":222},[35,103834,206],{},[110,103836,230],{"stretchy":112},[35,103838,4160],{},[110,103840,257],{"stretchy":112},[110,103842,4815],{},[35,103844,4102],{},[110,103846,4105],{},[195,103848,103849,103851],{},[35,103850,223],{"mathvariant":222},[35,103852,213],{},[110,103854,230],{"stretchy":112},[6816,103856,103857,103859],{},[35,103858,7569],{},[35,103860,4160],{},[110,103862,257],{"stretchy":112},[39,103864,103865],{"encoding":41},"\\mathbb E_P[g]-\\log\\mathbb E_Q[e^{g}]",[15,103867,103869,103933],{"className":103868,"ariaHidden":46},[45],[15,103870,103872,103875,103915,103918,103921,103924,103927,103930],{"className":103871},[50],[15,103873],{"className":103874,"style":139},[54],[15,103876,103878,103881],{"className":103877},[59],[15,103879,223],{"className":103880},[59,366],[15,103882,103884],{"className":103883},[281],[15,103885,103887,103907],{"className":103886},[285,286],[15,103888,103890,103904],{"className":103889},[290],[15,103891,103893],{"className":103892,"style":379},[294],[15,103894,103895,103898],{"style":382},[15,103896],{"className":103897,"style":303},[302],[15,103899,103901],{"className":103900},[307,308,309,310],[15,103902,206],{"className":103903,"style":334},[59,60,310],[15,103905,318],{"className":103906},[317],[15,103908,103910],{"className":103909},[290],[15,103911,103913],{"className":103912,"style":7970},[294],[15,103914],{},[15,103916,230],{"className":103917},[146],[15,103919,4160],{"className":103920,"style":981},[59,60],[15,103922,257],{"className":103923},[153],[15,103925],{"className":103926,"style":2589},[157],[15,103928,4815],{"className":103929},[2593],[15,103931],{"className":103932,"style":2589},[157],[15,103934,103936,103939,103944,103947,103987,103990,104022],{"className":103935},[50],[15,103937],{"className":103938,"style":270},[54],[15,103940,4156,103942],{"className":103941},[1486],[15,103943,4160],{"style":4159},[15,103945],{"className":103946,"style":406},[157],[15,103948,103950,103953],{"className":103949},[59],[15,103951,223],{"className":103952},[59,366],[15,103954,103956],{"className":103955},[281],[15,103957,103959,103979],{"className":103958},[285,286],[15,103960,103962,103976],{"className":103961},[290],[15,103963,103965],{"className":103964,"style":379},[294],[15,103966,103967,103970],{"style":382},[15,103968],{"className":103969,"style":303},[302],[15,103971,103973],{"className":103972},[307,308,309,310],[15,103974,213],{"className":103975},[59,60,310],[15,103977,318],{"className":103978},[317],[15,103980,103982],{"className":103981},[290],[15,103983,103985],{"className":103984,"style":325},[294],[15,103986],{},[15,103988,230],{"className":103989},[146],[15,103991,103993,103996],{"className":103992},[59],[15,103994,7569],{"className":103995},[59,60],[15,103997,103999],{"className":103998},[281],[15,104000,104002],{"className":104001},[285],[15,104003,104005],{"className":104004},[290],[15,104006,104008],{"className":104007,"style":50379},[294],[15,104009,104010,104013],{"style":6867},[15,104011],{"className":104012,"style":303},[302],[15,104014,104016],{"className":104015},[307,308,309,310],[15,104017,104019],{"className":104018},[59,310],[15,104020,4160],{"className":104021,"style":981},[59,60,310],[15,104023,257],{"className":104024},[153],"，二者同源但第二项不同：",[15,104027,104029,104052],{"className":104028,"translate":19},[18],[15,104030,104032],{"className":104031},[23],[25,104033,104034],{"xmlns":27},[29,104035,104036,104050],{},[32,104037,104038],{},[6816,104039,104040,104042],{},[35,104041,7569],{},[32,104043,104044,104046,104048],{},[35,104045,40263],{},[110,104047,4815],{},[115,104049,117],{},[39,104051,64939],{"encoding":41},[15,104053,104055],{"className":104054,"ariaHidden":46},[45],[15,104056,104058,104061],{"className":104057},[50],[15,104059],{"className":104060,"style":20054},[54],[15,104062,104064,104067],{"className":104063},[59],[15,104065,7569],{"className":104066},[59,60],[15,104068,104070],{"className":104069},[281],[15,104071,104073],{"className":104072},[285],[15,104074,104076],{"className":104075},[290],[15,104077,104079],{"className":104078,"style":20054},[294],[15,104080,104081,104084],{"style":6867},[15,104082],{"className":104083,"style":303},[302],[15,104085,104087],{"className":104086},[307,308,309,310],[15,104088,104090,104093,104096],{"className":104089},[59,310],[15,104091,40263],{"className":104092,"style":981},[59,60,310],[15,104094,4815],{"className":104095},[2593,310],[15,104097,117],{"className":104098},[59,310]," 是把 ",[15,104101,104103,104118],{"className":104102,"translate":19},[18],[15,104104,104106],{"className":104105},[23],[25,104107,104108],{"xmlns":27},[29,104109,104110,104116],{},[32,104111,104112,104114],{},[35,104113,4102],{},[110,104115,4105],{},[39,104117,21609],{"encoding":41},[15,104119,104121],{"className":104120,"ariaHidden":46},[45],[15,104122,104124,104127],{"className":104123},[50],[15,104125],{"className":104126,"style":55},[54],[15,104128,4156,104130],{"className":104129},[1486],[15,104131,4160],{"style":4159}," 移到最外层并做 ",[15,104134,104136,104149],{"className":104135,"translate":19},[18],[15,104137,104139],{"className":104138},[23],[25,104140,104141],{"xmlns":27},[29,104142,104143,104147],{},[32,104144,104145],{},[35,104146,37],{},[39,104148,37],{"encoding":41},[15,104150,104152],{"className":104151,"ariaHidden":46},[45],[15,104153,104155,104158],{"className":104154},[50],[15,104156],{"className":104157,"style":55},[54],[15,104159,37],{"className":104160,"style":61},[59,60],"-散度对偶后的结果。）",[10029,104163,104164,104390,104783,105214],{},[6667,104165,104166,104169,104170,49198,104203,104231,104232,104287,104288,104338,104339,104389],{},[998,104167,104168],{},"反向 KL \u002F α-GAN","：把 α 从 ",[15,104171,104173,104188],{"className":104172,"translate":19},[18],[15,104174,104176],{"className":104175},[23],[25,104177,104178],{"xmlns":27},[29,104179,104180,104186],{},[32,104181,104182,104184],{},[110,104183,4815],{},[115,104185,117],{},[39,104187,49182],{"encoding":41},[15,104189,104191],{"className":104190,"ariaHidden":46},[45],[15,104192,104194,104197,104200],{"className":104193},[50],[15,104195],{"className":104196,"style":12582},[54],[15,104198,4815],{"className":104199},[59],[15,104201,117],{"className":104202},[59],[15,104204,104206,104219],{"className":104205,"translate":19},[18],[15,104207,104209],{"className":104208},[23],[25,104210,104211],{"xmlns":27},[29,104212,104213,104217],{},[32,104214,104215],{},[115,104216,117],{},[39,104218,117],{"encoding":41},[15,104220,104222],{"className":104221,"ariaHidden":46},[45],[15,104223,104225,104228],{"className":104224},[50],[15,104226],{"className":104227,"style":172},[54],[15,104229,117],{"className":104230},[59],"，散度从反向 KL（",[15,104233,104235,104254],{"className":104234,"translate":19},[18],[15,104236,104238],{"className":104237},[23],[25,104239,104240],{"xmlns":27},[29,104241,104242,104252],{},[32,104243,104244,104246,104248,104250],{},[35,104245,44780],{},[110,104247,1032],{},[110,104249,4815],{},[115,104251,117],{},[39,104253,49065],{"encoding":41},[15,104255,104257,104275],{"className":104256,"ariaHidden":46},[45],[15,104258,104260,104263,104266,104269,104272],{"className":104259},[50],[15,104261],{"className":104262,"style":2347},[54],[15,104264,44780],{"className":104265,"style":44810},[59,60],[15,104267],{"className":104268,"style":158},[157],[15,104270,1032],{"className":104271},[162],[15,104273],{"className":104274,"style":158},[157],[15,104276,104278,104281,104284],{"className":104277},[50],[15,104279],{"className":104280,"style":12582},[54],[15,104282,4815],{"className":104283},[59],[15,104285,117],{"className":104286},[59],"，zero-forcing、模式崩塌倾向）经 Hellinger 型对称散度（",[15,104289,104291,104308],{"className":104290,"translate":19},[18],[15,104292,104294],{"className":104293},[23],[25,104295,104296],{"xmlns":27},[29,104297,104298,104306],{},[32,104299,104300,104302,104304],{},[35,104301,44780],{},[110,104303,123],{},[115,104305,126],{},[39,104307,46514],{"encoding":41},[15,104309,104311,104329],{"className":104310,"ariaHidden":46},[45],[15,104312,104314,104317,104320,104323,104326],{"className":104313},[50],[15,104315],{"className":104316,"style":2347},[54],[15,104318,44780],{"className":104319,"style":44810},[59,60],[15,104321],{"className":104322,"style":158},[157],[15,104324,123],{"className":104325},[162],[15,104327],{"className":104328,"style":158},[157],[15,104330,104332,104335],{"className":104331},[50],[15,104333],{"className":104334,"style":172},[54],[15,104336,126],{"className":104337},[59],"）过渡到正向 KL（",[15,104340,104342,104359],{"className":104341,"translate":19},[18],[15,104343,104345],{"className":104344},[23],[25,104346,104347],{"xmlns":27},[29,104348,104349,104357],{},[32,104350,104351,104353,104355],{},[35,104352,44780],{},[110,104354,1032],{},[115,104356,117],{},[39,104358,47241],{"encoding":41},[15,104360,104362,104380],{"className":104361,"ariaHidden":46},[45],[15,104363,104365,104368,104371,104374,104377],{"className":104364},[50],[15,104366],{"className":104367,"style":2347},[54],[15,104369,44780],{"className":104370,"style":44810},[59,60],[15,104372],{"className":104373,"style":158},[157],[15,104375,1032],{"className":104376},[162],[15,104378],{"className":104379,"style":158},[157],[15,104381,104383,104386],{"className":104382},[50],[15,104384],{"className":104385,"style":172},[54],[15,104387,117],{"className":104388},[59],"，zero-avoiding、模式覆盖倾向），可以在\"崩塌—覆盖\"之间连续调节（见 3.6 节的两端极限）。",[6667,104391,104392,9402,104394,2747,104638,104778,104779,104782],{},[998,104393,53150],{},[15,104395,104397,104447],{"className":104396,"translate":19},[18],[15,104398,104400],{"className":104399},[23],[25,104401,104402],{"xmlns":27},[29,104403,104404,104444],{},[32,104405,104406,104412,104414,104416,104422,104424,104426,104428,104430,104432,104442],{},[6816,104407,104408,104410],{},[35,104409,37],{},[110,104411,7575],{},[110,104413,123],{},[110,104415,4815],{},[238,104417,104418,104420],{},[115,104419,117],{},[115,104421,2465],{},[35,104423,4102],{},[110,104425,4105],{},[110,104427,113],{"stretchy":112},[115,104429,2465],{},[110,104431,4815],{},[6816,104433,104434,104436],{},[35,104435,7569],{},[32,104437,104438,104440],{},[115,104439,2465],{},[35,104441,40263],{},[110,104443,120],{"stretchy":112},[39,104445,104446],{"encoding":41},"f^*=-\\frac12\\log(2-e^{2y})",[15,104448,104450,104494,104594],{"className":104449,"ariaHidden":46},[45],[15,104451,104453,104456,104485,104488,104491],{"className":104452},[50],[15,104454],{"className":104455,"style":55},[54],[15,104457,104459,104462],{"className":104458},[59],[15,104460,37],{"className":104461,"style":61},[59,60],[15,104463,104465],{"className":104464},[281],[15,104466,104468],{"className":104467},[285],[15,104469,104471],{"className":104470},[290],[15,104472,104474],{"className":104473,"style":8501},[294],[15,104475,104476,104479],{"style":6867},[15,104477],{"className":104478,"style":303},[302],[15,104480,104482],{"className":104481},[307,308,309,310],[15,104483,7575],{"className":104484},[2593,310],[15,104486],{"className":104487,"style":158},[157],[15,104489,123],{"className":104490},[162],[15,104492],{"className":104493,"style":158},[157],[15,104495,104497,104500,104503,104571,104574,104579,104582,104585,104588,104591],{"className":104496},[50],[15,104498],{"className":104499,"style":2507},[54],[15,104501,4815],{"className":104502},[59],[15,104504,104506,104509,104568],{"className":104505},[59],[15,104507],{"className":104508},[146,441],[15,104510,104512],{"className":104511},[238],[15,104513,104515,104560],{"className":104514},[285,286],[15,104516,104518,104557],{"className":104517},[290],[15,104519,104521,104535,104543],{"className":104520,"style":2526},[294],[15,104522,104523,104526],{"style":2529},[15,104524],{"className":104525,"style":461},[302],[15,104527,104529],{"className":104528},[307,308,309,310],[15,104530,104532],{"className":104531},[59,310],[15,104533,2465],{"className":104534},[59,310],[15,104536,104537,104540],{"style":473},[15,104538],{"className":104539,"style":461},[302],[15,104541],{"className":104542,"style":481},[480],[15,104544,104545,104548],{"style":2552},[15,104546],{"className":104547,"style":461},[302],[15,104549,104551],{"className":104550},[307,308,309,310],[15,104552,104554],{"className":104553},[59,310],[15,104555,117],{"className":104556},[59,310],[15,104558,318],{"className":104559},[317],[15,104561,104563],{"className":104562},[290],[15,104564,104566],{"className":104565,"style":2574},[294],[15,104567],{},[15,104569],{"className":104570},[153,441],[15,104572],{"className":104573,"style":406},[157],[15,104575,4156,104577],{"className":104576},[1486],[15,104578,4160],{"style":4159},[15,104580,113],{"className":104581},[146],[15,104583,2465],{"className":104584},[59],[15,104586],{"className":104587,"style":2589},[157],[15,104589,4815],{"className":104590},[2593],[15,104592],{"className":104593,"style":2589},[157],[15,104595,104597,104600,104635],{"className":104596},[50],[15,104598],{"className":104599,"style":20029},[54],[15,104601,104603,104606],{"className":104602},[59],[15,104604,7569],{"className":104605},[59,60],[15,104607,104609],{"className":104608},[281],[15,104610,104612],{"className":104611},[285],[15,104613,104615],{"className":104614},[290],[15,104616,104618],{"className":104617,"style":20054},[294],[15,104619,104620,104623],{"style":6867},[15,104621],{"className":104622,"style":303},[302],[15,104624,104626],{"className":104625},[307,308,309,310],[15,104627,104629,104632],{"className":104628},[59,310],[15,104630,2465],{"className":104631},[59,310],[15,104633,40263],{"className":104634,"style":981},[59,60,310],[15,104636,120],{"className":104637},[153],[15,104639,104641,104668],{"className":104640,"translate":19},[18],[15,104642,104644],{"className":104643},[23],[25,104645,104646],{"xmlns":27},[29,104647,104648,104666],{},[32,104649,104650,104652,104654],{},[35,104651,40263],{},[110,104653,7253],{},[238,104655,104656,104664],{},[32,104657,104658,104660,104662],{},[35,104659,4102],{},[110,104661,4105],{},[115,104663,2465],{},[115,104665,2465],{},[39,104667,65780],{"encoding":41},[15,104669,104671,104689],{"className":104670,"ariaHidden":46},[45],[15,104672,104674,104677,104680,104683,104686],{"className":104673},[50],[15,104675],{"className":104676,"style":31718},[54],[15,104678,40263],{"className":104679,"style":981},[59,60],[15,104681],{"className":104682,"style":158},[157],[15,104684,7253],{"className":104685},[162],[15,104687],{"className":104688,"style":158},[157],[15,104690,104692,104695],{"className":104691},[50],[15,104693],{"className":104694,"style":65808},[54],[15,104696,104698,104701,104775],{"className":104697},[59],[15,104699],{"className":104700},[146,441],[15,104702,104704],{"className":104703},[238],[15,104705,104707,104767],{"className":104706},[285,286],[15,104708,104710,104764],{"className":104709},[290],[15,104711,104713,104727,104735],{"className":104712,"style":2827},[294],[15,104714,104715,104718],{"style":2529},[15,104716],{"className":104717,"style":461},[302],[15,104719,104721],{"className":104720},[307,308,309,310],[15,104722,104724],{"className":104723},[59,310],[15,104725,2465],{"className":104726},[59,310],[15,104728,104729,104732],{"style":473},[15,104730],{"className":104731,"style":461},[302],[15,104733],{"className":104734,"style":481},[480],[15,104736,104737,104740],{"style":2855},[15,104738],{"className":104739,"style":461},[302],[15,104741,104743],{"className":104742},[307,308,309,310],[15,104744,104746,104758,104761],{"className":104745},[59,310],[15,104747,104749,104752,104755],{"className":104748},[1486,310],[15,104750,47684],{"className":104751},[310],[15,104753,47688],{"className":104754},[310],[15,104756,4160],{"className":104757,"style":4159},[310],[15,104759],{"className":104760,"style":42392},[157,310],[15,104762,2465],{"className":104763},[59,310],[15,104765,318],{"className":104766},[317],[15,104768,104770],{"className":104769},[290],[15,104771,104773],{"className":104772,"style":2574},[294],[15,104774],{},[15,104776],{"className":104777},[153,441],"。代回即",[998,104780,104781],{},"原始 GAN 的对抗目标","（见 5.4）。",[6667,104784,104785,9402,104846,104986,104987,105209,105210,105213],{},[998,104786,104787,104845],{},[15,104788,104790,104807],{"className":104789,"translate":19},[18],[15,104791,104793],{"className":104792},[23],[25,104794,104795],{"xmlns":27},[29,104796,104797,104805],{},[32,104798,104799],{},[6816,104800,104801,104803],{},[35,104802,37475],{},[115,104804,2465],{},[39,104806,40420],{"encoding":41},[15,104808,104810],{"className":104809,"ariaHidden":46},[45],[15,104811,104813,104816],{"className":104812},[50],[15,104814],{"className":104815,"style":40430},[54],[15,104817,104819,104822],{"className":104818},[59],[15,104820,37475],{"className":104821},[59,60],[15,104823,104825],{"className":104824},[281],[15,104826,104828],{"className":104827},[285],[15,104829,104831],{"className":104830},[290],[15,104832,104834],{"className":104833,"style":20054},[294],[15,104835,104836,104839],{"style":6867},[15,104837],{"className":104838,"style":303},[302],[15,104840,104842],{"className":104841},[307,308,309,310],[15,104843,2465],{"className":104844},[59,310],"（LSGAN）",[15,104847,104849,104883],{"className":104848,"translate":19},[18],[15,104850,104852],{"className":104851},[23],[25,104853,104854],{"xmlns":27},[29,104855,104856,104880],{},[32,104857,104858,104864,104866,104868,104870,104876,104878],{},[6816,104859,104860,104862],{},[35,104861,37],{},[110,104863,7575],{},[110,104865,123],{},[35,104867,40263],{},[110,104869,2472],{},[6816,104871,104872,104874],{},[35,104873,40263],{},[115,104875,2465],{},[35,104877,941],{"mathvariant":209},[115,104879,39015],{},[39,104881,104882],{"encoding":41},"f^*=y+y^2\u002F4",[15,104884,104886,104930,104948],{"className":104885,"ariaHidden":46},[45],[15,104887,104889,104892,104921,104924,104927],{"className":104888},[50],[15,104890],{"className":104891,"style":55},[54],[15,104893,104895,104898],{"className":104894},[59],[15,104896,37],{"className":104897,"style":61},[59,60],[15,104899,104901],{"className":104900},[281],[15,104902,104904],{"className":104903},[285],[15,104905,104907],{"className":104906},[290],[15,104908,104910],{"className":104909,"style":8501},[294],[15,104911,104912,104915],{"style":6867},[15,104913],{"className":104914,"style":303},[302],[15,104916,104918],{"className":104917},[307,308,309,310],[15,104919,7575],{"className":104920},[2593,310],[15,104922],{"className":104923,"style":158},[157],[15,104925,123],{"className":104926},[162],[15,104928],{"className":104929,"style":158},[157],[15,104931,104933,104936,104939,104942,104945],{"className":104932},[50],[15,104934],{"className":104935,"style":40347},[54],[15,104937,40263],{"className":104938,"style":981},[59,60],[15,104940],{"className":104941,"style":2589},[157],[15,104943,2472],{"className":104944},[2593],[15,104946],{"className":104947,"style":2589},[157],[15,104949,104951,104954,104983],{"className":104950},[50],[15,104952],{"className":104953,"style":20029},[54],[15,104955,104957,104960],{"className":104956},[59],[15,104958,40263],{"className":104959,"style":981},[59,60],[15,104961,104963],{"className":104962},[281],[15,104964,104966],{"className":104965},[285],[15,104967,104969],{"className":104968},[290],[15,104970,104972],{"className":104971,"style":20054},[294],[15,104973,104974,104977],{"style":6867},[15,104975],{"className":104976,"style":303},[302],[15,104978,104980],{"className":104979},[307,308,309,310],[15,104981,2465],{"className":104982},[59,310],[15,104984,40398],{"className":104985},[59],"，目标变为 ",[15,104988,104990,105040],{"className":104989,"translate":19},[18],[15,104991,104993],{"className":104992},[23],[25,104994,104995],{"xmlns":27},[29,104996,104997,105037],{},[32,104998,104999,105005,105007,105009,105011,105013,105019,105021,105023,105025,105031,105033,105035],{},[195,105000,105001,105003],{},[35,105002,223],{"mathvariant":222},[35,105004,206],{},[110,105006,230],{"stretchy":112},[35,105008,4160],{},[110,105010,257],{"stretchy":112},[110,105012,4815],{},[195,105014,105015,105017],{},[35,105016,223],{"mathvariant":222},[35,105018,213],{},[110,105020,230],{"stretchy":112},[35,105022,4160],{},[110,105024,2472],{},[6816,105026,105027,105029],{},[35,105028,4160],{},[115,105030,2465],{},[35,105032,941],{"mathvariant":209},[115,105034,39015],{},[110,105036,257],{"stretchy":112},[39,105038,105039],{"encoding":41},"\\mathbb E_P[g]-\\mathbb E_Q[g+g^2\u002F4]",[15,105041,105043,105107,105168],{"className":105042,"ariaHidden":46},[45],[15,105044,105046,105049,105089,105092,105095,105098,105101,105104],{"className":105045},[50],[15,105047],{"className":105048,"style":139},[54],[15,105050,105052,105055],{"className":105051},[59],[15,105053,223],{"className":105054},[59,366],[15,105056,105058],{"className":105057},[281],[15,105059,105061,105081],{"className":105060},[285,286],[15,105062,105064,105078],{"className":105063},[290],[15,105065,105067],{"className":105066,"style":379},[294],[15,105068,105069,105072],{"style":382},[15,105070],{"className":105071,"style":303},[302],[15,105073,105075],{"className":105074},[307,308,309,310],[15,105076,206],{"className":105077,"style":334},[59,60,310],[15,105079,318],{"className":105080},[317],[15,105082,105084],{"className":105083},[290],[15,105085,105087],{"className":105086,"style":7970},[294],[15,105088],{},[15,105090,230],{"className":105091},[146],[15,105093,4160],{"className":105094,"style":981},[59,60],[15,105096,257],{"className":105097},[153],[15,105099],{"className":105100,"style":2589},[157],[15,105102,4815],{"className":105103},[2593],[15,105105],{"className":105106,"style":2589},[157],[15,105108,105110,105113,105153,105156,105159,105162,105165],{"className":105109},[50],[15,105111],{"className":105112,"style":270},[54],[15,105114,105116,105119],{"className":105115},[59],[15,105117,223],{"className":105118},[59,366],[15,105120,105122],{"className":105121},[281],[15,105123,105125,105145],{"className":105124},[285,286],[15,105126,105128,105142],{"className":105127},[290],[15,105129,105131],{"className":105130,"style":379},[294],[15,105132,105133,105136],{"style":382},[15,105134],{"className":105135,"style":303},[302],[15,105137,105139],{"className":105138},[307,308,309,310],[15,105140,213],{"className":105141},[59,60,310],[15,105143,318],{"className":105144},[317],[15,105146,105148],{"className":105147},[290],[15,105149,105151],{"className":105150,"style":325},[294],[15,105152],{},[15,105154,230],{"className":105155},[146],[15,105157,4160],{"className":105158,"style":981},[59,60],[15,105160],{"className":105161,"style":2589},[157],[15,105163,2472],{"className":105164},[2593],[15,105166],{"className":105167,"style":2589},[157],[15,105169,105171,105174,105203,105206],{"className":105170},[50],[15,105172],{"className":105173,"style":20029},[54],[15,105175,105177,105180],{"className":105176},[59],[15,105178,4160],{"className":105179,"style":981},[59,60],[15,105181,105183],{"className":105182},[281],[15,105184,105186],{"className":105185},[285],[15,105187,105189],{"className":105188},[290],[15,105190,105192],{"className":105191,"style":20054},[294],[15,105193,105194,105197],{"style":6867},[15,105195],{"className":105196,"style":303},[302],[15,105198,105200],{"className":105199},[307,308,309,310],[15,105201,2465],{"className":105202},[59,310],[15,105204,40398],{"className":105205},[59],[15,105207,257],{"className":105208},[153],"，即对判别器输出做",[998,105211,105212],{},"最小二乘","回归，梯度不会像 sigmoid 那样饱和。",[6667,105215,105216,9402,105219,2747,105342,10182],{},[998,105217,105218],{},"Hellinger²",[15,105220,105222,105256],{"className":105221,"translate":19},[18],[15,105223,105225],{"className":105224},[23],[25,105226,105227],{"xmlns":27},[29,105228,105229,105253],{},[32,105230,105231,105237,105239,105241,105243,105245,105247,105249,105251],{},[6816,105232,105233,105235],{},[35,105234,37],{},[110,105236,7575],{},[110,105238,123],{},[35,105240,40263],{},[35,105242,941],{"mathvariant":209},[110,105244,113],{"stretchy":112},[115,105246,117],{},[110,105248,4815],{},[35,105250,40263],{},[110,105252,120],{"stretchy":112},[39,105254,105255],{"encoding":41},"f^*=y\u002F(1-y)",[15,105257,105259,105303,105330],{"className":105258,"ariaHidden":46},[45],[15,105260,105262,105265,105294,105297,105300],{"className":105261},[50],[15,105263],{"className":105264,"style":55},[54],[15,105266,105268,105271],{"className":105267},[59],[15,105269,37],{"className":105270,"style":61},[59,60],[15,105272,105274],{"className":105273},[281],[15,105275,105277],{"className":105276},[285],[15,105278,105280],{"className":105279},[290],[15,105281,105283],{"className":105282,"style":8501},[294],[15,105284,105285,105288],{"style":6867},[15,105286],{"className":105287,"style":303},[302],[15,105289,105291],{"className":105290},[307,308,309,310],[15,105292,7575],{"className":105293},[2593,310],[15,105295],{"className":105296,"style":158},[157],[15,105298,123],{"className":105299},[162],[15,105301],{"className":105302,"style":158},[157],[15,105304,105306,105309,105312,105315,105318,105321,105324,105327],{"className":105305},[50],[15,105307],{"className":105308,"style":139},[54],[15,105310,40263],{"className":105311,"style":981},[59,60],[15,105313,941],{"className":105314},[59],[15,105316,113],{"className":105317},[146],[15,105319,117],{"className":105320},[59],[15,105322],{"className":105323,"style":2589},[157],[15,105325,4815],{"className":105326},[2593],[15,105328],{"className":105329,"style":2589},[157],[15,105331,105333,105336,105339],{"className":105332},[50],[15,105334],{"className":105335,"style":139},[54],[15,105337,40263],{"className":105338,"style":981},[59,60],[15,105340,120],{"className":105341},[153],[15,105343,105345,105362],{"className":105344,"translate":19},[18],[15,105346,105348],{"className":105347},[23],[25,105349,105350],{"xmlns":27},[29,105351,105352,105360],{},[32,105353,105354,105356,105358],{},[35,105355,40263],{},[110,105357,7253],{},[115,105359,117],{},[39,105361,66810],{"encoding":41},[15,105363,105365,105383],{"className":105364,"ariaHidden":46},[45],[15,105366,105368,105371,105374,105377,105380],{"className":105367},[50],[15,105369],{"className":105370,"style":31718},[54],[15,105372,40263],{"className":105373,"style":981},[59,60],[15,105375],{"className":105376,"style":158},[157],[15,105378,7253],{"className":105379},[162],[15,105381],{"className":105382,"style":158},[157],[15,105384,105386,105389],{"className":105385},[50],[15,105387],{"className":105388,"style":172},[54],[15,105390,117],{"className":105391},[59],[2313,105393,105395],{"id":105394},"_54-原始-gan-与-js-散度","5.4 原始 GAN 与 JS 散度",[11,105397,105398,105399,105581],{},"原始 GAN 的判别器输出经 sigmoid 得 ",[15,105400,105402,105454],{"className":105401,"translate":19},[18],[15,105403,105405],{"className":105404},[23],[25,105406,105407],{"xmlns":27},[29,105408,105409,105451],{},[32,105410,105411,105413,105415,105417,105419,105421,105423,105425,105427,105433,105435,105437,105439,105441,105443,105445,105447,105449],{},[35,105412,2334],{},[110,105414,113],{"stretchy":112},[35,105416,199],{},[110,105418,113],{"stretchy":112},[35,105420,774],{},[110,105422,120],{"stretchy":112},[110,105424,120],{"stretchy":112},[110,105426,123],{},[195,105428,105429,105431],{},[35,105430,199],{},[35,105432,7563],{},[110,105434,113],{"stretchy":112},[35,105436,774],{},[110,105438,120],{"stretchy":112},[110,105440,5911],{},[110,105442,113],{"stretchy":112},[115,105444,126],{},[110,105446,551],{"separator":46},[115,105448,117],{},[110,105450,120],{"stretchy":112},[39,105452,105453],{"encoding":41},"\\sigma(D(x))=D_{s}(x)\\in(0,1)",[15,105455,105457,105490,105557],{"className":105456,"ariaHidden":46},[45],[15,105458,105460,105463,105466,105469,105472,105475,105478,105481,105484,105487],{"className":105459},[50],[15,105461],{"className":105462,"style":139},[54],[15,105464,2334],{"className":105465,"style":981},[59,60],[15,105467,113],{"className":105468},[146],[15,105470,199],{"className":105471,"style":277},[59,60],[15,105473,113],{"className":105474},[146],[15,105476,774],{"className":105477},[59,60],[15,105479,19073],{"className":105480},[153],[15,105482],{"className":105483,"style":158},[157],[15,105485,123],{"className":105486},[162],[15,105488],{"className":105489,"style":158},[157],[15,105491,105493,105496,105539,105542,105545,105548,105551,105554],{"className":105492},[50],[15,105494],{"className":105495,"style":139},[54],[15,105497,105499,105502],{"className":105498},[59],[15,105500,199],{"className":105501,"style":277},[59,60],[15,105503,105505],{"className":105504},[281],[15,105506,105508,105531],{"className":105507},[285,286],[15,105509,105511,105528],{"className":105510},[290],[15,105512,105514],{"className":105513,"style":44955},[294],[15,105515,105516,105519],{"style":298},[15,105517],{"className":105518,"style":303},[302],[15,105520,105522],{"className":105521},[307,308,309,310],[15,105523,105525],{"className":105524},[59,310],[15,105526,7563],{"className":105527},[59,60,310],[15,105529,318],{"className":105530},[317],[15,105532,105534],{"className":105533},[290],[15,105535,105537],{"className":105536,"style":7970},[294],[15,105538],{},[15,105540,113],{"className":105541},[146],[15,105543,774],{"className":105544},[59,60],[15,105546,120],{"className":105547},[153],[15,105549],{"className":105550,"style":158},[157],[15,105552,5911],{"className":105553},[162],[15,105555],{"className":105556,"style":158},[157],[15,105558,105560,105563,105566,105569,105572,105575,105578],{"className":105559},[50],[15,105561],{"className":105562,"style":139},[54],[15,105564,113],{"className":105565},[146],[15,105567,126],{"className":105568},[59],[15,105570,551],{"className":105571},[573],[15,105573],{"className":105574,"style":406},[157],[15,105576,117],{"className":105577},[59],[15,105579,120],{"className":105580},[153],"，其目标等价于最小化 JS 散度下的对抗博弈。判别器写成 logit",[15,105583,105585],{"className":105584,"translate":19},[180],[15,105586,105588,105666],{"className":105587,"translate":19},[18],[15,105589,105591],{"className":105590},[23],[25,105592,105593],{"xmlns":27,"display":189},[29,105594,105595,105663],{},[32,105596,105597,105599,105601,105603,105605,105607,105609,105611,105617,105619,105621,105623,105625,105627,105629,105631,105633,105635,105641,105643,105645,105647,105649,105651,105653,105655,105661],{},[35,105598,4160],{},[110,105600,113],{"stretchy":112},[35,105602,774],{},[110,105604,120],{"stretchy":112},[110,105606,123],{},[35,105608,4102],{},[110,105610,4105],{},[195,105612,105613,105615],{},[35,105614,199],{},[35,105616,7563],{},[110,105618,113],{"stretchy":112},[35,105620,774],{},[110,105622,120],{"stretchy":112},[110,105624,4815],{},[35,105626,4102],{},[110,105628,4105],{},[110,105630,113],{"stretchy":112},[115,105632,117],{},[110,105634,4815],{},[195,105636,105637,105639],{},[35,105638,199],{},[35,105640,7563],{},[110,105642,113],{"stretchy":112},[35,105644,774],{},[110,105646,120],{"stretchy":112},[110,105648,120],{"stretchy":112},[110,105650,123],{},[35,105652,4102],{},[110,105654,4105],{},[238,105656,105657,105659],{},[35,105658,11],{},[35,105660,944],{},[110,105662,551],{"separator":46},[39,105664,105665],{"encoding":41},"g(x)=\\log D_s(x)-\\log(1-D_s(x))\n=\\log\\frac{p}{q},",[15,105667,105669,105696,105768,105794,105858],{"className":105668,"ariaHidden":46},[45],[15,105670,105672,105675,105678,105681,105684,105687,105690,105693],{"className":105671},[50],[15,105673],{"className":105674,"style":139},[54],[15,105676,4160],{"className":105677,"style":981},[59,60],[15,105679,113],{"className":105680},[146],[15,105682,774],{"className":105683},[59,60],[15,105685,120],{"className":105686},[153],[15,105688],{"className":105689,"style":158},[157],[15,105691,123],{"className":105692},[162],[15,105694],{"className":105695,"style":158},[157],[15,105697,105699,105702,105707,105710,105750,105753,105756,105759,105762,105765],{"className":105698},[50],[15,105700],{"className":105701,"style":139},[54],[15,105703,4156,105705],{"className":105704},[1486],[15,105706,4160],{"style":4159},[15,105708],{"className":105709,"style":406},[157],[15,105711,105713,105716],{"className":105712},[59],[15,105714,199],{"className":105715,"style":277},[59,60],[15,105717,105719],{"className":105718},[281],[15,105720,105722,105742],{"className":105721},[285,286],[15,105723,105725,105739],{"className":105724},[290],[15,105726,105728],{"className":105727,"style":44955},[294],[15,105729,105730,105733],{"style":298},[15,105731],{"className":105732,"style":303},[302],[15,105734,105736],{"className":105735},[307,308,309,310],[15,105737,7563],{"className":105738},[59,60,310],[15,105740,318],{"className":105741},[317],[15,105743,105745],{"className":105744},[290],[15,105746,105748],{"className":105747,"style":7970},[294],[15,105749],{},[15,105751,113],{"className":105752},[146],[15,105754,774],{"className":105755},[59,60],[15,105757,120],{"className":105758},[153],[15,105760],{"className":105761,"style":2589},[157],[15,105763,4815],{"className":105764},[2593],[15,105766],{"className":105767,"style":2589},[157],[15,105769,105771,105774,105779,105782,105785,105788,105791],{"className":105770},[50],[15,105772],{"className":105773,"style":139},[54],[15,105775,4156,105777],{"className":105776},[1486],[15,105778,4160],{"style":4159},[15,105780,113],{"className":105781},[146],[15,105783,117],{"className":105784},[59],[15,105786],{"className":105787,"style":2589},[157],[15,105789,4815],{"className":105790},[2593],[15,105792],{"className":105793,"style":2589},[157],[15,105795,105797,105800,105840,105843,105846,105849,105852,105855],{"className":105796},[50],[15,105798],{"className":105799,"style":139},[54],[15,105801,105803,105806],{"className":105802},[59],[15,105804,199],{"className":105805,"style":277},[59,60],[15,105807,105809],{"className":105808},[281],[15,105810,105812,105832],{"className":105811},[285,286],[15,105813,105815,105829],{"className":105814},[290],[15,105816,105818],{"className":105817,"style":44955},[294],[15,105819,105820,105823],{"style":298},[15,105821],{"className":105822,"style":303},[302],[15,105824,105826],{"className":105825},[307,308,309,310],[15,105827,7563],{"className":105828},[59,60,310],[15,105830,318],{"className":105831},[317],[15,105833,105835],{"className":105834},[290],[15,105836,105838],{"className":105837,"style":7970},[294],[15,105839],{},[15,105841,113],{"className":105842},[146],[15,105844,774],{"className":105845},[59,60],[15,105847,19073],{"className":105848},[153],[15,105850],{"className":105851,"style":158},[157],[15,105853,123],{"className":105854},[162],[15,105856],{"className":105857,"style":158},[157],[15,105859,105861,105864,105869,105872,105934],{"className":105860},[50],[15,105862],{"className":105863,"style":22036},[54],[15,105865,4156,105867],{"className":105866},[1486],[15,105868,4160],{"style":4159},[15,105870],{"className":105871,"style":406},[157],[15,105873,105875,105878,105931],{"className":105874},[59],[15,105876],{"className":105877},[146,441],[15,105879,105881],{"className":105880},[238],[15,105882,105884,105923],{"className":105883},[285,286],[15,105885,105887,105920],{"className":105886},[290],[15,105888,105890,105901,105909],{"className":105889,"style":3556},[294],[15,105891,105892,105895],{"style":457},[15,105893],{"className":105894,"style":461},[302],[15,105896,105898],{"className":105897},[59],[15,105899,944],{"className":105900,"style":981},[59,60],[15,105902,105903,105906],{"style":473},[15,105904],{"className":105905,"style":461},[302],[15,105907],{"className":105908,"style":481},[480],[15,105910,105911,105914],{"style":484},[15,105912],{"className":105913,"style":461},[302],[15,105915,105917],{"className":105916},[59],[15,105918,11],{"className":105919},[59,60],[15,105921,318],{"className":105922},[317],[15,105924,105926],{"className":105925},[290],[15,105927,105929],{"className":105928,"style":506},[294],[15,105930],{},[15,105932],{"className":105933},[153,441],[15,105935,551],{"className":105936},[573],[11,105938,105939],{},"它与 5.2 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E_P\\Big[\\log\\frac{2p}{p+q}\\Big]\n=\\mathbb E_P[\\log D_s]+\\log2,",[15,107702,107704,107740,107895,108004],{"className":107703,"ariaHidden":46},[45],[15,107705,107707,107710,107716,107719,107722,107725,107728,107731,107734,107737],{"className":107706},[50],[15,107708],{"className":107709,"style":139},[54],[15,107711,107713],{"className":107712},[59],[15,107714,32689],{"className":107715},[59,32688],[15,107717,113],{"className":107718},[146],[15,107720,206],{"className":107721,"style":334},[59,60],[15,107723,210],{"className":107724},[59],[15,107726,25593],{"className":107727,"style":25627},[59,60],[15,107729,120],{"className":107730},[153],[15,107732],{"className":107733,"style":158},[157],[15,107735,123],{"className":107736},[162],[15,107738],{"className":107739,"style":158},[157],[15,107741,107743,107746,107786,107792,107795,107800,107803,107880,107886,107889,107892],{"className":107742},[50],[15,107744],{"className":107745,"style":31329},[54],[15,107747,107749,107752],{"className":107748},[59],[15,107750,223],{"className":107751},[59,366],[15,107753,107755],{"className":107754},[281],[15,107756,107758,107778],{"className":107757},[285,286],[15,107759,107761,107775],{"className":107760},[290],[15,107762,107764],{"className":107763,"style":379},[294],[15,107765,107766,107769],{"style":382},[15,107767],{"className":107768,"style":303},[302],[15,107770,107772],{"className":107771},[307,308,309,310],[15,107773,206],{"className":107774,"style":334},[59,60,310],[15,107776,318],{"className":107777},[317],[15,107779,107781],{"className":107780},[290],[15,107782,107784],{"className":107783,"style":7970},[294],[15,107785],{},[15,107787,107789],{"className":107788},[59],[15,107790,230],{"className":107791},[419,3722],[15,107793],{"className":107794,"style":406},[157],[15,107796,4156,107798],{"className":107797},[1486],[15,107799,4160],{"style":4159},[15,107801],{"className":107802,"style":406},[157],[15,107804,107806,107809,107877],{"className":107805},[59],[15,107807],{"className":107808},[146,441],[15,107810,107812],{"className":107811},[238],[15,107813,107815,107869],{"className":107814},[285,286],[15,107816,107818,107866],{"className":107817},[290],[15,107819,107821,107844,107852],{"className":107820,"style":22915},[294],[15,107822,107823,107826],{"style":457},[15,107824],{"className":107825,"style":461},[302],[15,107827,107829,107832,107835,107838,107841],{"className":107828},[59],[15,107830,11],{"className":107831},[59,60],[15,107833],{"className":107834,"style":2589},[157],[15,107836,2472],{"className":107837},[2593],[15,107839],{"className":107840,"style":2589},[157],[15,107842,944],{"className":107843,"style":981},[59,60],[15,107845,107846,107849],{"style":473},[15,107847],{"className":107848,"style":461},[302],[15,107850],{"className":107851,"style":481},[480],[15,107853,107854,107857],{"style":484},[15,107855],{"className":107856,"style":461},[302],[15,107858,107860,107863],{"className":107859},[59],[15,107861,2465],{"className":107862},[59],[15,107864,11],{"className":107865},[59,60],[15,107867,318],{"className":107868},[317],[15,107870,107872],{"className":107871},[290],[15,107873,107875],{"className":107874,"style":506},[294],[15,107876],{},[15,107878],{"className":107879},[153,441],[15,107881,107883],{"className":107882},[59],[15,107884,257],{"className":107885},[419,3722],[15,107887],{"className":107888,"style":158},[157],[15,107890,123],{"className":107891},[162],[15,107893],{"className":107894,"style":158},[157],[15,107896,107898,107901,107941,107944,107949,107952,107992,107995,107998,108001],{"className":107897},[50],[15,107899],{"className":107900,"style":139},[54],[15,107902,107904,107907],{"className":107903},[59],[15,107905,223],{"className":107906},[59,366],[15,107908,107910],{"className":107909},[281],[15,107911,107913,107933],{"className":107912},[285,286],[15,107914,107916,107930],{"className":107915},[290],[15,107917,107919],{"className":107918,"style":379},[294],[15,107920,107921,107924],{"style":382},[15,107922],{"className":107923,"style":303},[302],[15,107925,107927],{"className":107926},[307,308,309,310],[15,107928,206],{"className":107929,"style":334},[59,60,310],[15,107931,318],{"className":107932},[317],[15,107934,107936],{"className":107935},[290],[15,107937,107939],{"className":107938,"style":7970},[294],[15,107940],{},[15,107942,230],{"className":107943},[146],[15,107945,4156,107947],{"className":107946},[1486],[15,107948,4160],{"style":4159},[15,107950],{"className":107951,"style":406},[157],[15,107953,107955,107958],{"className":107954},[59],[15,107956,199],{"className":107957,"style":277},[59,60],[15,107959,107961],{"className":107960},[281],[15,107962,107964,107984],{"className":107963},[285,286],[15,107965,107967,107981],{"className":107966},[290],[15,107968,107970],{"className":107969,"style":44955},[294],[15,107971,107972,107975],{"style":298},[15,107973],{"className":107974,"style":303},[302],[15,107976,107978],{"className":107977},[307,308,309,310],[15,107979,7563],{"className":107980},[59,60,310],[15,107982,318],{"className":107983},[317],[15,107985,107987],{"className":107986},[290],[15,107988,107990],{"className":107989,"style":7970},[294],[15,107991],{},[15,107993,257],{"className":107994},[153],[15,107996],{"className":107997,"style":2589},[157],[15,107999,2472],{"className":108000},[2593],[15,108002],{"className":108003,"style":2589},[157],[15,108005,108007,108010,108015,108018,108021],{"className":108006},[50],[15,108008],{"className":108009,"style":55},[54],[15,108011,4156,108013],{"className":108012},[1486],[15,108014,4160],{"style":4159},[15,108016],{"className":108017,"style":406},[157],[15,108019,2465],{"className":108020},[59],[15,108022,551],{"className":108023},[573],[11,108025,14265,108026,108054,108055,108109,108110,108224,108225,108283],{},[15,108027,108029,108042],{"className":108028,"translate":19},[18],[15,108030,108032],{"className":108031},[23],[25,108033,108034],{"xmlns":27},[29,108035,108036,108040],{},[32,108037,108038],{},[35,108039,213],{},[39,108041,213],{"encoding":41},[15,108043,108045],{"className":108044,"ariaHidden":46},[45],[15,108046,108048,108051],{"className":108047},[50],[15,108049],{"className":108050,"style":566},[54],[15,108052,213],{"className":108053},[59,60]," 同理，两式相加即得。于是「最小化 JS」与「在最优判别器下最大化 GAN 目标」是同一件事，而常数 ",[15,108056,108058,108080],{"className":108057,"translate":19},[18],[15,108059,108061],{"className":108060},[23],[25,108062,108063],{"xmlns":27},[29,108064,108065,108077],{},[32,108066,108067,108069,108071,108073,108075],{},[110,108068,4815],{},[115,108070,2465],{},[35,108072,4102],{},[110,108074,4105],{},[115,108076,2465],{},[39,108078,108079],{"encoding":41},"-2\\log2",[15,108081,108083],{"className":108082,"ariaHidden":46},[45],[15,108084,108086,108089,108092,108095,108098,108103,108106],{"className":108085},[50],[15,108087],{"className":108088,"style":55},[54],[15,108090,4815],{"className":108091},[59],[15,108093,2465],{"className":108094},[59],[15,108096],{"className":108097,"style":406},[157],[15,108099,4156,108101],{"className":108100},[1486],[15,108102,4160],{"style":4159},[15,108104],{"className":108105,"style":406},[157],[15,108107,2465],{"className":108108},[59]," 与生成器无关。这解释了「GAN 训练不收敛」的来源：JS 有界，一旦支撑集几近不重叠，",[15,108111,108113,108143],{"className":108112,"translate":19},[18],[15,108114,108116],{"className":108115},[23],[25,108117,108118],{"xmlns":27},[29,108119,108120,108140],{},[32,108121,108122,108132,108134,108136,108138],{},[195,108123,108124,108126],{},[35,108125,199],{},[32,108127,108128,108130],{},[35,108129,23769],{},[35,108131,25763],{},[110,108133,1032],{},[35,108135,4102],{},[110,108137,4105],{},[115,108139,2465],{},[39,108141,108142],{"encoding":41},"D_{JS}\\to\\log2",[15,108144,108146,108207],{"className":108145,"ariaHidden":46},[45],[15,108147,108149,108152,108198,108201,108204],{"className":108148},[50],[15,108150],{"className":108151,"style":15965},[54],[15,108153,108155,108158],{"className":108154},[59],[15,108156,199],{"className":108157,"style":277},[59,60],[15,108159,108161],{"className":108160},[281],[15,108162,108164,108190],{"className":108163},[285,286],[15,108165,108167,108187],{"className":108166},[290],[15,108168,108170],{"className":108169,"style":379},[294],[15,108171,108172,108175],{"style":298},[15,108173],{"className":108174,"style":303},[302],[15,108176,108178],{"className":108177},[307,308,309,310],[15,108179,108181,108184],{"className":108180},[59,310],[15,108182,23769],{"className":108183,"style":24179},[59,60,310],[15,108185,25763],{"className":108186,"style":25887},[59,60,310],[15,108188,318],{"className":108189},[317],[15,108191,108193],{"className":108192},[290],[15,108194,108196],{"className":108195,"style":7970},[294],[15,108197],{},[15,108199],{"className":108200,"style":158},[157],[15,108202,1032],{"className":108203},[162],[15,108205],{"className":108206,"style":158},[157],[15,108208,108210,108213,108218,108221],{"className":108209},[50],[15,108211],{"className":108212,"style":55},[54],[15,108214,4156,108216],{"className":108215},[1486],[15,108217,4160],{"style":4159},[15,108219],{"className":108220,"style":406},[157],[15,108222,2465],{"className":108223},[59],"，判别器梯度饱和，生成器梯度消失。反过来，换成 LSGAN（",[15,108226,108228,108245],{"className":108227,"translate":19},[18],[15,108229,108231],{"className":108230},[23],[25,108232,108233],{"xmlns":27},[29,108234,108235,108243],{},[32,108236,108237],{},[6816,108238,108239,108241],{},[35,108240,37475],{},[115,108242,2465],{},[39,108244,40420],{"encoding":41},[15,108246,108248],{"className":108247,"ariaHidden":46},[45],[15,108249,108251,108254],{"className":108250},[50],[15,108252],{"className":108253,"style":40430},[54],[15,108255,108257,108260],{"className":108256},[59],[15,108258,37475],{"className":108259},[59,60],[15,108261,108263],{"className":108262},[281],[15,108264,108266],{"className":108265},[285],[15,108267,108269],{"className":108268},[290],[15,108270,108272],{"className":108271,"style":20054},[294],[15,108273,108274,108277],{"style":6867},[15,108275],{"className":108276,"style":303},[302],[15,108278,108280],{"className":108279},[307,308,309,310],[15,108281,2465],{"className":108282},[59,310],"）、Wasserstein（非 f-散度）或 α-GAN 就能改变这一点。",[2313,108285,108287],{"id":108286},"_55-估计性质","5.5 估计性质",[11,108289,108290],{},"用 5.1 的 Monte Carlo 下界做估计时：",[10029,108292,108293,108438,108593],{},[6667,108294,108295,108296,33851,108365,108367,108368,108437],{},"它是 ",[15,108297,108299,108316],{"className":108298,"translate":19},[18],[15,108300,108302],{"className":108301},[23],[25,108303,108304],{"xmlns":27},[29,108305,108306,108314],{},[32,108307,108308],{},[195,108309,108310,108312],{},[35,108311,199],{},[35,108313,37],{},[39,108315,4024],{"encoding":41},[15,108317,108319],{"className":108318,"ariaHidden":46},[45],[15,108320,108322,108325],{"className":108321},[50],[15,108323],{"className":108324,"style":4034},[54],[15,108326,108328,108331],{"className":108327},[59],[15,108329,199],{"className":108330,"style":277},[59,60],[15,108332,108334],{"className":108333},[281],[15,108335,108337,108357],{"className":108336},[285,286],[15,108338,108340,108354],{"className":108339},[290],[15,108341,108343],{"className":108342,"style":295},[294],[15,108344,108345,108348],{"style":298},[15,108346],{"className":108347,"style":303},[302],[15,108349,108351],{"className":108350},[307,308,309,310],[15,108352,37],{"className":108353,"style":61},[59,60,310],[15,108355,318],{"className":108356},[317],[15,108358,108360],{"className":108359},[290],[15,108361,108363],{"className":108362,"style":325},[294],[15,108364],{},[998,108366,99884],{},"（偏差向下），且偏差来自 ",[15,108369,108371,108388],{"className":108370,"translate":19},[18],[15,108372,108374],{"className":108373},[23],[25,108375,108376],{"xmlns":27},[29,108377,108378,108386],{},[32,108379,108380],{},[195,108381,108382,108384],{},[35,108383,4160],{},[35,108385,99904],{},[39,108387,99907],{"encoding":41},[15,108389,108391],{"className":108390,"ariaHidden":46},[45],[15,108392,108394,108397],{"className":108393},[50],[15,108395],{"className":108396,"style":2278},[54],[15,108398,108400,108403],{"className":108399},[59],[15,108401,4160],{"className":108402,"style":981},[59,60],[15,108404,108406],{"className":108405},[281],[15,108407,108409,108429],{"className":108408},[285,286],[15,108410,108412,108426],{"className":108411},[290],[15,108413,108415],{"className":108414,"style":44955},[294],[15,108416,108417,108420],{"style":18869},[15,108418],{"className":108419,"style":303},[302],[15,108421,108423],{"className":108422},[307,308,309,310],[15,108424,99904],{"className":108425,"style":981},[59,60,310],[15,108427,318],{"className":108428},[317],[15,108430,108432],{"className":108431},[290],[15,108433,108435],{"className":108434,"style":7970},[294],[15,108436],{}," 未达到最优 + 样本有限；",[6667,108439,68169,108440,108470,108471,108529,108530,108592],{},[15,108441,108443,108458],{"className":108442,"translate":19},[18],[15,108444,108446],{"className":108445},[23],[25,108447,108448],{"xmlns":27},[29,108449,108450,108456],{},[32,108451,108452,108454],{},[35,108453,35458],{},[110,108455,4105],{},[39,108457,57177],{"encoding":41},[15,108459,108461],{"className":108460,"ariaHidden":46},[45],[15,108462,108464,108467],{"className":108463},[50],[15,108465],{"className":108466,"style":2278},[54],[15,108468,35458],{"className":108469},[1486]," 取到极值时，估计量方差随 ",[15,108472,108474,108491],{"className":108473,"translate":19},[18],[15,108475,108477],{"className":108476},[23],[25,108478,108479],{"xmlns":27},[29,108480,108481,108489],{},[32,108482,108483],{},[6816,108484,108485,108487],{},[35,108486,37],{},[110,108488,7575],{},[39,108490,60745],{"encoding":41},[15,108492,108494],{"className":108493,"ariaHidden":46},[45],[15,108495,108497,108500],{"className":108496},[50],[15,108498],{"className":108499,"style":55},[54],[15,108501,108503,108506],{"className":108502},[59],[15,108504,37],{"className":108505,"style":61},[59,60],[15,108507,108509],{"className":108508},[281],[15,108510,108512],{"className":108511},[285],[15,108513,108515],{"className":108514},[290],[15,108516,108518],{"className":108517,"style":8501},[294],[15,108519,108520,108523],{"style":6867},[15,108521],{"className":108522,"style":303},[302],[15,108524,108526],{"className":108525},[307,308,309,310],[15,108527,7575],{"className":108528},[2593,310]," 增长而增大（如 KL 的 ",[15,108531,108533,108551],{"className":108532,"translate":19},[18],[15,108534,108536],{"className":108535},[23],[25,108537,108538],{"xmlns":27},[29,108539,108540,108548],{},[32,108541,108542],{},[6816,108543,108544,108546],{},[35,108545,7569],{},[35,108547,4160],{},[39,108549,108550],{"encoding":41},"e^{g}",[15,108552,108554],{"className":108553,"ariaHidden":46},[45],[15,108555,108557,108560],{"className":108556},[50],[15,108558],{"className":108559,"style":50379},[54],[15,108561,108563,108566],{"className":108562},[59],[15,108564,7569],{"className":108565},[59,60],[15,108567,108569],{"className":108568},[281],[15,108570,108572],{"className":108571},[285],[15,108573,108575],{"className":108574},[290],[15,108576,108578],{"className":108577,"style":50379},[294],[15,108579,108580,108583],{"style":6867},[15,108581],{"className":108582,"style":303},[302],[15,108584,108586],{"className":108585},[307,308,309,310],[15,108587,108589],{"className":108588},[59,310],[15,108590,4160],{"className":108591,"style":981},[59,60,310]," 会重尾）；",[6667,108594,108595,108596,108624],{},"一个常用替代是「对称化」或「有界化」",[15,108597,108599,108612],{"className":108598,"translate":19},[18],[15,108600,108602],{"className":108601},[23],[25,108603,108604],{"xmlns":27},[29,108605,108606,108610],{},[32,108607,108608],{},[35,108609,37],{},[39,108611,37],{"encoding":41},[15,108613,108615],{"className":108614,"ariaHidden":46},[45],[15,108616,108618,108621],{"className":108617},[50],[15,108619],{"className":108620,"style":55},[54],[15,108622,37],{"className":108623,"style":61},[59,60],"（如 JS、TV），牺牲敏感性换取稳定梯度。这与第 6 节讨论的「有界\u002F无界」准确性—稳健性权衡是同一件事。",[528,108626,108628],{"id":108627},"_6-与其他散度的比较","6. 与其他散度的比较",[2313,108630,108632],{"id":108631},"_61-汇总","6.1 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12524,112527,112545],{"className":112525,"style":112526},[294],"height:0.8156em;",[15,112528,112530,112533],{"className":112529,"style":11867},[6309],[15,112531],{"className":112532,"style":461},[302],[15,112534,112536,112539,112542],{"className":112535,"style":34413},[59,310],[15,112537,117],{"className":112538},[59,310],[15,112540,4815],{"className":112541},[2593,310],[15,112543,40263],{"className":112544,"style":981},[59,60,310],[15,112546,112548,112551],{"style":112547},"top:-2.7756em;",[15,112549],{"className":112550,"style":461},[302],[15,112552,112554],{"className":112553,"style":34470},[34469,310],[6325,112555,112556],{"xmlns":6327,"width":6328,"height":34473,"viewBox":34474,"preserveAspectRatio":6331},[6333,112557],{"d":34477},[15,112559,318],{"className":112560},[317],[15,112562,112564],{"className":112563},[290],[15,112565,112568],{"className":112566,"style":112567},[294],"height:0.2244em;",[15,112569],{},[15,112571,112572,112575],{"style":473},[15,112573],{"className":112574,"style":461},[302],[15,112576],{"className":112577,"style":481},[480],[15,112579,112580,112583],{"style":2552},[15,112581],{"className":112582,"style":461},[302],[15,112584,112586],{"className":112585},[307,308,309,310],[15,112587,112589],{"className":112588},[59,310],[15,112590,117],{"className":112591},[59,310],[15,112593,318],{"className":112594},[317],[15,112596,112598],{"className":112597},[290],[15,112599,112601],{"className":112600,"style":98297},[294],[15,112602],{},[15,112604],{"className":112605},[153,441],[15,112607,112609,112626],{"className":112608,"translate":19},[18],[15,112610,112612],{"className":112611},[23],[25,112613,112614],{"xmlns":27},[29,112615,112616,112624],{},[32,112617,112618,112620,112622],{},[35,112619,40263],{},[110,112621,7253],{},[115,112623,117],{},[39,112625,66810],{"encoding":41},[15,112627,112629,112647],{"className":112628,"ariaHidden":46},[45],[15,112630,112632,112635,112638,112641,112644],{"className":112631},[50],[15,112633],{"className":112634,"style":31718},[54],[15,112636,40263],{"className":112637,"style":981},[59,60],[15,112639],{"className":112640,"style":158},[157],[15,112642,7253],{"className":112643},[162],[15,112645],{"className":112646,"style":158},[157],[15,112648,112650,112653],{"className":112649},[50],[15,112651],{"className":112652,"style":172},[54],[15,112654,117],{"className":112655},[59],"），代回得 ",[15,112658,112660,112752],{"className":112659,"translate":19},[18],[15,112661,112663],{"className":112662},[23],[25,112664,112665],{"xmlns":27},[29,112666,112667,112749],{},[32,112668,112669,112675,112677,112679,112681,112683,112685,112687,112689,112691,112693,112695,112701,112703,112705,112715,112717,112719,112721,112731,112733,112735,112737,112739],{},[6816,112670,112671,112673],{},[35,112672,37],{},[110,112674,7575],{},[110,112676,123],{},[35,112678,769],{},[110,112680,113],{"stretchy":112},[35,112682,40263],{},[110,112684,4815],{},[115,112686,117],{},[110,112688,120],{"stretchy":112},[110,112690,2472],{},[115,112692,2465],{},[110,112694,4815],{},[238,112696,112697,112699],{},[115,112698,117],{},[35,112700,769],{},[110,112702,123],{},[110,112704,4815],{},[34365,112706,112707],{},[32,112708,112709,112711,112713],{},[115,112710,117],{},[110,112712,4815],{},[35,112714,40263],{},[110,112716,2472],{},[115,112718,2465],{},[110,112720,4815],{},[34365,112722,112723],{},[32,112724,112725,112727,112729],{},[115,112726,117],{},[110,112728,4815],{},[35,112730,40263],{},[110,112732,123],{},[115,112734,2465],{},[110,112736,4815],{},[115,112738,2465],{},[34365,112740,112741],{},[32,112742,112743,112745,112747],{},[115,112744,117],{},[110,112746,4815],{},[35,112748,40263],{},[39,112750,112751],{"encoding":41},"f^*=t(y-1)+2-\\frac1t=-\\sqrt{1-y}+2-\\sqrt{1-y}=2-2\\sqrt{1-y}",[15,112753,112755,112799,112823,112844,112862,112945,113022,113040,113114,113132],{"className":112754,"ariaHidden":46},[45],[15,112756,112758,112761,112790,112793,112796],{"className":112757},[50],[15,112759],{"className":112760,"style":55},[54],[15,112762,112764,112767],{"className":112763},[59],[15,112765,37],{"className":112766,"style":61},[59,60],[15,112768,112770],{"className":112769},[281],[15,112771,112773],{"className":112772},[285],[15,112774,112776],{"className":112775},[290],[15,112777,112779],{"className":112778,"style":8501},[294],[15,112780,112781,112784],{"style":6867},[15,112782],{"className":112783,"style":303},[302],[15,112785,112787],{"className":112786},[307,308,309,310],[15,112788,7575],{"className":112789},[2593,310],[15,112791],{"className":112792,"style":158},[157],[15,112794,123],{"className":112795},[162],[15,112797],{"className":112798,"style":158},[157],[15,112800,112802,112805,112808,112811,112814,112817,112820],{"className":112801},[50],[15,112803],{"className":112804,"style":139},[54],[15,112806,769],{"className":112807},[59,60],[15,112809,113],{"className":112810},[146],[15,112812,40263],{"className":112813,"style":981},[59,60],[15,112815],{"className":112816,"style":2589},[157],[15,112818,4815],{"className":112819},[2593],[15,112821],{"className":112822,"style":2589},[157],[15,112824,112826,112829,112832,112835,112838,112841],{"className":112825},[50],[15,112827],{"className":112828,"style":139},[54],[15,112830,117],{"className":112831},[59],[15,112833,120],{"className":112834},[153],[15,112836],{"className":112837,"style":2589},[157],[15,112839,2472],{"className":112840},[2593],[15,112842],{"className":112843,"style":2589},[157],[15,112845,112847,112850,112853,112856,112859],{"className":112846},[50],[15,112848],{"className":112849,"style":12582},[54],[15,112851,2465],{"className":112852},[59],[15,112854],{"className":112855,"style":2589},[157],[15,112857,4815],{"className":112858},[2593],[15,112860],{"className":112861,"style":2589},[157],[15,112863,112865,112868,112936,112939,112942],{"className":112864},[50],[15,112866],{"className":112867,"style":2507},[54],[15,112869,112871,112874,112933],{"className":112870},[59],[15,112872],{"className":112873},[146,441],[15,112875,112877],{"className":112876},[238],[15,112878,112880,112925],{"className":112879},[285,286],[15,112881,112883,112922],{"className":112882},[290],[15,112884,112886,112900,112908],{"className":112885,"style":2526},[294],[15,112887,112888,112891],{"style":2529},[15,112889],{"className":112890,"style":461},[302],[15,112892,112894],{"className":112893},[307,308,309,310],[15,112895,112897],{"className":112896},[59,310],[15,112898,769],{"className":112899},[59,60,310],[15,112901,112902,112905],{"style":473},[15,112903],{"className":112904,"style":461},[302],[15,112906],{"className":112907,"style":481},[480],[15,112909,112910,112913],{"style":2552},[15,112911],{"className":112912,"style":461},[302],[15,112914,112916],{"className":112915},[307,308,309,310],[15,112917,112919],{"className":112918},[59,310],[15,112920,117],{"className":112921},[59,310],[15,112923,318],{"className":112924},[317],[15,112926,112928],{"className":112927},[290],[15,112929,112931],{"className":112930,"style":2574},[294],[15,112932],{},[15,112934],{"className":112935},[153,441],[15,112937],{"className":112938,"style":158},[157],[15,112940,123],{"className":112941},[162],[15,112943],{"className":112944,"style":158},[157],[15,112946,112948,112951,112954,113013,113016,113019],{"className":112947},[50],[15,112949],{"className":112950,"style":110440},[54],[15,112952,4815],{"className":112953},[59],[15,112955,112957],{"className":112956},[59,34393],[15,112958,112960,113005],{"className":112959},[285,286],[15,112961,112963,113002],{"className":112962},[290],[15,112964,112966,112990],{"className":112965,"style":110456},[294],[15,112967,112969,112972],{"className":112968,"style":11867},[6309],[15,112970],{"className":112971,"style":461},[302],[15,112973,112975,112978,112981,112984,112987],{"className":112974,"style":34413},[59],[15,112976,117],{"className":112977},[59],[15,112979],{"className":112980,"style":2589},[157],[15,112982,4815],{"className":112983},[2593],[15,112985],{"className":112986,"style":2589},[157],[15,112988,40263],{"className":112989,"style":981},[59,60],[15,112991,112992,112995],{"style":110483},[15,112993],{"className":112994,"style":461},[302],[15,112996,112998],{"className":112997,"style":34470},[34469],[6325,112999,113000],{"xmlns":6327,"width":6328,"height":34473,"viewBox":34474,"preserveAspectRatio":6331},[6333,113001],{"d":34477},[15,113003,318],{"className":113004},[317],[15,113006,113008],{"className":113007},[290],[15,113009,113011],{"className":113010,"style":110503},[294],[15,113012],{},[15,113014],{"className":113015,"style":2589},[157],[15,113017,2472],{"className":113018},[2593],[15,113020],{"className":113021,"style":2589},[157],[15,113023,113025,113028,113031,113034,113037],{"className":113024},[50],[15,113026],{"className":113027,"style":12582},[54],[15,113029,2465],{"className":113030},[59],[15,113032],{"className":113033,"style":2589},[157],[15,113035,4815],{"className":113036},[2593],[15,113038],{"className":113039,"style":2589},[157],[15,113041,113043,113046,113105,113108,113111],{"className":113042},[50],[15,113044],{"className":113045,"style":110440},[54],[15,113047,113049],{"className":113048},[59,34393],[15,113050,113052,113097],{"className":113051},[285,286],[15,113053,113055,113094],{"className":113054},[290],[15,113056,113058,113082],{"className":113057,"style":110456},[294],[15,113059,113061,113064],{"className":113060,"style":11867},[6309],[15,113062],{"className":113063,"style":461},[302],[15,113065,113067,113070,113073,113076,113079],{"className":113066,"style":34413},[59],[15,113068,117],{"className":113069},[59],[15,113071],{"className":113072,"style":2589},[157],[15,113074,4815],{"className":113075},[2593],[15,113077],{"className":113078,"style":2589},[157],[15,113080,40263],{"className":113081,"style":981},[59,60],[15,113083,113084,113087],{"style":110483},[15,113085],{"className":113086,"style":461},[302],[15,113088,113090],{"className":113089,"style":34470},[34469],[6325,113091,113092],{"xmlns":6327,"width":6328,"height":34473,"viewBox":34474,"preserveAspectRatio":6331},[6333,113093],{"d":34477},[15,113095,318],{"className":113096},[317],[15,113098,113100],{"className":113099},[290],[15,113101,113103],{"className":113102,"style":110503},[294],[15,113104],{},[15,113106],{"className":113107,"style":158},[157],[15,113109,123],{"className":113110},[162],[15,113112],{"className":113113,"style":158},[157],[15,113115,113117,113120,113123,113126,113129],{"className":113116},[50],[15,113118],{"className":113119,"style":12582},[54],[15,113121,2465],{"className":113122},[59],[15,113124],{"className":113125,"style":2589},[157],[15,113127,4815],{"className":113128},[2593],[15,113130],{"className":113131,"style":2589},[157],[15,113133,113135,113138,113141],{"className":113134},[50],[15,113136],{"className":113137,"style":110440},[54],[15,113139,2465],{"className":113140},[59],[15,113142,113144],{"className":113143},[59,34393],[15,113145,113147,113192],{"className":113146},[285,286],[15,113148,113150,113189],{"className":113149},[290],[15,113151,113153,113177],{"className":113152,"style":110456},[294],[15,113154,113156,113159],{"className":113155,"style":11867},[6309],[15,113157],{"className":113158,"style":461},[302],[15,113160,113162,113165,113168,113171,113174],{"className":113161,"style":34413},[59],[15,113163,117],{"className":113164},[59],[15,113166],{"className":113167,"style":2589},[157],[15,113169,4815],{"className":113170},[2593],[15,113172],{"className":113173,"style":2589},[157],[15,113175,40263],{"className":113176,"style":981},[59,60],[15,113178,113179,113182],{"style":110483},[15,113180],{"className":113181,"style":461},[302],[15,113183,113185],{"className":113184,"style":34470},[34469],[6325,113186,113187],{"xmlns":6327,"width":6328,"height":34473,"viewBox":34474,"preserveAspectRatio":6331},[6333,113188],{"d":34477},[15,113190,318],{"className":113191},[317],[15,113193,113195],{"className":113194},[290],[15,113196,113198],{"className":113197,"style":110503},[294],[15,113199],{},"。α 一行取 ",[15,113202,113204,113243],{"className":113203,"translate":19},[18],[15,113205,113207],{"className":113206},[23],[25,113208,113209],{"xmlns":27},[29,113210,113211,113241],{},[32,113212,113213,113215,113217,113229,113231,113233,113235,113237,113239],{},[35,113214,45206],{},[110,113216,123],{},[238,113218,113219,113227],{},[32,113220,113221,113223,113225],{},[115,113222,117],{},[110,113224,2472],{},[35,113226,44780],{},[115,113228,2465],{},[110,113230,5911],{},[110,113232,113],{"stretchy":112},[115,113234,126],{},[110,113236,551],{"separator":46},[115,113238,117],{},[110,113240,120],{"stretchy":112},[39,113242,45235],{"encoding":41},[15,113244,113246,113264,113353],{"className":113245,"ariaHidden":46},[45],[15,113247,113249,113252,113255,113258,113261],{"className":113248},[50],[15,113250],{"className":113251,"style":2347},[54],[15,113253,45206],{"className":113254},[59,60],[15,113256],{"className":113257,"style":158},[157],[15,113259,123],{"className":113260},[162],[15,113262],{"className":113263,"style":158},[157],[15,113265,113267,113270,113344,113347,113350],{"className":113266},[50],[15,113268],{"className":113269,"style":2507},[54],[15,113271,113273,113276,113341],{"className":113272},[59],[15,113274],{"className":113275},[146,441],[15,113277,113279],{"className":113278},[238],[15,113280,113282,113333],{"className":113281},[285,286],[15,113283,113285,113330],{"className":113284},[290],[15,113286,113288,113302,113310],{"className":113287,"style":2526},[294],[15,113289,113290,113293],{"style":2529},[15,113291],{"className":113292,"style":461},[302],[15,113294,113296],{"className":113295},[307,308,309,310],[15,113297,113299],{"className":113298},[59,310],[15,113300,2465],{"className":113301},[59,310],[15,113303,113304,113307],{"style":473},[15,113305],{"className":113306,"style":461},[302],[15,113308],{"className":113309,"style":481},[480],[15,113311,113312,113315],{"style":2552},[15,113313],{"className":113314,"style":461},[302],[15,113316,113318],{"className":113317},[307,308,309,310],[15,113319,113321,113324,113327],{"className":113320},[59,310],[15,113322,117],{"className":113323},[59,310],[15,113325,2472],{"className":113326},[2593,310],[15,113328,44780],{"className":113329,"style":44810},[59,60,310],[15,113331,318],{"className":113332},[317],[15,113334,113336],{"className":113335},[290],[15,113337,113339],{"className":113338,"style":2574},[294],[15,113340],{},[15,113342],{"className":113343},[153,441],[15,113345],{"className":113346,"style":158},[157],[15,113348,5911],{"className":113349},[162],[15,113351],{"className":113352,"style":158},[157],[15,113354,113356,113359,113362,113365,113368,113371,113374],{"className":113355},[50],[15,113357],{"className":113358,"style":139},[54],[15,113360,113],{"className":113361},[146],[15,113363,126],{"className":113364},[59],[15,113366,551],{"className":113367},[573],[15,113369],{"className":113370,"style":406},[157],[15,113372,117],{"className":113373},[59],[15,113375,120],{"className":113376},[153],"，验证：",[15,113379,113381,113457],{"className":113380,"translate":19},[18],[15,113382,113384],{"className":113383},[23],[25,113385,113386],{"xmlns":27},[29,113387,113388,113454],{},[32,113389,113390,113398,113400,113402,113404,113406,113416,113418,113420,113422,113424,113452],{},[7555,113391,113392,113394,113396],{},[35,113393,37],{},[35,113395,44780],{},[110,113397,7575],{},[110,113399,113],{"stretchy":112},[35,113401,40263],{},[110,113403,120],{"stretchy":112},[110,113405,123],{},[195,113407,113408,113414],{},[32,113409,113410,113412],{},[35,113411,35458],{},[110,113413,4105],{},[35,113415,769],{},[110,113417,2900],{"stretchy":112},[35,113419,769],{},[35,113421,40263],{},[110,113423,4815],{},[238,113425,113426,113438],{},[32,113427,113428,113430,113432],{},[115,113429,117],{},[110,113431,4815],{},[6816,113433,113434,113436],{},[35,113435,769],{},[35,113437,45206],{},[32,113439,113440,113442,113444,113446,113448,113450],{},[35,113441,45206],{},[110,113443,113],{"stretchy":112},[115,113445,117],{},[110,113447,4815],{},[35,113449,45206],{},[110,113451,120],{"stretchy":112},[110,113453,2910],{"stretchy":112},[39,113455,113456],{"encoding":41},"f_\\alpha^*(y)=\\sup_t\\{ty-\\frac{1-t^a}{a(1-a)}\\}",[15,113458,113460,113535,113599],{"className":113459,"ariaHidden":46},[45],[15,113461,113463,113466,113517,113520,113523,113526,113529,113532],{"className":113462},[50],[15,113464],{"className":113465,"style":139},[54],[15,113467,113469,113472],{"className":113468},[59],[15,113470,37],{"className":113471,"style":61},[59,60],[15,113473,113475],{"className":113474},[281],[15,113476,113478,113509],{"className":113477},[285,286],[15,113479,113481,113506],{"className":113480},[290],[15,113482,113484,113495],{"className":113483,"style":8501},[294],[15,113485,113486,113489],{"style":8253},[15,113487],{"className":113488,"style":303},[302],[15,113490,113492],{"className":113491},[307,308,309,310],[15,113493,44780],{"className":113494,"style":44810},[59,60,310],[15,113496,113497,113500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,[2593],[15,113790],{"className":113791,"style":2589},[157],[15,113793,113795,113799,113909,113912,113915],{"className":113794},[50],[15,113796],{"className":113797,"style":113798},[54],"height:1.4213em;vertical-align:-0.4033em;",[15,113800,113802,113805,113906],{"className":113801},[59],[15,113803],{"className":113804},[146,441],[15,113806,113808],{"className":113807},[238],[15,113809,113811,113898],{"className":113810},[285,286],[15,113812,113814,113895],{"className":113813},[290],[15,113815,113818,113838,113846],{"className":113816,"style":113817},[294],"height:1.0179em;",[15,113819,113820,113823],{"style":2529},[15,113821],{"className":113822,"style":461},[302],[15,113824,113826],{"className":113825},[307,308,309,310],[15,113827,113829,113832,113835],{"className":113828},[59,310],[15,113830,117],{"className":113831},[59,310],[15,113833,4815],{"className":113834},[2593,310],[15,113836,45206],{"className":113837},[59,60,310],[15,113839,113840,113843],{"style":473},[15,113841],{"className":113842,"style":461},[302],[15,113844],{"className":113845,"style":481},[480],[15,113847,113848,113851],{"style":2552},[15,113849],{"className":113850,"style":461},[302],[15,113852,113854],{"className":113853},[307,308,309,310],[15,113855,113857],{"className":113856},[59,310],[15,113858,113860,113863],{"className":113859},[59,310],[15,113861,769],{"className":113862},[59,60,310],[15,113864,113866],{"className":113865},[281],[15,113867,113869],{"className":113868},[285],[15,113870,113872],{"className":113871},[290],[15,113873,113875],{"className":113874,"style":38538},[294],[15,113876,113877,113880],{"style":38541},[15,113878],{"className":113879,"style":30232},[302],[15,113881,113883],{"className":113882},[307,30236,9676,310],[15,113884,113886,113889,113892],{"className":113885},[59,310],[15,113887,45206],{"className":113888},[59,60,310],[15,113890,4815],{"className":113891},[2593,310],[15,113893,117],{"className":113894},[59,310],[15,113896,318],{"className":113897},[317],[15,113899,113901],{"className":113900},[290],[15,113902,113904],{"className":113903,"style":16433},[294],[15,113905],{},[15,113907],{"className":113908},[153,441],[15,113910],{"className":113911,"style":158},[157],[15,113913,123],{"className":113914},[162],[15,113916],{"className":113917,"style":158},[157],[15,113919,113921,113924],{"className":113920},[50],[15,113922],{"className":113923,"style":172},[54],[15,113925,126],{"className":113926},[59],[15,113928,113930,113986],{"className":113929,"translate":19},[18],[15,113931,113933],{"className":113932},[23],[25,113934,113935],{"xmlns":27},[29,113936,113937,113983],{},[32,113938,113939,113941,113943,113945,113947,113949,113951,113953,113955,113957,113959,113961,113963],{},[35,113940,769],{},[110,113942,123],{},[110,113944,113],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[110,113946,113],{"stretchy":112},[115,113948,117],{},[110,113950,4815],{},[35,113952,45206],{},[110,113954,120],{"stretchy":112},[110,113956,113],{"stretchy":112},[110,113958,4815],{},[35,113960,40263],{},[110,113962,120],{"stretchy":112},[6816,113964,113965,113967],{},[110,113966,120],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[32,113968,113969,113971],{},[110,113970,4815],{},[238,113972,113973,113975],{},[115,113974,117],{},[32,113976,113977,113979,113981],{},[115,113978,117],{},[110,113980,4815],{},[35,113982,45206],{},[39,113984,113985],{"encoding":41},"t=\\big((1-a)(-y)\\big)^{-\\frac{1}{1-a}}",[15,113987,113989,114007,114034],{"className":113988,"ariaHidden":46},[45],[15,113990,113992,113995,113998,114001,114004],{"className":113991},[50],[15,113993],{"className":113994,"style":5345},[54],[15,113996,769],{"className":113997},[59,60],[15,113999],{"className":114000,"style":158},[157],[15,114002,123],{"className":114003},[162],[15,114005],{"className":114006,"style":158},[157],[15,114008,114010,114013,114019,114022,114025,114028,114031],{"className":114009},[50],[15,114011],{"className":114012,"style":20589},[54],[15,114014,114016],{"className":114015},[59],[15,114017,113],{"className":114018},[419,9676],[15,114020,113],{"className":114021},[146],[15,114023,117],{"className":114024},[59],[15,114026],{"className":114027,"style":2589},[157],[15,114029,4815],{"className":114030},[2593],[15,114032],{"className":114033,"style":2589},[157],[15,114035,114037,114041,114044,114047,114050,114053,114056,114059],{"className":114036},[50],[15,114038],{"className":114039,"style":114040},[54],"height:1.5439em;vertical-align:-0.35em;",[15,114042,45206],{"className":114043},[59,60],[15,114045,120],{"className":114046},[153],[15,114048,113],{"className":114049},[146],[15,114051,4815],{"className":114052},[59],[15,114054,40263],{"className":114055,"style":981},[59,60],[15,114057,120],{"className":114058},[153],[15,114060,114062,114068],{"className":114061},[59],[15,114063,114065],{"className":114064},[59],[15,114066,120],{"className":114067},[419,9676],[15,114069,114071],{"className":114070},[281],[15,114072,114074],{"clas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,"ariaHidden":46},[45],[15,114365,114367,114370,114373,114376,114379],{"className":114366},[50],[15,114368],{"className":114369,"style":2347},[54],[15,114371,44780],{"className":114372,"style":44810},[59,60],[15,114374],{"className":114375,"style":158},[157],[15,114377,123],{"className":114378},[162],[15,114380],{"className":114381,"style":158},[157],[15,114383,114385,114388],{"className":114384},[50],[15,114386],{"className":114387,"style":172},[54],[15,114389,126],{"className":114390},[59],[15,114392,114394,114424],{"className":114393,"translate":19},[18],[15,114395,114397],{"className":114396},[23],[25,114398,114399],{"xmlns":27},[29,114400,114401,114421],{},[32,114402,114403,114405,114407,114409,114411,114413,114415,114419],{},[35,114404,37],{},[110,114406,123],{},[115,114408,39015],{},[110,114410,113],{"stretchy":112},[115,114412,117],{},[110,114414,4815],{},[34365,114416,114417],{},[35,114418,769],{},[110,114420,120],{"stretchy":112},[39,114422,114423],{"encoding":41},"f=4(1-\\sqrt t)",[15,114425,114427,114445,114469],{"className":114426,"ariaHidden":46},[45],[15,114428,114430,114433,114436,114439,114442],{"className":114429},[50],[15,114431],{"className":114432,"style":55},[54],[15,114434,37],{"className":114435,"style":61},[59,60],[15,114437],{"className":114438,"style":158},[157],[15,114440,123],{"className":114441},[162],[15,114443],{"className":114444,"style":158},[157],[15,114446,114448,114451,114454,114457,114460,114463,114466],{"className":114447},[50],[15,114449],{"className":114450,"style":139},[54],[15,114452,39015],{"className":114453},[59],[15,114455,113],{"className":114456},[146],[15,114458,117],{"className":114459},[59],[15,114461],{"className":114462,"style":2589},[157],[15,114464,4815],{"className":114465},[2593],[15,114467],{"className":114468,"style":2589},[157],[15,114470,114472,114475,114519],{"className":114471},[50],[15,114473],{"className":114474,"style":42693},[54],[15,114476,114478],{"className":114477},[59,34393],[15,114479,114481,114511],{"className":114480},[285,286],[15,114482,114484,114508],{"className":114483},[290],[15,114485,114487,114496],{"className":114486,"style":42709},[294],[15,114488,114490,114493],{"className":114489,"style":11867},[6309],[15,114491],{"className":114492,"style":461},[302],[15,114494,769],{"className":114495,"style":34413},[59,60],[15,114497,114498,114501],{"style":42721},[15,114499],{"className":114500,"style":461},[302],[15,114502,114504],{"className":114503,"style":34470},[34469],[6325,114505,114506],{"xmlns":6327,"width":6328,"height":34473,"viewBox":34474,"preserveAspectRatio":6331},[6333,114507],{"d":34477},[15,114509,318],{"className":114510},[317],[15,114512,114514],{"className":114513},[290],[15,114515,114517],{"className":114516,"style":42741},[294],[15,114518],{},[15,114520,120],{"className":114521},[153],"）时公式给出 ",[15,114524,114526,114552],{"className":114525,"translate":19},[18],[15,114527,114529],{"className":114528},[23],[25,114530,114531],{"xmlns":27},[29,114532,114533,114549],{},[32,114534,114535,114545,114547],{},[238,114536,114537,114539],{},[115,114538,39015],{},[32,114540,114541,114543],{},[110,114542,4815],{},[35,114544,40263],{},[110,114546,4815],{},[115,114548,39015],{},[39,114550,114551],{"encoding":41},"\\frac{4}{-y}-4",[15,114553,114555,114642],{"className":114554,"ariaHidden":46},[45],[15,114556,114558,114562,114633,114636,114639],{"className":114557},[50],[15,114559],{"className":114560,"style":114561},[54],"height:1.3262em;vertical-align:-0.4811em;",[15,114563,114565,114568,114630],{"className":114564},[59],[15,114566],{"className":114567},[146,441],[15,114569,114571],{"className":114570},[238],[15,114572,114574,114622],{"className":114573},[285,286],[15,114575,114577,114619],{"className":114576},[290],[15,114578,114580,114597,114605],{"className":114579,"style":2526},[294],[15,114581,114582,114585],{"style":2529},[15,114583],{"className":114584,"style":461},[302],[15,114586,114588],{"className":114587},[307,308,309,310],[15,114589,114591,114594],{"className":114590},[59,310],[15,114592,4815],{"className":114593},[59,310],[15,114595,40263],{"className":114596,"style":981},[59,60,310],[15,114598,114599,114602],{"style":473},[15,114600],{"className":114601,"style":461},[302],[15,114603],{"className":114604,"style":481},[480],[15,114606,114607,114610],{"style":2552},[15,114608],{"className":114609,"style":461},[302],[15,114611,114613],{"className":114612},[307,308,309,310],[15,114614,114616],{"className":114615},[59,310],[15,114617,39015],{"className":114618},[59,310],[15,114620,318],{"className":114621},[317],[15,114623,114625],{"className":114624},[290],[15,114626,114628],{"className":114627,"style":2741},[294],[15,114629],{},[15,114631],{"className":114632},[153,441],[15,114634],{"className":114635,"style":2589},[157],[15,114637,4815],{"className":114638},[2593],[15,114640],{"className":114641,"style":2589},[157],[15,114643,114645,114648],{"className":114644},[50],[15,114646],{"className":114647,"style":172},[54],[15,114649,39015],{"className":114650},[59],"，可与直接展开互相验证。）",[11,114653,114654],{},"可见，f-散度族统一了「不对称\u002F对称」「有界\u002F无界」的多种散度。",[2313,114656,114658],{"id":114657},"_62-积分概率度量ipm","6.2 积分概率度量（IPM）",[11,114660,114661,114662,114711],{},"f-散度只依赖似然比 ",[15,114663,114665,114687],{"className":114664,"translate":19},[18],[15,114666,114668],{"className":114667},[23],[25,114669,114670],{"xmlns":27},[29,114671,114672,114684],{},[32,114673,114674,114676,114678,114680,114682],{},[35,114675,244],{},[35,114677,206],{},[35,114679,941],{"mathvariant":209},[35,114681,244],{},[35,114683,213],{},[39,114685,114686],{"encoding":41},"dP\u002FdQ",[15,114688,114690],{"className":114689,"ariaHidden":46},[45],[15,114691,114693,114696,114699,114702,114705,114708],{"className":114692},[50],[15,114694],{"className":114695,"style":139},[54],[15,114697,244],{"className":114698},[59,60],[15,114700,206],{"className":114701,"style":334},[59,60],[15,114703,941],{"className":114704},[59],[15,114706,244],{"className":114707},[59,60],[15,114709,213],{"className":114710},[59,60],"，因此把两个分布同时「搬到别处」值不变。这带来一个根本局限：",[15,114713,114715],{"className":114714,"translate":19},[180],[15,114716,114718,114773],{"className":114717,"translate":19},[18],[15,114719,114721],{"className":114720},[23],[25,114722,114723],{"xmlns":27,"display":189},[29,114724,114725,114770],{},[32,114726,114727,114729,114731,114733,114735,114737,114739,114742,114744,114746,114752,114754,114756,114758,114760,114766,114768],{},[1299,114728,4340],{},[35,114730,206],{},[110,114732,33573],{},[35,114734,213],{},[1299,114736,4285],{},[110,114738,113],{"stretchy":112},[1299,114740,114741],{},"支撑集不重叠，如 ",[35,114743,206],{},[110,114745,123],{},[195,114747,114748,114750],{},[35,114749,15716],{},[115,114751,126],{},[110,114753,551],{"separator":46},[1299,114755,4285],{},[35,114757,213],{},[110,114759,123],{},[195,114761,114762,114764],{},[35,114763,15716],{},[115,114765,117],{},[110,114767,120],{"stretchy":112},[110,114769,551],{"separator":46},[39,114771,114772],{"encoding":41},"\\text{若 } P\\perp Q\\\n(\\text{支撑集不重叠，如 } P=\\delta_0,\\ Q=\\delta_1),",[15,114774,114776,114803,114840,114907],{"className":114775,"ariaHidden":46},[45],[15,114777,114779,114782,114791,114794,114797,114800],{"className":114778},[50],[15,114780],{"className":114781,"style":12601},[54],[15,114783,114785,114788],{"className":114784},[59,4600],[15,114786,4651],{"className":114787},[59,4604],[15,114789,4285],{"className":114790},[59],[15,114792,206],{"className":114793,"style":334},[59,60],[15,114795],{"className":114796,"style":158},[157],[15,114798,33573],{"className":114799},[162],[15,114801],{"className":114802,"style":158},[157],[15,114804,114806,114809,114812,114815,114818,114828,114831,114834,114837],{"className":114805},[50],[15,114807],{"className":114808,"style":139},[54],[15,114810,213],{"className":114811},[59,60],[15,114813,4285],{"className":114814},[157],[15,114816,113],{"className":114817},[146],[15,114819,114821,114825],{"className":114820},[59,4600],[15,114822,114824],{"className":114823},[59,4604],"支撑集不重叠，如",[15,114826,4285],{"className":114827},[59],[15,114829,206],{"className":114830,"style":334},[59,60],[15,114832],{"className":114833,"style":158},[157],[15,114835,123],{"className":114836},[162],[15,114838],{"className":114839,"style":158},[157],[15,114841,114843,114846,114886,114889,114892,114895,114898,114901,114904],{"className":114842},[50],[15,114844],{"className":114845,"style":55},[54],[15,114847,114849,114852],{"className":114848},[59],[15,114850,15716],{"className":114851,"style":15765},[59,60],[15,114853,114855],{"className":114854},[281],[15,114856,114858,114878],{"className":114857},[285,286],[15,114859,114861,114875],{"className":114860},[290],[15,114862,114864],{"className":114863,"style":11137},[294],[15,114865,114866,114869],{"style":94786},[15,114867],{"className":114868,"style":303},[302],[15,114870,114872],{"className":114871},[307,308,309,310],[15,114873,126],{"className":114874},[59,310],[15,114876,318],{"className":114877},[317],[15,114879,114881],{"className":114880},[290],[15,114882,114884],{"className":114883,"style":7970},[294],[15,114885],{},[15,114887,551],{"className":114888},[573],[15,114890,4285],{"className":114891},[157],[15,114893],{"className":114894,"style":406},[157],[15,114896,213],{"className":114897},[59,60],[15,114899],{"className":114900,"style":158},[157],[15,114902,123],{"className":114903},[162],[15,114905],{"className":114906,"style":158},[157],[15,114908,114910,114913,114953,114956],{"className":114909},[50],[15,114911],{"className":114912,"style":139},[54],[15,114914,114916,114919],{"className":114915},[59],[15,114917,15716],{"className":114918,"style":15765},[59,60],[15,114920,114922],{"className":114921},[281],[15,114923,114925,114945],{"className":114924},[285,286],[15,114926,114928,114942],{"className":114927},[290],[15,114929,114931],{"className":114930,"style":11137},[294],[15,114932,114933,114936],{"style":94786},[15,114934],{"className":114935,"style":303},[302],[15,114937,114939],{"className":114938},[307,308,309,310],[15,114940,117],{"className":114941},[59,310],[15,114943,318],{"className":114944},[317],[15,114946,114948],{"className":114947},[290],[15,114949,114951],{"className":114950,"style":7970},[294],[15,114952],{},[15,114954,120],{"className":114955},[153],[15,114957,551],{"className":114958},[573],[15,114960,114962],{"className":114961,"translate":19},[180],[15,114963,114965,115015],{"className":114964,"translate":19},[18],[15,114966,114968],{"className":114967},[23],[25,114969,114970],{"xmlns":27,"display":189},[29,114971,114972,115012],{},[32,114973,114974,114984,114986,114988,114990,114992,114994,114996,115006,115008,115010],{},[195,114975,114976,114978],{},[35,114977,199],{},[32,114979,114980,114982],{},[35,114981,23769],{},[35,114983,25763],{},[110,114985,123],{},[35,114987,4102],{},[110,114989,4105],{},[115,114991,2465],{},[110,114993,551],{"separator":46},[157,114995],{"width":4317},[195,114997,114998,115000],{},[35,114999,199],{},[32,115001,115002,115004],{},[35,115003,5394],{},[35,115005,34876],{},[110,115007,123],{},[115,115009,117],{},[110,115011,551],{"separator":46},[39,115013,115014],{"encoding":41},"D_{JS}=\\log2,\\quad D_{TV}=1,",[15,115016,115018,115079,115160],{"className":115017,"ariaHidden":46},[45],[15,115019,115021,115024,115070,115073,115076],{"className":115020},[50],[15,115022],{"className":115023,"style":15965},[54],[15,115025,115027,115030],{"className":115026},[59],[15,115028,199],{"className":115029,"style":277},[59,60],[15,115031,115033],{"className":115032},[281],[15,115034,115036,115062],{"className":115035},[285,286],[15,115037,115039,115059],{"className":115038},[290],[15,115040,115042],{"className":115041,"style":379},[294],[15,115043,115044,115047],{"style":298},[15,115045],{"className":115046,"style":303},[302],[15,115048,115050],{"className":115049},[307,308,309,310],[15,115051,115053,115056],{"className":115052},[59,310],[15,115054,23769],{"className":115055,"style":24179},[59,60,310],[15,115057,25763],{"className":115058,"style":25887},[59,60,310],[15,115060,318],{"className":115061},[317],[15,115063,115065],{"className":115064},[290],[15,115066,115068],{"className":115067,"style":7970},[294],[15,115069],{},[15,115071],{"className":115072,"style":158},[157],[15,115074,123],{"className":115075},[162],[15,115077],{"className":115078,"style":158},[157],[15,115080,115082,115085,115090,115093,115096,115099,115102,115105,115151,115154,115157],{"className":115081},[50],[15,115083],{"className":115084,"style":55},[54],[15,115086,4156,115088],{"className":115087},[1486],[15,115089,4160],{"style":4159},[15,115091],{"className":115092,"style":406},[157],[15,115094,2465],{"className":115095},[59],[15,115097,551],{"className":115098},[573],[15,115100],{"className":115101,"style":88009},[157],[15,115103],{"className":115104,"style":406},[157],[15,115106,115108,115111],{"className":115107},[59],[15,115109,199],{"className":115110,"style":277},[59,60],[15,115112,115114],{"className":115113},[281],[15,115115,115117,115143],{"className":115116},[285,286],[15,115118,115120,115140],{"className":115119},[290],[15,115121,115123],{"className":115122,"style":379},[294],[15,115124,115125,115128],{"style":298},[15,115126],{"className":115127,"style":303},[302],[15,115129,115131],{"className":115130},[307,308,309,310],[15,115132,115134,115137],{"className":115133},[59,310],[15,115135,5394],{"className":115136,"style":334},[59,60,310],[15,115138,34876],{"className":115139,"style":2589},[59,60,310],[15,115141,318],{"className":115142},[317],[15,115144,115146],{"className":115145},[290],[15,115147,115149],{"className":115148,"style":7970},[294],[15,115150],{},[15,115152],{"className":115153,"style":158},[157],[15,115155,123],{"className":115156},[162],[15,115158],{"className":115159,"style":158},[157],[15,115161,115163,115166,115169],{"className":115162},[50],[15,115164],{"className":115165,"style":9360},[54],[15,115167,117],{"className":115168},[59],[15,115170,551],{"className":115171},[573],[11,115173,115174,115175,115178,115179,115243,115244,115294,115295,9402],{},"也就是说，f-散度在支撑分离时",[998,115176,115177],{},"饱和到常数上界，对生成器参数的微小改动失去梯度","：因为被积分的量 ",[15,115180,115182,115210],{"className":115181,"translate":19},[18],[15,115183,115185],{"className":115184},[23],[25,115186,115187],{"xmlns":27},[29,115188,115189,115207],{},[32,115190,115191,115193,115195,115197,115199,115201,115203,115205],{},[35,115192,944],{},[1299,115194,1301],{},[35,115196,37],{},[110,115198,113],{"stretchy":112},[35,115200,11],{},[35,115202,941],{"mathvariant":209},[35,115204,944],{},[110,115206,120],{"stretchy":112},[39,115208,115209],{"encoding":41},"q\\,f(p\u002Fq)",[15,115211,115213],{"className":115212,"ariaHidden":46},[45],[15,115214,115216,115219,115222,115225,115228,115231,115234,115237,115240],{"className":115215},[50],[15,115217],{"className":115218,"style":139},[54],[15,115220,944],{"className":115221,"style":981},[59,60],[15,115223],{"className":115224,"style":406},[157],[15,115226,37],{"className":115227,"style":61},[59,60],[15,115229,113],{"className":115230},[146],[15,115232,11],{"className":115233},[59,60],[15,115235,941],{"className":115236},[59],[15,115238,944],{"className":115239,"style":981},[59,60],[15,115241,120],{"className":115242},[153]," 只在 ",[15,115245,115247,115264],{"className":115246,"translate":19},[18],[15,115248,115250],{"className":115249},[23],[25,115251,115252],{"xmlns":27},[29,115253,115254,115262],{},[32,115255,115256,115258,115260],{},[35,115257,944],{},[110,115259,2905],{},[115,115261,126],{},[39,115263,31708],{"encoding":41},[15,115265,115267,115285],{"className":115266,"ariaHidden":46},[45],[15,115268,115270,115273,115276,115279,115282],{"className":115269},[50],[15,115271],{"className":115272,"style":31718},[54],[15,115274,944],{"className":115275,"style":981},[59,60],[15,115277],{"className":115278,"style":158},[157],[15,115280,2905],{"className":115281},[162],[15,115283],{"className":115284,"style":158},[157],[15,115286,115288,115291],{"className":115287},[50],[15,115289],{"className":115290,"style":172},[54],[15,115292,126],{"className":115293},[59]," 处有定义，当两个支撑几乎不重叠时，极小化散度的博弈只剩\"常数项 − 常数项\"，梯度信息全部丢失（这正是原 GAN 训练崩溃的理论根源）。Wasserstein 距离与 MMD 属于另一族——",[998,115296,115297],{},"积分概率度量",[15,115299,115301],{"className":115300,"translate":19},[180],[15,115302,115304,115379],{"className":115303,"translate":19},[18],[15,115305,115307],{"className":115306},[23],[25,115308,115309],{"xmlns":27,"display":189},[29,115310,115311,115376],{},[32,115312,115313,115320,115322,115324,115326,115328,115330,115332,115348,115350,115352,115354,115356,115358,115360,115362,115364,115366,115368,115370,115372,115374],{},[195,115314,115315,115318],{},[1299,115316,115317],{},"IPM",[35,115319,36909],{"mathvariant":597},[110,115321,113],{"stretchy":112},[35,115323,206],{},[110,115325,551],{"separator":46},[35,115327,213],{},[110,115329,120],{"stretchy":112},[110,115331,123],{},[6018,115333,115334,115340],{},[32,115335,115336,115338],{},[35,115337,35458],{},[110,115339,4105],{},[32,115341,115342,115344,115346],{},[35,115343,38691],{},[110,115345,5911],{},[35,115347,36909],{"mathvariant":597},[110,115349,4810],{"fence":112,"stretchy":46,"minsize":17443,"maxsize":17443},[110,115351,1289],{},[35,115353,38691],{},[1299,115355,1301],{},[35,115357,244],{},[35,115359,206],{},[110,115361,4815],{},[110,115363,1289],{},[35,115365,38691],{},[1299,115367,1301],{},[35,115369,244],{},[35,115371,213],{},[110,115373,4810],{"fence":112,"stretchy":46,"minsize":17443,"maxsize":17443},[35,115375,4377],{"mathvariant":209},[39,115377,115378],{"encoding":41},"\\text{IPM}_{\\mathcal F}(P,Q)\n=\\sup_{h\\in\\mathcal F}\\Big|\\int h\\,dP\n-\\int h\\,dQ\\Big| .",[15,115380,115382,115462,115595],{"className":115381,"ariaHidden":46},[45],[15,115383,115385,115388,115435,115438,115441,115444,115447,115450,115453,115456,115459],{"className":115384},[50],[15,115386],{"className":115387,"style":139},[54],[15,115389,115391,115397],{"className":115390},[59],[15,115392,115394],{"className":115393},[59,4600],[15,115395,115317],{"className":115396},[59],[15,115398,115400],{"className":115399},[281],[15,115401,115403,115427],{"className":115402},[285,286],[15,115404,115406,115424],{"className":115405},[290],[15,115407,115409],{"className":115408,"style":379},[294],[15,115410,115411,115414],{"style":52472},[15,115412],{"className":115413,"style":303},[302],[15,115415,115417],{"className":115416},[307,308,309,310],[15,115418,115420],{"className":115419},[59,310],[15,115421,36909],{"className":115422,"style":115423},[59,624,310],"margin-right:0.0993em;",[15,115425,318],{"className":115426},[317],[15,115428,115430],{"className":115429},[290],[15,115431,115433],{"className":115432,"style":7970},[294],[15,115434],{},[15,115436,113],{"className":115437},[146],[15,115439,206],{"className":115440,"style":334},[59,60],[15,115442,551],{"className":115443},[573],[15,115445],{"className":115446,"style":406},[157],[15,115448,213],{"className":115449},[59,60],[15,115451,120],{"className":115452},[153],[15,115454],{"className":115455,"style":158},[157],[15,115457,123],{"className":115458},[162],[15,115460],{"className":115461,"style":158},[157],[15,115463,115465,115469,115522,115525,115565,115568,115571,115574,115577,115580,115583,115586,115589,115592],{"className":115464},[50],[15,115466],{"className":115467,"style":115468},[54],"height:2.3339em;vertical-align:-0.9739em;",[15,115470,115472],{"className":115471},[1486,13720],[15,115473,115475,115514],{"className":115474},[285,286],[15,115476,115478,115511],{"className":115477},[290],[15,115479,115481,115501],{"className":115480,"style":2347},[294],[15,115482,115483,115486],{"style":51732},[15,115484],{"className":115485,"style":461},[302],[15,115487,115489],{"className":115488},[307,308,309,310],[15,115490,115492,115495,115498],{"className":115491},[59,310],[15,115493,38691],{"className":115494},[59,60,310],[15,115496,5911],{"className":115497},[162,310],[15,115499,36909],{"className":115500,"style":115423},[59,624,310],[15,115502,115503,115506],{"style":11867},[15,115504],{"className":115505,"style":461},[302],[15,115507,115508],{},[15,115509,35458],{"className":115510},[1486],[15,115512,318],{"className":115513},[317],[15,115515,115517],{"className":115516},[290],[15,115518,115520],{"className":115519,"style":51776},[294],[15,115521],{},[15,115523],{"className":115524,"style":406},[157],[15,115526,115528],{"className":115527},[59],[15,115529,115531],{"className":115530},[419,14833],[15,115532,115534,115557],{"className":115533},[285,286],[15,115535,115537,115554],{"className":115536},[290],[15,115538,115540],{"className":115539,"style":85556},[294],[15,115541,115542,115545],{"style":73561},[15,115543],{"className":115544,"style":82628},[302],[15,115546,115548],{"style":115547},"width:0.333em;height:1.8em;",[6325,115549,115551],{"xmlns":6327,"width":35077,"height":17443,"viewBox":115550},"0 0 333 1800",[6333,115552],{"d":115553},"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",[15,115555,318],{"className":115556},[317],[15,115558,115560],{"className":115559},[290],[15,115561,115563],{"className":115562,"style":85580},[294],[15,115564],{},[15,115566],{"className":115567,"style":406},[157],[15,115569,1289],{"className":115570,"style":1489},[1486,1487,1488],[15,115572],{"className":115573,"style":406},[157],[15,115575,38691],{"className":115576},[59,60],[15,115578],{"className":115579,"style":406},[157],[15,115581,244],{"className":115582},[59,60],[15,115584,206],{"className":115585,"style":334},[59,60],[15,115587],{"className":115588,"style":2589},[157],[15,115590,4815],{"className":115591},[2593],[15,115593],{"className":115594,"style":2589},[157],[15,115596,115598,115601,115604,115607,115610,115613,115616,115619,115656],{"className":115597},[50],[15,115599],{"className":115600,"style":5779},[54],[15,115602,1289],{"className":115603,"style":1489},[1486,1487,1488],[15,115605],{"className":115606,"style":406},[157],[15,115608,38691],{"className":115609},[59,60],[15,115611],{"className":115612,"style":406},[157],[15,115614,244],{"className":115615},[59,60],[15,115617,213],{"className":115618},[59,60],[15,115620,115622],{"className":115621},[59],[15,115623,115625],{"className":115624},[419,14833],[15,115626,115628,115648],{"className":115627},[285,286],[15,115629,115631,115645],{"className":115630},[290],[15,115632,115634],{"className":115633,"style":85556},[294],[15,115635,115636,115639],{"style":73561},[15,115637],{"className":115638,"style":82628},[302],[15,115640,115641],{"style":115547},[6325,115642,115643],{"xmlns":6327,"width":35077,"height":17443,"viewBox":115550},[6333,115644],{"d":115553},[15,115646,318],{"className":115647},[317],[15,115649,115651],{"className":115650},[290],[15,115652,115654],{"className":115653,"style":85580},[294],[15,115655],{},[15,115657,4377],{"className":115658},[59],[11,115660,115661,115662,115691,115692,115743,115744,115773,115774,115824,115825,115853,115854,115904,115905,115933],{},"它们比较的是「在函数类 ",[15,115663,115665,115679],{"className":115664,"translate":19},[18],[15,115666,115668],{"className":115667},[23],[25,115669,115670],{"xmlns":27},[29,115671,115672,115676],{},[32,115673,115674],{},[35,115675,36909],{"mathvariant":597},[39,115677,115678],{"encoding":41},"\\mathcal F",[15,115680,115682],{"className":115681,"ariaHidden":46},[45],[15,115683,115685,115688],{"className":115684},[50],[15,115686],{"className":115687,"style":664},[54],[15,115689,36909],{"className":115690,"style":115423},[59,624]," 上积分的差」，而不是似然比，因此利用了样本空间的度量\u002F几何结构。KS 距离（",[15,115693,115695,115713],{"className":115694,"translate":19},[18],[15,115696,115698],{"className":115697},[23],[25,115699,115700],{"xmlns":27},[29,115701,115702,115710],{},[32,115703,115704,115706,115708],{},[35,115705,36909],{"mathvariant":597},[110,115707,123],{},[110,115709,2900],{"stretchy":112},[39,115711,115712],{"encoding":41},"\\mathcal F=\\{",[15,115714,115716,115734],{"className":115715,"ariaHidden":46},[45],[15,115717,115719,115722,115725,115728,115731],{"className":115718},[50],[15,115720],{"className":115721,"style":664},[54],[15,115723,36909],{"className":115724,"style":115423},[59,624],[15,115726],{"className":115727,"style":158},[157],[15,115729,123],{"className":115730},[162],[15,115732],{"className":115733,"style":158},[157],[15,115735,115737,115740],{"className":115736},[50],[15,115738],{"className":115739,"style":139},[54],[15,115741,2900],{"className":115742},[146],"一维区间指示函数",[15,115745,115747,115761],{"className":115746,"translate":19},[18],[15,115748,115750],{"className":115749},[23],[25,115751,115752],{"xmlns":27},[29,115753,115754,115758],{},[32,115755,115756],{},[110,115757,2910],{"stretchy":112},[39,115759,115760],{"encoding":41},"\\}",[15,115762,115764],{"className":115763,"ariaHidden":46},[45],[15,115765,115767,115770],{"className":115766},[50],[15,115768],{"className":115769,"style":139},[54],[15,115771,2910],{"className":115772},[153],"）、Wasserstein（",[15,115775,115777,115794],{"className":115776,"translate":19},[18],[15,115778,115780],{"className":115779},[23],[25,115781,115782],{"xmlns":27},[29,115783,115784,115792],{},[32,115785,115786,115788,115790],{},[35,115787,36909],{"mathvariant":597},[110,115789,123],{},[110,115791,2900],{"stretchy":112},[39,115793,115712],{"encoding":41},[15,115795,115797,115815],{"className":115796,"ariaHidden":46},[45],[15,115798,115800,115803,115806,115809,115812],{"className":115799},[50],[15,115801],{"className":115802,"style":664},[54],[15,115804,36909],{"className":115805,"style":115423},[59,624],[15,115807],{"className":115808,"style":158},[157],[15,115810,123],{"className":115811},[162],[15,115813],{"className":115814,"style":158},[157],[15,115816,115818,115821],{"className":115817},[50],[15,115819],{"className":115820,"style":139},[54],[15,115822,2900],{"className":115823},[146],"1-Lipschitz 函数",[15,115826,115828,115841],{"className":115827,"translate":19},[18],[15,115829,115831],{"className":115830},[23],[25,115832,115833],{"xmlns":27},[29,115834,115835,115839],{},[32,115836,115837],{},[110,115838,2910],{"stretchy":112},[39,115840,115760],{"encoding":41},[15,115842,115844],{"className":115843,"ariaHidden":46},[45],[15,115845,115847,115850],{"className":115846},[50],[15,115848],{"className":115849,"style":139},[54],[15,115851,2910],{"className":115852},[153],"）、MMD（",[15,115855,115857,115874],{"className":115856,"translate":19},[18],[15,115858,115860],{"className":115859},[23],[25,115861,115862],{"xmlns":27},[29,115863,115864,115872],{},[32,115865,115866,115868,115870],{},[35,115867,36909],{"mathvariant":597},[110,115869,123],{},[110,115871,2900],{"stretchy":112},[39,115873,115712],{"encoding":41},[15,115875,115877,115895],{"className":115876,"ariaHidden":46},[45],[15,115878,115880,115883,115886,115889,115892],{"className":115879},[50],[15,115881],{"className":115882,"style":664},[54],[15,115884,36909],{"className":115885,"style":115423},[59,624],[15,115887],{"className":115888,"style":158},[157],[15,115890,123],{"className":115891},[162],[15,115893],{"className":115894,"style":158},[157],[15,115896,115898,115901],{"className":115897},[50],[15,115899],{"className":115900,"style":139},[54],[15,115902,2900],{"className":115903},[146],"RKHS 单位球",[15,115906,115908,115921],{"className":115907,"translate":19},[18],[15,115909,115911],{"className":115910},[23],[25,115912,115913],{"xmlns":27},[29,115914,115915,115919],{},[32,115916,115917],{},[110,115918,2910],{"stretchy":112},[39,115920,115760],{"encoding":41},[15,115922,115924],{"className":115923,"ariaHidden":46},[45],[15,115925,115927,115930],{"className":115926},[50],[15,115928],{"className":115929,"style":139},[54],[15,115931,2910],{"className":115932},[153],"）都是 IPM。上面对 f-散度的变分表示其实是「带惩罚的 IPM」：",[15,115935,115937],{"className":115936,"translate":19},[180],[15,115938,115940,116022],{"className":115939,"translate":19},[18],[15,115941,115943],{"className":115942},[23],[25,115944,115945],{"xmlns":27,"display":189},[29,115946,115947,116019],{},[32,115948,115949,115955,115957,115959,115961,115963,115965,115967,115977,115979,115985,115987,115989,115991,115993,115999,116001,116007,116009,116011,116013,116015,116017],{},[195,115950,115951,115953],{},[35,115952,199],{},[35,115954,37],{},[110,115956,113],{"stretchy":112},[35,115958,206],{},[35,115960,210],{"mathvariant":209},[35,115962,213],{},[110,115964,120],{"stretchy":112},[110,115966,123],{},[6018,115968,115969,115975],{},[32,115970,115971,115973],{},[35,115972,35458],{},[110,115974,4105],{},[35,115976,4160],{},[110,115978,2900],{"fence":112,"stretchy":46,"minsize":17443,"maxsize":17443},[195,115980,115981,115983],{},[35,115982,223],{"mathvariant":222},[35,115984,206],{},[110,115986,230],{"stretchy":112},[35,115988,4160],{},[110,115990,257],{"stretchy":112},[110,115992,4815],{},[195,115994,115995,115997],{},[35,115996,223],{"mathvariant":222},[35,115998,213],{},[110,116000,230],{"stretchy":112},[6816,116002,116003,116005],{},[35,116004,37],{},[110,116006,7575],{},[110,116008,113],{"stretchy":112},[35,116010,4160],{},[110,116012,120],{"stretchy":112},[110,116014,257],{"stretchy":112},[110,116016,2910],{"fence":112,"stretchy":46,"minsize":17443,"maxsize":17443},[110,116018,551],{"separator":46},[39,116020,116021],{"encoding":41},"D_f(P\\|Q)\n=\\sup_{g}\\Big\\{\\mathbb E_P[g]\n-\\mathbb E_Q[f^*(g)]\\Big\\},",[15,116023,116025,116095,116215],{"className":116024,"ariaHidden":46},[45],[15,116026,116028,116031,116071,116074,116077,116080,116083,116086,116089,116092],{"className":116027},[50],[15,116029],{"className":116030,"style":270},[54],[15,116032,116034,116037],{"className":116033},[59],[15,116035,199],{"className":116036,"style":277},[59,60],[15,116038,116040],{"className":116039},[281],[15,116041,116043,116063],{"className":116042},[285,286],[15,116044,116046,116060],{"className":116045},[290],[15,116047,116049],{"className":116048,"style":295},[294],[15,116050,116051,116054],{"style":298},[15,116052],{"className":116053,"style":303},[302],[15,116055,116057],{"className":116056},[307,308,309,310],[15,116058,37],{"className":116059,"style":61},[59,60,310],[15,116061,318],{"className":116062},[317],[15,116064,116066],{"className":116065},[290],[15,116067,116069],{"className":116068,"style":325},[294],[15,116070],{},[15,116072,113],{"className":116073},[146],[15,116075,206],{"className":116076,"style":334},[59,60],[15,116078,210],{"className":116079},[59],[15,116081,213],{"className":116082},[59,60],[15,116084,120],{"className":116085},[153],[15,116087],{"className":116088,"style":158},[157],[15,116090,123],{"className":116091},[162],[15,116093],{"className":116094,"style":158},[157],[15,116096,116098,116101,116148,116151,116157,116197,116200,116203,116206,116209,116212],{"className":116097},[50],[15,116099],{"className":116100,"style":99528},[54],[15,116102,116104],{"className":116103},[1486,13720],[15,116105,116107,116140],{"className":116106},[285,286],[15,116108,116110,116137],{"className":116109},[290],[15,116111,116113,116127],{"className":116112,"style":2347},[294],[15,116114,116115,116118],{"style":61227},[15,116116],{"className":116117,"style":461},[302],[15,116119,116121],{"className":116120},[307,308,309,310],[15,116122,116124],{"className":116123},[59,310],[15,116125,4160],{"className":116126,"style":981},[59,60,310],[15,116128,116129,116132],{"style":11867},[15,116130],{"className":116131,"style":461},[302],[15,116133,116134],{},[15,116135,35458],{"className":116136},[1486],[15,116138,318],{"className":116139},[317],[15,116141,116143],{"className":116142},[290],[15,116144,116146],{"className":116145,"style":61259},[294],[15,116147],{},[15,116149],{"className":116150,"style":406},[157],[15,116152,116154],{"className":116153},[59],[15,116155,2900],{"className":116156},[419,3722],[15,116158,116160,116163],{"className":116159},[59],[15,116161,223],{"className":116162},[59,366],[15,116164,116166],{"className":116165},[281],[15,116167,116169,116189],{"className":116168},[285,286],[15,116170,116172,116186],{"className":116171},[290],[15,116173,116175],{"className":116174,"style":379},[294],[15,116176,116177,116180],{"style":382},[15,116178],{"className":116179,"style":303},[302],[15,116181,116183],{"className":116182},[307,308,309,310],[15,116184,206],{"className":116185,"style":334},[59,60,310],[15,116187,318],{"className":116188},[317],[15,116190,116192],{"className":116191},[290],[15,116193,116195],{"className":116194,"style":7970},[294],[15,116196],{},[15,116198,230],{"className":116199},[146],[15,116201,4160],{"className":116202,"style":981},[59,60],[15,116204,257],{"className":116205},[153],[15,116207],{"className":116208,"style":2589},[157],[15,116210,4815],{"className":116211},[2593],[15,116213],{"className":116214,"style":2589},[157],[15,116216,116218,116221,116261,116264,116293,116296,116299,116302,116308],{"className":116217},[50],[15,116219],{"className":116220,"style":68154},[54],[15,116222,116224,116227],{"className":116223},[59],[15,116225,223],{"className":116226},[59,366],[15,116228,116230],{"className":116229},[281],[15,116231,116233,116253],{"className":116232},[285,286],[15,116234,116236,116250],{"className":116235},[290],[15,116237,116239],{"className":116238,"style":379},[294],[15,116240,116241,116244],{"style":382},[15,116242],{"className":116243,"style":303},[302],[15,116245,116247],{"className":116246},[307,308,309,310],[15,116248,213],{"className":116249},[59,60,310],[15,116251,318],{"className":116252},[317],[15,116254,116256],{"className":116255},[290],[15,116257,116259],{"className":116258,"style":325},[294],[15,116260],{},[15,116262,230],{"className":116263},[146],[15,116265,116267,116270],{"className":116266},[59],[15,116268,37],{"className":116269,"style":61},[59,60],[15,116271,116273],{"className":116272},[281],[15,116274,116276],{"className":116275},[285],[15,116277,116279],{"className":116278},[290],[15,116280,116282],{"className":116281,"style":8250},[294],[15,116283,116284,116287],{"style":8279},[15,116285],{"className":116286,"style":303},[302],[15,116288,116290],{"className":116289},[307,308,309,310],[15,116291,7575],{"className":116292},[2593,310],[15,116294,113],{"className":116295},[146],[15,116297,4160],{"className":116298,"style":981},[59,60],[15,116300,20416],{"className":116301},[153],[15,116303,116305],{"className":116304},[59],[15,116306,2910],{"className":116307},[419,3722],[15,116309,551],{"className":116310},[573],[11,116312,116313,116314,116388,116389,116431],{},"它的被估计量是 ",[15,116315,116317,116341],{"className":116316,"translate":19},[18],[15,116318,116320],{"className":116319},[23],[25,116321,116322],{"xmlns":27},[29,116323,116324,116338],{},[32,116325,116326,116332,116334,116336],{},[6816,116327,116328,116330],{},[35,116329,37],{},[110,116331,7575],{},[110,116333,113],{"stretchy":112},[35,116335,4160],{},[110,116337,120],{"stretchy":112},[39,116339,116340],{"encoding":41},"f^*(g)",[15,116342,116344],{"className":116343,"ariaHidden":46},[45],[15,116345,116347,116350,116379,116382,116385],{"className":116346},[50],[15,116348],{"className":116349,"style":139},[54],[15,116351,116353,116356],{"className":116352},[59],[15,116354,37],{"className":116355,"style":61},[59,60],[15,116357,116359],{"className":116358},[281],[15,116360,116362],{"className":116361},[285],[15,116363,116365],{"className":116364},[290],[15,116366,116368],{"className":116367,"style":8501},[294],[15,116369,116370,116373],{"style":6867},[15,116371],{"className":116372,"style":303},[302],[15,116374,116376],{"className":116375},[307,308,309,310],[15,116377,7575],{"className":116378},[2593,310],[15,116380,113],{"className":116381},[146],[15,116383,4160],{"className":116384,"style":981},[59,60],[15,116386,120],{"className":116387},[153],"，而不是 ",[15,116390,116392,116410],{"className":116391,"translate":19},[18],[15,116393,116395],{"className":116394},[23],[25,116396,116397],{"xmlns":27},[29,116398,116399,116407],{},[32,116400,116401,116403,116405],{},[35,116402,4810],{"mathvariant":209},[110,116404,3301],{"lspace":1249,"rspace":1249},[35,116406,4810],{"mathvariant":209},[39,116408,116409],{"encoding":41},"|{\\cdot}|",[15,116411,116413],{"className":116412,"ariaHidden":46},[45],[15,116414,116416,116419,116422,116428],{"className":116415},[50],[15,116417],{"className":116418,"style":139},[54],[15,116420,4810],{"className":116421},[59],[15,116423,116425],{"className":116424},[59],[15,116426,3301],{"className":116427},[59],[15,116429,4810],{"className":116430},[59],"，所以是「非对称 IPM」。",[11,116433,116434,116436],{},[998,116435,6495],{}," Wasserstein\u002FMMD 不是 f-散度，也不能写成 f-散度；它们的关键区别在于是否利用几何、是否在支撑分离时仍给梯度、以及是否满足数据处理不等式（Wasserstein 一般不满足数据处理不等式，因为传输方案可以随条件分布\"重新安排\"质量；MMD 由核特征映射的条件期望是收缩算子，沿核的具体结构可证其满足数据处理不等式）。",[2313,116438,116440],{"id":116439},"_63-支撑不重叠的极限","6.3 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e^{10^4}",[15,117306,117308,117320],{"className":117307,"ariaHidden":46},[45],[15,117309,117311,117314,117317],{"className":117310},[50],[15,117312],{"className":117313,"style":117206},[54],[15,117315,32985],{"className":117316},[162],[15,117318],{"className":117319,"style":158},[157],[15,117321,117323,117327],{"className":117322},[50],[15,117324],{"className":117325,"style":117326},[54],"height:0.9869em;",[15,117328,117330,117333],{"className":117329},[59],[15,117331,7569],{"className":117332},[59,60],[15,117334,117336],{"className":117335},[281],[15,117337,117339],{"className":117338},[285],[15,117340,117342],{"className":117341},[290],[15,117343,117345],{"className":117344,"style":117326},[294],[15,117346,117347,117350],{"style":6867},[15,117348],{"className":117349,"style":303},[302],[15,117351,117353],{"className":117352},[307,308,309,310],[15,117354,117356,117359],{"className":117355},[59,310],[15,117357,117],{"className":117358},[59,310],[15,117360,117362,117365],{"className":117361},[59,310],[15,117363,126],{"className":117364},[59,310],[15,117366,117368],{"className":117367},[281],[15,117369,117371],{"className":117370},[285],[15,117372,117374],{"className":117373},[290],[15,117375,117377],{"className":117376,"style":38538},[294],[15,117378,117379,117382],{"style":38541},[15,117380],{"className":117381,"style":30232},[302],[15,117383,117385],{"className":117384},[307,30236,9676,310],[15,117386,39015],{"className":117387},[59,310],[52621,117389,117390],{},[15,117391,117393,117409],{"className":117392,"translate":19},[18],[15,117394,117396],{"className":117395},[23],[25,117397,117398],{"xmlns":27},[29,117399,117400,117406],{},[32,117401,117402,117404],{},[110,117403,32985],{},[115,117405,2465],{},[39,117407,117408],{"encoding":41},"\\approx2",[15,117410,117412,117424],{"className":117411,"ariaHidden":46},[45],[15,117413,117415,117418,117421],{"className":117414},[50],[15,117416],{"className":117417,"style":117206},[54],[15,117419,32985],{"className":117420},[162],[15,117422],{"className":117423,"style":158},[157],[15,117425,117427,117430],{"className":117426},[50],[15,117428],{"className":117429,"style":172},[54],[15,117431,2465],{"className":117432},[59],[52621,117434,117435],{},[15,117436,117438,117454],{"className":117437,"translate":19},[18],[15,117439,117441],{"className":117440},[23],[25,117442,117443],{"xmlns":27},[29,117444,117445,117451],{},[32,117446,117447,117449],{},[110,117448,123],{},[115,117450,116979],{},[39,117452,117453],{"encoding":41},"=100",[15,117455,117457,117469],{"className":117456,"ariaHidden":46},[45],[15,117458,117460,117463,117466],{"className":117459},[50],[15,117461],{"className":117462,"style":11007},[54],[15,117464,123],{"className":117465},[162],[15,117467],{"className":117468,"style":158},[157],[15,117470,117472,117475],{"className":117471},[50],[15,117473],{"className":117474,"style":172},[54],[15,117476,116979],{"className":117477},[59],[11,117479,117480],{},"f-散度在第二行全部「饱和到下界\u002F上界」，而 Wasserstein 随距离线性增长——这正是词嵌入、最优传输、Wasserstein GAN 的动机。",[528,117482,117484],{"id":117483},"_7-机器学习应用","7. 机器学习应用",[11,117486,117487],{},"f-散度族之所以重要，是因为：",[6664,117489,117490,117496,117506,117541],{},[6667,117491,117492,117495],{},[998,117493,117494],{},"统一框架","：许多生成模型的目标函数可以写成 f-散度。",[6667,117497,117498,117501,117502,117505],{},[998,117499,117500],{},"变分估计","：通过凸共轭，可以用神经网络估计 f-散度，这就是 ",[998,117503,117504],{},"f-GAN"," 的核心思想。",[6667,117507,117508,117511,117512,117540],{},[998,117509,117510],{},"灵活选择","：不同 ",[15,117513,117515,117528],{"className":117514,"translate":19},[18],[15,117516,117518],{"className":117517},[23],[25,117519,117520],{"xmlns":27},[29,117521,117522,117526],{},[32,117523,117524],{},[35,117525,37],{},[39,117527,37],{"encoding":41},[15,117529,117531],{"className":117530,"ariaHidden":46},[45],[15,117532,117534,117537],{"className":117533},[50],[15,117535],{"className":117536,"style":55},[54],[15,117538,37],{"className":117539,"style":61},[59,60]," 对应不同训练性质。例如 JS 有界但可能梯度消失，KL 无界但敏感。",[6667,117542,117543,117546],{},[998,117544,117545],{},"理论分析","：数据处理不等式、凸性、对偶表示等为收敛性和泛化分析提供工具。",[11,117548,117549],{},"下面把这几条落到具体场景。",[2313,117551,117553],{"id":117552},"_71-f-gan","7.1 f-GAN",[11,117555,117556,117557,117685,117686,117714,117715,117743],{},"见第 5 节。核心是：",[15,117558,117560,117592],{"className":117559,"translate":19},[18],[15,117561,117563],{"className":117562},[23],[25,117564,117565],{"xmlns":27},[29,117566,117567,117589],{},[32,117568,117569,117579],{},[195,117570,117571,117577],{},[32,117572,117573,117575],{},[35,117574,51634],{},[110,117576,4105],{},[35,117578,100802],{},[195,117580,117581,117587],{},[32,117582,117583,117585],{},[35,117584,52413],{},[110,117586,4105],{},[35,117588,99904],{},[39,117590,117591],{"encoding":41},"\\min_\\theta\\max_\\omega",[15,117593,117595],{"className":117594,"ariaHidden":46},[45],[15,117596,117598,117602,117642,117645],{"className":117597},[50],[15,117599],{"className":117600,"style":117601},[54],"height:0.8179em;vertical-align:-0.15em;",[15,117603,117605,117608],{"className":117604},[1486],[15,117606,51634],{"className":117607},[1486],[15,117609,117611],{"className":117610},[281],[15,117612,117614,117634],{"className":117613},[285,286],[15,117615,117617,117631],{"className":117616},[290],[15,117618,117620],{"className":117619,"style":295},[294],[15,117621,117622,117625],{"style":52472},[15,117623],{"className":117624,"style":303},[302],[15,117626,117628],{"className":117627},[307,308,309,310],[15,117629,100802],{"className":117630,"style":277},[59,60,310],[15,117632,318],{"className":117633},[317],[15,117635,117637],{"className":117636},[290],[15,117638,117640],{"className":117639,"style":7970},[294],[15,117641],{},[15,117643],{"className":117644,"style":406},[157],[15,117646,117648,117651],{"className":117647},[1486],[15,117649,52413],{"className":117650},[1486],[15,117652,117654],{"className":117653},[281],[15,117655,117657,117677],{"className":117656},[285,286],[15,117658,117660,117674],{"className":117659},[290],[15,117661,117663],{"className":117662,"style":44955},[294],[15,117664,117665,117668],{"style":52472},[15,117666],{"className":117667,"style":303},[302],[15,117669,117671],{"className":117670},[307,308,309,310],[15,117672,99904],{"className":117673,"style":981},[59,60,310],[15,117675,318],{"className":117676},[317],[15,117678,117680],{"className":117679},[290],[15,117681,117683],{"className":117682,"style":7970},[294],[15,117684],{}," 的对抗目标 = 对 ",[15,117687,117689,117702],{"className":117688,"translate":19},[18],[15,117690,117692],{"className":117691},[23],[25,117693,117694],{"xmlns":27},[29,117695,117696,117700],{},[32,117697,117698],{},[35,117699,37],{},[39,117701,37],{"encoding":41},[15,117703,117705],{"className":117704,"ariaHidden":46},[45],[15,117706,117708,117711],{"className":117707},[50],[15,117709],{"className":117710,"style":55},[54],[15,117712,37],{"className":117713,"style":61},[59,60]," 做对偶后的 f-散度估计。选择 ",[15,117716,117718,117731],{"className":117717,"translate":19},[18],[15,117719,117721],{"className":117720},[23],[25,117722,117723],{"xmlns":27},[29,117724,117725,117729],{},[32,117726,117727],{},[35,117728,37],{},[39,117730,37],{"encoding":41},[15,117732,117734],{"className":117733,"ariaHidden":46},[45],[15,117735,117737,117740],{"className":117736},[50],[15,117738],{"className":117739,"style":55},[54],[15,117741,37],{"className":117742,"style":61},[59,60]," 就是选择「用哪种散度训练生成器」，这直接决定模式崩塌、梯度消失与训练稳定性的权衡。",[2313,117745,117747],{"id":117746},"_72-变分推断与-elbo","7.2 变分推断与 ELBO",[11,117749,117750],{},"变分推断里最大化证据下界",[15,117752,117754],{"className":117753,"translate":19},[180],[15,117755,117757,117916],{"className":117756,"translate":19},[18],[15,117758,117760],{"className":117759},[23],[25,117761,117762],{"xmlns":27,"display":189},[29,117763,117764,117913],{},[1245,117765,117766,117843],{"rowspacing":1247,"columnalign":1248,"columnspacing":1249},[1251,117767,117768,117782],{},[1254,117769,117770],{},[1257,117771,117772],{"scriptlevel":126,"displaystyle":46},[32,117773,117774,117776,117778,117780],{},[35,117775,21919],{"mathvariant":597},[110,117777,113],{"stretchy":112},[35,117779,944],{},[110,117781,120],{"stretchy":112},[1254,117783,117784],{},[1257,117785,117786],{"scriptlevel":126,"displaystyle":46},[32,117787,117788,117790,117792,117798,117800,117802,117804,117806,117808,117810,117812,117815,117817,117819,117821,117827,117829,117831,117833,117835,117837,117839,117841],{},[32,117789],{},[110,117791,123],{},[195,117793,117794,117796],{},[35,117795,223],{"mathvariant":222},[35,117797,944],{},[110,117799,230],{"stretchy":112},[35,117801,4102],{},[110,117803,4105],{},[35,117805,11],{},[110,117807,113],{"stretchy":112},[35,117809,774],{},[110,117811,551],{"separator":46},[35,117813,117814],{},"z",[110,117816,120],{"stretchy":112},[110,117818,257],{"stretchy":112},[110,117820,4815],{},[195,117822,117823,117825],{},[35,117824,223],{"mathvariant":222},[35,117826,944],{},[110,117828,230],{"stretchy":112},[35,117830,4102],{},[110,117832,4105],{},[35,117834,944],{},[110,117836,113],{"stretchy":112},[35,117838,117814],{},[110,117840,120],{"stretchy":112},[110,117842,257],{"stretchy":112},[1251,117844,117845,117851],{},[1254,117846,117847],{},[1257,117848,117849],{"scriptlevel":126,"displaystyle":46},[32,117850],{},[1254,117852,117853],{},[1257,117854,117855],{"scriptlevel":126,"displaystyle":46},[32,117856,117857,117859,117861,117863,117865,117867,117869,117871,117873,117875,117881,117883,117885,117887,117889,117891,117893,117895,117897,117899,117901,117903,117905,117907,117909,117911],{},[32,117858],{},[110,117860,123],{},[35,117862,4102],{},[110,117864,4105],{},[35,117866,11],{},[110,117868,113],{"stretchy":112},[35,117870,774],{},[110,117872,120],{"stretchy":112},[110,117874,4815],{},[32,117876,117877,117879],{},[35,117878,10698],{"mathvariant":209},[35,117880,21919],{"mathvariant":209},[110,117882,113],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[35,117884,944],{},[110,117886,113],{"stretchy":112},[35,117888,117814],{},[110,117890,120],{"stretchy":112},[1299,117892,1301],{},[35,117894,210],{"mathvariant":209},[1299,117896,1301],{},[35,117898,11],{},[110,117900,113],{"stretchy":112},[35,117902,117814],{},[35,117904,4810],{"mathvariant":209},[35,117906,774],{},[110,117908,120],{"stretchy":112},[110,117910,120],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[110,117912,551],{"separator":46},[39,117914,117915],{"encoding":41},"\\begin{aligned}\n\\mathcal L(q)\n&=\\mathbb E_q[\\log p(x,z)]\n-\\mathbb E_q[\\log q(z)]\\\\\n&=\\log p(x)\n-\\mathrm{KL}\\big(q(z)\\,\\|\\,p(z|x)\\big),\n\\end{aligned}",[15,117917,117919],{"className":117918,"ariaHidden":46},[45],[15,117920,117922,117925],{"className":117921},[50],[15,117923],{"className":117924,"style":14590},[54],[15,117926,117928],{"className":117927},[59],[15,117929,117931,117983],{"className":117930},[1245],[15,117932,117934],{"className":117933},[1362],[15,117935,117937,117975],{"className":117936},[285,286],[15,117938,117940,117972],{"className":117939},[290],[15,117941,117943,117964],{"className":117942,"style":14609},[294],[15,117944,117946,117949],{"style":117945},"top:-3.765em;",[15,117947],{"className":117948,"style":461},[302],[15,117950,117952,117955,117958,117961],{"className":117951},[59],[15,117953,21919],{"className":117954},[59,624],[15,117956,113],{"className":117957},[146],[15,117959,944],{"className":117960,"style":981},[59,60],[15,117962,120],{"className":117963},[153],[15,117965,117966,117969],{"style":14997},[15,117967],{"className":117968,"style":461},[302],[15,117970],{"className":117971},[59],[15,117973,318],{"className":117974},[317],[15,117976,117978],{"className":117977},[290],[15,117979,117981],{"className":117980,"style":14638},[294],[15,117982],{},[15,117984,117986],{"className":117985},[1453],[15,117987,117989,118272],{"className":117988},[285,286],[15,117990,117992,118269],{"className":117991},[290],[15,117993,117995,118160],{"className":117994,"style":14609},[294],[15,117996,117997,118000],{"style":117945},[15,117998],{"className":117999,"style":461},[302],[15,118001,118003,118006,118009,118012,118015,118055,118058,118063,118066,118069,118072,118075,118078,118081,118085,118088,118091,118094,118097,118137,118140,118145,118148,118151,118154,118157],{"className":118002},[59],[15,118004],{"className":118005},[59],[15,118007],{"className":118008,"style":158},[157],[15,118010,123],{"className":118011},[162],[15,118013],{"className":118014,"style":158},[157],[15,118016,118018,118021],{"className":118017},[59],[15,118019,223],{"className":118020},[59,366],[15,118022,118024],{"className":118023},[281],[15,118025,118027,118047],{"className":118026},[285,286],[15,118028,118030,118044],{"className":118029},[290],[15,118031,118033],{"className":118032,"style":44955},[294],[15,118034,118035,118038],{"style":382},[15,118036],{"className":118037,"style":303},[302],[15,118039,118041],{"className":118040},[307,308,309,310],[15,118042,944],{"className":118043,"style":981},[59,60,310],[15,118045,318],{"className":118046},[317],[15,118048,118050],{"className":118049},[290],[15,118051,118053],{"className":118052,"style":325},[294],[15,118054],{},[15,118056,230],{"className":118057},[146],[15,118059,4156,118061],{"className":118060},[1486],[15,118062,4160],{"style":4159},[15,118064],{"className":118065,"style":406},[157],[15,118067,11],{"className":118068},[59,60],[15,118070,113],{"className":118071},[146],[15,118073,774],{"className":118074},[59,60],[15,118076,551],{"className":118077},[573],[15,118079],{"className":118080,"style":406},[157],[15,118082,117814],{"className":118083,"style":118084},[59,60],"margin-right:0.044em;",[15,118086,20416],{"className":118087},[153],[15,118089],{"className":118090,"style":2589},[157],[15,118092,4815],{"className":118093},[2593],[15,118095],{"className":118096,"style":2589},[157],[15,118098,118100,118103],{"className":118099},[59],[15,118101,223],{"className":118102},[59,366],[15,118104,118106],{"className":118105},[281],[15,118107,118109,118129],{"className":118108},[285,286],[15,118110,118112,118126],{"className":118111},[290],[15,118113,118115],{"className":118114,"style":44955},[294],[15,118116,118117,118120],{"style":382},[15,118118],{"className":118119,"style":303},[302],[15,118121,118123],{"className":118122},[307,308,309,310],[15,118124,944],{"className":118125,"style":981},[59,60,310],[15,118127,318],{"className":118128},[317],[15,118130,118132],{"className":118131},[290],[15,118133,118135],{"className":118134,"style":325},[294],[15,118136],{},[15,118138,230],{"className":118139},[146],[15,118141,4156,118143],{"className":118142},[1486],[15,118144,4160],{"style":4159},[15,118146],{"className":118147,"style":406},[157],[15,118149,944],{"className":118150,"style":981},[59,60],[15,118152,113],{"className":118153},[146],[15,118155,117814],{"className":118156,"style":118084},[59,60],[15,118158,20416],{"className":118159},[153],[15,118161,118162,118165],{"style":14997},[15,118163],{"className":118164,"style":461},[302],[15,118166,118168,118171,118174,118177,118180,118185,118188,118191,118194,118197,118200,118203,118206,118209,118215,118221,118224,118227,118230,118233,118236,118239,118242,118245,118248,118251,118254,118257,118260,118266],{"className":118167},[59],[15,118169],{"className":118170},[59],[15,118172],{"className":118173,"style":158},[157],[15,118175,123],{"className":118176},[162],[15,118178],{"className":118179,"style":158},[157],[15,118181,4156,118183],{"className":118182},[1486],[15,118184,4160],{"style":4159},[15,11818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是 f-散度（",[15,118345,118347,118371],{"className":118346,"translate":19},[18],[15,118348,118350],{"className":118349},[23],[25,118351,118352],{"xmlns":27},[29,118353,118354,118368],{},[32,118355,118356,118358,118360,118362,118364,118366],{},[35,118357,37],{},[110,118359,123],{},[35,118361,769],{},[35,118363,4102],{},[110,118365,4105],{},[35,118367,769],{},[39,118369,118370],{"encoding":41},"f=t\\log t",[15,118372,118374,118392],{"className":118373,"ariaHidden":46},[45],[15,118375,118377,118380,118383,118386,118389],{"className":118376},[50],[15,118378],{"className":118379,"style":55},[54],[15,118381,37],{"className":118382,"style":61},[59,60],[15,118384],{"className":118385,"style":158},[157],[15,118387,123],{"className":118388},[162],[15,118390],{"className":118391,"style":158},[157],[15,118393,118395,118398,118401,118404,118409,118412],{"className":118394},[50],[15,118396],{"className":118397,"style":55},[54],[15,118399,769],{"className":118400},[59,60],[15,118402],{"className":118403,"style":406},[157],[15,118405,4156,118407],{"className":118406},[1486],[15,118408,4160],{"style":4159},[15,118410],{"className":118411,"style":406},[157],[15,118413,769],{"className":118414},[59,60],"）。反过来，任何以 ",[15,118417,118419,118451],{"className":118418,"translate":19},[18],[15,118420,118422],{"className":118421},[23],[25,118423,118424],{"xmlns":27},[29,118425,118426,118448],{},[32,118427,118428,118434,118436,118438,118440,118442,118444,118446],{},[195,118429,118430,118432],{},[35,118431,223],{"mathvariant":222},[35,118433,944],{},[110,118435,230],{"stretchy":112},[35,118437,4102],{},[110,118439,4105],{},[110,118441,113],{"stretchy":112},[110,118443,3301],{},[110,118445,120],{"stretchy":112},[110,118447,257],{"stretchy":112},[39,118449,118450],{"encoding":41},"\\mathbb E_q[\\log(\\cdot)]",[15,118452,118454],{"className":118453,"ariaHidden":46},[45],[15,118455,118457,118460,118500,118503,118508,118511,118514],{"className":118456},[50],[15,118458],{"className":118459,"style":270},[54],[15,118461,118463,118466],{"className":118462},[59],[15,118464,223],{"className":118465},[59,366],[15,118467,118469],{"className":118468},[281],[15,118470,118472,118492],{"className":118471},[285,286],[15,118473,118475,118489],{"className":118474},[290],[15,118476,118478],{"className":118477,"style":44955},[294],[15,118479,118480,118483],{"style":382},[15,118481],{"className":118482,"style":303},[302],[15,118484,118486],{"className":118485},[307,308,309,310],[15,118487,944],{"className":118488,"style":981},[59,60,310],[15,118490,318],{"className":118491},[317],[15,118493,118495],{"className":118494},[290],[15,118496,118498],{"className":118497,"style":325},[294],[15,118499],{},[15,118501,230],{"className":118502},[146],[15,118504,4156,118506],{"className":118505},[1486],[15,118507,4160],{"style":4159},[15,118509,113],{"className":118510},[146],[15,118512,3301],{"className":118513},[59],[15,118515,20416],{"className":118516},[153]," 为组成部分的目标都可以用 f-散度的语言重写；",[15,118519,118521,118534],{"className":118520,"translate":19},[18],[15,118522,118524],{"className":118523},[23],[25,118525,118526],{"xmlns":27},[29,118527,118528,118532],{},[32,118529,118530],{},[35,118531,44780],{},[39,118533,49116],{"encoding":41},[15,118535,118537],{"className":118536,"ariaHidden":46},[45],[15,118538,118540,118543],{"className":118539},[50],[15,118541],{"className":118542,"style":2347},[54],[15,118544,44780],{"className":118545,"style":44810},[59,60],"-VI、",[15,118548,118550,118567],{"className":118549,"translate":19},[18],[15,118551,118553],{"className":118552},[23],[25,118554,118555],{"xmlns":27},[29,118556,118557,118565],{},[32,118558,118559],{},[6816,118560,118561,118563],{},[35,118562,37475],{},[115,118564,2465],{},[39,118566,40420],{"encoding":41},[15,118568,118570],{"className":118569,"ariaHidden":46},[45],[15,118571,118573,118576],{"className":118572},[50],[15,118574],{"className":118575,"style":40430},[54],[15,118577,118579,118582],{"className":118578},[59],[15,118580,37475],{"className":118581},[59,60],[15,118583,118585],{"className":118584},[281],[15,118586,118588],{"className":118587},[285],[15,118589,118591],{"className":118590},[290],[15,118592,118594],{"className":118593,"style":20054},[294],[15,118595,118596,118599],{"style":6867},[15,118597],{"className":118598,"style":303},[302],[15,118600,118602],{"className":118601},[307,308,309,310],[15,118603,2465],{"className":118604},[59,310],"-VI 等就换用别的 ",[15,118607,118609,118622],{"className":118608,"translate":19},[18],[15,118610,118612],{"className":118611},[23],[25,118613,118614],{"xmlns":27},[29,118615,118616,118620],{},[32,118617,118618],{},[35,118619,37],{},[39,118621,37],{"encoding":41},[15,118623,118625],{"className":118624,"ariaHidden":46},[45],[15,118626,118628,118631],{"className":118627},[50],[15,118629],{"className":118630,"style":55},[54],[15,118632,37],{"className":118633,"style":61},[59,60],"，在 3.1 节所说的「零回避」（mass-covering）与「零强制」（mode-seeking）两种极端行为之间取舍。",[2313,118636,118638],{"id":118637},"_73-互信息估计","7.3 互信息估计",[11,118640,118641],{},"互信息可写成",[15,118643,118645],{"className":118644,"translate":19},[180],[15,118646,118648,118763],{"className":118647,"translate":19},[18],[15,118649,118651],{"className":118650},[23],[25,118652,118653],{"xmlns":27,"display":189},[29,118654,118655,118760],{},[32,118656,118657,118659,118661,118663,118665,118667,118669,118671,118677,118679,118689,118691,118693,118695,118701,118704,118710,118712,118714,118728,118730,118740,118742,118748,118750,118756,118758],{},[35,118658,33894],{},[110,118660,113],{"stretchy":112},[35,118662,598],{},[110,118664,33901],{"separator":46},[35,118666,5403],{},[110,118668,120],{"stretchy":112},[110,118670,123],{},[32,118672,118673,118675],{},[35,118674,10698],{"mathvariant":209},[35,118676,21919],{"mathvariant":209},[110,118678,113],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[195,118680,118681,118683],{},[35,118682,206],{},[32,118684,118685,118687],{},[35,118686,598],{},[35,118688,5403],{},[1299,118690,1301],{},[35,118692,210],{"mathvariant":209},[1299,118694,1301],{},[195,118696,118697,118699],{},[35,118698,206],{},[35,118700,598],{},[110,118702,118703],{},"⊗",[195,118705,118706,118708],{},[35,118707,206],{},[35,118709,5403],{},[110,118711,120],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[110,118713,123],{},[195,118715,118716,118718],{},[35,118717,199],{},[32,118719,118720,118722,118724,118726],{},[35,118721,769],{},[35,118723,4102],{},[110,118725,4105],{},[35,118727,769],{},[110,118729,113],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[195,118731,118732,118734],{},[35,118733,206],{},[32,118735,118736,118738],{},[35,118737,598],{},[35,118739,5403],{},[35,118741,210],{"mathvariant":209},[195,118743,118744,118746],{},[35,118745,206],{},[35,118747,598],{},[110,118749,118703],{},[195,118751,118752,118754],{},[35,118753,206],{},[35,118755,5403],{},[110,118757,120],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[110,118759,551],{"separator":46},[39,118761,118762],{"encoding":41},"I(X;Y)\n=\\mathrm{KL}\\big(P_{XY}\\,\\|\\,P_X\\otimes P_Y\\big)\n=D_{t\\log t}\\big(P_{XY}\\|P_X\\otimes P_Y\\big),",[15,118764,118766,118802,118924,118985,119159],{"className":118765,"ariaHidden":46},[45],[15,118767,118769,118772,118775,118778,118781,118784,118787,118790,118793,118796,118799],{"className":118768},[50],[15,118770],{"className":118771,"style":139},[54],[15,118773,33894],{"className":118774,"style":8889},[59,60],[15,118776,113],{"className":118777},[146],[15,118779,598],{"className":118780,"style":8889},[59,60],[15,118782,33901],{"className":118783},[573],[15,118785],{"className":118786,"style":406},[157],[15,118788,5403],{"className":118789,"style":2589},[59,60],[15,118791,120],{"className":118792},[153],[15,118794],{"className":118795,"style":158},[157],[15,118797,123],{"className":118798},[162],[15,118800],{"className":118801,"style":158},[157],[15,118803,118805,118808,118814,118820,118866,118869,118872,118875,118915,118918,118921],{"className":118804},[50],[15,118806],{"className":118807,"style":20589},[54],[15,118809,118811],{"className":118810},[59],[15,118812,32689],{"className":118813},[59,32688],[15,118815,118817],{"className":118816},[59],[15,118818,113],{"className":118819},[419,9676],[15,118821,118823,118826],{"className":118822},[59],[15,118824,206],{"className":118825,"style":334},[59,60],[15,118827,118829],{"className":118828},[281],[15,118830,118832,118858],{"className":118831},[285,286],[15,118833,118835,118855],{"className":118834},[290],[15,118836,118838],{"className":118837,"style":379},[294],[15,118839,118840,118843],{"style":11140},[15,118841],{"className":118842,"style":303},[302],[15,118844,118846],{"className":118845},[307,308,309,310],[15,118847,118849,118852],{"className":118848},[59,310],[15,118850,598],{"className":118851,"style":8889},[59,60,310],[15,118853,5403],{"className":118854,"style":2589},[59,60,310],[15,118856,318],{"className":118857},[317],[15,118859,118861],{"className":118860},[290],[15,118862,118864],{"className":118863,"style":7970},[294],[15,118865],{},[15,118867],{"className":118868,"style":406},[157],[15,118870,210],{"className":118871},[59],[15,118873],{"className":118874,"style":406},[157],[15,118876,118878,118881],{"className":118877},[59],[15,118879,206],{"className":118880,"style":334},[59,60],[15,118882,118884],{"className":118883},[281],[15,118885,118887,118907],{"className":118886},[285,286],[15,118888,118890,118904],{"className":118889},[290],[15,118891,118893],{"className":118892,"style":379},[294],[15,118894,118895,118898],{"style":11140},[15,118896],{"className":118897,"style":303},[302],[15,118899,118901],{"className":118900},[307,308,309,310],[15,118902,598],{"className":118903,"style":8889},[59,60,310],[15,118905,318],{"className":118906},[317],[15,118908,118910],{"className":118909},[290],[15,118911,118913],{"className":118912,"style":7970},[294],[15,118914],{},[15,118916],{"className":118917,"style":2589},[157],[15,118919,118703],{"className":118920},[2593],[15,118922],{"className":118923,"style":2589},[157],[15,118925,118927,118930,118970,118976,118979,118982],{"className":118926},[50],[15,118928],{"className":118929,"style":20589},[54],[15,118931,118933,118936],{"className":118932},[59],[15,118934,206],{"className":118935,"style":334},[59,60],[15,118937,118939],{"className":118938},[281],[15,118940,118942,118962],{"className":118941},[285,286],[15,118943,118945,118959],{"className":118944},[290],[15,118946,118948],{"className":118947,"style":379},[294],[15,118949,118950,118953],{"style":11140},[15,118951],{"className":118952,"style":303},[302],[15,118954,118956],{"className":118955},[307,308,309,310],[15,118957,5403],{"className":118958,"style":2589},[59,60,310],[15,118960,318],{"className":118961},[317],[15,118963,118965],{"className":118964},[290],[15,118966,118968],{"className":118967,"style":7970},[294],[15,118969],{},[15,118971,118973],{"className":118972},[59],[15,118974,120],{"className":118975},[419,9676],[15,118977],{"className":118978,"style":158},[157],[15,118980,123],{"className":118981},[162],[15,118983],{"className":118984,"style":158},[157],[15,118986,118988,118991,119055,119061,119107,119110,119150,119153,119156],{"className":118987},[50],[15,118989],{"className":118990,"style":20589},[54],[15,118992,118994,118997],{"className":118993},[59],[15,118995,199],{"className":118996,"style":277},[59,60],[15,118998,119000],{"className":118999},[281],[15,119001,119003,119047],{"className":119002},[285,286],[15,119004,119006,119044],{"className":119005},[290],[15,119007,119009],{"className":119008,"style":295},[294],[15,119010,119011,119014],{"style":298},[15,119012],{"className":119013,"style":303},[302],[15,119015,119017],{"className":119016},[307,308,309,310],[15,119018,119020,119023,119026,119038,119041],{"className":119019},[59,310],[15,119021,769],{"className":119022},[59,60,310],[15,119024],{"className":119025,"style":42392},[157,310],[15,119027,119029,119032,119035],{"className":119028},[1486,310],[15,119030,47684],{"className":119031},[310],[15,119033,47688],{"className":119034},[310],[15,119036,4160],{"className":119037,"style":4159},[310],[15,119039],{"className":119040,"style":42392},[157,310],[15,119042,769],{"className":119043},[59,60,310],[15,119045,318],{"className":119046},[317],[15,119048,119050],{"className":119049},[290],[15,119051,119053],{"className":119052,"style":325},[294],[15,119054],{},[15,119056,119058],{"className":119057},[59],[15,119059,113],{"className":119060},[419,9676],[15,119062,119064,119067],{"className":119063},[59],[15,119065,206],{"className":119066,"style":334},[59,60],[15,119068,119070],{"className":119069},[281],[15,119071,119073,119099],{"className":119072},[285,286],[15,119074,119076,119096],{"className":119075},[290],[15,119077,119079],{"className":119078,"style":379},[294],[15,119080,119081,119084],{"style":11140},[15,119082],{"className":119083,"style":303},[302],[15,119085,119087],{"className":119086},[307,308,309,310],[15,119088,119090,119093],{"className":119089},[59,310],[15,119091,598],{"className":119092,"style":8889},[59,60,310],[15,119094,5403],{"className":119095,"style":2589},[59,60,310],[15,119097,318],{"className":119098},[317],[15,119100,119102],{"className":119101},[290],[15,119103,119105],{"className":119104,"style":7970},[294],[15,119106],{},[15,119108,210],{"className":119109},[59],[15,119111,119113,119116],{"className":119112},[59],[15,119114,206],{"className":119115,"style":334},[59,60],[15,119117,119119],{"className":119118},[281],[15,119120,119122,119142],{"className":119121},[285,286],[15,119123,119125,119139],{"className":119124},[290],[15,119126,119128],{"className":119127,"style":379},[294],[15,119129,119130,119133],{"style":11140},[15,119131],{"className":119132,"style":303},[302],[15,119134,119136],{"className":119135},[307,308,309,310],[15,119137,598],{"className":119138,"style":8889},[59,60,310],[15,119140,318],{"className":119141},[317],[15,119143,119145],{"className":119144},[290],[15,119146,119148],{"className":119147,"style":7970},[294],[15,119149],{},[15,119151],{"className":119152,"style":2589},[157],[15,119154,118703],{"className":119155},[2593],[15,119157],{"className":119158,"style":2589},[157],[15,119160,119162,119165,119205,119211],{"className":119161},[50],[15,119163],{"className":119164,"style":20589},[54],[15,119166,119168,119171],{"className":119167},[59],[15,119169,206],{"className":119170,"style":334},[59,60],[15,119172,119174],{"className":119173},[281],[15,119175,119177,119197],{"className":119176},[285,286],[15,119178,119180,119194],{"className":119179},[290],[15,119181,119183],{"className":119182,"style":379},[294],[15,119184,119185,119188],{"style":11140},[15,119186],{"className":119187,"style":303},[302],[15,119189,119191],{"className":119190},[307,308,309,310],[15,119192,5403],{"className":119193,"style":2589},[59,60,310],[15,119195,318],{"className":119196},[317],[15,119198,119200],{"className":119199},[290],[15,119201,119203],{"className":119202,"style":7970},[294],[15,119204],{},[15,119206,119208],{"className":119207},[59],[15,119209,120],{"className":119210},[419,9676],[15,119212,551],{"className":119213},[573],[11,119215,119216,119217,119245,119246,119274,119275,119333],{},"因此它是 f-散度的特例。MINE 用 Donsker–Varadhan 对偶估计它，f-MINE 则对一般 ",[15,119218,119220,119233],{"className":119219,"translate":19},[18],[15,119221,119223],{"className":119222},[23],[25,119224,119225],{"xmlns":27},[29,119226,119227,119231],{},[32,119228,119229],{},[35,119230,37],{},[39,119232,37],{"encoding":41},[15,119234,119236],{"className":119235,"ariaHidden":46},[45],[15,119237,119239,119242],{"className":119238},[50],[15,119240],{"className":119241,"style":55},[54],[15,119243,37],{"className":119244,"style":61},[59,60]," 用 4.4 节的对偶；不同 ",[15,119247,119249,119262],{"className":119248,"translate":19},[18],[15,119250,119252],{"className":119251},[23],[25,119253,119254],{"xmlns":27},[29,119255,119256,119260],{},[32,119257,119258],{},[35,119259,37],{},[39,119261,37],{"encoding":41},[15,119263,119265],{"className":119264,"ariaHidden":46},[45],[15,119266,119268,119271],{"className":119267},[50],[15,119269],{"className":119270,"style":55},[54],[15,119272,37],{"className":119273,"style":61},[59,60],"（KL、JS、",[15,119276,119278,119295],{"className":119277,"translate":19},[18],[15,119279,119281],{"className":119280},[23],[25,119282,119283],{"xmlns":27},[29,119284,119285,119293],{},[32,119286,119287],{},[6816,119288,119289,119291],{},[35,119290,37475],{},[115,119292,2465],{},[39,119294,40420],{"encoding":41},[15,119296,119298],{"className":119297,"ariaHidden":46},[45],[15,119299,119301,119304],{"className":119300},[50],[15,119302],{"className":119303,"style":40430},[54],[15,119305,119307,119310],{"className":119306},[59],[15,119308,37475],{"className":119309},[59,60],[15,119311,119313],{"className":119312},[281],[15,119314,119316],{"className":119315},[285],[15,119317,119319],{"className":119318},[290],[15,119320,119322],{"className":119321,"style":20054},[294],[15,119323,119324,119327],{"style":6867},[15,119325],{"className":119326,"style":303},[302],[15,119328,119330],{"className":119329},[307,308,309,310],[15,119331,2465],{"className":119332},[59,310],"）给出不同的「MI 下界」和不同的方差—偏差折衷。对比式学习的 InfoNCE 也是 KL 下界的一种变体。",[2313,119335,119337],{"id":119336},"_74-表示学习","7.4 表示学习",[11,119339,119340,119341,119385],{},"任何表示 ",[15,119342,119344,119364],{"className":119343,"translate":19},[18],[15,119345,119347],{"className":119346},[23],[25,119348,119349],{"xmlns":27},[29,119350,119351,119361],{},[32,119352,119353,119355,119357,119359],{},[35,119354,5394],{},[110,119356,113],{"stretchy":112},[35,119358,598],{},[110,119360,120],{"stretchy":112},[39,119362,119363],{"encoding":41},"T(X)",[15,119365,119367],{"className":119366,"ariaHidden":46},[45],[15,119368,119370,119373,119376,119379,119382],{"className":119369},[50],[15,119371],{"className":119372,"style":139},[54],[15,119374,5394],{"className":119375,"style":334},[59,60],[15,119377,113],{"className":119378},[146],[15,119380,598],{"className":119381,"style":8889},[59,60],[15,119383,120],{"className":119384},[153]," 都满足（4.5 节）",[15,119387,119389],{"className":119388,"translate":19},[180],[15,119390,119392,119472],{"className":119391,"translate":19},[18],[15,119393,119395],{"className":119394},[23],[25,119396,119397],{"xmlns":27,"display":189},[29,119398,119399,119469],{},[32,119400,119401,119407,119409,119423,119425,119439,119441,119443,119449,119451,119457,119459,119465,119467],{},[195,119402,119403,119405],{},[35,119404,199],{},[35,119406,37],{},[110,119408,113],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[195,119410,119411,119413],{},[35,119412,206],{},[32,119414,119415,119417,119419,119421],{},[35,119416,5394],{},[110,119418,113],{"stretchy":112},[35,119420,598],{},[110,119422,120],{"stretchy":112},[35,119424,210],{"mathvariant":209},[195,119426,119427,119429],{},[35,119428,213],{},[32,119430,119431,119433,119435,119437],{},[35,119432,5394],{},[110,119434,113],{"stretchy":112},[35,119436,598],{},[110,119438,120],{"stretchy":112},[110,119440,120],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[110,119442,9954],{},[195,119444,119445,119447],{},[35,119446,199],{},[35,119448,37],{},[110,119450,113],{"stretchy":112},[195,119452,119453,119455],{},[35,119454,206],{},[35,119456,598],{},[35,119458,210],{"mathvariant":209},[195,119460,119461,119463],{},[35,119462,213],{},[35,119464,598],{},[110,119466,120],{"stretchy":112},[110,119468,551],{"separator":46},[39,119470,119471],{"encoding":41},"D_f\\big(P_{T(X)}\\|Q_{T(X)}\\big)\n\\le D_f(P_X\\|Q_X),",[15,119473,119475,119651],{"className":119474,"ariaHidden":46},[45],[15,119476,119478,119482,119522,119528,119581,119584,119636,119642,119645,119648],{"className":119477},[50],[15,119479],{"className":119480,"style":119481},[54],"height:1.2052em;vertical-align:-0.3552em;",[15,119483,119485,119488],{"className":119484},[59],[15,119486,199],{"className":119487,"style":277},[59,60],[15,119489,119491],{"className":119490},[281],[15,119492,119494,119514],{"className":119493},[285,286],[15,119495,119497,119511],{"className":119496},[290],[15,119498,119500],{"className":119499,"style":295},[294],[15,119501,119502,119505],{"style":298},[15,119503],{"className":119504,"style":303},[302],[15,119506,119508],{"className":119507},[307,308,309,310],[15,119509,37],{"className":119510,"style":61},[59,60,310],[15,119512,318],{"className":119513},[317],[15,119515,119517],{"className":119516},[290],[15,119518,119520],{"className":119519,"style":325},[294],[15,119521],{},[15,119523,119525],{"className":119524},[59],[15,119526,113],{"className":119527},[419,9676],[15,119529,119531,119534],{"className":119530},[59],[15,119532,206],{"className":119533,"style":334},[59,60],[15,119535,119537],{"className":119536},[281],[15,119538,119540,119573],{"className":119539},[285,286],[15,119541,119543,119570],{"className":119542},[290],[15,119544,119546],{"className":119545,"style":6100},[294],[15,119547,119549,119552],{"style":119548},"top:-2.5198em;margin-left:-0.1389em;margin-right:0.05em;",[15,119550],{"className":119551,"style":303},[302],[15,119553,119555],{"className":119554},[307,308,309,310],[15,119556,119558,119561,119564,119567],{"className":119557},[59,310],[15,119559,5394],{"className":119560,"style":334},[59,60,310],[15,119562,113],{"className":119563},[146,310],[15,119565,598],{"className":119566,"style":8889},[59,60,310],[15,119568,120],{"className":119569},[153,310],[15,119571,318],{"className":119572},[317],[15,119574,119576],{"className":119575},[290],[15,119577,119579],{"className":119578,"style":6146},[294],[15,119580],{},[15,119582,210],{"className":119583},[59],[15,119585,119587,119590],{"className":119586},[59],[15,119588,213],{"className":119589},[59,60],[15,119591,119593],{"className":119592},[281],[15,119594,119596,119628],{"className":119595},[285,286],[15,119597,119599,119625],{"className":119598},[290],[15,119600,119602],{"className":119601,"style":6100},[294],[15,119603,119604,119607],{"style":75612},[15,119605],{"className":119606,"style":303},[302],[15,119608,119610],{"className":119609},[307,308,309,310],[15,119611,119613,119616,119619,119622],{"className":119612},[59,310],[15,119614,5394],{"className":119615,"style":334},[59,60,310],[15,119617,113],{"className":119618},[146,310],[15,119620,598],{"className":119621,"style":8889},[59,60,310],[15,119623,120],{"className":119624},[153,310],[15,119626,318],{"className":119627},[317],[15,119629,119631],{"className":119630},[290],[15,119632,119634],{"className":119633,"style":6146},[294],[15,119635],{},[15,119637,119639],{"className":119638},[59],[15,119640,120],{"className":119641},[419,9676],[15,119643],{"className":119644,"style":158},[157],[15,119646,9954],{"className":119647},[162],[15,119649],{"className":119650,"style":158},[157],[15,119652,119654,119657,119697,119700,119740,119743,119783,119786],{"className":119653},[50],[15,119655],{"className":119656,"style":270},[54],[15,119658,119660,119663],{"className":119659},[59],[15,119661,199],{"className":119662,"style":277},[59,60],[15,119664,119666],{"className":119665},[281],[15,119667,119669,119689],{"className":119668},[285,286],[15,119670,119672,119686],{"className":119671},[290],[15,119673,119675],{"className":119674,"style":295},[294],[15,119676,119677,119680],{"style":298},[15,119678],{"className":119679,"style":303},[302],[15,119681,119683],{"className":119682},[307,308,309,310],[15,119684,37],{"className":119685,"style":61},[59,60,310],[15,119687,318],{"className":119688},[317],[15,119690,119692],{"className":119691},[290],[15,119693,119695],{"className":119694,"style":325},[294],[15,119696],{},[15,119698,113],{"className":119699},[146],[15,119701,119703,119706],{"className":119702},[59],[15,119704,206],{"className":119705,"style":334},[59,60],[15,119707,119709],{"className":119708},[281],[15,119710,119712,119732],{"className":119711},[285,286],[15,119713,119715,119729],{"className":119714},[290],[15,119716,119718],{"className":119717,"style":379},[294],[15,119719,119720,119723],{"style":11140},[15,119721],{"className":119722,"style":303},[302],[15,119724,119726],{"className":119725},[307,308,309,310],[15,119727,598],{"className":119728,"style":8889},[59,60,310],[15,119730,318],{"className":119731},[317],[15,119733,119735],{"className":119734},[290],[15,119736,119738],{"className":119737,"style":7970},[294],[15,119739],{},[15,119741,210],{"className":119742},[59],[15,119744,119746,119749],{"className":119745},[59],[15,119747,213],{"className":119748},[59,60],[15,119750,119752],{"className":119751},[281],[15,119753,119755,119775],{"className":119754},[285,286],[15,119756,119758,119772],{"className":119757},[290],[15,119759,119761],{"className":119760,"style":379},[294],[15,119762,119763,119766],{"style":382},[15,119764],{"className":119765,"style":303},[302],[15,119767,119769],{"className":119768},[307,308,309,310],[15,119770,598],{"className":119771,"style":8889},[59,60,310],[15,119773,318],{"className":119774},[317],[15,119776,119778],{"className":119777},[290],[15,119779,119781],{"className":119780,"style":7970},[294],[15,119782],{},[15,119784,120],{"className":119785},[153],[15,119787,551],{"className":119788},[573],[11,119790,119791],{},"即「编码不会创造信息」。这为对比学习、域对齐、特征选择提供了可证伪的约束：任何声称「表示后差异更大」的方法都必然在某个环节违反了 DPI，需要解释为什么。",[2313,119793,119795],{"id":119794},"_75-域适应","7.5 域适应",[11,119797,119798,119799,119829,119830,119860],{},"在源域 ",[15,119800,119802,119816],{"className":119801,"translate":19},[18],[15,119803,119805],{"className":119804},[23],[25,119806,119807],{"xmlns":27},[29,119808,119809,119813],{},[32,119810,119811],{},[35,119812,25763],{"mathvariant":597},[39,119814,119815],{"encoding":41},"\\mathcal S",[15,119817,119819],{"className":119818,"ariaHidden":46},[45],[15,119820,119822,119825],{"className":119821},[50],[15,119823],{"className":119824,"style":664},[54],[15,119826,25763],{"className":119827,"style":119828},[59,624],"margin-right:0.075em;"," 与目标域 ",[15,119831,119833,119847],{"className":119832,"translate":19},[18],[15,119834,119836],{"className":119835},[23],[25,119837,119838],{"xmlns":27},[29,119839,119840,119844],{},[32,119841,119842],{},[35,119843,5394],{"mathvariant":597},[39,119845,119846],{"encoding":41},"\\mathcal T",[15,119848,119850],{"className":119849,"ariaHidden":46},[45],[15,119851,119853,119856],{"className":119852},[50],[15,119854],{"className":119855,"style":664},[54],[15,119857,5394],{"className":119858,"style":119859},[59,624],"margin-right:0.2542em;"," 上，衡量分布漂移的现代方法（DANN、ADDA）本质上用 f-散度或 IPM 对齐边缘分布；其中用 JS 的（GAN 式）对齐与用 MMD\u002FWasserstein 的对齐，区别正是 6.2 节的几何与支撑问题。",[2313,119862,119864],{"id":119863},"_76-假设检验","7.6 假设检验",[11,119866,119867],{},"f-散度是假设检验的错误指数：",[10029,119869,119870,119937],{},[6667,119871,119872,119875,119876,19992],{},[998,119873,119874],{},"Stein 引理","：在固定第一类错误下，第二类错误的最优指数为 ",[15,119877,119879,119907],{"className":119878,"translate":19},[18],[15,119880,119882],{"className":119881},[23],[25,119883,119884],{"xmlns":27},[29,119885,119886,119904],{},[32,119887,119888,119894,119896,119898,119900,119902],{},[32,119889,119890,119892],{},[35,119891,10698],{"mathvariant":209},[35,119893,21919],{"mathvariant":209},[110,119895,113],{"stretchy":112},[35,119897,206],{},[35,119899,210],{"mathvariant":209},[35,119901,213],{},[110,119903,120],{"stretchy":112},[39,119905,119906],{"encoding":41},"\\mathrm{KL}(P\\|Q)",[15,119908,119910],{"className":119909,"ariaHidden":46},[45],[15,119911,119913,119916,119922,119925,119928,119931,119934],{"className":119912},[50],[15,119914],{"className":119915,"style":139},[54],[15,119917,119919],{"className":119918},[59],[15,119920,32689],{"className":119921},[59,32688],[15,119923,113],{"className":119924},[146],[15,119926,206],{"className":119927,"style":334},[59,60],[15,119929,210],{"className":119930},[59],[15,119932,213],{"className":119933},[59,60],[15,119935,120],{"className":119936},[153],[6667,119938,119939,9402,119942,120221],{},[998,119940,119941],{},"Chernoff 信息",[15,119943,119945,120019],{"className":119944,"translate":19},[18],[15,119946,119948],{"className":119947},[23],[25,119949,119950],{"xmlns":27},[29,119951,119952,120016],{},[32,119953,119954,119956,119958,119960,119962,119964,119966,119968,119988,119990,119992,119994,119996,119998,120006,120008,120010,120012,120014],{},[35,119955,39320],{},[110,119957,113],{"stretchy":112},[35,119959,206],{},[110,119961,551],{"separator":46},[35,119963,213],{},[110,119965,120],{"stretchy":112},[110,119967,123],{},[195,119969,119970,119976],{},[32,119971,119972,119974],{},[35,119973,35458],{},[110,119975,4105],{},[32,119977,119978,119980,119982,119984,119986],{},[115,119979,126],{},[110,119981,7253],{},[35,119983,11343],{},[110,119985,7253],{},[115,119987,117],{},[110,119989,113],{"stretchy":112},[115,119991,117],{},[110,119993,4815],{},[35,119995,11343],{},[110,119997,120],{"stretchy":112},[7555,119999,120000,120002,120004],{},[35,120001,199],{},[35,120003,11343],{},[35,120005,1035],{},[110,120007,113],{"stretchy":112},[35,120009,206],{},[35,120011,210],{"mathvariant":209},[35,120013,213],{},[110,120015,120],{"stretchy":112},[39,120017,120018],{"encoding":41},"C(P,Q)=\\sup_{0\u003C\\lambda\u003C1}(1-\\lambda)D^{R}_{\\lambda}(P\\|Q)",[15,120020,120022,120058,120135],{"className":120021,"ariaHidden":46},[45],[15,120023,120025,120028,120031,120034,120037,120040,120043,120046,120049,120052,120055],{"className":120024},[50],[15,120026],{"className":120027,"style":139},[54],[15,120029,39320],{"className":120030,"style":10786},[59,60],[15,120032,113],{"className":120033},[146],[15,120035,206],{"className":120036,"style":334},[59,60],[15,120038,551],{"className":120039},[573],[15,120041],{"className":120042,"style":406},[157],[15,120044,213],{"className":120045},[59,60],[15,120047,120],{"className":120048},[153],[15,120050],{"className":120051,"style":158},[157],[15,120053,123],{"className":120054},[162],[15,120056],{"className":120057,"style":158},[157],[15,120059,120061,120064,120120,120123,120126,120129,120132],{"className":120060},[50],[15,120062],{"className":120063,"style":35582},[54],[15,120065,120067,120070],{"className":120066},[1486],[15,120068,35458],{"className":120069},[1486],[15,120071,120073],{"className":120072},[281],[15,120074,120076,120112],{"className":120075},[285,286],[15,120077,120079,120109],{"className":120078},[290],[15,120080,120083],{"className":120081,"style":120082},[294],"height:0.242em;",[15,120084,120085,120088],{"style":35604},[15,120086],{"className":120087,"style":303},[302],[15,120089,120091],{"className":120090},[307,308,309,310],[15,120092,120094,120097,120100,120103,120106],{"className":120093},[59,310],[15,120095,126],{"className":120096},[59,310],[15,120098,7253],{"className":120099},[162,310],[15,120101,11343],{"className":120102},[59,60,310],[15,120104,7253],{"className":120105},[162,310],[15,120107,117],{"className":120108},[59,310],[15,120110,318],{"className":120111},[317],[15,120113,120115],{"className":120114},[290],[15,120116,120118],{"className":120117,"style":35632},[294],[15,120119],{},[15,120121,113],{"className":120122},[146],[15,120124,117],{"className":120125},[59],[15,120127],{"className":120128,"style":2589},[157],[15,120130,4815],{"className":120131},[2593],[15,120133],{"className":120134,"style":2589},[157],[15,120136,120138,120142,120145,120148,120206,120209,120212,120215,120218],{"className":120137},[50],[15,120139],{"className":120140,"style":120141},[54],"height:1.1244em;vertical-align:-0.2831em;",[15,120143,11343],{"className":120144},[59,60],[15,120146,120],{"className":120147},[153],[15,120149,120151,120154],{"className":120150},[59],[15,120152,199],{"className":120153,"style":277},[59,60],[15,120155,120157],{"className":120156},[281],[15,120158,120160,120198],{"className":120159},[285,286],[15,120161,120163,120195],{"className":120162},[290],[15,120164,120166,120181],{"className":120165,"style":52111},[294],[15,120167,120169,120172],{"style":120168},"top:-2.4169em;margin-left:-0.0278em;margin-right:0.05em;",[15,120170],{"className":120171,"style":303},[302],[15,120173,120175],{"className":120174},[307,308,309,310],[15,120176,120178],{"className":120177},[59,310],[15,120179,11343],{"className":120180},[59,60,310],[15,120182,120183,120186],{"style":6867},[15,120184],{"className":120185,"style":303},[302],[15,120187,120189],{"className":120188},[307,308,309,310],[15,120190,120192],{"className":120191},[59,310],[15,120193,1035],{"className":120194,"style":49625},[59,60,310],[15,120196,318],{"className":120197},[317],[15,120199,120201],{"className":120200},[290],[15,120202,120204],{"className":120203,"style":8549},[294],[15,120205],{},[15,120207,113],{"className":120208},[146],[15,120210,206],{"className":120211,"style":334},[59,60],[15,120213,210],{"className":120214},[59],[15,120216,213],{"className":120217},[59,60],[15,120219,120],{"className":120220},[153],"，即 Rényi\u002Fα-族中的最优点；对称检验的指数由它决定。",[11,120223,120224],{},"因此 f-散度不只是「训练目标」，也是统计判决的极限性能。",[528,120226,120228],{"id":120227},"_8-差分隐私f-dp","8. 差分隐私（f-DP）",[11,120230,120231,120232,120235,120236,6496,120239,120267,120268,120309],{},"一个较新的应用是把 f-散度的语言搬到隐私保护。差分隐私（DP）要求相邻数据集的输出分布「难以区分」。Dong–Roth–Su 提出 ",[998,120233,120234],{},"f-差分隐私（f-DP）","：用一个",[998,120237,120238],{},"权衡函数（tradeoff function）",[15,120240,120242,120255],{"className":120241,"translate":19},[18],[15,120243,120245],{"className":120244},[23],[25,120246,120247],{"xmlns":27},[29,120248,120249,120253],{},[32,120250,120251],{},[35,120252,37],{},[39,120254,37],{"encoding":41},[15,120256,120258],{"className":120257,"ariaHidden":46},[45],[15,120259,120261,120264],{"className":120260},[50],[15,120262],{"className":120263,"style":55},[54],[15,120265,37],{"className":120266,"style":61},[59,60]," 刻画两个输出分布 ",[15,120269,120271,120288],{"className":120270,"translate":19},[18],[15,120272,120274],{"className":120273},[23],[25,120275,120276],{"xmlns":27},[29,120277,120278,120286],{},[32,120279,120280,120282,120284],{},[35,120281,206],{},[110,120283,551],{"separator":46},[35,120285,213],{},[39,120287,556],{"encoding":41},[15,120289,120291],{"className":120290,"ariaHidden":46},[45],[15,120292,120294,120297,120300,120303,120306],{"className":120293},[50],[15,120295],{"className":120296,"style":566},[54],[15,120298,206],{"className":120299,"style":334},[59,60],[15,120301,551],{"className":120302},[573],[15,120304],{"className":120305,"style":406},[157],[15,120307,213],{"className":120308},[59,60]," 上「漏报率—误报率」的权衡曲线：",[15,120311,120313],{"className":120312,"translate":19},[180],[15,120314,120316,120413],{"className":120315,"translate":19},[18],[15,120317,120319],{"className":120318},[23],[25,120320,120321],{"xmlns":27,"display":189},[29,120322,120323,120410],{},[32,120324,120325,120327,120329,120331,120333,120335,120337,120339,120341,120343,120345,120356,120358,120360,120362,120364,120366,120368,120370,120372,120374,120376,120378,120380,120382,120384,120386,120388,120390,120392,120394,120396,120398,120400,120402,120404,120406,120408],{},[35,120326,5394],{},[110,120328,113],{"stretchy":112},[35,120330,206],{},[110,120332,551],{"separator":46},[35,120334,213],{},[110,120336,120],{"stretchy":112},[110,120338,113],{"stretchy":112},[35,120340,44780],{},[110,120342,120],{"stretchy":112},[110,120344,123],{},[6018,120346,120347,120354],{},[32,120348,120349,120352],{},[35,120350,120351],{},"inf",[110,120353,4105],{},[35,120355,603],{},[110,120357,2900],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[35,120359,206],{},[110,120361,113],{"stretchy":112},[35,120363,603],{},[110,120365,120],{"stretchy":112},[1299,120367,4285],{},[110,120369,1018],{},[1299,120371,4285],{},[35,120373,213],{},[110,120375,113],{"stretchy":112},[35,120377,603],{},[110,120379,120],{"stretchy":112},[110,120381,6930],{},[115,120383,117],{},[110,120385,4815],{},[35,120387,44780],{},[110,120389,2910],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[110,120391,551],{"separator":46},[157,120393],{"width":3264},[35,120395,44780],{},[110,120397,5911],{},[110,120399,230],{"stretchy":112},[115,120401,126],{},[110,120403,551],{"separator":46},[115,120405,117],{},[110,120407,257],{"stretchy":112},[35,120409,4377],{"mathvariant":209},[39,120411,120412],{"encoding":41},"T(P,Q)(\\alpha)\n=\\inf_{A}\\big\\{P(A)\\ :\\ Q(A)\\ge1-\\alpha\\big\\},\n\\qquad \\alpha\\in[0,1] .",[15,120414,120416,120461,120556,120583,120601,120637],{"className":120415,"ariaHidden":46},[45],[15,120417,120419,120422,120425,120428,120431,120434,120437,120440,120443,120446,120449,120452,120455,120458],{"className":120418},[50],[15,120420],{"className":120421,"style":139},[54],[15,120423,5394],{"className":120424,"style":334},[59,60],[15,120426,113],{"className":120427},[146],[15,120429,206],{"className":120430,"style":334},[59,60],[15,120432,551],{"className":120433},[573],[15,120435],{"className":120436,"style":406},[157],[15,120438,213],{"className":120439},[59,60],[15,120441,120],{"className":120442},[153],[15,120444,113],{"className":120445},[146],[15,120447,44780],{"className":120448,"style":44810},[59,60],[15,120450,120],{"className":120451},[153],[15,120453],{"className":120454,"style":158},[157],[15,120456,123],{"className":120457},[162],[15,120459],{"className":120460,"style":158},[157],[15,120462,120464,120468,120520,120523,120529,120532,120535,120538,120541,120544,120547,120550,120553],{"className":120463},[50],[15,120465],{"className":120466,"style":120467},[54],"height:1.5943em;vertical-align:-0.7443em;",[15,120469,120471],{"className":120470},[1486,13720],[15,120472,120474,120511],{"className":120473},[285,286],[15,120475,120477,120508],{"className":120476},[290],[15,120478,120480,120495],{"className":120479,"style":12601},[294],[15,120481,120483,120486],{"style":120482},"top:-2.3557em;margin-left:0em;",[15,120484],{"className":120485,"style":461},[302],[15,120487,120489],{"className":120488},[307,308,309,310],[15,120490,120492],{"className":120491},[59,310],[15,120493,603],{"className":120494},[59,60,310],[15,120496,120497,120500],{"style":11867},[15,120498],{"className":120499,"style":461},[302],[15,120501,120502],{},[15,120503,120505,120506],{"className":120504},[1486],"in",[15,120507,37],{"style":94877},[15,120509,318],{"className":120510},[317],[15,120512,120514],{"className":120513},[290],[15,120515,120518],{"className":120516,"style":120517},[294],"height:0.7443em;",[15,120519],{},[15,120521],{"className":120522,"style":406},[157],[15,120524,120526],{"className":120525},[59],[15,120527,2900],{"className":120528},[419,9676],[15,120530,206],{"className":120531,"style":334},[59,60],[15,120533,113],{"className":120534},[146],[15,120536,603],{"className":120537},[59,60],[15,120539,120],{"className":120540},[153],[15,120542,4285],{"className":120543},[157],[15,120545],{"className":120546,"style":158},[157],[15,120548,1018],{"className":120549},[162],[15,120551,4285],{"className":120552},[157],[15,120554],{"className":120555,"style":158},[157],[15,120557,120559,120562,120565,120568,120571,120574,120577,120580],{"className":120558},[50],[15,120560],{"className":120561,"style":139},[54],[15,120563,213],{"className":120564},[59,60],[15,120566,113],{"className":120567},[146],[15,120569,603],{"className":120570},[59,60],[15,120572,120],{"className":120573},[153],[15,120575],{"className":120576,"style":158},[157],[15,120578,6930],{"className":120579},[162],[15,120581],{"className":120582,"style":158},[157],[15,120584,120586,120589,120592,120595,120598],{"className":120585},[50],[15,120587],{"className":120588,"style":12582},[54],[15,120590,117],{"className":120591},[59],[15,120593],{"className":120594,"style":2589},[157],[15,120596,4815],{"className":120597},[2593],[15,120599],{"className":120600,"style":2589},[157],[15,120602,120604,120607,120610,120616,120619,120622,120625,120628,120631,120634],{"className":120603},[50],[15,120605],{"className":120606,"style":20589},[54],[15,120608,44780],{"className":120609,"style":44810},[59,60],[15,120611,120613],{"className":120612},[59],[15,120614,2910],{"className":120615},[419,9676],[15,120617,551],{"className":120618},[573],[15,120620],{"className":120621,"style":3636},[157],[15,120623],{"className":120624,"style":406},[157],[15,120626,44780],{"className":120627,"style":44810},[59,60],[15,120629],{"className":120630,"style":158},[157],[15,120632,5911],{"className":120633},[162],[15,120635],{"className":120636,"style":158},[157],[15,120638,120640,120643,120646,120649,120652,120655,120658,120661],{"className":120639},[50],[15,120641],{"className":120642,"style":139},[54],[15,120644,230],{"className":120645},[146],[15,120647,126],{"className":120648},[59],[15,120650,551],{"className":120651},[573],[15,120653],{"className":120654,"style":406},[157],[15,120656,117],{"className":120657},[59],[15,120659,257],{"className":120660},[153],[15,120662,4377],{"className":120663},[59],[11,120665,120666,120667,78107,120695,120723,120724,120799,120800,120939],{},"机制 ",[15,120668,120670,120683],{"className":120669,"translate":19},[18],[15,120671,120673],{"className":120672},[23],[25,120674,120675],{"xmlns":27},[29,120676,120677,120681],{},[32,120678,120679],{},[35,120680,25593],{},[39,120682,25593],{"encoding":41},[15,120684,120686],{"className":120685,"ariaHidden":46},[45],[15,120687,120689,120692],{"className":120688},[50],[15,120690],{"className":120691,"style":664},[54],[15,120693,25593],{"className":120694,"style":25627},[59,60],[15,120696,120698,120711],{"className":120697,"translate":19},[18],[15,120699,120701],{"className":120700},[23],[25,120702,120703],{"xmlns":27},[29,120704,120705,120709],{},[32,120706,120707],{},[35,120708,37],{},[39,120710,37],{"encoding":41},[15,120712,120714],{"className":120713,"ariaHidden":46},[45],[15,120715,120717,120720],{"className":120716},[50],[15,120718],{"className":120719,"style":55},[54],[15,120721,37],{"className":120722,"style":61},[59,60],"-DP，若对所有相邻 ",[15,120725,120727,120749],{"className":120726,"translate":19},[18],[15,120728,120730],{"className":120729},[23],[25,120731,120732],{"xmlns":27},[29,120733,120734,120746],{},[32,120735,120736,120738,120740],{},[35,120737,199],{},[110,120739,551],{"separator":46},[6816,120741,120742,120744],{},[35,120743,199],{},[110,120745,6822],{"mathvariant":209,"lspace":1249,"rspace":1249},[39,120747,120748],{"encoding":41},"D,D'",[15,120750,120752],{"className":120751,"ariaHidden":46},[45],[15,120753,120755,120758,120761,120764,120767],{"className":120754},[50],[15,120756],{"className":120757,"style":64198},[54],[15,120759,199],{"className":120760,"style":277},[59,60],[15,120762,551],{"className":120763},[573],[15,120765],{"className":120766,"style":406},[157],[15,120768,120770,120773],{"className":120769},[59],[15,120771,199],{"className":120772,"style":277},[59,60],[15,120774,120776],{"className":120775},[281],[15,120777,120779],{"className":120778},[285],[15,120780,120782],{"className":120781},[290],[15,120783,120785],{"className":120784,"style":6864},[294],[15,120786,120787,120790],{"style":6867},[15,120788],{"className":120789,"style":303},[302],[15,120791,120793],{"className":120792},[307,308,309,310],[15,120794,120796],{"className":120795},[59,310],[15,120797,6822],{"className":120798},[59,310],"，恒有 ",[15,120801,120803,120847],{"className":120802,"translate":19},[18],[15,120804,120806],{"className":120805},[23],[25,120807,120808],{"xmlns":27},[29,120809,120810,120844],{},[32,120811,120812,120814,120816,120818,120820,120822,120824,120826,120828,120830,120836,120838,120840,120842],{},[35,120813,5394],{},[110,120815,113],{"stretchy":112},[35,120817,25593],{},[110,120819,113],{"stretchy":112},[35,120821,199],{},[110,120823,120],{"stretchy":112},[110,120825,551],{"separator":46},[35,120827,25593],{},[110,120829,113],{"stretchy":112},[6816,120831,120832,120834],{},[35,120833,199],{},[110,120835,6822],{"mathvariant":209,"lspace":1249,"rspace":1249},[110,120837,120],{"stretchy":112},[110,120839,120],{"stretchy":112},[110,120841,6930],{},[35,120843,37],{},[39,120845,120846],{"encoding":41},"T(M(D),M(D'))\\ge f",[15,120848,120850,120930],{"className":120849,"ariaHidden":46},[45],[15,120851,120853,120856,120859,120862,120865,120868,120871,120874,120877,120880,120883,120886,120918,120921,120924,120927],{"className":120852},[50],[15,120854],{"className":120855,"style":6845},[54],[15,120857,5394],{"className":120858,"style":334},[59,60],[15,120860,113],{"className":120861},[146],[15,120863,25593],{"className":120864,"style":25627},[59,60],[15,120866,113],{"className":120867},[146],[15,120869,199],{"className":120870,"style":277},[59,60],[15,120872,120],{"className":120873},[153],[15,120875,551],{"className":120876},[573],[15,120878],{"className":120879,"style":406},[157],[15,120881,25593],{"className":120882,"style":25627},[59,60],[15,120884,113],{"className":120885},[146],[15,120887,120889,120892],{"className":120888},[59],[15,120890,199],{"className":120891,"style":277},[59,60],[15,120893,120895],{"className":120894},[281],[15,120896,120898],{"className":120897},[285],[15,120899,120901],{"className":120900},[290],[15,120902,120904],{"className":120903,"style":6864},[294],[15,120905,120906,120909],{"style":6867},[15,120907],{"className":120908,"style":303},[302],[15,120910,120912],{"className":120911},[307,308,309,310],[15,120913,120915],{"className":120914},[59,310],[15,120916,6822],{"className":120917},[59,310],[15,120919,19073],{"className":120920},[153],[15,120922],{"className":120923,"style":158},[157],[15,120925,6930],{"className":120926},[162],[15,120928],{"className":120929,"style":158},[157],[15,120931,120933,120936],{"className":120932},[50],[15,120934],{"className":120935,"style":55},[54],[15,120937,37],{"className":120938,"style":61},[59,60],"。可以证明：",[10029,120941,120942],{},[6667,120943,120944,32115,120947,121019,121020,121068,121069,121097],{},[998,120945,120946],{},"权衡曲线与 f-散度互相决定。",[15,120948,120950,120980],{"className":120949,"translate":19},[18],[15,120951,120953],{"className":120952},[23],[25,120954,120955],{"xmlns":27},[29,120956,120957,120977],{},[32,120958,120959,120961,120963,120965,120967,120969,120971,120973,120975],{},[35,120960,5394],{},[110,120962,113],{"stretchy":112},[35,120964,206],{},[110,120966,551],{"separator":46},[35,120968,213],{},[110,120970,120],{"stretchy":112},[110,120972,113],{"stretchy":112},[35,120974,44780],{},[110,120976,120],{"stretchy":112},[39,120978,120979],{"encoding":41},"T(P,Q)(\\alpha)",[15,120981,120983],{"className":120982,"ariaHidden":46},[45],[15,120984,120986,120989,120992,120995,120998,121001,121004,121007,121010,121013,121016],{"className":120985},[50],[15,120987],{"className":120988,"style":139},[54],[15,120990,5394],{"className":120991,"style":334},[59,60],[15,120993,113],{"className":120994},[146],[15,120996,206],{"className":120997,"style":334},[59,60],[15,120999,551],{"className":121000},[573],[15,121002],{"className":121003,"style":406},[157],[15,121005,213],{"className":121006},[59,60],[15,121008,120],{"className":121009},[153],[15,121011,113],{"className":121012},[146],[15,121014,44780],{"className":121015,"style":44810},[59,60],[15,121017,120],{"className":121018},[153]," 只依赖似然比 ",[15,121021,121023,121044],{"className":121022,"translate":19},[18],[15,121024,121026],{"className":121025},[23],[25,121027,121028],{"xmlns":27},[29,121029,121030,121042],{},[32,121031,121032,121034,121036,121038,121040],{},[35,121033,244],{},[35,121035,206],{},[35,121037,941],{"mathvariant":209},[35,121039,244],{},[35,121041,213],{},[39,121043,114686],{"encoding":41},[15,121045,121047],{"className":121046,"ariaHidden":46},[45],[15,121048,121050,121053,121056,121059,121062,121065],{"className":121049},[50],[15,121051],{"className":121052,"style":139},[54],[15,121054,244],{"className":121055},[59,60],[15,121057,206],{"className":121058,"style":334},[59,60],[15,121060,941],{"className":121061},[59],[15,121063,244],{"className":121064},[59,60],[15,121066,213],{"className":121067},[59,60]," 的分布，因此 ",[15,121070,121072,121085],{"className":121071,"translate":19},[18],[15,121073,121075],{"className":121074},[23],[25,121076,121077],{"xmlns":27},[29,121078,121079,121083],{},[32,121080,121081],{},[35,121082,5394],{},[39,121084,5394],{"encoding":41},[15,121086,121088],{"className":121087,"ariaHidden":46},[45],[15,121089,121091,121094],{"className":121090},[50],[15,121092],{"className":121093,"style":664},[54],[15,121095,5394],{"className":121096,"style":334},[59,60]," 与\"全体 f-散度\"携带同样的信息。对 hockey-stick 生成元族",[15,121099,121101],{"className":121100,"translate":19},[180],[15,121102,121104,121171],{"className":121103,"translate":19},[18],[15,121105,121107],{"className":121106},[23],[25,121108,121109],{"xmlns":27,"display":189},[29,121110,121111,121168],{},[32,121112,121113,121120,121122,121124,121126,121128,121130,121132,121134,121136,121138,121140,121142,121144,121146,121148,121150,121152,121154,121156,121158,121160,121162,121164,121166],{},[195,121114,121115,121117],{},[35,121116,4160],{},[35,121118,121119],{},"γ",[110,121121,113],{"stretchy":112},[35,121123,769],{},[110,121125,120],{"stretchy":112},[110,121127,123],{},[35,121129,52413],{},[110,121131,4105],{},[110,121133,2900],{"stretchy":112},[115,121135,126],{},[110,121137,551],{"separator":46},[1299,121139,1301],{},[35,121141,769],{},[110,121143,4815],{},[35,121145,121119],{},[110,121147,2910],{"stretchy":112},[110,121149,551],{"separator":46},[157,121151],{"width":3264},[35,121153,121119],{},[110,121155,5911],{},[110,121157,230],{"stretchy":112},[115,121159,126],{},[110,121161,551],{"separator":46},[115,121163,117],{},[110,121165,257],{"stretchy":112},[110,121167,551],{"separator":46},[39,121169,121170],{"encoding":41},"g_\\gamma(t)=\\max\\{0,\\,t-\\gamma\\},\n\\qquad \\gamma\\in[0,1],",[15,121172,121174,121239,121275,121308],{"className":121173,"ariaHidden":46},[45],[15,121175,121177,121180,121221,121224,121227,121230,121233,121236],{"className":121176},[50],[15,121178],{"className":121179,"style":270},[54],[15,121181,121183,121186],{"className":121182},[59],[15,121184,4160],{"className":121185,"style":981},[59,60],[15,121187,121189],{"className":121188},[281],[15,121190,121192,121213],{"className":121191},[285,286],[15,121193,121195,121210],{"className":121194},[290],[15,121196,121198],{"className":121197,"style":44955},[294],[15,121199,121200,121203],{"style":18869},[15,121201],{"className":121202,"style":303},[302],[15,121204,121206],{"className":121205},[307,308,309,310],[15,121207,121119],{"className":121208,"style":121209},[59,60,310],"margin-right:0.0556em;",[15,121211,318],{"className":121212},[317],[15,121214,121216],{"className":121215},[290],[15,121217,121219],{"className":121218,"style":325},[294],[15,121220],{},[15,121222,113],{"className":121223},[146],[15,121225,769],{"className":121226},[59,60],[15,121228,120],{"className":121229},[153],[15,121231],{"className":121232,"style":158},[157],[15,121234,123],{"className":121235},[162],[15,121237],{"className":121238,"style":158},[157],[15,121240,121242,121245,121248,121251,121254,121257,121260,121263,121266,121269,121272],{"className":121241},[50],[15,121243],{"className":121244,"style":139},[54],[15,121246,52413],{"className":121247},[1486],[15,121249,2900],{"className":121250},[146],[15,121252,126],{"className":121253},[59],[15,121255,551],{"className":121256},[573],[15,121258],{"className":121259,"style":406},[157],[15,121261],{"className":121262,"style":406},[157],[15,121264,769],{"className":121265},[59,60],[15,121267],{"className":121268,"style":2589},[157],[15,121270,4815],{"className":121271},[2593],[15,121273],{"className":121274,"style":2589},[157],[15,121276,121278,121281,121284,121287,121290,121293,121296,121299,121302,121305],{"className":121277},[50],[15,121279],{"className":121280,"style":139},[54],[15,121282,121119],{"className":121283,"style":121209},[59,60],[15,121285,2910],{"className":121286},[153],[15,121288,551],{"className":121289},[573],[15,121291],{"className":121292,"style":3636},[157],[15,121294],{"className":121295,"style":406},[157],[15,121297,121119],{"className":121298,"style":121209},[59,60],[15,121300],{"className":121301,"style":158},[157],[15,121303,5911],{"className":121304},[162],[15,121306],{"className":121307,"style":158},[157],[15,121309,121311,121314,121317,121320,121323,121326,121329,121332],{"className":121310},[50],[15,121312],{"className":121313,"style":139},[54],[15,121315,230],{"className":121316},[146],[15,121318,126],{"className":121319},[59],[15,121321,551],{"className":121322},[573],[15,121324],{"className":121325,"style":406},[157],[15,121327,117],{"className":121328},[59],[15,121330,257],{"className":121331},[153],[15,121333,551],{"className":121334},[573],[11,121336,121337,121338,121519,121520,8669,121549,121600],{},"（Dong–Roth–Su 的 ",[15,121339,121341,121389],{"className":121340,"translate":19},[18],[15,121342,121344],{"className":121343},[23],[25,121345,121346],{"xmlns":27},[29,121347,121348,121386],{},[32,121349,121350,121356,121358,121360,121362,121364,121366,121368,121370,121372,121374,121376,121378,121380,121382,121384],{},[195,121351,121352,121354],{},[35,121353,38691],{},[35,121355,121119],{},[110,121357,113],{"stretchy":112},[35,121359,769],{},[110,121361,120],{"stretchy":112},[110,121363,123],{},[35,121365,52413],{},[110,121367,4105],{},[110,121369,2900],{"stretchy":112},[115,121371,126],{},[110,121373,551],{"separator":46},[35,121375,769],{},[110,121377,4815],{},[115,121379,117],{},[110,121381,2472],{},[35,121383,121119],{},[110,121385,2910],{"stretchy":112},[39,121387,121388],{"encoding":41},"h_\\gamma(t)=\\max\\{0,t-1+\\gamma\\}",[15,121390,121392,121456,121489,121507],{"className":121391,"ariaHidden":46},[45],[15,121393,121395,121398,121438,121441,121444,121447,121450,121453],{"className":121394},[50],[15,121396],{"className":121397,"style":270},[54],[15,121399,121401,121404],{"className":121400},[59],[15,121402,38691],{"className":121403},[59,60],[15,121405,121407],{"className":121406},[281],[15,121408,121410,121430],{"className":121409},[285,286],[15,121411,121413,121427],{"className":121412},[290],[15,121414,121416],{"className":121415,"style":44955},[294],[15,121417,121418,121421],{"style":382},[15,121419],{"className":121420,"style":303},[302],[15,121422,121424],{"className":121423},[307,308,309,310],[15,121425,121119],{"className":121426,"style":121209},[59,60,310],[15,121428,318],{"className":121429},[317],[15,121431,121433],{"className":121432},[290],[15,121434,121436],{"className":121435,"style":325},[294],[15,121437],{},[15,121439,113],{"className":121440},[146],[15,121442,769],{"className":121443},[59,60],[15,121445,120],{"className":121446},[153],[15,121448],{"className":121449,"style":158},[157],[15,121451,123],{"className":121452},[162],[15,121454],{"className":121455,"style":158},[157],[15,121457,121459,121462,121465,121468,121471,121474,121477,121480,121483,121486],{"className":121458},[50],[15,121460],{"className":121461,"style":139},[54],[15,121463,52413],{"className":121464},[1486],[15,121466,2900],{"className":121467},[146],[15,121469,126],{"className":121470},[59],[15,121472,551],{"className":121473},[573],[15,121475],{"className":121476,"style":406},[157],[15,121478,769],{"className":121479},[59,60],[15,121481],{"className":121482,"style":2589},[157],[15,121484,4815],{"className":121485},[2593],[15,121487],{"className":121488,"style":2589},[157],[15,121490,121492,121495,121498,121501,121504],{"className":121491},[50],[15,121493],{"className":121494,"style":12582},[54],[15,121496,117],{"className":121497},[59],[15,121499],{"className":121500,"style":2589},[157],[15,121502,2472],{"className":121503},[2593],[15,121505],{"className":121506,"style":2589},[157],[15,121508,121510,121513,121516],{"className":121509},[50],[15,121511],{"className":121512,"style":139},[54],[15,121514,121119],{"className":121515,"style":121209},[59,60],[15,121517,2910],{"className":121518},[153]," 就是把 ",[15,121521,121523,121537],{"className":121522,"translate":19},[18],[15,121524,121526],{"className":121525},[23],[25,121527,121528],{"xmlns":27},[29,121529,121530,121534],{},[32,121531,121532],{},[35,121533,121119],{},[39,121535,121536],{"encoding":41},"\\gamma",[15,121538,121540],{"className":121539,"ariaHidden":46},[45],[15,121541,121543,121546],{"className":121542},[50],[15,121544],{"className":121545,"style":2278},[54],[15,121547,121119],{"className":121548,"style":121209},[59,60],[15,121550,121552,121570],{"className":121551,"translate":19},[18],[15,121553,121555],{"className":121554},[23],[25,121556,121557],{"xmlns":27},[29,121558,121559,121567],{},[32,121560,121561,121563,121565],{},[115,121562,117],{},[110,121564,4815],{},[35,121566,121119],{},[39,121568,121569],{"encoding":41},"1-\\gamma",[15,121571,121573,121591],{"className":121572,"ariaHidden":46},[45],[15,121574,121576,121579,121582,121585,121588],{"className":121575},[50],[15,121577],{"className":121578,"style":12582},[54],[15,121580,117],{"className":121581},[59],[15,121583],{"className":121584,"style":2589},[157],[15,121586,4815],{"className":121587},[2593],[15,121589],{"className":121590,"style":2589},[157],[15,121592,121594,121597],{"className":121593},[50],[15,121595],{"className":121596,"style":2278},[54],[15,121598,121119],{"className":121599,"style":121209},[59,60]," 的同一族），变分表示给出干净的检验集形式：",[15,121602,121604],{"className":121603,"translate":19},[180],[15,121605,121607,121685],{"className":121606,"translate":19},[18],[15,121608,121610],{"className":121609},[23],[25,121611,121612],{"xmlns":27,"display":189},[29,121613,121614,121682],{},[32,121615,121616,121626,121628,121630,121632,121634,121636,121638,121654,121656,121658,121660,121662,121664,121666,121668,121670,121672,121674,121676,121678,121680],{},[195,121617,121618,121620],{},[35,121619,199],{},[195,121621,121622,121624],{},[35,121623,4160],{},[35,121625,121119],{},[110,121627,113],{"stretchy":112},[35,121629,206],{},[35,121631,210],{"mathvariant":209},[35,121633,213],{},[110,121635,120],{"stretchy":112},[110,121637,123],{},[6018,121639,121640,121646],{},[32,121641,121642,121644],{},[35,121643,35458],{},[110,121645,4105],{},[32,121647,121648,121650,121652],{},[35,121649,603],{},[110,121651,5911],{},[35,121653,603],{"mathvariant":597},[110,121655,2900],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[35,121657,206],{},[110,121659,113],{"stretchy":112},[35,121661,603],{},[110,121663,120],{"stretchy":112},[110,121665,4815],{},[35,121667,121119],{},[1299,121669,1301],{},[35,121671,213],{},[110,121673,113],{"stretchy":112},[35,121675,603],{},[110,121677,120],{"stretchy":112},[110,121679,2910],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[35,121681,4377],{"mathvariant":209},[39,121683,121684],{"encoding":41},"D_{g_\\gamma}(P\\|Q)\n=\\sup_{A\\in\\mathcal A}\\big\\{P(A)\n-\\gamma\\,Q(A)\\big\\}.",[15,121686,121688,121802,121891],{"className":121687,"ariaHidden":46},[45],[15,121689,121691,121695,121778,121781,121784,121787,121790,121793,121796,121799],{"className":121690},[50],[15,121692],{"className":121693,"style":121694},[54],"height:1.0973em;vertical-align:-0.3473em;",[15,121696,121698,121701],{"className":121697},[59],[15,121699,199],{"className":121700,"style":277},[59,60],[15,121702,121704],{"className":121703},[281],[15,121705,121707,121769],{"className":121706},[285,286],[15,121708,121710,121766],{"className":121709},[290],[15,121711,121713],{"className":121712,"style":44955},[294],[15,121714,121715,121718],{"style":298},[15,121716],{"className":121717,"style":303},[302],[15,121719,121721],{"className":121720},[307,308,309,310],[15,121722,121724],{"className":121723},[59,310],[15,121725,121727,121730],{"className":121726},[59,310],[15,121728,4160],{"className":121729,"style":981},[59,60,310],[15,121731,121733],{"className":121732},[281],[15,121734,121736,121757],{"className":121735},[285,286],[15,121737,121739,121754],{"className":121738},[290],[15,121740,121742],{"className":121741,"style":74299},[294],[15,121743,121745,121748],{"style":121744},"top:-2.357em;margin-left:-0.0359em;margin-right:0.0714em;",[15,121746],{"className":121747,"style":30232},[302],[15,121749,121751],{"className":121750},[307,30236,9676,310],[15,121752,121119],{"className":121753,"style":121209},[59,60,310],[15,121755,318],{"className":121756},[317],[15,121758,121760],{"className":121759},[290],[15,121761,121764],{"className":121762,"style":121763},[294],"height:0.2819em;",[15,121765],{},[15,121767,318],{"className":121768},[317],[15,121770,121772],{"className":121771},[290],[15,121773,121776],{"className":121774,"style":121775},[294],"height:0.3473em;",[15,121777],{},[15,121779,113],{"className":121780},[146],[15,121782,206],{"className":121783,"style":334},[59,60],[15,121785,210],{"className":121786},[59],[15,121788,213],{"className":121789},[59,60],[15,121791,120],{"className":121792},[153],[15,121794],{"className":121795,"style":158},[157],[15,121797,123],{"className":121798},[162],[15,121800],{"className":121801,"style":158},[157],[15,121803,121805,121808,121861,121864,121870,121873,121876,121879,121882,121885,121888],{"className":121804},[50],[15,121806],{"className":121807,"style":36379},[54],[15,121809,121811],{"className":121810},[1486,13720],[15,121812,121814,121853],{"className":121813},[285,286],[15,121815,121817,121850],{"className":121816},[290],[15,121818,121820,121840],{"className":121819,"style":2347},[294],[15,121821,121822,121825],{"style":36394},[15,121823],{"className":121824,"style":461},[302],[15,121826,121828],{"className":121827},[307,308,309,310],[15,121829,121831,121834,121837],{"className":121830},[59,310],[15,121832,603],{"className":121833},[59,60,310],[15,121835,5911],{"className":121836},[162,310],[15,121838,603],{"className":121839},[59,624,310],[15,121841,121842,121845],{"style":11867},[15,121843],{"className":121844,"style":461},[302],[15,121846,121847],{},[15,121848,35458],{"className":121849},[1486],[15,121851,318],{"className":121852},[317],[15,121854,121856],{"className":121855},[290],[15,121857,121859],{"className":121858,"style":36432},[294],[15,121860],{},[15,121862],{"className":121863,"style":406},[157],[15,121865,121867],{"className":121866},[59],[15,121868,2900],{"className":121869},[419,9676],[15,121871,206],{"className":121872,"style":334},[59,60],[15,121874,113],{"className":121875},[146],[15,121877,603],{"className":121878},[59,60],[15,121880,120],{"className":121881},[153],[15,121883],{"className":121884,"style":2589},[157],[15,121886,4815],{"className":121887},[2593],[15,121889],{"className":121890,"style":2589},[157],[15,121892,121894,121897,121900,121903,121906,121909,121912,121915,121921],{"className":121893},[50],[15,121895],{"className":121896,"style":20589},[54],[15,121898,121119],{"className":121899,"style":121209},[59,60],[15,121901],{"className":121902,"style":406},[157],[15,121904,213],{"className":121905},[59,60],[15,121907,113],{"className":121908},[146],[15,121910,603],{"className":121911},[59,60],[15,121913,120],{"className":121914},[153],[15,121916,121918],{"className":121917},[59],[15,121919,2910],{"className":121920},[419,9676],[15,121922,4377],{"className":121923},[59],[11,121925,121926,121928,121929,122003,122004,2282,122173,122201,122202,122295],{},[998,121927,107322],{}," 一方面，任意 ",[15,121930,121932,121954],{"className":121931,"translate":19},[18],[15,121933,121935],{"className":121934},[23],[25,121936,121937],{"xmlns":27},[29,121938,121939,121951],{},[32,121940,121941,121943,121945,121947,121949],{},[115,121942,126],{},[110,121944,9954],{},[35,121946,4160],{},[110,121948,9954],{},[115,121950,117],{},[39,121952,121953],{"encoding":41},"0\\le g\\le1",[15,121955,121957,121976,121994],{"className":121956,"ariaHidden":46},[45],[15,121958,121960,121964,121967,121970,121973],{"className":121959},[50],[15,121961],{"className":121962,"style":121963},[54],"height:0.7804em;vertical-align:-0.136em;",[15,121965,126],{"className":121966},[59],[15,121968],{"className":121969,"style":158},[157],[15,121971,9954],{"className":121972},[162],[15,121974],{"className":121975,"style":158},[157],[15,121977,121979,121982,121985,121988,121991],{"className":121978},[50],[15,121980],{"className":121981,"style":15421},[54],[15,121983,4160],{"className":121984,"style":981},[59,60],[15,121986],{"className":121987,"style":158},[157],[15,121989,9954],{"className":121990},[162],[15,121992],{"className":121993,"style":158},[157],[15,121995,121997,122000],{"className":121996},[50],[15,121998],{"className":121999,"style":172},[54],[15,122001,117],{"className":122002},[59]," 可写为指示函数的线性组合（层叠公式），而 ",[15,122005,122007,122047],{"className":122006,"translate":19},[18],[15,122008,122010],{"className":122009},[23],[25,122011,122012],{"xmlns":27},[29,122013,122014,122044],{},[32,122015,122016,122022,122024,122026,122028,122030,122032,122038,122040,122042],{},[195,122017,122018,122020],{},[35,122019,223],{},[35,122021,206],{},[110,122023,230],{"stretchy":112},[35,122025,4160],{},[110,122027,257],{"stretchy":112},[110,122029,4815],{},[35,122031,121119],{},[195,122033,122034,122036],{},[35,122035,223],{},[35,122037,213],{},[110,122039,230],{"stretchy":112},[35,122041,4160],{},[110,122043,257],{"stretchy":112},[39,122045,122046],{"encoding":41},"E_P[g]-\\gamma E_Q[g]",[15,122048,122050,122115],{"className":122049,"ariaHidden":46},[45],[15,122051,122053,122056,122097,122100,122103,122106,122109,122112],{"className":122052},[50],[15,122054],{"className":122055,"style":139},[54],[15,122057,122059,122062],{"className":122058},[59],[15,122060,223],{"className":122061,"style":25887},[59,60],[15,122063,122065],{"className":122064},[281],[15,122066,122068,122089],{"className":122067},[285,286],[15,122069,122071,122086],{"className":122070},[290],[15,122072,122074],{"className":122073,"style":379},[294],[15,122075,122077,122080],{"style":122076},"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;",[15,122078],{"className":122079,"style":303},[302],[15,122081,122083],{"className":122082},[307,308,309,310],[15,122084,206],{"className":122085,"style":334},[59,60,310],[15,122087,318],{"className":122088},[317],[15,122090,122092],{"className":122091},[290],[15,122093,122095],{"className":122094,"style":7970},[294],[15,122096],{},[15,122098,230],{"className":122099},[146],[15,122101,4160],{"className":122102,"style":981},[59,60],[15,122104,257],{"className":122105},[153],[15,122107],{"className":122108,"style":2589},[157],[15,122110,4815],{"className":122111},[2593],[15,122113],{"className":122114,"style":2589},[157],[15,122116,122118,122121,122124,122164,122167,122170],{"className":122117},[50],[15,122119],{"className":122120,"style":270},[54],[15,122122,121119],{"className":122123,"style":121209},[59,60],[15,122125,122127,122130],{"className":122126},[59],[15,122128,223],{"className":122129,"style":25887},[59,60],[15,122131,122133],{"className":122132},[281],[15,122134,122136,122156],{"className":122135},[285,286],[15,122137,122139,122153],{"className":122138},[290],[15,122140,122142],{"className":122141,"style":379},[294],[15,122143,122144,122147],{"style":122076},[15,122145],{"className":122146,"style":303},[302],[15,122148,122150],{"className":122149},[307,308,309,310],[15,122151,213],{"className":122152},[59,60,310],[15,122154,318],{"className":122155},[317],[15,122157,122159],{"className":122158},[290],[15,122160,122162],{"className":122161,"style":325},[294],[15,122163],{},[15,122165,230],{"className":122166},[146],[15,122168,4160],{"className":122169,"style":981},[59,60],[15,122171,257],{"className":122172},[153],[15,122174,122176,122189],{"className":122175,"translate":19},[18],[15,122177,122179],{"className":122178},[23],[25,122180,122181],{"xmlns":27},[29,122182,122183,122187],{},[32,122184,122185],{},[35,122186,4160],{},[39,122188,4160],{"encoding":41},[15,122190,122192],{"className":122191,"ariaHidden":46},[45],[15,122193,122195,122198],{"className":122194},[50],[15,122196],{"className":122197,"style":2278},[54],[15,122199,4160],{"className":122200,"style":981},[59,60]," 的线性泛函，故其上确界在指示函数 ",[15,122203,122205,122227],{"className":122204,"translate":19},[18],[15,122206,122208],{"className":122207},[23],[25,122209,122210],{"xmlns":27},[29,122211,122212,122224],{},[32,122213,122214,122216,122218],{},[35,122215,4160],{},[110,122217,123],{},[195,122219,122220,122222],{},[115,122221,117],{"mathvariant":75252},[35,122223,603],{},[39,122225,122226],{"encoding":41},"g=\\mathbf1_A",[15,122228,122230,122248],{"className":122229,"ariaHidden":46},[45],[15,122231,122233,122236,122239,122242,122245],{"className":122232},[50],[15,122234],{"className":122235,"style":2278},[54],[15,122237,4160],{"className":122238,"style":981},[59,60],[15,122240],{"className":122241,"style":158},[157],[15,122243,123],{"className":122244},[162],[15,122246],{"className":122247,"style":158},[157],[15,122249,122251,122255],{"className":122250},[50],[15,122252],{"className":122253,"style":122254},[54],"height:0.7944em;vertical-align:-0.15em;",[15,122256,122258,122261],{"className":122257},[59],[15,122259,117],{"className":122260},[59,75597],[15,122262,122264],{"className":122263},[281],[15,122265,122267,122287],{"className":122266},[285,286],[15,122268,122270,122284],{"className":122269},[290],[15,122271,122273],{"className":122272,"style":379},[294],[15,122274,122275,122278],{"style":382},[15,122276],{"className":122277,"style":303},[302],[15,122279,122281],{"className":122280},[307,308,309,310],[15,122282,603],{"className":122283},[59,60,310],[15,122285,318],{"className":122286},[317],[15,122288,122290],{"className":122289},[290],[15,122291,122293],{"className":122292,"style":7970},[294],[15,122294],{}," 上达到：",[15,122297,122299],{"className":122298,"translate":19},[180],[15,122300,122302,122404],{"className":122301,"translate":19},[18],[15,122303,122305],{"className":122304},[23],[25,122306,122307],{"xmlns":27,"display":189},[29,122308,122309,122401],{},[32,122310,122311,122331,122333,122339,122341,122343,122345,122347,122349,122355,122357,122359,122361,122363,122365,122375,122377,122379,122381,122383,122385,122387,122389,122391,122393,122395,122397,122399],{},[6018,122312,122313,122319],{},[32,122314,122315,122317],{},[35,122316,35458],{},[110,122318,4105],{},[32,122320,122321,122323,122325,122327,122329],{},[115,122322,126],{},[110,122324,9954],{},[35,122326,4160],{},[110,122328,9954],{},[115,122330,117],{},[110,122332,2900],{"stretchy":112},[195,122334,122335,122337],{},[35,122336,223],{},[35,122338,206],{},[110,122340,230],{"stretchy":112},[35,122342,4160],{},[110,122344,257],{"stretchy":112},[110,122346,4815],{},[35,122348,121119],{},[195,122350,122351,122353],{},[35,122352,223],{},[35,122354,213],{},[110,122356,230],{"stretchy":112},[35,122358,4160],{},[110,122360,257],{"stretchy":112},[110,122362,2910],{"stretchy":112},[110,122364,123],{},[6018,122366,122367,122373],{},[32,122368,122369,122371],{},[35,122370,35458],{},[110,122372,4105],{},[35,122374,603],{},[110,122376,2900],{"stretchy":112},[35,122378,206],{},[110,122380,113],{"stretchy":112},[35,122382,603],{},[110,122384,120],{"stretchy":112},[110,122386,4815],{},[35,122388,121119],{},[35,122390,213],{},[110,122392,113],{"stretchy":112},[35,122394,603],{},[110,122396,120],{"stretchy":112},[110,122398,2910],{"stretchy":112},[110,122400,551],{"separator":46},[39,122402,122403],{"encoding":41},"\\sup_{0\\le g\\le1}\\{E_P[g]-\\gamma E_Q[g]\\}\n=\\sup_{A}\\{P(A)-\\gamma Q(A)\\},",[15,122405,122407,122535,122603,122682],{"className":122406,"ariaHidden":46},[45],[15,122408,122410,122414,122474,122477,122517,122520,122523,122526,122529,122532],{"className":122409},[50],[15,122411],{"className":122412,"style":122413},[54],"height:1.7977em;vertical-align:-1.0477em;",[15,122415,122417],{"className":122416},[1486,13720],[15,122418,122420,122465],{"className":122419},[285,286],[15,122421,122423,122462],{"className":122422},[290],[15,122424,122426,122452],{"className":122425,"style":2347},[294],[15,122427,122428,122431],{"style":60959},[15,122429],{"className":122430,"style":461},[302],[15,122432,122434],{"className":122433},[307,308,309,310],[15,122435,122437,122440,122443,122446,122449],{"className":122436},[59,310],[15,122438,126],{"className":122439},[59,310],[15,122441,9954],{"className":122442},[162,310],[15,122444,4160],{"className":122445,"style":981},[59,60,310],[15,122447,9954],{"className":122448},[162,310],[15,122450,117],{"className":122451},[59,310],[15,122453,122454,122457],{"style":11867},[15,122455],{"className":122456,"style":461},[302],[15,122458,122459],{},[15,122460,35458],{"className":122461},[1486],[15,122463,318],{"className":122464},[317],[15,122466,122468],{"className":122467},[290],[15,122469,122472],{"className":122470,"style":122471},[294],"height:1.0477em;",[15,122473],{},[15,122475,2900],{"className":122476},[146],[15,122478,122480,122483],{"className":122479},[59],[15,122481,223],{"className":122482,"style":25887},[59,60],[15,122484,122486],{"className":122485},[281],[15,122487,122489,122509],{"className":122488},[285,286],[15,122490,122492,122506],{"className":122491},[290],[15,122493,122495],{"className":122494,"style":379},[294],[15,122496,122497,122500],{"style":122076},[15,122498],{"className":122499,"style":303},[302],[15,122501,122503],{"className":122502},[307,308,309,310],[15,122504,206],{"className":122505,"style":334},[59,60,310],[15,122507,318],{"className":122508},[317],[15,122510,122512],{"className":122511},[290],[15,122513,122515],{"className":122514,"style":7970},[294],[15,122516],{},[15,122518,230],{"className":122519},[146],[15,122521,4160],{"className":122522,"style":981},[59,60],[15,122524,257],{"className":122525},[153],[15,122527],{"className":122528,"style":2589},[157],[15,122530,4815],{"className":122531},[2593],[15,122533],{"className":122534,"style":2589},[157],[15,122536,122538,122541,122544,122584,122587,122590,122594,122597,122600],{"className":122537},[50],[15,122539],{"className":122540,"style":270},[54],[15,122542,121119],{"className":122543,"style":121209},[59,60],[15,122545,122547,122550],{"className":122546},[59],[15,122548,223],{"className":122549,"style":25887},[59,60],[15,122551,122553],{"className":122552},[281],[15,122554,122556,122576],{"className":122555},[285,286],[15,122557,122559,122573],{"className":122558},[290],[15,122560,122562],{"className":122561,"style":379},[294],[15,122563,122564,122567],{"style":122076},[15,122565],{"className":122566,"style":303},[302],[15,122568,122570],{"className":122569},[307,308,309,310],[15,122571,213],{"className":122572},[59,60,310],[15,122574,318],{"className":122575},[317],[15,122577,122579],{"className":122578},[290],[15,122580,122582],{"className":122581,"style":325},[294],[15,122583],{},[15,122585,230],{"className":122586},[146],[15,122588,4160],{"className":122589,"style":981},[59,60],[15,122591,122593],{"className":122592},[153],"]}",[15,122595],{"className":122596,"style":158},[157],[15,122598,123],{"className":122599},[162],[15,122601],{"className":122602,"style":158},[157],[15,122604,122606,122610,122658,122661,122664,122667,122670,122673,122676,122679],{"className":122605},[50],[15,122607],{"className":122608,"style":122609},[54],"height:1.6888em;vertical-align:-0.9388em;",[15,122611,122613],{"className":122612},[1486,13720],[15,122614,122616,122649],{"className":122615},[285,286],[15,122617,122619,122646],{"className":122618},[290],[15,122620,122622,122636],{"className":122621,"style":2347},[294],[15,122623,122624,122627],{"style":36394},[15,122625],{"className":122626,"style":461},[302],[15,122628,122630],{"className":122629},[307,308,309,310],[15,122631,122633],{"className":122632},[59,310],[15,122634,603],{"className":122635},[59,60,310],[15,122637,122638,122641],{"style":11867},[15,122639],{"className":122640,"style":461},[302],[15,122642,122643],{},[15,122644,35458],{"className":122645},[1486],[15,122647,318],{"className":122648},[317],[15,122650,122652],{"className":122651},[290],[15,122653,122656],{"className":122654,"style":122655},[294],"height:0.9388em;",[15,122657],{},[15,122659,2900],{"className":122660},[146],[15,122662,206],{"className":122663,"style":334},[59,60],[15,122665,113],{"className":122666},[146],[15,122668,603],{"className":122669},[59,60],[15,122671,120],{"className":122672},[153],[15,122674],{"className":122675,"style":2589},[157],[15,122677,4815],{"className":122678},[2593],[15,122680],{"className":122681,"style":2589},[157],[15,122683,122685,122688,122691,122694,122697,122700,122703],{"className":122684},[50],[15,122686],{"className":122687,"style":139},[54],[15,122689,121119],{"className":122690,"style":121209},[59,60],[15,122692,213],{"className":122693},[59,60],[15,122695,113],{"className":122696},[146],[15,122698,603],{"className":122699},[59,60],[15,122701,61036],{"className":122702},[153],[15,122704,551],{"className":122705},[573],[11,122707,122708,122709,122882,122883,18734,123029,123144,123145,123293,123294,123580],{},"且由 ",[15,122710,122712,122754],{"className":122711,"translate":19},[18],[15,122713,122715],{"className":122714},[23],[25,122716,122717],{"xmlns":27},[29,122718,122719,122751],{},[32,122720,122721,122727,122729,122731,122733,122735,122741,122743,122745,122747,122749],{},[195,122722,122723,122725],{},[35,122724,223],{},[35,122726,206],{},[110,122728,230],{"stretchy":112},[35,122730,4160],{},[110,122732,257],{"stretchy":112},[110,122734,123],{},[195,122736,122737,122739],{},[35,122738,223],{},[35,122740,213],{},[110,122742,230],{"stretchy":112},[35,122744,4160],{},[1299,122746,1301],{},[35,122748,769],{},[110,122750,257],{"stretchy":112},[39,122752,122753],{"encoding":41},"E_P[g]=E_Q[g\\,t]",[15,122755,122757,122821],{"className":122756,"ariaHidden":46},[45],[15,122758,122760,122763,122803,122806,122809,122812,122815,122818],{"className":122759},[50],[15,122761],{"className":122762,"style":139},[54],[15,122764,122766,122769],{"className":122765},[59],[15,122767,223],{"className":122768,"style":25887},[59,60],[15,122770,122772],{"className":122771},[281],[15,122773,122775,122795],{"className":122774},[285,286],[15,122776,122778,122792],{"className":122777},[290],[15,122779,122781],{"className":122780,"style":379},[294],[15,122782,122783,122786],{"style":122076},[15,122784],{"className":122785,"style":303},[302],[15,122787,122789],{"className":122788},[307,308,309,310],[15,122790,206],{"className":122791,"style":334},[59,60,310],[15,122793,318],{"className":122794},[317],[15,122796,122798],{"className":122797},[290],[15,122799,122801],{"className":122800,"style":7970},[294],[15,122802],{},[15,122804,230],{"className":122805},[146],[15,122807,4160],{"className":122808,"style":981},[59,60],[15,122810,257],{"className":122811},[153],[15,122813],{"className":122814,"style":158},[157],[15,122816,123],{"className":122817},[162],[15,122819],{"className":122820,"style":158},[157],[15,122822,122824,122827,122867,122870,122873,122876,122879],{"className":122823},[50],[15,122825],{"className":122826,"style":270},[54],[15,122828,122830,122833],{"className":122829},[59],[15,122831,223],{"className":122832,"style":25887},[59,60],[15,122834,122836],{"className":122835},[281],[15,122837,122839,122859],{"className":122838},[285,286],[15,122840,122842,122856],{"className":122841},[290],[15,122843,122845],{"className":122844,"style":379},[294],[15,122846,122847,122850],{"style":122076},[15,122848],{"className":122849,"style":303},[302],[15,122851,122853],{"className":122852},[307,308,309,310],[15,122854,213],{"className":122855},[59,60,310],[15,122857,318],{"className":122858},[317],[15,122860,122862],{"className":122861},[290],[15,122863,122865],{"className":122864,"style":325},[294],[15,122866],{},[15,122868,230],{"className":122869},[146],[15,122871,4160],{"className":122872,"style":981},[59,60],[15,122874],{"className":122875,"style":406},[157],[15,122877,769],{"className":122878},[59,60],[15,122880,257],{"className":122881},[153],"，左端点态上确界为 ",[15,122884,122886,122922],{"className":122885,"translate":19},[18],[15,122887,122889],{"className":122888},[23],[25,122890,122891],{"xmlns":27},[29,122892,122893,122919],{},[32,122894,122895,122901,122903,122905,122907,122909,122911,122917],{},[195,122896,122897,122899],{},[35,122898,223],{},[35,122900,213],{},[110,122902,230],{"stretchy":112},[110,122904,113],{"stretchy":112},[35,122906,769],{},[110,122908,4815],{},[35,122910,121119],{},[6816,122912,122913,122915],{},[110,122914,120],{"stretchy":112},[110,122916,2472],{},[110,122918,257],{"stretchy":112},[39,122920,122921],{"encoding":41},"E_Q[(t-\\gamma)^+]",[15,122923,122925,122987],{"className":122924,"ariaHidden":46},[45],[15,122926,122928,122931,122971,122975,122978,122981,122984],{"className":122927},[50],[15,122929],{"className":122930,"style":270},[54],[15,122932,122934,122937],{"className":122933},[59],[15,122935,223],{"className":122936,"style":25887},[59,60],[15,122938,122940],{"className":122939},[281],[15,122941,122943,122963],{"className":122942},[285,286],[15,122944,122946,122960],{"className":122945},[290],[15,122947,122949],{"className":122948,"style":379},[294],[15,122950,122951,122954],{"style":122076},[15,122952],{"className":122953,"style":303},[302],[15,122955,122957],{"className":122956},[307,308,309,310],[15,122958,213],{"className":122959},[59,60,310],[15,122961,318],{"className":122962},[317],[15,122964,122966],{"className":122965},[290],[15,122967,122969],{"className":122968,"style":325},[294],[15,122970],{},[15,122972,122974],{"className":122973},[146],"[(",[15,122976,769],{"className":122977},[59,60],[15,122979],{"className":122980,"style":2589},[157],[15,122982,4815],{"className":122983},[2593],[15,122985],{"className":122986,"style":2589},[157],[15,122988,122990,122994,122997,123026],{"className":122989},[50],[15,122991],{"className":122992,"style":122993},[54],"height:1.0213em;vertical-align:-0.25em;",[15,122995,121119],{"className":122996,"style":121209},[59,60],[15,122998,123000,123003],{"className":122999},[153],[15,123001,120],{"className":123002},[153],[15,123004,123006],{"className":123005},[281],[15,123007,123009],{"className":123008},[285],[15,123010,123012],{"className":123011},[290],[15,123013,123015],{"className":123014,"style":21035},[294],[15,123016,123017,123020],{"style":6867},[15,123018],{"className":123019,"style":303},[302],[15,123021,123023],{"className":123022},[307,308,309,310],[15,123024,2472],{"className":123025},[2593,310],[15,123027,257],{"className":123028},[153],[15,123030,123032,123054],{"className":123031,"translate":19},[18],[15,123033,123035],{"className":123034},[23],[25,123036,123037],{"xmlns":27},[29,123038,123039,123051],{},[32,123040,123041],{},[195,123042,123043,123045],{},[35,123044,199],{},[195,123046,123047,123049],{},[35,123048,4160],{},[35,123050,121119],{},[39,123052,123053],{"encoding":41},"D_{g_\\gamma}",[15,123055,123057],{"className":123056,"ariaHidden":46},[45],[15,123058,123060,123064],{"className":123059},[50],[15,123061],{"className":123062,"style":123063},[54],"height:1.0307em;vertical-align:-0.3473em;",[15,123065,123067,123070],{"className":123066},[59],[15,123068,199],{"className":123069,"style":277},[59,60],[15,123071,123073],{"className":123072},[281],[15,123074,123076,123136],{"className":123075},[285,286],[15,123077,123079,123133],{"className":123078},[290],[15,123080,123082],{"className":123081,"style":44955},[294],[15,123083,123084,123087],{"style":298},[15,123085],{"className":123086,"style":303},[302],[15,123088,123090],{"className":123089},[307,308,309,310],[15,123091,123093],{"className":123092},[59,310],[15,123094,123096,123099],{"className":123095},[59,310],[15,123097,4160],{"className":123098,"style":981},[59,60,310],[15,123100,123102],{"className":123101},[281],[15,123103,123105,123125],{"className":123104},[285,286],[15,123106,123108,123122],{"className":123107},[290],[15,123109,123111],{"className":123110,"style":74299},[294],[15,123112,123113,123116],{"style":121744},[15,123114],{"className":123115,"style":30232},[302],[15,123117,123119],{"className":123118},[307,30236,9676,310],[15,123120,121119],{"className":123121,"style":121209},[59,60,310],[15,123123,318],{"className":123124},[317],[15,123126,123128],{"className":123127},[290],[15,123129,123131],{"className":123130,"style":121763},[294],[15,123132],{},[15,123134,318],{"className":123135},[317],[15,123137,123139],{"className":123138},[290],[15,123140,123142],{"className":123141,"style":121775},[294],[15,123143],{},"。另一方面取 ",[15,123146,123148,123184],{"className":123147,"translate":19},[18],[15,123149,123151],{"className":123150},[23],[25,123152,123153],{"xmlns":27},[29,123154,123155,123181],{},[32,123156,123157,123163,123165],{},[6816,123158,123159,123161],{},[35,123160,4160],{},[110,123162,7575],{},[110,123164,123],{},[195,123166,123167,123169],{},[115,123168,117],{"mathvariant":75252},[32,123170,123171,123173,123175,123177,123179],{},[110,123172,2900],{"stretchy":112},[35,123174,769],{},[110,123176,2905],{},[35,123178,121119],{},[110,123180,2910],{"stretchy":112},[39,123182,123183],{"encoding":41},"g^*=\\mathbf 1_{\\{t>\\gamma\\}}",[15,123185,123187,123231],{"className":123186,"ariaHidden":46},[45],[15,123188,123190,123193,123222,123225,123228],{"className":123189},[50],[15,123191],{"className":123192,"style":64052},[54],[15,123194,123196,123199],{"className":123195},[59],[15,123197,4160],{"className":123198,"style":981},[59,60],[15,123200,123202],{"className":123201},[281],[15,123203,123205],{"className":123204},[285],[15,123206,123208],{"className":123207},[290],[15,123209,123211],{"className":123210,"style":8501},[294],[15,123212,123213,123216],{"style":6867},[15,123214],{"className":123215,"style":303},[302],[15,123217,123219],{"className":123218},[307,308,309,310],[15,123220,7575],{"className":123221},[2593,310],[15,123223],{"className":123224,"style":158},[157],[15,123226,123],{"className":123227},[162],[15,123229],{"className":123230,"style":158},[157],[15,123232,123234,123238],{"className":123233},[50],[15,123235],{"className":123236,"style":123237},[54],"height:0.9996em;vertical-align:-0.3552em;",[15,123239,123241,123244],{"className":123240},[59],[15,123242,117],{"className":123243},[59,75597],[15,123245,123247],{"className":123246},[281],[15,123248,123250,123285],{"className":123249},[285,286],[15,123251,123253,123282],{"className":123252},[290],[15,123254,123256],{"className":123255,"style":6100},[294],[15,123257,123258,123261],{"style":75612},[15,123259],{"className":123260,"style":303},[302],[15,123262,123264],{"className":123263},[307,308,309,310],[15,123265,123267,123270,123273,123276,123279],{"className":123266},[59,310],[15,123268,2900],{"className":123269},[146,310],[15,123271,769],{"className":123272},[59,60,310],[15,123274,2905],{"className":123275},[162,310],[15,123277,121119],{"className":123278,"style":121209},[59,60,310],[15,123280,2910],{"className":123281},[153,310],[15,123283,318],{"className":123284},[317],[15,123286,123288],{"className":123287},[290],[15,123289,123291],{"className":123290,"style":6146},[294],[15,123292],{}," 恰有 ",[15,123295,123297,123361],{"className":123296,"translate":19},[18],[15,123298,123300],{"className":123299},[23],[25,123301,123302],{"xmlns":27},[29,123303,123304,123358],{},[32,123305,123306,123312,123314,123320,123322,123324,123326,123328,123330,123332,123334,123340,123342,123344,123346,123348,123350,123356],{},[195,123307,123308,123310],{},[35,123309,223],{},[35,123311,213],{},[110,123313,230],{"stretchy":112},[6816,123315,123316,123318],{},[35,123317,4160],{},[110,123319,7575],{},[110,123321,113],{"stretchy":112},[35,123323,769],{},[110,123325,4815],{},[35,123327,121119],{},[110,123329,120],{"stretchy":112},[110,123331,257],{"stretchy":112},[110,123333,123],{},[195,123335,123336,123338],{},[35,123337,223],{},[35,123339,213],{},[110,123341,230],{"stretchy":112},[110,123343,113],{"stretchy":112},[35,123345,769],{},[110,123347,4815],{},[35,123349,121119],{},[6816,123351,123352,123354],{},[110,123353,120],{"stretchy":112},[110,123355,2472],{},[110,123357,257],{"stretchy":112},[39,123359,123360],{"encoding":41},"E_Q[g^*(t-\\gamma)]=E_Q[(t-\\gamma)^+]",[15,123362,123364,123457,123478,123539],{"className":123363,"ariaHidden":46},[45],[15,123365,123367,123370,123410,123413,123442,123445,123448,123451,123454],{"className":123366},[50],[15,123368],{"className":123369,"style":270},[54],[15,123371,123373,123376],{"className":123372},[59],[15,123374,223],{"className":123375,"style":25887},[59,60],[15,123377,123379],{"className":123378},[281],[15,123380,123382,123402],{"className":123381},[285,286],[15,123383,123385,123399],{"className":123384},[290],[15,123386,123388],{"className":123387,"style":379},[294],[15,123389,123390,123393],{"style":122076},[15,123391],{"className":123392,"style":303},[302],[15,123394,123396],{"className":123395},[307,308,309,310],[15,123397,213],{"className":123398},[59,60,310],[15,123400,318],{"className":123401},[317],[15,123403,123405],{"className":123404},[290],[15,123406,123408],{"className":123407,"style":325},[294],[15,123409],{},[15,123411,230],{"className":123412},[146],[15,123414,123416,123419],{"className":123415},[59],[15,123417,4160],{"className":123418,"style":981},[59,60],[15,123420,123422],{"className":123421},[281],[15,123423,123425],{"className":123424},[285],[15,123426,123428],{"className":123427},[290],[15,123429,123431],{"className":123430,"style":8501},[294],[15,123432,123433,123436],{"style":6867},[15,123434],{"className":123435,"style":303},[302],[15,123437,123439],{"className":123438},[307,308,309,310],[15,123440,7575],{"className":123441},[2593,310],[15,123443,113],{"className":123444},[146],[15,123446,769],{"className":123447},[59,60],[15,123449],{"className":123450,"style":2589},[157],[15,123452,4815],{"className":123453},[2593],[15,123455],{"className":123456,"style":2589},[157],[15,123458,123460,123463,123466,123469,123472,123475],{"className":123459},[50],[15,123461],{"className":123462,"style":139},[54],[15,123464,121119],{"className":123465,"style":121209},[59,60],[15,123467,20416],{"className":123468},[153],[15,123470],{"className":123471,"style":158},[157],[15,123473,123],{"className":123474},[162],[15,123476],{"className":123477,"style":158},[157],[15,123479,123481,123484,123524,123527,123530,123533,123536],{"className":123480},[50],[15,123482],{"className":123483,"style":270},[54],[15,123485,123487,123490],{"className":123486},[59],[15,123488,223],{"className":123489,"style":25887},[59,60],[15,123491,123493],{"className":123492},[281],[15,123494,123496,123516],{"className":123495},[285,286],[15,123497,123499,123513],{"className":123498},[290],[15,123500,123502],{"className":123501,"style":379},[294],[15,123503,123504,123507],{"style":122076},[15,123505],{"className":123506,"style":303},[302],[15,123508,123510],{"className":123509},[307,308,309,310],[15,123511,213],{"className":123512},[59,60,310],[15,123514,318],{"className":123515},[317],[15,123517,123519],{"className":123518},[290],[15,123520,123522],{"className":123521,"style":325},[294],[15,123523],{},[15,123525,122974],{"className":123526},[146],[15,123528,769],{"className":123529},[59,60],[15,123531],{"className":123532,"style":2589},[157],[15,123534,4815],{"className":123535},[2593],[15,123537],{"className":123538,"style":2589},[157],[15,123540,123542,123545,123548,123577],{"className":123541},[50],[15,123543],{"className":123544,"style":122993},[54],[15,123546,121119],{"className":123547,"style":121209},[59,60],[15,123549,123551,123554],{"className":123550},[153],[15,123552,120],{"className":123553},[153],[15,123555,123557],{"className":123556},[281],[15,123558,123560],{"className":123559},[285],[15,123561,123563],{"className":123562},[290],[15,123564,123566],{"className":123565,"style":21035},[294],[15,123567,123568,123571],{"style":6867},[15,123569],{"className":123570,"style":303},[302],[15,123572,123574],{"className":123573},[307,308,309,310],[15,123575,2472],{"className":123576},[2593,310],[15,123578,257],{"className":123579},[153],"，故等号可取。",[11,123582,123583,123584,123612,123613,123664,123665,123701,123702,15799,123805,123856,123857,124024,124025,124083,124084,124112],{},"几何解释：",[15,123585,123587,123600],{"className":123586,"translate":19},[18],[15,123588,123590],{"className":123589},[23],[25,123591,123592],{"xmlns":27},[29,123593,123594,123598],{},[32,123595,123596],{},[35,123597,121119],{},[39,123599,121536],{"encoding":41},[15,123601,123603],{"className":123602,"ariaHidden":46},[45],[15,123604,123606,123609],{"className":123605},[50],[15,123607],{"className":123608,"style":2278},[54],[15,123610,121119],{"className":123611,"style":121209},[59,60]," 扫过 ",[15,123614,123616,123637],{"className":123615,"translate":19},[18],[15,123617,123619],{"className":123618},[23],[25,123620,123621],{"xmlns":27},[29,123622,123623,123635],{},[32,123624,123625,123627,123629,123631,123633],{},[110,123626,230],{"stretchy":112},[115,123628,126],{},[110,123630,551],{"separator":46},[115,123632,117],{},[110,123634,257],{"stretchy":112},[39,123636,36840],{"encoding":41},[15,123638,123640],{"className":123639,"ariaHidden":46},[45],[15,123641,123643,123646,123649,123652,123655,123658,123661],{"className":123642},[50],[15,123644],{"className":123645,"style":139},[54],[15,123647,230],{"className":123648},[146],[15,123650,126],{"className":123651},[59],[15,123653,551],{"className":123654},[573],[15,123656],{"className":123657,"style":406},[157],[15,123659,117],{"className":123660},[59],[15,123662,257],{"className":123663},[153]," 时，右端就是权衡曲线（ROC 下边界）各条斜率为 ",[15,123666,123668,123686],{"className":123667,"translate":19},[18],[15,123669,123671],{"className":123670},[23],[25,123672,123673],{"xmlns":27},[29,123674,123675,123683],{},[32,123676,123677,123679,123681],{},[115,123678,117],{},[35,123680,941],{"mathvariant":209},[35,123682,121119],{},[39,123684,123685],{"encoding":41},"1\u002F\\gamma",[15,123687,123689],{"className":123688,"ariaHidden":46},[45],[15,123690,123692,123695,123698],{"className":123691},[50],[15,123693],{"className":123694,"style":139},[54],[15,123696,8591],{"className":123697},[59],[15,123699,121119],{"className":123700,"style":121209},[59,60]," 的支撑线的截距——",[998,123703,123704,123705,123733,123734,123804],{},"知道整条曲线 ",[15,123706,123708,123721],{"className":123707,"translate":19},[18],[15,123709,123711],{"className":123710},[23],[25,123712,123713],{"xmlns":27},[29,123714,123715,123719],{},[32,123716,123717],{},[35,123718,5394],{},[39,123720,5394],{"encoding":41},[15,123722,123724],{"className":123723,"ariaHidden":46},[45],[15,123725,123727,123730],{"className":123726},[50],[15,123728],{"className":123729,"style":664},[54],[15,123731,5394],{"className":123732,"style":334},[59,60]," 就知道全部 ",[15,123735,123737,123755],{"className":123736,"translate":19},[18],[15,123738,123740],{"className":123739},[23],[25,123741,123742],{"xmlns":27},[29,123743,123744,123752],{},[32,123745,123746],{},[195,123747,123748,123750],{},[35,123749,4160],{},[35,123751,121119],{},[39,123753,123754],{"encoding":41},"g_\\gamma",[15,123756,123758],{"className":123757,"ariaHidden":46},[45],[15,123759,123761,123764],{"className":123760},[50],[15,123762],{"className":123763,"style":93001},[54],[15,123765,123767,123770],{"className":123766},[59],[15,123768,4160],{"className":123769,"style":981},[59,60],[15,123771,123773],{"className":123772},[281],[15,123774,123776,123796],{"className":123775},[285,286],[15,123777,123779,123793],{"className":123778},[290],[15,123780,123782],{"className":123781,"style":44955},[294],[15,123783,123784,123787],{"style":18869},[15,123785],{"className":123786,"style":303},[302],[15,123788,123790],{"className":123789},[307,308,309,310],[15,123791,121119],{"className":123792,"style":121209},[59,60,310],[15,123794,318],{"className":123795},[317],[15,123797,123799],{"className":123798},[290],[15,123800,123802],{"className":123801,"style":325},[294],[15,123803],{},"-散度",[15,123806,123808,123826],{"className":123807,"translate":19},[18],[15,123809,123811],{"className":123810},[23],[25,123812,123813],{"xmlns":27},[29,123814,123815,123823],{},[32,123816,123817,123819,123821],{},[35,123818,121119],{},[110,123820,123],{},[115,123822,117],{},[39,123824,123825],{"encoding":41},"\\gamma=1",[15,123827,123829,123847],{"className":123828,"ariaHidden":46},[45],[15,123830,123832,123835,123838,123841,123844],{"className":123831},[50],[15,123833],{"className":123834,"style":2278},[54],[15,123836,121119],{"className":123837,"style":121209},[59,60],[15,123839],{"className":123840,"style":158},[157],[15,123842,123],{"className":123843},[162],[15,123845],{"className":123846,"style":158},[157],[15,123848,123850,123853],{"className":123849},[50],[15,123851],{"className":123852,"style":172},[54],[15,123854,117],{"className":123855},[59]," 时立刻给出 ",[15,123858,123860,123912],{"className":123859,"translate":19},[18],[15,123861,123863],{"className":123862},[23],[25,123864,123865],{"xmlns":27},[29,123866,123867,123909],{},[32,123868,123869,123875,123877,123887,123889,123891,123893,123895,123897,123899,123901,123903,123905,123907],{},[32,123870,123871,123873],{},[35,123872,5394],{"mathvariant":209},[35,123874,34876],{"mathvariant":209},[110,123876,123],{},[195,123878,123879,123885],{},[32,123880,123881,123883],{},[35,123882,35458],{},[110,123884,4105],{},[35,123886,603],{},[110,123888,2900],{"stretchy":112},[35,123890,206],{},[110,123892,113],{"stretchy":112},[35,123894,603],{},[110,123896,120],{"stretchy":112},[110,123898,4815],{},[35,123900,213],{},[110,123902,113],{"stretchy":112},[35,123904,603],{},[110,123906,120],{"stretchy":112},[110,123908,2910],{"stretchy":112},[39,123910,123911],{"encoding":41},"\\mathrm{TV}=\\sup_A\\{P(A)-Q(A)\\}",[15,123913,123915,123936,124006],{"className":123914,"ariaHidden":46},[45],[15,123916,123918,123921,123927,123930,123933],{"className":123917},[50],[15,123919],{"className":123920,"style":664},[54],[15,123922,123924],{"className":123923},[59],[15,123925,86803],{"className":123926,"style":4159},[59,32688],[15,123928],{"className":123929,"style":158},[157],[15,123931,123],{"className":123932},[162],[15,123934],{"className":123935,"style":158},[157],[15,123937,123939,123942,123982,123985,123988,123991,123994,123997,124000,124003],{"className":123938},[50],[15,123940],{"className":123941,"style":139},[54],[15,123943,123945,123948],{"className":123944},[1486],[15,123946,35458],{"className":123947},[1486],[15,123949,123951],{"className":123950},[281],[15,123952,123954,123974],{"className":123953},[285,286],[15,123955,123957,123971],{"className":123956},[290],[15,123958,123960],{"className":123959,"style":35601},[294],[15,123961,123962,123965],{"style":35604},[15,123963],{"className":123964,"style":303},[302],[15,123966,123968],{"className":123967},[307,308,309,310],[15,123969,603],{"className":123970},[59,60,310],[15,123972,318],{"className":123973},[317],[15,123975,123977],{"className":123976},[290],[15,123978,123980],{"className":123979,"style":37046},[294],[15,123981],{},[15,123983,2900],{"className":123984},[146],[15,123986,206],{"className":123987,"style":334},[59,60],[15,123989,113],{"className":123990},[146],[15,123992,603],{"className":123993},[59,60],[15,123995,120],{"className":123996},[153],[15,123998],{"className":123999,"style":2589},[157],[15,124001,4815],{"className":124002},[2593],[15,124004],{"className":124005,"style":2589},[157],[15,124007,124009,124012,124015,124018,124021],{"className":124008},[50],[15,124010],{"className":124011,"style":139},[54],[15,124013,213],{"className":124014},[59,60],[15,124016,113],{"className":124017},[146],[15,124019,603],{"className":124020},[59,60],[15,124022,61036],{"className":124023},[153],"）。反过来，知道单个标量散度（如 KL 或 ",[15,124026,124028,124045],{"className":124027,"translate":19},[18],[15,124029,124031],{"className":124030},[23],[25,124032,124033],{"xmlns":27},[29,124034,124035,124043],{},[32,124036,124037],{},[6816,124038,124039,124041],{},[35,124040,37475],{},[115,124042,2465],{},[39,124044,40420],{"encoding":41},[15,124046,124048],{"className":124047,"ariaHidden":46},[45],[15,124049,124051,124054],{"className":124050},[50],[15,124052],{"className":124053,"style":40430},[54],[15,124055,124057,124060],{"className":124056},[59],[15,124058,37475],{"className":124059},[59,60],[15,124061,124063],{"className":124062},[281],[15,124064,124066],{"className":124065},[285],[15,124067,124069],{"className":124068},[290],[15,124070,124072],{"className":124071,"style":20054},[294],[15,124073,124074,124077],{"style":6867},[15,124075],{"className":124076,"style":303},[302],[15,124078,124080],{"className":124079},[307,308,309,310],[15,124081,2465],{"className":124082},[59,310],"）只对 ",[15,124085,124087,124100],{"className":124086,"translate":19},[18],[15,124088,124090],{"className":124089},[23],[25,124091,124092],{"xmlns":27},[29,124093,124094,124098],{},[32,124095,124096],{},[35,124097,5394],{},[39,124099,5394],{"encoding":41},[15,124101,124103],{"className":124102,"ariaHidden":46},[45],[15,124104,124106,124109],{"className":124105},[50],[15,124107],{"className":124108,"style":664},[54],[15,124110,5394],{"className":124111,"style":334},[59,60]," 施加一个切向约束（4.7 节的不等式网络），信息严格更少；",[10029,124114,124115,124347],{},[6667,124116,124117,124118,19992],{},"在「小隐私预算」下，所有 f-DP 都由**高斯差分隐私（GDP）**统一逼近，其权衡函数 ",[15,124119,124121,124179],{"className":124120,"translate":19},[18],[15,124122,124124],{"className":124123},[23],[25,124125,124126],{"xmlns":27},[29,124127,124128,124176],{},[32,124129,124130,124137,124139,124141,124143,124145,124148,124150,124160,124162,124164,124166,124168,124170,124172,124174],{},[195,124131,124132,124135],{},[35,124133,124134],{},"G",[35,124136,2366],{},[110,124138,113],{"stretchy":112},[35,124140,44780],{},[110,124142,120],{"stretchy":112},[110,124144,123],{},[35,124146,124147],{"mathvariant":209},"Φ",[110,124149,113],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[6816,124151,124152,124154],{},[35,124153,124147],{"mathvariant":209},[32,124155,124156,124158],{},[110,124157,4815],{},[115,124159,117],{},[110,124161,113],{"stretchy":112},[115,124163,117],{},[110,124165,4815],{},[35,124167,44780],{},[110,124169,120],{"stretchy":112},[110,124171,4815],{},[35,124173,2366],{},[110,124175,120],{"fence":112,"stretchy":46,"minsize":9452,"maxsize":9452},[39,124177,124178],{"encoding":41},"G_\\mu(\\alpha)=\\Phi\\big(\\Phi^{-1}(1-\\alpha)-\\mu\\big)",[15,124180,124182,124246,124311,124332],{"className":124181,"ariaHidden":46},[45],[15,124183,124185,124188,124228,124231,124234,124237,124240,124243],{"className":124184},[50],[15,124186],{"className":124187,"style":270},[54],[15,124189,124191,124194],{"className":124190},[59],[15,124192,124134],{"className":124193},[59,60],[15,124195,124197],{"className":124196},[281],[15,124198,124200,124220],{"className":124199},[285,286],[15,124201,124203,124217],{"className":124202},[290],[15,124204,124206],{"className":124205,"style":44955},[294],[15,124207,124208,124211],{"style":382},[15,124209],{"className":124210,"style":303},[302],[15,124212,124214],{"className":124213},[307,308,309,310],[15,124215,2366],{"className":124216},[59,60,310],[15,124218,318],{"className":124219},[317],[15,124221,124223],{"className":124222},[290],[15,124224,124226],{"className":124225,"style":325},[294],[15,124227],{},[15,124229,113],{"className":124230},[146],[15,124232,44780],{"className":124233,"style":44810},[59,60],[15,124235,120],{"className":124236},[153],[15,124238],{"className":124239,"style":158},[157],[15,124241,123],{"className":124242},[162],[15,124244],{"className":124245,"style":158},[157],[15,124247,124249,124252,124255,124261,124296,124299,124302,124305,124308],{"className":124248},[50],[15,124250],{"className":124251,"style":20589},[54],[15,124253,124147],{"className":124254},[59],[15,124256,124258],{"className":124257},[59],[15,124259,113],{"className":124260},[419,9676],[15,124262,124264,124267],{"className":124263},[59],[15,124265,124147],{"className":124266},[59],[15,124268,124270],{"className":124269},[281],[15,124271,124273],{"className":124272},[285],[15,124274,124276],{"className":124275},[290],[15,124277,124279],{"className":124278,"style":20054},[294],[15,124280,124281,124284],{"style":6867},[15,124282],{"className":124283,"style":303},[302],[15,124285,124287],{"className":124286},[307,308,309,310],[15,124288,124290,124293],{"className":124289},[59,310],[15,124291,4815],{"className":124292},[59,310],[15,124294,117],{"className":124295},[59,310],[15,124297,113],{"className":124298},[146],[15,124300,117],{"className":124301},[59],[15,124303],{"className":124304,"style":2589},[157],[15,124306,4815],{"className":124307},[2593],[15,124309],{"className":124310,"style":2589},[157],[15,124312,124314,124317,124320,124323,124326,124329],{"className":124313},[50],[15,124315],{"className":124316,"style":139},[54],[15,124318,44780],{"className":124319,"style":44810},[59,60],[15,124321,120],{"className":124322},[153],[15,124324],{"className":124325,"style":2589},[157],[15,124327,4815],{"className":124328},[2593],[15,124330],{"className":124331,"style":2589},[157],[15,124333,124335,124338,124341],{"className":124334},[50],[15,124336],{"className":124337,"style":20589},[54],[15,124339,2366],{"className":124340},[59,60],[15,124342,124344],{"className":124343},[59],[15,124345,120],{"className":124346},[419,9676],[6667,124348,124349,124350,124353,124354,124382],{},"权衡函数族在",[998,124351,124352],{},"组合","（post-processing、串联）下封闭，并装备了一个与 f-散度\"散度序\"同构的偏序——这与 f-散度族的\"",[15,124355,124357,124370],{"className":124356,"translate":19},[18],[15,124358,124360],{"className":124359},[23],[25,124361,124362],{"xmlns":27},[29,124363,124364,124368],{},[32,124365,124366],{},[110,124367,6930],{},[39,124369,55517],{"encoding":41},[15,124371,124373],{"className":124372,"ariaHidden":46},[45],[15,124374,124376,124379],{"className":124375},[50],[15,124377],{"className":124378,"style":43214},[54],[15,124380,6930],{"className":124381},[162]," 比较\"在结构上完全平行。",[11,124384,124385],{},"这说明 f-散度不仅统一了信息论度量，也统一了现代隐私的定义框架：用一条凸曲线代替一个标量数字。",[528,124387,124389],{"id":124388},"_9-计算实例","9. 计算实例",[2313,124391,124393],{"id":124392},"_91-正态分布","9.1 正态分布",[11,124395,534,124396,5647,124571,10182],{},[15,124397,124399,124437],{"className":124398,"translate":19},[18],[15,124400,124402],{"className":124401},[23],[25,124403,124404],{"xmlns":27},[29,124405,124406,124434],{},[32,124407,124408,124410,124412,124414,124416,124422,124424,124432],{},[35,124409,206],{},[110,124411,123],{},[35,124413,38310],{},[110,124415,113],{"stretchy":112},[195,124417,124418,124420],{},[35,124419,2366],{},[115,124421,117],{},[110,124423,551],{"separator":46},[7555,124425,124426,124428,124430],{},[35,124427,2334],{},[115,124429,117],{},[115,124431,2465],{},[110,124433,120],{"stretchy":112},[39,124435,124436],{"encoding":41},"P=N(\\mu_1,\\sigma_1^2)",[15,124438,124440,124458],{"className":124439,"ariaHidden":46},[45],[15,124441,124443,124446,124449,124452,124455],{"className":124442},[50],[15,124444],{"className":124445,"style":664},[54],[15,124447,206],{"className":124448,"style":334},[59,60],[15,124450],{"className":124451,"style":158},[157],[15,124453,123],{"className":124454},[162],[15,124456],{"className":124457,"style":158},[157],[15,124459,124461,124464,124467,124470,124510,124513,124516,124568],{"className":124460},[50],[15,124462],{"className":124463,"style":20029},[54],[15,124465,38310],{"className":124466,"style":25627},[59,60],[15,124468,113],{"className":124469},[146],[15,124471,124473,124476],{"className":124472},[59],[15,124474,2366],{"className":124475},[59,60],[15,124477,124479],{"className":124478},[281],[15,124480,124482,124502],{"className":124481},[285,286],[15,124483,124485,124499],{"className":124484},[290],[15,124486,124488],{"className":124487,"style":11137},[294],[15,124489,124490,124493],{"style":382},[15,124491],{"className":124492,"style":303},[302],[15,124494,124496],{"className":124495},[307,308,309,310],[15,124497,117],{"className":124498},[59,310],[15,124500,318],{"className":124501},[317],[15,124503,124505],{"className":124504},[290],[15,124506,124508],{"className":124507,"style":7970},[294],[15,124509],{},[15,124511,551],{"className":124512},[573],[15,124514],{"className":124515,"style":406},[157],[15,124517,124519,124522],{"className":124518},[59],[15,124520,2334],{"className":124521,"style":981},[59,60],[15,124523,124525],{"className":124524},[281],[15,124526,124528,124560],{"className":124527},[285,286],[15,124529,124531,124557],{"className":124530},[290],[15,124532,124534,124546],{"className":124533,"style":20054},[294],[15,124535,124537,124540],{"style":124536},"top:-2.4519em;margin-left:-0.0359em;margin-right:0.05em;",[15,124538],{"className":124539,"style":303},[302],[15,124541,124543],{"className":124542},[307,308,309,310],[15,124544,117],{"className":124545},[59,310],[15,124547,124548,124551],{"style":6867},[15,124549],{"className":124550,"style":303},[302],[15,124552,124554],{"className":124553},[307,308,309,310],[15,124555,2465],{"className":124556},[59,310],[15,124558,318],{"className":124559},[317],[15,124561,124563],{"className":124562},[290],[15,124564,124566],{"className":124565,"style":52150},[294],[15,124567],{},[15,124569,120],{"className":124570},[153],[15,124572,124574,124612],{"className":124573,"translate":19},[18],[15,124575,124577],{"className":124576},[23],[25,124578,124579],{"xmlns":27},[29,124580,124581,124609],{},[32,124582,124583,124585,124587,124589,124591,124597,124599,124607],{},[35,124584,213],{},[110,124586,123],{},[35,124588,38310],{},[110,124590,113],{"stretchy":112},[195,124592,124593,124595],{},[35,124594,2366],{},[115,124596,2465],{},[110,124598,551],{"separator":46},[7555,124600,124601,124603,124605],{},[35,124602,2334],{},[115,124604,2465],{},[115,124606,2465],{},[110,124608,120],{"stretchy":112},[39,124610,124611],{"encoding":41},"Q=N(\\mu_2,\\sigma_2^2)",[15,124613,124615,124633],{"className":124614,"ariaHidden":46},[45],[15,124616,124618,124621,124624,124627,124630],{"className":124617},[50],[15,124619],{"className":124620,"style":566},[54],[15,124622,213],{"className":124623},[59,60],[15,124625],{"className":124626,"style":158},[157],[15,124628,123],{"className":124629},[162],[15,124631],{"className":124632,"style":158},[157],[15,124634,124636,124639,124642,124645,124685,124688,124691,124742],{"className":124635},[50],[15,124637],{"className":124638,"style":20029},[54],[15,124640,38310],{"className":124641,"style":25627},[59,60],[15,124643,113],{"className":124644},[146],[15,124646,124648,124651],{"className":124647},[59],[15,124649,2366],{"className":124650},[59,60],[15,124652,124654],{"className":124653},[281],[15,124655,124657,124677],{"className":124656},[285,286],[15,124658,124660,124674],{"className":124659},[290],[15,124661,124663],{"className":124662,"style":11137},[294],[15,124664,124665,124668],{"style":382},[15,124666],{"className":124667,"style":303},[302],[15,124669,124671],{"className":124670},[307,308,309,310],[15,124672,2465],{"className":124673},[59,310],[15,124675,318],{"className":124676},[317],[15,124678,124680],{"className":124679},[290],[15,124681,124683],{"className":124682,"sty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计算 ",[15,125865,125867,125891],{"className":125866,"translate":19},[18],[15,125868,125870],{"className":125869},[23],[25,125871,125872],{"xmlns":27},[29,125873,125874,125888],{},[32,125875,125876,125878,125884,125886],{},[110,125877,1289],{},[6816,125879,125880,125882],{},[35,125881,11],{},[115,125883,2465],{},[35,125885,941],{"mathvariant":209},[35,125887,944],{},[39,125889,125890],{"encoding":41},"\\int p^2\u002Fq",[15,125892,125894],{"className":125893,"ariaHidden":46},[45],[15,125895,125897,125900,125903,125906,125935,125938],{"className":125896},[50],[15,125898],{"className":125899,"style":50354},[54],[15,125901,1289],{"className":125902,"style":5619},[1486,1487,5618],[15,125904],{"className":125905,"style":406},[157],[15,125907,125909,125912],{"className":125908},[59],[15,125910,11],{"className":125911},[59,60],[15,125913,125915],{"className":125914},[281],[15,125916,125918],{"className":125917},[285],[15,125919,125921],{"className":125920},[290],[15,125922,125924],{"className":125923,"style":20054},[294],[15,125925,125926,125929],{"style":6867},[15,125927],{"className":125928,"style":303},[302],[15,125930,125932],{"className":125931},[307,308,309,310],[15,125933,2465],{"className":125934},[59,310],[15,125936,941],{"className":125937},[59],[15,125939,944],{"className":125940,"style":981},[59,60],"：由配方可得",[15,125943,125945],{"className":125944,"translate":19},[180],[15,125946,125948,126009],{"className":125947,"translate":19},[18],[15,125949,125951],{"className":125950},[23],[25,125952,125953],{"xmlns":27,"display":189},[29,125954,125955,126006],{},[32,125956,125957,125959,125969,125971,125973,125975,125977,125980,125982,125984,126004],{},[110,125958,1289],{},[238,125960,125961,125967],{},[6816,125962,125963,125965],{},[35,125964,11],{},[115,125966,2465],{},[35,125968,944],{},[1299,125970,1301],{},[35,125972,244],{},[35,125974,774],{},[110,125976,123],{},[35,125978,125979],{},"exp",[110,125981,4105],{},[1299,125983,1306],{},[32,125985,125986,125988,126002],{},[110,125987,113],{"fence":46},[238,125989,125990,125996],{},[6816,125991,125992,125994],{},[35,125993,125721],{"mathvariant":209},[115,125995,2465],{},[6816,125997,125998,126000],{},[35,125999,2334],{},[115,126001,2465],{},[110,126003,120],{"fence":46},[110,126005,551],{"separator":46},[39,126007,126008],{"encoding":41},"\\int \\frac{p^2}{q}\\,dx\n=\\exp\\!\\left(\\frac{\\Delta^2}{\\sigma^2}\\right),",[15,126010,126012,126130],{"className":126011,"ariaHidden":46},[45],[15,126013,126015,126018,126021,126024,126112,126115,126118,126121,126124,126127],{"className":126014},[50],[15,126016],{"className":126017,"style":37724},[54],[15,126019,1289],{"className":126020,"style":1489},[1486,1487,1488],[15,126022],{"className":126023,"style":406},[157],[15,126025,126027,126030,126109],{"className":126026},[59],[15,126028],{"className":126029},[146,441],[15,126031,126033],{"className":126032},[238],[15,126034,126036,126101],{"className":126035},[285,286],[15,126037,126039,126098],{"className":126038},[290],[15,126040,126042,126053,126061],{"className":126041,"style":37749},[294],[15,126043,126044,126047],{"style":457},[15,126045],{"className":126046,"style":461},[302],[15,126048,126050],{"className":126049},[59],[15,126051,944],{"className":126052,"style":981},[59,60],[15,126054,126055,126058],{"style":473},[15,126056],{"className":126057,"style":461},[302],[15,126059],{"className":126060,"style":481},[480],[15,126062,126063,126066],{"style":484},[15,126064],{"className":126065,"style":461},[302],[15,126067,126069],{"className":126068},[59],[15,126070,126072,126075],{"className":126071},[59],[15,126073,11],{"className":126074},[59,60],[15,126076,126078],{"className":126077},[281],[15,126079,126081],{"className":126080},[285],[15,126082,126084],{"className":126083},[290],[15,126085,126087],{"className":126086,"style":20054},[294],[15,126088,126089,126092],{"style":6867},[15,126090],{"className":126091,"style":303},[302],[15,126093,126095],{"className":126094},[307,308,309,310],[15,126096,2465],{"className":126097},[59,310],[15,126099,318],{"className":126100},[317],[15,126102,126104],{"className":126103},[290],[15,126105,126107],{"className":126106,"style":506},[294],[15,126108],{},[15,126110],{"className":126111},[153,441],[15,126113],{"className":126114,"style":406},[157],[15,126116,244],{"className":126117},[59,60],[15,126119,774],{"className":126120},[59,60],[15,126122],{"className":126123,"style":158},[157],[15,126125,123],{"className":126126},[162],[15,126128],{"className":126129,"style":158},[157],[15,126131,126133,126136,126139,126142,126145,126274,126277],{"className":126132},[50],[15,126134],{"className":126135,"style":87299},[54],[15,126137,125979],{"className":126138},[1486],[15,126140],{"className":126141,"style":1514},[157],[15,126143],{"className":126144,"style":406},[157],[15,126146,126148,126154,126268],{"className":126147},[410],[15,126149,126151],{"className":126150,"style":415},[146,414],[15,126152,113],{"className":126153},[419,309],[15,126155,126157,126160,126265],{"className":126156},[59],[15,126158],{"className":126159},[146,441],[15,126161,126163],{"className":126162},[238],[15,126164,126166,126257],{"className":126165},[285,286],[15,126167,126169,126254],{"className":126168},[290],[15,126170,126172,126209,126217],{"className":126171,"style":37749},[294],[15,126173,126174,126177],{"style":457},[15,126175],{"className":126176,"style":461},[302],[15,126178,126180],{"className":126179},[59],[15,126181,126183,126186],{"className":126182},[59],[15,126184,2334],{"className":126185,"style":981},[59,60],[15,126187,126189],{"className":126188},[281],[15,126190,126192],{"className":126191},[285],[15,126193,126195],{"className":126194},[290],[15,126196,126198],{"className":126197,"style":45058},[294],[15,126199,126200,126203],{"style":45061},[15,126201],{"className":126202,"style":303},[302],[15,126204,126206],{"className":126205},[307,308,309,310],[15,126207,2465],{"className":126208},[59,310],[15,126210,126211,126214],{"style":473},[15,126212],{"className":126213,"style":461},[302],[15,126215],{"className":126216,"style":481},[480],[15,126218,126219,126222],{"style":484},[15,126220],{"className":126221,"style":461},[302],[15,126223,126225],{"className":126224},[59],[15,126226,126228,126231],{"className":126227},[59],[15,126229,125721],{"className":126230},[59],[15,126232,126234],{"className":126233},[281],[15,126235,126237],{"className":126236},[285],[15,126238,126240],{"className":126239},[290],[15,126241,126243],{"className":126242,"style":20054},[294],[15,126244,126245,126248],{"style":6867},[15,126246],{"className":126247,"style":303},[302],[15,126249,126251],{"className":126250},[307,308,309,310],[15,126252,2465],{"className":126253},[59,310],[15,126255,318],{"className":126256},[317],[15,126258,126260],{"className":126259},[290],[15,126261,126263],{"className":126262,"style":3780},[294],[15,126264],{},[15,126266],{"className":126267},[153,441],[15,126269,126271],{"className":126270,"style":415},[153,414],[15,126272,120],{"className":126273},[419,309],[15,126275],{"className":126276,"style":406},[157],[15,126278,551],{"className":126279},[573],[11,126281,48185],{},[15,126283,126285],{"className":126284,"translate":19},[180],[15,126286,126288,126396],{"className":126287,"translate":19},[18],[15,126289,126291],{"className":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.",[15,126397,126399,126458,126594,126703,126721,126874],{"className":126398,"ariaHidden":46},[45],[15,126400,126402,126405,126434,126437,126440,126443,126446,126449,126452,126455],{"className":126401},[50],[15,126403],{"className":126404,"style":37277},[54],[15,126406,126408,126411],{"className":126407},[59],[15,126409,37475],{"className":126410},[59,60],[15,126412,126414],{"className":126413},[281],[15,126415,126417],{"className":126416},[285],[15,126418,126420],{"className":126419},[290],[15,126421,126423],{"className":126422,"style":19761},[294],[15,126424,126425,126428],{"style":8279},[15,126426],{"className":126427,"style":303},[302],[15,126429,126431],{"className":126430},[307,308,309,310],[15,126432,2465],{"className":126433},[59,310],[15,126435,113],{"className":126436},[146],[15,126438,206],{"className":126439,"style":334},[59,60],[15,126441,210],{"className":126442},[59],[15,126444,213],{"className":126445},[59,60],[15,126447,120],{"className":126448},[153],[15,126450],{"className":126451,"style":158},[157],[15,126453,123],{"className":126454},[162],[15,126456],{"className":126457,"style":158},[157],[15,126459,126461,126464,126467,126470,126576,126579,126582,126585,126588,126591],{"className":126460},[50],[15,126462],{"className":126463,"style":37724},[54],[15,126465,1289],{"className":126466,"style":1489},[1486,1487,1488],[15,126468],{"className":126469,"style":406},[157],[15,126471,126473,126476,126573],{"className":126472},[59],[15,126474],{"className":126475},[146,441],[15,126477,126479],{"className":126478},[238],[15,126480,126482,126565],{"className":126481},[285,286],[15,126483,126485,126562],{"className":126484},[290],[15,126486,126488,126499,126507],{"className":126487,"style":37749},[294],[15,126489,126490,126493],{"style":457},[15,126491],{"className":126492,"style":461},[302],[15,126494,126496],{"className":126495},[59],[15,126497,944],{"className":126498,"style":981},[59,60],[15,126500,126501,126504],{"style":473},[15,126502],{"className":126503,"style":461},[302],[15,126505],{"className":126506,"style":481},[480],[15,126508,126509,126512],{"style":484},[15,126510],{"className":126511,"style":461},[302],[15,126513,126515,126518,126521,126524,126527,126530,126533],{"className":126514},[59],[15,126516,113],{"className":126517},[146],[15,126519,11],{"className":126520},[59,60],[15,126522],{"className":126523,"style":2589},[157],[15,126525,4815],{"className":126526},[2593],[15,126528],{"className":126529,"style":2589},[157],[15,126531,944],{"className":126532,"style":981},[59,60],[15,126534,126536,126539],{"className":126535},[153],[15,126537,120],{"className":126538},[153],[15,126540,126542],{"className":126541},[281],[15,126543,126545],{"className":126544},[285],[15,126546,126548],{"className":126547},[290],[15,126549,126551],{"className":126550,"style":20054},[294],[15,126552,126553,126556],{"style":6867},[15,126554],{"className":126555,"style":303},[302],[15,126557,126559],{"className":126558},[307,308,309,310],[15,126560,2465],{"className":126561},[59,310],[15,126563,318],{"className":126564},[317],[15,126566,126568],{"className":126567},[290],[15,126569,126571],{"className":126570,"style":506},[294],[15,126572],{},[15,126574],{"className":126575},[153,441],[15,126577],{"className":126578,"style":406},[157],[15,126580,244],{"className":126581},[59,60],[15,126583,774],{"className":126584},[59,60],[15,126586],{"className":126587,"style":158},[157],[15,126589,123],{"className":126590},[162],[15,126592],{"className":126593,"style":158},[157],[15,126595,126597,126600,126603,126606,126694,126697,126700],{"className":126596},[50],[15,126598],{"className":126599,"style":37724},[54],[15,126601,1289],{"className":126602,"style":1489},[1486,1487,1488],[15,126604],{"className":126605,"style":406},[157],[15,126607,126609,126612,126691],{"className":126608},[59],[15,126610],{"className":126611},[146,441],[15,126613,126615],{"className":126614},[238],[15,126616,126618,126683],{"className":126617},[285,286],[15,126619,126621,126680],{"className":126620},[290],[15,126622,126624,126635,126643],{"className":126623,"style":37749},[294],[15,126625,126626,126629],{"style":457},[15,126627],{"className":126628,"style":461},[302],[15,126630,126632],{"className":126631},[59],[15,126633,944],{"className":126634,"style":981},[59,60],[15,126636,126637,126640],{"style":473},[15,126638],{"className":126639,"style":461},[302],[15,126641],{"className":126642,"style":481},[480],[15,126644,126645,126648],{"style":484},[15,126646],{"className":126647,"style":461},[302],[15,126649,126651],{"className":126650},[59],[15,126652,126654,126657],{"className":126653},[59],[15,126655,11],{"className":126656},[59,60],[15,126658,126660],{"className":126659},[281],[15,126661,126663],{"className":126662},[285],[15,126664,126666],{"className":126665},[290],[15,126667,126669],{"className":126668,"style":20054},[294],[15,126670,126671,126674],{"style":6867},[15,126672],{"className":126673,"style":303},[302],[15,126675,126677],{"className":126676},[307,308,309,310],[15,126678,2465],{"className":126679},[59,310],[15,126681,318],{"className":126682},[317],[15,126684,126686],{"className":126685},[290],[15,126687,126689],{"className":126688,"style":506},[294],[15,126690],{},[15,126692],{"className":126693},[153,441],[15,126695],{"className":126696,"style":2589},[157],[15,126698,4815],{"className":126699},[2593],[15,126701],{"className":126702,"style":2589},[157],[15,126704,126706,126709,126712,126715,126718],{"className":126705},[50],[15,126707],{"className":126708,"style":172},[54],[15,126710,117],{"className":126711},[59],[15,126713],{"className":126714,"style":158},[157],[15,126716,123],{"className":126717},[162],[15,126719],{"className":126720,"style":158},[157],[15,126722,126724,126727,126730,126733,126736,126865,126868,126871],{"className":126723},[50],[15,126725],{"className":126726,"style":87299},[54],[15,126728,125979],{"className":126729},[1486],[15,126731],{"className":126732,"style":1514},[157],[15,126734],{"className":126735,"style":406},[157],[15,126737,126739,126745,126859],{"className":126738},[410],[15,126740,126742],{"className":126741,"style":415},[146,414],[15,126743,113],{"className":126744},[419,309],[15,126746,126748,126751,126856],{"className":126747},[59],[15,126749],{"className":126750},[146,441],[15,126752,126754],{"className":126753},[238],[15,126755,126757,126848],{"className":126756},[285,286],[15,126758,126760,126845],{"className":126759},[290],[15,126761,126763,126800,126808],{"className":126762,"style":37749},[294],[15,126764,126765,126768],{"style":457},[15,126766],{"className":126767,"style":461},[302],[15,126769,126771],{"className":126770},[59],[15,126772,126774,126777],{"className":126773},[59],[15,126775,2334],{"className":126776,"style":981},[59,60],[15,126778,126780],{"className":126779},[281],[15,126781,126783],{"className":126782},[285],[15,126784,126786],{"className":126785},[290],[15,126787,126789],{"className":126788,"style":45058},[294],[15,126790,126791,126794],{"style":45061},[15,126792],{"className":126793,"style":303},[302],[15,126795,126797],{"className":126796},[307,308,309,310],[15,126798,2465],{"className":126799},[59,310],[15,126801,126802,126805],{"style":473},[15,126803],{"className":126804,"style":461},[302],[15,126806],{"className":126807,"style":481},[480],[15,126809,126810,126813],{"style":484},[15,126811],{"className":126812,"style":461},[302],[15,126814,126816],{"className":126815},[59],[15,126817,126819,126822],{"className":126818},[59],[15,126820,125721],{"className":126821},[59],[15,126823,126825],{"className":126824},[281],[15,126826,126828],{"className":126827},[285],[15,126829,126831],{"className":126830},[290],[15,126832,126834],{"className":126833,"style":20054},[294],[15,126835,126836,126839],{"style":6867},[15,126837],{"className":126838,"style":303},[302],[15,126840,126842],{"className":126841},[307,308,309,310],[15,126843,2465],{"className":126844},[59,310],[15,126846,318],{"className":126847},[317],[15,126849,126851],{"className":126850},[290],[15,126852,126854],{"className":126853,"style":3780},[294],[15,126855],{},[15,126857],{"className":126858},[153,441],[15,126860,126862],{"className":126861,"style":415},[153,414],[15,126863,120],{"className":126864},[419,309],[15,126866],{"className":126867,"style":2589},[157],[15,126869,4815],{"className":126870},[2593],[15,126872],{"className":126873,"style":2589},[157],[15,126875,126877,126880],{"className":126876},[50],[15,126878],{"className":126879,"style":172},[54],[15,126881,49536],{"className":126882},[59],[11,126884,126885,126886,126944,126945,127044],{},"这印证了「无界」：两个虽很接近但支撑重叠的正态，",[15,126887,126889,126906],{"className":126888,"translate":19},[18],[15,126890,126892],{"className":126891},[23],[25,126893,126894],{"xmlns":27},[29,126895,126896,126904],{},[32,126897,126898],{},[6816,126899,126900,126902],{},[35,126901,37475],{},[115,126903,2465],{},[39,126905,40420],{"encoding":41},[15,126907,126909],{"className":126908,"ariaHidden":46},[45],[15,126910,126912,126915],{"className":126911},[50],[15,126913],{"className":126914,"style":40430},[54],[15,126916,126918,126921],{"className":126917},[59],[15,126919,37475],{"className":126920},[59,60],[15,126922,126924],{"className":126923},[281],[15,126925,126927],{"className":126926},[285],[15,126928,126930],{"className":126929},[290],[15,126931,126933],{"className":126932,"style":20054},[294],[15,126934,126935,126938],{"style":6867},[15,126936],{"className":126937,"style":303},[302],[15,126939,126941],{"className":126940},[307,308,309,310],[15,126942,2465],{"className":126943},[59,310]," 也会随 ",[15,126946,126948,126974],{"className":126947,"translate":19},[18],[15,126949,126951],{"className":126950},[23],[25,126952,126953],{"xmlns":27},[29,126954,126955,126971],{},[32,126956,126957,126963,126965],{},[6816,126958,126959,126961],{},[35,126960,125721],{"mathvariant":209},[115,126962,2465],{},[35,126964,941],{"mathvariant":209},[6816,126966,126967,126969],{},[35,126968,2334],{},[115,126970,2465],{},[39,126972,126973],{"encoding":41},"\\Delta^2\u002F\\sigma^2",[15,126975,126977],{"className":126976,"ariaHidden":46},[45],[15,126978,126980,126983,127012,127015],{"className":126979},[50],[15,126981],{"className":126982,"style":20029},[54],[15,126984,126986,126989],{"className":126985},[59],[15,126987,125721],{"className":126988},[59],[15,126990,126992],{"className":126991},[281],[15,126993,126995],{"className":126994},[285],[15,126996,126998],{"className":126997},[290],[15,126999,127001],{"className":127000,"style":20054},[294],[15,127002,127003,127006],{"style":6867},[15,127004],{"className":127005,"style":303},[302],[15,127007,127009],{"className":127008},[307,308,309,310],[15,127010,2465],{"className":127011},[59,310],[15,127013,941],{"className":127014},[59],[15,127016,127018,127021],{"className":127017},[59],[15,127019,2334],{"className":127020,"style":981},[59,60],[15,127022,127024],{"className":127023},[281],[15,127025,127027],{"className":127026},[285],[15,127028,127030],{"className":127029},[290],[15,127031,127033],{"className":127032,"style":20054},[294],[15,127034,127035,127038],{"style":6867},[15,127036],{"className":127037,"style":303},[302],[15,127039,127041],{"className":127040},[307,308,309,310],[15,127042,2465],{"className":127043},[59,310]," 指数爆炸。",[11,127046,127047,127050],{},[998,127048,127049],{},"Hellinger²（等方差）。"," Bhattacharyya 系数",[15,127052,127054],{"className":127053,"translate":19},[180],[15,127055,127057,127131],{"className":127056,"translate":19},[18],[15,127058,127060],{"className":127059},[23],[25,127061,127062],{"xmlns":27,"display":189},[29,127063,127064,127128],{},[32,127065,127066,127068,127070,127072,127074,127076,127078,127080,127082,127084,127092,127094,127096,127098,127100,127126],{},[35,127067,39317],{},[35,127069,39320],{},[110,127071,113],{"stretchy":112},[35,127073,206],{},[110,127075,551],{"separator":46},[35,127077,213],{},[110,127079,120],{"stretchy":112},[110,127081,123],{},[110,127083,1289],{},[34365,127085,127086],{},[32,127087,127088,127090],{},[35,127089,11],{},[35,127091,944],{},[110,127093,123],{},[35,127095,125979],{},[110,127097,4105],{},[1299,127099,1306],{},[32,127101,127102,127104,127106,127124],{},[110,127103,113],{"fence":46},[110,127105,4815],{},[238,127107,127108,127114],{},[6816,127109,127110,127112],{},[35,127111,125721],{"mathvariant":209},[115,127113,2465],{},[32,127115,127116,127118],{},[115,127117,39329],{},[6816,127119,127120,127122],{},[35,127121,2334],{},[115,127123,2465],{},[110,127125,120],{"fence":46},[110,127127,551],{"separator":46},[39,127129,127130],{"encoding":41},"BC(P,Q)=\\int\\sqrt{pq}\n=\\exp\\!\\left(-\\frac{\\Delta^2}{8\\sigma^2}\\right),",[15,127132,127134,127173,127241],{"className":127133,"ariaHidden":46},[45],[15,127135,127137,127140,127143,127146,127149,127152,127155,127158,127161,127164,127167,127170],{"className":127136},[50],[15,127138],{"className":127139,"style":139},[54],[15,127141,39317],{"className":127142,"style":39413},[59,60],[15,127144,39320],{"className":127145,"style":10786},[59,60],[15,127147,113],{"className":127148},[146],[15,127150,206],{"className":127151,"style":334},[59,60],[15,127153,551],{"className":127154},[573],[15,127156],{"className":127157,"style":406},[157],[15,127159,213],{"className":127160},[59,60],[15,127162,120],{"className":127163},[153],[15,127165],{"className":127166,"style":158},[157],[15,127168,123],{"className":127169},[162],[15,127171],{"className":127172,"style":158},[157],[15,127174,127176,127179,127182,127185,127232,127235,127238],{"className":127175},[50],[15,127177],{"className":127178,"style":5779},[54],[15,127180,1289],{"className":127181,"style":1489},[1486,1487,1488],[15,127183],{"className":127184,"style":406},[157],[15,127186,127188],{"className":127187},[59,34393],[15,127189,127191,127224],{"className":127190},[285,286],[15,127192,127194,127221],{"className":127193},[290],[15,127195,127197,127209],{"className":127196,"style":6864},[294],[15,127198,127200,127203],{"className":127199,"style":11867},[6309],[15,127201],{"className":127202,"style":461},[302],[15,127204,127206],{"className":127205,"style":34413},[59],[15,127207,41631],{"className":127208,"style":981},[59,60],[15,127210,127211,127214],{"style":41376},[15,127212],{"className":127213,"style":461},[302],[15,127215,127217],{"className":127216,"style":34470},[34469],[6325,127218,127219],{"xmlns":6327,"width":6328,"height":34473,"viewBox":34474,"preserveAspectRatio":6331},[6333,127220],{"d":34477},[15,127222,318],{"className":127223},[317],[15,127225,127227],{"className":127226},[290],[15,127228,127230],{"className":127229,"style":41396},[294],[15,127231],{},[15,127233],{"className":127234,"style":158},[157],[15,127236,123],{"className":127237},[162],[15,127239],{"className":127240,"style":158},[157],[15,127242,127244,127247,127250,127253,127256,127391,127394],{"className":127243},[50],[15,127245],{"className":127246,"style":87299},[54],[15,127248,125979],{"className":127249},[1486],[15,127251],{"className":127252,"style":1514},[157],[15,127254],{"className":127255,"style":406},[157],[15,127257,127259,127265,127268,127385],{"className":127258},[410],[15,127260,127262],{"className":127261,"style":415},[146,414],[15,127263,113],{"className":127264},[419,309],[15,127266,4815],{"className":127267},[59],[15,127269,127271,127274,127382],{"className":127270},[59],[15,127272],{"className":127273},[146,441],[15,127275,127277],{"className":127276},[238],[15,127278,127280,127374],{"className":127279},[285,286],[15,127281,127283,127371],{"className":127282},[290],[15,127284,127286,127326,127334],{"className":127285,"style":37749},[294],[15,127287,127288,127291],{"style":457},[15,127289],{"className":127290,"style":461},[302],[15,127292,127294,127297],{"className":127293},[59],[15,127295,39329],{"className":127296},[59],[15,127298,127300,127303],{"className":127299},[59],[15,127301,2334],{"className":127302,"style":981},[59,60],[15,127304,127306],{"className":127305},[281],[15,127307,127309],{"className":127308},[285],[15,127310,127312],{"className":127311},[290],[15,127313,127315],{"className":127314,"style":45058},[294],[15,127316,127317,127320],{"style":45061},[15,127318],{"className":127319,"style":303},[302],[15,127321,127323],{"className":127322},[307,308,309,310],[15,127324,2465],{"className":127325},[59,310],[15,127327,127328,127331],{"style":473},[15,127329],{"c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由 3.6 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a=\\frac{1+\\alpha}{2},",[15,128034,128036,128091,128198,128410],{"className":128035,"ariaHidden":46},[45],[15,128037,128039,128042,128082,128085,128088],{"className":128038},[50],[15,128040],{"className":128041,"style":15965},[54],[15,128043,128045,128048],{"className":128044},[59],[15,128046,199],{"className":128047,"style":277},[59,60],[15,128049,128051],{"className":128050},[281],[15,128052,128054,128074],{"className":128053},[285,286],[15,128055,128057,128071],{"className":128056},[290],[15,128058,128060],{"className":128059,"style":44955},[294],[15,128061,128062,128065],{"style":298},[15,128063],{"className":128064,"style":303},[302],[15,128066,128068],{"className":128067},[307,308,309,310],[15,128069,44780],{"className":128070,"style":44810},[59,60,310],[15,128072,318],{"className":128073},[317],[15,128075,128077],{"className":128076},[290],[15,128078,128080],{"className":128079,"style":7970},[294],[15,128081],{},[15,128083],{"className":128084,"style":158},[157],[15,128086,123],{"className":128087},[162],[15,128089],{"className":128090,"style":158},[157],[15,128092,128094,128097,128180,128186,128189,128192,128195],{"className":128093},[50],[15,128095],{"className":128096,"style":68606},[54],[15,128098,128100,128103,128177],{"className":128099},[59],[15,128101],{"className":128102},[146,441],[15,128104,128106],{"className":128105},[238],[15,128107,128109,128169],{"className":128108},[285,286],[15,128110,128112,128166],{"className":128111},[290],[15,128113,128115,128147,128155],{"className":128114,"style":22915},[294],[15,128116,128117,128120],{"style":457},[15,128118],{"className":128119,"style":461},[302],[15,128121,128123,128126,128129,128132,128135,128138,128141,128144],{"className":128122},[59],[15,128124,45206],{"className":128125},[59,60],[15,128127,113],{"className":128128},[146],[15,128130,117],{"className":128131},[59],[15,128133],{"className":128134,"style":2589},[157],[15,128136,4815],{"className":128137},[2593],[15,128139],{"className":128140,"style":2589},[157],[15,128142,45206],{"className":128143},[59,60],[15,128145,120],{"className":128146},[153],[15,128148,128149,128152],{"style":473},[15,128150],{"className":128151,"style":461},[302],[15,128153],{"className":128154,"style":481},[480],[15,128156,128157,128160],{"style":484},[15,128158],{"className":128159,"style":461},[302],[15,128161,128163],{"className":128162},[59],[15,128164,117],{"className":128165},[59],[15,128167,318],{"className":128168},[317],[15,128170,128172],{"className":128171},[290],[15,128173,128175],{"className":128174,"style":1603},[294],[15,128176],{},[15,128178],{"className":128179},[153,441],[15,128181,128183],{"className":128182},[59],[15,128184,113],{"className":128185},[419,3722],[15,128187,117],{"className":128188},[59],[15,128190],{"className":128191,"style":2589},[157],[15,128193,4815],{"className":128194},[2593],[15,128196],{"className":128197,"style":2589},[157],[15,128199,128201,128205,128383,128389,128392,128395,128398,128401,128404,128407],{"className":128200},[50],[15,128202],{"className":128203,"style":128204},[54],"height:1.9791em;vertical-align:-0.65em;",[15,128206,128208,128211],{"className":128207},[59],[15,128209,7569],{"className":128210},[59,60],[15,128212,128214],{"className":128213},[281],[15,128215,128217],{"className":128216},[285],[15,128218,128220],{"className":128219},[290],[15,128221,128224],{"className":128222,"style":128223},[294],"height:1.3291em;",[15,128225,128227,128230],{"style":128226},"top:-3.4534em;margin-right:0.05em;",[15,128228],{"className":128229,"style":461},[302],[15,128231,128233],{"className":128232},[307,308,309,310],[15,128234,128236,128239],{"className":128235},[59,310],[15,128237,4815],{"className":128238},[59,310],[15,128240,128242,128245,128380],{"className":128241},[59,310],[15,128243],{"className":128244},[146,441,307,30236,111368],[15,128246,128248],{"className":128247},[238],[15,128249,128251,128371],{"className":128250},[285,286],[15,128252,128254,128368],{"className":128253},[290],[15,128255,128258,128302,128310],{"className":128256,"style":128257},[294],"height:1.251em;",[15,128259,128261,128264],{"style":128260},"top:-2.5062em;",[15,128262],{"className":128263,"style":461},[302],[15,128265,128267],{"className":128266},[307,30236,9676,310],[15,128268,128270,128273],{"className":128269},[59,310],[15,128271,2465],{"className":128272},[59,310],[15,128274,128276,128279],{"className":128275},[59,310],[15,128277,2334],{"className":128278,"style":981},[59,60,310],[15,128280,128282],{"className":128281},[281],[15,128283,128285],{"className":128284},[285],[15,128286,128288],{"className":128287},[290],[15,128289,128292],{"className":128290,"style":128291},[294],"height:0.9384em;",[15,128293,128295,128299],{"style":128294},"top:-2.9384em;margin-right:0.1em;",[15,128296],{"className":128297,"style":128298},[302],"height:2.6444em;",[15,128300,2465],{"className":128301},[59,310],[15,128303,128304,128307],{"style":111405},[15,128305],{"className":128306,"style":461},[30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",[15,128497,128499,128513],{"className":128498,"translate":19},[18],[15,128500,128502],{"className":128501},[23],[25,128503,128504],{"xmlns":27},[29,128505,128506,128510],{},[32,128507,128508],{},[35,128509,125721],{"mathvariant":209},[39,128511,128512],{"encoding":41},"\\Delta",[15,128514,128516],{"className":128515,"ariaHidden":46},[45],[15,128517,128519,128522],{"className":128518},[50],[15,128520],{"className":128521,"style":664},[54],[15,128523,125721],{"className":128524},[59],[15,128526,128528,128560],{"className":128527,"translate":19},[18],[15,128529,128531],{"className":128530},[23],[25,128532,128533],{"xmlns":27},[29,128534,128535,128557],{},[32,128536,128537,128539],{},[110,128538,32985],{},[238,128540,128541,128547],{},[6816,128542,128543,128545],{},[35,128544,125721],{"mathvariant":209},[115,128546,2465],{},[32,128548,128549,128551],{},[115,128550,2465],{},[6816,128552,128553,128555],{},[35,128554,2334],{},[115,128556,2465],{},[39,128558,128559],{"encoding":41},"\\approx\\frac{\\Delta^2}{2\\sigma^2}",[15,128561,128563,128575],{"className":128562,"ariaHidden":46},[45],[15,128564,128566,128569,128572],{"className":128565},[50],[15,128567],{"className":128568,"style":117206},[54],[15,128570,32985],{"className":128571},[162],[15,128573],{"className":128574,"style":158},[157],[15,128576,128578,128582],{"className":128577},[50],[15,128579],{"className":128580,"style":128581},[54],"height:1.3629em;vertical-align:-0.345em;",[15,128583,128585,128588,128702],{"className":128584},[59],[15,128586],{"className":128587},[146,441],[15,128589,128591],{"className":128590},[238],[15,128592,128594,128694],{"className":128593},[285,286],[15,128595,128597,128691],{"className":128596},[290],[15,128598,128600,128643,128651],{"className":128599,"style":113817},[294],[15,128601,128602,128605],{"style":2529},[15,128603],{"className":128604,"style":461},[302],[15,128606,128608],{"className":128607},[307,308,309,310],[15,128609,128611,128614],{"className":128610},[59,310],[15,128612,2465],{"className":128613},[59,310],[15,128615,128617,128620],{"className":128616},[59,310],[15,128618,2334],{"className":128619,"style":981},[59,60,310],[15,128621,128623],{"className":128622},[281],[15,128624,128626],{"className":128625},[285],[15,128627,128629],{"className":128628},[290],[15,128630,128632],{"className":128631,"style":45485},[294],[15,128633,128634,128637],{"style":45488},[15,128635],{"className":128636,"style":30232},[302],[15,128638,128640],{"className":128639},[307,30236,9676,310],[15,128641,2465],{"className":128642},[59,310],[15,128644,128645,128648],{"style":473},[15,128646],{"className":128647,"style":461},[302],[15,128649],{"className":128650,"style":481},[480],[15,128652,128653,128656],{"style":2552},[15,128654],{"className":128655,"style":461},[302],[15,128657,128659],{"className":128658},[307,308,309,310],[15,128660,128662],{"className":128661},[59,310],[15,128663,128665,128668],{"className":128664},[59,310],[15,128666,125721],{"className":128667},[59,310],[15,128669,128671],{"className":128670},[281],[15,128672,128674],{"className":128673},[285],[15,128675,128677],{"className":128676},[290],[15,128678,128680],{"className":128679,"style":38538},[294],[15,128681,128682,128685],{"style":38541},[15,128683],{"className":128684,"style":30232},[302],[15,128686,128688],{"className":128687},[307,30236,9676,310],[15,128689,2465],{"className":128690},[59,310],[15,128692,318],{"className":128693},[317],[15,128695,128697],{"className":128696},[290],[15,128698,128700],{"className":128699,"style":2574},[294],[15,128701],{},[15,128703],{"className":128704},[153,441],"，与 4.6 节的局部展开一致（且此时与 ",[15,128707,128709,128722],{"className":128708,"translate":19},[18],[15,128710,128712],{"className":128711},[23],[25,128713,128714],{"xmlns":27},[29,128715,128716,128720],{},[32,128717,128718],{},[35,128719,44780],{},[39,128721,49116],{"encoding":41},[15,128723,128725],{"className":128724,"ariaHidden":46},[45],[15,128726,128728,128731],{"className":128727},[50],[15,128729],{"className":128730,"style":2347},[54],[15,128732,44780],{"className":128733,"style":44810},[59,60]," 无关）。",[2313,128736,128738],{"id":128737},"_92-伯努利分布","9.2 伯努利分布",[11,128740,534,128741,5647,128818,15799,128895,128982,128983,129085,129086,129114,129115,129085,129281,991],{},[15,128742,128744,128776],{"className":128743,"translate":19},[18],[15,128745,128747],{"className":128746},[23],[25,128748,128749],{"xmlns":27},[29,128750,128751,128773],{},[32,128752,128753,128755,128757,128767,128769,128771],{},[35,128754,206],{},[110,128756,123],{},[32,128758,128759,128761,128763,128765],{},[35,128760,39317],{"mathvariant":209},[35,128762,7569],{"mathvariant":209},[35,128764,16454],{"mathvariant":209},[35,128766,82337],{"mathvariant":209},[110,128768,113],{"stretchy":112},[35,128770,11],{},[110,128772,120],{"stretchy":112},[39,128774,128775],{"encoding":41},"P=\\mathrm{Bern}(p)",[15,128777,128779,128797],{"className":128778,"ariaHidden":46},[45],[15,128780,128782,128785,128788,128791,128794],{"className":128781},[50],[15,128783],{"className":128784,"style":664},[54],[15,128786,206],{"className":128787,"style":334},[59,60],[15,128789],{"className":128790,"style":158},[157],[15,128792,123],{"className":128793},[162],[15,128795],{"className":128796,"style":158},[157],[15,128798,128800,128803,128809,128812,128815],{"className":128799},[50],[15,128801],{"className":128802,"style":139},[54],[15,128804,128806],{"className":128805},[59],[15,128807,84349],{"className":128808},[59,32688],[15,128810,113],{"className":128811},[146],[15,128813,11],{"className":128814},[59,60],[15,128816,120],{"className":128817},[153],[15,128819,128821,128853],{"className":128820,"translate":19},[18],[15,128822,128824],{"className":128823},[23],[25,128825,128826],{"xmlns":27},[29,128827,128828,128850],{},[32,128829,128830,128832,128834,128844,128846,128848],{},[35,128831,213],{},[110,128833,123],{},[32,128835,128836,128838,128840,128842],{},[35,128837,39317],{"mathvariant":209},[35,128839,7569],{"mathvariant":209},[35,128841,16454],{"mathvariant":209},[35,128843,82337],{"mathvariant":209},[110,128845,113],{"stretchy":112},[35,128847,944],{},[110,128849,120],{"stretchy":112},[39,128851,128852],{"encoding":41},"Q=\\mathrm{Bern}(q)",[15,128854,128856,128874],{"className":128855,"ariaHidden":46},[45],[15,128857,128859,128862,128865,128868,128871],{"className":128858},[50],[15,128860],{"className":128861,"style":566},[54],[15,128863,213],{"className":128864},[59,60],[15,128866],{"className":128867,"style":158},[157],[15,128869,123],{"className":128870},[162],[15,128872],{"className":128873,"style":158},[157],[15,128875,128877,128880,128886,128889,128892],{"className":128876},[50],[15,128878],{"className":128879,"style":139},[54],[15,128881,128883],{"className":128882},[59],[15,128884,84349],{"className":128885},[59,32688],[15,128887,113],{"className":128888},[146],[15,128890,944],{"className":128891,"style":981},[59,60],[15,128893,120],{"className":128894},[153],[15,128896,128898,128928],{"className":128897,"translate":19},[18],[15,128899,128901],{"className":128900},[23],[25,128902,128903],{"xmlns":27},[29,128904,128905,128925],{},[32,128906,128907,128909,128911,128913,128915,128917,128919,128921,128923],{},[35,128908,11],{},[110,128910,551],{"separator":46},[35,128912,944],{},[110,128914,5911],{},[110,128916,113],{"stretchy":112},[115,128918,126],{},[110,128920,551],{"separator":46},[115,128922,117],{},[110,128924,120],{"stretchy":112},[39,128926,128927],{"encoding":41},"p,q\\in(0,1)",[15,128929,128931,128958],{"className":128930,"ariaHidden":46},[45],[15,128932,128934,128937,128940,128943,128946,128949,128952,128955],{"className":128933},[50],[15,128935],{"className":128936,"style":31718},[54],[15,128938,11],{"className":128939},[59,60],[15,128941,551],{"className":128942},[573],[15,128944],{"className":128945,"style":406},[157],[15,128947,944],{"className":128948,"style":981},[59,60],[15,128950],{"className":128951,"style":158},[157],[15,128953,5911],{"className":128954},[162],[15,128956],{"className":128957,"style":158},[157],[15,128959,128961,128964,128967,128970,128973,128976,128979],{"className":128960},[50],[15,128962],{"className":128963,"style":139},[54],[15,128965,113],{"className":128966},[146],[15,128968,126],{"className":128969},[59],[15,128971,551],{"className":128972},[573],[15,128974],{"className":128975,"style":406},[157],[15,128977,117],{"className":128978},[59],[15,128980,120],{"className":128981},[153],"），似然比取值 ",[15,128984,128986,129012],{"className":128985,"translate":19},[18],[15,128987,128989],{"className":128988},[23],[25,128990,128991],{"xmlns":27},[29,128992,128993,129009],{},[32,128994,128995,129001,129003,129005,129007],{},[195,128996,128997,128999],{},[35,128998,769],{},[115,129000,117],{},[110,129002,123],{},[35,129004,11],{},[35,129006,941],{"mathvariant":209},[35,129008,944],{},[39,129010,129011],{"encoding":41},"t_1=p\u002Fq",[15,129013,129015,129070],{"className":129014,"ariaHidden":46},[45],[15,129016,129018,129021,129061,129064,129067],{"className":129017},[50],[15,129019],{"className":129020,"style":95968},[54],[15,129022,129024,129027],{"className":129023},[59],[15,129025,769],{"className":129026},[59,60],[15,129028,129030],{"className":129029},[281],[15,129031,129033,129053],{"className":129032},[285,286],[15,129034,129036,129050],{"className":129035},[290],[15,129037,129039],{"className":129038,"style":11137},[294],[15,129040,129041,129044],{"style":382},[15,129042],{"className":129043,"style":303},[302],[15,129045,129047],{"className":129046},[307,308,309,310],[15,129048,117],{"className":129049},[59,310],[15,129051,318],{"className":129052},[317],[15,129054,129056],{"className":129055},[290],[15,129057,129059],{"className":129058,"style":7970},[294],[15,129060],{},[15,129062],{"className":129063,"style":158},[157],[15,129065,123],{"className":129066},[162],[15,129068],{"className":129069,"style":158},[157],[15,129071,129073,129076,129079,129082],{"className":129072},[50],[15,129074],{"className":129075,"style":139},[54],[15,129077,11],{"className":129078},[59,60],[15,129080,941],{"className":129081},[59],[15,129083,944],{"className":129084,"style":981},[59,60],"（概率 ",[15,129087,129089,129102],{"className":129088,"translate":19},[18],[15,129090,129092],{"className":129091},[23],[25,129093,129094],{"xmlns":27},[29,129095,129096,129100],{},[32,129097,129098],{},[35,129099,944],{},[39,129101,944],{"encoding":41},[15,129103,129105],{"className":129104,"ariaHidden":46},[45],[15,129106,129108,129111],{"className":129107},[50],[15,129109],{"className":129110,"style":2278},[54],[15,129112,944],{"className":129113,"style":981},[59,60],"）与 ",[15,129116,129118,129160],{"className":129117,"translate":19},[18],[15,129119,129121],{"className":129120},[23],[25,129122,129123],{"xmlns":27},[29,129124,129125,129157],{},[32,129126,129127,129133,129135,129137,129139,129141,129143,129145,129147,129149,129151,129153,129155],{},[195,129128,129129,129131],{},[35,129130,769],{},[115,129132,126],{},[110,129134,123],{},[110,129136,113],{"stretchy":112},[115,129138,117],{},[110,129140,4815],{},[35,129142,11],{},[110,129144,120],{"stretchy":112},[35,129146,941],{"mathvariant":209},[110,129148,113],{"stretchy":112},[115,129150,117],{},[110,129152,4815],{},[35,129154,944],{},[110,129156,120],{"stretchy":112},[39,129158,129159],{"encoding":41},"t_0=(1-p)\u002F(1-q)",[15,129161,129163,129218,129239,129269],{"className":129162,"ariaHidden":46},[45],[15,129164,129166,129169,129209,129212,129215],{"className":129165},[50],[15,129167],{"className":129168,"style":95968},[54],[15,129170,129172,129175],{"className":129171},[59],[15,129173,769],{"className":129174},[59,60],[15,129176,129178],{"className":129177},[281],[15,129179,129181,129201],{"className":129180},[285,286],[15,129182,129184,129198],{"className":129183},[290],[15,129185,129187],{"className":129186,"style":11137},[294],[15,129188,129189,129192],{"style":382},[15,129190],{"className":129191,"style":303},[302],[15,129193,129195],{"className":129194},[307,308,309,310],[15,129196,126],{"className":129197},[59,310],[15,129199,318],{"className":129200},[317],[15,129202,129204],{"className":129203},[290],[15,129205,129207],{"className":129206,"style":7970},[294],[15,129208],{},[15,129210],{"className":129211,"style":158},[157],[15,129213,123],{"className":129214},[162],[15,129216],{"className":129217,"style":158},[157],[15,129219,129221,129224,129227,129230,129233,129236],{"className":129220},[50],[15,129222],{"className":129223,"style":139},[54],[15,129225,113],{"className":129226},[146],[15,129228,117],{"className":129229},[59],[15,129231],{"className":129232,"style":2589},[157],[15,129234,4815],{"className":129235},[2593],[15,129237],{"className":129238,"style":2589},[157],[15,129240,129242,129245,129248,129251,129254,129257,129260,129263,129266],{"className":129241},[50],[15,129243],{"className":129244,"style":139},[54],[15,129246,11],{"className":129247},[59,60],[15,129249,120],{"className":129250},[153],[15,129252,941],{"className":129253},[59],[15,129255,113],{"className":129256},[146],[15,129258,117],{"className":129259},[59],[15,129261],{"className":129262,"style":2589},[157],[15,129264,4815],{"className":129265},[2593],[15,129267],{"className":129268,"style":2589},[157],[15,129270,129272,129275,129278],{"className":129271},[50],[15,129273],{"className":129274,"style":139},[54],[15,129276,944],{"className":129277,"style":981},[59,60],[15,129279,120],{"className":129280},[153],[15,129282,129284,129302],{"className":129283,"translate":19},[18],[15,129285,129287],{"className":129286},[23],[25,129288,129289],{"xmlns":27},[29,129290,129291,129299],{},[32,129292,129293,129295,129297],{},[115,129294,117],{},[110,129296,4815],{},[35,129298,944],{},[39,129300,129301],{"encoding":41},"1-q",[15,129303,129305,129323],{"className":129304,"ariaHidden":46},[45],[15,129306,129308,129311,129314,129317,129320],{"className":129307},[50],[15,129309],{"className":129310,"style":12582},[54],[15,129312,117],{"className":129313},[59],[15,129315],{"className":129316,"style":2589},[157],[15,129318,4815],{"className":129319},[2593],[15,129321],{"className":129322,"style":2589},[157],[15,129324,129326,129329],{"className":129325},[50],[15,129327],{"className":129328,"style":2278},[54],[15,129330,944],{"className":129331,"style":981},[59,60],[15,129333,129335],{"className":129334,"translate":19},[180],[15,129336,129338,129420],{"className":129337,"translate":19},[18],[15,129339,129341],{"className":129340},[23],[25,129342,129343],{"xmlns":27,"display":189},[29,129344,129345,129417],{},[32,129346,129347,129357,129359,129361,129363,129365,129367,129369,129371,129373,129375,129381,129383,129385,129387,129389,129391,129393,129395,129397,129415],{},[195,129348,129349,129351],{},[35,129350,199],{},[32,129352,129353,129355],{},[35,129354,10698],{},[35,129356,21919],{},[110,129358,113],{"stretchy":112},[35,129360,206],{},[35,129362,210],{"mathvariant":209},[35,129364,213],{},[110,129366,120],{"stretchy":112},[110,129368,123],{},[35,129370,11],{},[35,129372,4102],{},[110,129374,4105],{},[238,129376,129377,129379],{},[35,129378,11],{},[35,129380,944],{},[110,129382,2472],{},[110,129384,113],{"stretchy":112},[115,129386,117],{},[110,129388,4815],{},[35,129390,11],{},[110,129392,120],{"stretchy":112},[35,129394,4102],{},[110,129396,4105],{},[238,129398,129399,129407],{},[32,129400,129401,129403,129405],{},[115,129402,117],{},[110,129404,4815],{},[35,129406,11],{},[32,129408,129409,129411,129413],{},[115,129410,117],{},[110,129412,4815],{},[35,129414,944],{},[110,129416,551],{"separator":46},[39,129418,129419],{"encoding":41},"D_{KL}(P\\|Q)=p\\lo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",[15,130495,130497,130519],{"className":130496,"translate":19},[18],[15,130498,130500],{"className":130499},[23],[25,130501,130502],{"xmlns":27},[29,130503,130504,130516],{},[32,130505,130506],{},[195,130507,130508,130510],{},[35,130509,199],{},[32,130511,130512,130514],{},[35,130513,10698],{},[35,130515,21919],{},[39,130517,130518],{"encoding":41},"D_{KL}",[15,130520,130522],{"className":130521,"ariaHidden":46},[45],[15,130523,130525,130528],{"className":130524},[50],[15,130526],{"className":130527,"style":15965},[54],[15,130529,130531,130534],{"className":130530},[59],[15,130532,199],{"className":130533,"style":277},[59,60],[15,130535,130537],{"className":130536},[281],[15,130538,130540,130566],{"className":130539},[285,286],[15,130541,130543,130563],{"className":130542},[290],[15,130544,130546],{"className":130545,"style":379},[294],[15,130547,130548,130551],{"style":298},[15,130549],{"className":130550,"style":303},[302],[15,130552,130554],{"className":130553},[307,308,309,310],[15,130555,130557,130560],{"className":130556},[59,310],[15,130558,10698],{"className":130559,"style":10786},[59,60,310],[15,130561,21919],{"className":130562},[59,60,310],[15,130564,318],{"className":130565},[317],[15,130567,130569],{"className":130568},[290],[15,130570,130572],{"className":130571,"style":7970},[294],[15,130573],{},[15,130575,130577,130590],{"className":130576,"translate":19},[18],[15,130578,130580],{"className":130579},[23],[25,130581,130582],{"xmlns":27},[29,130583,130584,130588],{},[32,130585,130586],{},[35,130587,11],{},[39,130589,11],{"encoding":41},[15,130591,130593],{"className":130592,"ariaHidden":46},[45],[15,130594,130596,130599],{"className":130595},[50],[15,130597],{"className":130598,"style":2278},[54],[15,130600,11],{"className":130601},[59,60],[15,130603,130605,130623],{"className":130604,"translate":19},[18],[15,130606,130608],{"className":130607},[23],[25,130609,130610],{"xmlns":27},[29,130611,130612,130620],{},[32,130613,130614,130616,130618],{},[35,130615,11],{},[110,130617,123],{},[35,130619,944],{},[39,130621,130622],{"encoding":41},"p=q",[15,130624,130626,130644],{"className":130625,"ariaHidden":46},[45],[15,130627,130629,130632,130635,130638,130641],{"className":130628},[50],[15,130630],{"className":130631,"style":2278},[54],[15,130633,11],{"className":130634},[59,60],[15,130636],{"className":130637,"style":158},[157],[15,130639,123],{"className":130640},[162],[15,130642],{"className":130643,"style":158},[157],[15,130645,130647,130650],{"className":130646},[50],[15,130648],{"className":130649,"style":2278},[54],[15,130651,944],{"className":130652,"style":981},[59,60]," 处做二阶展开，应回到 ",[15,130655,130657,130717],{"className":130656,"translate":19},[18],[15,130658,130660],{"className":130659},[23],[25,130661,130662],{"xmlns":27},[29,130663,130664,130714],{},[32,130665,130666,130700,130702,130708],{},[238,130667,130668,130684],{},[32,130669,130670,130672,130674,130676,130678],{},[110,130671,113],{"stretchy":112},[35,130673,11],{},[110,130675,4815],{},[35,130677,944],{},[6816,130679,130680,130682],{},[110,130681,120],{"stretchy":112},[115,130683,2465],{},[32,130685,130686,130688,130690,130692,130694,130696,130698],{},[115,130687,2465],{},[35,130689,944],{},[110,130691,113],{"stretchy":112},[115,130693,117],{},[110,130695,4815],{},[35,130697,944],{},[110,130699,120],{"stretchy":112},[110,130701,123],{},[238,130703,130704,130706],{},[115,130705,117],{},[115,130707,2465],{},[6816,130709,130710,130712],{},[35,130711,37475],{},[115,130713,2465],{},[39,130715,130716],{"encoding":41},"\\frac{(p-q)^2}{2q(1-q)}=\\frac12\\chi^2",[15,130718,130720,130860],{"className":130719,"ariaHidden":46},[45],[15,130721,130723,130727,130851,130854,130857],{"className":130722},[50],[15,130724],{"className":130725,"style":130726},[54],"height:1.6289em;vertical-align:-0.52em;",[15,130728,130730,130733,130848],{"className":130729},[59],[15,130731],{"className":130732},[146,441],[15,130734,130736],{"className":130735},[238],[15,130737,130739,130840],{"className":130738},[285,286],[15,130740,130742,130837],{"className":130741},[290],[15,130743,130745,130777,130785],{"className":130744,"style":38474},[294],[15,130746,130747,130750],{"style":2529},[15,130748],{"className":130749,"style":461},[302],[15,130751,130753],{"className":130752},[307,308,309,310],[15,130754,130756,130759,130762,130765,130768,130771,130774],{"className":130755},[59,310],[15,130757,2465],{"className":130758},[59,310],[15,130760,944],{"className":130761,"style":981},[59,60,310],[15,130763,113],{"className":130764},[146,310],[15,130766,117],{"className":130767},[59,310],[15,130769,4815],{"className":130770},[2593,310],[15,130772,944],{"className":130773,"style":981},[59,60,310],[15,130775,120],{"className":130776},[153,310],[15,130778,130779,130782],{"style":473},[15,130780],{"className":130781,"style":461},[302],[15,130783],{"className":130784,"style":481},[480],[15,130786,130787,130790],{"style":26444},[15,130788],{"className":130789,"style":461},[302],[15,130791,130793],{"className":130792},[307,308,309,310],[15,130794,130796,130799,130802,130805,130808],{"className":130795},[59,310],[15,130797,113],{"className":130798},[146,310],[15,130800,11],{"className":130801},[59,60,310],[15,130803,4815],{"className":130804},[2593,310],[15,130806,944],{"className":130807,"style":981},[59,60,310],[15,130809,130811,130814],{"className":130810},[153,310],[15,130812,120],{"className":130813},[153,310],[15,130815,130817],{"className":130816},[281],[15,130818,130820],{"className":130819},[285],[15,130821,130823],{"className":130822},[290],[15,130824,130826],{"className":130825,"style":38538},[294],[15,130827,130828,130831],{"style":38541},[15,130829],{"className":130830,"style":30232},[302],[15,130832,130834],{"className":130833},[307,30236,9676,310],[15,130835,2465],{"className":130836},[59,310],[15,130838,318],{"className":130839},[317],[15,130841,130843],{"className":130842},[290],[15,130844,130846],{"className":130845,"style":45628},[294],[15,130847],{},[15,130849],{"className":130850},[153,441],[15,130852],{"className":130853,"style":158},[157],[15,130855,123],{"className":130856},[162],[15,130858],{"className":130859,"style":158},[157],[15,130861,130863,130866,130934],{"className":130862},[50],[15,130864],{"className":130865,"style":2507},[54],[15,130867,130869,130872,130931],{"className":130868},[59],[15,130870],{"className":130871},[146,441],[15,130873,130875],{"className":130874},[238],[15,130876,130878,130923],{"className":130877},[285,286],[15,130879,130881,130920],{"className":130880},[290],[15,130882,130884,130898,130906],{"className":130883,"style":2526},[294],[15,130885,130886,130889],{"style":2529},[15,130887],{"className":130888,"style":461},[302],[15,130890,130892],{"className":130891},[307,308,309,310],[15,130893,130895],{"className":130894},[59,310],[15,130896,2465],{"className":130897},[59,310],[15,130899,130900,130903],{"style":473},[15,130901],{"className":130902,"style":461},[302],[15,130904],{"className":130905,"style":481},[480],[15,130907,130908,130911],{"style":2552},[15,130909],{"className":130910,"style":461},[302],[15,130912,130914],{"className":130913},[307,308,309,310],[15,130915,130917],{"className":130916},[59,310],[15,130918,117],{"className":130919},[59,310],[15,130921,318],{"className":130922},[317],[15,130924,130926],{"className":130925},[290],[15,130927,130929],{"className":130928,"style":2574},[294],[15,130930],{},[15,130932],{"className":130933},[153,441],[15,130935,130937,130940],{"className":130936},[59],[15,130938,37475],{"className":130939},[59,60],[15,130941,130943],{"className":130942},[281],[15,130944,130946],{"className":130945},[285],[15,130947,130949],{"className":130948},[290],[15,130950,130952],{"className":130951,"style":20054},[294],[15,130953,130954,130957],{"style":6867},[15,130955],{"className":130956,"style":303},[302],[15,130958,130960],{"className":130959},[307,308,309,310],[15,130961,2465],{"className":130962},[59,310],"，与 4.6 节吻合。",[2313,130965,130967],{"id":130966},"_93-用对偶表示反推-kl","9.3 用对偶表示反推 KL",[11,130969,130970,130971,5647,131056,131197,131198,9402],{},"取 ",[15,130972,130974,131003],{"className":130973,"translate":19},[18],[15,130975,130977],{"className":130976},[23],[25,130978,130979],{"xmlns":27},[29,130980,130981,131001],{},[32,130982,130983,130985,130987,130989,130991,130993,130995,130997,130999],{},[35,130984,37],{},[110,130986,113],{"stretchy":112},[35,130988,769],{},[110,130990,120],{"stretchy":112},[110,130992,123],{},[35,130994,769],{},[35,130996,4102],{},[110,130998,4105],{},[35,131000,769],{},[39,131002,4110],{"encoding":41},[15,131004,131006,131033],{"className":131005,"ariaHidden":46},[45],[15,131007,131009,131012,131015,131018,131021,131024,131027,131030],{"className":131008},[50],[15,131010],{"className":131011,"style":139},[54],[15,131013,37],{"className":131014,"style":61},[59,60],[15,131016,113],{"className":131017},[146],[15,131019,769],{"className":131020},[59,60],[15,131022,120],{"className":131023},[153],[15,131025],{"className":131026,"style":158},[157],[15,131028,123],{"className":131029},[162],[15,131031],{"className":131032,"style":158},[157],[15,131034,131036,131039,131042,131045,131050,131053],{"className":131035},[50],[15,131037],{"className":131038,"style":55},[54],[15,131040,769],{"className":131041},[59,60],[15,131043],{"className":131044,"style":406},[157],[15,131046,4156,131048],{"className":131047},[1486],[15,131049,4160],{"style":4159},[15,131051],{"className":131052,"style":406},[157],[15,131054,769],{"className":131055},[59,60],[15,131057,131059,131097],{"className":131058,"translate":19},[18],[15,131060,131062],{"className":131061},[23],[25,131063,131064],{"xmlns":27},[29,131065,131066,131094],{},[32,131067,131068,131074,131076,131078,131080,131082],{},[6816,131069,131070,131072],{},[35,131071,37],{},[110,131073,7575],{},[110,131075,113],{"stretchy":112},[35,131077,40263],{},[110,131079,120],{"stretchy":112},[110,131081,123],{},[6816,131083,131084,131086],{},[35,131085,7569],{},[32,131087,131088,131090,131092],{},[35,131089,40263],{},[110,131091,4815],{},[115,131093,117],{},[39,131095,131096],{"encoding":41},"f^*(y)=e^{y-1}",[15,131098,131100,131153],{"className":131099,"ariaHidden":46},[45],[15,131101,131103,131106,131135,131138,131141,131144,131147,131150],{"className":131102},[50],[15,131104],{"className":131105,"style":139},[54],[15,131107,131109,131112],{"className":131108},[59],[15,131110,37],{"className":131111,"style":61},[59,60],[15,131113,131115],{"className":131114},[281],[15,131116,131118],{"className":131117},[285],[15,131119,131121],{"className":131120},[290],[15,131122,131124],{"className":131123,"style":8501},[294],[15,131125,131126,131129],{"style":6867},[15,131127],{"className":131128,"style":303},[302],[15,131130,131132],{"className":131131},[307,308,309,310],[15,131133,7575],{"className":131134},[2593,310],[15,131136,113],{"className":131137},[146],[15,131139,40263],{"className":131140,"style":981},[59,60],[15,131142,120],{"className":131143},[153],[15,131145],{"className":131146,"style":158},[157],[15,131148,123],{"className":131149},[162],[15,131151],{"className":131152,"style":158},[157],[15,131154,131156,131159],{"className":131155},[50],[15,131157],{"className":131158,"style":20054},[54],[15,131160,131162,131165],{"className":131161},[59],[15,131163,7569],{"className":131164},[59,60],[15,131166,131168],{"className":131167},[281],[15,131169,131171],{"className":131170},[285],[15,131172,131174],{"className":131173},[290],[15,131175,131177],{"className":131176,"style":20054},[294],[15,131178,131179,131182],{"style":6867},[15,131180],{"className":131181,"style":303},[302],[15,131183,131185],{"className":131184},[307,308,309,310],[15,131186,131188,131191,131194],{"className":131187},[59,310],[15,131189,40263],{"className":131190,"style":981},[59,60,310],[15,131192,4815],{"className":131193},[2593,310],[15,131195,117],{"className":131196},[59,310],"，则对任意 ",[15,131199,131201,131214],{"className":131200,"translate":19},[18],[15,131202,131204],{"className":131203},[23],[25,131205,131206],{"xmlns":27},[29,131207,131208,131212],{},[32,131209,131210],{},[35,131211,4160],{},[39,131213,4160],{"encoding":41},[15,131215,131217],{"className":131216,"ariaHidden":46},[45],[15,131218,131220,131223],{"className":131219},[50],[15,131221],{"className":131222,"style":2278},[54],[15,131224,4160],{"className":131225,"style":981},[59,60],[15,131227,131229],{"className":131228,"translate":19},[180],[15,131230,131232,131304],{"className":131231,"translate":19},[18],[15,131233,131235],{"className":131234},[23],[25,131236,131237],{"xmlns":27,"display":189},[29,131238,131239,131301],{},[32,131240,131241,131251,131253,131255,131257,131259,131261,131263,131269,131271,131273,131275,131277,131283,131285,131297,131299],{},[195,131242,131243,131245],{},[35,131244,199],{},[32,131246,131247,131249],{},[35,131248,10698],{},[35,131250,21919],{},[110,131252,113],{"stretchy":112},[35,131254,206],{},[35,131256,210],{"mathvariant":209},[35,131258,213],{},[110,131260,120],{"stretchy":112},[110,131262,6930],{},[195,131264,131265,131267],{},[35,131266,223],{"mathvariant":222},[35,131268,206],{},[110,131270,230],{"stretchy":112},[35,131272,4160],{},[110,131274,257],{"stretchy":112},[110,131276,4815],{},[195,131278,131279,131281],{},[35,131280,223],{"mathvariant":222},[35,131282,213],{},[110,131284,230],{"stretchy":112},[6816,131286,131287,131289],{},[35,131288,7569],{},[32,131290,131291,131293,131295],{},[35,131292,4160],{},[110,131294,4815],{},[115,131296,117],{},[110,131298,257],{"stretchy":112},[110,131300,551],{"separator":46},[39,131302,131303],{"encoding":41},"D_{KL}(P\\|Q)\\ge \\mathbb E_P[g]\n-\\mathbb E_Q[e^{g-1}],",[15,131305,131307,131383,131447],{"className":131306,"ariaHidden":46},[45],[15,131308,131310,131313,131359,131362,131365,131368,131371,131374,131377,131380],{"className":131309},[50],[15,131311],{"className":131312,"style":139},[54],[15,131314,131316,131319],{"className":131315},[59],[15,131317,199],{"className":131318,"style":277},[59,60],[15,131320,131322],{"className":131321},[281],[15,131323,131325,131351],{"className":131324},[285,286],[15,131326,131328,131348],{"className":131327},[290],[15,131329,131331],{"className":131330,"style":379},[294],[15,131332,131333,131336],{"style":298},[15,131334],{"className":131335,"style":303},[302],[15,131337,131339],{"className":131338},[307,308,309,310],[15,131340,131342,131345],{"className":131341},[59,310],[15,131343,10698],{"className":131344,"style":10786},[59,60,310],[15,131346,21919],{"className":131347},[59,60,310],[15,131349,318],{"className":131350},[317],[15,131352,131354],{"className":131353},[290],[15,131355,131357],{"className":131356,"style":7970},[294],[15,131358],{},[15,131360,113],{"className":131361},[146],[15,131363,206],{"className":131364,"style":334},[59,60],[15,131366,210],{"className":131367},[59],[15,131369,213],{"className":131370},[59,60],[15,131372,120],{"className":131373},[153],[15,131375],{"className":131376,"style":158},[157],[15,131378,6930],{"className":131379},[162],[15,131381],{"className":131382,"style":158},[157],[15,131384,131386,131389,131429,131432,131435,131438,131441,131444],{"className":131385},[50],[15,131387],{"className":131388,"style":139},[54],[15,131390,131392,131395],{"className":131391},[59],[15,131393,223],{"className":131394},[59,366],[15,131396,131398],{"className":131397},[281],[15,131399,131401,131421],{"className":131400},[285,286],[15,131402,131404,131418],{"className":131403},[290],[15,131405,131407],{"className":131406,"style":379},[294],[15,131408,131409,131412],{"style":382},[15,131410],{"className":131411,"style":303},[302],[15,131413,131415],{"className":131414},[307,308,309,310],[15,131416,206],{"className":131417,"style":334},[59,60,310],[15,131419,318],{"className":131420},[317],[15,131422,131424],{"className":131423},[290],[15,131425,131427],{"className":131426,"style":7970},[294],[15,131428],{},[15,131430,230],{"className":131431},[146],[15,131433,4160],{"className":131434,"style":981},[59,60],[15,131436,257],{"className":131437},[153],[15,131439],{"className":131440,"style":2589},[157],[15,131442,4815],{"className":131443},[2593],[15,131445],{"className":131446,"style":2589},[157],[15,131448,131450,131454,131494,131497,131535,131538],{"className":131449},[50],[15,131451],{"className":131452,"style":131453},[54],"height:1.1502em;vertical-align:-0.2861em;",[15,131455,131457,131460],{"className":131456},[59],[15,131458,223],{"className":131459},[59,366],[15,131461,131463],{"className":131462},[281],[15,131464,131466,131486],{"className":131465},[285,286],[15,131467,131469,131483],{"className":131468},[290],[15,131470,131472],{"className":131471,"style":379},[294],[15,131473,131474,131477],{"style":382},[15,131475],{"className":131476,"style":303},[302],[15,131478,131480],{"className":131479},[307,308,309,310],[15,131481,213],{"className":131482},[59,60,310],[15,131484,318],{"className":131485},[317],[15,131487,131489],{"className":131488},[290],[15,131490,131492],{"className":131491,"style":325},[294],[15,131493],{},[15,131495,230],{"className":131496},[146],[15,131498,131500,131503],{"className":131499},[59],[15,131501,7569],{"className":131502},[59,60],[15,131504,131506],{"className":131505},[281],[15,131507,131509],{"className":131508},[285],[15,131510,131512],{"className":131511},[290],[15,131513,131515],{"className":131514,"style":19761},[294],[15,131516,131517,131520],{"style":8279},[15,131518],{"className":131519,"style":303},[302],[15,131521,131523],{"className":131522},[307,308,309,310],[15,131524,131526,131529,131532],{"className":131525},[59,310],[15,131527,4160],{"className":131528,"style":981},[59,60,310],[15,131530,4815],{"className":131531},[2593,310],[15,131533,117],{"className":131534},[59,310],[15,131536,257],{"className":131537},[153],[15,131539,551],{"className":131540},[573],[11,131542,131543,131544,131646],{},"等号在 ",[15,131545,131547,131581],{"className":131546,"translate":19},[18],[15,131548,131550],{"className":131549},[23],[25,131551,131552],{"xmlns":27},[29,131553,131554,131578],{},[32,131555,131556,131558,131560,131562,131564,131566,131568,131570,131572,131574,131576],{},[35,131557,4160],{},[110,131559,123],{},[35,131561,4102],{},[110,131563,4105],{},[110,131565,113],{"stretchy":112},[35,131567,11],{},[35,131569,941],{"mathvariant":209},[35,131571,944],{},[110,131573,120],{"stretchy":112},[110,131575,2472],{},[115,131577,117],{},[39,131579,131580],{"encoding":41},"g=\\log(p\u002Fq)+1",[15,131582,131584,131602,131637],{"className":131583,"ariaHidden":46},[45],[15,131585,131587,131590,131593,131596,131599],{"className":131586},[50],[15,131588],{"className":131589,"style":2278},[54],[15,131591,4160],{"className":131592,"style":981},[59,60],[15,131594],{"className":131595,"style":158},[157],[15,131597,123],{"className":131598},[162],[15,131600],{"className":131601,"style":158},[157],[15,131603,131605,131608,131613,131616,131619,131622,131625,131628,131631,131634],{"className":131604},[50],[15,131606],{"className":131607,"style":139},[54],[15,131609,4156,131611],{"className":131610},[1486],[15,131612,4160],{"style":4159},[15,131614,113],{"className":131615},[146],[15,131617,11],{"className":131618},[59,60],[15,131620,941],{"className":131621},[59],[15,131623,944],{"className":131624,"style":981},[59,60],[15,131626,120],{"className":131627},[153],[15,131629],{"className":131630,"style":2589},[157],[15,131632,2472],{"className":131633},[2593],[15,131635],{"className":131636,"style":2589},[157],[15,131638,131640,131643],{"className":131639},[50],[15,131641],{"className":131642,"style":172},[54],[15,131644,117],{"className":131645},[59]," 处取到。代回：",[15,131648,131650],{"className":131649,"translate":19},[180],[15,131651,131653,131779],{"className":131652,"translate":19},[18],[15,131654,131656],{"className":131655},[23],[25,131657,131658],{"xmlns":27,"display":189},[29,131659,131660,131776],{},[32,131661,131662,131668,131670,131672,131674,131676,131678,131680,131682,131684,131686,131688,131690,131692,131698,131700,131720,131722,131724,131734,131736,131738,131740,131742,131744,131746,131748,131750,131752,131754,131764,131766,131768,131770,131772,131774],{},[195,131663,131664,131666],{},[35,131665,223],{"mathvariant":222},[35,131667,206],{},[110,131669,230],{"stretchy":112},[35,131671,4102],{},[110,131673,4105],{},[110,131675,113],{"stretchy":112},[35,131677,11],{},[35,131679,941],{"mathvariant":209},[35,131681,944],{},[110,131683,120],{"stretchy":112},[110,131685,2472],{},[115,131687,117],{},[110,131689,257],{"stretchy":112},[110,131691,4815],{},[195,131693,131694,131696],{},[35,131695,223],{"mathvariant":222},[35,131697,213],{},[110,131699,230],{"stretchy":112},[6816,131701,131702,131704],{},[35,131703,7569],{},[32,131705,131706,131708,131710,131712,131714,131716,131718],{},[35,131707,4102],{},[110,131709,4105],{},[110,131711,113],{"stretchy":112},[35,131713,11],{},[35,131715,941],{"mathvariant":209},[35,131717,944],{},[110,131719,120],{"stretchy":112},[110,131721,257],{"stretchy":112},[110,131723,123],{},[195,131725,131726,131728],{},[35,131727,199],{},[32,131729,131730,131732],{},[35,131731,10698],{},[35,131733,21919],{},[110,131735,113],{"stretchy":112},[35,131737,206],{},[35,131739,210],{"mathvariant":209},[35,131741,213],{},[110,131743,120],{"stretchy":112},[110,131745,2472],{},[115,131747,117],{},[110,131749,4815],{},[115,131751,117],{},[110,131753,123],{},[195,131755,131756,131758],{},[35,131757,199],{},[32,131759,131760,131762],{},[35,131761,10698],{},[35,131763,21919],{},[110,131765,113],{"stretchy":112},[35,131767,206],{},[35,131769,210],{"mathvariant":209},[35,131771,213],{},[110,131773,120],{"stretchy":112},[110,131775,551],{"separator":46},[39,131777,131778],{"encoding":41},"\\mathbb E_P[\\log(p\u002Fq)+1]\n-\\mathbb E_Q[e^{\\log(p\u002Fq)}]\n=D_{KL}(P\\|Q)+1-1=D_{KL}(P\\|Q),",[15,131780,131782,131860,131881,131999,132075,132093,132111],{"className":131781,"ariaHidden":46},[45],[15,131783,131785,131788,131828,131831,131836,131839,131842,131845,131848,131851,131854,131857],{"className":131784},[50],[15,131786],{"className":131787,"style":139},[54],[15,131789,131791,131794],{"className":131790},[59],[15,131792,223],{"className":131793},[59,366],[15,131795,131797],{"className":131796},[281],[15,131798,131800,131820],{"className":131799},[285,286],[15,131801,131803,131817],{"className":131802},[290],[15,131804,131806],{"className":131805,"style":379},[294],[15,131807,131808,131811],{"style":382},[15,131809],{"className":131810,"style":303},[302],[15,131812,131814],{"className":131813},[307,308,309,310],[15,131815,206],{"className":131816,"style":334},[59,60,310],[15,131818,318],{"className":131819},[317],[15,131821,131823],{"className":131822},[290],[15,131824,131826],{"className":131825,"style":7970},[294],[15,131827],{},[15,131829,230],{"className":131830},[146],[15,131832,4156,131834],{"className":131833},[1486],[15,131835,4160],{"style":4159},[15,131837,113],{"className":131838},[146],[15,131840,11],{"className":131841},[59,60],[15,131843,941],{"className":131844},[59],[15,131846,944],{"className":131847,"style":981},[59,60],[15,131849,120],{"className":131850},[153],[15,131852],{"className":131853,"style":2589},[157],[15,131855,2472],{"className":131856},[2593],[15,131858],{"className":131859,"style":2589},[157],[15,131861,131863,131866,131869,131872,131875,131878],{"className":131862},[50],[15,131864],{"className":131865,"style":139},[54],[15,131867,117],{"className":131868},[59],[15,131870,257],{"className":131871},[153],[15,131873],{"className":131874,"style":2589},[157],[15,131876,4815],{"className":131877},[2593],[15,131879],{"className":131880,"style":2589},[157],[15,131882,131884,131888,131928,131931,131987,131990,131993,131996],{"className":131883},[50],[15,131885],{"className":131886,"style":131887},[54],"height:1.2241em;vertical-align:-0.2861em;",[15,131889,131891,131894],{"className":131890},[59],[15,131892,223],{"className":131893},[59,366],[15,131895,131897],{"className":131896},[281],[15,131898,131900,131920],{"className":131899},[285,286],[15,131901,131903,131917],{"className":131902},[290],[15,131904,131906],{"className":131905,"style":379},[294],[15,131907,131908,131911],{"style":382},[15,131909],{"className":131910,"style":303},[302],[15,131912,131914],{"className":131913},[307,308,309,310],[15,131915,213],{"className":131916},[59,60,310],[15,131918,318],{"className":131919},[317],[15,131921,131923],{"className":131922},[290],[15,131924,131926],{"className":131925,"style":325},[294],[15,131927],{},[15,131929,230],{"className":131930},[146],[15,131932,131934,131937],{"className":131933},[59],[15,131935,7569],{"className":131936},[59,60],[15,131938,131940],{"className":131939},[281],[15,131941,131943],{"className":131942},[285],[15,131944,131946],{"className":131945},[290],[15,131947,131949],{"className":131948,"style":45146},[294],[15,131950,131951,131954],{"style":8279},[15,131952],{"className":131953,"style":303},[302],[15,131955,131957],{"className":131956},[307,308,309,310],[15,131958,131960,131972,131975,131978,131981,131984],{"className":131959},[59,310],[15,131961,131963,131966,131969],{"className":131962},[1486,310],[15,131964,47684],{"className":131965},[310],[15,131967,47688],{"className":131968},[310],[15,131970,4160],{"className":131971,"style":4159},[310],[15,131973,113],{"className":131974},[146,310],[15,131976,11],{"className":131977},[59,60,310],[15,131979,941],{"className":131980},[59,310],[15,131982,944],{"className":131983,"style":981},[59,60,310],[15,131985,120],{"className":131986},[153,310],[15,131988,257],{"className":131989},[153],[15,131991],{"className":131992,"style":158},[157],[15,131994,123],{"className":131995},[162],[15,131997],{"className":131998,"style":158},[157],[15,132000,132002,132005,132051,132054,132057,132060,132063,132066,132069,132072],{"className":132001},[50],[15,132003],{"className":132004,"style":139},[54],[15,132006,132008,132011],{"className":132007},[59],[15,132009,199],{"className":132010,"style":277},[59,60],[15,132012,132014],{"className":132013},[281],[15,132015,132017,132043],{"className":132016},[285,286],[15,132018,132020,132040],{"className":132019},[290],[15,132021,132023],{"className":132022,"style":379},[294],[15,132024,132025,132028],{"style":298},[15,132026],{"className":132027,"style":303},[302],[15,132029,132031],{"className":132030},[307,308,309,310],[15,132032,132034,132037],{"className":132033},[59,310],[15,132035,10698],{"className":132036,"style":10786},[59,60,310],[15,132038,21919],{"className":132039},[59,60,310],[15,132041,318],{"className":132042},[317],[15,132044,132046],{"className":132045},[290],[15,132047,132049],{"className":132048,"style":7970},[294],[15,132050],{},[15,132052,113],{"className":132053},[146],[15,132055,206],{"className":132056,"style":334},[59,60],[15,132058,210],{"className":132059},[59],[15,132061,213],{"className":132062},[59,60],[15,132064,120],{"className":132065},[153],[15,132067],{"className":132068,"style":2589},[157],[15,132070,2472],{"className":132071},[2593],[15,132073],{"className":132074,"style":2589},[157],[15,132076,132078,132081,132084,132087,132090],{"className":132077},[50],[15,132079],{"className":132080,"style":12582},[54],[15,132082,117],{"className":132083},[59],[15,132085],{"className":132086,"style":2589},[157],[15,132088,4815],{"className":132089},[2593],[15,132091],{"className":132092,"style":2589},[157],[15,132094,132096,132099,132102,132105,132108],{"className":132095},[50],[15,132097],{"className":132098,"style":172},[54],[15,132100,117],{"className":132101},[59],[15,132103],{"className":132104,"style":158},[157],[15,132106,123],{"className":132107},[162],[15,132109],{"className":132110,"style":158},[157],[15,132112,132114,132117,132163,132166,132169,132172,132175,132178],{"className":132113},[50],[15,132115],{"className":132116,"style":139},[54],[15,132118,132120,132123],{"className":132119},[59],[15,132121,199],{"className":132122,"style":277},[59,60],[15,132124,132126],{"className":132125},[281],[15,132127,132129,132155],{"className":132128},[285,286],[15,132130,132132,132152],{"className":132131},[290],[15,132133,132135],{"className":132134,"style":379},[294],[15,132136,132137,132140],{"style":298},[15,132138],{"className":132139,"style":303},[302],[15,132141,132143],{"className":132142},[307,308,309,310],[15,132144,132146,132149],{"className":132145},[59,310],[15,132147,10698],{"className":132148,"style":10786},[59,60,310],[15,132150,21919],{"className":132151},[59,60,310],[15,132153,318],{"className":132154},[317],[15,132156,132158],{"className":132157},[290],[15,132159,132161],{"className":132160,"style":7970},[294],[15,132162],{},[15,132164,113],{"className":132165},[146],[15,132167,206],{"className":132168,"style":334},[59,60],[15,132170,210],{"className":132171},[59],[15,132173,213],{"className":132174},[59,60],[15,132176,120],{"className":132177},[153],[15,132179,551],{"className":132180},[573],[11,132182,132183],{},"完全自洽。这给出了一个「可验证」的对偶表示练习。",[528,132185,132187],{"id":132186},"_10-结论","10. 结论",[11,132189,132190],{},"f-散度族的故事可以浓缩成三句话：",[6664,132192,132193,132496,132662],{},[6667,132194,132195,9402,132198,2747,132403,92512,132431,10182],{},[998,132196,132197],{},"一个模板",[15,132199,132201,132257],{"className":132200,"translate":19},[18],[15,132202,132204],{"className":132203},[23],[25,132205,132206],{"xmlns":27},[29,132207,132208,132254],{},[32,132209,132210,132216,132218,132220,132222,132224,132226,132228,132234,132236,132238,132240,132242,132244,132246,132248,132250,132252],{},[195,132211,132212,132214],{},[35,132213,199],{},[35,132215,37],{},[110,132217,113],{"stretchy":112},[35,132219,206],{},[35,132221,210],{"mathvariant":209},[35,132223,213],{},[110,132225,120],{"stretchy":112},[110,132227,123],{},[195,132229,132230,132232],{},[35,132231,223],{"mathvariant":222},[35,132233,213],{},[110,132235,230],{"stretchy":112},[35,132237,37],{},[110,132239,113],{"stretchy":112},[35,132241,244],{},[35,132243,206],{},[35,132245,941],{"mathvariant":209},[35,132247,244],{},[35,132249,213],{},[110,132251,120],{"stretchy":112},[110,132253,257],{"stretchy":112},[39,132255,132256],{"encoding":41},"D_f(P\\|Q)=\\mathbb E_Q[f(dP\u002FdQ)]",[15,132258,132260,132330],{"className":132259,"ariaHidden":46},[45],[15,132261,132263,132266,132306,132309,132312,132315,132318,132321,132324,132327],{"className":132262},[50],[15,132264],{"className":132265,"style":270},[54],[15,132267,132269,132272],{"className":132268},[59],[15,132270,199],{"className":132271,"style":277},[59,60],[15,132273,132275],{"className":132274},[281],[15,132276,132278,132298],{"className":132277},[285,286],[15,132279,132281,132295],{"className":132280},[290],[15,132282,132284],{"className":132283,"style":295},[294],[15,132285,132286,132289],{"style":298},[15,132287],{"className":132288,"style":303},[302],[15,132290,132292],{"className":132291},[307,308,309,310],[15,132293,37],{"className":132294,"style":61},[59,60,310],[15,132296,318],{"className":132297},[317],[15,132299,132301],{"className":132300},[290],[15,132302,132304],{"className":132303,"style":325},[294],[15,132305],{},[15,132307,113],{"className":132308},[146],[15,132310,206],{"className":132311,"style":334},[59,60],[15,132313,210],{"className":132314},[59],[15,132316,213],{"className":132317},[59,60],[15,132319,120],{"className":132320},[153],[15,132322],{"className":132323,"style":158},[157],[15,132325,123],{"className":132326},[162],[15,132328],{"className":132329,"style":158},[157],[15,132331,132333,132336,132376,132379,132382,132385,132388,132391,132394,132397,132400],{"className":132332},[50],[15,132334],{"className":132335,"style":270},[54],[15,132337,132339,132342],{"className":132338},[59],[15,132340,223],{"className":132341},[59,366],[15,132343,132345],{"className":132344},[281],[15,132346,132348,132368],{"className":132347},[285,286],[15,132349,132351,132365],{"className":132350},[290],[15,132352,132354],{"className":132353,"style":379},[294],[15,132355,132356,132359],{"style":382},[15,132357],{"className":132358,"style":303},[302],[15,132360,132362],{"className":132361},[307,308,309,310],[15,132363,213],{"className":132364},[59,60,310],[15,132366,318],{"className":132367},[317],[15,132369,132371],{"className":132370},[290],[15,132372,132374],{"className":132373,"style":325},[294],[15,132375],{},[15,132377,230],{"className":132378},[146],[15,132380,37],{"className":132381,"style":61},[59,60],[15,132383,113],{"className":132384},[146],[15,132386,244],{"className":132387},[59,60],[15,132389,206],{"className":132390,"style":334},[59,60],[15,132392,941],{"className":132393},[59],[15,132395,244],{"className":132396},[59,60],[15,132398,213],{"className":132399},[59,60],[15,132401,20416],{"className":132402},[153],[15,132404,132406,132419],{"className":132405,"translate":19},[18],[15,132407,132409],{"className":132408},[23],[25,132410,132411],{"xmlns":27},[29,132412,132413,132417],{},[32,132414,132415],{},[35,132416,37],{},[39,132418,37],{"encoding":41},[15,132420,132422],{"className":132421,"ariaHidden":46},[45],[15,132423,132425,132428],{"className":132424},[50],[15,132426],{"className":132427,"style":55},[54],[15,132429,37],{"className":132430,"style":61},[59,60],[15,132432,132434,132457],{"className":132433,"translate":19},[18],[15,132435,132437],{"className":132436},[23],[25,132438,132439],{"xmlns":27},[29,132440,132441,132455],{},[32,132442,132443,132445,132447,132449,132451,132453],{},[35,132444,37],{},[110,132446,113],{"stretchy":112},[115,132448,117],{},[110,132450,120],{"stretchy":112},[110,132452,123],{},[115,132454,126],{},[39,132456,129],{"encoding":41},[15,132458,132460,132487],{"className":132459,"ariaHidden":46},[45],[15,132461,132463,132466,132469,132472,132475,132478,132481,132484],{"className":132462},[50],[15,132464],{"className":132465,"style":139},[54],[15,132467,37],{"className":132468,"style":61},[59,60],[15,132470,113],{"className":132471},[146],[15,132473,117],{"className":132474},[59],[15,132476,120],{"className":132477},[153],[15,132479],{"className":132480,"style":158},[157],[15,132482,123],{"className":132483},[162],[15,132485],{"className":132486,"style":158},[157],[15,132488,132490,132493],{"className":132489},[50],[15,132491],{"className":132492,"style":172},[54],[15,132494,126],{"className":132495},[59],[6667,132497,132498,9402,132501,132582,132583,132661],{},[998,132499,132500],{},"两组对偶",[15,132502,132504,132526],{"className":132503,"translate":19},[18],[15,132505,132507],{"className":132506},[23],[25,132508,132509],{"xmlns":27},[29,132510,132511,132523],{},[32,132512,132513,132515,132517],{},[35,132514,37],{},[110,132516,21385],{},[6816,132518,132519,132521],{},[35,132520,37],{},[110,132522,7575],{},[39,132524,132525],{"encoding":41},"f\\leftrightarrow f^*",[15,132527,132529,132547],{"className":132528,"ariaHidden":46},[45],[15,132530,132532,132535,132538,132541,132544],{"className":132531},[50],[15,132533],{"className":132534,"style":55},[54],[15,132536,37],{"className":132537,"style":61},[59,60],[15,132539],{"className":132540,"style":158},[157],[15,132542,21385],{"className":132543},[162],[15,132545],{"className":132546,"style":158},[157],[15,132548,132550,132553],{"className":132549},[50],[15,132551],{"className":132552,"style":55},[54],[15,132554,132556,132559],{"className":132555},[59],[15,132557,37],{"className":132558,"style":61},[59,60],[15,132560,132562],{"className":132561},[281],[15,132563,132565],{"className":132564},[285],[15,132566,132568],{"className":132567},[290],[15,132569,132571],{"className":132570,"style":8501},[294],[15,132572,132573,132576],{"style":6867},[15,132574],{"className":132575,"style":303},[302],[15,132577,132579],{"className":132578},[307,308,309,310],[15,132580,7575],{"className":132581},[2593,310]," 给出变分表示（可用于估计与对抗训练）；",[15,132584,132586,132616],{"className":132585,"translate":19},[18],[15,132587,132589],{"className":132588},[23],[25,132590,132591],{"xmlns":27},[29,132592,132593,132613],{},[32,132594,132595,132597,132599,132601,132603,132605,132607,132609,132611],{},[35,132596,37],{},[110,132598,21385],{},[35,132600,769],{},[35,132602,37],{},[110,132604,113],{"stretchy":112},[115,132606,117],{},[35,132608,941],{"mathvariant":209},[35,132610,769],{},[110,132612,120],{"stretchy":112},[39,132614,132615],{"encoding":41},"f\\leftrightarrow t f(1\u002Ft)",[15,132617,132619,132637],{"className":132618,"ariaHidden":46},[45],[15,132620,132622,132625,132628,132631,132634],{"className":132621},[50],[15,132623],{"className":132624,"style":55},[54],[15,132626,37],{"className":132627,"style":61},[59,60],[15,132629],{"className":132630,"style":158},[157],[15,132632,21385],{"className":132633},[162],[15,132635],{"className":132636,"style":158},[157],[15,132638,132640,132643,132646,132649,132652,132655,132658],{"className":132639},[50],[15,132641],{"className":132642,"style":139},[54],[15,132644,769],{"className":132645},[59,60],[15,132647,37],{"className":132648,"style":61},[59,60],[15,132650,113],{"className":132651},[146],[15,132653,8591],{"className":132654},[59],[15,132656,769],{"className":132657},[59,60],[15,132659,120],{"className":132660},[153]," 给出方向对称性（可用于构造对称散度）。",[6667,132663,132664,132667],{},[998,132665,132666],{},"一个不变量","：数据处理不等式，说明它是「信息量」而非「几何量」。正因如此，它与 Wasserstein\u002FMMD 在支撑分离时行为迥异。",[11,132669,132670],{},"经典文献可参考：Csiszár (1967)、Ali–Silvey (1966) 的原始定义；Liese–Vajda 的专著对 f-散度的对偶、拓扑与充分性做了系统处理；Amari 的信息几何给出 α-族与 Fisher 度量；Nowozin 等 (2016) 的 f-GAN、Belghazi 等 (2018) 的 MINE 把对偶表示变成可训练算法；Dong–Roth–Su (2022) 说明 f-散度的语言如何统一差分隐私。把这几条线索接起来，读者就能在「定义—性质—估计—应用」四个层面上自由穿梭。",{"title":132672,"searchDepth":132673,"depth":132673,"links":132674},"",2,[132675,132682,132687,132697,132707,132714,132719,132727,132728,132733],{"id":530,"depth":132673,"text":531,"children":132676},[132677,132679,132680,132681],{"id":2315,"depth":132678,"text":2316},3,{"id":5206,"depth":132678,"text":5207},{"id":5575,"depth":132678,"text":5576},{"id":7363,"depth":132678,"text":7364},{"id":8753,"depth":132673,"text":132683,"children":132684},"2. 为何 fff 凸且 f(1)=0f(1)=0f(1)=0",[132685,132686],{"id":8851,"depth":132678,"text":8852},{"id":10562,"depth":132678,"text":10563},{"id":21433,"depth":132673,"text":21434,"children":132688},[132689,132690,132691,132692,132693,132694,132695,132696],{"id":21495,"depth":132678,"text":21496},{"id":25574,"depth":132678,"text":25575},{"id":34535,"depth":132678,"text":34536},{"id":37174,"depth":132678,"text":37175},{"id":40463,"depth":132678,"text":40464},{"id":44762,"depth":132678,"text":44763},{"id":49401,"depth":132678,"text":49402},{"id":52515,"depth":132678,"text":52516},{"id":54938,"depth":132673,"text":54939,"children":132698},[132699,132700,132701,132702,132703,132704,132705,132706],{"id":55237,"depth":132678,"text":55238},{"id":56164,"depth":132678,"text":56165},{"id":58320,"depth":132678,"text":58321},{"id":60716,"depth":132678,"text":60717},{"id":74487,"depth":132678,"text":74488},{"id":78070,"depth":132678,"text":78071},{"id":82098,"depth":132678,"text":82099},{"id":96992,"depth":132678,"text":96993},{"id":99340,"depth":132673,"text":99341,"children":132708},[132709,132710,132711,132712,132713],{"id":99344,"depth":132678,"text":99345},{"id":101388,"depth":132678,"text":101389},{"id":103232,"depth":132678,"text":103233},{"id":105394,"depth":132678,"text":105395},{"id":108286,"depth":132678,"text":108287},{"id":108627,"depth":132673,"text":108628,"children":132715},[132716,132717,132718],{"id":108631,"depth":132678,"text":108632},{"id":114657,"depth":132678,"text":114658},{"id":116439,"depth":132678,"text":116440},{"id":117483,"depth":132673,"text":117484,"children":132720},[132721,132722,132723,132724,132725,132726],{"id":117552,"depth":132678,"text":117553},{"id":117746,"depth":132678,"text":117747},{"id":118637,"depth":132678,"text":118638},{"id":119336,"depth":132678,"text":119337},{"id":119794,"depth":132678,"text":119795},{"id":119863,"depth":132678,"text":119864},{"id":120227,"depth":132673,"text":120228},{"id":124388,"depth":132673,"text":124389,"children":132729},[132730,132731,132732],{"id":124392,"depth":132678,"text":124393},{"id":128737,"depth":132678,"text":128738},{"id":130966,"depth":132678,"text":130967},{"id":132186,"depth":132673,"text":132187},"f-散度族是概率分布之间差异的一大类度量，它用一个凸函数 fff 来统一描述 KL 散度、JS 散度、全变差、卡方散度等许多常见散度。可以把它理解为「散度」的通用模板。",false,true,"md",null,{},"\u002Fblog\u002F2026\u002F0011-about-the-f-divergence",{"title":6,"description":132734},"blog\u002F2026\u002F0011-about-the-f-divergence","f-散度族（f-divergence）用一个凸函数 f 统一刻画两个概率分布之间的差异，KL、JS、全变差、卡方、Hellinger、α-散度都是它的特例。",[132745,132746],"statistics","mathematics","2026-10-17T04:47:08+08:00","2WAe5fQsIoDK6gYqN03WvWAWQ1lJEHGnxsbnKfYymdg",{"id":132750,"title":132751,"body":132752,"description":132756,"draft":132735,"enableComment":132736,"extension":132737,"image":132672,"important":132735,"location":132738,"meta":133080,"navigation":132736,"ogImage":132738,"path":133081,"seo":133082,"stem":133083,"summary":133084,"tags":133085,"time":133087,"__hash__":133088},"blog\u002Fblog\u002F2026\u002F0010-biography-of-claude-shannon.md","克劳德·香农小传",{"type":8,"value":132753,"toc":133070},[132754,132757,132760,132763,132766,132769,132772,132775,132778,132781,132784,132787,132790,132793,132796,132799,132802,132805,132808,132811,132818,132821,132824,132827,132830,132833,132836,132839,132842,132845,132848,132851,132854,132857,132860,132863,132866,132869,132872,132875,132878,132881,132884,132889,132892,132895,132898,132901,132904,132907,132910,132913,132916,132920,132923,132926,132929,132932,132935,132938,132941,132944,132947,132950,132953,132956,132959,132962,132967,132970,132973,132977,132983,132986,132989,132992,132995,132998,133001,133004,133007,133010,133013,133016,133020,133026,133029,133032,133035,133038,133041,133044,133047,133050,133053,133056,133059,133062,133065],[11,132755,132756],{},"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。",[11,132758,132759],{},"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。",[528,132761,132762],{"id":132762},"密歇根小镇的修理少年",[11,132764,132765],{},"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。",[11,132767,132768],{},"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」。",[11,132770,132771],{},"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。",[11,132773,132774],{},"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」。",[11,132776,132777],{},"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。",[11,132779,132780],{},"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。",[528,132782,132783],{"id":132783},"一门选修课和一篇论文",[11,132785,132786],{},"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。",[11,132788,132789],{},"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？",[11,132791,132792],{},"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？",[11,132794,132795],{},"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。",[11,132797,132798],{},"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。",[11,132800,132801],{},"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。」",[11,132803,132804],{},"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。",[11,132806,132807],{},"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。",[11,132809,132810],{},"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。",[11,132812,132813],{},[132814,132815],"img",{"alt":132816,"src":132817},"Claude Shannon in 1932","https:\u002F\u002Fimage-assets.dreams.plus\u002F202610162230267.png",[11,132819,132820],{},"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。",[528,132822,132823],{"id":132823},"普林斯顿的拜访与贝尔实验室的夜晚",[11,132825,132826],{},"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。",[11,132828,132829],{},"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。",[11,132831,132832],{},"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。",[11,132834,132835],{},"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。",[11,132837,132838],{},"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？」",[11,132840,132841],{},"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。」",[11,132843,132844],{},"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。",[11,132846,132847],{},"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？",[11,132849,132850],{},"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。」",[11,132852,132853],{},"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。",[11,132855,132856],{},"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。",[11,132858,132859],{},"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。",[528,132861,132862],{"id":132862},"一篇论文与一个安静的家",[11,132864,132865],{},"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。",[11,132867,132868],{},"「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。",[11,132870,132871],{},"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。",[11,132873,132874],{},"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。",[11,132876,132877],{},"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」。",[11,132879,132880],{},"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」。",[11,132882,132883],{},"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：",[993,132885,132886],{},[11,132887,132888],{},"「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。」",[11,132890,132891],{},"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」。",[11,132893,132894],{},"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。",[11,132896,132897],{},"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。",[11,132899,132900],{},"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。",[11,132902,132903],{},"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。",[11,132905,132906],{},"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。」",[11,132908,132909],{},"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。",[11,132911,132912],{},"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。",[11,132914,132915],{},"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。",[528,132917,132919],{"id":132918},"玩具机器与赌场","玩具、机器与赌场",[11,132921,132922],{},"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。",[11,132924,132925],{},"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。」",[11,132927,132928],{},"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」。",[11,132930,132931],{},"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」。",[11,132933,132934],{},"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。",[11,132936,132937],{},"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。",[11,132939,132940],{},"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！」",[11,132942,132943],{},"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。」",[11,132945,132946],{},"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。",[11,132948,132949],{},"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。",[11,132951,132952],{},"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。",[11,132954,132955],{},"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。",[11,132957,132958],{},"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。",[11,132960,132961],{},"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：",[993,132963,132964],{},[11,132965,132966],{},"「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。」",[11,132968,132969],{},"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。",[11,132971,132972],{},"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。」",[528,132974,132976],{"id":132975},"麻省理工的怪教授","麻省理工的「怪教授」",[11,132978,132979],{},[132814,132980],{"alt":132981,"src":132982},"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",[11,132984,132985],{},"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」。",[11,132987,132988],{},"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。",[11,132990,132991],{},"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。」",[11,132993,132994],{},"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。」",[11,132996,132997],{},"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。",[11,132999,133000],{},"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。",[11,133002,133003],{},"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。」",[11,133005,133006],{},"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。",[11,133008,133009],{},"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。",[11,133011,133012],{},"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。",[11,133014,133015],{},"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。",[528,133017,133019],{"id":133018},"熵之屋里的晚年","「熵之屋」里的晚年",[11,133021,133022],{},[132814,133023],{"alt":133024,"src":133025},"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",[11,133027,133028],{},"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。",[11,133030,133031],{},"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」。",[11,133033,133034],{},"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。",[11,133036,133037],{},"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。",[11,133039,133040],{},"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。」",[11,133042,133043],{},"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。",[11,133045,133046],{},"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。",[11,133048,133049],{},"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。",[528,133051,133052],{"id":133052},"一个把工作当成游戏的人",[11,133054,133055],{},"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。",[11,133057,133058],{},"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。」",[11,133060,133061],{},"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。」",[11,133063,133064],{},"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。",[993,133066,133067],{},[11,133068,133069],{},"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":132672,"searchDepth":132673,"depth":132673,"links":133071},[133072,133073,133074,133075,133076,133077,133078,133079],{"id":132762,"depth":132673,"text":132762},{"id":132783,"depth":132673,"text":132783},{"id":132823,"depth":132673,"text":132823},{"id":132862,"depth":132673,"text":132862},{"id":132918,"depth":132673,"text":132919},{"id":132975,"depth":132673,"text":132976},{"id":133018,"depth":132673,"text":133019},{"id":133052,"depth":132673,"text":133052},{},"\u002Fblog\u002F2026\u002F0010-biography-of-claude-shannon",{"title":132751,"description":132756},"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。",[133086],"thoughts","2026-10-16T22:03:15+08:00","lG9XuZP_DA5IpAhSLPz6guqBwMoOwfOl7mPBMsPtti8",1792183806064]