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P}\\big[L(Y,\\delta(X))\\big].",[23,952,954,982],{"className":953,"ariaHidden":122},[121],[23,955,957,960,964,967,970,973,976,979],{"className":956},[126],[23,958],{"className":959,"style":131},[130],[23,961,572],{"className":962,"style":963},[135,136],"margin-right:0.0077em;",[23,965,54],{"className":966},[140],[23,968,736],{"className":969,"style":783},[135,136],[23,971,66],{"className":972},[169],[23,974],{"className":975,"style":149},[148],[23,977,69],{"className":978},[153],[23,980],{"className":981,"style":149},[148],[23,983,985,989,1060,1068,1071,1074,1077,1080,1083,1086,1089,1092,1096,1102],{"className":984},[126],[23,986],{"className":987,"style":988},[130],"height:1.2052em;vertical-align:-0.3552em;",[23,990,992,995],{"className":991},[135],[23,993,902],{"className":994},[135,644],[23,996,999],{"className":997},[998],"msupsub",[23,1000,1002,1051],{"className":1001},[199,200],[23,1003,1005,1048],{"className":1004},[204],[23,1006,1009],{"className":1007,"style":1008},[208],"height:0.3448em;",[23,1010,1012,1016],{"style":1011},"top:-2.5198em;margin-left:0em;margin-right:0.05em;",[23,1013],{"className":1014,"style":1015},[216],"height:2.7em;",[23,1017,1023],{"className":1018},[1019,1020,1021,1022],"sizing","reset-size6","size3","mtight",[23,1024,1026,1029,1032,1035,1038,1041,1044],{"className":1025},[135,1022],[23,1027,54],{"className":1028},[140,1022],[23,1030,434],{"className":1031,"style":454},[135,136,1022],[23,1033,747],{"className":1034},[824,1022],[23,1036,369],{"className":1037,"style":392},[135,136,1022],[23,1039,66],{"className":1040},[169,1022],[23,1042,917],{"className":1043},[153,1022],[23,1045,920],{"className":1046,"style":1047},[135,136,1022],"margin-right:0.1389em;",[23,1049,297],{"className":1050},[296],[23,1052,1054],{"className":1053},[204],[23,1055,1058],{"className":1056,"style":1057},[208],"height:0.3552em;",[23,1059],{},[23,1061,1063],{"className":1062},[135],[23,1064,924],{"className":1065},[1066,1067],"delimsizing","size1",[23,1069,555],{"className":1070},[135,136],[23,1072,54],{"className":1073},[140],[23,1075,369],{"className":1076,"style":392},[135,136],[23,1078,747],{"className":1079},[824],[23,1081],{"className":1082,"style":280},[148],[23,1084,736],{"className":1085,"style":783},[135,136],[23,1087,54],{"className":1088},[140],[23,1090,434],{"className":1091,"style":454},[135,136],[23,1093,1095],{"className":1094},[169],"))",[23,1097,1099],{"className":1098},[135],[23,1100,945],{"className":1101},[1066,1067],[23,1103,112],{"className":1104},[135],[11,1106,1107,1108,1165],{},"这里期望是对联合分布 ",[23,1109,1111,1135],{"className":1110,"translate":27},[31],[23,1112,1114],{"className":1113},[35],[37,1115,1116],{"xmlns":39},[42,1117,1118,1132],{},[45,1119,1120,1122,1124,1126,1128,1130],{},[48,1121,920],{},[51,1123,54],{"stretchy":53},[48,1125,434],{},[51,1127,747],{"separator":122},[48,1129,369],{},[51,1131,66],{"stretchy":53},[114,1133,1134],{"encoding":116},"P(X,Y)",[23,1136,1138],{"className":1137,"ariaHidden":122},[121],[23,1139,1141,1144,1147,1150,1153,1156,1159,1162],{"className":1140},[126],[23,1142],{"className":1143,"style":131},[130],[23,1145,920],{"className":1146,"style":1047},[135,136],[23,1148,54],{"className":1149},[140],[23,1151,434],{"className":1152,"style":454},[135,136],[23,1154,747],{"className":1155},[824],[23,1157],{"className":1158,"style":280},[148],[23,1160,369],{"className":1161,"style":392},[135,136],[23,1163,66],{"className":1164},[169]," 取的。这个分布在实践中未知，于是我们退而求其次：用样本来估计它，或者直接在一个受限的函数类里最小化经验风险。这个「退而求其次」的过程，就是统计学习理论。",[11,1167,1168,1169,1197,1198,1226,1227,1255,1256,1284,1285,1313,1314,1342,1343,1371,1372,1471],{},"同一套语言可以覆盖很多看似不同的任务。医疗诊断中，",[23,1170,1172,1185],{"className":1171,"translate":27},[31],[23,1173,1175],{"className":1174},[35],[37,1176,1177],{"xmlns":39},[42,1178,1179,1183],{},[45,1180,1181],{},[48,1182,369],{},[114,1184,369],{"encoding":116},[23,1186,1188],{"className":1187,"ariaHidden":122},[121],[23,1189,1191,1194],{"className":1190},[126],[23,1192],{"className":1193,"style":470},[130],[23,1195,369],{"className":1196,"style":392},[135,136]," 是是否患病，",[23,1199,1201,1214],{"className":1200,"translate":27},[31],[23,1202,1204],{"className":1203},[35],[37,1205,1206],{"xmlns":39},[42,1207,1208,1212],{},[45,1209,1210],{},[48,1211,495],{},[114,1213,495],{"encoding":116},[23,1215,1217],{"className":1216,"ariaHidden":122},[121],[23,1218,1220,1223],{"className":1219},[126],[23,1221],{"className":1222,"style":703},[130],[23,1224,495],{"className":1225},[135,136]," 是是否治疗，损失把漏诊与误诊的代价写成两个不同的数；垃圾邮件过滤中，",[23,1228,1230,1243],{"className":1229,"translate":27},[31],[23,1231,1233],{"className":1232},[35],[37,1234,1235],{"xmlns":39},[42,1236,1237,1241],{},[45,1238,1239],{},[48,1240,369],{},[114,1242,369],{"encoding":116},[23,1244,1246],{"className":1245,"ariaHidden":122},[121],[23,1247,1249,1252],{"className":1248},[126],[23,1250],{"className":1251,"style":470},[130],[23,1253,369],{"className":1254,"style":392},[135,136]," 是邮件类别，",[23,1257,1259,1272],{"className":1258,"translate":27},[31],[23,1260,1262],{"className":1261},[35],[37,1263,1264],{"xmlns":39},[42,1265,1266,1270],{},[45,1267,1268],{},[48,1269,495],{},[114,1271,495],{"encoding":116},[23,1273,1275],{"className":1274,"ariaHidden":122},[121],[23,1276,1278,1281],{"className":1277},[126],[23,1279],{"className":1280,"style":703},[130],[23,1282,495],{"className":1283},[135,136]," 是放行或拦截；回归中，",[23,1286,1288,1301],{"className":1287,"translate":27},[31],[23,1289,1291],{"className":1290},[35],[37,1292,1293],{"xmlns":39},[42,1294,1295,1299],{},[45,1296,1297],{},[48,1298,495],{},[114,1300,495],{"encoding":116},[23,1302,1304],{"className":1303,"ariaHidden":122},[121],[23,1305,1307,1310],{"className":1306},[126],[23,1308],{"className":1309,"style":703},[130],[23,1311,495],{"className":1312},[135,136]," 是一个实数，损失通常是平方误差；强化学习中，",[23,1315,1317,1330],{"className":1316,"translate":27},[31],[23,1318,1320],{"className":1319},[35],[37,1321,1322],{"xmlns":39},[42,1323,1324,1328],{},[45,1325,1326],{},[48,1327,434],{},[114,1329,434],{"encoding":116},[23,1331,1333],{"className":1332,"ariaHidden":122},[121],[23,1334,1336,1339],{"className":1335},[126],[23,1337],{"className":1338,"style":470},[130],[23,1340,434],{"className":1341,"style":454},[135,136]," 是历史状态，",[23,1344,1346,1359],{"className":1345,"translate":27},[31],[23,1347,1349],{"className":1348},[35],[37,1350,1351],{"xmlns":39},[42,1352,1353,1357],{},[45,1354,1355],{},[48,1356,495],{},[114,1358,495],{"encoding":116},[23,1360,1362],{"className":1361,"ariaHidden":122},[121],[23,1363,1365,1368],{"className":1364},[126],[23,1366],{"className":1367,"style":703},[130],[23,1369,495],{"className":1370},[135,136]," 是动作，损失是负回报。它们的共同点是：",[18,1373,1374,1375,1441,1442,1470],{},"不确定性由 ",[23,1376,1378,1402],{"className":1377,"translate":27},[31],[23,1379,1381],{"className":1380},[35],[37,1382,1383],{"xmlns":39},[42,1384,1385,1399],{},[45,1386,1387,1389,1391,1393,1395,1397],{},[48,1388,920],{},[51,1390,54],{"stretchy":53},[48,1392,369],{},[51,1394,60],{},[48,1396,434],{},[51,1398,66],{"stretchy":53},[114,1400,1401],{"encoding":116},"P(Y\\mid X)",[23,1403,1405,1429],{"className":1404,"ariaHidden":122},[121],[23,1406,1408,1411,1414,1417,1420,1423,1426],{"className":1407},[126],[23,1409],{"className":1410,"style":131},[130],[23,1412,920],{"className":1413,"style":1047},[135,136],[23,1415,54],{"className":1416},[140],[23,1418,369],{"className":1419,"style":392},[135,136],[23,1421],{"className":1422,"style":149},[148],[23,1424,60],{"className":1425},[153],[23,1427],{"className":1428,"style":149},[148],[23,1430,1432,1435,1438],{"className":1431},[126],[23,1433],{"className":1434,"style":131},[130],[23,1436,434],{"className":1437,"style":454},[135,136],[23,1439,66],{"className":1440},[169]," 描述，偏好由 ",[23,1443,1445,1458],{"className":1444,"translate":27},[31],[23,1446,1448],{"className":1447},[35],[37,1449,1450],{"xmlns":39},[42,1451,1452,1456],{},[45,1453,1454],{},[48,1455,555],{},[114,1457,555],{"encoding":116},[23,1459,1461],{"className":1460,"ariaHidden":122},[121],[23,1462,1464,1467],{"className":1463},[126],[23,1465],{"className":1466,"style":470},[130],[23,1468,555],{"className":1469},[135,136]," 描述，决策则是两者的交汇","。",[11,1473,1474,1475,1503],{},"把联合风险按 ",[23,1476,1478,1491],{"className":1477,"translate":27},[31],[23,1479,1481],{"className":1480},[35],[37,1482,1483],{"xmlns":39},[42,1484,1485,1489],{},[45,1486,1487],{},[48,1488,434],{},[114,1490,434],{"encoding":116},[23,1492,1494],{"className":1493,"ariaHidden":122},[121],[23,1495,1497,1500],{"className":1496},[126],[23,1498],{"className":1499,"style":470},[130],[23,1501,434],{"className":1502,"style":454},[135,136]," 分解，可以得到一个更有用的形式：",[23,1505,1507],{"className":1506,"translate":27},[26],[23,1508,1510,1677],{"className":1509,"translate":27},[31],[23,1511,1513],{"className":1512},[35],[37,1514,1515],{"xmlns":39,"display":40},[42,1516,1517,1674],{},[1518,1519,1523,1625],"mtable",{"rowspacing":1520,"columnalign":1521,"columnspacing":1522},"0.25em","right left","0em",[1524,1525,1526,1543],"mtr",{},[1527,1528,1529],"mtd",{},[1530,1531,1533],"mstyle",{"scriptlevel":1532,"displaystyle":122},"0",[45,1534,1535,1537,1539,1541],{},[48,1536,572],{},[51,1538,54],{"stretchy":53},[48,1540,736],{},[51,1542,66],{"stretchy":53},[1527,1544,1545],{},[1530,1546,1547],{"scriptlevel":1532,"displaystyle":122},[45,1548,1549,1551,1553,1560,1611,1613,1615,1617,1619,1621,1623],{},[45,1550],{},[51,1552,69],{},[898,1554,1555,1558],{},[51,1556,1557],{},"∫",[48,1559,434],{"mathvariant":375},[45,1561,1562,1564,1570,1572,1574,1576,1578,1580,1582,1584,1586,1588,1590,1592,1594,1596,1598,1600,1602,1604,1607,1609],{},[51,1563,54],{"fence":122},[898,1565,1566,1568],{},[51,1567,1557],{},[48,1569,369],{"mathvariant":375},[48,1571,555],{},[51,1573,54],{"stretchy":53},[48,1575,660],{},[51,1577,747],{"separator":122},[48,1579,736],{},[51,1581,54],{"stretchy":53},[48,1583,753],{},[51,1585,66],{"stretchy":53},[51,1587,66],{"stretchy":53},[88,1589,90],{},[48,1591,11],{},[51,1593,54],{"stretchy":53},[48,1595,660],{},[51,1597,60],{},[48,1599,753],{},[51,1601,66],{"stretchy":53},[88,1603,90],{},[48,1605,1606],{},"d",[48,1608,660],{},[51,1610,66],{"fence":122},[48,1612,11],{},[51,1614,54],{"stretchy":53},[48,1616,753],{},[51,1618,66],{"stretchy":53},[88,1620,90],{},[48,1622,1606],{},[48,1624,753],{},[1524,1626,1627,1633],{},[1527,1628,1629],{},[1530,1630,1631],{"scriptlevel":1532,"displaystyle":122},[45,1632],{},[1527,1634,1635],{},[1530,1636,1637],{"scriptlevel":1532,"displaystyle":122},[45,1638,1639,1641,1643,1649,1651,1654,1656,1658,1660,1662,1664,1666,1668,1670,1672],{},[45,1640],{},[51,1642,69],{},[898,1644,1645,1647],{},[48,1646,902],{"mathvariant":571},[48,1648,434],{},[51,1650,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,1652,1653],{},"r",[51,1655,54],{"stretchy":53},[48,1657,736],{},[51,1659,54],{"stretchy":53},[48,1661,434],{},[51,1663,66],{"stretchy":53},[51,1665,747],{"separator":122},[48,1667,434],{},[51,1669,66],{"stretchy":53},[51,1671,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,1673,747],{"separator":122},[114,1675,1676],{"encoding":116},"\\begin{aligned}\nR(\\delta)\n&=\\int_{\\mathcal X}\\left(\\int_{\\mathcal Y}L(y,\\delta(x))\\,p(y\\mid x)\\,dy\\right)p(x)\\,dx\\\\\n&=\\mathbb E_X\\big[r(\\delta(X),X)\\big],\n\\end{aligned}",[23,1678,1680],{"className":1679,"ariaHidden":122},[121],[23,1681,1683,1687],{"className":1682},[126],[23,1684],{"className":1685,"style":1686},[130],"height:3.94em;vertical-align:-1.72em;",[23,1688,1690],{"className":1689},[135],[23,1691,1693,1750],{"className":1692},[1518],[23,1694,1697],{"className":1695},[1696],"col-align-r",[23,1698,1700,1741],{"className":1699},[199,200],[23,1701,1703,1738],{"className":1702},[204],[23,1704,1707,1729],{"className":1705,"style":1706},[208],"height:2.22em;",[23,1708,1710,1714],{"style":1709},"top:-4.22em;",[23,1711],{"className":1712,"style":1713},[216],"height:3.45em;",[23,1715,1717,1720,1723,1726],{"className":1716},[135],[23,1718,572],{"className":1719,"style":963},[135,136],[23,1721,54],{"className":1722},[140],[23,1724,736],{"className":1725,"style":783},[135,136],[23,1727,66],{"className":1728},[169],[23,1730,1732,1735],{"style":1731},"top:-2.09em;",[23,1733],{"className":1734,"style":1713},[216],[23,1736],{"className":1737},[135],[23,1739,297],{"className":1740},[296],[23,1742,1744],{"className":1743},[204],[23,1745,1748],{"className":1746,"style":1747},[208],"height:1.72em;",[23,1749],{},[23,1751,1754],{"className":1752},[1753],"col-align-l",[23,1755,1757,2099],{"className":1756},[199,200],[23,1758,1760,2096],{"className":1759},[204],[23,1761,1763,1988],{"className":1762,"style":1706},[208],[23,1764,1765,1768],{"style":1709},[23,1766],{"className":1767,"style":1713},[216],[23,1769,1771,1774,1777,1780,1783,1833,1836,1964,1967,1970,1973,1976,1979,1982,1985],{"className":1770},[135],[23,1772],{"className":1773},[135],[23,1775],{"className":1776,"style":149},[148],[23,1778,69],{"className":1779},[153],[23,1781],{"className":1782,"style":149},[148],[23,1784,1787,1793],{"className":1785},[1786],"mop",[23,1788,1557],{"className":1789,"style":1792},[1786,1790,1791],"op-symbol","large-op","margin-right:0.4445em;position:relative;top:-0.0011em;",[23,1794,1796],{"className":1795},[998],[23,1797,1799,1824],{"className":1798},[199,200],[23,1800,1802,1821],{"className":1801},[204],[23,1803,1806],{"className":1804,"style":1805},[208],"height:-0.4336em;",[23,1807,1809,1812],{"style":1808},"top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;",[23,1810],{"className":1811,"style":1015},[216],[23,1813,1815],{"className":1814},[1019,1020,1021,1022],[23,1816,1818],{"className":1817},[135,1022],[23,1819,434],{"className":1820,"style":474},[135,412,1022],[23,1822,297],{"className":1823},[296],[23,1825,1827],{"className":1826},[204],[23,1828,1831],{"className":1829,"style":1830},[208],"height:0.9119em;",[23,1832],{},[23,1834],{"className":1835,"style":280},[148],[23,1837,1840,1848,1892,1895,1898,1901,1904,1907,1910,1913,1916,1919,1922,1925,1928,1931,1934,1937,1940,1943,1946,1949,1952,1955,1958],{"className":1838},[1839],"minner",[23,1841,1845],{"className":1842,"style":1844},[140,1843],"delimcenter","top:0em;",[23,1846,54],{"className":1847},[1066,1021],[23,1849,1851,1854],{"className":1850},[1786],[23,1852,1557],{"className":1853,"style":1792},[1786,1790,1791],[23,1855,1857],{"className":1856},[998],[23,1858,1860,1883],{"className":1859},[199,200],[23,1861,1863,1880],{"className":1862},[204],[23,1864,1866],{"className":1865,"style":1805},[208],[23,1867,1868,1871],{"style":1808},[23,1869],{"className":1870,"style":1015},[216],[23,1872,1874],{"className":1873},[1019,1020,1021,1022],[23,1875,1877],{"className":1876},[135,1022],[23,1878,369],{"className":1879,"style":413},[135,412,1022],[23,1881,297],{"className":1882},[296],[23,1884,1886],{"className":1885},[204],[23,1887,1890],{"className":1888,"style":1889},[208],"height:0.98em;",[23,1891],{},[23,1893],{"className":1894,"style":280},[148],[23,1896,555],{"className":1897},[135,136],[23,1899,54],{"className":1900},[140],[23,1902,660],{"className":1903,"style":676},[135,136],[23,1905,747],{"className":1906},[824],[23,1908],{"className":1909,"style":280},[148],[23,1911,736],{"className":1912,"style":783},[135,136],[23,1914,54],{"className":1915},[140],[23,1917,753],{"className":1918},[135,136],[23,1920,1095],{"className":1921},[169],[23,1923],{"className":1924,"style":280},[148],[23,1926,11],{"className":1927},[135,136],[23,1929,54],{"className":1930},[140],[23,1932,660],{"className":1933,"style":676},[135,136],[23,1935],{"className":1936,"style":149},[148],[23,1938,60],{"className":1939},[153],[23,1941],{"className":1942,"style":149},[148],[23,1944,753],{"className":1945},[135,136],[23,1947,66],{"className":1948},[169],[23,1950],{"className":1951,"style":280},[148],[23,1953,1606],{"className":1954},[135,136],[23,1956,660],{"className":1957,"style":676},[135,136],[23,1959,1961],{"className":1960,"style":1844},[169,1843],[23,1962,66],{"className":1963},[1066,1021],[23,1965],{"className":1966,"style":280},[148],[23,1968,11],{"className":1969},[135,136],[23,1971,54],{"className":1972},[140],[23,1974,753],{"className":1975},[135,136],[23,1977,66],{"className":1978},[169],[23,1980],{"className":1981,"style":280},[148],[23,1983,1606],{"className":1984},[135,136],[23,1986,753],{"className":1987},[135,136],[23,1989,1990,1993],{"style":1731},[23,1991],{"className":1992,"style":1713},[216],[23,1994,1996,1999,2002,2005,2008,2051,2057,2060,2063,2066,2069,2072,2075,2078,2081,2084,2087,2093],{"className":1995},[135],[23,1997],{"className":1998},[135],[23,2000],{"className":2001,"style":149},[148],[23,2003,69],{"className":2004},[153],[23,2006],{"className":2007,"style":149},[148],[23,2009,2011,2014],{"className":2010},[135],[23,2012,902],{"className":2013},[135,644],[23,2015,2017],{"className":2016},[998],[23,2018,2020,2042],{"className":2019},[199,200],[23,2021,2023,2039],{"className":2022},[204],[23,2024,2027],{"className":2025,"style":2026},[208],"height:0.3283em;",[23,2028,2030,2033],{"style":2029},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[23,2031],{"className":2032,"style":1015},[216],[23,2034,2036],{"className":2035},[1019,1020,1021,1022],[23,2037,434],{"className":2038,"style":454},[135,136,1022],[23,2040,297],{"className":2041},[296],[23,2043,2045],{"className":2044},[204],[23,2046,2049],{"className":2047,"style":2048},[208],"height:0.15em;",[23,2050],{},[23,2052,2054],{"className":2053},[135],[23,2055,924],{"className":2056},[1066,1067],[23,2058,1653],{"className":2059,"style":144},[135,136],[23,2061,54],{"className":2062},[140],[23,2064,736],{"className":2065,"style":783},[135,136],[23,2067,54],{"className":2068},[140],[23,2070,434],{"className":2071,"style":454},[135,136],[23,2073,66],{"className":2074},[169],[23,2076,747],{"className":2077},[824],[23,2079],{"className":2080,"style":280},[148],[23,2082,434],{"className":2083,"style":454},[135,136],[23,2085,66],{"className":2086},[169],[23,2088,2090],{"className":2089},[135],[23,2091,945],{"className":2092},[1066,1067],[23,2094,747],{"className":2095},[824],[23,2097,297],{"className":2098},[296],[23,2100,2102],{"className":2101},[204],[23,2103,2105],{"className":2104,"style":1747},[208],[23,2106],{},[11,2108,2109],{},"其中",[23,2111,2113],{"className":2112,"translate":27},[26],[23,2114,2116,2168],{"className":2115,"translate":27},[31],[23,2117,2119],{"className":2118},[35],[37,2120,2121],{"xmlns":39,"display":40},[42,2122,2123,2165],{},[45,2124,2125,2127,2129,2131,2133,2135,2137,2139,2141,2143,2145,2147,2149,2151,2153,2155,2157,2159,2161,2163],{},[48,2126,1653],{},[51,2128,54],{"stretchy":53},[48,2130,495],{},[51,2132,747],{"separator":122},[48,2134,753],{},[51,2136,66],{"stretchy":53},[51,2138,69],{},[48,2140,902],{"mathvariant":571},[51,2142,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,2144,555],{},[51,2146,54],{"stretchy":53},[48,2148,369],{},[51,2150,747],{"separator":122},[48,2152,495],{},[51,2154,66],{"stretchy":53},[51,2156,60],{},[48,2158,434],{},[51,2160,69],{},[48,2162,753],{},[51,2164,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[114,2166,2167],{"encoding":116},"r(a,x)=\\mathbb E\\big[L(Y,a)\\mid X=x\\big]",[23,2169,2171,2207,2253,2271],{"className":2170,"ariaHidden":122},[121],[23,2172,2174,2177,2180,2183,2186,2189,2192,2195,2198,2201,2204],{"className":2173},[126],[23,2175],{"className":2176,"style":131},[130],[23,2178,1653],{"className":2179,"style":144},[135,136],[23,2181,54],{"className":2182},[140],[23,2184,495],{"className":2185},[135,136],[23,2187,747],{"className":2188},[824],[23,2190],{"className":2191,"style":280},[148],[23,2193,753],{"className":2194},[135,136],[23,2196,66],{"className":2197},[169],[23,2199],{"className":2200,"style":149},[148],[23,2202,69],{"className":2203},[153],[23,2205],{"className":2206,"style":149},[148],[23,2208,2210,2214,2217,2223,2226,2229,2232,2235,2238,2241,2244,2247,2250],{"className":2209},[126],[23,2211],{"className":2212,"style":2213},[130],"height:1.2em;vertical-align:-0.35em;",[23,2215,902],{"className":2216},[135,644],[23,2218,2220],{"className":2219},[135],[23,2221,924],{"className":2222},[1066,1067],[23,2224,555],{"className":2225},[135,136],[23,2227,54],{"className":2228},[140],[23,2230,369],{"className":2231,"style":392},[135,136],[23,2233,747],{"className":2234},[824],[23,2236],{"className":2237,"style":280},[148],[23,2239,495],{"className":2240},[135,136],[23,2242,66],{"className":2243},[169],[23,2245],{"className":2246,"style":149},[148],[23,2248,60],{"className":2249},[153],[23,2251],{"className":2252,"style":149},[148],[23,2254,2256,2259,2262,2265,2268],{"className":2255},[126],[23,2257],{"className":2258,"style":470},[130],[23,2260,434],{"className":2261,"style":454},[135,136],[23,2263],{"className":2264,"style":149},[148],[23,2266,69],{"className":2267},[153],[23,2269],{"className":2270,"style":149},[148],[23,2272,2274,2277,2280],{"className":2273},[126],[23,2275],{"className":2276,"style":2213},[130],[23,2278,753],{"className":2279},[135,136],[23,2281,2283],{"className":2282},[135],[23,2284,945],{"className":2285},[1066,1067],[11,2287,2288,2289,2317,2318,2346,2347,2350,2351,2379,2380,2408,2409,2466,2467,2495],{},"称为在 ",[23,2290,2292,2305],{"className":2291,"translate":27},[31],[23,2293,2295],{"className":2294},[35],[37,2296,2297],{"xmlns":39},[42,2298,2299,2303],{},[45,2300,2301],{},[48,2302,753],{},[114,2304,753],{"encoding":116},[23,2306,2308],{"className":2307,"ariaHidden":122},[121],[23,2309,2311,2314],{"className":2310},[126],[23,2312],{"className":2313,"style":703},[130],[23,2315,753],{"className":2316},[135,136]," 处采取行动 ",[23,2319,2321,2334],{"className":2320,"translate":27},[31],[23,2322,2324],{"className":2323},[35],[37,2325,2326],{"xmlns":39},[42,2327,2328,2332],{},[45,2329,2330],{},[48,2331,495],{},[114,2333,495],{"encoding":116},[23,2335,2337],{"className":2336,"ariaHidden":122},[121],[23,2338,2340,2343],{"className":2339},[126],[23,2341],{"className":2342,"style":703},[130],[23,2344,495],{"className":2345},[135,136]," 的",[18,2348,2349],{},"条件风险","。这个分解是整个理论的第一块基石：联合风险被拆成了「每个观测点上的局部风险」再对 ",[23,2352,2354,2367],{"className":2353,"translate":27},[31],[23,2355,2357],{"className":2356},[35],[37,2358,2359],{"xmlns":39},[42,2360,2361,2365],{},[45,2362,2363],{},[48,2364,434],{},[114,2366,434],{"encoding":116},[23,2368,2370],{"className":2369,"ariaHidden":122},[121],[23,2371,2373,2376],{"className":2372},[126],[23,2374],{"className":2375,"style":470},[130],[23,2377,434],{"className":2378,"style":454},[135,136]," 平均。于是，只要在每个 ",[23,2381,2383,2396],{"className":2382,"translate":27},[31],[23,2384,2386],{"className":2385},[35],[37,2387,2388],{"xmlns":39},[42,2389,2390,2394],{},[45,2391,2392],{},[48,2393,753],{},[114,2395,753],{"encoding":116},[23,2397,2399],{"className":2398,"ariaHidden":122},[121],[23,2400,2402,2405],{"className":2401},[126],[23,2403],{"className":2404,"style":703},[130],[23,2406,753],{"className":2407},[135,136]," 上把 ",[23,2410,2412,2436],{"className":2411,"translate":27},[31],[23,2413,2415],{"className":2414},[35],[37,2416,2417],{"xmlns":39},[42,2418,2419,2433],{},[45,2420,2421,2423,2425,2427,2429,2431],{},[48,2422,1653],{},[51,2424,54],{"stretchy":53},[48,2426,495],{},[51,2428,747],{"separator":122},[48,2430,753],{},[51,2432,66],{"stretchy":53},[114,2434,2435],{"encoding":116},"r(a,x)",[23,2437,2439],{"className":2438,"ariaHidden":122},[121],[23,2440,2442,2445,2448,2451,2454,2457,2460,2463],{"className":2441},[126],[23,2443],{"className":2444,"style":131},[130],[23,2446,1653],{"className":2447,"style":144},[135,136],[23,2449,54],{"className":2450},[140],[23,2452,495],{"className":2453},[135,136],[23,2455,747],{"className":2456},[824],[23,2458],{"className":2459,"style":280},[148],[23,2461,753],{"className":2462},[135,136],[23,2464,66],{"className":2465},[169]," 压到最低，全局风险就自动最低，不同 ",[23,2468,2470,2483],{"className":2469,"translate":27},[31],[23,2471,2473],{"className":2472},[35],[37,2474,2475],{"xmlns":39},[42,2476,2477,2481],{},[45,2478,2479],{},[48,2480,753],{},[114,2482,753],{"encoding":116},[23,2484,2486],{"className":2485,"ariaHidden":122},[121],[23,2487,2489,2492],{"className":2488},[126],[23,2490],{"className":2491,"style":703},[130],[23,2493,753],{"className":2494},[135,136]," 之间没有耦合。",[338,2497,2499],{"id":2498},"_2-概率的语言","2. 概率的语言",[2501,2502,2504],"h3",{"id":2503},"_21-条件概率信息如何改变概率","2.1 条件概率：信息如何改变概率",[11,2506,2507],{},"条件概率的定义是",[23,2509,2511],{"className":2510,"translate":27},[26],[23,2512,2514,2587],{"className":2513,"translate":27},[31],[23,2515,2517],{"className":2516},[35],[37,2518,2519],{"xmlns":39,"display":40},[42,2520,2521,2584],{},[45,2522,2523,2525,2527,2529,2531,2534,2536,2538,2565,2567,2569,2571,2573,2575,2577,2580],{},[48,2524,920],{},[51,2526,54],{"stretchy":53},[48,2528,500],{},[51,2530,60],{},[48,2532,2533],{},"B",[51,2535,66],{"stretchy":53},[51,2537,69],{},[71,2539,2540,2555],{},[45,2541,2542,2544,2546,2548,2551,2553],{},[48,2543,920],{},[51,2545,54],{"stretchy":53},[48,2547,500],{},[51,2549,2550],{},"∩",[48,2552,2533],{},[51,2554,66],{"stretchy":53},[45,2556,2557,2559,2561,2563],{},[48,2558,920],{},[51,2560,54],{"stretchy":53},[48,2562,2533],{},[51,2564,66],{"stretchy":53},[51,2566,747],{"separator":122},[148,2568],{"width":750},[48,2570,920],{},[51,2572,54],{"stretchy":53},[48,2574,2533],{},[51,2576,66],{"stretchy":53},[51,2578,2579],{},">",[2581,2582,2583],"mn",{},"0.",[114,2585,2586],{"encoding":116},"P(A\\mid B)=\\frac{P(A\\cap B)}{P(B)},\\qquad P(B)>0.",[23,2588,2590,2614,2636,2764],{"className":2589,"ariaHidden":122},[121],[23,2591,2593,2596,2599,2602,2605,2608,2611],{"className":2592},[126],[23,2594],{"className":2595,"style":131},[130],[23,2597,920],{"className":2598,"style":1047},[135,136],[23,2600,54],{"className":2601},[140],[23,2603,500],{"className":2604},[135,136],[23,2606],{"className":2607,"style":149},[148],[23,2609,60],{"className":2610},[153],[23,2612],{"className":2613,"style":149},[148],[23,2615,2617,2620,2624,2627,2630,2633],{"className":2616},[126],[23,2618],{"className":2619,"style":131},[130],[23,2621,2533],{"className":2622,"style":2623},[135,136],"margin-right:0.0502em;",[23,2625,66],{"className":2626},[169],[23,2628],{"className":2629,"style":149},[148],[23,2631,69],{"className":2632},[153],[23,2634],{"className":2635,"style":149},[148],[23,2637,2639,2642,2734,2737,2740,2743,2746,2749,2752,2755,2758,2761],{"className":2638},[126],[23,2640],{"className":2641,"style":185},[130],[23,2643,2645,2648,2731],{"className":2644},[135],[23,2646],{"className":2647},[140,192],[23,2649,2651],{"className":2650},[71],[23,2652,2654,2723],{"className":2653},[199,200],[23,2655,2657,2720],{"className":2656},[204],[23,2658,2660,2680,2688],{"className":2659,"style":209},[208],[23,2661,2662,2665],{"style":212},[23,2663],{"className":2664,"style":217},[216],[23,2666,2668,2671,2674,2677],{"className":2667},[135],[23,2669,920],{"className":2670,"style":1047},[135,136],[23,2672,54],{"className":2673},[140],[23,2675,2533],{"className":2676,"style":2623},[135,136],[23,2678,66],{"className":2679},[169],[23,2681,2682,2685],{"style":235},[23,2683],{"className":2684,"style":217},[216],[23,2686],{"className":2687,"style":243},[242],[23,2689,2690,2693],{"style":246},[23,2691],{"className":2692,"style":217},[216],[23,2694,2696,2699,2702,2705,2708,2711,2714,2717],{"className":2695},[135],[23,2697,920],{"className":2698,"style":1047},[135,136],[23,2700,54],{"className":2701},[140],[23,2703,500],{"className":2704},[135,136],[23,2706],{"className":2707,"style":392},[148],[23,2709,2550],{"className":2710},[612],[23,2712],{"className":2713,"style":392},[148],[23,2715,2533],{"className":2716,"style":2623},[135,136],[23,2718,66],{"className":2719},[169],[23,2721,297],{"className":2722},[296],[23,2724,2726],{"className":2725},[204],[23,2727,2729],{"className":2728,"style":304},[208],[23,2730],{},[23,2732],{"className":2733},[169,192],[23,2735,747],{"className":2736},[824],[23,2738],{"className":2739,"style":828},[148],[23,2741],{"className":2742,"style":280},[148],[23,2744,920],{"className":2745,"style":1047},[135,136],[23,2747,54],{"className":2748},[140],[23,2750,2533],{"className":2751,"style":2623},[135,136],[23,2753,66],{"className":2754},[169],[23,2756],{"className":2757,"style":149},[148],[23,2759,2579],{"className":2760},[153],[23,2762],{"className":2763,"style":149},[148],[23,2765,2767,2771],{"className":2766},[126],[23,2768],{"className":2769,"style":2770},[130],"height:0.6444em;",[23,2772,2583],{"className":2773},[135],[11,2775,2776,2777,2805,2806,2834,2835,2977],{},"它不是在「重新定义」概率，而是在缩小样本空间：一旦知道 ",[23,2778,2780,2793],{"className":2779,"translate":27},[31],[23,2781,2783],{"className":2782},[35],[37,2784,2785],{"xmlns":39},[42,2786,2787,2791],{},[45,2788,2789],{},[48,2790,2533],{},[114,2792,2533],{"encoding":116},[23,2794,2796],{"className":2795,"ariaHidden":122},[121],[23,2797,2799,2802],{"className":2798},[126],[23,2800],{"className":2801,"style":470},[130],[23,2803,2533],{"className":2804,"style":2623},[135,136]," 发生，所有不含 ",[23,2807,2809,2822],{"className":2808,"translate":27},[31],[23,2810,2812],{"className":2811},[35],[37,2813,2814],{"xmlns":39},[42,2815,2816,2820],{},[45,2817,2818],{},[48,2819,2533],{},[114,2821,2533],{"encoding":116},[23,2823,2825],{"className":2824,"ariaHidden":122},[121],[23,2826,2828,2831],{"className":2827},[126],[23,2829],{"className":2830,"style":470},[130],[23,2832,2533],{"className":2833,"style":2623},[135,136]," 的结果都被排除，原来的概率需要在剩下的世界里重新归一化。连续情形下，条件分布由条件密度 ",[23,2836,2838,2887],{"className":2837,"translate":27},[31],[23,2839,2841],{"className":2840},[35],[37,2842,2843],{"xmlns":39},[42,2844,2845,2884],{},[45,2846,2847,2849,2851,2853,2855,2857,2859,2861,2863,2865,2867,2869,2871,2873,2876,2878,2880,2882],{},[48,2848,11],{},[51,2850,54],{"stretchy":53},[48,2852,660],{},[51,2854,60],{},[48,2856,753],{},[51,2858,66],{"stretchy":53},[51,2860,69],{},[48,2862,11],{},[51,2864,54],{"stretchy":53},[48,2866,753],{},[51,2868,747],{"separator":122},[48,2870,660],{},[51,2872,66],{"stretchy":53},[48,2874,2875],{"mathvariant":111},"\u002F",[48,2877,11],{},[51,2879,54],{"stretchy":53},[48,2881,753],{},[51,2883,66],{"stretchy":53},[114,2885,2886],{"encoding":116},"p(y\\mid x)=p(x,y)\u002Fp(x)",[23,2888,2890,2914,2935],{"className":2889,"ariaHidden":122},[121],[23,2891,2893,2896,2899,2902,2905,2908,2911],{"className":2892},[126],[23,2894],{"className":2895,"style":131},[130],[23,2897,11],{"className":2898},[135,136],[23,2900,54],{"className":2901},[140],[23,2903,660],{"className":2904,"style":676},[135,136],[23,2906],{"className":2907,"style":149},[148],[23,2909,60],{"className":2910},[153],[23,2912],{"className":2913,"style":149},[148],[23,2915,2917,2920,2923,2926,2929,2932],{"className":2916},[126],[23,2918],{"className":2919,"style":131},[130],[23,2921,753],{"className":2922},[135,136],[23,2924,66],{"className":2925},[169],[23,2927],{"className":2928,"style":149},[148],[23,2930,69],{"className":2931},[153],[23,2933],{"className":2934,"style":149},[148],[23,2936,2938,2941,2944,2947,2950,2953,2956,2959,2962,2965,2968,2971,2974],{"className":2937},[126],[23,2939],{"className":2940,"style":131},[130],[23,2942,11],{"className":2943},[135,136],[23,2945,54],{"className":2946},[140],[23,2948,753],{"className":2949},[135,136],[23,2951,747],{"className":2952},[824],[23,2954],{"className":2955,"style":280},[148],[23,2957,660],{"className":2958,"style":676},[135,136],[23,2960,66],{"className":2961},[169],[23,2963,2875],{"className":2964},[135],[23,2966,11],{"className":2967},[135,136],[23,2969,54],{"className":2970},[140],[23,2972,753],{"className":2973},[135,136],[23,2975,66],{"className":2976},[169]," 给出；更一般地，它是测度论意义下的正则条件分布，本文不展开技术细节。",[11,2979,2980],{},"由条件概率的定义立刻得到乘法公式",[23,2982,2984],{"className":2983,"translate":27},[26],[23,2985,2987,3057],{"className":2986,"translate":27},[31],[23,2988,2990],{"className":2989},[35],[37,2991,2992],{"xmlns":39,"display":40},[42,2993,2994,3054],{},[45,2995,2996,2998,3000,3002,3004,3006,3008,3010,3012,3014,3016,3018,3020,3022,3024,3026,3028,3030,3032,3034,3036,3038,3040,3042,3044,3046,3048,3050,3052],{},[48,2997,920],{},[51,2999,54],{"stretchy":53},[48,3001,500],{},[51,3003,2550],{},[48,3005,2533],{},[51,3007,66],{"stretchy":53},[51,3009,69],{},[48,3011,920],{},[51,3013,54],{"stretchy":53},[48,3015,500],{},[51,3017,60],{},[48,3019,2533],{},[51,3021,66],{"stretchy":53},[48,3023,920],{},[51,3025,54],{"stretchy":53},[48,3027,2533],{},[51,3029,66],{"stretchy":53},[51,3031,69],{},[48,3033,920],{},[51,3035,54],{"stretchy":53},[48,3037,2533],{},[51,3039,60],{},[48,3041,500],{},[51,3043,66],{"stretchy":53},[48,3045,920],{},[51,3047,54],{"stretchy":53},[48,3049,500],{},[51,3051,66],{"stretchy":53},[51,3053,747],{"separator":122},[114,3055,3056],{"encoding":116},"P(A\\cap B)=P(A\\mid B)P(B)=P(B\\mid A)P(A),",[23,3058,3060,3084,3105,3129,3162,3186],{"className":3059,"ariaHidden":122},[121],[23,3061,3063,3066,3069,3072,3075,3078,3081],{"className":3062},[126],[23,3064],{"className":3065,"style":131},[130],[23,3067,920],{"className":3068,"style":1047},[135,136],[23,3070,54],{"className":3071},[140],[23,3073,500],{"className":3074},[135,136],[23,3076],{"className":3077,"style":392},[148],[23,3079,2550],{"className":3080},[612],[23,3082],{"className":3083,"style":392},[148],[23,3085,3087,3090,3093,3096,3099,3102],{"className":3086},[126],[23,3088],{"className":3089,"style":131},[130],[23,3091,2533],{"className":3092,"style":2623},[135,136],[23,3094,66],{"className":3095},[169],[23,3097],{"className":3098,"style":149},[148],[23,3100,69],{"className":3101},[153],[23,3103],{"className":3104,"style":149},[148],[23,3106,3108,3111,3114,3117,3120,3123,3126],{"className":3107},[126],[23,3109],{"className":3110,"style":131},[130],[23,3112,920],{"className":3113,"style":1047},[135,136],[23,3115,54],{"className":3116},[140],[23,3118,500],{"className":3119},[135,136],[23,3121],{"className":3122,"style":149},[148],[23,3124,60],{"className":3125},[153],[23,3127],{"className":3128,"style":149},[148],[23,3130,3132,3135,3138,3141,3144,3147,3150,3153,3156,3159],{"className":3131},[126],[23,3133],{"className":3134,"style":131},[130],[23,3136,2533],{"className":3137,"style":2623},[135,136],[23,3139,66],{"className":3140},[169],[23,3142,920],{"className":3143,"style":1047},[135,136],[23,3145,54],{"className":3146},[140],[23,3148,2533],{"className":3149,"style":2623},[135,136],[23,3151,66],{"className":3152},[169],[23,3154],{"className":3155,"style":149},[148],[23,3157,69],{"className":3158},[153],[23,3160],{"className":3161,"style":149},[148],[23,3163,3165,3168,3171,3174,3177,3180,3183],{"className":3164},[126],[23,3166],{"className":3167,"style":131},[130],[23,3169,920],{"className":3170,"style":1047},[135,136],[23,3172,54],{"className":3173},[140],[23,3175,2533],{"className":3176,"style":2623},[135,136],[23,3178],{"className":3179,"style":149},[148],[23,3181,60],{"className":3182},[153],[23,3184],{"className":3185,"style":149},[148],[23,3187,3189,3192,3195,3198,3201,3204,3207,3210],{"className":3188},[126],[23,3190],{"className":3191,"style":131},[130],[23,3193,500],{"className":3194},[135,136],[23,3196,66],{"className":3197},[169],[23,3199,920],{"className":3200,"style":1047},[135,136],[23,3202,54],{"className":3203},[140],[23,3205,500],{"className":3206},[135,136],[23,3208,66],{"className":3209},[169],[23,3211,747],{"className":3212},[824],[11,3214,3215,3216,3260],{},"两边同除以 ",[23,3217,3219,3239],{"className":3218,"translate":27},[31],[23,3220,3222],{"className":3221},[35],[37,3223,3224],{"xmlns":39},[42,3225,3226,3236],{},[45,3227,3228,3230,3232,3234],{},[48,3229,920],{},[51,3231,54],{"stretchy":53},[48,3233,2533],{},[51,3235,66],{"stretchy":53},[114,3237,3238],{"encoding":116},"P(B)",[23,3240,3242],{"className":3241,"ariaHidden":122},[121],[23,3243,3245,3248,3251,3254,3257],{"className":3244},[126],[23,3246],{"className":3247,"style":131},[130],[23,3249,920],{"className":3250,"style":1047},[135,136],[23,3252,54],{"className":3253},[140],[23,3255,2533],{"className":3256,"style":2623},[135,136],[23,3258,66],{"className":3259},[169],"，就是贝叶斯公式。",[2501,3262,3264],{"id":3263},"_22-贝叶斯公式","2.2 贝叶斯公式",[11,3266,3267,3268,3296,3297,3325,3326,3354,3355,3384],{},"把 ",[23,3269,3271,3284],{"className":3270,"translate":27},[31],[23,3272,3274],{"className":3273},[35],[37,3275,3276],{"xmlns":39},[42,3277,3278,3282],{},[45,3279,3280],{},[48,3281,2533],{},[114,3283,2533],{"encoding":116},[23,3285,3287],{"className":3286,"ariaHidden":122},[121],[23,3288,3290,3293],{"className":3289},[126],[23,3291],{"className":3292,"style":470},[130],[23,3294,2533],{"className":3295,"style":2623},[135,136]," 换成数据 ",[23,3298,3300,3313],{"className":3299,"translate":27},[31],[23,3301,3303],{"className":3302},[35],[37,3304,3305],{"xmlns":39},[42,3306,3307,3311],{},[45,3308,3309],{},[48,3310,63],{},[114,3312,63],{"encoding":116},[23,3314,3316],{"className":3315,"ariaHidden":122},[121],[23,3317,3319,3322],{"className":3318},[126],[23,3320],{"className":3321,"style":470},[130],[23,3323,63],{"className":3324,"style":144},[135,136],"，把 ",[23,3327,3329,3342],{"className":3328,"translate":27},[31],[23,3330,3332],{"className":3331},[35],[37,3333,3334],{"xmlns":39},[42,3335,3336,3340],{},[45,3337,3338],{},[48,3339,500],{},[114,3341,500],{"encoding":116},[23,3343,3345],{"className":3344,"ariaHidden":122},[121],[23,3346,3348,3351],{"className":3347},[126],[23,3349],{"className":3350,"style":470},[130],[23,3352,500],{"className":3353},[135,136]," 换成参数或隐变量 ",[23,3356,3358,3372],{"className":3357,"translate":27},[31],[23,3359,3361],{"className":3360},[35],[37,3362,3363],{"xmlns":39},[42,3364,3365,3369],{},[45,3366,3367],{},[48,3368,57],{},[114,3370,3371],{"encoding":116},"\\theta",[23,3373,3375],{"className":3374,"ariaHidden":122},[121],[23,3376,3378,3381],{"className":3377},[126],[23,3379],{"className":3380,"style":779},[130],[23,3382,57],{"className":3383,"style":144},[135,136],"，得到",[23,3386,3388],{"className":3387,"translate":27},[26],[23,3389,3391,3499],{"className":3390,"translate":27},[31],[23,3392,3394],{"className":3393},[35],[37,3395,3396],{"xmlns":39,"display":40},[42,3397,3398,3496],{},[45,3399,3400,3402,3404,3406,3408,3410,3412,3414,3450,3452,3454,3456,3458,3460,3462,3464,3466,3468,3470,3472,3474,3476,3478,3480,3482,3484,3486,3488,3490,3492,3494],{},[48,3401,11],{},[51,3403,54],{"stretchy":53},[48,3405,57],{},[51,3407,60],{},[48,3409,63],{},[51,3411,66],{"stretchy":53},[51,3413,69],{},[71,3415,3416,3440],{},[45,3417,3418,3420,3422,3424,3426,3428,3430,3432,3434,3436,3438],{},[48,3419,11],{},[51,3421,54],{"stretchy":53},[48,3423,63],{},[51,3425,60],{},[48,3427,57],{},[51,3429,66],{"stretchy":53},[88,3431,90],{},[48,3433,11],{},[51,3435,54],{"stretchy":53},[48,3437,57],{},[51,3439,66],{"stretchy":53},[45,3441,3442,3444,3446,3448],{},[48,3443,11],{},[51,3445,54],{"stretchy":53},[48,3447,63],{},[51,3449,66],{"stretchy":53},[51,3451,747],{"separator":122},[148,3453],{"width":750},[48,3455,11],{},[51,3457,54],{"stretchy":53},[48,3459,63],{},[51,3461,66],{"stretchy":53},[51,3463,69],{},[51,3465,1557],{},[48,3467,11],{},[51,3469,54],{"stretchy":53},[48,3471,63],{},[51,3473,60],{},[48,3475,57],{},[51,3477,66],{"stretchy":53},[88,3479,90],{},[48,3481,11],{},[51,3483,54],{"stretchy":53},[48,3485,57],{},[51,3487,66],{"stretchy":53},[88,3489,90],{},[48,3491,1606],{},[48,3493,57],{},[48,3495,112],{"mathvariant":111},[114,3497,3498],{"encoding":116},"p(\\theta\\mid D)=\\frac{p(D\\mid\\theta)\\,p(\\theta)}{p(D)},\n\\qquad\np(D)=\\int p(D\\mid\\theta)\\,p(\\theta)\\,d\\theta .",[23,3500,3502,3526,3547,3690,3721],{"className":3501,"ariaHidden":122},[121],[23,3503,3505,3508,3511,3514,3517,3520,3523],{"className":3504},[126],[23,3506],{"className":3507,"style":131},[130],[23,3509,11],{"className":3510},[135,136],[23,3512,54],{"className":3513},[140],[23,3515,57],{"className":3516,"style":144},[135,136],[23,3518],{"className":3519,"style":149},[148],[23,3521,60],{"className":3522},[153],[23,3524],{"className":3525,"style":149},[148],[23,3527,3529,3532,3535,3538,3541,3544],{"className":3528},[126],[23,3530],{"className":3531,"style":131},[130],[23,3533,63],{"className":3534,"style":144},[135,136],[23,3536,66],{"className":3537},[169],[23,3539],{"className":3540,"style":149},[148],[23,3542,69],{"className":3543},[153],[23,3545],{"className":3546,"style":149},[148],[23,3548,3550,3553,3660,3663,3666,3669,3672,3675,3678,3681,3684,3687],{"className":3549},[126],[23,3551],{"className":3552,"style":185},[130],[23,3554,3556,3559,3657],{"className":3555},[135],[23,3557],{"className":3558},[140,192],[23,3560,3562],{"className":3561},[71],[23,3563,3565,3649],{"className":3564},[199,200],[23,3566,3568,3646],{"className":3567},[204],[23,3569,3571,3591,3599],{"className":3570,"style":209},[208],[23,3572,3573,3576],{"style":212},[23,3574],{"className":3575,"style":217},[216],[23,3577,3579,3582,3585,3588],{"className":3578},[135],[23,3580,11],{"className":3581},[135,136],[23,3583,54],{"className":3584},[140],[23,3586,63],{"className":3587,"style":144},[135,136],[23,3589,66],{"className":3590},[169],[23,3592,3593,3596],{"style":235},[23,3594],{"className":3595,"style":217},[216],[23,3597],{"className":3598,"style":243},[242],[23,3600,3601,3604],{"style":246},[23,3602],{"className":3603,"style":217},[216],[23,3605,3607,3610,3613,3616,3619,3622,3625,3628,3631,3634,3637,3640,3643],{"className":3606},[135],[23,3608,11],{"className":3609},[135,136],[23,3611,54],{"className":3612},[140],[23,3614,63],{"className":3615,"style":144},[135,136],[23,3617],{"className":3618,"style":149},[148],[23,3620,60],{"className":3621},[153],[23,3623],{"className":3624,"style":149},[148],[23,3626,57],{"className":3627,"style":144},[135,136],[23,3629,66],{"className":3630},[169],[23,3632],{"className":3633,"style":280},[148],[23,3635,11],{"className":3636},[135,136],[23,3638,54],{"className":3639},[140],[23,3641,57],{"className":3642,"style":144},[135,136],[23,3644,66],{"className":3645},[169],[23,3647,297],{"className":3648},[296],[23,3650,3652],{"className":3651},[204],[23,3653,3655],{"className":3654,"style":304},[208],[23,3656],{},[23,3658],{"className":3659},[169,192],[23,3661,747],{"className":3662},[824],[23,3664],{"className":3665,"style":828},[148],[23,3667],{"className":3668,"style":280},[148],[23,3670,11],{"className":3671},[135,136],[23,3673,54],{"className":3674},[140],[23,3676,63],{"className":3677,"style":144},[135,136],[23,3679,66],{"className":3680},[169],[23,3682],{"className":3683,"style":149},[148],[23,3685,69],{"className":3686},[153],[23,3688],{"className":3689,"style":149},[148],[23,3691,3693,3697,3700,3703,3706,3709,3712,3715,3718],{"className":3692},[126],[23,3694],{"className":3695,"style":3696},[130],"height:2.2222em;vertical-align:-0.8622em;",[23,3698,1557],{"className":3699,"style":1792},[1786,1790,1791],[23,3701],{"className":3702,"style":280},[148],[23,3704,11],{"className":3705},[135,136],[23,3707,54],{"className":3708},[140],[23,3710,63],{"className":3711,"style":144},[135,136],[23,3713],{"className":3714,"style":149},[148],[23,3716,60],{"className":3717},[153],[23,3719],{"className":3720,"style":149},[148],[23,3722,3724,3727,3730,3733,3736,3739,3742,3745,3748,3751,3754,3757],{"className":3723},[126],[23,3725],{"className":3726,"style":131},[130],[23,3728,57],{"className":3729,"style":144},[135,136],[23,3731,66],{"className":3732},[169],[23,3734],{"className":3735,"style":280},[148],[23,3737,11],{"className":3738},[135,136],[23,3740,54],{"className":3741},[140],[23,3743,57],{"className":3744,"style":144},[135,136],[23,3746,66],{"className":3747},[169],[23,3749],{"className":3750,"style":280},[148],[23,3752,1606],{"className":3753},[135,136],[23,3755,57],{"className":3756,"style":144},[135,136],[23,3758,112],{"className":3759},[135],[11,3761,3762],{},"公式的四个部分各有名字：",[3764,3765,3766,3782],"table",{},[3767,3768,3769],"thead",{},[3770,3771,3772,3776,3779],"tr",{},[3773,3774,3775],"th",{},"名称",[3773,3777,3778],{},"记号",[3773,3780,3781],{},"含义",[3783,3784,3785,3840,3916,3999],"tbody",{},[3770,3786,3787,3791,3837],{},[3788,3789,3790],"td",{},"先验",[3788,3792,3793],{},[23,3794,3796,3816],{"className":3795,"translate":27},[31],[23,3797,3799],{"className":3798},[35],[37,3800,3801],{"xmlns":39},[42,3802,3803,3813],{},[45,3804,3805,3807,3809,3811],{},[48,3806,11],{},[51,3808,54],{"stretchy":53},[48,3810,57],{},[51,3812,66],{"stretchy":53},[114,3814,3815],{"encoding":116},"p(\\theta)",[23,3817,3819],{"className":3818,"ariaHidden":122},[121],[23,3820,3822,3825,3828,3831,3834],{"className":3821},[126],[23,3823],{"className":3824,"style":131},[130],[23,3826,11],{"className":3827},[135,136],[23,3829,54],{"className":3830},[140],[23,3832,57],{"className":3833,"style":144},[135,136],[23,3835,66],{"className":3836},[169],[3788,3838,3839],{},"看到数据前的信念",[3770,3841,3842,3845,3913],{},[3788,3843,3844],{},"似然",[3788,3846,3847],{},[23,3848,3850,3874],{"className":3849,"translate":27},[31],[23,3851,3853],{"className":3852},[35],[37,3854,3855],{"xmlns":39},[42,3856,3857,3871],{},[45,3858,3859,3861,3863,3865,3867,3869],{},[48,3860,11],{},[51,3862,54],{"stretchy":53},[48,3864,63],{},[51,3866,60],{},[48,3868,57],{},[51,3870,66],{"stretchy":53},[114,3872,3873],{"encoding":116},"p(D\\mid\\theta)",[23,3875,3877,3901],{"className":3876,"ariaHidden":122},[121],[23,3878,3880,3883,3886,3889,3892,3895,3898],{"className":3879},[126],[23,3881],{"className":3882,"style":131},[130],[23,3884,11],{"className":3885},[135,136],[23,3887,54],{"className":3888},[140],[23,3890,63],{"className":3891,"style":144},[135,136],[23,3893],{"className":3894,"style":149},[148],[23,3896,60],{"className":3897},[153],[23,3899],{"className":3900,"style":149},[148],[23,3902,3904,3907,3910],{"className":3903},[126],[23,3905],{"className":3906,"style":131},[130],[23,3908,57],{"className":3909,"style":144},[135,136],[23,3911,66],{"className":3912},[169],[3788,3914,3915],{},"给定参数时数据的概率",[3770,3917,3918,3921,3967],{},[3788,3919,3920],{},"证据",[3788,3922,3923],{},[23,3924,3926,3946],{"className":3925,"translate":27},[31],[23,3927,3929],{"className":3928},[35],[37,3930,3931],{"xmlns":39},[42,3932,3933,3943],{},[45,3934,3935,3937,3939,3941],{},[48,3936,11],{},[51,3938,54],{"stretchy":53},[48,3940,63],{},[51,3942,66],{"stretchy":53},[114,3944,3945],{"encoding":116},"p(D)",[23,3947,3949],{"className":3948,"ariaHidden":122},[121],[23,3950,3952,3955,3958,3961,3964],{"className":3951},[126],[23,3953],{"className":3954,"style":131},[130],[23,3956,11],{"className":3957},[135,136],[23,3959,54],{"className":3960},[140],[23,3962,63],{"className":3963,"style":144},[135,136],[23,3965,66],{"className":3966},[169],[3788,3968,3969,3970,3998],{},"与 ",[23,3971,3973,3986],{"className":3972,"translate":27},[31],[23,3974,3976],{"className":3975},[35],[37,3977,3978],{"xmlns":39},[42,3979,3980,3984],{},[45,3981,3982],{},[48,3983,57],{},[114,3985,3371],{"encoding":116},[23,3987,3989],{"className":3988,"ariaHidden":122},[121],[23,3990,3992,3995],{"className":3991},[126],[23,3993],{"className":3994,"style":779},[130],[23,3996,57],{"className":3997,"style":144},[135,136]," 无关的归一化常数，也是模型比较的关键",[3770,4000,4001,4004,4072],{},[3788,4002,4003],{},"后验",[3788,4005,4006],{},[23,4007,4009,4033],{"className":4008,"translate":27},[31],[23,4010,4012],{"className":4011},[35],[37,4013,4014],{"xmlns":39},[42,4015,4016,4030],{},[45,4017,4018,4020,4022,4024,4026,4028],{},[48,4019,11],{},[51,4021,54],{"stretchy":53},[48,4023,57],{},[51,4025,60],{},[48,4027,63],{},[51,4029,66],{"stretchy":53},[114,4031,4032],{"encoding":116},"p(\\theta\\mid D)",[23,4034,4036,4060],{"className":4035,"ariaHidden":122},[121],[23,4037,4039,4042,4045,4048,4051,4054,4057],{"className":4038},[126],[23,4040],{"className":4041,"style":131},[130],[23,4043,11],{"className":4044},[135,136],[23,4046,54],{"className":4047},[140],[23,4049,57],{"className":4050,"style":144},[135,136],[23,4052],{"className":4053,"style":149},[148],[23,4055,60],{"className":4056},[153],[23,4058],{"className":4059,"style":149},[148],[23,4061,4063,4066,4069],{"className":4062},[126],[23,4064],{"className":4065,"style":131},[130],[23,4067,63],{"className":4068,"style":144},[135,136],[23,4070,66],{"className":4071},[169],[3788,4073,4074],{},"看到数据后的信念",[11,4076,4077],{},"下面的图把这条链画全。注意「后验」不是终点：它只是决策的输入。",[4079,4080,4081],"div",{},[4082,4083],"img",{"height":4084,"width":4085,"style":4086,"data-d2-theme":4087,"alt":4088,"src":4089},257,720,"max-width:100%;height:auto","default","贝叶斯决策流程：先验与似然合成后验，后验与损失共同决定最优行动","data:image\u002Fsvg+xml;base64,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\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\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",[11,4091,4092,4093,4136,4137,4165],{},"公式右端的 ",[23,4094,4096,4115],{"className":4095,"translate":27},[31],[23,4097,4099],{"className":4098},[35],[37,4100,4101],{"xmlns":39},[42,4102,4103,4113],{},[45,4104,4105,4107,4109,4111],{},[48,4106,11],{},[51,4108,54],{"stretchy":53},[48,4110,63],{},[51,4112,66],{"stretchy":53},[114,4114,3945],{"encoding":116},[23,4116,4118],{"className":4117,"ariaHidden":122},[121],[23,4119,4121,4124,4127,4130,4133],{"className":4120},[126],[23,4122],{"className":4123,"style":131},[130],[23,4125,11],{"className":4126},[135,136],[23,4128,54],{"className":4129},[140],[23,4131,63],{"className":4132,"style":144},[135,136],[23,4134,66],{"className":4135},[169]," 与 ",[23,4138,4140,4153],{"className":4139,"translate":27},[31],[23,4141,4143],{"className":4142},[35],[37,4144,4145],{"xmlns":39},[42,4146,4147,4151],{},[45,4148,4149],{},[48,4150,57],{},[114,4152,3371],{"encoding":116},[23,4154,4156],{"className":4155,"ariaHidden":122},[121],[23,4157,4159,4162],{"className":4158},[126],[23,4160],{"className":4161,"style":779},[130],[23,4163,57],{"className":4164,"style":144},[135,136]," 无关，所以在求后验时常常被略去，写成",[23,4167,4169],{"className":4168,"translate":27},[26],[23,4170,4172,4223],{"className":4171,"translate":27},[31],[23,4173,4175],{"className":4174},[35],[37,4176,4177],{"xmlns":39,"display":40},[42,4178,4179,4220],{},[45,4180,4181,4183,4185,4187,4189,4191,4193,4196,4198,4200,4202,4204,4206,4208,4210,4212,4214,4216,4218],{},[48,4182,11],{},[51,4184,54],{"stretchy":53},[48,4186,57],{},[51,4188,60],{},[48,4190,63],{},[51,4192,66],{"stretchy":53},[51,4194,4195],{},"∝",[48,4197,11],{},[51,4199,54],{"stretchy":53},[48,4201,63],{},[51,4203,60],{},[48,4205,57],{},[51,4207,66],{"stretchy":53},[88,4209,90],{},[48,4211,11],{},[51,4213,54],{"stretchy":53},[48,4215,57],{},[51,4217,66],{"stretchy":53},[48,4219,112],{"mathvariant":111},[114,4221,4222],{"encoding":116},"p(\\theta\\mid D)\\propto p(D\\mid\\theta)\\,p(\\theta).",[23,4224,4226,4250,4271,4295],{"className":4225,"ariaHidden":122},[121],[23,4227,4229,4232,4235,4238,4241,4244,4247],{"className":4228},[126],[23,4230],{"className":4231,"style":131},[130],[23,4233,11],{"className":4234},[135,136],[23,4236,54],{"className":4237},[140],[23,4239,57],{"className":4240,"style":144},[135,136],[23,4242],{"className":4243,"style":149},[148],[23,4245,60],{"className":4246},[153],[23,4248],{"className":4249,"style":149},[148],[23,4251,4253,4256,4259,4262,4265,4268],{"className":4252},[126],[23,4254],{"className":4255,"style":131},[130],[23,4257,63],{"className":4258,"style":144},[135,136],[23,4260,66],{"className":4261},[169],[23,4263],{"className":4264,"style":149},[148],[23,4266,4195],{"className":4267},[153],[23,4269],{"className":4270,"style":149},[148],[23,4272,4274,4277,4280,4283,4286,4289,4292],{"className":4273},[126],[23,4275],{"className":4276,"style":131},[130],[23,4278,11],{"className":4279},[135,136],[23,4281,54],{"className":4282},[140],[23,4284,63],{"className":4285,"style":144},[135,136],[23,4287],{"className":4288,"style":149},[148],[23,4290,60],{"className":4291},[153],[23,4293],{"className":4294,"style":149},[148],[23,4296,4298,4301,4304,4307,4310,4313,4316,4319,4322],{"className":4297},[126],[23,4299],{"className":4300,"style":131},[130],[23,4302,57],{"className":4303,"style":144},[135,136],[23,4305,66],{"className":4306},[169],[23,4308],{"className":4309,"style":280},[148],[23,4311,11],{"className":4312},[135,136],[23,4314,54],{"className":4315},[140],[23,4317,57],{"className":4318,"style":144},[135,136],[23,4320,66],{"className":4321},[169],[23,4323,112],{"className":4324},[135],[11,4326,4327,4328,4371],{},"但这个省略只在「给定模型」内部成立。一旦要比较不同模型，",[23,4329,4331,4350],{"className":4330,"translate":27},[31],[23,4332,4334],{"className":4333},[35],[37,4335,4336],{"xmlns":39},[42,4337,4338,4348],{},[45,4339,4340,4342,4344,4346],{},[48,4341,11],{},[51,4343,54],{"stretchy":53},[48,4345,63],{},[51,4347,66],{"stretchy":53},[114,4349,3945],{"encoding":116},[23,4351,4353],{"className":4352,"ariaHidden":122},[121],[23,4354,4356,4359,4362,4365,4368],{"className":4355},[126],[23,4357],{"className":4358,"style":131},[130],[23,4360,11],{"className":4361},[135,136],[23,4363,54],{"className":4364},[140],[23,4366,63],{"className":4367,"style":144},[135,136],[23,4369,66],{"className":4370},[169]," 就是主角，第 6.5 节会回到这一点。",[11,4373,4374,4375,4403],{},"还要强调一点：贝叶斯公式本身只是概率论的一条定理，接受它不需要接受任何统计哲学。真正带有立场的是下一步——把 ",[23,4376,4378,4391],{"className":4377,"translate":27},[31],[23,4379,4381],{"className":4380},[35],[37,4382,4383],{"xmlns":39},[42,4384,4385,4389],{},[45,4386,4387],{},[48,4388,57],{},[114,4390,3371],{"encoding":116},[23,4392,4394],{"className":4393,"ariaHidden":122},[121],[23,4395,4397,4400],{"className":4396},[126],[23,4398],{"className":4399,"style":779},[130],[23,4401,57],{"className":4402,"style":144},[135,136]," 看成随机变量，并为它指定先验。",[2501,4405,4407],{"id":4406},"_23-betabernoulli一个可以手算的更新","2.3 Beta–Bernoulli：一个可以手算的更新",[11,4409,4410,4411,4487],{},"设 ",[23,4412,4414,4441],{"className":4413,"translate":27},[31],[23,4415,4417],{"className":4416},[35],[37,4418,4419],{"xmlns":39},[42,4420,4421,4438],{},[45,4422,4423,4425,4427,4429,4431,4433,4436],{},[48,4424,57],{},[51,4426,372],{},[51,4428,924],{"stretchy":53},[2581,4430,1532],{},[51,4432,747],{"separator":122},[2581,4434,4435],{},"1",[51,4437,945],{"stretchy":53},[114,4439,4440],{"encoding":116},"\\theta\\in[0,1]",[23,4442,4444,4463],{"className":4443,"ariaHidden":122},[121],[23,4445,4447,4451,4454,4457,4460],{"className":4446},[126],[23,4448],{"className":4449,"style":4450},[130],"height:0.7335em;vertical-align:-0.0391em;",[23,4452,57],{"className":4453,"style":144},[135,136],[23,4455],{"className":4456,"style":149},[148],[23,4458,372],{"className":4459},[153],[23,4461],{"className":4462,"style":149},[148],[23,4464,4466,4469,4472,4475,4478,4481,4484],{"className":4465},[126],[23,4467],{"className":4468,"style":131},[130],[23,4470,924],{"className":4471},[140],[23,4473,1532],{"className":4474},[135],[23,4476,747],{"className":4477},[824],[23,4479],{"className":4480,"style":280},[148],[23,4482,4435],{"className":4483},[135],[23,4485,945],{"className":4486},[169]," 是一枚硬币正面向上的概率，先验取 Beta 分布",[23,4489,4491],{"className":4490,"translate":27},[26],[23,4492,4494,4572],{"className":4493,"translate":27},[31],[23,4495,4497],{"className":4496},[35],[37,4498,4499],{"xmlns":39,"display":40},[42,4500,4501,4569],{},[45,4502,4503,4505,4507,4509,4511,4513,4567],{},[48,4504,11],{},[51,4506,54],{"stretchy":53},[48,4508,57],{},[51,4510,66],{"stretchy":53},[51,4512,69],{},[71,4514,4515,4553],{},[45,4516,4517,4532,4534,4536,4538,4540],{},[4518,4519,4520,4522],"msup",{},[48,4521,57],{},[45,4523,4524,4527,4530],{},[48,4525,4526],{},"α",[51,4528,4529],{},"−",[2581,4531,4435],{},[51,4533,54],{"stretchy":53},[2581,4535,4435],{},[51,4537,4529],{},[48,4539,57],{},[4518,4541,4542,4544],{},[51,4543,66],{"stretchy":53},[45,4545,4546,4549,4551],{},[48,4547,4548],{},"β",[51,4550,4529],{},[2581,4552,4435],{},[45,4554,4555,4557,4559,4561,4563,4565],{},[48,4556,2533],{},[51,4558,54],{"stretchy":53},[48,4560,4526],{},[51,4562,747],{"separator":122},[48,4564,4548],{},[51,4566,66],{"stretchy":53},[48,4568,112],{"mathvariant":111},[114,4570,4571],{"encoding":116},"p(\\theta)=\\frac{\\theta^{\\alpha-1}(1-\\theta)^{\\beta-1}}{B(\\alpha,\\beta)}.",[23,4573,4575,4602],{"className":4574,"ariaHidden":122},[121],[23,4576,4578,4581,4584,4587,4590,4593,4596,4599],{"className":4577},[126],[23,4579],{"className":4580,"style":131},[130],[23,4582,11],{"className":4583},[135,136],[23,4585,54],{"className":4586},[140],[23,4588,57],{"className":4589,"style":144},[135,136],[23,4591,66],{"className":4592},[169],[23,4594],{"className":4595,"style":149},[148],[23,4597,69],{"className":4598},[153],[23,4600],{"className":4601,"style":149},[148],[23,4603,4605,4609,4786],{"className":4604},[126],[23,4606],{"className":4607,"style":4608},[130],"height:2.4621em;vertical-align:-0.936em;",[23,4610,4612,4615,4783],{"className":4611},[135],[23,4613],{"className":4614},[140,192],[23,4616,4618],{"className":4617},[71],[23,4619,4621,4775],{"className":4620},[199,200],[23,4622,4624,4772],{"className":4623},[204],[23,4625,4628,4659,4667],{"className":4626,"style":4627},[208],"height:1.5261em;",[23,4629,4630,4633],{"style":212},[23,4631],{"className":4632,"style":217},[216],[23,4634,4636,4639,4642,4646,4649,4652,4656],{"className":4635},[135],[23,4637,2533],{"className":4638,"style":2623},[135,136],[23,4640,54],{"className":4641},[140],[23,4643,4526],{"className":4644,"style":4645},[135,136],"margin-right:0.0037em;",[23,4647,747],{"className":4648},[824],[23,4650],{"className":4651,"style":280},[148],[23,4653,4548],{"className":4654,"style":4655},[135,136],"margin-right:0.0528em;",[23,4657,66],{"className":4658},[169],[23,4660,4661,4664],{"style":235},[23,4662],{"className":4663,"style":217},[216],[23,4665],{"className":4666,"style":243},[242],[23,4668,4669,4672],{"style":246},[23,4670],{"className":4671,"style":217},[216],[23,4673,4675,4715,4718,4721,4724,4727,4730,4733],{"className":4674},[135],[23,4676,4678,4681],{"className":4677},[135],[23,4679,57],{"className":4680,"style":144},[135,136],[23,4682,4684],{"className":4683},[998],[23,4685,4687],{"className":4686},[199],[23,4688,4690],{"className":4689},[204],[23,4691,4694],{"className":4692,"style":4693},[208],"height:0.8141em;",[23,4695,4697,4700],{"style":4696},"top:-3.063em;margin-right:0.05em;",[23,4698],{"className":4699,"style":1015},[216],[23,4701,4703],{"className":4702},[1019,1020,1021,1022],[23,4704,4706,4709,4712],{"className":4705},[135,1022],[23,4707,4526],{"className":4708,"style":4645},[135,136,1022],[23,4710,4529],{"className":4711},[612,1022],[23,4713,4435],{"className":4714},[135,1022],[23,4716,54],{"className":4717},[140],[23,4719,4435],{"className":4720},[135],[23,4722],{"className":4723,"style":392},[148],[23,4725,4529],{"className":4726},[612],[23,4728],{"className":4729,"style":392},[148],[23,4731,57],{"className":4732,"style":144},[135,136],[23,4734,4736,4739],{"className":4735},[169],[23,4737,66],{"className":4738},[169],[23,4740,4742],{"className":4741},[998],[23,4743,4745],{"className":4744},[199],[23,4746,4748],{"className":4747},[204],[23,4749,4752],{"className":4750,"style":4751},[208],"height:0.8491em;",[23,4753,4754,4757],{"style":4696},[23,4755],{"className":4756,"style":1015},[216],[23,4758,4760],{"className":4759},[1019,1020,1021,1022],[23,4761,4763,4766,4769],{"className":4762},[135,1022],[23,4764,4548],{"className":4765,"style":4655},[135,136,1022],[23,4767,4529],{"className":4768},[612,1022],[23,4770,4435],{"className":4771},[135,1022],[23,4773,297],{"className":4774},[296],[23,4776,4778],{"className":4777},[204],[23,4779,4781],{"className":4780,"style":304},[208],[23,4782],{},[23,4784],{"className":4785},[169,192],[23,4787,112],{"className":4788},[135],[11,4790,4791,4792,4821,4822,4852],{},"观测到 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次正面，似然为",[23,4854,4856],{"className":4855,"translate":27},[26],[23,4857,4859,4913],{"className":4858,"translate":27},[31],[23,4860,4862],{"className":4861},[35],[37,4863,4864],{"xmlns":39,"display":40},[42,4865,4866,4910],{},[45,4867,4868,4870,4872,4874,4876,4878,4880,4882,4888,4890,4892,4894,4896,4908],{},[48,4869,11],{},[51,4871,54],{"stretchy":53},[48,4873,63],{},[51,4875,60],{},[48,4877,57],{},[51,4879,66],{"stretchy":53},[51,4881,69],{},[4518,4883,4884,4886],{},[48,4885,57],{},[48,4887,4836],{},[51,4889,54],{"stretchy":53},[2581,4891,4435],{},[51,4893,4529],{},[48,4895,57],{},[4518,4897,4898,4900],{},[51,4899,66],{"stretchy":53},[45,4901,4902,4904,4906],{},[48,4903,4806],{},[51,4905,4529],{},[48,4907,4836],{},[48,4909,112],{"mathvariant":111},[114,4911,4912],{"encoding":116},"p(D\\mid\\theta)=\\theta^k(1-\\theta)^{n-k}.",[23,4914,4916,4940,4961,5014],{"className":4915,"ariaHidden":122},[121],[23,4917,4919,4922,4925,4928,4931,4934,4937],{"className":4918},[126],[23,4920],{"className":4921,"style":131},[130],[23,4923,11],{"className":4924},[135,136],[23,4926,54],{"className":4927},[140],[23,4929,63],{"className":4930,"style":144},[135,136],[23,4932],{"className":4933,"style":149},[148],[23,4935,60],{"className":4936},[153],[23,4938],{"className":4939,"style":149},[148],[23,4941,4943,4946,4949,4952,4955,4958],{"className":4942},[126],[23,4944],{"className":4945,"style":131},[130],[23,4947,57],{"className":4948,"style":144},[135,136],[23,4950,66],{"className":4951},[169],[23,4953],{"className":4954,"style":149},[148],[23,4956,69],{"className":4957},[153],[23,4959],{"className":4960,"style":149},[148],[23,4962,4964,4968,4999,5002,5005,5008,5011],{"className":4963},[126],[23,4965],{"className":4966,"style":4967},[130],"height:1.1491em;vertical-align:-0.25em;",[23,4969,4971,4974],{"className":4970},[135],[23,4972,57],{"className":4973,"style":144},[135,136],[23,4975,4977],{"className":4976},[998],[23,4978,4980],{"className":4979},[199],[23,4981,4983],{"className":4982},[204],[23,4984,4987],{"className":4985,"style":4986},[208],"height:0.8991em;",[23,4988,4990,4993],{"style":4989},"top:-3.113em;margin-right:0.05em;",[23,4991],{"className":4992,"style":1015},[216],[23,4994,4996],{"className":4995},[1019,1020,1021,1022],[23,4997,4836],{"className":4998,"style":4851},[135,136,1022],[23,5000,54],{"className":5001},[140],[23,5003,4435],{"className":5004},[135],[23,5006],{"className":5007,"style":392},[148],[23,5009,4529],{"className":5010},[612],[23,5012],{"className":5013,"style":392},[148],[23,5015,5017,5020,5023,5061],{"className":5016},[126],[23,5018],{"className":5019,"style":4967},[130],[23,5021,57],{"className":5022,"style":144},[135,136],[23,5024,5026,5029],{"className":5025},[169],[23,5027,66],{"className":5028},[169],[23,5030,5032],{"className":5031},[998],[23,5033,5035],{"className":5034},[199],[23,5036,5038],{"className":5037},[204],[23,5039,5041],{"className":5040,"style":4986},[208],[23,5042,5043,5046],{"style":4989},[23,5044],{"className":5045,"style":1015},[216],[23,5047,5049],{"className":5048},[1019,1020,1021,1022],[23,5050,5052,5055,5058],{"className":5051},[135,1022],[23,5053,4806],{"className":5054},[135,136,1022],[23,5056,4529],{"className":5057},[612,1022],[23,5059,4836],{"className":5060,"style":4851},[135,136,1022],[23,5062,112],{"className":5063},[135],[11,5065,5066],{},"两者相乘，后验的核为",[23,5068,5070],{"className":5069,"translate":27},[26],[23,5071,5073,5146],{"className":5072,"translate":27},[31],[23,5074,5076],{"className":5075},[35],[37,5077,5078],{"xmlns":39,"display":40},[42,5079,5080,5143],{},[45,5081,5082,5084,5086,5088,5090,5092,5094,5096,5113,5115,5117,5119,5121,5141],{},[48,5083,11],{},[51,5085,54],{"stretchy":53},[48,5087,57],{},[51,5089,60],{},[48,5091,63],{},[51,5093,66],{"stretchy":53},[51,5095,4195],{},[4518,5097,5098,5100],{},[48,5099,57],{},[45,5101,5102,5104,5107,5109,5111],{},[48,5103,4526],{},[51,5105,5106],{},"+",[48,5108,4836],{},[51,5110,4529],{},[2581,5112,4435],{},[51,5114,54],{"stretchy":53},[2581,5116,4435],{},[51,5118,4529],{},[48,5120,57],{},[4518,5122,5123,5125],{},[51,5124,66],{"stretchy":53},[45,5126,5127,5129,5131,5133,5135,5137,5139],{},[48,5128,4548],{},[51,5130,5106],{},[48,5132,4806],{},[51,5134,4529],{},[48,5136,4836],{},[51,5138,4529],{},[2581,5140,4435],{},[51,5142,747],{"separator":122},[114,5144,5145],{"encoding":116},"p(\\theta\\mid 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Beta 分布的形式已经把它确定了。这就是",[18,5472,5473],{},"共轭先验","的好处：后验与先验属于同一个分布族，更新只需要改两个参数。",[11,5476,5477],{},"后验均值是",[23,5479,5481],{"className":5480,"translate":27},[26],[23,5482,5484,5534],{"className":5483,"translate":27},[31],[23,5485,5487],{"className":5486},[35],[37,5488,5489],{"xmlns":39,"display":40},[42,5490,5491,5531],{},[45,5492,5493,5495,5497,5499,5501,5503,5505,5507,5529],{},[48,5494,902],{"mathvariant":571},[51,5496,924],{"stretchy":53},[48,5498,57],{},[51,5500,60],{},[48,5502,63],{},[51,5504,945],{"stretchy":53},[51,5506,69],{},[71,5508,5509,5517],{},[45,5510,5511,5513,5515],{},[48,5512,4526],{},[51,5514,5106],{},[48,5516,4836],{},[45,5518,5519,5521,5523,5525,5527],{},[48,5520,4526],{},[51,5522,5106],{},[48,5524,4548],{},[51,5526,5106],{},[48,5528,4806],{},[51,5530,747],{"separator":122},[114,5532,5533],{"encoding":116},"\\mathbb E[\\theta\\mid D]=\\frac{\\alpha+k}{\\alpha+\\beta+n},",[23,5535,5537,5561,5582],{"className":5536,"ariaHidden":122},[121],[23,5538,5540,5543,5546,5549,5552,5555,5558],{"className":5539},[126],[23,5541],{"className":5542,"style":131},[130],[23,5544,902],{"className":5545},[135,644],[23,5547,924],{"className":5548},[140],[23,5550,57],{"className":5551,"style":144},[135,136],[23,5553],{"className":5554,"style":149},[148],[23,5556,60],{"className":5557},[153],[23,5559],{"className":5560,"style":149},[148],[23,5562,5564,5567,5570,5573,5576,5579],{"className":5563},[126],[23,5565],{"className":5566,"style":131},[130],[23,5568,63],{"className":5569,"style":144},[135,136],[23,5571,945],{"className":5572},[169],[23,5574],{"className":5575,"style":149},[148],[23,5577,69],{"className":5578},[153],[23,5580],{"className":5581,"style":149},[148],[23,5583,5585,5589,5689],{"className":5584},[126],[23,5586],{"className":5587,"style":5588},[130],"height:2.2519em;vertical-align:-0.8804em;",[23,5590,5592,5595,5686],{"className":5591},[135],[23,5593],{"className":5594},[140,192],[23,5596,5598],{"className":5597},[71],[23,5599,5601,5677],{"className":5600},[199,200],[23,5602,5604,5674],{"className":5603},[204],[23,5605,5608,5643,5651],{"className":5606,"style":5607},[208],"height:1.3714em;",[23,5609,5610,5613],{"style":212},[23,5611],{"className":5612,"style":217},[216],[23,5614,5616,5619,5622,5625,5628,5631,5634,5637,5640],{"className":5615},[135],[23,5617,4526],{"className":5618,"style":4645},[135,136],[23,5620],{"className":5621,"style":392},[148],[23,5623,5106],{"className":5624},[612],[23,5626],{"className":5627,"style":392},[148],[23,5629,4548],{"className":5630,"style":4655},[135,136],[23,5632],{"className":5633,"style":392},[148],[23,5635,5106],{"className":5636},[612],[23,5638],{"className":5639,"style":392},[148],[23,5641,4806],{"className":5642},[135,136],[23,5644,5645,5648],{"style":235},[23,5646],{"className":5647,"style":217},[216],[23,5649],{"className":5650,"style":243},[242],[23,5652,5653,5656],{"style":246},[23,5654],{"className":5655,"style":217},[216],[23,5657,5659,5662,5665,5668,5671],{"className":5658},[135],[23,5660,4526],{"className":5661,"style":4645},[135,136],[23,5663],{"className":5664,"style":392},[148],[23,5666,5106],{"className":5667},[612],[23,5669],{"className":5670,"style":392},[148],[23,5672,4836],{"className":5673,"style":4851},[135,136],[23,5675,297],{"className":5676},[296],[23,5678,5680],{"className":5679},[204],[23,5681,5684],{"className":5682,"style":5683},[208],"height:0.8804em;",[23,5685],{},[23,5687],{"className":5688},[169,192],[23,5690,747],{"className":5691},[824],[11,5693,5694],{},"把它改写成加权平均：",[23,5696,5698],{"className":5697,"translate":27},[26],[23,5699,5701,5802],{"className":5700,"translate":27},[31],[23,5702,5704],{"className":5703},[35],[37,5705,5706],{"xmlns":39,"display":40},[42,5707,5708,5799],{},[45,5709,5710,5732,5734,5756,5759,5771,5773,5789,5791,5797],{},[71,5711,5712,5720],{},[45,5713,5714,5716,5718],{},[48,5715,4526],{},[51,5717,5106],{},[48,5719,4836],{},[45,5721,5722,5724,5726,5728,5730],{},[48,5723,4526],{},[51,5725,5106],{},[48,5727,4548],{},[51,5729,5106],{},[48,5731,4806],{},[51,5733,69],{},[71,5735,5736,5744],{},[45,5737,5738,5740,5742],{},[48,5739,4526],{},[51,5741,5106],{},[48,5743,4548],{},[45,5745,5746,5748,5750,5752,5754],{},[48,5747,4526],{},[51,5749,5106],{},[48,5751,4548],{},[51,5753,5106],{},[48,5755,4806],{},[51,5757,5758],{},"⋅",[71,5760,5761,5763],{},[48,5762,4526],{},[45,5764,5765,5767,5769],{},[48,5766,4526],{},[51,5768,5106],{},[48,5770,4548],{},[51,5772,5106],{},[71,5774,5775,5777],{},[48,5776,4806],{},[45,5778,5779,5781,5783,5785,5787],{},[48,5780,4526],{},[51,5782,5106],{},[48,5784,4548],{},[51,5786,5106],{},[48,5788,4806],{},[51,5790,5758],{},[71,5792,5793,5795],{},[48,5794,4836],{},[48,5796,4806],{},[48,5798,112],{"mathvariant":111},[114,5800,5801],{"encoding":116},"\\frac{\\alpha+k}{\\alpha+\\beta+n}\n=\n\\frac{\\alpha+\\beta}{\\alpha+\\beta+n}\\cdot\\frac{\\alpha}{\\alpha+\\beta}\n+\n\\frac{n}{\\alpha+\\beta+n}\\cdot\\frac{k}{n}.",[23,5803,5805,5918,6031,6122,6223],{"className":5804,"ariaHidden":122},[121],[23,5806,5808,5811,5909,5912,5915],{"className":5807},[126],[23,5809],{"className":5810,"style":5588},[130],[23,5812,5814,5817,5906],{"className":5813},[135],[23,5815],{"className":5816},[140,192],[23,5818,5820],{"className":5819},[71],[23,5821,5823,5898],{"className":5822},[199,200],[23,5824,5826,5895],{"className":5825},[204],[23,5827,5829,5864,5872],{"className":5828,"style":5607},[208],[23,5830,5831,5834],{"style":212},[23,5832],{"className":5833,"style":217},[216],[23,5835,5837,5840,5843,5846,5849,5852,5855,5858,5861],{"className":5836},[135],[23,5838,4526],{"className":5839,"style":4645},[135,136],[23,5841],{"className":5842,"style":392},[148],[23,5844,5106],{"className":5845},[612],[23,5847],{"className":5848,"style":392},[148],[23,5850,4548],{"className":5851,"style":4655},[135,136],[23,5853],{"className":5854,"style":392},[148],[23,5856,5106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sName":6168,"style":4655},[135,136],[23,6170],{"className":6171,"style":392},[148],[23,6173,5106],{"className":6174},[612],[23,6176],{"className":6177,"style":392},[148],[23,6179,4806],{"className":6180},[135,136],[23,6182,6183,6186],{"style":235},[23,6184],{"className":6185,"style":217},[216],[23,6187],{"className":6188,"style":243},[242],[23,6190,6191,6194],{"style":246},[23,6192],{"className":6193,"style":217},[216],[23,6195,6197],{"className":6196},[135],[23,6198,4806],{"className":6199},[135,136],[23,6201,297],{"className":6202},[296],[23,6204,6206],{"className":6205},[204],[23,6207,6209],{"className":6208,"style":5683},[208],[23,6210],{},[23,6212],{"className":6213},[169,192],[23,6215],{"className":6216,"style":392},[148],[23,6218,5758],{"className":6219},[612],[23,6221],{"className":6222,"style":392},[148],[23,6224,6226,6230,6293],{"className":6225},[126],[23,6227],{"className":6228,"style":6229},[130],"height:2.0574em;vertical-align:-0.686em;",[23,6231,6233,6236,6290],{"className":6232},[135],[23,6234],{"className":6235},[140,192],[23,6237,6239],{"className":6238},[71],[23,6240,6242,6281],{"className":6241},[199,200],[23,6243,6245,6278],{"className":6244},[204],[23,6246,6248,6259,6267],{"className":6247,"style":5607},[208],[23,6249,6250,6253],{"style":212},[23,6251],{"className":6252,"style":217},[216],[23,6254,6256],{"className":6255},[135],[23,6257,4806],{"className":6258},[135,136],[23,6260,6261,6264],{"style":235},[23,6262],{"className":6263,"style":217},[216],[23,6265],{"className":6266,"style":243},[242],[23,6268,6269,6272],{"style":246},[23,6270],{"className":6271,"style":217},[216],[23,6273,6275],{"className":6274},[135],[23,6276,4836],{"className":6277,"style":4851},[135,136],[23,6279,297],{"className":6280},[296],[23,6282,6284],{"className":6283},[204],[23,6285,6288],{"className":6286,"style":6287},[208],"height:0.686em;",[23,6289],{},[23,6291],{"className":6292},[169,192],[23,6294,112],{"className":6295},[135],[11,6297,6298,6299,6350,6351,6379,6380,6383],{},"右边第一项是先验均值，第二项是样本频率。权重由「先验有多强」（",[23,6300,6302,6320],{"className":6301,"translate":27},[31],[23,6303,6305],{"className":6304},[35],[37,6306,6307],{"xmlns":39},[42,6308,6309,6317],{},[45,6310,6311,6313,6315],{},[48,6312,4526],{},[51,6314,5106],{},[48,6316,4548],{},[114,6318,6319],{"encoding":116},"\\alpha+\\beta",[23,6321,6323,6341],{"className":6322,"ariaHidden":122},[121],[23,6324,6326,6329,6332,6335,6338],{"className":6325},[126],[23,6327],{"className":6328,"style":5445},[130],[23,6330,4526],{"className":6331,"style":4645},[135,136],[23,6333],{"className":6334,"style":392},[148],[23,6336,5106],{"className":6337},[612],[23,6339],{"className":6340,"style":392},[148],[23,6342,6344,6347],{"className":6343},[126],[23,6345],{"className":6346,"style":5414},[130],[23,6348,4548],{"className":6349,"style":4655},[135,136],"）与「数据有多少」（",[23,6352,6354,6367],{"className":6353,"translate":27},[31],[23,6355,6357],{"className":6356},[35],[37,6358,6359],{"xmlns":39},[42,6360,6361,6365],{},[45,6362,6363],{},[48,6364,4806],{},[114,6366,4806],{"encoding":116},[23,6368,6370],{"className":6369,"ariaHidden":122},[121],[23,6371,6373,6376],{"className":6372},[126],[23,6374],{"className":6375,"style":703},[130],[23,6377,4806],{"className":6378},[135,136],"）决定。数据越多，先验越被稀释。这就是最朴素的",[18,6381,6382],{},"收缩","（shrinkage）。",[11,6385,6386],{},"如果下一步要预测新的一次投掷，贝叶斯答案是后验预测分布",[23,6388,6390],{"className":6389,"translate":27},[26],[23,6391,6393,6466],{"className":6392,"translate":27},[31],[23,6394,6396],{"className":6395},[35],[37,6397,6398],{"xmlns":39,"display":40},[42,6399,6400,6463],{},[45,6401,6402,6404,6406,6413,6415,6417,6419,6421,6423,6425,6427,6429,6431,6433,6435,6437,6439,6461],{},[48,6403,11],{},[51,6405,54],{"stretchy":53},[898,6407,6408,6410],{},[48,6409,753],{},[88,6411,6412],{},"new",[51,6414,69],{},[2581,6416,4435],{},[51,6418,60],{},[48,6420,63],{},[51,6422,66],{"stretchy":53},[51,6424,69],{},[48,6426,902],{"mathvariant":571},[51,6428,924],{"stretchy":53},[48,6430,57],{},[51,6432,60],{},[48,6434,63],{},[51,6436,945],{"stretchy":53},[51,6438,69],{},[71,6440,6441,6449],{},[45,6442,6443,6445,6447],{},[48,6444,4526],{},[51,6446,5106],{},[48,6448,4836],{},[45,6450,6451,6453,6455,6457,6459],{},[48,6452,4526],{},[51,6454,5106],{},[48,6456,4548],{},[51,6458,5106],{},[48,6460,4806],{},[48,6462,112],{"mathvariant":111},[114,6464,6465],{"encoding":116},"p(x_{\\text{new}}=1\\mid D)=\\mathbb E[\\theta\\mid D]=\\frac{\\alpha+k}{\\alpha+\\beta+n}.",[23,6467,6469,6538,6556,6577,6601,6622],{"className":6468,"ariaHidden":122},[121],[23,6470,6472,6475,6478,6481,6529,6532,6535],{"className":6471},[126],[23,6473],{"className":6474,"style":131},[130],[23,6476,11],{"className":6477},[135,136],[23,6479,54],{"className":6480},[140],[23,6482,6484,6487],{"className":6483},[135],[23,6485,753],{"className":6486},[135,136],[23,6488,6490],{"className":6489},[998],[23,6491,6493,6521],{"className":6492},[199,200],[23,6494,6496,6518],{"className":6495},[204],[23,6497,6500],{"className":6498,"style":6499},[208],"height:0.1514em;",[23,6501,6502,6505],{"style":2029},[23,6503],{"className":6504,"style":1015},[216],[23,6506,6508],{"className":6507},[1019,1020,1021,1022],[23,6509,6511],{"className":6510},[135,1022],[23,6512,6515],{"className":6513},[135,6514,1022],"text",[23,6516,6412],{"className":6517},[135,1022],[23,6519,297],{"className":6520},[296],[23,6522,6524],{"className":6523},[204],[23,6525,6527],{"className":6526,"style":2048},[208],[23,6528],{},[23,6530],{"className":6531,"style":149},[148],[23,6533,69],{"className":6534},[153],[23,6536],{"className":6537,"style":149},[148],[23,6539,6541,6544,6547,6550,6553],{"className":6540},[126],[23,6542],{"className":6543,"style":131},[130],[23,6545,4435],{"className":6546},[135],[23,6548],{"className":6549,"style":149},[148],[23,6551,60],{"className":6552},[153],[23,6554],{"className":6555,"style":149},[148],[23,6557,6559,6562,6565,6568,6571,6574],{"className":6558},[126],[23,6560],{"className":6561,"style":131},[130],[23,6563,63],{"className":6564,"style":144},[135,136],[23,6566,66],{"className":6567},[169],[23,6569],{"className":6570,"style":149},[148],[23,6572,69],{"className":6573},[153],[23,6575],{"className":6576,"style":149},[148],[23,6578,6580,6583,6586,6589,6592,6595,6598],{"className":6579},[126],[23,6581],{"className":6582,"style":131},[130],[23,6584,902],{"className":6585},[135,644],[23,6587,924],{"className":6588},[140],[23,6590,57],{"className":6591,"style":144},[135,136],[23,6593],{"className":6594,"style":149},[148],[23,6596,60],{"className":6597},[153],[23,6599],{"className":6600,"style":149},[148],[23,6602,6604,6607,6610,6613,6616,6619],{"className":6603},[126],[23,6605],{"className":6606,"style":131},[130],[23,6608,63],{"className":6609,"style":144},[135,136],[23,6611,945],{"className":6612},[169],[23,6614],{"className":6615,"style":149},[148],[23,6617,69],{"className":6618},[153],[23,6620],{"className":6621,"style":149},[148],[23,6623,6625,6628,6726],{"className":6624},[126],[23,6626],{"className":6627,"style":5588},[130],[23,6629,6631,6634,6723],{"className":6630},[135],[23,6632],{"className":6633},[140,192],[23,6635,6637],{"className":6636},[71],[23,6638,6640,6715],{"className":6639},[199,200],[23,6641,6643,6712],{"className":6642},[204],[23,6644,6646,6681,6689],{"className":6645,"style":5607},[208],[23,6647,6648,6651],{"style":212},[23,6649],{"className":6650,"style":217},[216],[23,6652,6654,6657,6660,6663,6666,6669,6672,6675,6678],{"className":6653},[135],[23,6655,4526],{"className":6656,"style":4645},[135,136],[23,6658],{"className":6659,"style":392},[148],[23,6661,5106],{"className":6662},[612],[23,6664],{"className":6665,"style":392},[148],[23,6667,4548],{"className":6668,"style":4655},[135,136],[23,6670],{"className":6671,"style":392},[148],[23,6673,5106],{"className":6674},[612],[23,6676],{"className":6677,"style":392},[148],[23,6679,4806],{"className":6680},[135,136],[23,6682,6683,6686],{"style":235},[23,6684],{"className":6685,"style":217},[216],[23,6687],{"className":6688,"style":243},[242],[23,6690,6691,6694],{"style":246},[23,6692],{"className":6693,"style":217},[216],[23,6695,6697,6700,6703,6706,6709],{"className":6696},[135],[23,6698,4526],{"className":6699,"style":4645},[135,136],[23,6701],{"className":6702,"style":392},[148],[23,6704,5106],{"className":6705},[612],[23,6707],{"className":6708,"style":392},[148],[23,6710,4836],{"className":6711,"style":4851},[135,136],[23,6713,297],{"className":6714},[296],[23,6716,6718],{"className":6717},[204],[23,6719,6721],{"className":6720,"style":5683},[208],[23,6722],{},[23,6724],{"className":6725},[169,192],[23,6727,112],{"className":6728},[135],[11,6730,6731,6732,6760],{},"这里恰好等于后验均值，是因为伯努利概率是 ",[23,6733,6735,6748],{"className":6734,"translate":27},[31],[23,6736,6738],{"className":6737},[35],[37,6739,6740],{"xmlns":39},[42,6741,6742,6746],{},[45,6743,6744],{},[48,6745,57],{},[114,6747,3371],{"encoding":116},[23,6749,6751],{"className":6750,"ariaHidden":122},[121],[23,6752,6754,6757],{"className":6753},[126],[23,6755],{"className":6756,"style":779},[130],[23,6758,57],{"className":6759,"style":144},[135,136]," 的线性函数。对一般模型，后验预测分布并不等于「把后验均值代进去」，这个区别在第 6 节会反复出现。",[2501,6762,6764],{"id":6763},"_24-先验是什么不是什么","2.4 先验是什么，不是什么",[11,6766,6767],{},"先验常被批评为「主观」。这个批评有一半是对的：先验确实是一个假设，而且不同先验会给出不同后验。但「主观」不等于「任意」。先验可以被数据稀释：在正则条件下，后验的渐近行为与先验无关（见 6.4）。先验也可以层级化：给超参数再加一层先验，让数据同时决定参数与超参数。先验还可以被检验：通过后验预测检验、敏感性分析，看结论是否依赖于某个特定选择。",[11,6769,6770],{},"更重要的是，非贝叶斯方法同样有假设。模型类、正则项、损失函数、优化算法，都是假设，只是它们不以「先验」的名字出现。贝叶斯的优点不是没有假设，而是把假设写在了明处，并且提供了一个用概率语言修改假设的机制。",[11,6772,6773,6774,6777],{},"第 7.8 节会给出一个更强的说法：在相当一般的条件下，所有",[18,6775,6776],{},"可容许","（admissible）的决策规则都可以看成某个先验（或先验序列）下的贝叶斯规则。换句话说，贝叶斯规则不是一个「流派」，而是可容许决策规则的一般形式。这个结论把「为什么要用先验」从个人偏好提升到了决策论的结构性事实。",[338,6779,6781],{"id":6780},"_3-从后验到行动","3. 从后验到行动",[2501,6783,6785],{"id":6784},"_31-条件风险与逐点最优","3.1 条件风险与逐点最优",[11,6787,6788,6789,6855],{},"有了后验 ",[23,6790,6792,6816],{"className":6791,"translate":27},[31],[23,6793,6795],{"className":6794},[35],[37,6796,6797],{"xmlns":39},[42,6798,6799,6813],{},[45,6800,6801,6803,6805,6807,6809,6811],{},[48,6802,11],{},[51,6804,54],{"stretchy":53},[48,6806,660],{},[51,6808,60],{},[48,6810,753],{},[51,6812,66],{"stretchy":53},[114,6814,6815],{"encoding":116},"p(y\\mid x)",[23,6817,6819,6843],{"className":6818,"ariaHidden":122},[121],[23,6820,6822,6825,6828,6831,6834,6837,6840],{"className":6821},[126],[23,6823],{"className":6824,"style":131},[130],[23,6826,11],{"className":6827},[135,136],[23,6829,54],{"className":6830},[140],[23,6832,660],{"className":6833,"style":676},[135,136],[23,6835],{"className":6836,"style":149},[148],[23,6838,60],{"className":6839},[153],[23,6841],{"className":6842,"style":149},[148],[23,6844,6846,6849,6852],{"className":6845},[126],[23,6847],{"className":6848,"style":131},[130],[23,6850,753],{"className":6851},[135,136],[23,6853,66],{"className":6854},[169],"，决策问题就变成了一个逐点的优化问题。",[6857,6858,6860,6925,7249],"theorem",{"title":6859},"定义 3.1（贝叶斯决策规则）",[11,6861,6862,6863,6924],{},"若决策规则 ",[23,6864,6866,6885],{"className":6865,"translate":27},[31],[23,6867,6869],{"className":6868},[35],[37,6870,6871],{"xmlns":39},[42,6872,6873,6882],{},[45,6874,6875],{},[4518,6876,6877,6879],{},[48,6878,736],{},[51,6880,6881],{},"∗",[114,6883,6884],{"encoding":116},"\\delta^*",[23,6886,6888],{"className":6887,"ariaHidden":122},[121],[23,6889,6891,6894],{"className":6890},[126],[23,6892],{"className":6893,"style":779},[130],[23,6895,6897,6900],{"className":6896},[135],[23,6898,736],{"className":6899,"style":783},[135,136],[23,6901,6903],{"className":6902},[998],[23,6904,6906],{"className":6905},[199],[23,6907,6909],{"className":6908},[204],[23,6910,6913],{"className":6911,"style":6912},[208],"height:0.6887em;",[23,6914,6915,6918],{"style":4696},[23,6916],{"className":6917,"style":1015},[216],[23,6919,6921],{"className":6920},[1019,1020,1021,1022],[23,6922,6881],{"className":6923},[612,1022]," 满足",[23,6926,6928],{"className":6927,"translate":27},[26],[23,6929,6931,7018],{"className":6930,"translate":27},[31],[23,6932,6934],{"className":6933},[35],[37,6935,6936],{"xmlns":39,"display":40},[42,6937,6938,7015],{},[45,6939,6940,6946,6948,6950,6952,6954,6957,6960,6978,6980,6982,6984,6986,6988,6990,6992,6994,6996,6998,7000,7002,7004,7006,7008,7011,7013],{},[4518,6941,6942,6944],{},[48,6943,736],{},[51,6945,6881],{},[51,6947,54],{"stretchy":53},[48,6949,753],{},[51,6951,66],{"stretchy":53},[51,6953,372],{},[48,6955,6956],{},"arg",[51,6958,6959],{},"⁡",[6961,6962,6963,6970],"munder",{},[45,6964,6965,6968],{},[48,6966,6967],{},"min",[51,6969,6959],{},[45,6971,6972,6974,6976],{},[48,6973,495],{},[51,6975,372],{},[48,6977,500],{"mathvariant":375},[88,6979,5360],{},[48,6981,902],{"mathvariant":571},[51,6983,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,6985,555],{},[51,6987,54],{"stretchy":53},[48,6989,369],{},[51,6991,747],{"separator":122},[48,6993,495],{},[51,6995,66],{"stretchy":53},[51,6997,60],{},[48,6999,434],{},[51,7001,69],{},[48,7003,753],{},[51,7005,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[148,7007],{"width":750},[88,7009,7010],{},"对几乎所有的 ",[48,7012,753],{},[51,7014,747],{"separator":122},[114,7016,7017],{"encoding":116},"\\delta^*(x)\\in\\arg\\min_{a\\in\\mathcal A}\\ \\mathbb E\\big[L(Y,a)\\mid X=x\\big]\n\\qquad \\text{对几乎所有的 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",[23,7253,7255,7272],{"className":7254,"translate":27},[31],[23,7256,7258],{"className":7257},[35],[37,7259,7260],{"xmlns":39},[42,7261,7262,7270],{},[45,7263,7264],{},[4518,7265,7266,7268],{},[48,7267,736],{},[51,7269,6881],{},[114,7271,6884],{"encoding":116},[23,7273,7275],{"className":7274,"ariaHidden":122},[121],[23,7276,7278,7281],{"className":7277},[126],[23,7279],{"className":7280,"style":779},[130],[23,7282,7284,7287],{"className":7283},[135],[23,7285,736],{"className":7286,"style":783},[135,136],[23,7288,7290],{"className":7289},[998],[23,7291,7293],{"className":7292},[199],[23,7294,7296],{"className":7295},[204],[23,7297,7299],{"className":7298,"style":6912},[208],[23,7300,7301,7304],{"style":4696},[23,7302],{"className":7303,"style":1015},[216],[23,7305,7307],{"className":7306},[1019,1020,1021,1022],[23,7308,6881],{"className":7309},[612,1022]," 为",[18,7312,7313],{},"贝叶斯决策规则","（Bayes decision rule）。",[6857,7316,7318,7351],{"title":7317},"定理 3.1（逐点最优即全局最优）",[11,7319,7320,7321,7350],{},"对任意决策规则 ",[23,7322,7324,7338],{"className":7323,"translate":27},[31],[23,7325,7327],{"className":7326},[35],[37,7328,7329],{"xmlns":39},[42,7330,7331,7335],{},[45,7332,7333],{},[48,7334,736],{},[114,7336,7337],{"encoding":116},"\\delta",[23,7339,7341],{"className":7340,"ariaHidden":122},[121],[23,7342,7344,7347],{"className":7343},[126],[23,7345],{"className":7346,"style":779},[130],[23,7348,736],{"className":7349,"style":783},[135,136],"，都有",[23,7352,7354],{"className":7353,"translate":27},[26],[23,7355,7357,7439],{"className":7356,"translate":27},[31],[23,7358,7360],{"className":7359},[35],[37,7361,7362],{"xmlns":39,"display":40},[42,7363,7364,7436],{},[45,7365,7366,7368,7370,7372,7374,7376,7379,7381,7383,7385,7391,7393,7395,7401,7404,7420,7422,7424,7426,7428,7430,7432,7434],{},[48,7367,572],{},[51,7369,54],{"stretchy":53},[48,7371,736],{},[51,7373,66],{"stretchy":53},[88,7375,5360],{},[51,7377,7378],{},"≥",[88,7380,5360],{},[48,7382,572],{},[51,7384,54],{"stretchy":53},[4518,7386,7387,7389],{},[48,7388,736],{},[51,7390,6881],{},[51,7392,66],{"stretchy":53},[51,7394,69],{},[898,7396,7397,7399],{},[48,7398,902],{"mathvariant":571},[48,7400,434],{},[51,7402,924],{"fence":53,"stretchy":122,"minsize":7403,"maxsize":7403},"1.8em",[6961,7405,7406,7412],{},[45,7407,7408,7410],{},[48,7409,6967],{},[51,7411,6959],{},[45,7413,7414,7416,7418],{},[48,7415,495],{},[51,7417,372],{},[48,7419,500],{"mathvariant":375},[48,7421,1653],{},[51,7423,54],{"stretchy":53},[48,7425,495],{},[51,7427,747],{"separator":122},[48,7429,434],{},[51,7431,66],{"stretchy":53},[51,7433,945],{"fence":53,"stretchy":122,"minsize":7403,"maxsize":7403},[48,7435,112],{"mathvariant":111},[114,7437,7438],{"encoding":116},"R(\\delta)\\ \\ge\\ R(\\delta^*)=\\mathbb E_X\\Big[\\min_{a\\in\\mathcal A}r(a,X)\\Big].",[23,7440,7442,7475,7528],{"className":7441,"ariaHidden":122},[121],[23,7443,7445,7448,7451,7454,7457,7460,7463,7466,7469,7472],{"className":7444},[126],[23,7446],{"className":7447,"style":131},[130],[23,7449,572],{"className":7450,"style":963},[135,136],[23,7452,54],{"className":7453},[140],[23,7455,736],{"className":7456,"style":783},[135,136],[23,7458,66],{"className":7459},[169],[23,7461,5360],{"className":7462},[148],[23,7464],{"className":7465,"style":149},[148],[23,7467,7378],{"className":7468},[153],[23,7470,5360],{"className":7471},[148],[23,7473],{"className":7474,"style":149},[148],[23,7476,7478,7481,7484,7487,7516,7519,7522,7525],{"className":7477},[126],[23,7479],{"className":7480,"style":131},[130],[23,7482,572],{"className":7483,"style":963},[135,136],[23,7485,54],{"className":7486},[140],[23,7488,7490,7493],{"className":7489},[135],[23,7491,736],{"className":7492,"style":783},[135,136],[23,7494,7496],{"className":7495},[998],[23,7497,7499],{"className":7498},[199],[23,7500,7502],{"className":7501},[204],[23,7503,7505],{"className":7504,"style":7045},[208],[23,7506,7507,7510],{"style":4989},[23,7508],{"className":7509,"style":1015},[216],[23,7511,7513],{"className":7512},[1019,1020,1021,1022],[23,7514,6881],{"className":7515},[612,1022],[23,7517,66],{"className":7518},[169],[23,7520],{"className":7521,"style":149},[148],[23,7523,69],{"className":7524},[153],[23,7526],{"className":7527,"style":149},[148],[23,7529,7531,7535,7575,7582,7585,7638,7641,7644,7647,7650,7653,7656,7659,7662,7668],{"className":7530},[126],[23,7532],{"className":7533,"style":7534},[130],"height:1.9217em;vertical-align:-0.7717em;",[23,7536,7538,7541],{"className":7537},[135],[23,7539,902],{"className":7540},[135,644],[23,7542,7544],{"className":7543},[998],[23,7545,7547,7567],{"className":7546},[199,200],[23,7548,7550,7564],{"className":7549},[204],[23,7551,7553],{"className":7552,"style":2026},[208],[23,7554,7555,7558],{"style":2029},[23,7556],{"className":7557,"style":1015},[216],[23,7559,7561],{"className":7560},[1019,1020,1021,1022],[23,7562,434],{"className":7563,"style":454},[135,136,1022],[23,7565,297],{"className":7566},[296],[23,7568,7570],{"className":7569},[204],[23,7571,7573],{"className":7572,"style":2048},[208],[23,7574],{},[23,7576,7578],{"className":7577},[135],[23,7579,924],{"className":7580},[1066,7581],"size2",[23,7583],{"className":7584,"style":280},[148],[23,7586,7588],{"className":7587},[1786,7096],[23,7589,7591,7630],{"className":7590},[199,200],[23,7592,7594,7627],{"className":7593},[204],[23,7595,7597,7617],{"className":7596,"style":7106},[208],[23,7598,7599,7602],{"style":7109},[23,7600],{"className":7601,"style":217},[216],[23,7603,7605],{"className":7604},[1019,1020,1021,1022],[23,7606,7608,7611,7614],{"className":7607},[135,1022],[23,7609,495],{"className":7610},[135,136,1022],[23,7612,372],{"className":7613},[153,1022],[23,7615,500],{"className":7616},[135,412,1022],[23,7618,7619,7622],{"style":7130},[23,7620],{"className":7621,"style":217},[216],[23,7623,7624],{},[23,7625,6967],{"className":7626},[1786],[23,7628,297],{"className":7629},[296],[23,7631,7633],{"className":7632},[204],[23,7634,7636],{"className":7635,"style":7148},[208],[23,7637],{},[23,7639],{"className":7640,"style":280},[148],[23,7642,1653],{"className":7643,"style":144},[135,136],[23,7645,54],{"className":7646},[140],[23,7648,495],{"className":7649},[135,136],[23,7651,747],{"className":7652},[824],[23,7654],{"className":7655,"style":280},[148],[23,7657,434],{"className":7658,"style":454},[135,136],[23,7660,66],{"className":7661},[169],[23,7663,7665],{"className":7664},[135],[23,7666,945],{"className":7667},[1066,7581],[23,7669,112],{"className":7670},[135],[11,7672,7673,7676],{},[18,7674,7675],{},"证明。"," 由塔性质，",[23,7678,7680],{"className":7679,"translate":27},[26],[23,7681,7683,7775],{"className":7682,"translate":27},[31],[23,7684,7686],{"className":7685},[35],[37,7687,7688],{"xmlns":39,"display":40},[42,7689,7690,7772],{},[45,7691,7692,7694,7696,7698,7700,7702,7708,7710,7712,7714,7716,7718,7720,7722,7724,7726,7728,7730,7732,7734,7736,7738,7740,7742,7748,7750,7752,7754,7756,7758,7760,7762,7764,7766,7768,7770],{},[48,7693,572],{},[51,7695,54],{"stretchy":53},[48,7697,736],{},[51,7699,66],{"stretchy":53},[51,7701,69],{},[898,7703,7704,7706],{},[48,7705,902],{"mathvariant":571},[48,7707,434],{},[51,7709,924],{"fence":53,"stretchy":122,"minsize":7403,"maxsize":7403},[48,7711,902],{"mathvariant":571},[51,7713,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,7715,555],{},[51,7717,54],{"stretchy":53},[48,7719,369],{},[51,7721,747],{"separator":122},[48,7723,736],{},[51,7725,54],{"stretchy":53},[48,7727,434],{},[51,7729,66],{"stretchy":53},[51,7731,66],{"stretchy":53},[51,7733,60],{},[48,7735,434],{},[51,7737,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,7739,945],{"fence":53,"stretchy":122,"minsize":7403,"maxsize":7403},[51,7741,69],{},[898,7743,7744,7746],{},[48,7745,902],{"mathvariant":571},[48,7747,434],{},[51,7749,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,7751,1653],{},[51,7753,54],{"stretchy":53},[48,7755,736],{},[51,7757,54],{"stretchy":53},[48,7759,434],{},[51,7761,66],{"stretchy":53},[51,7763,747],{"separator":122},[48,7765,434],{},[51,7767,66],{"stretchy":53},[51,7769,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,7771,112],{"mathvariant":111},[114,7773,7774],{"encoding":116},"R(\\delta)=\\mathbb E_X\\Big[\\mathbb E\\big[L(Y,\\delta(X))\\mid X\\big]\\Big]\n=\\mathbb 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≤ 先对点取最小再平均」这样一个显而易见的事实。它不要求损失凸、不要求后验有密度、也不要求行动空间是有限的。这种一般性是贝叶斯决策理论最漂亮的地方之一。严格地说，若逐点下确界不可达，把 ",[23,8269,8271,8287],{"className":8270,"translate":27},[31],[23,8272,8274],{"className":8273},[35],[37,8275,8276],{"xmlns":39},[42,8277,8278,8284],{},[45,8279,8280,8282],{},[48,8281,6967],{},[51,8283,6959],{},[114,8285,8286],{"encoding":116},"\\min",[23,8288,8290],{"className":8289,"ariaHidden":122},[121],[23,8291,8293,8296],{"className":8292},[126],[23,8294],{"className":8295,"style":7106},[130],[23,8297,6967],{"className":8298},[1786]," 换成 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换成「取一列逼近的行动」即可，证明不变。",[11,8384,8385],{},"把上式改写成差值，可以得到一个更有解释力的恒等式：",[23,8387,8389],{"className":8388,"translate":27},[26],[23,8390,8392,8491],{"className":8391,"translate":27},[31],[23,8393,8395],{"className":8394},[35],[37,8396,8397],{"xmlns":39,"display":40},[42,8398,8399,8488],{},[45,8400,8401,8403,8405,8407,8409,8411,8417,8419,8425,8427,8484,8486],{},[48,8402,572],{},[51,8404,54],{"stretchy":53},[48,8406,736],{},[51,8408,66],{"stretchy":53},[51,8410,4529],{},[4518,8412,8413,8415],{},[48,8414,572],{},[51,8416,6881],{},[51,8418,69],{},[898,8420,8421,8423],{},[48,8422,902],{"mathvariant":571},[48,8424,434],{},[51,8426,924],{"fence":53,"stretchy":122,"minsize":7403,"maxsize":7403},[6961,8428,8429,8478],{},[6961,8430,8431,8475],{},[45,8432,8433,8435,8437,8439,8441,8443,8445,8447,8449,8451,8453,8463,8465,8467,8469,8471,8473],{},[48,8434,1653],{},[51,8436,54],{"stretchy":53},[48,8438,736],{},[51,8440,54],{"stretchy":53},[48,8442,434],{},[51,8444,66],{"stretchy":53},[51,8446,747],{"separator":122},[48,8448,434],{},[51,8450,66],{"stretchy":53},[51,8452,4529],{},[6961,8454,8455,8461],{},[45,8456,8457,8459],{},[48,8458,6967],{},[51,8460,6959],{},[48,8462,495],{},[48,8464,1653],{},[51,8466,54],{"stretchy":53},[48,8468,495],{},[51,8470,747],{"separator":122},[48,8472,434],{},[51,8474,66],{"stretchy":53},[51,8476,8477],{"stretchy":122},"⏟",[45,8479,8480,8482],{},[51,8481,7378],{},[2581,8483,1532],{},[51,8485,945],{"fence":53,"stretchy":122,"minsize":7403,"maxsize":7403},[48,8487,112],{"mathvariant":111},[114,8489,8490],{"encoding":116},"R(\\delta)-R^*\n=\\mathbb 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20-1h17z",[23,8720,8721,8724],{"style":7130},[23,8722],{"className":8723,"style":217},[216],[23,8725,8727,8730,8733,8736,8739,8742,8745,8748,8751,8754,8757,8760,8763,8766,8815,8818,8821,8824,8827,8830,8833,8836],{"className":8726},[135],[23,8728,1653],{"className":8729,"style":144},[135,136],[23,8731,54],{"className":8732},[140],[23,8734,736],{"className":8735,"style":783},[135,136],[23,8737,54],{"className":8738},[140],[23,8740,434],{"className":8741,"style":454},[135,136],[23,8743,66],{"className":8744},[169],[23,8746,747],{"className":8747},[824],[23,8749],{"className":8750,"style":280},[148],[23,8752,434],{"className":8753,"style":454},[135,136],[23,8755,66],{"className":8756},[169],[23,8758],{"className":8759,"style":392},[148],[23,8761,4529],{"className":8762},[612],[23,8764],{"className":8765,"style":392},[148],[23,8767,8769],{"className":8768},[1786,7096],[23,8770,8772,8806],{"className":8771},[199,200],[23,8773,8775,8803],{"className":8774},[204],[23,8776,8778,8793],{"className":8777,"style":7106},[208],[23,8779,8781,8784],{"style":8780},"top:-2.4em;margin-left:0em;",[23,8782],{"className":8783,"style":217},[216],[23,8785,8787],{"className":8786},[1019,1020,1021,1022],[23,8788,8790],{"className":8789},[135,1022],[23,8791,495],{"className":8792},[135,136,1022],[23,8794,8795,8798],{"style":7130},[23,8796],{"className":8797,"style":217},[216],[23,8799,8800],{},[23,8801,6967],{"className":8802},[1786],[23,8804,297],{"className":8805},[296],[23,8807,8809],{"className":8808},[204],[23,8810,8813],{"className":8811,"style":8812},[208],"height:0.7em;",[23,8814],{},[23,8816],{"className":8817,"style":280},[148],[23,8819,1653],{"className":8820,"style":144},[135,136],[23,8822,54],{"className":8823},[140],[23,8825,495],{"className":8826},[135,136],[23,8828,747],{"className":8829},[824],[23,8831],{"className":8832,"style":280},[148],[23,8834,434],{"className":8835,"style":454},[135,136],[23,8837,66],{"className":8838},[169],[23,8840,297],{"className":8841},[296],[23,8843,8845],{"className":8844},[204],[23,8846,8849],{"className":8847,"style":8848},[208],"height:1.348em;",[23,8850],{},[23,8852,297],{"className":8853},[296],[23,8855,8857],{"className":8856},[204],[23,8858,8861],{"className":8859,"style":8860},[208],"height:2.0943em;",[23,8862],{},[23,8864],{"className":8865,"style":280},[148],[23,8867,8869],{"className":8868},[135],[23,8870,945],{"className":8871},[1066,7581],[23,8873,112],{"className":8874},[135],[11,8876,8877,8878,8906,8907,8910,8911,8939,8940,8968],{},"右边每一项都是非负的，称为在 ",[23,8879,8881,8894],{"className":8880,"translate":27},[31],[23,8882,8884],{"className":8883},[35],[37,8885,8886],{"xmlns":39},[42,8887,8888,8892],{},[45,8889,8890],{},[48,8891,753],{},[114,8893,753],{"encoding":116},[23,8895,8897],{"className":8896,"ariaHidden":122},[121],[23,8898,8900,8903],{"className":8899},[126],[23,8901],{"className":8902,"style":703},[130],[23,8904,753],{"className":8905},[135,136]," 处的",[18,8908,8909],{},"逐点遗憾","（pointwise regret）。它说明：一个决策规则的次优性完全来自它在某些 ",[23,8912,8914,8927],{"className":8913,"translate":27},[31],[23,8915,8917],{"className":8916},[35],[37,8918,8919],{"xmlns":39},[42,8920,8921,8925],{},[45,8922,8923],{},[48,8924,753],{},[114,8926,753],{"encoding":116},[23,8928,8930],{"className":8929,"ariaHidden":122},[121],[23,8931,8933,8936],{"className":8932},[126],[23,8934],{"className":8935,"style":703},[130],[23,8937,753],{"className":8938},[135,136]," 上没有选到局部最优行动。不同 ",[23,8941,8943,8956],{"className":8942,"translate":27},[31],[23,8944,8946],{"className":8945},[35],[37,8947,8948],{"xmlns":39},[42,8949,8950,8954],{},[45,8951,8952],{},[48,8953,753],{},[114,8955,753],{"encoding":116},[23,8957,8959],{"className":8958,"ariaHidden":122},[121],[23,8960,8962,8965],{"className":8961},[126],[23,8963],{"className":8964,"style":703},[130],[23,8966,753],{"className":8967},[135,136]," 之间的错误不会相互抵消，也不会相互放大。",[2501,8970,8972],{"id":8971},"_32-0-1-损失最大后验","3.2 0-1 损失：最大后验",[11,8974,8975],{},"分类问题最常用的损失是 0-1 损失：",[23,8977,8979],{"className":8978,"translate":27},[26],[23,8980,8982,9026],{"className":8981,"translate":27},[31],[23,8983,8985],{"className":8984},[35],[37,8986,8987],{"xmlns":39,"display":40},[42,8988,8989,9023],{},[45,8990,8991,8993,8995,8997,8999,9001,9003,9005,9008,9011,9013,9016,9018,9021],{},[48,8992,555],{},[51,8994,54],{"stretchy":53},[48,8996,660],{},[51,8998,747],{"separator":122},[48,9000,495],{},[51,9002,66],{"stretchy":53},[51,9004,69],{},[2581,9006,4435],{"mathvariant":9007},"bold",[51,9009,9010],{"stretchy":53},"{",[48,9012,660],{},[51,9014,9015],{"mathvariant":111},"≠",[48,9017,495],{},[51,9019,9020],{"stretchy":53},"}",[48,9022,112],{"mathvariant":111},[114,9024,9025],{"encoding":116},"L(y,a)=\\mathbf 1\\{y\\ne a\\}.",[23,9027,9029,9065,9130],{"className":9028,"ariaHidden":122},[121],[23,9030,9032,9035,9038,9041,9044,9047,9050,9053,9056,9059,9062],{"className":9031},[126],[23,9033],{"className":9034,"style":131},[130],[23,9036,555],{"className":9037},[135,136],[23,9039,54],{"className":9040},[140],[23,9042,660],{"className":9043,"style":676},[135,136],[23,9045,747],{"className":9046},[824],[23,9048],{"className":9049,"style":280},[148],[23,9051,495],{"className":9052},[135,136],[23,9054,66],{"className":9055},[169],[23,9057],{"className":9058,"style":149},[148],[23,9060,69],{"className":9061},[153],[23,9063],{"className":9064,"style":149},[148],[23,9066,9068,9071,9075,9078,9081,9084,9127],{"className":9067},[126],[23,9069],{"className":9070,"style":131},[130],[23,9072,4435],{"className":9073},[135,9074],"mathbf",[23,9076,9010],{"className":9077},[140],[23,9079,660],{"className":9080,"style":676},[135,136],[23,9082],{"className":9083,"style":149},[148],[23,9085,9087,9120,9124],{"className":9086},[153],[23,9088,9090],{"className":9089},[153],[23,9091,9094],{"className":9092},[135,9093],"vbox",[23,9095,9098],{"className":9096},[9097],"thinbox",[23,9099,9102,9105,9116],{"className":9100},[9101],"rlap",[23,9103],{"className":9104,"style":5414},[130],[23,9106,9109],{"className":9107},[9108],"inner",[23,9110,9112],{"className":9111},[135],[23,9113,9115],{"className":9114},[153],"",[23,9117],{"className":9118},[9119],"fix",[23,9121],{"className":9122},[148,9123],"nobreak",[23,9125,69],{"className":9126},[153],[23,9128],{"className":9129,"style":149},[148],[23,9131,9133,9136,9139,9142],{"className":9132},[126],[23,9134],{"className":9135,"style":131},[130],[23,9137,495],{"className":9138},[135,136],[23,9140,9020],{"className":9141},[169],[23,9143,112],{"className":9144},[135],[11,9146,9147],{},"此时条件风险为",[23,9149,9151],{"className":9150,"translate":27},[26],[23,9152,9154,9226],{"className":9153,"translate":27},[31],[23,9155,9157],{"className":9156},[35],[37,9158,9159],{"xmlns":39,"display":40},[42,9160,9161,9223],{},[45,9162,9163,9165,9167,9169,9171,9173,9175,9177,9179,9181,9183,9185,9187,9189,9191,9193,9195,9197,9199,9201,9203,9205,9207,9209,9211,9213,9215,9217,9219,9221],{},[48,9164,1653],{},[51,9166,54],{"stretchy":53},[48,9168,495],{},[51,9170,747],{"separator":122},[48,9172,753],{},[51,9174,66],{"stretchy":53},[51,9176,69],{},[48,9178,902],{"mathvariant":571},[51,9180,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[2581,9182,4435],{"mathvariant":9007},[51,9184,9010],{"stretchy":53},[48,9186,369],{},[51,9188,9015],{"mathvariant":111},[48,9190,495],{},[51,9192,9020],{"stretchy":53},[51,9194,60],{},[48,9196,434],{},[51,9198,69],{},[48,9200,753],{},[51,9202,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,9204,69],{},[2581,9206,4435],{},[51,9208,4529],{},[48,9210,11],{},[51,9212,54],{"stretchy":53},[48,9214,495],{},[51,9216,60],{},[48,9218,753],{},[51,9220,66],{"stretchy":53},[51,9222,747],{"separator":122},[114,9224,9225],{"encoding":116},"r(a,x)=\\mathbb E\\big[\\mathbf 1\\{Y\\ne a\\}\\mid X=x\\big]=1-p(a\\mid x),",[23,9227,9229,9265,9331,9352,9370,9394,9413,9437],{"className":9228,"ariaHidden":122},[121],[23,9230,9232,9235,9238,9241,9244,9247,9250,9253,9256,9259,9262],{"className":9231},[126],[23,9233],{"className":9234,"style":131},[130],[23,9236,1653],{"className":9237,"style":144},[135,136],[23,9239,54],{"className":9240},[140],[23,9242,495],{"className":9243},[135,136],[23,9245,747],{"className":9246},[824],[23,9248],{"className":9249,"style":280},[148],[23,9251,753],{"className":9252},[135,136],[23,9254,66],{"className":9255},[169],[23,9257],{"className":9258,"style":149},[148],[23,9260,69],{"className":9261},[153],[23,9263],{"className":9264,"style":149},[148],[23,9266,9268,9271,9274,9280,9283,9286,9289,9292,9328],{"className":9267},[126],[23,9269],{"className":9270,"style":2213},[130],[23,9272,902],{"className":9273},[135,644],[23,9275,9277],{"className":9276},[135],[23,9278,924],{"className":9279},[1066,1067],[23,9281,4435],{"className":9282},[135,9074],[23,9284,9010],{"className":9285},[140],[23,9287,369],{"className":9288,"style":392},[135,136],[23,9290],{"className":9291,"style":149},[148],[23,9293,9295,9322,9325],{"className":9294},[153],[23,9296,9298],{"className":9297},[153],[23,9299,9301],{"className":9300},[135,9093],[23,9302,9304],{"className":9303},[9097],[23,9305,9307,9310,9319],{"className":9306},[9101],[23,9308],{"className":9309,"style":5414},[130],[23,9311,9313],{"className":9312},[9108],[23,9314,9316],{"className":9315},[135],[23,9317,9115],{"className":9318},[153],[23,9320],{"className":9321},[9119],[23,9323],{"className":9324},[148,9123],[23,9326,69],{"className":9327},[153],[23,9329],{"className":9330,"style":149},[148],[23,9332,9334,9337,9340,9343,9346,9349],{"className":9333},[126],[23,9335],{"className":9336,"style":131},[130],[23,9338,495],{"className":9339},[135,136],[23,9341,9020],{"className":9342},[169],[23,9344],{"className":9345,"style":149},[148],[23,9347,60],{"className":9348},[153],[23,9350],{"className":9351,"style":149},[148],[23,9353,9355,9358,9361,9364,9367],{"className":9354},[126],[23,9356],{"className":9357,"style":470},[130],[23,9359,434],{"className":9360,"style":454},[135,136],[23,9362],{"className":9363,"style":149},[148],[23,9365,69],{"className":9366},[153],[23,9368],{"className":9369,"style":149},[148],[23,9371,9373,9376,9379,9385,9388,9391],{"className":9372},[126],[23,9374],{"className":9375,"style":2213},[130],[23,9377,753],{"className":9378},[135,136],[23,9380,9382],{"className":9381},[135],[23,9383,945],{"className":9384},[1066,1067],[23,9386],{"className":9387,"style":149},[148],[23,9389,69],{"className":9390},[153],[23,9392],{"className":9393,"style":149},[148],[23,9395,9397,9401,9404,9407,9410],{"className":9396},[126],[23,9398],{"className":9399,"style":9400},[130],"height:0.7278em;vertical-align:-0.0833em;",[23,9402,4435],{"className":9403},[135],[23,9405],{"className":9406,"style":392},[148],[23,9408,4529],{"className":9409},[612],[23,9411],{"className":9412,"style":392},[148],[23,9414,9416,9419,9422,9425,9428,9431,9434],{"className":9415},[126],[23,9417],{"className":9418,"style":131},[130],[23,9420,11],{"className":9421},[135,136],[23,9423,54],{"className":9424},[140],[23,9426,495],{"className":9427},[135,136],[23,9429],{"className":9430,"style":149},[148],[23,9432,60],{"className":9433},[153],[23,9435],{"className":9436,"style":149},[148],[23,9438,9440,9443,9446,9449],{"className":9439},[126],[23,9441],{"className":9442,"style":131},[130],[23,9444,753],{"className":9445},[135,136],[23,9447,66],{"className":9448},[169],[23,9450,747],{"className":9451},[824],[11,9453,9454,9455,9511],{},"最小化 ",[23,9456,9458,9481],{"className":9457,"translate":27},[31],[23,9459,9461],{"className":9460},[35],[37,9462,9463],{"xmlns":39},[42,9464,9465,9479],{},[45,9466,9467,9469,9471,9473,9475,9477],{},[48,9468,1653],{},[51,9470,54],{"stretchy":53},[48,9472,495],{},[51,9474,747],{"separator":122},[48,9476,753],{},[51,9478,66],{"stretchy":53},[114,9480,2435],{"encoding":116},[23,9482,9484],{"className":9483,"ariaHidden":122},[121],[23,9485,9487,9490,9493,9496,9499,9502,9505,9508],{"className":9486},[126],[23,9488],{"className":9489,"style":131},[130],[23,9491,1653],{"className":9492,"style":144},[135,136],[23,9494,54],{"className":9495},[140],[23,9497,495],{"className":9498},[135,136],[23,9500,747],{"className":9501},[824],[23,9503],{"className":9504,"style":280},[148],[23,9506,753],{"className":9507},[135,136],[23,9509,66],{"className":9510},[169]," 等价于最大化后验概率：",[23,9513,9515],{"className":9514,"translate":27},[26],[23,9516,9518,9581],{"className":9517,"translate":27},[31],[23,9519,9521],{"className":9520},[35],[37,9522,9523],{"xmlns":39,"display":40},[42,9524,9525,9578],{},[45,9526,9527,9533,9535,9537,9539,9541,9543,9545,9562,9564,9566,9568,9570,9572,9574,9576],{},[4518,9528,9529,9531],{},[48,9530,736],{},[51,9532,6881],{},[51,9534,54],{"stretchy":53},[48,9536,753],{},[51,9538,66],{"stretchy":53},[51,9540,69],{},[48,9542,6956],{},[51,9544,6959],{},[6961,9546,9547,9554],{},[45,9548,9549,9552],{},[48,9550,9551],{},"max",[51,9553,6959],{},[45,9555,9556,9558,9560],{},[48,9557,660],{},[51,9559,372],{},[48,9561,369],{"mathvariant":375},[88,9563,5360],{},[48,9565,11],{},[51,9567,54],{"stretchy":53},[48,9569,660],{},[51,9571,60],{},[48,9573,753],{},[51,9575,66],{"stretchy":53},[48,9577,112],{"mathvariant":111},[114,9579,9580],{"encoding":116},"\\delta^*(x)=\\arg\\max_{y\\in\\mathcal Y}\\ p(y\\mid x).",[23,9582,9584,9637,9729],{"className":9583,"ariaHidden":122},[121],[23,9585,9587,9590,9619,9622,9625,9628,9631,9634],{"className":9586},[126],[23,9588],{"className":9589,"style":131},[130],[23,9591,9593,9596],{"className":9592},[135],[23,9594,736],{"className":9595,"style":783},[135,136],[23,9597,9599],{"className":9598},[998],[23,9600,9602],{"className":9601},[199],[23,9603,9605],{"className":9604},[204],[23,9606,9608],{"className":9607,"style":7045},[208],[23,9609,9610,9613],{"style":4989},[23,9611],{"className":9612,"style":1015},[216],[23,9614,9616],{"className":9615},[1019,1020,1021,1022],[23,9617,6881],{"className":9618},[612,1022],[23,9620,54],{"className":9621},[140],[23,9623,753],{"className":9624},[135,136],[23,9626,66],{"className":9627},[169],[23,9629],{"className":9630,"style":149},[148],[23,9632,69],{"className":9633},[153],[23,9635],{"className":9636,"style":149},[148],[23,9638,9640,9644,9649,9652,9705,9708,9711,9714,9717,9720,9723,9726],{"className":9639},[126],[23,9641],{"className":9642,"style":9643},[130],"height:1.6304em;vertical-align:-0.8804em;",[23,9645,7085,9647],{"className":9646},[1786],[23,9648,7089],{"style":7088},[23,9650],{"className":9651,"style":280},[148],[23,9653,9655],{"className":9654},[1786,7096],[23,9656,9658,9697],{"className":9657},[199,200],[23,9659,9661,9694],{"className":9660},[204],[23,9662,9664,9684],{"className":9663,"style":703},[208],[23,9665,9666,9669],{"style":7109},[23,9667],{"className":9668,"style":217},[216],[23,9670,9672],{"className":9671},[1019,1020,1021,1022],[23,9673,9675,9678,9681],{"className":9674},[135,1022],[23,9676,660],{"className":9677,"style":676},[135,136,1022],[23,9679,372],{"className":9680},[153,1022],[23,9682,369],{"className":9683,"style":413},[135,412,1022],[23,9685,9686,9689],{"style":7130},[23,9687],{"className":9688,"style":217},[216],[23,9690,9691],{},[23,9692,9551],{"className":9693},[1786],[23,9695,297],{"className":9696},[296],[23,9698,9700],{"className":9699},[204],[23,9701,9703],{"className":9702,"style":5683},[208],[23,9704],{},[23,9706,5360],{"className":9707},[148],[23,9709],{"className":9710,"style":280},[148],[23,9712,11],{"className":9713},[135,136],[23,9715,54],{"className":9716},[140],[23,9718,660],{"className":9719,"style":676},[135,136],[23,9721],{"className":9722,"style":149},[148],[23,9724,60],{"className":9725},[153],[23,9727],{"className":9728,"style":149},[148],[23,9730,9732,9735,9738,9741],{"className":9731},[126],[23,9733],{"className":9734,"style":131},[130],[23,9736,753],{"className":9737},[135,136],[23,9739,66],{"className":9740},[169],[23,9742,112],{"className":9743},[135],[11,9745,9746,9747,9750,9751,9877],{},"这就是",[18,9748,9749],{},"最大后验决策","（maximum 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& \\eta(x)\\ge \\tfrac12,\\\\[2pt]\n0, & 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",[23,10371,10373,10409],{"className":10372,"translate":27},[31],[23,10374,10376],{"className":10375},[35],[37,10377,10378],{"xmlns":39},[42,10379,10380,10406],{},[45,10381,10382,10384,10386,10388,10390,10392,10394,10396,10398,10400,10402,10404],{},[51,10383,9010],{"stretchy":53},[48,10385,753],{},[51,10387,558],{},[48,10389,9765],{},[51,10391,54],{"stretchy":53},[48,10393,753],{},[51,10395,66],{"stretchy":53},[51,10397,69],{},[2581,10399,4435],{},[48,10401,2875],{"mathvariant":111},[2581,10403,9952],{},[51,10405,9020],{"stretchy":53},[114,10407,10408],{"encoding":116},"\\{x:\\eta(x)=1\u002F2\\}",[23,10410,10412,10433,10460],{"className":10411,"ariaHidden":122},[121],[23,10413,10415,10418,10421,10424,10427,10430],{"className":10414},[126],[23,10416],{"className":10417,"style":131},[130],[23,10419,9010],{"className":10420},[140],[23,10422,753],{"className":10423},[135,136],[23,10425],{"className":10426,"style":149},[148],[23,10428,558],{"className":10429},[153],[23,10431],{"className":10432,"style":149},[148],[23,10434,10436,10439,10442,10445,10448,10451,10454,10457],{"className":10435},[126],[23,10437],{"className":10438,"style":131},[130],[23,10440,9765],{"className":10441,"style":676},[135,136],[23,10443,54],{"className":10444},[140],[23,10446,753],{"className":10447},[135,136],[23,10449,66],{"className":10450},[169],[23,10452],{"className":10453,"style":149},[148],[23,10455,69],{"className":10456},[153],[23,10458],{"className":10459,"style":149},[148],[23,10461,10463,10466,10470],{"className":10462},[126],[23,10464],{"className":10465,"style":131},[130],[23,10467,10469],{"className":10468},[135],"1\u002F2",[23,10471,9020],{"className":10472},[169],"。当 ",[23,10475,10477,10505],{"className":10476,"translate":27},[31],[23,10478,10480],{"className":10479},[35],[37,10481,10482],{"xmlns":39},[42,10483,10484,10502],{},[45,10485,10486,10488,10490,10492,10494,10496,10498,10500],{},[48,10487,9765],{},[51,10489,54],{"stretchy":53},[48,10491,753],{},[51,10493,66],{"stretchy":53},[51,10495,69],{},[2581,10497,4435],{},[48,10499,2875],{"mathvariant":111},[2581,10501,9952],{},[114,10503,10504],{"encoding":116},"\\eta(x)=1\u002F2",[23,10506,10508,10535],{"className":10507,"ariaHidden":122},[121],[23,10509,10511,10514,10517,10520,10523,10526,10529,10532],{"className":10510},[126],[23,10512],{"className":10513,"style":131},[130],[23,10515,9765],{"className":10516,"style":676},[135,136],[23,10518,54],{"className":10519},[140],[23,10521,753],{"className":10522},[135,136],[23,10524,66],{"className":10525},[169],[23,10527],{"className":10528,"style":149},[148],[23,10530,69],{"className":10531},[153],[23,10533],{"className":10534,"style":149},[148],[23,10536,10538,10541],{"className":10537},[126],[23,10539],{"className":10540,"style":131},[130],[23,10542,10469],{"className":10543},[135]," 时两种行动同样好，实践中用固定的打破平局规则即可。",[2501,10546,10548],{"id":10547},"_33-平方损失后验均值","3.3 平方损失：后验均值",[11,10550,10551,10552,10694],{},"回归问题常用平方损失 ",[23,10553,10555,10595],{"className":10554,"translate":27},[31],[23,10556,10558],{"className":10557},[35],[37,10559,10560],{"xmlns":39},[42,10561,10562,10592],{},[45,10563,10564,10566,10568,10570,10572,10574,10576,10578,10580,10582,10584,10586],{},[48,10565,555],{},[51,10567,54],{"stretchy":53},[48,10569,660],{},[51,10571,747],{"separator":122},[48,10573,495],{},[51,10575,66],{"stretchy":53},[51,10577,69],{},[51,10579,54],{"stretchy":53},[48,10581,660],{},[51,10583,4529],{},[48,10585,495],{},[4518,10587,10588,10590],{},[51,10589,66],{"stretchy":53},[2581,10591,9952],{},[114,10593,10594],{"encoding":116},"L(y,a)=(y-a)^2",[23,10596,10598,10634,10655],{"className":10597,"ariaHidden":122},[121],[23,10599,10601,10604,10607,10610,10613,10616,10619,10622,10625,10628,10631],{"className":10600},[126],[23,10602],{"className":10603,"style":131},[130],[23,10605,555],{"className":10606},[135,136],[23,10608,54],{"className":10609},[140],[23,10611,660],{"className":10612,"style":676},[135,136],[23,10614,747],{"className":10615},[824],[23,10617],{"className":10618,"style":280},[148],[23,10620,495],{"className":10621},[135,136],[23,10623,66],{"className":10624},[169],[23,10626],{"className":10627,"style":149},[148],[23,10629,69],{"className":10630},[153],[23,10632],{"className":10633,"style":149},[148],[23,10635,10637,10640,10643,10646,10649,10652],{"className":10636},[126],[23,10638],{"className":10639,"style":131},[130],[23,10641,54],{"className":10642},[140],[23,10644,660],{"className":10645,"style":676},[135,136],[23,10647],{"className":10648,"style":392},[148],[23,10650,4529],{"className":10651},[612],[23,10653],{"className":10654,"style":392},[148],[23,10656,10658,10662,10665],{"className":10657},[126],[23,10659],{"className":10660,"style":10661},[130],"height:1.0641em;vertical-align:-0.25em;",[23,10663,495],{"className":10664},[135,136],[23,10666,10668,10671],{"className":10667},[169],[23,10669,66],{"className":10670},[169],[23,10672,10674],{"className":10673},[998],[23,10675,10677],{"className":10676},[199],[23,10678,10680],{"className":10679},[204],[23,10681,10683],{"className":10682,"style":4693},[208],[23,10684,10685,10688],{"style":4696},[23,10686],{"className":10687,"style":1015},[216],[23,10689,10691],{"className":10690},[1019,1020,1021,1022],[23,10692,9952],{"className":10693},[135,1022],"。此时",[23,10696,10698],{"className":10697,"translate":27},[26],[23,10699,10701,10844],{"className":10700,"translate":27},[31],[23,10702,10704],{"className":10703},[35],[37,10705,10706],{"xmlns":39,"display":40},[42,10707,10708,10841],{},[1518,10709,10710,10768],{"rowspacing":1520,"columnalign":1521,"columnspacing":1522},[1524,10711,10712,10730],{},[1527,10713,10714],{},[1530,10715,10716],{"scriptlevel":1532,"displaystyle":122},[45,10717,10718,10720,10722,10724,10726,10728],{},[48,10719,1653],{},[51,10721,54],{"stretchy":53},[48,10723,495],{},[51,10725,747],{"separator":122},[48,10727,753],{},[51,10729,66],{"stretchy":53},[1527,10731,10732],{},[1530,10733,10734],{"scriptlevel":1532,"displaystyle":122},[45,10735,10736,10738,10740,10742,10744,10746,10748,10750,10752,10758,10760,10762,10764,10766],{},[45,10737],{},[51,10739,69],{},[48,10741,902],{"mathvariant":571},[51,10743,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,10745,54],{"stretchy":53},[48,10747,369],{},[51,10749,4529],{},[48,10751,495],{},[4518,10753,10754,10756],{},[51,10755,66],{"stretchy":53},[2581,10757,9952],{},[51,10759,60],{},[48,10761,434],{},[51,10763,69],{},[48,10765,753],{},[51,10767,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[1524,10769,10770,10776],{},[1527,10771,10772],{},[1530,10773,10774],{"scriptlevel":1532,"displaystyle":122},[45,10775],{},[1527,10777,10778],{},[1530,10779,10780],{"scriptlevel":1532,"displaystyle":122},[45,10781,10782,10784,10786,10795,10797,10799,10801,10803,10805,10807,10809,10811,10813,10815,10817,10819,10821,10823,10825,10827,10829,10831,10833,10839],{},[45,10783],{},[51,10785,69],{},[45,10787,10788,10791,10793],{},[48,10789,10790],{"mathvariant":111},"V",[48,10792,495],{"mathvariant":111},[48,10794,1653],{"mathvariant":111},[51,10796,54],{"stretchy":53},[48,10798,369],{},[51,10800,60],{},[48,10802,434],{},[51,10804,69],{},[48,10806,753],{},[51,10808,66],{"stretchy":53},[51,10810,5106],{},[51,10812,54],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,10814,495],{},[51,10816,4529],{},[48,10818,902],{"mathvariant":571},[51,10820,924],{"stretchy":53},[48,10822,369],{},[51,10824,60],{},[48,10826,434],{},[51,10828,69],{},[48,10830,753],{},[51,10832,945],{"stretchy":53},[4518,10834,10835,10837],{},[51,10836,66],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[2581,10838,9952],{},[51,10840,747],{"separator":122},[114,10842,10843],{"encoding":116},"\\begin{aligned}\nr(a,x)\n&=\\mathbb E\\big[(Y-a)^2\\mid X=x\\big]\\\\\n&=\\mathrm{Var}(Y\\mid X=x)+\\big(a-\\mathbb E[Y\\mid X=x]\\big)^2,\n\\end{aligned}",[23,10845,10847],{"className":10846,"ariaHidden":122},[121],[23,10848,10850,10854],{"className":10849},[126],[23,10851],{"className":10852,"style":10853},[130],"height:2.9381em;vertical-align:-1.2191em;",[23,10855,10857],{"className":10856},[135],[23,10858,10860,10925],{"className":10859},[1518],[23,10861,10863],{"className":10862},[1696],[23,10864,10866,10916],{"className":10865},[199,200],[23,10867,10869,10913],{"className":10868},[204],[23,10870,10873,10904],{"className":10871,"style":10872},[208],"height:1.7191em;",[23,10874,10876,10880],{"style":10875},"top:-3.909em;",[23,10877],{"className":10878,"style":10879},[216],"height:3.054em;",[23,10881,10883,10886,10889,10892,10895,10898,10901],{"className":10882},[135],[23,10884,1653],{"className":10885,"style":144},[135,136],[23,10887,54],{"className":10888},[140],[23,10890,495],{"className":10891},[135,136],[23,10893,747],{"className":10894},[824],[23,10896],{"className":10897,"style":280},[148],[23,10899,753],{"className":10900},[135,136],[23,10902,66],{"className":10903},[169],[23,10905,10907,10910],{"style":10906},"top:-2.1949em;",[23,10908],{"className":10909,"style":10879},[216],[23,10911],{"className":10912},[135],[23,10914,297],{"className":10915},[296],[23,10917,10919],{"className":10918},[204],[23,10920,10923],{"className":10921,"style":10922},[208],"height:1.2191em;",[23,10924],{},[23,10926,10928],{"className":10927},[1753],[23,10929,10931,11207],{"className":10930},[199,200],[23,10932,10934,11204],{"className":10933},[204],[23,10935,10937,11044],{"className":10936,"style":10872},[208],[23,10938,10939,10942],{"style":10875},[23,10940],{"className":10941,"style":10879},[216],[23,10943,10945,10948,10951,10954,10957,10960,10966,10969,10972,10975,10978,10981,10984,11014,11017,11020,11023,11026,11029,11032,11035,11038],{"className":10944},[135],[23,10946],{"className":10947},[135],[23,10949],{"className":10950,"style":149},[148],[23,10952,69],{"className":10953},[153],[23,10955],{"className":10956,"style":149},[148],[23,10958,902],{"className":10959},[135,644],[23,10961,10963],{"className":10962},[135],[23,10964,924],{"className":10965},[1066,1067],[23,10967,54],{"className":10968},[140],[23,10970,369],{"className":10971,"style":392},[135,136],[23,10973],{"className":10974,"style":392},[148],[23,10976,4529],{"className":10977},[612],[23,10979],{"className":10980,"style":392},[148],[23,10982,495],{"className":10983},[135,136],[23,10985,10987,10990],{"className":10986},[169],[23,10988,66],{"className":10989},[169],[23,10991,10993],{"className":10992},[998],[23,10994,10996],{"className":10995},[199],[23,10997,10999],{"className":10998},[204],[23,11000,11003],{"className":11001,"style":11002},[208],"height:0.8641em;",[23,11004,11005,11008],{"style":4989},[23,11006],{"className":11007,"style":1015},[216],[23,11009,11011],{"className":11010},[1019,1020,1021,1022],[23,11012,9952],{"className":11013},[135,1022],[23,11015],{"className":11016,"style":149},[148],[23,11018,60],{"className":11019},[153],[23,11021],{"className":11022,"style":149},[148],[23,11024,434],{"className":11025,"style":454},[135,136],[23,11027],{"className":11028,"style":149},[148],[23,11030,69],{"className":11031},[153],[23,11033],{"className":11034,"style":149},[148],[23,11036,753],{"className":11037},[135,136],[23,11039,11041],{"className":11040},[135],[23,11042,945],{"className":11043},[1066,1067],[23,11045,11046,11049],{"style":10906},[23,11047],{"className":11048,"style":10879},[216],[23,11050,11052,11055,11058,11061,11064,11071,11074,11077,11080,11083,11086,11089,11092,11095,11098,11101,11104,11107,11110,11113,11119,11122,11125,11128,11131,11134,11137,11140,11143,11146,11149,11152,11155,11158,11161,11164,11167,11201],{"className":11051},[135],[23,11053],{"className":11054},[135],[23,11056],{"className":11057,"style":149},[148],[23,11059,69],{"className":11060},[153],[23,11062],{"className":11063,"style":149},[148],[23,11065,11067],{"className":11066},[135],[23,11068,11070],{"className":11069},[135,5391],"Var",[23,11072,54],{"className":11073},[140],[23,11075,369],{"className":11076,"style":392},[135,136],[23,11078],{"className":11079,"style":149},[148],[23,11081,60],{"className":11082},[153],[23,11084],{"className":11085,"style":149},[148],[23,11087,434],{"className":11088,"style":454},[135,136],[23,11090],{"className":11091,"style":149},[148],[23,11093,69],{"className":11094},[153],[23,11096],{"className":11097,"style":149},[148],[23,11099,753],{"className":11100},[135,136],[23,11102,66],{"className":11103},[169],[23,11105],{"className":11106,"style":392},[148],[23,11108,5106],{"className":11109},[612],[23,11111],{"className":11112,"style":392},[148],[23,11114,11116],{"className":11115},[135],[23,11117,54],{"className":11118},[1066,1067],[23,11120,495],{"className":11121},[135,136],[23,11123],{"className":11124,"style":392},[148],[23,11126,4529],{"className":11127},[612],[23,11129],{"className":11130,"style":392},[148],[23,11132,902],{"className":11133},[135,644],[23,11135,924],{"className":11136},[140],[23,11138,369],{"className":11139,"style":392},[135,136],[23,11141],{"className":11142,"style":149},[148],[23,11144,60],{"className":11145},[153],[23,11147],{"className":11148,"style":149},[148],[23,11150,434],{"className":11151,"style":454},[135,136],[23,11153],{"className":11154,"style":149},[148],[23,11156,69],{"className":11157},[153],[23,11159],{"className":11160,"style":149},[148],[23,11162,753],{"className":11163},[135,136],[23,11165,945],{"className":11166},[169],[23,11168,11170,11176],{"className":11169},[135],[23,11171,11173],{"className":11172},[135],[23,11174,66],{"className":11175},[1066,1067],[23,11177,11179],{"className":11178},[998],[23,11180,11182],{"className":11181},[199],[23,11183,11185],{"className":11184},[204],[23,11186,11189],{"className":11187,"style":11188},[208],"height:1.054em;",[23,11190,11192,11195],{"style":11191},"top:-3.3029em;margin-right:0.05em;",[23,11193],{"className":11194,"style":1015},[216],[23,11196,11198],{"className":11197},[1019,1020,1021,1022],[23,11199,9952],{"className":11200},[135,1022],[23,11202,747],{"className":11203},[824],[23,11205,297],{"className":11206},[296],[23,11208,11210],{"className":11209},[204],[23,11211,11213],{"className":11212,"style":10922},[208],[23,11214],{},[11,11216,11217,11218,11328,11329,11332],{},"第二项在 ",[23,11219,11221,11253],{"className":11220,"translate":27},[31],[23,11222,11224],{"className":11223},[35],[37,11225,11226],{"xmlns":39},[42,11227,11228,11250],{},[45,11229,11230,11232,11234,11236,11238,11240,11242,11244,11246,11248],{},[48,11231,495],{},[51,11233,69],{},[48,11235,902],{"mathvariant":571},[51,11237,924],{"stretchy":53},[48,11239,369],{},[51,11241,60],{},[48,11243,434],{},[51,11245,69],{},[48,11247,753],{},[51,11249,945],{"stretchy":53},[114,11251,11252],{"encoding":116},"a=\\mathbb E[Y\\mid X=x]",[23,11254,11256,11274,11298,11316],{"className":11255,"ariaHidden":122},[121],[23,11257,11259,11262,11265,11268,11271],{"className":11258},[126],[23,11260],{"className":11261,"style":703},[130],[23,11263,495],{"className":11264},[135,136],[23,11266],{"className":11267,"style":149},[148],[23,11269,69],{"className":11270},[153],[23,11272],{"className":11273,"style":149},[148],[23,11275,11277,11280,11283,11286,11289,11292,11295],{"className":11276},[126],[23,11278],{"className":11279,"style":131},[130],[23,11281,902],{"className":11282},[135,644],[23,11284,924],{"className":11285},[140],[23,11287,369],{"className":11288,"style":392},[135,136],[23,11290],{"className":11291,"style":149},[148],[23,11293,60],{"className":11294},[153],[23,11296],{"className":11297,"style":149},[148],[23,11299,11301,11304,11307,11310,11313],{"className":11300},[126],[23,11302],{"className":11303,"style":470},[130],[23,11305,434],{"className":11306,"style":454},[135,136],[23,11308],{"className":11309,"style":149},[148],[23,11311,69],{"className":11312},[153],[23,11314],{"className":11315,"style":149},[148],[23,11317,11319,11322,11325],{"className":11318},[126],[23,11320],{"className":11321,"style":131},[130],[23,11323,753],{"className":11324},[135,136],[23,11326,945],{"className":11327},[169]," 时取到最小值。因此平方损失下的贝叶斯行动是",[18,11330,11331],{},"后验均值","，也就是回归函数",[23,11334,11336],{"className":11335,"translate":27},[26],[23,11337,11339,11383],{"className":11338,"translate":27},[31],[23,11340,11342],{"className":11341},[35],[37,11343,11344],{"xmlns":39,"display":40},[42,11345,11346,11380],{},[45,11347,11348,11354,11356,11358,11360,11362,11364,11366,11368,11370,11372,11374,11376,11378],{},[4518,11349,11350,11352],{},[48,11351,8336],{},[51,11353,6881],{},[51,11355,54],{"stretchy":53},[48,11357,753],{},[51,11359,66],{"stretchy":53},[51,11361,69],{},[48,11363,902],{"mathvariant":571},[51,11365,924],{"stretchy":53},[48,11367,369],{},[51,11369,60],{},[48,11371,434],{},[51,11373,69],{},[48,11375,753],{},[51,11377,945],{"stretchy":53},[48,11379,112],{"mathvariant":111},[114,11381,11382],{"encoding":116},"f^*(x)=\\mathbb E[Y\\mid X=x].",[23,11384,11386,11440,11464,11482],{"className":11385,"ariaHidden":122},[121],[23,11387,11389,11392,11422,11425,11428,11431,11434,11437],{"className":11388},[126],[23,11390],{"className":11391,"style":131},[130],[23,11393,11395,11399],{"className":11394},[135],[23,11396,8336],{"className":11397,"style":11398},[135,136],"margin-right:0.1076em;",[23,11400,11402],{"className":11401},[998],[23,11403,11405],{"className":11404},[199],[23,11406,11408],{"className":11407},[204],[23,11409,11411],{"className":11410,"style":7045},[208],[23,11412,11413,11416],{"style":4989},[23,11414],{"className":11415,"style":1015},[216],[23,11417,11419],{"className":11418},[1019,1020,1021,1022],[23,11420,6881],{"className":11421},[612,1022],[23,11423,54],{"className":11424},[140],[23,11426,753],{"className":11427},[135,136],[23,11429,66],{"className":11430},[169],[23,11432],{"className":11433,"style":149},[148],[23,11435,69],{"className":11436},[153],[23,11438],{"className":11439,"style":149},[148],[23,11441,11443,11446,11449,11452,11455,11458,11461],{"className":11442},[126],[23,11444],{"className":11445,"style":131},[130],[23,11447,902],{"className":11448},[135,644],[23,11450,924],{"className":11451},[140],[23,11453,369],{"className":11454,"style":392},[135,136],[23,11456],{"className":11457,"style":149},[148],[23,11459,60],{"className":11460},[153],[23,11462],{"className":11463,"style":149},[148],[23,11465,11467,11470,11473,11476,11479],{"className":11466},[126],[23,11468],{"className":11469,"style":470},[130],[23,11471,434],{"className":11472,"style":454},[135,136],[23,11474],{"className":11475,"style":149},[148],[23,11477,69],{"className":11478},[153],[23,11480],{"className":11481,"style":149},[148],[23,11483,11485,11488,11491,11494],{"className":11484},[126],[23,11486],{"className":11487,"style":131},[130],[23,11489,753],{"className":11490},[135,136],[23,11492,945],{"className":11493},[169],[23,11495,112],{"className":11496},[135],[11,11498,11499,11500,11528,11529,11588],{},"这个结论还可以从另一个角度理解：后验均值是 ",[23,11501,11503,11516],{"className":11502,"translate":27},[31],[23,11504,11506],{"className":11505},[35],[37,11507,11508],{"xmlns":39},[42,11509,11510,11514],{},[45,11511,11512],{},[48,11513,369],{},[114,11515,369],{"encoding":116},[23,11517,11519],{"className":11518,"ariaHidden":122},[121],[23,11520,11522,11525],{"className":11521},[126],[23,11523],{"className":11524,"style":470},[130],[23,11526,369],{"className":11527,"style":392},[135,136]," 在 ",[23,11530,11532,11550],{"className":11531,"translate":27},[31],[23,11533,11535],{"className":11534},[35],[37,11536,11537],{"xmlns":39},[42,11538,11539,11547],{},[45,11540,11541],{},[4518,11542,11543,11545],{},[48,11544,555],{},[2581,11546,9952],{},[114,11548,11549],{"encoding":116},"L^2",[23,11551,11553],{"className":11552,"ariaHidden":122},[121],[23,11554,11556,11559],{"className":11555},[126],[23,11557],{"className":11558,"style":4693},[130],[23,11560,11562,11565],{"className":11561},[135],[23,11563,555],{"className":11564},[135,136],[23,11566,11568],{"className":11567},[998],[23,11569,11571],{"className":11570},[199],[23,11572,11574],{"className":11573},[204],[23,11575,11577],{"className":11576,"style":4693},[208],[23,11578,11579,11582],{"style":4696},[23,11580],{"className":11581,"style":1015},[216],[23,11583,11585],{"className":11584},[1019,1020,1021,1022],[23,11586,9952],{"className":11587},[135,1022]," 意义下最好的预测，任何其他预测都会多出一项方差。",[2501,11590,11592],{"id":11591},"_34-绝对损失后验中位数","3.4 绝对损失：后验中位数",[11,11594,11595,11596,11707,11708,11878,11879,11907,11908,11936],{},"若损失为 ",[23,11597,11599,11635],{"className":11598,"translate":27},[31],[23,11600,11602],{"className":11601},[35],[37,11603,11604],{"xmlns":39},[42,11605,11606,11632],{},[45,11607,11608,11610,11612,11614,11616,11618,11620,11622,11624,11626,11628,11630],{},[48,11609,555],{},[51,11611,54],{"stretchy":53},[48,11613,660],{},[51,11615,747],{"separator":122},[48,11617,495],{},[51,11619,66],{"stretchy":53},[51,11621,69],{},[48,11623,60],{"mathvariant":111},[48,11625,660],{},[51,11627,4529],{},[48,11629,495],{},[48,11631,60],{"mathvariant":111},[114,11633,11634],{"encoding":116},"L(y,a)=|y-a|",[23,11636,11638,11674,11695],{"className":11637,"ariaHidden":122},[121],[23,11639,11641,11644,11647,11650,11653,11656,11659,11662,11665,11668,11671],{"className":11640},[126],[23,11642],{"className":11643,"style":131},[130],[23,11645,555],{"className":11646},[135,136],[23,11648,54],{"className":11649},[140],[23,11651,660],{"className":11652,"style":676},[135,136],[23,11654,747],{"className":11655},[824],[23,11657],{"className":11658,"style":280},[148],[23,11660,495],{"className":11661},[135,136],[23,11663,66],{"className":11664},[169],[23,11666],{"className":11667,"style":149},[148],[23,11669,69],{"className":11670},[153],[23,11672],{"className":11673,"style":149},[148],[23,11675,11677,11680,11683,11686,11689,11692],{"className":11676},[126],[23,11678],{"className":11679,"style":131},[130],[23,11681,60],{"className":11682},[135],[23,11684,660],{"className":11685,"style":676},[135,136],[23,11687],{"className":11688,"style":392},[148],[23,11690,4529],{"className":11691},[612],[23,11693],{"className":11694,"style":392},[148],[23,11696,11698,11701,11704],{"className":11697},[126],[23,11699],{"className":11700,"style":131},[130],[23,11702,495],{"className":11703},[135,136],[23,11705,60],{"className":11706},[135],"，条件风险为 ",[23,11709,11711,11761],{"className":11710,"translate":27},[31],[23,11712,11714],{"className":11713},[35],[37,11715,11716],{"xmlns":39},[42,11717,11718,11758],{},[45,11719,11720,11722,11724,11726,11728,11730,11732,11734,11736,11738,11740,11742,11744,11746,11748,11750,11752,11754,11756],{},[48,11721,1653],{},[51,11723,54],{"stretchy":53},[48,11725,495],{},[51,11727,747],{"separator":122},[48,11729,753],{},[51,11731,66],{"stretchy":53},[51,11733,69],{},[48,11735,902],{"mathvariant":571},[51,11737,924],{"stretchy":53},[48,11739,60],{"mathvariant":111},[48,11741,369],{},[51,11743,4529],{},[48,11745,495],{},[48,11747,60],{"mathvariant":111},[51,11749,60],{},[48,11751,434],{},[51,11753,69],{},[48,11755,753],{},[51,11757,945],{"stretchy":53},[114,11759,11760],{"encoding":116},"r(a,x)=\\mathbb E[|Y-a|\\mid X=x]",[23,11762,11764,11800,11827,11848,11866],{"className":11763,"ariaHidden":122},[121],[23,11765,11767,11770,11773,11776,11779,11782,11785,11788,11791,11794,11797],{"className":11766},[126],[23,11768],{"className":11769,"style":131},[130],[23,11771,1653],{"className":11772,"style":144},[135,136],[23,11774,54],{"className":11775},[140],[23,11777,495],{"className":11778},[135,136],[23,11780,747],{"className":11781},[824],[23,11783],{"className":11784,"style":280},[148],[23,11786,753],{"className":11787},[135,136],[23,11789,66],{"className":11790},[169],[23,11792],{"className":11793,"style":149},[148],[23,11795,69],{"className":11796},[153],[23,11798],{"className":11799,"style":149},[148],[23,11801,11803,11806,11809,11812,11815,11818,11821,11824],{"className":11802},[126],[23,11804],{"className":11805,"style":131},[130],[23,11807,902],{"className":11808},[135,644],[23,11810,924],{"className":11811},[140],[23,11813,60],{"className":11814},[135],[23,11816,369],{"className":11817,"style":392},[135,136],[23,11819],{"className":11820,"style":392},[148],[23,11822,4529],{"className":11823},[612],[23,11825],{"className":11826,"style":392},[148],[23,11828,11830,11833,11836,11839,11842,11845],{"className":11829},[126],[23,11831],{"className":11832,"style":131},[130],[23,11834,495],{"className":11835},[135,136],[23,11837,60],{"className":11838},[135],[23,11840],{"className":11841,"style":149},[148],[23,11843,60],{"className":11844},[153],[23,11846],{"className":11847,"style":149},[148],[23,11849,11851,11854,11857,11860,11863],{"className":11850},[126],[23,11852],{"className":11853,"style":470},[130],[23,11855,434],{"className":11856,"style":454},[135,136],[23,11858],{"className":11859,"style":149},[148],[23,11861,69],{"className":11862},[153],[23,11864],{"className":11865,"style":149},[148],[23,11867,11869,11872,11875],{"className":11868},[126],[23,11870],{"className":11871,"style":131},[130],[23,11873,753],{"className":11874},[135,136],[23,11876,945],{"className":11877},[169],"。对 ",[23,11880,11882,11895],{"className":11881,"translate":27},[31],[23,11883,11885],{"className":11884},[35],[37,11886,11887],{"xmlns":39},[42,11888,11889,11893],{},[45,11890,11891],{},[48,11892,495],{},[114,11894,495],{"encoding":116},[23,11896,11898],{"className":11897,"ariaHidden":122},[121],[23,11899,11901,11904],{"className":11900},[126],[23,11902],{"className":11903,"style":703},[130],[23,11905,495],{"className":11906},[135,136]," 求次梯度，在 ",[23,11909,11911,11924],{"className":11910,"translate":27},[31],[23,11912,11914],{"className":11913},[35],[37,11915,11916],{"xmlns":39},[42,11917,11918,11922],{},[45,11919,11920],{},[48,11921,369],{},[114,11923,369],{"encoding":116},[23,11925,11927],{"className":11926,"ariaHidden":122},[121],[23,11928,11930,11933],{"className":11929},[126],[23,11931],{"className":11932,"style":470},[130],[23,11934,369],{"className":11935,"style":392},[135,136]," 的分布连续时它就是导数：",[23,11938,11940],{"className":11939,"translate":27},[26],[23,11941,11943,12012],{"className":11942,"translate":27},[31],[23,11944,11946],{"className":11945},[35],[37,11947,11948],{"xmlns":39,"display":40},[42,11949,11950,12009],{},[45,11951,11952,11959,11961,11963,11965,11967,11969,11971,11973,11975,11977,11979,11981,11983,11985,11987,11989,11991,11993,11995,11997,11999,12001,12003,12005,12007],{},[898,11953,11954,11957],{},[48,11955,11956],{"mathvariant":111},"∂",[48,11958,495],{},[48,11960,1653],{},[51,11962,54],{"stretchy":53},[48,11964,495],{},[51,11966,747],{"separator":122},[48,11968,753],{},[51,11970,66],{"stretchy":53},[51,11972,69],{},[48,11974,920],{},[51,11976,54],{"stretchy":53},[48,11978,369],{},[51,11980,9983],{},[48,11982,495],{},[51,11984,60],{},[48,11986,753],{},[51,11988,66],{"stretchy":53},[51,11990,4529],{},[48,11992,920],{},[51,11994,54],{"stretchy":53},[48,11996,369],{},[51,11998,2579],{},[48,12000,495],{},[51,12002,60],{},[48,12004,753],{},[51,12006,66],{"stretchy":53},[51,12008,747],{"separator":122},[114,12010,12011],{"encoding":116},"\\partial_a r(a,x)=P(Y\u003Ca\\mid x)-P(Y>a\\mid x),",[23,12013,12015,12093,12117,12135,12156,12180,12198],{"className":12014,"ariaHidden":122},[121],[23,12016,12018,12021,12063,12066,12069,12072,12075,12078,12081,12084,12087,12090],{"className":12017},[126],[23,12019],{"className":12020,"style":131},[130],[23,12022,12024,12028],{"className":12023},[135],[23,12025,11956],{"className":12026,"style":12027},[135],"margin-right:0.0556em;",[23,12029,12031],{"className":12030},[998],[23,12032,12034,12055],{"className":12033},[199,200],[23,12035,12037,12052],{"className":12036},[204],[23,12038,12040],{"className":12039,"style":6499},[208],[23,12041,12043,12046],{"style":12042},"top:-2.55em;margin-left:-0.0556em;margin-right:0.05em;",[23,12044],{"className":12045,"style":1015},[216],[23,12047,12049],{"className":12048},[1019,1020,1021,1022],[23,12050,495],{"className":12051},[135,136,1022],[23,12053,297],{"className":12054},[296],[23,12056,12058],{"className":12057},[204],[23,12059,12061],{"className":12060,"style":2048},[208],[23,12062],{},[23,12064,1653],{"className":12065,"style":144},[135,136],[23,12067,54],{"className":12068},[140],[23,12070,495],{"className":12071},[135,136],[23,12073,747],{"className":12074},[824],[23,12076],{"className":12077,"style":280},[148],[23,12079,753],{"className":12080},[135,136],[23,12082,66],{"className":12083},[169],[23,12085],{"className":12086,"style":149},[148],[23,12088,69],{"className":12089},[153],[23,12091],{"className":12092,"style":149},[148],[23,12094,12096,12099,12102,12105,12108,12111,12114],{"className":12095},[126],[23,12097],{"className":12098,"style":131},[130],[23,12100,920],{"className":12101,"style":1047},[135,136],[23,12103,54],{"className":12104},[140],[23,12106,369],{"className":12107,"style":392},[135,136],[23,12109],{"className":12110,"style":149},[148],[23,12112,9983],{"className":12113},[153],[23,12115],{"className":12116,"style":149},[148],[23,12118,12120,12123,12126,12129,12132],{"className":12119},[126],[23,12121],{"className":12122,"style":131},[130],[23,12124,495],{"className":12125},[135,136],[23,12127],{"className":12128,"style":149},[148],[23,12130,60],{"className":12131},[153],[23,12133],{"className":12134,"style":149},[148],[23,12136,12138,12141,12144,12147,12150,12153],{"className":12137},[126],[23,12139],{"className":12140,"style":131},[130],[23,12142,753],{"className":12143},[135,136],[23,12145,66],{"className":12146},[169],[23,12148],{"className":12149,"style":392},[148],[23,12151,4529],{"className":12152},[612],[23,12154],{"className":12155,"style":392},[148],[23,12157,12159,12162,12165,12168,12171,12174,12177],{"className":12158},[126],[23,12160],{"className":12161,"style":131},[130],[23,12163,920],{"className":12164,"style":1047},[135,136],[23,12166,54],{"className":12167},[140],[23,12169,369],{"className":12170,"style":392},[135,136],[23,12172],{"className":12173,"style":149},[148],[23,12175,2579],{"className":12176},[153],[23,12178],{"className":12179,"style":149},[148],[23,12181,12183,12186,12189,12192,12195],{"className":12182},[126],[23,12184],{"className":12185,"style":131},[130],[23,12187,495],{"className":12188},[135,136],[23,12190],{"className":12191,"style":149},[148],[23,12193,60],{"className":12194},[153],[23,12196],{"className":12197,"style":149},[148],[23,12199,12201,12204,12207,12210],{"className":12200},[126],[23,12202],{"className":12203,"style":131},[130],[23,12205,753],{"className":12206},[135,136],[23,12208,66],{"className":12209},[169],[23,12211,747],{"className":12212},[824],[11,12214,12215,12216,12244],{},"令 ",[23,12217,12219,12232],{"className":12218,"translate":27},[31],[23,12220,12222],{"className":12221},[35],[37,12223,12224],{"xmlns":39},[42,12225,12226,12230],{},[45,12227,12228],{},[2581,12229,1532],{},[114,12231,1532],{"encoding":116},[23,12233,12235],{"className":12234,"ariaHidden":122},[121],[23,12236,12238,12241],{"className":12237},[126],[23,12239],{"className":12240,"style":2770},[130],[23,12242,1532],{"className":12243},[135]," 属于次梯度，等价于",[23,12246,12248],{"className":12247,"translate":27},[26],[23,12249,12251,12318],{"className":12250,"translate":27},[31],[23,12252,12254],{"className":12253},[35],[37,12255,12256],{"xmlns":39,"display":40},[42,12257,12258,12315],{},[45,12259,12260,12262,12264,12266,12269,12271,12273,12275,12277,12279,12285,12287,12289,12291,12293,12295,12297,12299,12301,12303,12305,12307,12313],{},[48,12261,920],{},[51,12263,54],{"stretchy":53},[48,12265,369],{},[51,12267,12268],{},"≤",[48,12270,495],{},[51,12272,60],{},[48,12274,753],{},[51,12276,66],{"stretchy":53},[51,12278,7378],{},[71,12280,12281,12283],{},[2581,12282,4435],{},[2581,12284,9952],{},[51,12286,747],{"separator":122},[148,12288],{"width":750},[48,12290,920],{},[51,12292,54],{"stretchy":53},[48,12294,369],{},[51,12296,7378],{},[48,12298,495],{},[51,12300,60],{},[48,12302,753],{},[51,12304,66],{"stretchy":53},[51,12306,7378],{},[71,12308,12309,12311],{},[2581,12310,4435],{},[2581,12312,9952],{},[48,12314,112],{"mathvariant":111},[114,12316,12317],{"encoding":116},"P(Y\\le a\\mid x)\\ge \\frac12,\\qquad P(Y\\ge a\\mid x)\\ge \\frac12 .",[23,12319,12321,12345,12363,12384,12481,12499,12520],{"className":12320,"ariaHidden":122},[121],[23,12322,12324,12327,12330,12333,12336,12339,12342],{"className":12323},[126],[23,12325],{"className":12326,"style":131},[130],[23,12328,920],{"className":12329,"style":1047},[135,136],[23,12331,54],{"className":12332},[140],[23,12334,369],{"className":12335,"style":392},[135,136],[23,12337],{"className":12338,"style":149},[148],[23,12340,12268],{"className":12341},[153],[23,12343],{"className":12344,"style":149},[148],[23,12346,12348,12351,12354,12357,12360],{"className":12347},[126],[23,12349],{"className":12350,"style":131},[130],[23,12352,495],{"className":12353},[135,136],[23,12355],{"className":12356,"style":149},[148],[23,12358,60],{"className":12359},[153],[23,12361],{"className":12362,"style":149},[148],[23,12364,12366,12369,12372,12375,12378,12381],{"className":12365},[126],[23,12367],{"className":12368,"style":131},[130],[23,12370,753],{"className":12371},[135,136],[23,12373,66],{"className":12374},[169],[23,12376],{"className":12377,"style":149},[148],[23,12379,7378],{"className":12380},[153],[23,12382],{"className":12383,"style":149},[148],[23,12385,12387,12391,12454,12457,12460,12463,12466,12469,12472,12475,12478],{"className":12386},[126],[23,12388],{"className":12389,"style":12390},[130],"height:2.0074em;vertical-align:-0.686em;",[23,12392,12394,12397,12451],{"className":12393},[135],[23,12395],{"className":12396},[140,192],[23,12398,12400],{"className":12399},[71],[23,12401,12403,12443],{"className":12402},[199,200],[23,12404,12406,12440],{"className":12405},[204],[23,12407,12410,12421,12429],{"className":12408,"style":12409},[208],"height:1.3214em;",[23,12411,12412,12415],{"style":212},[23,12413],{"className":12414,"style":217},[216],[23,12416,12418],{"className":12417},[135],[23,12419,9952],{"className":12420},[135],[23,12422,12423,12426],{"style":235},[23,12424],{"className":12425,"style":217},[216],[23,12427],{"className":12428,"style":243},[242],[23,12430,12431,12434],{"style":246},[23,12432],{"className":12433,"style":217},[216],[23,12435,12437],{"className":12436},[135],[23,12438,4435],{"className":12439},[135],[23,12441,297],{"className":12442},[296],[23,12444,12446],{"className":12445},[204],[23,12447,12449],{"className":12448,"style":6287},[208],[23,12450],{},[23,12452],{"className":12453},[169,192],[23,12455,747],{"className":12456},[824],[23,12458],{"className":12459,"style":828},[148],[23,12461],{"className":12462,"style":280},[148],[23,12464,920],{"className":12465,"style":1047},[135,136],[23,12467,54],{"className":12468},[140],[23,12470,369],{"className":12471,"style":392},[135,136],[23,12473],{"className":12474,"style":149},[148],[23,12476,7378],{"className":12477},[153],[23,12479],{"className":12480,"style":149},[148],[23,12482,12484,12487,12490,12493,12496],{"className":12483},[126],[23,12485],{"className":12486,"style":131},[130],[23,12488,495],{"className":12489},[135,136],[23,12491],{"className":12492,"style":149},[148],[23,12494,60],{"className":12495},[153],[23,12497],{"className":12498,"style":149},[148],[23,12500,12502,12505,12508,12511,12514,12517],{"className":12501},[126],[23,12503],{"className":12504,"style":131},[130],[23,12506,753],{"className":12507},[135,136],[23,12509,66],{"className":12510},[169],[23,12512],{"className":12513,"style":149},[148],[23,12515,7378],{"className":12516},[153],[23,12518],{"className":12519,"style":149},[148],[23,12521,12523,12526,12588],{"className":12522},[126],[23,12524],{"className":12525,"style":12390},[130],[23,12527,12529,12532,12585],{"className":12528},[135],[23,12530],{"className":12531},[140,192],[23,12533,12535],{"className":12534},[71],[23,12536,12538,12577],{"className":12537},[199,200],[23,12539,12541,12574],{"className":12540},[204],[23,12542,12544,12555,12563],{"className":12543,"style":12409},[208],[23,12545,12546,12549],{"style":212},[23,12547],{"className":12548,"style":217},[216],[23,12550,12552],{"className":12551},[135],[23,12553,9952],{"className":12554},[135],[23,12556,12557,12560],{"style":235},[23,12558],{"className":12559,"style":217},[216],[23,12561],{"className":12562,"style":243},[242],[23,12564,12565,12568],{"style":246},[23,12566],{"className":12567,"style":217},[216],[23,12569,12571],{"className":12570},[135],[23,12572,4435],{"className":12573},[135],[23,12575,297],{"className":12576},[296],[23,12578,12580],{"className":12579},[204],[23,12581,12583],{"className":12582,"style":6287},[208],[23,12584],{},[23,12586],{"className":12587},[169,192],[23,12589,112],{"className":12590},[135],[11,12592,12593,12594,12622,12623,12651],{},"满足这两个不等式的 ",[23,12595,12597,12610],{"className":12596,"translate":27},[31],[23,12598,12600],{"className":12599},[35],[37,12601,12602],{"xmlns":39},[42,12603,12604,12608],{},[45,12605,12606],{},[48,12607,495],{},[114,12609,495],{"encoding":116},[23,12611,12613],{"className":12612,"ariaHidden":122},[121],[23,12614,12616,12619],{"className":12615},[126],[23,12617],{"className":12618,"style":703},[130],[23,12620,495],{"className":12621},[135,136]," 就是后验中位数。直观上，中位数把后验概率一分为二，左右两侧的「质量」相等，因此向左或向右移动 ",[23,12624,12626,12639],{"className":12625,"translate":27},[31],[23,12627,12629],{"className":12628},[35],[37,12630,12631],{"xmlns":39},[42,12632,12633,12637],{},[45,12634,12635],{},[48,12636,495],{},[114,12638,495],{"encoding":116},[23,12640,12642],{"className":12641,"ariaHidden":122},[121],[23,12643,12645,12648],{"className":12644},[126],[23,12646],{"className":12647,"style":703},[130],[23,12649,495],{"className":12650},[135,136]," 都不会降低期望绝对误差。",[11,12653,12654],{},"平方损失对异常值敏感，因为误差被平方；绝对损失只看见误差的大小。这个差异在决策层面的表现，就是均值与中位数的差异。",[2501,12656,12658,12659],{"id":12657},"_35-非对称损失阈值不再在-121212","3.5 非对称损失：阈值不再在 ",[23,12660,12662,12679],{"className":12661,"translate":27},[31],[23,12663,12665],{"className":12664},[35],[37,12666,12667],{"xmlns":39},[42,12668,12669,12677],{},[45,12670,12671,12673,12675],{},[2581,12672,4435],{},[48,12674,2875],{"mathvariant":111},[2581,12676,9952],{},[114,12678,10469],{"encoding":116},[23,12680,12682],{"className":12681,"ariaHidden":122},[121],[23,12683,12685,12688],{"className":12684},[126],[23,12686],{"className":12687,"style":131},[130],[23,12689,10469],{"className":12690},[135],[11,12692,12693,12694,12778,12779,12861,12862,10694],{},"现实中的两类错误很少同等昂贵。设预测为正类的代价为 ",[23,12695,12697,12721],{"className":12696,"translate":27},[31],[23,12698,12700],{"className":12699},[35],[37,12701,12702],{"xmlns":39},[42,12703,12704,12718],{},[45,12705,12706],{},[898,12707,12708,12711],{},[48,12709,12710],{},"c",[45,12712,12713,12716],{},[48,12714,12715],{"mathvariant":111},"F",[48,12717,920],{"mathvariant":111},[114,12719,12720],{"encoding":116},"c_{\\mathrm{FP}}",[23,12722,12724],{"className":12723,"ariaHidden":122},[121],[23,12725,12727,12731],{"className":12726},[126],[23,12728],{"className":12729,"style":12730},[130],"height:0.5806em;vertical-align:-0.15em;",[23,12732,12734,12737],{"className":12733},[135],[23,12735,12710],{"className":12736},[135,136],[23,12738,12740],{"className":12739},[998],[23,12741,12743,12770],{"className":12742},[199,200],[23,12744,12746,12767],{"className":12745},[204],[23,12747,12749],{"className":12748,"style":2026},[208],[23,12750,12751,12754],{"style":2029},[23,12752],{"className":12753,"style":1015},[216],[23,12755,12757],{"className":12756},[1019,1020,1021,1022],[23,12758,12760],{"className":12759},[135,1022],[23,12761,12763],{"className":12762},[135,1022],[23,12764,12766],{"className":12765},[135,5391,1022],"FP",[23,12768,297],{"className":12769},[296],[23,12771,12773],{"className":12772},[204],[23,12774,12776],{"className":12775,"style":2048},[208],[23,12777],{},"（假阳性），预测为负类的代价为 ",[23,12780,12782,12805],{"className":12781,"translate":27},[31],[23,12783,12785],{"className":12784},[35],[37,12786,12787],{"xmlns":39},[42,12788,12789,12802],{},[45,12790,12791],{},[898,12792,12793,12795],{},[48,12794,12710],{},[45,12796,12797,12799],{},[48,12798,12715],{"mathvariant":111},[48,12800,12801],{"mathvariant":111},"N",[114,12803,12804],{"encoding":116},"c_{\\mathrm{FN}}",[23,12806,12808],{"className":12807,"ariaHidden":122},[121],[23,12809,12811,12814],{"className":12810},[126],[23,12812],{"className":12813,"style":12730},[130],[23,12815,12817,12820],{"className":12816},[135],[23,12818,12710],{"className":12819},[135,136],[23,12821,12823],{"className":12822},[998],[23,12824,12826,12853],{"className":12825},[199,200],[23,12827,12829,12850],{"className":12828},[204],[23,12830,12832],{"className":12831,"style":2026},[208],[23,12833,12834,12837],{"style":2029},[23,12835],{"className":12836,"style":1015},[216],[23,12838,12840],{"className":12839},[1019,1020,1021,1022],[23,12841,12843],{"className":12842},[135,1022],[23,12844,12846],{"className":12845},[135,1022],[23,12847,12849],{"className":12848},[135,5391,1022],"FN",[23,12851,297],{"className":12852},[296],[23,12854,12856],{"className":12855},[204],[23,12857,12859],{"className":12858,"style":2048},[208],[23,12860],{},"（假阴性），正确分类的代价为 ",[23,12863,12865,12878],{"className":12864,"translate":27},[31],[23,12866,12868],{"className":12867},[35],[37,12869,12870],{"xmlns":39},[42,12871,12872,12876],{},[45,12873,12874],{},[2581,12875,1532],{},[114,12877,1532],{"encoding":116},[23,12879,12881],{"className":12880,"ariaHidden":122},[121],[23,12882,12884,12887],{"className":12883},[126],[23,12885],{"className":12886,"style":2770},[130],[23,12888,1532],{"className":12889},[135],[23,12891,12893],{"className":12892,"translate":27},[26],[23,12894,12896,12990],{"className":12895,"translate":27},[31],[23,12897,12899],{"className":12898},[35],[37,12900,12901],{"xmlns":39,"display":40},[42,12902,12903,12987],{},[45,12904,12905,12907,12909,12911,12913,12915,12917,12919,12929,12931,12933,12935,12937,12939,12941,12943,12945,12947,12949,12951,12953,12955,12957,12959,12961,12971,12973,12975,12977,12979,12981,12983,12985],{},[48,12906,1653],{},[51,12908,54],{"stretchy":53},[2581,12910,4435],{},[51,12912,747],{"separator":122},[48,12914,753],{},[51,12916,66],{"stretchy":53},[51,12918,69],{},[898,12920,12921,12923],{},[48,12922,12710],{},[45,12924,12925,12927],{},[48,12926,12715],{"mathvariant":111},[48,12928,920],{"mathvariant":111},[88,12930,90],{},[48,12932,11],{},[51,12934,54],{"stretchy":53},[2581,12936,1532],{},[51,12938,60],{},[48,12940,753],{},[51,12942,66],{"stretchy":53},[51,12944,747],{"separator":122},[148,12946],{"width":750},[48,12948,1653],{},[51,12950,54],{"stretchy":53},[2581,12952,1532],{},[51,12954,747],{"separator":122},[48,12956,753],{},[51,12958,66],{"stretchy":53},[51,12960,69],{},[898,12962,12963,12965],{},[48,12964,12710],{},[45,12966,12967,12969],{},[48,12968,12715],{"mathvariant":111},[48,12970,12801],{"mathvariant":111},[88,12972,90],{},[48,12974,11],{},[51,12976,54],{"stretchy":53},[2581,12978,4435],{},[51,12980,60],{},[48,12982,753],{},[51,12984,66],{"stretchy":53},[48,12986,112],{"mathvariant":111},[114,12988,12989],{"encoding":116},"r(1,x)=c_{\\mathrm{FP}}\\,p(0\\mid x),\\qquad\nr(0,x)=c_{\\mathrm{FN}}\\,p(1\\mid x).",[23,12991,12993,13029,13102,13153,13226],{"className":12992,"ariaHidden":122},[121],[23,12994,12996,12999,13002,13005,13008,13011,13014,13017,13020,13023,13026],{"className":12995},[126],[23,12997],{"className":12998,"style":131},[130],[23,13000,1653],{"className":13001,"style":144},[135,136],[23,13003,54],{"className":13004},[140],[23,13006,4435],{"className":13007},[135],[23,13009,747],{"className":13010},[824],[23,13012],{"className":13013,"style":280},[148],[23,13015,753],{"className":13016},[135,136],[23,13018,66],{"className":13019},[169],[23,13021],{"className":13022,"style":149},[148],[23,13024,69],{"className":13025},[153],[23,13027],{"className":13028,"style":149},[148],[23,13030,13032,13035,13081,13084,13087,13090,13093,13096,13099],{"className":13031},[126],[23,13033],{"className":13034,"style":131},[130],[23,13036,13038,13041],{"className":13037},[135],[23,13039,12710],{"className":13040},[135,136],[23,13042,13044],{"className":13043},[998],[23,13045,13047,13073],{"className":13046},[199,200],[23,13048,13050,13070],{"className":13049},[204],[23,13051,13053],{"className":13052,"style":2026},[208],[23,13054,13055,13058],{"style":2029},[23,13056],{"className":13057,"style":1015},[216],[23,13059,13061],{"className":13060},[1019,1020,1021,1022],[23,13062,13064],{"className":13063},[135,1022],[23,13065,13067],{"className":13066},[135,1022],[23,13068,12766],{"className":13069},[135,5391,1022],[23,13071,297],{"className":13072},[296],[23,13074,13076],{"className":13075},[204],[23,13077,13079],{"className":13078,"style":2048},[208],[23,13080],{},[23,13082],{"className":13083,"style":280},[148],[23,13085,11],{"className":13086},[135,136],[23,13088,54],{"className":13089},[140],[23,13091,1532],{"className":13092},[135],[23,13094],{"className":13095,"style":149},[148],[23,13097,60],{"className":13098},[153],[23,13100],{"className":13101,"style":149},[148],[23,13103,13105,13108,13111,13114,13117,13120,13123,13126,13129,13132,13135,13138,13141,13144,13147,13150],{"className":13104},[126],[23,13106],{"className":13107,"style":131},[130],[23,13109,753],{"className":13110},[135,136],[23,13112,66],{"className":13113},[169],[23,13115,747],{"className":13116},[824],[23,13118],{"className":13119,"style":828},[148],[23,13121],{"className":13122,"style":280},[148],[23,13124,1653],{"className":13125,"style":144},[135,136],[23,13127,54],{"className":13128},[140],[23,13130,1532],{"className":13131},[135],[23,13133,747],{"className":13134},[824],[23,13136],{"className":13137,"style":280},[148],[23,13139,753],{"className":13140},[135,136],[23,13142,66],{"className":13143},[169],[23,13145],{"className":13146,"style":149},[148],[23,13148,69],{"className":13149},[153],[23,13151],{"className":13152,"style":149},[148],[23,13154,13156,13159,13205,13208,13211,13214,13217,13220,13223],{"className":13155},[126],[23,13157],{"className":13158,"style":131},[130],[23,13160,13162,13165],{"className":13161},[135],[23,13163,12710],{"className":13164},[135,136],[23,13166,13168],{"className":13167},[998],[23,13169,13171,13197],{"className":13170},[199,200],[23,13172,13174,13194],{"className":13173},[204],[23,13175,13177],{"className":13176,"style":2026},[208],[23,13178,13179,13182],{"style":2029},[23,13180],{"className":13181,"style":1015},[216],[23,13183,13185],{"className":13184},[1019,1020,1021,1022],[23,13186,13188],{"className":13187},[135,1022],[23,13189,13191],{"className":13190},[135,1022],[23,13192,12849],{"className":13193},[135,5391,1022],[23,13195,297],{"className":13196},[296],[23,13198,13200],{"className":13199},[204],[23,13201,13203],{"className":13202,"style":2048},[208],[23,13204],{},[23,13206],{"className":13207,"style":280},[148],[23,13209,11],{"className":13210},[135,136],[23,13212,54],{"className":13213},[140],[23,13215,4435],{"className":13216},[135],[23,13218],{"className":13219,"style":149},[148],[23,13221,60],{"className":13222},[153],[23,13224],{"className":13225,"style":149},[148],[23,13227,13229,13232,13235,13238],{"className":13228},[126],[23,13230],{"className":13231,"style":131},[130],[23,13233,753],{"className":13234},[135,136],[23,13236,66],{"className":13237},[169],[23,13239,112],{"className":13240},[135],[11,13242,13243,13244,13351],{},"选择正类当且仅当 ",[23,13245,13247,13285],{"className":13246,"translate":27},[31],[23,13248,13250],{"className":13249},[35],[37,13251,13252],{"xmlns":39},[42,13253,13254,13282],{},[45,13255,13256,13258,13260,13262,13264,13266,13268,13270,13272,13274,13276,13278,13280],{},[48,13257,1653],{},[51,13259,54],{"stretchy":53},[2581,13261,4435],{},[51,13263,747],{"separator":122},[48,13265,753],{},[51,13267,66],{"stretchy":53},[51,13269,12268],{},[48,13271,1653],{},[51,13273,54],{"stretchy":53},[2581,13275,1532],{},[51,13277,747],{"separator":122},[48,13279,753],{},[51,13281,66],{"stretchy":53},[114,13283,13284],{"encoding":116},"r(1,x)\\le r(0,x)",[23,13286,13288,13324],{"className":13287,"ariaHidden":122},[121],[23,13289,13291,13294,13297,13300,13303,13306,13309,13312,13315,13318,13321],{"className":13290},[126],[23,13292],{"className":13293,"style":131},[130],[23,13295,1653],{"className":13296,"style":144},[135,136],[23,13298,54],{"className":13299},[140],[23,13301,4435],{"className":13302},[135],[23,13304,747],{"className":13305},[824],[23,13307],{"className":13308,"style":280},[148],[23,13310,753],{"className":13311},[135,136],[23,13313,66],{"className":13314},[169],[23,13316],{"className":13317,"style":149},[148],[23,13319,12268],{"className":13320},[153],[23,13322],{"className":13323,"style":149},[148],[23,13325,13327,13330,13333,13336,13339,13342,13345,13348],{"className":13326},[126],[23,13328],{"className":13329,"style":131},[130],[23,13331,1653],{"className":13332,"style":144},[135,136],[23,13334,54],{"className":13335},[140],[23,13337,1532],{"className":13338},[135],[23,13340,747],{"className":13341},[824],[23,13343],{"className":13344,"style":280},[148],[23,13346,753],{"className":13347},[135,136],[23,13349,66],{"className":13350},[169],"，即",[23,13353,13355],{"className":13354,"translate":27},[26],[23,13356,13358,13426],{"className":13357,"translate":27},[31],[23,13359,13361],{"className":13360},[35],[37,13362,13363],{"xmlns":39,"display":40},[42,13364,13365,13423],{},[45,13366,13367,13369,13371,13373,13375,13377,13379,13381,13383,13385,13421],{},[48,13368,11],{},[51,13370,54],{"stretchy":53},[2581,13372,4435],{},[51,13374,60],{},[48,13376,753],{},[51,13378,66],{"stretchy":53},[88,13380,5360],{},[51,13382,7378],{},[88,13384,5360],{},[71,13386,13387,13397],{},[898,13388,13389,13391],{},[48,13390,12710],{},[45,13392,13393,13395],{},[48,13394,12715],{"mathvariant":111},[48,13396,920],{"mathvariant":111},[45,13398,13399,13409,13411],{},[898,13400,13401,13403],{},[48,13402,12710],{},[45,13404,13405,13407],{},[48,13406,12715],{"mathvariant":111},[48,13408,920],{"mathvariant":111},[51,13410,5106],{},[898,13412,13413,13415],{},[48,13414,12710],{},[45,13416,13417,13419],{},[48,13418,12715],{"mathvariant":111},[48,13420,12801],{"mathvariant":111},[48,13422,112],{"mathvariant":111},[114,13424,13425],{"encoding":116},"p(1\\mid x)\\ \\ge\\ \\frac{c_{\\mathrm{FP}}}{c_{\\mathrm{FP}}+c_{\\mathrm{FN}}}.",[23,13427,13429,13453,13480],{"className":13428,"ariaHidden":122},[121],[23,13430,13432,13435,13438,13441,13444,13447,13450],{"className":13431},[126],[23,13433],{"className":13434,"style":131},[130],[23,13436,11],{"className":13437},[135,136],[23,13439,54],{"className":13440},[140],[23,13442,4435],{"className":13443},[135],[23,13445],{"className":13446,"style":149},[148],[23,13448,60],{"className":13449},[153],[23,13451],{"className":13452,"style":149},[148],[23,13454,13456,13459,13462,13465,13468,13471,13474,13477],{"className":13455},[126],[23,13457],{"className":13458,"style":131},[130],[23,13460,753],{"className":13461},[135,136],[23,13463,66],{"className":13464},[169],[23,13466,5360],{"className":13467},[148],[23,13469],{"className":13470,"style":149},[148],[23,13472,7378],{"className":13473},[153],[23,13475,5360],{"className":13476},[148],[23,13478],{"className":13479,"style":149},[148],[23,13481,13483,13487,13691],{"className":13482},[126],[23,13484],{"className":13485,"style":13486},[130],"height:1.9436em;vertical-align:-0.836em;",[23,13488,13490,13493,13688],{"className":13489},[135],[23,13491],{"className":13492},[140,192],[23,13494,13496],{"className":13495},[71],[23,13497,13499,13679],{"className":13498},[199,200],[23,13500,13502,13676],{"className":13501},[204],[23,13503,13505,13614,13622],{"className":13504,"style":6056},[208],[23,13506,13507,13510],{"style":212},[23,13508],{"className":13509,"style":217},[216],[23,13511,13513,13559,13562,13565,13568],{"className":13512},[135],[23,13514,13516,13519],{"className":13515},[135],[23,13517,12710],{"className":13518},[135,136],[23,13520,13522],{"className":13521},[998],[23,13523,13525,13551],{"className":13524},[199,200],[23,13526,13528,13548],{"className":13527},[204],[23,13529,13531],{"className":13530,"style":2026},[208],[23,13532,13533,13536],{"style":2029},[23,13534],{"className":13535,"style":1015},[216],[23,13537,13539],{"className":13538},[1019,1020,1021,1022],[23,13540,13542],{"className":13541},[135,1022],[23,13543,13545],{"className":13544},[135,1022],[23,13546,12766],{"className":13547},[135,5391,1022],[23,13549,297],{"className":13550},[296],[23,13552,13554],{"className":13553},[204],[23,13555,13557],{"className":13556,"style":2048},[208],[23,13558],{},[23,13560],{"className":13561,"style":392},[148],[23,13563,5106],{"className":13564},[612],[23,13566],{"className":13567,"style":392},[148],[23,13569,13571,13574],{"className":13570},[135],[23,13572,12710],{"className":13573},[135,136],[23,13575,13577],{"className":13576},[998],[23,13578,13580,13606],{"className":13579},[199,200],[23,13581,13583,13603],{"className":13582},[204],[23,13584,13586],{"className":13585,"style":2026},[208],[23,13587,13588,13591],{"style":2029},[23,13589],{"className":13590,"style":1015},[216],[23,13592,13594],{"className":13593},[1019,1020,1021,1022],[23,13595,13597],{"className":13596},[135,1022],[23,13598,13600],{"className":13599},[135,1022],[23,13601,12849],{"className":13602},[135,5391,1022],[23,13604,297],{"className":13605},[296],[23,13607,13609],{"className":13608},[204],[23,13610,13612],{"className":13611,"style":2048},[208],[23,13613],{},[23,13615,13616,13619],{"style":235},[23,13617],{"className":13618,"style":217},[216],[23,13620],{"className":13621,"style":243},[242],[23,13623,13624,13627],{"style":246},[23,13625],{"className":13626,"style":217},[216],[23,13628,13630],{"className":13629},[135],[23,13631,13633,13636],{"className":13632},[135],[23,13634,12710],{"className":13635},[135,136],[23,13637,13639],{"className":13638},[998],[23,13640,13642,13668],{"className":13641},[199,200],[23,13643,13645,13665],{"className":13644},[204],[23,13646,13648],{"className":13647,"style":2026},[208],[23,13649,13650,13653],{"style":2029},[23,13651],{"className":13652,"style":1015},[216],[23,13654,13656],{"className":13655},[1019,1020,1021,1022],[23,13657,13659],{"className":13658},[135,1022],[23,13660,13662],{"className":13661},[135,1022],[23,13663,12766],{"className":13664},[135,5391,1022],[23,13666,297],{"className":13667},[296],[23,13669,13671],{"className":13670},[204],[23,13672,13674],{"className":13673,"style":2048},[208],[23,13675],{},[23,13677,297],{"className":13678},[296],[23,13680,13682],{"className":13681},[204],[23,13683,13686],{"className":13684,"style":13685},[208],"height:0.836em;",[23,13687],{},[23,13689],{"className":13690},[169,192],[23,13692,112],{"className":13693},[135],[11,13695,13696,13697,13729,13730,13764],{},"阈值不再固定在 ",[23,13698,13700,13717],{"className":13699,"translate":27},[31],[23,13701,13703],{"className":13702},[35],[37,13704,13705],{"xmlns":39},[42,13706,13707,13715],{},[45,13708,13709,13711,13713],{},[2581,13710,4435],{},[48,13712,2875],{"mathvariant":111},[2581,13714,9952],{},[114,13716,10469],{"encoding":116},[23,13718,13720],{"className":13719,"ariaHidden":122},[121],[23,13721,13723,13726],{"className":13722},[126],[23,13724],{"className":13725,"style":131},[130],[23,13727,10469],{"className":13728},[135],"，而是由两类代价的比值决定。若漏诊的代价是误诊的十倍，阈值就降到 ",[23,13731,13733,13752],{"className":13732,"translate":27},[31],[23,13734,13736],{"className":13735},[35],[37,13737,13738],{"xmlns":39},[42,13739,13740,13749],{},[45,13741,13742,13744,13746],{},[2581,13743,4435],{},[48,13745,2875],{"mathvariant":111},[2581,13747,13748],{},"11",[114,13750,13751],{"encoding":116},"1\u002F11",[23,13753,13755],{"className":13754,"ariaHidden":122},[121],[23,13756,13758,13761],{"className":13757},[126],[23,13759],{"className":13760,"style":131},[130],[23,13762,13751],{"className":13763},[135]," 左右，模型会倾向于把更多样本判为正类。",[11,13766,13767,13768,13771],{},"这个式子也说明：类别不平衡本身不是问题，",[18,13769,13770],{},"损失不对称","才是问题。重采样改变了训练分布，相当于改变了先验；如果最终要解决的是原分布上的决策问题，那么重采样之后还需要把阈值校正回来。",[11,13773,13774],{},"把常见损失与对应的贝叶斯行动放在一起：",[3764,13776,13777,13786],{},[3767,13778,13779],{},[3770,13780,13781,13783],{},[3773,13782,540],{},[3773,13784,13785],{},"贝叶斯行动",[3783,13787,13788,13796,13803,13811,14043],{},[3770,13789,13790,13793],{},[3788,13791,13792],{},"0-1 损失",[3788,13794,13795],{},"后验众数",[3770,13797,13798,13801],{},[3788,13799,13800],{},"平方损失",[3788,13802,11331],{},[3770,13804,13805,13808],{},[3788,13806,13807],{},"绝对损失",[3788,13809,13810],{},"后验中位数",[3770,13812,13813,13816],{},[3788,13814,13815],{},"非对称 0-1 损失",[3788,13817,13818,13819],{},"阈值 ",[23,13820,13822,13872],{"className":13821,"translate":27},[31],[23,13823,13825],{"className":13824},[35],[37,13826,13827],{"xmlns":39},[42,13828,13829,13869],{},[45,13830,13831,13841,13843,13845,13855,13857,13867],{},[898,13832,13833,13835],{},[48,13834,12710],{},[45,13836,13837,13839],{},[48,13838,12715],{"mathvariant":111},[48,13840,920],{"mathvariant":111},[48,13842,2875],{"mathvariant":111},[51,13844,54],{"stretchy":53},[898,13846,13847,13849],{},[48,13848,12710],{},[45,13850,13851,13853],{},[48,13852,12715],{"mathvariant":111},[48,13854,920],{"mathvariant":111},[51,13856,5106],{},[898,13858,13859,13861],{},[48,13860,12710],{},[45,13862,13863,13865],{},[48,13864,12715],{"mathvariant":111},[48,13866,12801],{"mathvariant":111},[51,13868,66],{"stretchy":53},[114,13870,13871],{"encoding":116},"c_{\\mathrm{FP}}\u002F(c_{\\mathrm{FP}}+c_{\\mathrm{FN}})",[23,13873,13875,13988],{"className":13874,"ariaHidden":122},[121],[23,13876,13878,13881,13927,13930,13933,13979,13982,13985],{"className":13877},[126],[23,13879],{"className":13880,"style":131},[130],[23,13882,13884,13887],{"className":13883},[135],[23,13885,12710],{"className":13886},[135,136],[23,13888,13890],{"className":13889},[998],[23,13891,13893,13919],{"className":13892},[199,200],[23,13894,13896,13916],{"className":13895},[204],[23,13897,13899],{"className":13898,"style":2026},[208],[23,13900,13901,13904],{"style":2029},[23,13902],{"className":13903,"style":1015},[216],[23,13905,13907],{"className":13906},[1019,1020,1021,1022],[23,13908,13910],{"className":13909},[135,1022],[23,13911,13913],{"className":13912},[135,1022],[23,13914,12766],{"className":13915},[135,5391,1022],[23,13917,297],{"className":13918},[296],[23,13920,13922],{"className":13921},[204],[23,13923,13925],{"className":13924,"style":2048},[208],[23,13926],{},[23,13928,2875],{"className":13929},[135],[23,13931,54],{"className":13932},[140],[23,13934,13936,13939],{"className":13935},[135],[23,13937,12710],{"className":13938},[135,136],[23,13940,13942],{"className":13941},[998],[23,13943,13945,13971],{"className":13944},[199,200],[23,13946,13948,13968],{"className":13947},[204],[23,13949,13951],{"className":13950,"style":2026},[208],[23,13952,13953,13956],{"style":2029},[23,13954],{"className":13955,"style":1015},[216],[23,13957,13959],{"className":13958},[1019,1020,1021,1022],[23,13960,13962],{"className":13961},[135,1022],[23,13963,13965],{"className":13964},[135,1022],[23,13966,12766],{"className":13967},[135,5391,1022],[23,13969,297],{"className":13970},[296],[23,13972,13974],{"className":13973},[204],[23,13975,13977],{"className":13976,"style":2048},[208],[23,13978],{},[23,13980],{"className":13981,"style":392},[148],[23,13983,5106],{"className":13984},[612],[23,13986],{"className":13987,"style":392},[148],[23,13989,13991,13994,14040],{"className":13990},[126],[23,13992],{"className":13993,"style":131},[130],[23,13995,13997,14000],{"className":13996},[135],[23,13998,12710],{"className":13999},[135,136],[23,14001,14003],{"className":14002},[998],[23,14004,14006,14032],{"className":14005},[199,200],[23,14007,14009,14029],{"className":14008},[204],[23,14010,14012],{"className":14011,"style":2026},[208],[23,14013,14014,14017],{"style":2029},[23,14015],{"className":14016,"style":1015},[216],[23,14018,14020],{"className":14019},[1019,1020,1021,1022],[23,14021,14023],{"className":14022},[135,1022],[23,14024,14026],{"className":14025},[135,1022],[23,14027,12849],{"className":14028},[135,5391,1022],[23,14030,297],{"className":14031},[296],[23,14033,14035],{"className":14034},[204],[23,14036,14038],{"className":14037,"style":2048},[208],[23,14039],{},[23,14041,66],{"className":14042},[169],[3770,14044,14045,14048],{},[3788,14046,14047],{},"带拒绝的 0-1 损失",[3788,14049,14050,14051,14102],{},"最大后验低于 ",[23,14052,14054,14072],{"className":14053,"translate":27},[31],[23,14055,14057],{"className":14056},[35],[37,14058,14059],{"xmlns":39},[42,14060,14061,14069],{},[45,14062,14063,14065,14067],{},[2581,14064,4435],{},[51,14066,4529],{},[48,14068,12710],{},[114,14070,14071],{"encoding":116},"1-c",[23,14073,14075,14093],{"className":14074,"ariaHidden":122},[121],[23,14076,14078,14081,14084,14087,14090],{"className":14077},[126],[23,14079],{"className":14080,"style":9400},[130],[23,14082,4435],{"className":14083},[135],[23,14085],{"className":14086,"style":392},[148],[23,14088,4529],{"className":14089},[612],[23,14091],{"className":14092,"style":392},[148],[23,14094,14096,14099],{"className":14095},[126],[23,14097],{"className":14098,"style":703},[130],[23,14100,12710],{"className":14101},[135,136]," 时拒绝",[11,14104,14105,14106,14109,14110,14184,14185,14328,14329,14357],{},"最后一行是",[18,14107,14108],{},"拒绝选项","（reject option）：除了预测类别，模型还可以说「我不确定」。若拒绝的代价为 ",[23,14111,14113,14139],{"className":14112,"translate":27},[31],[23,14114,14116],{"className":14115},[35],[37,14117,14118],{"xmlns":39},[42,14119,14120,14136],{},[45,14121,14122,14124,14126,14128,14130,14132,14134],{},[48,14123,12710],{},[51,14125,372],{},[51,14127,54],{"stretchy":53},[2581,14129,1532],{},[51,14131,747],{"separator":122},[2581,14133,4435],{},[51,14135,66],{"stretchy":53},[114,14137,14138],{"encoding":116},"c\\in(0,1)",[23,14140,14142,14160],{"className":14141,"ariaHidden":122},[121],[23,14143,14145,14148,14151,14154,14157],{"className":14144},[126],[23,14146],{"className":14147,"style":513},[130],[23,14149,12710],{"className":14150},[135,136],[23,14152],{"className":14153,"style":149},[148],[23,14155,372],{"className":14156},[153],[23,14158],{"className":14159,"style":149},[148],[23,14161,14163,14166,14169,14172,14175,14178,14181],{"className":14162},[126],[23,14164],{"className":14165,"style":131},[130],[23,14167,54],{"className":14168},[140],[23,14170,1532],{"className":14171},[135],[23,14173,747],{"className":14174},[824],[23,14176],{"className":14177,"style":280},[148],[23,14179,4435],{"className":14180},[135],[23,14182,66],{"className":14183},[169],"，预测的期望代价为 ",[23,14186,14188,14226],{"className":14187,"translate":27},[31],[23,14189,14191],{"className":14190},[35],[37,14192,14193],{"xmlns":39},[42,14194,14195,14223],{},[45,14196,14197,14199,14201,14211,14213,14215,14217,14219,14221],{},[2581,14198,4435],{},[51,14200,4529],{},[898,14202,14203,14209],{},[45,14204,14205,14207],{},[48,14206,9551],{},[51,14208,6959],{},[48,14210,660],{},[48,14212,11],{},[51,14214,54],{"stretchy":53},[48,14216,660],{},[51,14218,60],{},[48,14220,753],{},[51,14222,66],{"stretchy":53},[114,14224,14225],{"encoding":116},"1-\\max_y p(y\\mid x)",[23,14227,14229,14247,14316],{"className":14228,"ariaHidden":122},[121],[23,14230,14232,14235,14238,14241,14244],{"className":14231},[126],[23,14233],{"className":14234,"style":9400},[130],[23,14236,4435],{"className":14237},[135],[23,14239],{"className":14240,"style":392},[148],[23,14242,4529],{"className":14243},[612],[23,14245],{"className":14246,"style":392},[148],[23,14248,14250,14254,14295,14298,14301,14304,14307,14310,14313],{"className":14249},[126],[23,14251],{"className":14252,"style":14253},[130],"height:1.0361em;vertical-align:-0.2861em;",[23,14255,14257,14260],{"className":14256},[1786],[23,14258,9551],{"className":14259},[1786],[23,14261,14263],{"className":14262},[998],[23,14264,14266,14286],{"className":14265},[199,200],[23,14267,14269,14283],{"className":14268},[204],[23,14270,14272],{"className":14271,"style":6499},[208],[23,14273,14274,14277],{"style":8187},[23,14275],{"className":14276,"style":1015},[216],[23,14278,14280],{"className":14279},[1019,1020,1021,1022],[23,14281,660],{"className":14282,"style":676},[135,136,1022],[23,14284,297],{"className":14285},[296],[23,14287,14289],{"className":14288},[204],[23,14290,14293],{"className":14291,"style":14292},[208],"height:0.2861em;",[23,14294],{},[23,14296],{"className":14297,"style":280},[148],[23,14299,11],{"className":14300},[135,136],[23,14302,54],{"className":14303},[140],[23,14305,660],{"className":14306,"style":676},[135,136],[23,14308],{"className":14309,"style":149},[148],[23,14311,60],{"className":14312},[153],[23,14314],{"className":14315,"style":149},[148],[23,14317,14319,14322,14325],{"className":14318},[126],[23,14320],{"className":14321,"style":131},[130],[23,14323,753],{"className":14324},[135,136],[23,14326,66],{"className":14327},[169],"，拒绝的代价为 ",[23,14330,14332,14345],{"className":14331,"translate":27},[31],[23,14333,14335],{"className":14334},[35],[37,14336,14337],{"xmlns":39},[42,14338,14339,14343],{},[45,14340,14341],{},[48,14342,12710],{},[114,14344,12710],{"encoding":116},[23,14346,14348],{"className":14347,"ariaHidden":122},[121],[23,14349,14351,14354],{"className":14350},[126],[23,14352],{"className":14353,"style":703},[130],[23,14355,12710],{"className":14356},[135,136],"，因此最优规则是",[23,14359,14361],{"className":14360,"translate":27},[26],[23,14362,14364,14418],{"className":14363,"translate":27},[31],[23,14365,14367],{"className":14366},[35],[37,14368,14369],{"xmlns":39,"display":40},[42,14370,14371,14415],{},[45,14372,14373,14376,14378,14381,14383,14393,14395,14397,14399,14401,14403,14405,14407,14409,14411,14413],{},[88,14374,14375],{},"拒绝",[148,14377],{"width":9915},[51,14379,14380],{},"⟺",[148,14382],{"width":9915},[6961,14384,14385,14391],{},[45,14386,14387,14389],{},[48,14388,9551],{},[51,14390,6959],{},[48,14392,660],{},[48,14394,11],{},[51,14396,54],{"stretchy":53},[48,14398,660],{},[51,14400,60],{},[48,14402,753],{},[51,14404,66],{"stretchy":53},[51,14406,9983],{},[2581,14408,4435],{},[51,14410,4529],{},[48,14412,12710],{},[48,14414,112],{"mathvariant":111},[114,14416,14417],{"encoding":116},"\\text{拒绝}\\quad\\Longleftrightarrow\\quad \\max_y p(y\\mid x)\u003C1-c.",[23,14419,14421,14450,14523,14544,14562],{"className":14420,"ariaHidden":122},[121],[23,14422,14424,14428,14434,14438,14441,14444,14447],{"className":14423},[126],[23,14425],{"className":14426,"style":14427},[130],"height:0.7073em;vertical-align:-0.024em;",[23,14429,14431],{"className":14430},[135,6514],[23,14432,14375],{"className":14433},[135,7238],[23,14435],{"className":14436,"style":14437},[148],"margin-right:1em;",[23,14439],{"className":14440,"style":149},[148],[23,14442,14380],{"className":14443},[153],[23,14445],{"className":14446,"style":14437},[148],[23,14448],{"className":14449,"style":149},[148],[23,14451,14453,14457,14502,14505,14508,14511,14514,14517,14520],{"className":14452},[126],[23,14454],{"className":14455,"style":14456},[130],"height:1.5861em;vertical-align:-0.8361em;",[23,14458,14460],{"className":14459},[1786,7096],[23,14461,14463,14493],{"className":14462},[199,200],[23,14464,14466,14490],{"className":14465},[204],[23,14467,14469,14480],{"className":14468,"style":703},[208],[23,14470,14471,14474],{"style":8780},[23,14472],{"className":14473,"style":217},[216],[23,14475,14477],{"className":14476},[1019,1020,1021,1022],[23,14478,660],{"className":14479,"style":676},[135,136,1022],[23,14481,14482,14485],{"style":7130},[23,14483],{"className":14484,"style":217},[216],[23,14486,14487],{},[23,14488,9551],{"className":14489},[1786],[23,14491,297],{"className":14492},[296],[23,14494,14496],{"className":14495},[204],[23,14497,14500],{"className":14498,"style":14499},[208],"height:0.8361em;",[23,14501],{},[23,14503],{"className":14504,"style":280},[148],[23,14506,11],{"className":14507},[135,136],[23,14509,54],{"className":14510},[140],[23,14512,660],{"className":14513,"style":676},[135,136],[23,14515],{"className":14516,"style":149},[148],[23,14518,60],{"className":14519},[153],[23,14521],{"className":14522,"style":149},[148],[23,14524,14526,14529,14532,14535,14538,14541],{"className":14525},[126],[23,14527],{"className":14528,"style":131},[130],[23,14530,753],{"className":14531},[135,136],[23,14533,66],{"className":14534},[169],[23,14536],{"className":14537,"style":149},[148],[23,14539,9983],{"className":14540},[153],[23,14542],{"className":14543,"style":149},[148],[23,14545,14547,14550,14553,14556,14559],{"className":14546},[126],[23,14548],{"className":14549,"style":9400},[130],[23,14551,4435],{"className":14552},[135],[23,14554],{"className":14555,"style":392},[148],[23,14557,4529],{"className":14558},[612],[23,14560],{"className":14561,"style":392},[148],[23,14563,14565,14568,14571],{"className":14564},[126],[23,14566],{"className":14567,"style":703},[130],[23,14569,12710],{"className":14570},[135,136],[23,14572,112],{"className":14573},[135],[11,14575,14576],{},"这就是 Chow 规则。它把「不确定性」直接翻译成了一个可执行的动作，而不是一个只用来画图的分数。",[2501,14578,14580],{"id":14579},"_36-贝叶斯风险","3.6 贝叶斯风险",[11,14582,14583],{},"把定理 3.1 中的最优值单独命名：",[23,14585,14587],{"className":14586,"translate":27},[26],[23,14588,14590,14670],{"className":14589,"translate":27},[31],[23,14591,14593],{"className":14592},[35],[37,14594,14595],{"xmlns":39,"display":40},[42,14596,14597,14667],{},[45,14598,14599,14605,14607,14617,14619,14621,14623,14625,14627,14633,14635,14651,14653,14655,14657,14659,14661,14663,14665],{},[4518,14600,14601,14603],{},[48,14602,572],{},[51,14604,6881],{},[51,14606,69],{},[6961,14608,14609,14615],{},[45,14610,14611,14613],{},[48,14612,8314],{},[51,14614,6959],{},[48,14616,736],{},[48,14618,572],{},[51,14620,54],{"stretchy":53},[48,14622,736],{},[51,14624,66],{"stretchy":53},[51,14626,69],{},[898,14628,14629,14631],{},[48,14630,902],{"mathvariant":571},[48,14632,434],{},[51,14634,924],{"fence":53,"stretchy":122,"minsize":7403,"maxsize":7403},[6961,14636,14637,14643],{},[45,14638,14639,14641],{},[48,14640,6967],{},[51,14642,6959],{},[45,14644,14645,14647,14649],{},[48,14646,495],{},[51,14648,372],{},[48,14650,500],{"mathvariant":375},[48,14652,1653],{},[51,14654,54],{"stretchy":53},[48,14656,495],{},[51,14658,747],{"separator":122},[48,14660,434],{},[51,14662,66],{"stretchy":53},[51,14664,945],{"fence":53,"stretchy":122,"minsize":7403,"maxsize":7403},[48,14666,112],{"mathvariant":111},[114,14668,14669],{"encoding":116},"R^*=\\inf_{\\delta}R(\\delta)\n=\\mathbb E_X\\Big[\\min_{a\\in\\mathcal A}r(a,X)\\Big].",[23,14671,14673,14717,14799],{"className":14672,"ariaHidden":122},[121],[23,14674,14676,14679,14708,14711,14714],{"className":14675},[126],[23,14677],{"className":14678,"style":7045},[130],[23,14680,14682,14685],{"className":14681},[135],[23,14683,572],{"className":14684,"style":963},[135,136],[23,14686,14688],{"className":14687},[998],[23,14689,14691],{"className":14690},[199],[23,14692,14694],{"className":14693},[204],[23,14695,14697],{"className":14696,"style":7045},[208],[23,14698,14699,14702],{"style":4989},[23,14700],{"className":14701,"style":1015},[216],[23,14703,14705],{"className":14704},[1019,1020,1021,1022],[23,14706,6881],{"className":14707},[612,1022],[23,14709],{"className":14710,"style":149},[148],[23,14712,69],{"className":14713},[153],[23,14715],{"className":14716,"style":149},[148],[23,14718,14720,14724,14775,14778,14781,14784,14787,14790,14793,14796],{"className":14719},[126],[23,14721],{"className":14722,"style":14723},[130],"height:1.5021em;vertical-align:-0.7521em;",[23,14725,14727],{"className":14726},[1786,7096],[23,14728,14730,14766],{"className":14729},[199,200],[23,14731,14733,14763],{"className":14732},[204],[23,14734,14736,14751],{"className":14735,"style":779},[208],[23,14737,14739,14742],{"style":14738},"top:-2.3479em;margin-left:0em;",[23,14740],{"className":14741,"style":217},[216],[23,14743,14745],{"className":14744},[1019,1020,1021,1022],[23,14746,14748],{"className":14747},[135,1022],[23,14749,736],{"className":14750,"style":783},[135,136,1022],[23,14752,14753,14756],{"style":7130},[23,14754],{"className":14755,"style":217},[216],[23,14757,14758],{},[23,14759,8332,14761],{"className":14760},[1786],[23,14762,8336],{"style":8335},[23,14764,297],{"className":14765},[296],[23,14767,14769],{"className":14768},[204],[23,14770,14773],{"className":14771,"style":14772},[208],"height:0.7521em;",[23,14774],{},[23,14776],{"className":14777,"style":280},[148],[23,14779,572],{"className":14780,"style":963},[135,136],[23,14782,54],{"className":14783},[140],[23,14785,736],{"className":14786,"style":783},[135,136],[23,14788,66],{"className":14789},[169],[23,14791],{"className":14792,"style":149},[148],[23,14794,69],{"className":14795},[153],[23,14797],{"className":14798,"style":149},[148],[23,14800,14802,14805,14845,14851,14854,14907,14910,14913,14916,14919,14922,14925,14928,14931,14937],{"className":14801},[126],[23,14803],{"className":14804,"style":7534},[130],[23,14806,14808,14811],{"className":14807},[135],[23,14809,902],{"className":14810},[135,644],[23,14812,14814],{"className":14813},[998],[23,14815,14817,14837],{"className":14816},[199,200],[23,14818,14820,14834],{"className":14819},[204],[23,14821,14823],{"className":14822,"style":2026},[208],[23,14824,14825,14828],{"style":2029},[23,14826],{"className":14827,"style":1015},[216],[23,14829,14831],{"className":14830},[1019,1020,1021,1022],[23,14832,434],{"className":14833,"style":454},[135,136,1022],[23,14835,297],{"className":14836},[296],[23,14838,14840],{"className":14839},[204],[23,14841,14843],{"className":14842,"style":2048},[208],[23,14844],{},[23,14846,14848],{"className":14847},[135],[23,14849,924],{"className":14850},[1066,7581],[23,14852],{"className":14853,"style":280},[148],[23,14855,14857],{"className":14856},[1786,7096],[23,14858,14860,14899],{"className":14859},[199,200],[23,14861,14863,14896],{"className":14862},[204],[23,14864,14866,14886],{"className":14865,"style":7106},[208],[23,14867,14868,14871],{"style":7109},[23,14869],{"className":14870,"style":217},[216],[23,14872,14874],{"className":14873},[1019,1020,1021,1022],[23,14875,14877,14880,14883],{"className":14876},[135,1022],[23,14878,495],{"className":14879},[135,136,1022],[23,14881,372],{"className":14882},[153,1022],[23,14884,500],{"className":14885},[135,412,1022],[23,14887,14888,14891],{"style":7130},[23,14889],{"className":14890,"style":217},[216],[23,14892,14893],{},[23,14894,6967],{"className":14895},[1786],[23,14897,297],{"className":14898},[296],[23,14900,14902],{"className":14901},[204],[23,14903,14905],{"className":14904,"style":7148},[208],[23,14906],{},[23,14908],{"className":14909,"style":280},[148],[23,14911,1653],{"className":14912,"style":144},[135,136],[23,14914,54],{"className":14915},[140],[23,14917,495],{"className":14918},[135,136],[23,14920,747],{"className":14921},[824],[23,14923],{"className":14924,"style":280},[148],[23,14926,434],{"className":14927,"style":454},[135,136],[23,14929,66],{"className":14930},[169],[23,14932,14934],{"className":14933},[135],[23,14935,945],{"className":14936},[1066,7581],[23,14938,112],{"className":14939},[135],[11,14941,14942,15001,15002,15004,15005,15033],{},[23,14943,14945,14963],{"className":14944,"translate":27},[31],[23,14946,14948],{"className":14947},[35],[37,14949,14950],{"xmlns":39},[42,14951,14952,14960],{},[45,14953,14954],{},[4518,14955,14956,14958],{},[48,14957,572],{},[51,14959,6881],{},[114,14961,14962],{"encoding":116},"R^*",[23,14964,14966],{"className":14965,"ariaHidden":122},[121],[23,14967,14969,14972],{"className":14968},[126],[23,14970],{"className":14971,"style":6912},[130],[23,14973,14975,14978],{"className":14974},[135],[23,14976,572],{"className":14977,"style":963},[135,136],[23,14979,14981],{"className":14980},[998],[23,14982,14984],{"className":14983},[199],[23,14985,14987],{"className":14986},[204],[23,14988,14990],{"className":14989,"style":6912},[208],[23,14991,14992,14995],{"style":4696},[23,14993],{"className":14994,"style":1015},[216],[23,14996,14998],{"className":14997},[1019,1020,1021,1022],[23,14999,6881],{"className":15000},[612,1022]," 称为",[18,15003,335],{},"（Bayes risk），也叫贝叶斯错误率（分类情形）。它是这个问题在信息层面的下限：即使知道真实分布、知道每个 ",[23,15006,15008,15021],{"className":15007,"translate":27},[31],[23,15009,15011],{"className":15010},[35],[37,15012,15013],{"xmlns":39},[42,15014,15015,15019],{},[45,15016,15017],{},[48,15018,753],{},[114,15020,753],{"encoding":116},[23,15022,15024],{"className":15023,"ariaHidden":122},[121],[23,15025,15027,15030],{"className":15026},[126],[23,15028],{"className":15029,"style":703},[130],[23,15031,753],{"className":15032},[135,136]," 上的后验，也只能做到这里。它不是某个算法的性能指标，而是问题本身的属性。",[11,15035,15036],{},"对二分类 0-1 损失，把 3.2 节的规则代回去，得到",[23,15038,15040],{"className":15039,"translate":27},[26],[23,15041,15043,15101],{"className":15042,"translate":27},[31],[23,15044,15046],{"className":15045},[35],[37,15047,15048],{"xmlns":39,"display":40},[42,15049,15050,15098],{},[45,15051,15052,15058,15060,15062,15064,15066,15068,15070,15072,15074,15076,15078,15080,15082,15084,15086,15088,15090,15092,15094,15096],{},[4518,15053,15054,15056],{},[48,15055,572],{},[51,15057,6881],{},[51,15059,69],{},[48,15061,902],{"mathvariant":571},[51,15063,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,15065,6967],{},[51,15067,6959],{},[51,15069,9010],{"stretchy":53},[48,15071,9765],{},[51,15073,54],{"stretchy":53},[48,15075,434],{},[51,15077,66],{"stretchy":53},[51,15079,747],{"separator":122},[2581,15081,4435],{},[51,15083,4529],{},[48,15085,9765],{},[51,15087,54],{"stretchy":53},[48,15089,434],{},[51,15091,66],{"stretchy":53},[51,15093,9020],{"stretchy":53},[51,15095,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,15097,112],{"mathvariant":111},[114,15099,15100],{"encoding":116},"R^*=\\mathbb E\\big[\\min\\{\\eta(X),1-\\eta(X)\\}\\big].",[23,15102,15104,15148,15202],{"className":15103,"ariaHidden":122},[121],[23,15105,15107,15110,15139,15142,15145],{"className":15106},[126],[23,15108],{"className":15109,"style":7045},[130],[23,15111,15113,15116],{"className":15112},[135],[23,15114,572],{"className":15115,"style":963},[135,136],[23,15117,15119],{"className":15118},[998],[23,15120,15122],{"className":15121},[199],[23,15123,15125],{"className":15124},[204],[23,15126,15128],{"className":15127,"style":7045},[208],[23,15129,15130,15133],{"style":4989},[23,15131],{"className":15132,"style":1015},[216],[23,15134,15136],{"className":15135},[1019,1020,1021,1022],[23,15137,6881],{"className":15138},[612,1022],[23,15140],{"className":15141,"style":149},[148],[23,15143,69],{"className":15144},[153],[23,15146],{"className":15147,"style":149},[148],[23,15149,15151,15154,15157,15163,15166,15169,15172,15175,15178,15181,15184,15187,15190,15193,15196,15199],{"className":15150},[126],[23,15152],{"className":15153,"style":2213},[130],[23,15155,902],{"className":15156},[135,644],[23,15158,15160],{"className":15159},[135],[23,15161,924],{"className":15162},[1066,1067],[23,15164],{"className":15165,"style":280},[148],[23,15167,6967],{"className":15168},[1786],[23,15170,9010],{"className":15171},[140],[23,15173,9765],{"className":15174,"style":676},[135,136],[23,15176,54],{"className":15177},[140],[23,15179,434],{"className":15180,"style":454},[135,136],[23,15182,66],{"className":15183},[169],[23,15185,747],{"className":15186},[824],[23,15188],{"className":15189,"style":280},[148],[23,15191,4435],{"className":15192},[135],[23,15194],{"className":15195,"style":392},[148],[23,15197,4529],{"className":15198},[612],[23,15200],{"className":15201,"style":392},[148],[23,15203,15205,15208,15211,15214,15217,15221,15227],{"className":15204},[126],[23,15206],{"className":15207,"style":2213},[130],[23,15209,9765],{"className":15210,"style":676},[135,136],[23,15212,54],{"className":15213},[140],[23,15215,434],{"className":15216,"style":454},[135,136],[23,15218,15220],{"className":15219},[169],")}",[23,15222,15224],{"className":15223},[135],[23,15225,945],{"className":15226},[1066,1067],[23,15228,112],{"className":15229},[135],[11,15231,15232],{},"对多分类，则是",[23,15234,15236],{"className":15235,"translate":27},[26],[23,15237,15239,15299],{"className":15238,"translate":27},[31],[23,15240,15242],{"className":15241},[35],[37,15243,15244],{"xmlns":39,"display":40},[42,15245,15246,15296],{},[45,15247,15248,15254,15256,15258,15260,15262,15264,15280,15282,15284,15286,15288,15290,15292,15294],{},[4518,15249,15250,15252],{},[48,15251,572],{},[51,15253,6881],{},[51,15255,69],{},[2581,15257,4435],{},[51,15259,4529],{},[48,15261,902],{"mathvariant":571},[51,15263,924],{"fence":53,"stretchy":122,"minsize":7403,"maxsize":7403},[6961,15265,15266,15272],{},[45,15267,15268,15270],{},[48,15269,9551],{},[51,15271,6959],{},[45,15273,15274,15276,15278],{},[48,15275,660],{},[51,15277,372],{},[48,15279,369],{"mathvariant":375},[48,15281,11],{},[51,15283,54],{"stretchy":53},[48,15285,660],{},[51,15287,60],{},[48,15289,434],{},[51,15291,66],{"stretchy":53},[51,15293,945],{"fence":53,"stretchy":122,"minsize":7403,"maxsize":7403},[48,15295,112],{"mathvariant":111},[114,15297,15298],{"encoding":116},"R^*=1-\\mathbb E\\Big[\\max_{y\\in\\mathcal Y}p(y\\mid X)\\Big].",[23,15300,15302,15346,15364,15457],{"className":15301,"ariaHidden":122},[121],[23,15303,15305,15308,15337,15340,15343],{"className":15304},[126],[23,15306],{"className":15307,"style":7045},[130],[23,15309,15311,15314],{"className":15310},[135],[23,15312,572],{"className":15313,"style":963},[135,136],[23,15315,15317],{"className":15316},[998],[23,15318,15320],{"className":15319},[199],[23,15321,15323],{"className":15322},[204],[23,15324,15326],{"className":15325,"style":7045},[208],[23,15327,15328,15331],{"style":4989},[23,15329],{"className":15330,"style":1015},[216],[23,15332,15334],{"className":15333},[1019,1020,1021,1022],[23,15335,6881],{"className":15336},[612,1022],[23,15338],{"className":15339,"style":149},[148],[23,15341,69],{"className":15342},[153],[23,15344],{"className":15345,"style":149},[148],[23,15347,15349,15352,15355,15358,15361],{"className":15348},[126],[23,15350],{"className":15351,"style":9400},[130],[23,15353,4435],{"className":15354},[135],[23,15356],{"className":15357,"style":392},[148],[23,15359,4529],{"className":15360},[612],[23,15362],{"className":15363,"style":392},[148],[23,15365,15367,15371,15374,15380,15383,15436,15439,15442,15445,15448,15451,15454],{"className":15366},[126],[23,15368],{"className":15369,"style":15370},[130],"height:2.0304em;vertical-align:-0.8804em;",[23,15372,902],{"className":15373},[135,644],[23,15375,15377],{"className":15376},[135],[23,15378,924],{"className":15379},[1066,7581],[23,15381],{"className":15382,"style":280},[148],[23,15384,15386],{"className":15385},[1786,7096],[23,15387,15389,15428],{"className":15388},[199,200],[23,15390,15392,15425],{"className":15391},[204],[23,15393,15395,15415],{"className":15394,"style":703},[208],[23,15396,15397,15400],{"style":7109},[23,15398],{"className":15399,"style":217},[216],[23,15401,15403],{"className":15402},[1019,1020,1021,1022],[23,15404,15406,15409,15412],{"className":15405},[135,1022],[23,15407,660],{"className":15408,"style":676},[135,136,1022],[23,15410,372],{"className":15411},[153,1022],[23,15413,369],{"className":15414,"style":413},[135,412,1022],[23,15416,15417,15420],{"style":7130},[23,15418],{"className":15419,"style":217},[216],[23,15421,15422],{},[23,15423,9551],{"className":15424},[1786],[23,15426,297],{"className":15427},[296],[23,15429,15431],{"className":15430},[204],[23,15432,15434],{"className":15433,"style":5683},[208],[23,15435],{},[23,15437],{"className":15438,"style":280},[148],[23,15440,11],{"className":15441},[135,136],[23,15443,54],{"className":15444},[140],[23,15446,660],{"className":15447,"style":676},[135,136],[23,15449],{"className":15450,"style":149},[148],[23,15452,60],{"className":15453},[153],[23,15455],{"className":15456,"style":149},[148],[23,15458,15460,15463,15466,15469,15475],{"className":15459},[126],[23,15461],{"className":15462,"style":7811},[130],[23,15464,434],{"className":15465,"style":454},[135,136],[23,15467,66],{"className":15468},[169],[23,15470,15472],{"className":15471},[135],[23,15473,945],{"className":15474},[1066,7581],[23,15476,112],{"className":15477},[135],[11,15479,15480],{},"这两个式子在理论里出现得非常多，因为它们把「分类有多难」压缩成了一个标量。",[11,15482,15483,15484,15487],{},"进一步，二分类的",[18,15485,15486],{},"超额风险","有一个很干净的形式：",[23,15489,15491],{"className":15490,"translate":27},[26],[23,15492,15494,15580],{"className":15493,"translate":27},[31],[23,15495,15497],{"className":15496},[35],[37,15498,15499],{"xmlns":39,"display":40},[42,15500,15501,15577],{},[45,15502,15503,15505,15507,15509,15511,15513,15519,15521,15523,15525,15527,15529,15531,15533,15535,15537,15539,15541,15543,15545,15547,15549,15551,15553,15555,15557,15559,15565,15567,15569,15571,15573,15575],{},[48,15504,572],{},[51,15506,54],{"stretchy":53},[48,15508,736],{},[51,15510,66],{"stretchy":53},[51,15512,4529],{},[4518,15514,15515,15517],{},[48,15516,572],{},[51,15518,6881],{},[51,15520,69],{},[48,15522,902],{"mathvariant":571},[51,15524,924],{"fence":53,"stretchy":122,"minsize":7403,"maxsize":7403},[48,15526,60],{"mathvariant":111},[2581,15528,9952],{},[48,15530,9765],{},[51,15532,54],{"stretchy":53},[48,15534,434],{},[51,15536,66],{"stretchy":53},[51,15538,4529],{},[2581,15540,4435],{},[48,15542,60],{"mathvariant":111},[88,15544,90],{},[2581,15546,4435],{"mathvariant":9007},[51,15548,9010],{"stretchy":53},[48,15550,736],{},[51,15552,54],{"stretchy":53},[48,15554,434],{},[51,15556,66],{"stretchy":53},[51,15558,9015],{"mathvariant":111},[4518,15560,15561,15563],{},[48,15562,736],{},[51,15564,6881],{},[51,15566,54],{"stretchy":53},[48,15568,434],{},[51,15570,66],{"stretchy":53},[51,15572,9020],{"stretchy":53},[51,15574,945],{"fence":53,"stretchy":122,"minsize":7403,"maxsize":7403},[48,15576,112],{"mathvariant":111},[114,15578,15579],{"encoding":116},"R(\\delta)-R^*\n=\\mathbb E\\Big[|2\\eta(X)-1|\\,\\mathbf 1\\{\\delta(X)\\ne\\delta^*(X)\\}\\Big].",[23,15581,15583,15610,15654,15694,15767],{"className":15582,"ariaHidden":122},[121],[23,15584,15586,15589,15592,15595,15598,15601,15604,15607],{"className":15585},[126],[23,15587],{"className":15588,"style":131},[130],[23,15590,572],{"className":15591,"style":963},[135,136],[23,15593,54],{"className":15594},[140],[23,15596,736],{"className":15597,"style":783},[135,136],[23,15599,66],{"className":15600},[169],[23,15602],{"className":15603,"style":392},[148],[23,15605,4529],{"className":15606},[612],[23,15608],{"className":15609,"style":392},[148],[23,15611,15613,15616,15645,15648,15651],{"className":15612},[126],[23,15614],{"className":15615,"style":7045},[130],[23,15617,15619,15622],{"className":15618},[135],[23,15620,572],{"className":15621,"style":963},[135,136],[23,15623,15625],{"className":15624},[998],[23,15626,15628],{"className":15627},[199],[23,15629,15631],{"className":15630},[204],[23,15632,15634],{"className":15633,"style":7045},[208],[23,15635,15636,15639],{"style":4989},[23,15637],{"className":15638,"style":1015},[216],[23,15640,15642],{"className":15641},[1019,1020,1021,1022],[23,15643,6881],{"className":15644},[612,1022],[23,15646],{"className":15647,"style":149},[148],[23,15649,69],{"className":15650},[153],[23,15652],{"className":15653,"style":149},[148],[23,15655,15657,15660,15663,15669,15673,15676,15679,15682,15685,15688,15691],{"className":15656},[126],[23,15658],{"className":15659,"style":7811},[130],[23,15661,902],{"className":15662},[135,644],[23,15664,15666],{"className":15665},[135],[23,15667,924],{"className":15668},[1066,7581],[23,15670,15672],{"className":15671},[135],"∣2",[23,15674,9765],{"className":15675,"style":676},[135,136],[23,15677,54],{"className":15678},[140],[23,15680,434],{"className":15681,"style":454},[135,136],[23,15683,66],{"className":15684},[169],[23,15686],{"className":15687,"style":392},[148],[23,15689,4529],{"className":15690},[612],[23,15692],{"className":15693,"style":392},[148],[23,15695,15697,15700,15704,15707,15710,15713,15716,15719,15722,15725,15728,15764],{"className":15696},[126],[23,15698],{"className":15699,"style":131},[130],[23,15701,15703],{"className":15702},[135],"1∣",[23,15705],{"className":15706,"style":280},[148],[23,15708,4435],{"className":15709},[135,9074],[23,15711,9010],{"className":15712},[140],[23,15714,736],{"className":15715,"style":783},[135,136],[23,15717,54],{"className":15718},[140],[23,15720,434],{"className":15721,"style":454},[135,136],[23,15723,66],{"className":15724},[169],[23,15726],{"className":15727,"style":149},[148],[23,15729,15731,15758,15761],{"className":15730},[153],[23,15732,15734],{"className":15733},[153],[23,15735,15737],{"className":15736},[135,9093],[23,15738,15740],{"className":15739},[9097],[23,15741,15743,15746,15755],{"className":15742},[9101],[23,15744],{"className":15745,"style":5414},[130],[23,15747,15749],{"className":15748},[9108],[23,15750,15752],{"className":15751},[135],[23,15753,9115],{"className":15754},[153],[23,15756],{"className":15757},[9119],[23,15759],{"className":15760},[148,9123],[23,15762,69],{"className":15763},[153],[23,15765],{"className":15766,"style":149},[148],[23,15768,15770,15773,15802,15805,15808,15811,15817],{"className":15769},[126],[23,15771],{"className":15772,"style":7811},[130],[23,15774,15776,15779],{"className":15775},[135],[23,15777,736],{"className":15778,"style":783},[135,136],[23,15780,15782],{"className":15781},[998],[23,15783,15785],{"className":15784},[199],[23,15786,15788],{"className":15787},[204],[23,15789,15791],{"className":15790,"style":7045},[208],[23,15792,15793,15796],{"style":4989},[23,15794],{"className":15795,"style":1015},[216],[23,15797,15799],{"className":15798},[1019,1020,1021,1022],[23,15800,6881],{"className":15801},[612,1022],[23,15803,54],{"className":15804},[140],[23,15806,434],{"className":15807,"style":454},[135,136],[23,15809,15220],{"className":15810},[169],[23,15812,15814],{"className":15813},[135],[23,15815,945],{"className":15816},[1066,7581],[23,15818,112],{"className":15819},[135],[11,15821,15822,15824,15825,15853,15854,15965,15966,15994,15995,16076,16077,16158],{},[18,15823,7675],{}," 对固定的 ",[23,15826,15828,15841],{"className":15827,"translate":27},[31],[23,15829,15831],{"className":15830},[35],[37,15832,15833],{"xmlns":39},[42,15834,15835,15839],{},[45,15836,15837],{},[48,15838,753],{},[114,15840,753],{"encoding":116},[23,15842,15844],{"className":15843,"ariaHidden":122},[121],[23,15845,15847,15850],{"className":15846},[126],[23,15848],{"className":15849,"style":703},[130],[23,15851,753],{"className":15852},[135,136],"，若 ",[23,15855,15857,15891],{"className":15856,"translate":27},[31],[23,15858,15860],{"className":15859},[35],[37,15861,15862],{"xmlns":39},[42,15863,15864,15888],{},[45,15865,15866,15868,15870,15872,15874,15876,15882,15884,15886],{},[48,15867,736],{},[51,15869,54],{"stretchy":53},[48,15871,753],{},[51,15873,66],{"stretchy":53},[51,15875,69],{},[4518,15877,15878,15880],{},[48,15879,736],{},[51,15881,6881],{},[51,15883,54],{"stretchy":53},[48,15885,753],{},[51,15887,66],{"stretchy":53},[114,15889,15890],{"encoding":116},"\\delta(x)=\\delta^*(x)",[23,15892,15894,15921],{"className":15893,"ariaHidden":122},[121],[23,15895,15897,15900,15903,15906,15909,15912,15915,15918],{"className":15896},[126],[23,15898],{"className":15899,"style":131},[130],[23,15901,736],{"className":15902,"style":783},[135,136],[23,15904,54],{"className":15905},[140],[23,15907,753],{"className":15908},[135,136],[23,15910,66],{"className":15911},[169],[23,15913],{"className":15914,"style":149},[148],[23,15916,69],{"className":15917},[153],[23,15919],{"className":15920,"style":149},[148],[23,15922,15924,15927,15956,15959,15962],{"className":15923},[126],[23,15925],{"className":15926,"style":131},[130],[23,15928,15930,15933],{"className":15929},[135],[23,15931,736],{"className":15932,"style":783},[135,136],[23,15934,15936],{"className":15935},[998],[23,15937,15939],{"className":15938},[199],[23,15940,15942],{"className":15941},[204],[23,15943,15945],{"className":15944,"style":6912},[208],[23,15946,15947,15950],{"style":4696},[23,15948],{"className":15949,"style":1015},[216],[23,15951,15953],{"className":15952},[1019,1020,1021,1022],[23,15954,6881],{"className":15955},[612,1022],[23,15957,54],{"className":15958},[140],[23,15960,753],{"className":15961},[135,136],[23,15963,66],{"className":15964},[169],"，逐点遗憾为 ",[23,15967,15969,15982],{"className":15968,"translate":27},[31],[23,15970,15972],{"className":15971},[35],[37,15973,15974],{"xmlns":39},[42,15975,15976,15980],{},[45,15977,15978],{},[2581,15979,1532],{},[114,15981,1532],{"encoding":116},[23,15983,15985],{"className":15984,"ariaHidden":122},[121],[23,15986,15988,15991],{"className":15987},[126],[23,15989],{"className":15990,"style":2770},[130],[23,15992,1532],{"className":15993},[135],"；若不等，则所选行动的后验概率是 ",[23,15996,15998,16028],{"className":15997,"translate":27},[31],[23,15999,16001],{"className":16000},[35],[37,16002,16003],{"xmlns":39},[42,16004,16005,16025],{},[45,16006,16007,16009,16011,16013,16015,16017,16019,16021,16023],{},[48,16008,9551],{},[51,16010,6959],{},[51,16012,9010],{"stretchy":53},[48,16014,9765],{},[51,16016,747],{"separator":122},[2581,16018,4435],{},[51,16020,4529],{},[48,16022,9765],{},[51,16024,9020],{"stretchy":53},[114,16026,16027],{"encoding":116},"\\max\\{\\eta,1-\\eta\\}",[23,16029,16031,16064],{"className":16030,"ariaHidden":122},[121],[23,16032,16034,16037,16040,16043,16046,16049,16052,16055,16058,16061],{"className":16033},[126],[23,16035],{"className":16036,"style":131},[130],[23,16038,9551],{"className":16039},[1786],[23,16041,9010],{"className":16042},[140],[23,16044,9765],{"className":16045,"style":676},[135,136],[23,16047,747],{"className":16048},[824],[23,16050],{"className":16051,"style":280},[148],[23,16053,4435],{"className":16054},[135],[23,16056],{"className":16057,"style":392},[148],[23,16059,4529],{"className":16060},[612],[23,16062],{"className":16063,"style":392},[148],[23,16065,16067,16070,16073],{"className":16066},[126],[23,16068],{"className":16069,"style":131},[130],[23,16071,9765],{"className":16072,"style":676},[135,136],[23,16074,9020],{"className":16075},[169],"，最优行动的后验概率是 ",[23,16078,16080,16110],{"className":16079,"translate":27},[31],[23,16081,16083],{"className":16082},[35],[37,16084,16085],{"xmlns":39},[42,16086,16087,16107],{},[45,16088,16089,16091,16093,16095,16097,16099,16101,16103,16105],{},[48,16090,6967],{},[51,16092,6959],{},[51,16094,9010],{"stretchy":53},[48,16096,9765],{},[51,16098,747],{"separator":122},[2581,16100,4435],{},[51,16102,4529],{},[48,16104,9765],{},[51,16106,9020],{"stretchy":53},[114,16108,16109],{"encoding":116},"\\min\\{\\eta,1-\\eta\\}",[23,16111,16113,16146],{"className":16112,"ariaHidden":122},[121],[23,16114,16116,16119,16122,16125,16128,16131,16134,16137,16140,16143],{"className":16115},[126],[23,16117],{"className":16118,"style":131},[130],[23,16120,6967],{"className":16121},[1786],[23,16123,9010],{"className":16124},[140],[23,16126,9765],{"className":16127,"style":676},[135,136],[23,16129,747],{"className":16130},[824],[23,16132],{"className":16133,"style":280},[148],[23,16135,4435],{"className":16136},[135],[23,16138],{"className":16139,"style":392},[148],[23,16141,4529],{"className":16142},[612],[23,16144],{"className":16145,"style":392},[148],[23,16147,16149,16152,16155],{"className":16148},[126],[23,16150],{"className":16151,"style":131},[130],[23,16153,9765],{"className":16154,"style":676},[135,136],[23,16156,9020],{"className":16157},[169],"，两者之差为",[23,16160,16162],{"className":16161,"translate":27},[26],[23,16163,16165,16231],{"className":16164,"translate":27},[31],[23,16166,16168],{"className":16167},[35],[37,16169,16170],{"xmlns":39,"display":40},[42,16171,16172,16228],{},[45,16173,16174,16176,16178,16180,16182,16184,16186,16188,16190,16192,16194,16196,16198,16200,16202,16204,16206,16208,16210,16212,16214,16216,16218,16220,16222,16224,16226],{},[48,16175,9551],{},[51,16177,6959],{},[51,16179,9010],{"stretchy":53},[48,16181,9765],{},[51,16183,747],{"separator":122},[2581,16185,4435],{},[51,16187,4529],{},[48,16189,9765],{},[51,16191,9020],{"stretchy":53},[51,16193,4529],{},[48,16195,6967],{},[51,16197,6959],{},[51,16199,9010],{"stretchy":53},[48,16201,9765],{},[51,16203,747],{"separator":122},[2581,16205,4435],{},[51,16207,4529],{},[48,16209,9765],{},[51,16211,9020],{"stretchy":53},[51,16213,69],{},[48,16215,60],{"mathvariant":111},[2581,16217,9952],{},[48,16219,9765],{},[51,16221,4529],{},[2581,16223,4435],{},[48,16225,60],{"mathvariant":111},[48,16227,112],{"mathvariant":111},[114,16229,16230],{"encoding":116},"\\max\\{\\eta,1-\\eta\\}-\\min\\{\\eta,1-\\eta\\}=|2\\eta-1|.",[23,16232,16234,16267,16288,16321,16342,16363],{"className":16233,"ariaHidden":122},[121],[23,16235,16237,16240,16243,16246,16249,16252,16255,16258,16261,16264],{"className":16236},[126],[23,16238],{"className":16239,"style":131},[130],[23,16241,9551],{"className":16242},[1786],[23,16244,9010],{"className":16245},[140],[23,16247,9765],{"className":16248,"style":676},[135,136],[23,16250,747],{"className":16251},[824],[23,16253],{"className":16254,"style":280},[148],[23,16256,4435],{"className":16257},[135],[23,16259],{"className":16260,"style":392},[148],[23,16262,4529],{"className":16263},[612],[23,16265],{"className":16266,"style":392},[148],[23,16268,16270,16273,16276,16279,16282,16285],{"className":16269},[126],[23,16271],{"className":16272,"style":131},[130],[23,16274,9765],{"className":16275,"style":676},[135,136],[23,16277,9020],{"className":16278},[169],[23,16280],{"className":16281,"style":392},[148],[23,16283,4529],{"className":16284},[612],[23,16286],{"className":16287,"style":392},[148],[23,16289,16291,16294,16297,16300,16303,16306,16309,16312,16315,16318],{"className":16290},[126],[23,16292],{"className":16293,"style":131},[130],[23,16295,6967],{"className":16296},[1786],[23,16298,9010],{"className":16299},[140],[23,16301,9765],{"className":16302,"style":676},[135,136],[23,16304,747],{"className":16305},[824],[23,16307],{"className":16308,"style":280},[148],[23,16310,4435],{"className":16311},[135],[23,16313],{"className":16314,"style":392},[148],[23,16316,4529],{"className":16317},[612],[23,16319],{"className":16320,"style":392},[148],[23,16322,16324,16327,16330,16333,16336,16339],{"className":16323},[126],[23,16325],{"className":16326,"style":131},[130],[23,16328,9765],{"className":16329,"style":676},[135,136],[23,16331,9020],{"className":16332},[169],[23,16334],{"className":16335,"style":149},[148],[23,16337,69],{"className":16338},[153],[23,16340],{"className":16341,"style":149},[148],[23,16343,16345,16348,16351,16354,16357,16360],{"className":16344},[126],[23,16346],{"className":16347,"style":131},[130],[23,16349,15672],{"className":16350},[135],[23,16352,9765],{"className":16353,"style":676},[135,136],[23,16355],{"className":16356,"style":392},[148],[23,16358,4529],{"className":16359},[612],[23,16361],{"className":16362,"style":392},[148],[23,16364,16366,16369],{"className":16365},[126],[23,16367],{"className":16368,"style":131},[130],[23,16370,16372],{"className":16371},[135],"1∣.",[11,16374,16375,16376,16404,16405],{},"对 ",[23,16377,16379,16392],{"className":16378,"translate":27},[31],[23,16380,16382],{"className":16381},[35],[37,16383,16384],{"xmlns":39},[42,16385,16386,16390],{},[45,16387,16388],{},[48,16389,434],{},[114,16391,434],{"encoding":116},[23,16393,16395],{"className":16394,"ariaHidden":122},[121],[23,16396,16398,16401],{"className":16397},[126],[23,16399],{"className":16400,"style":470},[130],[23,16402,434],{"className":16403,"style":454},[135,136]," 取期望即得。",[23,16406,16408,16421],{"className":16407,"translate":27},[31],[23,16409,16411],{"className":16410},[35],[37,16412,16413],{"xmlns":39},[42,16414,16415,16419],{},[45,16416,16417],{},[48,16418,8247],{"mathvariant":111},[114,16420,8250],{"encoding":116},[23,16422,16424],{"className":16423,"ariaHidden":122},[121],[23,16425,16427,16430],{"className":16426},[126],[23,16428],{"className":16429,"style":8260},[130],[23,16431,8247],{"className":16432},[135,8264],[11,16434,16435,16436,16511,16512,16544,16545,16573,16574,16602,16603,16631,16632,1471],{},"这个恒等式很有解释力：只有越过决策边界的样本才贡献错误，而且每个样本的权重是 ",[23,16437,16439,16469],{"className":16438,"translate":27},[31],[23,16440,16442],{"className":16441},[35],[37,16443,16444],{"xmlns":39},[42,16445,16446,16466],{},[45,16447,16448,16450,16452,16454,16456,16458,16460,16462,16464],{},[48,16449,60],{"mathvariant":111},[2581,16451,9952],{},[48,16453,9765],{},[51,16455,54],{"stretchy":53},[48,16457,434],{},[51,16459,66],{"stretchy":53},[51,16461,4529],{},[2581,16463,4435],{},[48,16465,60],{"mathvariant":111},[114,16467,16468],{"encoding":116},"|2\\eta(X)-1|",[23,16470,16472,16502],{"className":16471,"ariaHidden":122},[121],[23,16473,16475,16478,16481,16484,16487,16490,16493,16496,16499],{"className":16474},[126],[23,16476],{"className":16477,"style":131},[130],[23,16479,15672],{"className":16480},[135],[23,16482,9765],{"className":16483,"style":676},[135,136],[23,16485,54],{"className":16486},[140],[23,16488,434],{"className":16489,"style":454},[135,136],[23,16491,66],{"className":16492},[169],[23,16494],{"className":16495,"style":392},[148],[23,16497,4529],{"className":16498},[612],[23,16500],{"className":16501,"style":392},[148],[23,16503,16505,16508],{"className":16504},[126],[23,16506],{"className":16507,"style":131},[130],[23,16509,15703],{"className":16510},[135],"。后验接近 ",[23,16513,16515,16532],{"className":16514,"translate":27},[31],[23,16516,16518],{"className":16517},[35],[37,16519,16520],{"xmlns":39},[42,16521,16522,16530],{},[45,16523,16524,16526,16528],{},[2581,16525,4435],{},[48,16527,2875],{"mathvariant":111},[2581,16529,9952],{},[114,16531,10469],{"encoding":116},[23,16533,16535],{"className":16534,"ariaHidden":122},[121],[23,16536,16538,16541],{"className":16537},[126],[23,16539],{"className":16540,"style":131},[130],[23,16542,10469],{"className":16543},[135]," 的样本位于边界附近，即使分错，代价也小；后验接近 ",[23,16546,16548,16561],{"className":16547,"translate":27},[31],[23,16549,16551],{"className":16550},[35],[37,16552,16553],{"xmlns":39},[42,16554,16555,16559],{},[45,16556,16557],{},[2581,16558,1532],{},[114,16560,1532],{"encoding":116},[23,16562,16564],{"className":16563,"ariaHidden":122},[121],[23,16565,16567,16570],{"className":16566},[126],[23,16568],{"className":16569,"style":2770},[130],[23,16571,1532],{"className":16572},[135]," 或 ",[23,16575,16577,16590],{"className":16576,"translate":27},[31],[23,16578,16580],{"className":16579},[35],[37,16581,16582],{"xmlns":39},[42,16583,16584,16588],{},[45,16585,16586],{},[2581,16587,4435],{},[114,16589,4435],{"encoding":116},[23,16591,16593],{"className":16592,"ariaHidden":122},[121],[23,16594,16596,16599],{"className":16595},[126],[23,16597],{"className":16598,"style":2770},[130],[23,16600,4435],{"className":16601},[135]," 的样本一旦分错，代价就接近 ",[23,16604,16606,16619],{"className":16605,"translate":27},[31],[23,16607,16609],{"className":16608},[35],[37,16610,16611],{"xmlns":39},[42,16612,16613,16617],{},[45,16614,16615],{},[2581,16616,4435],{},[114,16618,4435],{"encoding":116},[23,16620,16622],{"className":16621,"ariaHidden":122},[121],[23,16623,16625,16628],{"className":16624},[126],[23,16626],{"className":16627,"style":2770},[130],[23,16629,4435],{"className":16630},[135],"。换句话说，",[18,16633,16634],{},"贝叶斯错误率不是由边界的长度决定的，而是由边界附近的后验质量决定的",[11,16636,16637,16638,16793,16794,8055],{},"回归情形也有类似的恒等式。设 ",[23,16639,16641,16683],{"className":16640,"translate":27},[31],[23,16642,16644],{"className":16643},[35],[37,16645,16646],{"xmlns":39},[42,16647,16648,16680],{},[45,16649,16650,16656,16658,16660,16662,16664,16666,16668,16670,16672,16674,16676,16678],{},[4518,16651,16652,16654],{},[48,16653,8336],{},[51,16655,6881],{},[51,16657,54],{"stretchy":53},[48,16659,753],{},[51,16661,66],{"stretchy":53},[51,16663,69],{},[48,16665,902],{"mathvariant":571},[51,16667,924],{"stretchy":53},[48,16669,369],{},[51,16671,60],{},[48,16673,434],{},[51,16675,69],{},[48,16677,753],{},[51,16679,945],{"stretchy":53},[114,16681,16682],{"encoding":116},"f^*(x)=\\mathbb E[Y\\mid X=x]",[23,16684,16686,16739,16763,16781],{"className":16685,"ariaHidden":122},[121],[23,16687,16689,16692,16721,16724,16727,16730,16733,16736],{"className":16688},[126],[23,16690],{"className":16691,"style":131},[130],[23,16693,16695,16698],{"className":16694},[135],[23,16696,8336],{"className":16697,"style":11398},[135,136],[23,16699,16701],{"className":16700},[998],[23,16702,16704],{"className":16703},[199],[23,16705,16707],{"className":16706},[204],[23,16708,16710],{"className":16709,"style":6912},[208],[23,16711,16712,16715],{"style":4696},[23,16713],{"className":16714,"style":1015},[216],[23,16716,16718],{"className":16717},[1019,1020,1021,1022],[23,16719,6881],{"className":16720},[612,1022],[23,16722,54],{"className":16723},[140],[23,16725,753],{"className":16726},[135,136],[23,16728,66],{"className":16729},[169],[23,16731],{"className":16732,"style":149},[148],[23,16734,69],{"className":16735},[153],[23,16737],{"className":16738,"style":149},[148],[23,16740,16742,16745,16748,16751,16754,16757,16760],{"className":16741},[126],[23,16743],{"className":16744,"style":131},[130],[23,16746,902],{"className":16747},[135,644],[23,16749,924],{"className":16750},[140],[23,16752,369],{"className":16753,"style":392},[135,136],[23,16755],{"className":16756,"style":149},[148],[23,16758,60],{"className":16759},[153],[23,16761],{"className":16762,"style":149},[148],[23,16764,16766,16769,16772,16775,16778],{"className":16765},[126],[23,16767],{"className":16768,"style":470},[130],[23,16770,434],{"className":16771,"style":454},[135,136],[23,16773],{"className":16774,"style":149},[148],[23,16776,69],{"className":16777},[153],[23,16779],{"className":16780,"style":149},[148],[23,16782,16784,16787,16790],{"className":16783},[126],[23,16785],{"className":16786,"style":131},[130],[23,16788,753],{"className":16789},[135,136],[23,16791,945],{"className":16792},[169],"，则对任意 ",[23,16795,16797,16817],{"className":16796,"translate":27},[31],[23,16798,16800],{"className":16799},[35],[37,16801,16802],{"xmlns":39},[42,16803,16804,16814],{},[45,16805,16806],{},[16807,16808,16809,16811],"mover",{"accent":122},[48,16810,8336],{},[51,16812,16813],{},"^",[114,16815,16816],{"encoding":116},"\\hat f",[23,16818,16820],{"className":16819,"ariaHidden":122},[121],[23,16821,16823,16827],{"className":16822},[126],[23,16824],{"className":16825,"style":16826},[130],"height:1.1523em;vertical-align:-0.1944em;",[23,16828,16831],{"className":16829},[135,16830],"accent",[23,16832,16834,16866],{"className":16833},[199,200],[23,16835,16837,16863],{"className":16836},[204],[23,16838,16841,16849],{"className":16839,"style":16840},[208],"height:0.9579em;",[23,16842,16843,16846],{"style":7130},[23,16844],{"className":16845,"style":217},[216],[23,16847,8336],{"className":16848,"style":11398},[135,136],[23,16850,16852,16855],{"style":16851},"top:-3.2634em;",[23,16853],{"className":16854,"style":217},[216],[23,16856,16860],{"className":16857,"style":16859},[16858],"accent-body","left:-0.0833em;",[23,16861,16813],{"className":16862},[135],[23,16864,297],{"className":16865},[296],[23,16867,16869],{"className":16868},[204],[23,16870,16873],{"className":16871,"style":16872},[208],"height:0.1944em;",[23,16874],{},[23,16876,16878],{"className":16877,"translate":27},[26],[23,16879,16881,16957],{"className":16880,"translate":27},[31],[23,16882,16884],{"className":16883},[35],[37,16885,16886],{"xmlns":39,"display":40},[42,16887,16888,16954],{},[45,16889,16890,16892,16894,16900,16902,16904,16910,16912,16914,16916,16918,16924,16926,16928,16930,16932,16938,16940,16942,16944,16950,16952],{},[48,16891,572],{},[51,16893,54],{"stretchy":53},[16807,16895,16896,16898],{"accent":122},[48,16897,8336],{},[51,16899,16813],{},[51,16901,66],{"stretchy":53},[51,16903,4529],{},[4518,16905,16906,16908],{},[48,16907,572],{},[51,16909,6881],{},[51,16911,69],{},[48,16913,902],{"mathvariant":571},[51,16915,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,16917,54],{"stretchy":53},[16807,16919,16920,16922],{"accent":122},[48,16921,8336],{},[51,16923,16813],{},[51,16925,54],{"stretchy":53},[48,16927,434],{},[51,16929,66],{"stretchy":53},[51,16931,4529],{},[4518,16933,16934,16936],{},[48,16935,8336],{},[51,16937,6881],{},[51,16939,54],{"stretchy":53},[48,16941,434],{},[51,16943,66],{"stretchy":53},[4518,16945,16946,16948],{},[51,16947,66],{"stretchy":53},[2581,16949,9952],{},[51,16951,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,16953,112],{"mathvariant":111},[114,16955,16956],{"encoding":116},"R(\\hat f)-R^*=\\mathbb E\\big[(\\hat 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E\\big[(Y-\\hat f(X))^2\\big]-\\mathbb E\\big[(Y-f^*(X))^2\\big]\\\\\n&=\\mathbb E\\big[(\\hat f(X)-f^*(X))^2\\big]\n+2\\,\\mathbb E\\big[(Y-f^*(X))(f^*(X)-\\hat f(X))\\big].\n\\end{aligned}",[23,17457,17459],{"className":17458,"ariaHidden":122},[121],[23,17460,17462,17466],{"className":17461},[126],[23,17463],{"className":17464,"style":17465},[130],"height:2.9358em;vertical-align:-1.2179em;",[23,17467,17469],{"className":17468},[135],[23,17470,17472,17516],{"className":17471},[1518],[23,17473,17475],{"className":17474},[1696],[23,17476,17478,17507],{"className":17477},[199,200],[23,17479,17481,17504],{"className":17480},[204],[23,17482,17485,17495],{"className":17483,"style":17484},[208],"height:1.7179em;",[23,17486,17488,17492],{"style":17487},"top:-3.7179em;",[23,17489],{"className":17490,"style":17491},[216],"height:2.9579em;",[23,17493],{"className":17494},[135],[23,17496,17498,17501],{"style":17497},"top:-2.1em;",[23,17499],{"className":17500,"style":17491},[216],[23,17502],{"className":17503},[135],[23,17505,297],{"className":17506},[296],[23,17508,17510],{"className":17509},[204],[23,17511,17514],{"className":17512,"style":17513},[208],"height:1.2179em;",[23,17515],{},[23,17517,17519],{"className":17518},[1753],[23,17520,17522,18112],{"className":17521},[199,200],[23,17523,17525,18109],{"className":17524},[204],[23,17526,17528,17756],{"className":17527,"style":17484},[208],[23,17529,17531,17534],{"style":17530},"top:-3.76em;",[23,17532],{"className":17533,"style":217},[216],[23,17535,17537,17540,17543,17549,17552,17555,17558,17561,17564,17606,17609,17612,17615,17644,17650,17653,17656,17659,17662,17668,17671,17674,17677,17680,17683,17712,17715,17718,17721,17750],{"className":17536},[135],[23,17538],{"className":17539},[135],[23,17541,902],{"className":17542},[135,644],[23,17544,17546],{"className":17545},[135],[23,17547,924],{"className":17548},[1066,1067],[23,17550,54],{"className":17551},[140],[23,17553,369],{"className":17554,"style":392},[135,136],[23,17556],{"className":17557,"style":392},[148],[23,17559,4529],{"className":17560},[612],[23,17562],{"className":17563,"style":392},[148],[23,17565,17567],{"className":17566},[135,16830],[23,17568,17570,17598],{"className":17569},[199,200],[23,17571,17573,17595],{"className":17572},[204],[23,17574,17576,17584],{"className":17575,"style":16840},[208],[23,17577,17578,17581],{"style":7130},[23,17579],{"className":17580,"style":217},[216],[23,17582,8336],{"className":17583,"style":11398},[135,136],[23,17585,17586,17589],{"style":16851},[23,17587],{"className":17588,"style":217},[216],[23,17590,17592],{"className":17591,"style":16859},[16858],[23,17593,16813],{"className":17594},[135],[23,17596,297],{"className":17597},[296],[23,17599,17601],{"className":17600},[204],[23,17602,17604],{"className":17603,"style":16872},[208],[23,17605],{},[23,17607,54],{"className":17608},[140],[23,17610,434],{"className":17611,"style":454},[135,136],[23,17613,66],{"className":17614},[169],[23,17616,17618,17621],{"className":17617},[169],[23,17619,66],{"className":17620},[169],[23,17622,17624],{"className":17623},[998],[23,17625,17627],{"className":17626},[199],[23,17628,17630],{"className":17629},[204],[23,17631,17633],{"className":17632,"style":11002},[208],[23,17634,17635,17638],{"style":4989},[23,17636],{"className":17637,"style":1015},[216],[23,17639,17641],{"className":17640},[1019,1020,1021,1022],[23,17642,9952],{"className":17643},[135,1022],[23,17645,17647],{"className":17646},[135],[23,17648,945],{"className":17649},[1066,1067],[23,17651],{"className":17652,"style":392},[148],[23,17654,4529],{"className":17655},[612],[23,17657],{"className":17658,"style":392},[148],[23,17660,902],{"className":17661},[135,644],[23,17663,17665],{"className":17664},[135],[23,17666,924],{"className":17667},[1066,1067],[23,17669,54],{"className":17670},[140],[23,17672,369],{"className":17673,"style":392},[135,136],[23,17675],{"className":17676,"style":392},[148],[23,17678,4529],{"className":17679},[612],[23,17681],{"className":17682,"style":392},[148],[23,17684,17686,17689],{"className":17685},[135],[23,17687,8336],{"className":17688,"style":11398},[135,136],[23,17690,17692],{"className":17691},[998],[23,17693,17695],{"className":17694},[199],[23,17696,17698],{"className":17697},[204],[23,17699,17701],{"className":17700,"style":7045},[208],[23,17702,17703,17706],{"style":4989},[23,17704],{"className":17705,"style":1015},[216],[23,17707,17709],{"className":17708},[1019,1020,1021,1022],[23,17710,6881],{"className":17711},[612,1022],[23,17713,54],{"className":17714},[140],[23,17716,434],{"className":17717,"style":454},[135,136],[23,17719,66],{"className":17720},[169],[23,17722,17724,17727],{"className":17723},[169],[23,17725,66],{"className":17726},[169],[23,17728,17730],{"className":17729},[998],[23,17731,17733],{"className":17732},[199],[23,17734,17736],{"className":17735},[204],[23,17737,17739],{"className":17738,"style":11002},[208],[23,17740,17741,17744],{"style":4989},[23,17742],{"className":17743,"style":1015},[216],[23,17745,17747],{"className":17746},[1019,1020,1021,1022],[23,17748,9952],{"className":17749},[135,1022],[23,17751,17753],{"className":17752},[135],[23,17754,945],{"className":17755},[1066,1067],[23,17757,17759,17762],{"style":17758},"top:-2.1421em;",[23,17760],{"className":17761,"style":217},[216],[23,17763,17765,17768,17771,17774,17777,17780,17786,17789,17831,17834,17837,17840,17843,17846,17849,17878,17881,17884,17887,17916,17922,17925,17928,17931,17934,17937,17940,17946,17949,17952,17955,17958,17961,17990,17993,17996,17999,18002,18031,18034,18037,18040,18043,18046,18049,18091,18094,18097,18100,18106],{"className":17764},[135],[23,17766],{"className":17767},[135],[23,17769],{"className":17770,"style":149},[148],[23,17772,69],{"className":17773},[153],[23,17775],{"className":17776,"style":149},[148],[23,17778,902],{"className":17779},[135,644],[23,17781,17783],{"className":17782},[135],[23,17784,924],{"className":17785},[1066,1067],[23,17787,54],{"className":17788},[140],[23,17790,17792],{"className":17791},[135,16830],[23,17793,17795,17823],{"className":17794},[199,200],[23,17796,17798,17820],{"className":17797},[204],[23,17799,17801,17809],{"className":17800,"style":16840},[208],[23,17802,17803,17806],{"style":7130},[23,17804],{"className":17805,"style":217},[216],[23,17807,8336],{"className":17808,"style":11398},[135,136],[23,17810,17811,17814],{"style":16851},[23,17812],{"className":17813,"style":217},[216],[23,17815,17817],{"className":17816,"style":16859},[16858],[23,17818,16813],{"className":17819},[135],[23,17821,297],{"className":17822},[296],[23,17824,17826],{"className":17825},[204],[23,17827,17829],{"className":17828,"style":16872},[208],[23,17830],{},[23,17832,54],{"className":17833},[140],[23,17835,434],{"className":17836,"style":454},[135,136],[23,17838,66],{"className":17839},[169],[23,17841],{"className":17842,"style":392},[148],[23,17844,4529],{"className":17845},[612],[23,17847],{"className":17848,"style":392},[148],[23,17850,17852,17855],{"className":17851},[135],[23,17853,8336],{"className":17854,"style":11398},[135,136],[23,17856,17858],{"className":17857},[998],[23,17859,17861],{"className":17860},[199],[23,17862,17864],{"className":17863},[204],[23,17865,17867],{"className":17866,"style":7045},[208],[23,17868,17869,17872],{"style":4989},[23,17870],{"className":17871,"style":1015},[216],[23,17873,17875],{"className":17874},[1019,1020,1021,1022],[23,17876,6881],{"className":17877},[612,1022],[23,17879,54],{"className":17880},[140],[23,17882,434],{"className":17883,"style":454},[135,136],[23,17885,66],{"className":17886},[169],[23,17888,17890,17893],{"className":17889},[169],[23,17891,66],{"className":17892},[169],[23,17894,17896],{"className":17895},[998],[23,17897,17899],{"className":17898},[199],[23,17900,17902],{"className":17901},[204],[23,17903,17905],{"className":17904,"style":11002},[208],[23,17906,17907,17910],{"style":4989},[23,17908],{"className":17909,"style":1015},[216],[23,17911,17913],{"className":17912},[1019,1020,1021,1022],[23,17914,9952],{"className":17915},[135,1022],[23,17917,17919],{"className":17918},[135],[23,17920,945],{"className":17921},[1066,1067],[23,17923],{"className":17924,"style":392},[148],[23,17926,5106],{"className":17927},[612],[23,17929],{"className":17930,"style":392},[148],[23,17932,9952],{"className":17933},[135],[23,17935],{"className":17936,"style":280},[148],[23,17938,902],{"className":17939},[135,644],[23,17941,17943],{"className":17942},[135],[23,17944,924],{"className":17945},[1066,1067],[23,17947,54],{"className":17948},[140],[23,17950,369],{"className":17951,"style":392},[135,136],[23,17953],{"className":17954,"style":392},[148],[23,17956,4529],{"className":17957},[612],[23,17959],{"className":17960,"style":392},[148],[23,17962,17964,17967],{"className":17963},[135],[23,17965,8336],{"className":17966,"style":11398},[135,136],[23,17968,17970],{"className":17969},[998],[23,17971,17973],{"className":17972},[199],[23,17974,17976],{"className":17975},[204],[23,17977,17979],{"className":17978,"style":7045},[208],[23,17980,17981,17984],{"style":4989},[23,17982],{"className":17983,"style":1015},[216],[23,17985,17987],{"className":17986},[1019,1020,1021,1022],[23,17988,6881],{"className":17989},[612,1022],[23,17991,54],{"className":17992},[140],[23,17994,434],{"className":17995,"style":454},[135,136],[23,17997,1095],{"className":17998},[169],[23,18000,54],{"className":18001},[140],[23,18003,18005,18008],{"className":18004},[135],[23,18006,8336],{"className":18007,"style":11398},[135,136],[23,18009,18011],{"className":18010},[998],[23,18012,18014],{"className":18013},[199],[23,18015,18017],{"className":18016},[204],[23,18018,18020],{"className":18019,"style":7045},[208],[23,18021,18022,18025],{"style":4989},[23,18023],{"className":18024,"style":1015},[216],[23,18026,18028],{"className":18027},[1019,1020,1021,1022],[23,18029,6881],{"className":18030},[612,1022],[23,18032,54],{"className":18033},[140],[23,18035,434],{"className":18036,"style":454},[135,136],[23,18038,66],{"className":18039},[169],[23,18041],{"className":18042,"style":392},[148],[23,18044,4529],{"className"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",[23,18124,18126,18139],{"className":18125,"translate":27},[31],[23,18127,18129],{"className":18128},[35],[37,18130,18131],{"xmlns":39},[42,18132,18133,18137],{},[45,18134,18135],{},[48,18136,434],{},[114,18138,434],{"encoding":116},[23,18140,18142],{"className":18141,"ariaHidden":122},[121],[23,18143,18145,18148],{"className":18144},[126],[23,18146],{"className":18147,"style":470},[130],[23,18149,434],{"className":18150,"style":454},[135,136]," 取条件期望后，",[23,18153,18155,18197],{"className":18154,"translate":27},[31],[23,18156,18158],{"className":18157},[35],[37,18159,18160],{"xmlns":39},[42,18161,18162,18194],{},[45,18163,18164,18166,18168,18170,18172,18178,18180,18182,18184,18186,18188,18190,18192],{},[48,18165,902],{"mathvariant":571},[51,18167,924],{"stretchy":53},[48,18169,369],{},[51,18171,4529],{},[4518,18173,18174,18176],{},[48,18175,8336],{},[51,18177,6881],{},[51,18179,54],{"stretchy":53},[48,18181,434],{},[51,18183,66],{"stretchy":53},[51,18185,60],{},[48,18187,434],{},[51,18189,945],{"stretchy":53},[51,18191,69],{},[2581,18193,1532],{},[114,18195,18196],{"encoding":116},"\\mathbb E[Y-f^*(X)\\mid X]=0",[23,18198,18200,18224,18277,18298],{"className":18199,"ariaHidden":122},[121],[23,18201,18203,18206,18209,18212,18215,18218,18221],{"className":18202},[126],[23,18204],{"className":18205,"style":131},[130],[23,18207,902],{"className":18208},[135,644],[23,18210,924],{"className":18211},[140],[23,18213,369],{"className":18214,"style":392},[135,136],[23,18216],{"className":18217,"style":392},[148],[23,18219,4529],{"className":18220},[612],[23,18222],{"className":18223,"style":392},[148],[23,18225,18227,18230,18259,18262,18265,18268,18271,18274],{"className":18226},[126],[23,18228],{"className":18229,"style":131},[130],[23,18231,18233,18236],{"className":18232},[135],[23,18234,8336],{"className":18235,"style":11398},[135,136],[23,18237,18239],{"className":18238},[998],[23,18240,18242],{"className":18241},[199],[23,18243,18245],{"className":18244},[204],[23,18246,18248],{"className":18247,"style":6912},[208],[23,18249,18250,18253],{"style":4696},[23,18251],{"className":18252,"style":1015},[216],[23,18254,18256],{"className":18255},[1019,1020,1021,1022],[23,18257,6881],{"className":18258},[612,1022],[23,18260,54],{"className":18261},[140],[23,18263,434],{"className":18264,"style":454},[135,136],[23,18266,66],{"className":18267},[169],[23,18269],{"className":18270,"style":149},[148],[23,18272,60],{"className":18273},[153],[23,18275],{"className":18276,"style":149},[148],[23,18278,18280,18283,18286,18289,18292,18295],{"className":18279},[126],[23,18281],{"className":18282,"style":131},[130],[23,18284,434],{"className":18285,"style":454},[135,136],[23,18287,945],{"className":18288},[169],[23,18290],{"className":18291,"style":149},[148],[23,18293,69],{"className":18294},[153],[23,18296],{"className":18297,"style":149},[148],[23,18299,18301,18304],{"className":18300},[126],[23,18302],{"className":18303,"style":2770},[130],[23,18305,1532],{"className":18306},[135],"，因此消失。",[23,18309,18311,18324],{"className":18310,"translate":27},[31],[23,18312,18314],{"className":18313},[35],[37,18315,18316],{"xmlns":39},[42,18317,18318,18322],{},[45,18319,18320],{},[48,18321,8247],{"mathvariant":111},[114,18323,8250],{"encoding":116},[23,18325,18327],{"className":18326,"ariaHidden":122},[121],[23,18328,18330,18333],{"className":18329},[126],[23,18331],{"className":18332,"style":8260},[130],[23,18334,8247],{"className":18335},[135,8264],[11,18337,18338],{},"这两个恒等式把「模型误差」翻译成了「决策遗憾」：分类器的错误只发生在越过边界的点上，回归器的误差则是函数逼近误差的平方。贝叶斯风险就是那个无法被任何模型消除的起点。",[2501,18340,18342],{"id":18341},"_37-map-不是贝叶斯行动","3.7 MAP 不是贝叶斯行动",[11,18344,18345],{},"这是最常见的混淆之一。MAP 是后验的众数，是一个「点」；贝叶斯行动是使后验期望损失最小的动作，依赖损失函数。两者只有在参数空间上取 0-1 损失时才重合。",[11,18347,18348,18349,18417,18418,18483,18484,18512,18513,18579],{},"更麻烦的是，MAP 对重参数化不协变。若 ",[23,18350,18352,18378],{"className":18351,"translate":27},[31],[23,18353,18355],{"className":18354},[35],[37,18356,18357],{"xmlns":39},[42,18358,18359,18375],{},[45,18360,18361,18364,18366,18369,18371,18373],{},[48,18362,18363],{},"ϕ",[51,18365,69],{},[48,18367,18368],{},"h",[51,18370,54],{"stretchy":53},[48,18372,57],{},[51,18374,66],{"stretchy":53},[114,18376,18377],{"encoding":116},"\\phi=h(\\theta)",[23,18379,18381,18399],{"className":18380,"ariaHidden":122},[121],[23,18382,18384,18387,18390,18393,18396],{"className":18383},[126],[23,18385],{"className":18386,"style":5414},[130],[23,18388,18363],{"className":18389},[135,136],[23,18391],{"className":18392,"style":149},[148],[23,18394,69],{"className":18395},[153],[23,18397],{"className":18398,"style":149},[148],[23,18400,18402,18405,18408,18411,18414],{"className":18401},[126],[23,18403],{"className":18404,"style":131},[130],[23,18406,18368],{"className":18407},[135,136],[23,18409,54],{"className":18410},[140],[23,18412,57],{"className":18413,"style":144},[135,136],[23,18415,66],{"className":18416},[169]," 是一个非线性变换，",[23,18419,18421,18444],{"className":18420,"translate":27},[31],[23,18422,18424],{"className":18423},[35],[37,18425,18426],{"xmlns":39},[42,18427,18428,18442],{},[45,18429,18430,18432,18434,18436,18438,18440],{},[48,18431,11],{},[51,18433,54],{"stretchy":53},[48,18435,57],{},[51,18437,60],{},[48,18439,63],{},[51,18441,66],{"stretchy":53},[114,18443,4032],{"encoding":116},[23,18445,18447,18471],{"className":18446,"ariaHidden":122},[121],[23,18448,18450,18453,18456,18459,18462,18465,18468],{"className":18449},[126],[23,18451],{"className":18452,"style":131},[130],[23,18454,11],{"className":18455},[135,136],[23,18457,54],{"className":18458},[140],[23,18460,57],{"className":18461,"style":144},[135,136],[23,18463],{"className":18464,"style":149},[148],[23,18466,60],{"className":18467},[153],[23,18469],{"className":18470,"style":149},[148],[23,18472,18474,18477,18480],{"className":18473},[126],[23,18475],{"className":18476,"style":131},[130],[23,18478,63],{"className":18479,"style":144},[135,136],[23,18481,66],{"className":18482},[169]," 的众数经过 ",[23,18485,18487,18500],{"className":18486,"translate":27},[31],[23,18488,18490],{"className":18489},[35],[37,18491,18492],{"xmlns":39},[42,18493,18494,18498],{},[45,18495,18496],{},[48,18497,18368],{},[114,18499,18368],{"encoding":116},[23,18501,18503],{"className":18502,"ariaHidden":122},[121],[23,18504,18506,18509],{"className":18505},[126],[23,18507],{"className":18508,"style":779},[130],[23,18510,18368],{"className":18511},[135,136]," 之后，一般不是 ",[23,18514,18516,18540],{"className":18515,"translate":27},[31],[23,18517,18519],{"className":18518},[35],[37,18520,18521],{"xmlns":39},[42,18522,18523,18537],{},[45,18524,18525,18527,18529,18531,18533,18535],{},[48,18526,11],{},[51,18528,54],{"stretchy":53},[48,18530,18363],{},[51,18532,60],{},[48,18534,63],{},[51,18536,66],{"stretchy":53},[114,18538,18539],{"encoding":116},"p(\\phi\\mid D)",[23,18541,18543,18567],{"className":18542,"ariaHidden":122},[121],[23,18544,18546,18549,18552,18555,18558,18561,18564],{"className":18545},[126],[23,18547],{"className":18548,"style":131},[130],[23,18550,11],{"className":18551},[135,136],[23,18553,54],{"className":18554},[140],[23,18556,18363],{"className":18557},[135,136],[23,18559],{"className":18560,"style":149},[148],[23,18562,60],{"className":18563},[153],[23,18565],{"className":18566,"style":149},[148],[23,18568,18570,18573,18576],{"className":18569},[126],[23,18571],{"className":18572,"style":131},[130],[23,18574,63],{"className":18575,"style":144},[135,136],[23,18577,66],{"className":18578},[169]," 的众数，因为密度会多出一个 Jacobian 因子。后验中位数在单调变换下等变，后验均值在仿射变换下等变，而 MAP 连这个性质都没有。这也是为什么在贝叶斯推断里，除非有明确的 0-1 损失，否则「取众数」并不是一个默认正确的选择。",[338,18581,18583],{"id":18582},"_4-贝叶斯错误率","4. 贝叶斯错误率",[2501,18585,18587],{"id":18586},"_41-两类情形全变差与重叠面积","4.1 两类情形：全变差与重叠面积",[11,18589,18590,18591,18716,18717,18786],{},"设两类先验相等，类条件密度为 ",[23,18592,18594,18620],{"className":18593,"translate":27},[31],[23,18595,18597],{"className":18596},[35],[37,18598,18599],{"xmlns":39},[42,18600,18601,18617],{},[45,18602,18603,18609,18611],{},[898,18604,18605,18607],{},[48,18606,11],{},[2581,18608,1532],{},[51,18610,747],{"separator":122},[898,18612,18613,18615],{},[48,18614,11],{},[2581,18616,4435],{},[114,18618,18619],{"encoding":116},"p_0,p_1",[23,18621,18623],{"className":18622,"ariaHidden":122},[121],[23,18624,18626,18629,18670,18673,18676],{"className":18625},[126],[23,18627],{"className":18628,"style":672},[130],[23,18630,18632,18635],{"className":18631},[135],[23,18633,11],{"className":18634},[135,136],[23,18636,18638],{"className":18637},[998],[23,18639,18641,18662],{"className":18640},[199,200],[23,18642,18644,18659],{"className":18643},[204],[23,18645,18648],{"className":18646,"style":18647},[208],"height:0.3011em;",[23,18649,18650,18653],{"style":2029},[23,18651],{"className":18652,"style":1015},[216],[23,18654,18656],{"className":18655},[1019,1020,1021,1022],[23,18657,1532],{"className":18658},[135,1022],[23,18660,297],{"className":18661},[296],[23,18663,18665],{"className":18664},[204],[23,18666,18668],{"className":18667,"style":2048},[208],[23,18669],{},[23,18671,747],{"className":18672},[824],[23,18674],{"className":18675,"style":280},[148],[23,18677,18679,18682],{"className":18678},[135],[23,18680,11],{"className":18681},[135,136],[23,18683,18685],{"className":18684},[998],[23,18686,18688,18708],{"className":18687},[199,200],[23,18689,18691,18705],{"className":18690},[204],[23,18692,18694],{"className":18693,"style":18647},[208],[23,18695,18696,18699],{"style":2029},[23,18697],{"className":18698,"style":1015},[216],[23,18700,18702],{"className":18701},[1019,1020,1021,1022],[23,18703,4435],{"className":18704},[135,1022],[23,18706,297],{"className":18707},[296],[23,18709,18711],{"className":18710},[204],[23,18712,18714],{"className":18713,"style":2048},[208],[23,18715],{},"。由 3.2 节，贝叶斯规则在 ",[23,18718,18720,18747],{"className":18719,"translate":27},[31],[23,18721,18723],{"className":18722},[35],[37,18724,18725],{"xmlns":39},[42,18726,18727,18745],{},[45,18728,18729,18731,18733,18735,18737,18739,18741,18743],{},[48,18730,9765],{},[51,18732,54],{"stretchy":53},[48,18734,753],{},[51,18736,66],{"stretchy":53},[51,18738,69],{},[2581,18740,4435],{},[48,18742,2875],{"mathvariant":111},[2581,18744,9952],{},[114,18746,10504],{"encoding":116},[23,18748,18750,18777],{"className":18749,"ariaHidden":122},[121],[23,18751,18753,18756,18759,18762,18765,18768,18771,18774],{"className":18752},[126],[23,18754],{"className":18755,"style":131},[130],[23,18757,9765],{"className":18758,"style":676},[135,136],[23,18760,54],{"className":18761},[140],[23,18763,753],{"className":18764},[135,136],[23,18766,66],{"className":18767},[169],[23,18769],{"className":18770,"style":149},[148],[23,18772,69],{"className":18773},[153],[23,18775],{"className":18776,"style":149},[148],[23,18778,18780,18783],{"className":18779},[126],[23,18781],{"className":18782,"style":131},[130],[23,18784,10469],{"className":18785},[135]," 处切换，其中",[23,18788,18790],{"className":18789,"translate":27},[26],[23,18791,18793,18861],{"className":18792,"translate":27},[31],[23,18794,18796],{"className":18795},[35],[37,18797,18798],{"xmlns":39,"display":40},[42,18799,18800,18858],{},[45,18801,18802,18804,18806,18808,18810,18812,18856],{},[48,18803,9765],{},[51,18805,54],{"stretchy":53},[48,18807,753],{},[51,18809,66],{"stretchy":53},[51,18811,69],{},[71,18813,18814,18828],{},[45,18815,18816,18822,18824,18826],{},[898,18817,18818,18820],{},[48,18819,11],{},[2581,18821,4435],{},[51,18823,54],{"stretchy":53},[48,18825,753],{},[51,18827,66],{"stretchy":53},[45,18829,18830,18836,18838,18840,18842,18844,18850,18852,18854],{},[898,18831,18832,18834],{},[48,18833,11],{},[2581,18835,1532],{},[51,18837,54],{"stretchy":53},[48,18839,753],{},[51,18841,66],{"stretchy":53},[51,18843,5106],{},[898,18845,18846,18848],{},[48,18847,11],{},[2581,18849,4435],{},[51,18851,54],{"stretchy":53},[48,18853,753],{},[51,18855,66],{"stretchy":53},[48,18857,112],{"mathvariant":111},[114,18859,18860],{"encoding":116},"\\eta(x)=\\frac{p_1(x)}{p_0(x)+p_1(x)}.",[23,18862,18864,18891],{"className":18863,"ariaHidden":122},[121],[23,18865,18867,18870,18873,18876,18879,18882,18885,18888],{"className":18866},[126],[23,18868],{"className":18869,"style":131},[130],[23,18871,9765],{"className":18872,"style":676},[135,136],[23,18874,54],{"className":18875},[140],[23,18877,753],{"className":18878},[135,136],[23,18880,66],{"className":18881},[169],[23,18883],{"className":18884,"style":149},[148],[23,18886,69],{"className":18887},[153],[23,18889],{"className":18890,"style":149},[148],[23,18892,18894,18897,19109],{"className":18893},[126],[23,18895],{"className":18896,"style":185},[130],[23,18898,18900,18903,19106],{"className":18899},[135],[23,18901],{"className":18902},[140,192],[23,18904,18906],{"className":18905},[71],[23,18907,18909,19098],{"className":18908},[199,200],[23,18910,18912,19095],{"className":18911},[204],[23,18913,18915,19030,19038],{"className":18914,"style":209},[208],[23,18916,18917,18920],{"style":212},[23,18918],{"className":18919,"style":217},[216],[23,18921,18923,18963,18966,18969,18972,18975,18978,18981,19021,19024,19027],{"className":18922},[135],[23,18924,18926,18929],{"className":18925},[135],[23,18927,11],{"className":18928},[135,136],[23,18930,18932],{"className":18931},[998],[23,18933,18935,18955],{"className":18934},[199,200],[23,18936,18938,18952],{"className":18937},[204],[23,18939,18941],{"className":18940,"style":18647},[208],[23,18942,18943,18946],{"style":2029},[23,18944],{"className":18945,"style":1015},[216],[23,18947,18949],{"className":18948},[1019,1020,1021,1022],[23,18950,1532],{"className":18951},[135,1022],[23,18953,297],{"className":18954},[296],[23,18956,18958],{"className":18957},[204],[23,18959,18961],{"className":18960,"style":2048},[208],[23,18962],{},[23,18964,54],{"className":18965},[140],[23,18967,753],{"className":18968},[135,136],[23,18970,66],{"className":18971},[169],[23,18973],{"className":18974,"style":392},[148],[23,18976,5106],{"className":18977},[612],[23,18979],{"className":18980,"style":392},[148],[23,18982,18984,18987],{"className":18983},[135],[23,18985,11],{"className":18986},[135,136],[23,18988,18990],{"className":18989},[998],[23,18991,18993,19013],{"className":18992},[199,200],[23,18994,18996,19010],{"className":18995},[204],[23,18997,18999],{"className":18998,"style":18647},[208],[23,19000,19001,19004],{"style":2029},[23,19002],{"className":19003,"style":1015},[216],[23,19005,19007],{"className":19006},[1019,1020,1021,1022],[23,19008,4435],{"className":19009},[135,1022],[23,19011,297],{"className":19012},[296],[23,19014,19016],{"className":19015},[204],[23,19017,19019],{"className":19018,"style":2048},[208],[23,19020],{},[23,19022,54],{"className":19023},[140],[23,19025,753],{"className":19026},[135,136],[23,19028,66],{"className":19029},[169],[23,19031,19032,19035],{"style":235},[23,19033],{"className":19034,"style":217},[216],[23,19036],{"className":19037,"style":243},[242],[23,19039,19040,19043],{"style":246},[23,19041],{"className":19042,"style":217},[216],[23,19044,19046,19086,19089,19092],{"className":19045},[135],[23,19047,19049,19052],{"className":19048},[135],[23,19050,11],{"className":19051},[135,136],[23,19053,19055],{"className":19054},[998],[23,19056,19058,19078],{"className":19057},[199,200],[23,19059,19061,19075],{"className":19060},[204],[23,19062,19064],{"className":19063,"style":18647},[208],[23,19065,19066,19069],{"style":2029},[23,19067],{"className":19068,"style":1015},[216],[23,19070,19072],{"className":19071},[1019,1020,1021,1022],[23,19073,4435],{"className":19074},[135,1022],[23,19076,297],{"className":19077},[296],[23,19079,19081],{"className":19080},[204],[23,19082,19084],{"className":19083,"style":2048},[208],[23,19085],{},[23,19087,54],{"className":19088},[140],[23,19090,753],{"className":19091},[135,136],[23,19093,66],{"className":19094},[169],[23,19096,297],{"className":19097},[296],[23,19099,1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\\min\\left\\{\\tfrac12 p_0(x),\\ \\tfrac12 p_1(x)\\right\\}dx\\\\\n&=\\frac12\\int \\min\\{p_0(x),p_1(x)\\}\\,dx 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\\big(1-\\|P_0-P_1\\|_{\\mathrm{TV}}\\big).",[23,20714,20716,20783,21084,21170,21229],{"className":20715,"ariaHidden":122},[121],[23,20717,20719,20723,20774,20777,20780],{"className":20718},[126],[23,20720],{"className":20721,"style":20722},[130],"height:0.9857em;vertical-align:-0.247em;",[23,20724,20726,20729],{"className":20725},[135],[23,20727,11],{"className":20728},[135,136],[23,20730,20732],{"className":20731},[998],[23,20733,20735,20766],{"className":20734},[199,200],[23,20736,20738,20763],{"className":20737},[204],[23,20739,20741,20752],{"className":20740,"style":7045},[208],[23,20742,20743,20746],{"style":19345},[23,20744],{"className":20745,"style":1015},[216],[23,20747,20749],{"className":20748},[1019,1020,1021,1022],[23,20750,5342],{"className":20751},[135,136,1022],[23,20753,20754,20757],{"style":4989},[23,20755],{"className":20756,"style":1015},[216],[23,20758,20760],{"className":20759},[1019,1020,1021,1022],[23,20761,6881],{"className":20762},[612,1022],[23,20764,297],{"className":20765},[296],[23,20767,20769],{"className":20768},[204],[23,20770,20772],{"className":20771,"style":19375},[208],[23,20773],{},[23,20775],{"className":20776,"style":149},[148],[23,20778,69],{"className":20779},[153],[23,20781],{"className":20782,"style":149},[148],[23,20784,20786,20790,20852,20855,21075,21078,21081],{"className":20785},[126],[23,20787],{"className":20788,"style":20789},[130],"height:2.4em;vertical-align:-0.95em;",[23,20791,20793,20796,20849],{"className":20792},[135],[23,20794],{"className":20795},[140,192],[23,20797,20799],{"className":20798},[71],[23,20800,20802,20841],{"className":20801},[199,200],[23,20803,20805,20838],{"className":20804},[204],[23,20806,20808,20819,20827],{"className":20807,"style":12409},[208],[23,20809,20810,20813],{"style":212},[23,20811],{"className":20812,"style":217},[216],[23,20814,20816],{"className":20815},[135],[23,20817,9952],{"className":20818},[135],[23,20820,20821,20824],{"style":235},[23,20822],{"className":20823,"style":217},[216],[23,20825],{"className":20826,"style":243},[242],[23,20828,20829,20832],{"style":246},[23,20830],{"className":20831,"style":217},[216],[23,20833,20835],{"className":20834},[135],[23,20836,4435],{"className":20837},[135],[23,20839,297],{"className":20840},[296],[23,20842,20844],{"className":20843},[204],[23,20845,20847],{"className":20846,"style":6287},[208],[23,20848],{},[23,20850],{"className":20851},[169,192],[23,20853],{"className":20854,"style":280},[148],[23,20856,20858,20864,20867,20870,20873,20876,20938,20941,20944,20947,20950,20990,20993,20996,20999,21002,21005,21008,21048,21051,21054,21057,21060,21063,21066,21069],{"className":20857},[1839],[23,20859,20861],{"className":20860,"style":1844},[140,1843],[23,20862,54],{"className":20863},[1066,1021],[23,20865,4435],{"className":20866},[135],[23,20868],{"className":20869,"style":392},[148],[23,20871,4529],{"className":20872},[612],[23,20874],{"className":20875,"style":392},[148],[23,20877,20879,20882,20935],{"className":20878},[135],[23,20880],{"className":20881},[140,192],[23,20883,20885],{"className":20884},[71],[23,20886,20888,20927],{"className":20887},[199,200],[23,20889,20891,20924],{"className":20890},[204],[23,20892,20894,20905,20913],{"className":20893,"style":12409},[208],[23,20895,20896,20899],{"style":212},[23,20897],{"className":20898,"style":217},[216],[23,20900,20902],{"className":20901},[135],[23,20903,9952],{"className":20904},[135],[23,20906,20907,20910],{"style":235},[23,20908],{"className":20909,"style":217},[216],[23,20911],{"className":20912,"style":243},[242],[23,20914,20915,20918],{"style":246},[23,20916],{"className":20917,"style":217},[216],[23,20919,20921],{"className":20920},[135],[23,20922,4435],{"className":20923},[135],[23,20925,297],{"className":20926},[296],[23,20928,20930],{"className":20929},[204],[23,20931,20933],{"className":20932,"style":6287},[208],[23,20934],{},[23,20936],{"className":20937},[169,192],[23,20939],{"className":20940,"style":280},[148],[23,20942,1557],{"className":20943,"style":1792},[1786,1790,1791],[23,20945],{"className":20946,"style":280},[148],[23,20948,60],{"className":20949},[135],[23,20951,20953,20956],{"className":20952},[135],[23,20954,11],{"className":20955},[135,136],[23,20957,20959],{"className":20958},[998],[23,20960,20962,20982],{"className":20961},[199,200],[23,20963,20965,20979],{"className":20964},[204],[23,20966,20968],{"className":20967,"style":18647},[208],[23,20969,20970,20973],{"style":2029},[23,20971],{"className":20972,"style":1015},[216],[23,20974,20976],{"className":20975},[1019,1020,1021,1022],[23,20977,1532],{"className":20978},[135,1022],[23,20980,297],{"className":20981},[296],[23,20983,20985],{"className":20984},[204],[23,20986,20988],{"className":20987,"style":2048},[208],[23,20989],{},[23,20991,54],{"className":20992},[140],[23,20994,753],{"className":20995},[135,136],[23,20997,66],{"className":20998},[169],[23,21000],{"className":21001,"style":392},[148],[23,21003,4529],{"className":21004},[612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",[23,21335,21337,21363],{"className":21336,"translate":27},[31],[23,21338,21340],{"className":21339},[35],[37,21341,21342],{"xmlns":39},[42,21343,21344,21360],{},[45,21345,21346,21348,21350],{},[48,21347,20680],{"mathvariant":111},[51,21349,5758],{},[898,21351,21352,21354],{},[48,21353,20680],{"mathvariant":111},[45,21355,21356,21358],{},[48,21357,20703],{"mathvariant":111},[48,21359,10790],{"mathvariant":111},[114,21361,21362],{"encoding":116},"\\|\\cdot\\|_{\\mathrm{TV}}",[23,21364,21366,21384],{"className":21365,"ariaHidden":122},[121],[23,21367,21369,21372,21375,21378,21381],{"className":21368},[126],[23,21370],{"className":21371,"style":131},[130],[23,21373,20680],{"className":21374},[135],[23,21376],{"className":21377,"style":392},[148],[23,21379,5758],{"className":21380},[612],[23,21382],{"className":21383,"style":392},[148],[23,21385,21387,21390],{"className":21386},[126],[23,21388],{"className":21389,"style":131},[130],[23,21391,21393,21396],{"className":21392},[135],[23,21394,20680],{"className":21395},[135],[23,21397,21399],{"className":21398},[998],[23,21400,21402,21428],{"className":21401},[199,200],[23,21403,21405,21425],{"className":21404},[204],[23,21406,21408],{"className":21407,"style":2026},[208],[23,21409,21410,21413],{"style":2029},[23,21411],{"className":21412,"style":1015},[216],[23,21414,21416],{"className":21415},[1019,1020,1021,1022],[23,21417,21419],{"className":21418},[135,1022],[23,21420,21422],{"className":21421},[135,1022],[23,21423,21310],{"className":21424,"style":7088},[135,5391,1022],[23,21426,297],{"className":21427},[296],[23,21429,21431],{"className":21430},[204],[23,21432,21434],{"className":21433,"style":2048},[208],[23,21435],{}," 是全变差距离。公式的几何意义很直接：两个类条件分布的重叠面积越大，错误率越高；当两个分布完全重合时 ",[23,21438,21440,21468],{"className":21439,"translate":27},[31],[23,21441,21443],{"className":21442},[35],[37,21444,21445],{"xmlns":39},[42,21446,21447,21465],{},[45,21448,21449,21457,21459,21461,21463],{},[19137,21450,21451,21453,21455],{},[48,21452,11],{},[48,21454,5342],{},[51,21456,6881],{},[51,21458,69],{},[2581,21460,4435],{},[48,21462,2875],{"mathvariant":111},[2581,21464,9952],{},[114,21466,21467],{"encoding":116},"p_e^*=1\u002F2",[23,21469,21471,21538],{"className":21470,"ariaHidden":122},[121],[23,21472,21474,21478,21529,21532,21535],{"className":21473},[126],[23,21475],{"className":21476,"style":21477},[130],"height:0.9357em;vertical-align:-0.247em;",[23,21479,21481,21484],{"className":21480},[135],[23,21482,11],{"className":21483},[135,136],[23,21485,21487],{"className":21486},[998],[23,21488,21490,21521],{"className":21489},[199,200],[23,21491,21493,21518],{"className":21492},[204],[23,21494,21496,21507],{"className":21495,"style":6912},[208],[23,21497,21498,21501],{"style":19345},[23,21499],{"className":21500,"style":1015},[216],[23,21502,21504],{"className":21503},[1019,1020,1021,1022],[23,21505,5342],{"className":21506},[135,136,1022],[23,21508,21509,21512],{"style":4696},[23,21510],{"className":21511,"style":1015},[216],[23,21513,21515],{"className":21514},[1019,1020,1021,1022],[23,21516,6881],{"className":21517},[612,1022],[23,21519,297],{"className":21520},[296],[23,21522,21524],{"className":21523},[204],[23,21525,21527],{"className":21526,"style":19375},[208],[23,21528],{},[23,21530],{"className":21531,"style":149},[148],[23,21533,69],{"className":21534},[153],[23,21536],{"className":21537,"style":149},[148],[23,21539,21541,21544],{"className":21540},[126],[23,21542],{"className":21543,"style":131},[130],[23,21545,10469],{"className":21546},[135],"，当支撑集不相交时 ",[23,21549,21551,21575],{"className":21550,"translate":27},[31],[23,21552,21554],{"className":21553},[35],[37,21555,21556],{"xmlns":39},[42,21557,21558,21572],{},[45,21559,21560,21568,21570],{},[19137,21561,21562,21564,21566],{},[48,21563,11],{},[48,21565,5342],{},[51,21567,6881],{},[51,21569,69],{},[2581,21571,1532],{},[114,21573,21574],{"encoding":116},"p_e^*=0",[23,21576,21578,21644],{"className":21577,"ariaHidden":122},[121],[23,21579,21581,21584,21635,21638,21641],{"className":21580},[126],[23,21582],{"className":21583,"style":21477},[130],[23,21585,21587,21590],{"className":21586},[135],[23,21588,11],{"className":21589},[135,136],[23,21591,21593],{"className":21592},[998],[23,21594,21596,21627],{"className":21595},[199,200],[23,21597,21599,21624],{"className":21598},[204],[23,21600,21602,21613],{"className":21601,"style":6912},[208],[23,21603,21604,21607],{"style":19345},[23,21605],{"className":21606,"style":1015},[216],[23,21608,21610],{"className":21609},[1019,1020,1021,1022],[23,21611,5342],{"className":21612},[135,136,1022],[23,21614,21615,21618],{"style":4696},[23,21616],{"className":21617,"style":1015},[216],[23,21619,21621],{"className":21620},[1019,1020,1021,1022],[23,21622,6881],{"className":21623},[612,1022],[23,21625,297],{"className":21626},[296],[23,21628,21630],{"className":21629},[204],[23,21631,21633],{"className":21632,"style":19375},[208],[23,21634],{},[23,21636],{"className":21637,"style":149},[148],[23,21639,69],{"className":21640},[153],[23,21642],{"className":21643,"style":149},[148],[23,21645,21647,21650],{"className":21646},[126],[23,21648],{"className":21649,"style":2770},[130],[23,21651,1532],{"className":21652},[135],[11,21654,21655],{},"若先验不相等，公式改为",[23,21657,21659],{"className":21658,"translate":27},[26],[23,21660,21662,21743],{"className":21661,"translate":27},[31],[23,21663,21665],{"className":21664},[35],[37,21666,21667],{"xmlns":39,"display":40},[42,21668,21669,21740],{},[45,21670,21671,21679,21681,21683,21685,21687,21689,21696,21702,21704,21706,21708,21710,21712,21718,21724,21726,21728,21730,21732,21734,21736,21738],{},[19137,21672,21673,21675,21677],{},[48,21674,11],{},[48,21676,5342],{},[51,21678,6881],{},[51,21680,69],{},[51,21682,1557],{},[48,21684,6967],{},[51,21686,6959],{},[51,21688,9010],{"stretchy":53},[898,21690,21691,21694],{},[48,21692,21693],{},"π",[2581,21695,1532],{},[898,21697,21698,21700],{},[48,21699,11],{},[2581,21701,1532],{},[51,21703,54],{"stretchy":53},[48,21705,753],{},[51,21707,66],{"stretchy":53},[51,21709,747],{"separator":122},[88,21711,5360],{},[898,21713,21714,21716],{},[48,21715,21693],{},[2581,21717,4435],{},[898,21719,21720,21722],{},[48,21721,11],{},[2581,21723,4435],{},[51,21725,54],{"stretchy":53},[48,21727,753],{},[51,21729,66],{"stretchy":53},[51,21731,9020],{"stretchy":53},[88,21733,90],{},[48,21735,1606],{},[48,21737,753],{},[51,21739,747],{"separator":122},[114,21741,21742],{"encoding":116},"p_e^*=\\int \\min\\{\\pi_0p_0(x),\\ 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高斯情形：马氏距离与 ",[23,22038,22040,22054],{"className":22039,"translate":27},[31],[23,22041,22043],{"className":22042},[35],[37,22044,22045],{"xmlns":39},[42,22046,22047,22052],{},[45,22048,22049],{},[48,22050,22051],{},"Q",[114,22053,22051],{"encoding":116},[23,22055,22057],{"className":22056,"ariaHidden":122},[121],[23,22058,22060,22063],{"className":22059},[126],[23,22061],{"className":22062,"style":817},[130],[23,22064,22051],{"className":22065},[135,136]," 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",[23,23757,23759,23777],{"className":23758,"translate":27},[31],[23,23760,23762],{"className":23761},[35],[37,23763,23764],{"xmlns":39},[42,23765,23766,23774],{},[45,23767,23768,23770,23772],{},[48,23769,369],{},[51,23771,69],{},[2581,23773,4435],{},[114,23775,23776],{"encoding":116},"Y=1",[23,23778,23780,23798],{"className":23779,"ariaHidden":122},[121],[23,23781,23783,23786,23789,23792,23795],{"className":23782},[126],[23,23784],{"className":23785,"style":470},[130],[23,23787,369],{"className":23788,"style":392},[135,136],[23,23790],{"className":23791,"style":149},[148],[23,23793,69],{"className":23794},[153],[23,23796],{"className":23797,"style":149},[148],[23,23799,23801,23804],{"className":23800},[126],[23,23802],{"className":23803,"style":2770},[130],[23,23805,4435],{"className":23806},[135]," 下，对数似然比服从 ",[23,23809,23811,23849],{"className":23810,"translate":27},[31],[23,23812,23814],{"className":23813},[35],[37,23815,23816],{"xmlns":39},[42,23817,23818,23846],{},[45,23819,23820,23822,23824,23830,23832,23834,23836,23838,23844],{},[48,23821,12801],{"mathvariant":375},[51,23823,54],{"stretchy":53},[4518,23825,23826,23828],{},[48,23827,23374],{"mathvariant":111},[2581,23829,9952],{},[48,23831,2875],{"mathvariant":111},[2581,23833,9952],{},[51,23835,747],{"separator":122},[88,23837,5360],{},[4518,23839,23840,23842],{},[48,23841,23374],{"mathvariant":111},[2581,23843,9952],{},[51,23845,66],{"stretchy":53},[114,23847,23848],{"encoding":116},"\\mathcal N(\\Delta^2\u002F2,\\ \\Delta^2)",[23,23850,23852],{"className":23851,"ariaHidden":122},[121],[23,23853,23855,23858,23861,23864,23893,23896,23899,23902,23905,23934],{"className":23854},[126],[23,23856],{"className":23857,"style":10661},[130],[23,23859,12801],{"className":23860,"style":22218},[135,412],[23,23862,54],{"className":23863},[140],[23,23865,23867,23870],{"className":23866},[135],[23,23868,23374],{"className":23869},[135],[23,23871,23873],{"className":23872},[998],[23,23874,23876],{"className":23875},[199],[23,23877,23879],{"className":23878},[204],[23,23880,23882],{"className":23881,"style":4693},[208],[23,23883,23884,23887],{"style":4696},[23,23885],{"className":23886,"style":1015},[216],[23,23888,23890],{"className":23889},[1019,1020,1021,1022],[23,23891,9952],{"className":23892},[135,1022],[23,23894,20560],{"className":23895},[135],[23,23897,747],{"className":23898},[824],[23,23900,5360],{"className":23901},[148],[23,23903],{"className":23904,"style":280},[148],[23,23906,23908,23911],{"className":23907},[135],[23,23909,23374],{"className":23910},[135],[23,23912,23914],{"className":23913},[998],[23,23915,23917],{"className":23916},[199],[23,23918,23920],{"className":23919},[204],[23,23921,23923],{"className":23922,"style":4693},[208],[23,23924,23925,23928],{"style":4696},[23,23926],{"className":23927,"style":1015},[216],[23,23929,23931],{"className":23930},[1019,1020,1021,1022],[23,23932,9952],{"className":23933},[135,1022],[23,23935,66],{"className":23936},[169],"。贝叶斯规则在似然比为零处切换，因此",[23,23939,23941],{"className":23940,"translate":27},[26],[23,23942,23944,24037],{"className":23943,"translate":27},[31],[23,23945,23947],{"className":23946},[35],[37,23948,23949],{"xmlns":39,"display":40},[42,23950,23951,24034],{},[45,23952,23953,23961,23963,23965,23967,23969,23971,23977,23979,23981,23983,23989,23991,23993,23995,23997,23999,24002,24016,24018,24020,24032],{},[19137,23954,23955,23957,23959],{},[48,23956,11],{},[48,23958,5342],{},[51,23960,6881],{},[51,23962,69],{},[48,23964,920],{},[51,23966,54],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,23968,12801],{"mathvariant":375},[51,23970,54],{"stretchy":53},[4518,23972,23973,23975],{},[48,23974,23374],{"mathvariant":111},[2581,23976,9952],{},[48,23978,2875],{"mathvariant":111},[2581,23980,9952],{},[51,23982,747],{"separator":122},[4518,23984,23985,23987],{},[48,23986,23374],{"mathvariant":111},[2581,23988,9952],{},[51,23990,66],{"stretchy":53},[51,23992,9983],{},[2581,23994,1532],{},[51,23996,66],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,23998,69],{},[48,24000,24001],{"mathvariant":111},"Φ",[45,24003,24004,24006,24008,24014],{},[51,24005,54],{"fence":122},[51,24007,4529],{},[71,24009,24010,24012],{},[48,24011,23374],{"mathvariant":111},[2581,24013,9952],{},[51,24015,66],{"fence":122},[51,24017,69],{},[48,24019,22051],{},[45,24021,24022,24024,24030],{},[51,24023,54],{"fence":122},[71,24025,24026,24028],{},[48,24027,23374],{"mathvariant":111},[2581,24029,9952],{},[51,24031,66],{"fence":122},[51,24033,747],{"separator":122},[114,24035,24036],{"encoding":116},"p_e^*=P\\big(\\mathcal N(\\Delta^2\u002F2,\\Delta^2)\u003C0\\big)\n=\\Phi\\left(-\\frac{\\Delta}{2}\\right)\n=Q\\left(\\frac{\\Delta}{2}\\right),",[23,24038,24040,24106,24206,24230,24332],{"className":24039,"ariaHidden":122},[121],[23,24041,24043,24046,24097,24100,24103],{"className":24042},[126],[23,24044],{"className":24045,"style":20722},[130],[23,24047,24049,24052],{"className":24048},[135],[23,24050,11],{"className":24051},[135,136],[23,24053,24055],{"className":24054},[998],[23,24056,24058,24089],{"className":24057},[199,200],[23,24059,24061,24086],{"className":24060},[204],[23,24062,24064,24075],{"className":24063,"style":7045},[208],[23,24065,24066,24069],{"style":19345},[23,24067],{"className":24068,"style":1015},[216],[23,24070,24072],{"className":24071},[1019,1020,1021,1022],[23,24073,5342],{"className":24074},[135,136,1022],[23,24076,24077,24080],{"style":4989},[23,24078],{"className":24079,"style":1015},[216],[23,24081,24083],{"className":24082},[1019,1020,1021,1022],[23,24084,6881],{"className":24085},[612,1022],[23,24087,297],{"className":24088},[296],[23,24090,24092],{"className":24091},[204],[23,24093,24095],{"className":24094,"style":19375},[208],[23,24096],{},[23,24098],{"className":24099,"style":149},[148],[23,24101,69],{"className":24102},[153],[23,24104],{"className":24105,"style":149},[148],[23,24107,24109,24112,24115,24121,24124,24127,24156,24159,24162,24165,24194,24197,24200,24203],{"className":24108},[126],[23,24110],{"className":24111,"style":17156},[130],[23,24113,920],{"className":24114,"style":1047},[135,136],[23,24116,24118],{"className":24117},[135],[23,24119,54],{"className":24120},[1066,1067],[23,24122,12801],{"className":24123,"style":22218},[135,412],[23,24125,54],{"className":24126},[140],[23,24128,24130,24133],{"className":24129},[135],[23,24131,23374],{"className":24132},[135],[23,24134,24136],{"className":24135},[998],[23,24137,24139],{"className":24138},[199],[23,24140,24142],{"className":24141},[204],[23,24143,24145],{"className":24144,"style":11002},[208],[23,24146,24147,24150],{"style":4989},[23,24148],{"className":24149,"style":1015},[216],[23,24151,24153],{"className":24152},[1019,1020,1021,1022],[23,24154,9952],{"className":24155},[135,1022],[23,24157,20560],{"className":24158},[135],[23,24160,747],{"className":24161},[824],[23,24163],{"className":24164,"style":280},[148],[23,24166,24168,24171],{"className":24167},[135],[23,24169,23374],{"className":24170},[135],[23,24172,24174],{"className":24173},[998],[23,24175,24177],{"className":24176},[199],[23,24178,24180],{"className":24179},[204],[23,24181,24183],{"className":24182,"style":11002},[208],[23,24184,24185,24188],{"style":4989},[23,24186],{"className":24187,"style":1015},[216],[23,24189,24191],{"className":24190},[1019,1020,1021,1022],[23,24192,9952],{"className":24193},[135,1022],[23,24195,66],{"className":24196},[169],[23,24198],{"className":24199,"style":149},[148],[23,24201,9983],{"className":24202},[153],[23,24204],{"className":24205,"style":149},[148],[23,24207,24209,24212,24215,24221,24224,24227],{"className":24208},[126],[23,24210],{"className":24211,"style":2213},[130],[23,24213,1532],{"className":24214},[135],[23,24216,24218],{"className":24217},[135],[23,24219,66],{"className":24220},[1066,1067],[23,24222],{"className":24223,"style":149},[148],[23,24225,69],{"className":24226},[153],[23,24228],{"className":24229,"style":149},[148],[23,24231,24233,24236,24239,24242,24323,24326,24329],{"className":24232},[126],[23,24234],{"className":24235,"style":20789},[130],[23,24237,24001],{"className":24238},[135],[23,24240],{"className":24241,"style":280},[148],[23,24243,24245,24251,24254,24317],{"className":24244},[1839],[23,24246,24248],{"className":24247,"style":1844},[140,1843],[23,24249,54],{"className":24250},[1066,1021],[23,24252,4529],{"className":24253},[135],[23,24255,24257,24260,24314],{"className":24256},[135],[23,24258],{"className":24259},[140,192],[23,24261,24263],{"className":24262},[71],[23,24264,24266,24306],{"className":24265},[199,200],[23,24267,24269,24303],{"className":24268},[204],[23,24270,24273,24284,24292],{"className":24271,"style":24272},[208],"height:1.3603em;",[23,24274,24275,24278],{"style":212},[23,24276],{"className":24277,"style":217},[216],[23,24279,24281],{"className":24280},[135],[23,24282,9952],{"className":24283},[135],[23,24285,24286,24289],{"style":235},[23,24287],{"className":24288,"style":217},[216],[23,24290],{"className":24291,"style":243},[242],[23,24293,24294,24297],{"style":246},[23,24295],{"className":24296,"style":217},[216],[23,24298,24300],{"className":24299},[135],[23,24301,23374],{"className":24302},[135],[23,24304,297],{"className":24305},[296],[23,24307,24309],{"className":24308},[204],[23,24310,24312],{"className":24311,"style":6287},[208],[23,24313],{},[23,24315],{"className":24316},[169,192],[23,24318,24320],{"className":24319,"style":1844},[169,1843],[23,24321,66],{"className":24322},[1066,1021],[23,24324],{"className":24325,"style":149},[148],[23,24327,69],{"className":24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",[23,24431,24433,24447],{"className":24432,"translate":27},[31],[23,24434,24436],{"className":24435},[35],[37,24437,24438],{"xmlns":39},[42,24439,24440,24444],{},[45,24441,24442],{},[48,24443,24001],{"mathvariant":111},[114,24445,24446],{"encoding":116},"\\Phi",[23,24448,24450],{"className":24449,"ariaHidden":122},[121],[23,24451,24453,24456],{"className":24452},[126],[23,24454],{"className":24455,"style":470},[130],[23,24457,24001],{"className":24458},[135]," 是标准正态分布函数，",[23,24461,24463,24485],{"className":24462,"translate":27},[31],[23,24464,24466],{"className":24465},[35],[37,24467,24468],{"xmlns":39},[42,24469,24470,24482],{},[45,24471,24472,24474,24476,24478,24480],{},[48,24473,22051],{},[51,24475,69],{},[2581,24477,4435],{},[51,24479,4529],{},[48,24481,24001],{"mathvariant":111},[114,24483,24484],{"encoding":116},"Q=1-\\Phi",[23,24486,24488,24506,24524],{"className":24487,"ariaHidden":122},[121],[23,24489,24491,24494,24497,24500,24503],{"className":24490},[126],[23,24492],{"className":24493,"style":817},[130],[23,24495,22051],{"className":24496},[135,136],[23,24498],{"className":24499,"style":149},[148],[23,24501,69],{"className":24502},[153],[23,24504],{"className":24505,"style":149},[148],[23,24507,24509,24512,24515,24518,24521],{"className":24508},[126],[23,24510],{"className":24511,"style":9400},[130],[23,24513,4435],{"className":24514},[135],[23,24516],{"className":24517,"style":392},[148],[23,24519,4529],{"className":24520},[612],[23,24522],{"className":24523,"style":392},[148],[23,24525,24527,24530],{"className":24526},[126],[23,24528],{"className":24529,"style":470},[130],[23,24531,24001],{"className":24532},[135],[11,24534,24535],{},"几个数值可以帮助建立直觉：",[3764,24537,24538,24893],{},[3767,24539,24540],{},[3770,24541,24542,24573,24700],{},[3773,24543,24544],{},[23,24545,24547,24561],{"className":24546,"translate":27},[31],[23,24548,24550],{"className":24549},[35],[37,24551,24552],{"xmlns":39},[42,24553,24554,24558],{},[45,24555,24556],{},[48,24557,23374],{"mathvariant":111},[114,24559,24560],{"encoding":116},"\\Delta",[23,24562,24564],{"className":24563,"ariaHidden":122},[121],[23,24565,24567,24570],{"className":24566},[126],[23,24568],{"className":24569,"style":470},[130],[23,24571,23374],{"className":24572},[135],[3773,24574,24575],{},[23,24576,24578,24612],{"className":24577,"translate":27},[31],[23,24579,24581],{"className":24580},[35],[37,24582,24583],{"xmlns":39},[42,24584,24585,24609],{},[45,24586,24587,24595,24597,24599,24601,24603,24605,24607],{},[19137,24588,24589,24591,24593],{},[48,24590,11],{},[48,24592,5342],{},[51,24594,6881],{},[51,24596,69],{},[48,24598,22051],{},[51,24600,54],{"stretchy":53},[48,24602,23374],{"mathvariant":111},[48,24604,2875],{"mathvariant":111},[2581,24606,9952],{},[51,24608,66],{"stretchy":53},[114,24610,24611],{"encoding":116},"p_e^*=Q(\\Delta\u002F2)",[23,24613,24615,24681],{"className":24614,"ariaHidden":122},[121],[23,24616,24618,24621,24672,24675,24678],{"className":24617},[126],[23,24619],{"className":24620,"style":21477},[130],[23,24622,24624,24627],{"className":24623},[135],[23,24625,11],{"className":24626},[135,136],[23,24628,24630],{"className":24629},[998],[23,24631,24633,24664],{"className":24632},[199,200],[23,24634,24636,24661],{"className":24635},[204],[23,24637,24639,24650],{"className":24638,"style":6912},[208],[23,24640,24641,24644],{"style":19345},[23,24642],{"className":24643,"style":1015},[216],[23,24645,24647],{"className":24646},[1019,1020,1021,1022],[23,24648,5342],{"className":24649},[135,136,1022],[23,24651,24652,24655],{"style":4696},[23,24653],{"className":24654,"style":1015},[216],[23,24656,24658],{"className":24657},[1019,1020,1021,1022],[23,24659,6881],{"className":24660},[612,1022],[23,24662,297],{"className":24663},[296],[23,24665,24667],{"className":24666},[204],[23,24668,24670],{"className":24669,"style":19375},[208],[23,24671],{},[23,24673],{"className":24674,"style":149},[148],[23,24676,69],{"className":24677},[153],[23,24679],{"className":24680,"style":149},[148],[23,24682,24684,24687,24690,24693,24697],{"className":24683},[126],[23,24685],{"className":24686,"style":131},[130],[23,24688,22051],{"className":24689},[135,136],[23,24691,54],{"className":24692},[140],[23,24694,24696],{"className":24695},[135],"Δ\u002F2",[23,24698,66],{"className":24699},[169],[3773,24701,24702,24703],{},"Bhattacharyya 上界 ",[23,24704,24706,24745],{"className":24705,"translate":27},[31],[23,24707,24709],{"className":24708},[35],[37,24710,24711],{"xmlns":39},[42,24712,24713,24742],{},[45,24714,24715,24723],{},[1530,24716,24717],{"scriptlevel":1532,"displaystyle":53},[71,24718,24719,24721],{},[2581,24720,4435],{},[2581,24722,9952],{},[4518,24724,24725,24727],{},[48,24726,5342],{},[45,24728,24729,24731,24737,24739],{},[51,24730,4529],{},[4518,24732,24733,24735],{},[48,24734,23374],{"mathvariant":111},[2581,24736,9952],{},[48,24738,2875],{"mathvariant":111},[2581,24740,24741],{},"8",[114,24743,24744],{"encoding":116},"\\tfrac12e^{-\\Delta^2\u002F8}",[23,24746,24748],{"className":24747,"ariaHidden":122},[121],[23,24749,24751,24755,24823],{"className":24750},[126],[23,24752],{"className":24753,"style":24754},[130],"height:1.3319em;vertical-align:-0.345em;",[23,24756,24758,24761,24820],{"className":24757},[135],[23,24759],{"className":24760},[140,192],[23,24762,24764],{"className":24763},[71],[23,24765,24767,24812],{"className":24766},[199,200],[23,24768,24770,24809],{"className":24769},[204],[23,24771,24773,24787,24795],{"className":24772,"style":10196},[208],[23,24774,24775,24778],{"style":10199},[23,24776],{"className":24777,"style":217},[216],[23,24779,24781],{"className":24780},[1019,1020,1021,1022],[23,24782,24784],{"className":24783},[135,1022],[23,24785,9952],{"className":24786},[135,1022],[23,24788,24789,24792],{"style":235},[23,24790],{"className":24791,"style":217},[216],[23,24793],{"className":24794,"style":243},[242],[23,24796,24797,24800],{"style":10222},[23,24798],{"className":24799,"style":217},[216],[23,24801,24803],{"className":24802},[1019,1020,1021,1022],[23,24804,24806],{"className":24805},[135,1022],[23,24807,4435],{"className":24808},[135,1022],[23,24810,297],{"className":24811},[296],[23,24813,24815],{"className":24814},[204],[23,24816,24818],{"className":24817,"style":10244},[208],[23,24819],{},[23,24821],{"className":24822},[169,192],[23,24824,24826,24829],{"className":24825},[135],[23,24827,5342],{"className":24828},[135,136],[23,24830,24832],{"className":24831},[998],[23,24833,24835],{"className":24834},[199],[23,24836,24838],{"className":24837},[204],[23,24839,24842],{"className":24840,"style":24841},[208],"height:0.9869em;",[23,24843,24844,24847],{"style":4696},[23,24845],{"className":24846,"style":1015},[216],[23,24848,24850],{"className":24849},[1019,1020,1021,1022],[23,24851,24853,24856,24889],{"className":24852},[135,1022],[23,24854,4529],{"className":24855},[135,1022],[23,24857,24859,24862],{"className":24858},[135,1022],[23,24860,23374],{"className":24861},[135,1022],[23,24863,24865],{"className":24864},[998],[23,24866,24868],{"className":24867},[199],[23,24869,24871],{"className":24870},[204],[23,24872,24875],{"className":24873,"style":24874},[208],"height:0.8913em;",[23,24876,24878,24882],{"style":24877},"top:-2.931em;margin-right:0.0714em;",[23,24879],{"className":24880,"style":24881},[216],"height:2.5em;",[23,24883,24886],{"className":24884},[1019,24885,1067,1022],"reset-size3",[23,24887,9952],{"className":24888},[135,1022],[23,24890,24892],{"className":24891},[135,1022],"\u002F8",[3783,24894,24895,24905,24915,24926],{},[3770,24896,24897,24899,24902],{},[3788,24898,4435],{},[3788,24900,24901],{},"0.3085",[3788,24903,24904],{},"0.4412",[3770,24906,24907,24909,24912],{},[3788,24908,9952],{},[3788,24910,24911],{},"0.1587",[3788,24913,24914],{},"0.3033",[3770,24916,24917,24920,24923],{},[3788,24918,24919],{},"3",[3788,24921,24922],{},"0.0668",[3788,24924,24925],{},"0.1623",[3770,24927,24928,24931,24934],{},[3788,24929,24930],{},"4",[3788,24932,24933],{},"0.0228",[3788,24935,24936],{},"0.0677",[11,24938,24939],{},"上界来自",[23,24941,24943],{"className":24942,"translate":27},[26],[23,24944,24946,25034],{"className":24945,"translate":27},[31],[23,24947,24949],{"className":24948},[35],[37,24950,24951],{"xmlns":39,"display":40},[42,24952,24953,25031],{},[45,24954,24955,24963,24965,24971,24973,24980,24982,24988,24990,24996,24998,25000,25006,25009,25011,25029],{},[19137,24956,24957,24959,24961],{},[48,24958,11],{},[48,24960,5342],{},[51,24962,6881],{},[51,24964,12268],{},[71,24966,24967,24969],{},[2581,24968,4435],{},[2581,24970,9952],{},[88,24972,90],{},[45,24974,24975,24977],{},[48,24976,2533],{"mathvariant":111},[48,24978,24979],{"mathvariant":111},"C",[51,24981,54],{"stretchy":53},[898,24983,24984,24986],{},[48,24985,920],{},[2581,24987,1532],{},[51,24989,747],{"separator":122},[898,24991,24992,24994],{},[48,24993,920],{},[2581,24995,4435],{},[51,24997,66],{"stretchy":53},[51,24999,69],{},[71,25001,25002,25004],{},[2581,25003,4435],{},[2581,25005,9952],{},[48,25007,25008],{},"exp",[51,25010,6959],{},[45,25012,25013,25015,25017,25027],{},[51,25014,54],{"fence":122},[51,25016,4529],{},[71,25018,25019,25025],{},[4518,25020,25021,25023],{},[48,25022,23374],{"mathvariant":111},[2581,25024,9952],{},[2581,25026,24741],{},[51,25028,66],{"fence":122},[51,25030,747],{"separator":122},[114,25032,25033],{"encoding"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\\frac{2}{\\Delta\\sqrt{2\\pi}}e^{-\\Delta^2\u002F8},",[23,25899,25901,25999],{"className":25900,"ariaHidden":122},[121],[23,25902,25904,25907,25910,25913,25990,25993,25996],{"className":25903},[126],[23,25905],{"className":25906,"style":20789},[130],[23,25908,22051],{"className":25909},[135,136],[23,25911],{"className":25912,"style":280},[148],[23,25914,25916,25922,25984],{"className":25915},[1839],[23,25917,25919],{"className":25918,"style":1844},[140,1843],[23,25920,54],{"className":25921},[1066,1021],[23,25923,25925,25928,25981],{"className":25924},[135],[23,25926],{"className":25927},[140,192],[23,25929,25931],{"className":25930},[71],[23,25932,25934,25973],{"className":25933},[199,200],[23,25935,25937,25970],{"className":25936},[204],[23,25938,25940,25951,25959],{"className":25939,"style":24272},[208],[23,25941,25942,25945],{"style":212},[23,25943],{"className":25944,"style":217},[216],[23,25946,25948],{"className":25947},[135],[23,25949,9952],{"className":25950},[135],[23,25952,25953,25956],{"style":235},[23,25954],{"className":25955,"style":217},[216],[23,25957],{"className":25958,"style":243},[242],[23,25960,25961,25964],{"style":246},[23,25962],{"className":25963,"style":217},[216],[23,25965,25967],{"className":25966},[135],[23,25968,23374],{"className":25969},[135],[23,25971,297],{"className":25972},[296],[23,25974,25976],{"className":25975},[204],[23,25977,25979],{"className":25978,"style":6287},[208],[23,25980],{},[23,25982],{"className":25983},[169,192],[23,25985,25987],{"className":25986,"style":1844},[169,1843],[23,25988,66],{"className":25989},[1066,1021],[23,25991],{"className":25992,"style":149},[148],[23,25994,917],{"className":25995},[153],[23,25997],{"className":25998,"style":149},[148],[23,26000,26002,26006,26123,26188],{"className":26001},[126],[23,26003],{"className":26004,"style":26005},[130],"height:2.2514em;vertical-align:-0.93em;",[23,26007,26009,26012,26120],{"className":26008},[135],[23,26010],{"className":26011},[140,192],[23,26013,26015],{"className":26014},[71],[23,26016,26018,26111],{"className":26017},[199,200],[23,26019,26021,26108],{"className":26020},[204],[23,26022,26024,26089,26097],{"className":26023,"style":12409},[208],[23,26025,26027,26030],{"style":26026},"top:-2.2028em;",[23,26028],{"className":26029,"style":217},[216],[23,26031,26033,26036],{"className":26032},[135],[23,26034,23374],{"className":26035},[135],[23,26037,26039],{"className":26038},[135,25667],[23,26040,26042,26080],{"className":26041},[199,200],[23,26043,26045,26077],{"className":26044},[204],[23,26046,26049,26064],{"className":26047,"style":26048},[208],"height:0.9072em;",[23,26050,26052,26055],{"className":26051,"style":7130},[8672],[23,26053],{"className":26054,"style":217},[216],[23,26056,26058,26061],{"className":26057,"style":25687},[135],[23,26059,9952],{"className":26060},[135],[23,26062,21693],{"className":26063,"style":676},[135,136],[23,26065,26067,26070],{"style":26066},"top:-2.8672em;",[23,26068],{"className":26069,"style":217},[216],[23,26071,26073],{"className":26072,"style":25778},[25777],[8688,26074,26075],{"xmlns":8690,"width":8691,"height":25781,"viewBox":25782,"preserveAspectRatio":8694},[8696,26076],{"d":25785},[23,26078,297],{"className":26079},[296],[23,26081,26083],{"className":26082},[204],[23,26084,26087],{"className":26085,"style":26086},[208],"height:0.1328em;",[23,26088],{},[23,26090,26091,26094],{"style":235},[23,26092],{"className":26093,"style":217},[216],[23,26095],{"className":26096,"style":243},[242],[23,26098,26099,26102],{"style":246},[23,26100],{"className":26101,"style":217},[216],[23,26103,26105],{"className":26104},[135],[23,26106,9952],{"className":26107},[135],[23,26109,297],{"className":26110},[296],[23,26112,26114],{"className":26113},[204],[23,26115,26118],{"className":26116,"style":26117},[208],"height:0.93em;",[23,26119],{},[23,26121],{"className":26122},[169,192],[23,26124,26126,26129],{"className":26125},[135],[23,26127,5342],{"className":26128},[135,136],[23,26130,26132],{"className":26131},[998],[23,26133,26135],{"className":26134},[199],[23,26136,26138],{"className":26137},[204],[23,26139,26142],{"className":26140,"style":26141},[208],"height:1.0369em;",[23,26143,26144,26147],{"style":4989},[23,26145],{"className":26146,"style":1015},[216],[23,26148,26150],{"className":26149},[1019,1020,1021,1022],[23,26151,26153,26156,26185],{"className":26152},[135,1022],[23,26154,4529],{"className":26155},[135,1022],[23,26157,26159,26162],{"className":26158},[135,1022],[23,26160,23374],{"className":26161},[135,1022],[23,26163,26165],{"className":26164},[998],[23,26166,26168],{"className":26167},[199],[23,26169,26171],{"className":26170},[204],[23,26172,26174],{"className":26173,"style":24874},[208],[23,26175,26176,26179],{"style":24877},[23,26177],{"className":26178,"style":24881},[216],[23,26180,26182],{"className":26181},[1019,24885,1067,1022],[23,26183,9952],{"className":26184},[135,1022],[23,26186,24892],{"className":26187},[135,1022],[23,26189,747],{"className":26190},[824],[11,26192,26193,26194,1471],{},"与上界只差一个多项式因子。指数部分完全一致，这正是 Chernoff 界想要抓住的东西：错误率衰减的",[18,26195,26196],{},"速率",[2501,26198,26200],{"id":26199},"_43-用散度给错误率定界","4.3 用散度给错误率定界",[11,26202,26203],{},"全变差是精确的，但它通常难以估计。实践中更常用的是 Hellinger 距离与 KL 散度。它们与全变差之间有一条标准的不等式链：",[23,26205,26207],{"className":26206,"translate":27},[26],[23,26208,26210,26331],{"className":26209,"translate":27},[31],[23,26211,26213],{"className":26212},[35],[37,26214,26215],{"xmlns":39,"display":40},[42,26216,26217,26328],{},[45,26218,26219,26221,26227,26229,26235,26245,26247,26250,26252,26258,26260,26266,26268,26270,26272,26278,26280,26286,26288,26294,26296,26298,26300,26302,26308,26310,26316,26318,26324,26326],{},[48,26220,20680],{"mathvariant":111},[898,26222,26223,26225],{},[48,26224,920],{},[2581,26226,1532],{},[51,26228,4529],{},[898,26230,26231,26233],{},[48,26232,920],{},[2581,26234,4435],{},[898,26236,26237,26239],{},[48,26238,20680],{"mathvariant":111},[45,26240,26241,26243],{},[48,26242,20703],{"mathvariant":111},[48,26244,10790],{"mathvariant":111},[51,26246,12268],{},[48,26248,26249],{},"H",[51,26251,54],{"stretchy":53},[898,26253,26254,26256],{},[48,26255,920],{},[2581,26257,1532],{},[51,26259,747],{"separator":122},[898,26261,26262,26264],{},[48,26263,920],{},[2581,26265,4435],{},[51,26267,66],{"stretchy":53},[51,26269,747],{"separator":122},[148,26271],{"width":750},[4518,26273,26274,26276],{},[48,26275,26249],{},[2581,26277,9952],{},[51,26279,54],{"stretchy":53},[898,26281,26282,26284],{},[48,26283,920],{},[2581,26285,1532],{},[51,26287,747],{"separator":122},[898,26289,26290,26292],{},[48,26291,920],{},[2581,26293,4435],{},[51,26295,66],{"stretchy":53},[51,26297,69],{},[2581,26299,4435],{},[51,26301,4529],{},[45,26303,26304,26306],{},[48,26305,2533],{"mathvariant":111},[48,26307,24979],{"mathvariant":111},[51,26309,54],{"stretchy":53},[898,26311,26312,26314],{},[48,26313,920],{},[2581,26315,1532],{},[51,26317,747],{"separator":122},[898,26319,26320,26322],{},[48,26321,920],{},[2581,26323,4435],{},[51,26325,66],{"stretchy":53},[51,26327,747],{"separator":122},[114,26329,26330],{"encoding":116},"\\|P_0-P_1\\|_{\\mathrm{TV}}\\le H(P_0,P_1),\n\\qquad\nH^2(P_0,P_1)=1-\\mathrm{BC}(P_0,P_1),",[23,26332,26334,26392,26493,26735,26753],{"className":26333,"ariaHidden":122},[121],[23,26335,26337,26340,26343,26383,26386,26389],{"className":26336},[126],[23,26338],{"className":26339,"style":131},[130],[23,26341,20680],{"className":26342},[135],[23,26344,26346,26349],{"className":26345},[135],[23,26347,920],{"className":26348,"style":1047},[135,136],[23,26350,26352],{"className":26351},[998],[23,26353,26355,26375],{"className":26354},[199,200],[23,26356,26358,26372],{"className":26357},[204],[23,26359,26361],{"className":26360,"style":18647},[208],[23,26362,26363,26366],{"style":21199},[23,26364],{"className":26365,"style":1015},[216],[23,26367,26369],{"className":26368},[1019,1020,1021,1022],[23,26370,1532],{"className":26371},[135,1022],[23,26373,297],{"className":26374},[296],[23,26376,26378],{"className":26377},[204],[23,26379,26381],{"className":26380,"style":2048},[208],[23,26382],{},[23,26384],{"className":26385,"style":392},[148],[23,26387,4529],{"className":26388},[612],[23,26390],{"className":26391,"style":392},[148],[23,26393,26395,26398,26438,26484,26487,26490],{"className":26394},[126],[23,26396],{"className":26397,"style":131},[130],[23,26399,26401,26404],{"className":26400},[135],[23,26402,920],{"className":26403,"style":1047},[135,136],[23,26405,26407],{"className":26406},[998],[23,26408,26410,26430],{"className":26409},[199,200],[23,26411,26413,26427],{"className":26412},[204],[23,26414,26416],{"className":26415,"style":18647},[208],[23,26417,26418,26421],{"style":21199},[23,26419],{"className":26420,"style":1015},[216],[23,26422,26424],{"className":26423},[1019,1020,1021,1022],[23,26425,4435],{"className":26426},[135,1022],[23,26428,297],{"className":26429},[296],[23,26431,26433],{"className":26432},[204],[23,26434,26436],{"className":26435,"style":2048},[208],[23,26437],{},[23,26439,26441,26444],{"className":26440},[135],[23,26442,20680],{"className":26443},[135],[23,26445,26447],{"className":26446},[998],[23,26448,26450,26476],{"className":26449},[199,200],[23,26451,26453,26473],{"className":26452},[204],[23,26454,26456],{"className":26455,"style":2026},[208],[23,26457,26458,26461],{"style":2029},[23,26459],{"className":26460,"style":1015},[216],[23,26462,26464],{"className":26463},[1019,1020,1021,1022],[23,26465,26467],{"className":26466},[135,1022],[23,26468,26470],{"className":26469},[135,1022],[23,26471,21310],{"className":26472,"style":7088},[135,5391,1022],[23,26474,297],{"className":26475},[296],[23,26477,26479],{"className":26478},[204],[23,26480,26482],{"className":26481,"style":2048},[208],[23,26483],{},[23,26485],{"className":26486,"style":149},[148],[23,26488,12268],{"className":26489},[153],[23,26491],{"className":26492,"style":149},[148],[23,26494,26496,26500,26504,26507,26547,26550,26553,26593,26596,26599,26602,26605,26634,26637,26677,26680,26683,26723,26726,26729,26732],{"className":26495},[126],[23,26497],{"className":26498,"style":26499},[130],"height:1.1141em;vertical-align:-0.25em;",[23,26501,26249],{"className":26502,"style":26503},[135,136],"margin-right:0.0813em;",[23,26505,54],{"className":26506},[140],[23,26508,26510,26513],{"className":26509},[135],[23,26511,920],{"className":26512,"style":1047},[135,136],[23,26514,26516],{"className":26515},[998],[23,26517,26519,26539],{"className":26518},[199,200],[23,26520,26522,26536],{"className":26521},[204],[23,26523,26525],{"className":26524,"style":18647},[208],[23,26526,26527,26530],{"style":21199},[23,26528],{"className":26529,"style":1015},[216],[23,26531,26533],{"className":26532},[1019,1020,1021,1022],[23,26534,1532],{"className":26535},[135,1022],[23,26537,297],{"className":26538},[296],[23,26540,26542],{"className":26541},[204],[23,26543,26545],{"className":26544,"style":2048},[208],[23,26546],{},[23,26548,747],{"className":26549},[824],[23,26551],{"className":26552,"style":280},[148],[23,26554,26556,26559],{"className":26555},[135],[23,26557,920],{"className":26558,"style":1047},[135,136],[23,26560,26562],{"className":26561},[998],[23,26563,26565,26585],{"className":26564},[199,200],[23,26566,26568,26582],{"className":26567},[204],[23,26569,26571],{"className":26570,"style":18647},[208],[23,26572,26573,26576],{"style":21199},[23,26574],{"className":26575,"style":1015},[216],[23,26577,26579],{"className":26578},[1019,1020,1021,1022],[23,26580,4435],{"className":26581},[135,1022],[23,26583,297],{"className":26584},[296],[23,26586,26588],{"className":26587},[204],[23,26589,26591],{"className":26590,"style":2048},[208],[23,26592],{},[23,26594,66],{"className":26595},[169],[23,26597,747],{"className":26598},[824],[23,26600],{"className":26601,"style":828},[148],[23,26603],{"className":26604,"style":280},[148],[23,26606,26608,26611],{"className":26607},[135],[23,26609,26249],{"className":26610,"style":26503},[135,136],[23,26612,26614],{"className":26613},[998],[23,26615,26617],{"className":26616},[199],[23,26618,26620],{"className":26619},[204],[23,26621,26623],{"className":26622,"style":11002},[208],[23,26624,26625,26628],{"style":4989},[23,26626],{"className":26627,"style":1015},[216],[23,26629,26631],{"className":26630},[1019,1020,1021,1022],[23,26632,9952],{"className":26633},[135,1022],[23,26635,54],{"className":26636},[140],[23,26638,26640,26643],{"className":26639},[135],[23,26641,920],{"className":26642,"style":1047},[135,136],[23,26644,26646],{"className":26645},[998],[23,26647,26649,26669],{"className":26648},[199,200],[23,26650,26652,26666],{"className":26651},[204],[23,26653,26655],{"className":26654,"style":18647},[208],[23,26656,26657,26660],{"style":21199},[23,26658],{"className":26659,"style":1015},[216],[23,26661,26663],{"className":26662},[1019,1020,1021,1022],[23,26664,1532],{"className":26665},[135,1022],[23,26667,297],{"className":26668},[296],[23,26670,26672],{"className":26671},[204],[23,26673,26675],{"className":26674,"style":2048},[208],[23,26676],{},[23,26678,747],{"className":26679},[824],[23,26681],{"className":26682,"style":280},[148],[23,26684,26686,26689],{"className":26685},[135],[23,26687,920],{"className":26688,"style":1047},[135,136],[23,26690,26692],{"className":26691},[998],[23,26693,26695,26715],{"className":26694},[199,200],[23,26696,26698,26712],{"className":26697},[204],[23,26699,26701],{"className":26700,"style":18647},[208],[23,26702,26703,26706],{"style":21199},[23,26704],{"className":26705,"style":1015},[216],[23,26707,26709],{"className":26708},[1019,1020,1021,1022],[23,26710,4435],{"className":26711},[135,1022],[23,26713,297],{"className":26714},[296],[23,26716,26718],{"className":26717},[204],[23,26719,26721],{"className":26720,"style":2048},[208],[23,26722],{},[23,26724,66],{"className":26725},[169],[23,26727],{"className":26728,"style":149},[148],[23,26730,69],{"className":26731},[153],[23,26733],{"className":26734,"style":149},[148],[23,26736,26738,26741,26744,26747,26750],{"className":26737},[126],[23,26739],{"className":26740,"style":9400},[130],[23,26742,4435],{"className":26743},[135],[23,26745],{"className":26746,"style":392},[148],[23,26748,4529],{"className":26749},[612],[23,26751],{"className":26752,"style":392},[148],[23,26754,26756,26759,26765,26768,26808,26811,26814,26854,26857],{"className":26755},[126],[23,26757],{"className":26758,"style":131},[130],[23,26760,26762],{"className":26761},[135],[23,26763,25180],{"className":26764},[135,5391],[23,26766,54],{"className":26767},[140],[23,26769,26771,26774],{"className":26770},[135],[23,26772,920],{"className":26773,"style":1047},[135,136],[23,26775,26777],{"className":26776},[998],[23,26778,26780,26800],{"className":26779},[199,200],[23,26781,26783,26797],{"className":26782},[204],[23,26784,26786],{"className":26785,"style":18647},[208],[23,26787,26788,26791],{"style":21199},[23,26789],{"className":26790,"style":1015},[216],[23,26792,26794],{"className":26793},[1019,1020,1021,1022],[23,26795,1532],{"className":26796},[135,1022],[23,26798,297],{"className":26799},[296],[23,26801,26803],{"className":26802},[204],[23,26804,26806],{"className":26805,"style":2048},[208],[23,26807],{},[23,26809,747],{"className":26810},[824],[23,26812],{"className":26813,"style":280},[148],[23,26815,26817,26820],{"className":26816},[135],[23,26818,920],{"className":26819,"style":1047},[135,136],[23,26821,26823],{"className":26822},[998],[23,26824,26826,26846],{"className":26825},[199,200],[23,26827,26829,26843],{"className":26828},[204],[23,26830,26832],{"className":26831,"style":18647},[208],[23,26833,26834,26837],{"style":21199},[23,26835],{"className":26836,"style":1015},[216],[23,26838,26840],{"className":26839},[1019,1020,1021,1022],[23,26841,4435],{"className":26842},[135,1022],[23,26844,297],{"className":26845},[296],[23,26847,26849],{"className":26848},[204],[23,26850,26852],{"className":26851,"style":2048},[208],[23,26853],{},[23,26855,66],{"className":26856},[169],[23,26858,747],{"className":26859},[824],[11,26861,26862],{},"以及",[23,26864,26866],{"className":26865,"translate":27},[26],[23,26867,26869,26944],{"className":26868,"translate":27},[31],[23,26870,26872],{"className":26871},[35],[37,26873,26874],{"xmlns":39,"display":40},[42,26875,26876,26941],{},[45,26877,26878,26884,26886,26892,26894,26900,26902,26904,26910,26921,26923,26929,26931,26937,26939],{},[4518,26879,26880,26882],{},[48,26881,26249],{},[2581,26883,9952],{},[51,26885,54],{"stretchy":53},[898,26887,26888,26890],{},[48,26889,920],{},[2581,26891,1532],{},[51,26893,747],{"separator":122},[898,26895,26896,26898],{},[48,26897,920],{},[2581,26899,4435],{},[51,26901,66],{"stretchy":53},[51,26903,12268],{},[71,26905,26906,26908],{},[2581,26907,4435],{},[2581,26909,9952],{},[898,26911,26912,26914],{},[48,26913,63],{},[45,26915,26916,26919],{},[48,26917,26918],{"mathvariant":111},"K",[48,26920,555],{"mathvariant":111},[51,26922,54],{"stretchy":53},[898,26924,26925,26927],{},[48,26926,920],{},[2581,26928,1532],{},[48,26930,20680],{"mathvariant":111},[898,26932,26933,26935],{},[48,26934,920],{},[2581,26936,4435],{},[51,26938,66],{"stretchy":53},[48,26940,112],{"mathvariant":111},[114,26942,26943],{"encoding":116},"H^2(P_0,P_1)\\le \\frac12 D_{\\mathrm{KL}}(P_0\\|P_1).",[23,26945,26947,27083],{"className":26946,"ariaHidden":122},[121],[23,26948,26950,26953,26982,26985,27025,27028,27031,27071,27074,27077,27080],{"className":26949},[126],[23,26951],{"className":26952,"style":26499},[130],[23,26954,26956,26959],{"className":26955},[135],[23,26957,26249],{"className":26958,"style":26503},[135,136],[23,26960,26962],{"className":26961},[998],[23,26963,26965],{"className":26964},[199],[23,26966,26968],{"className":26967},[204],[23,26969,26971],{"className":26970,"style":11002},[208],[23,26972,26973,26976],{"style":4989},[23,26974],{"className":26975,"style":1015},[216],[23,26977,26979],{"className":26978},[1019,1020,1021,1022],[23,26980,9952],{"className":26981},[135,1022],[23,26983,54],{"className":26984},[140],[23,26986,26988,26991],{"className":26987},[135],[23,26989,920],{"className":26990,"style":1047},[135,136],[23,26992,26994],{"className":26993},[998],[23,26995,26997,27017],{"className":26996},[199,200],[23,26998,27000,27014],{"className":26999},[204],[23,27001,27003],{"className":27002,"style":18647},[208],[23,27004,27005,27008],{"style":21199},[23,27006],{"className":27007,"style":1015},[216],[23,27009,27011],{"className":27010},[1019,1020,1021,1022],[23,27012,1532],{"className":27013},[135,1022],[23,27015,297],{"className":27016},[296],[23,27018,27020],{"className":27019},[204],[23,27021,27023],{"className":27022,"style":2048},[208],[23,27024],{},[23,27026,747],{"className":27027},[824],[23,27029],{"className":27030,"style":280},[148],[23,27032,27034,27037],{"className":27033},[135],[23,27035,920],{"className":27036,"style":1047},[135,136],[23,27038,27040],{"className":27039},[998],[23,27041,27043,27063],{"className":27042},[199,200],[23,27044,27046,27060],{"className":27045},[204],[23,27047,27049],{"className":27048,"style":18647},[208],[23,27050,27051,27054],{"style":21199},[23,27052],{"className":27053,"style":1015},[216],[23,27055,27057],{"className":27056},[1019,1020,1021,1022],[23,27058,4435],{"className":27059},[135,1022],[23,27061,297],{"className":27062},[296],[23,27064,27066],{"className":27065},[204],[23,27067,27069],{"className":27068,"style":2048},[208],[23,27070],{},[23,27072,66],{"className":27073},[169],[23,27075],{"className":27076,"style":149},[148],[23,27078,12268],{"className":27079},[153],[23,27081],{"className":27082,"style":149},[148],[23,27084,27086,27089,27151,27199,27202,27242,27245,27285,27288],{"className":27085},[126],[23,27087],{"className":27088,"style":12390},[130],[23,27090,27092,27095,27148],{"className":27091},[135],[23,27093],{"className":27094},[140,192],[23,27096,27098],{"className":27097},[71],[23,27099,27101,27140],{"className":27100},[199,200],[23,27102,27104,27137],{"className":27103},[204],[23,27105,27107,27118,27126],{"className":27106,"style":12409},[208],[23,27108,27109,27112],{"style":212},[23,27110],{"className":27111,"style":217},[216],[23,27113,27115],{"className":27114},[135],[23,27116,9952],{"className":27117},[135],[23,27119,27120,27123],{"style":235},[23,27121],{"className":27122,"style":217},[216],[23,27124],{"className":27125,"style":243},[242],[23,27127,27128,27131],{"style":246},[23,27129],{"className":27130,"style":217},[216],[23,27132,27134],{"className":27133},[135],[23,27135,4435],{"className":27136},[135],[23,27138,297],{"className":27139},[296],[23,27141,27143],{"className":27142},[204],[23,27144,27146],{"className":27145,"style":6287},[208],[23,27147],{},[23,27149],{"className":27150},[169,192],[23,27152,27154,27157],{"className":27153},[135],[23,27155,63],{"className":27156,"style":144},[135,136],[23,27158,27160],{"className":27159},[998],[23,27161,27163,27191],{"className":27162},[199,200],[23,27164,27166,27188],{"className":27165},[204],[23,27167,27169],{"className":27168,"style":2026},[208],[23,27170,27172,27175],{"style":27171},"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;",[23,27173],{"className":27174,"style":1015},[216],[23,27176,27178],{"className":27177},[1019,1020,1021,1022],[23,27179,27181],{"className":27180},[135,1022],[23,27182,27184],{"className":27183},[135,1022],[23,27185,27187],{"className":27186},[135,5391,1022],"KL",[23,27189,297],{"className":27190},[296],[23,27192,27194],{"className":27193},[204],[23,27195,27197],{"className":27196,"style":2048},[208],[23,27198],{},[23,27200,54],{"className":27201},[140],[23,27203,27205,27208],{"className":27204},[135],[23,27206,920],{"className":27207,"style":1047},[135,136],[23,27209,27211],{"className":27210},[998],[23,27212,27214,27234],{"className":27213},[199,200],[23,27215,27217,27231],{"className":27216},[204],[23,27218,27220],{"className":27219,"style":18647},[208],[23,27221,27222,27225],{"style":21199},[23,27223],{"className":27224,"style":1015},[216],[23,27226,27228],{"className":27227},[1019,1020,1021,1022],[23,27229,1532],{"className":27230},[135,1022],[23,27232,297],{"className":27233},[296],[23,27235,27237],{"className":27236},[204],[23,27238,27240],{"className":27239,"style":2048},[208],[23,272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",[23,27295,27297,27341],{"className":27296,"translate":27},[31],[23,27298,27300],{"className":27299},[35],[37,27301,27302],{"xmlns":39},[42,27303,27304,27338],{},[45,27305,27306,27314,27316,27324,27326,27328,27330,27336],{},[19137,27307,27308,27310,27312],{},[48,27309,11],{},[48,27311,5342],{},[51,27313,6881],{},[51,27315,69],{},[1530,27317,27318],{"scriptlevel":1532,"displaystyle":53},[71,27319,27320,27322],{},[2581,27321,4435],{},[2581,27323,9952],{},[51,27325,54],{"stretchy":53},[2581,27327,4435],{},[51,27329,4529],{},[45,27331,27332,27334],{},[48,27333,20703],{"mathvariant":111},[48,27335,10790],{"mathvariant":111},[51,27337,66],{"stretchy":53},[114,27339,27340],{"encoding":116},"p_e^*=\\tfrac12(1-\\mathrm{TV})",[23,27342,27344,27410,27500],{"className":27343,"ariaHidden":122},[121],[23,27345,27347,27350,27401,27404,27407],{"className":27346},[126],[23,27348],{"className":27349,"style":21477},[130],[23,27351,27353,27356],{"className":27352},[135],[23,27354,11],{"className":27355},[135,136],[23,27357,27359],{"className":27358},[998],[23,27360,27362,27393],{"className":27361},[199,200],[23,27363,27365,27390],{"className":27364},[204],[23,27366,27368,27379],{"className":27367,"style":6912},[208],[23,27369,27370,27373],{"style":19345},[23,27371],{"className":27372,"style":1015},[216],[23,27374,27376],{"className":27375},[1019,1020,1021,1022],[23,27377,5342],{"className":27378},[135,136,1022],[23,27380,27381,27384],{"style":4696},[23,27382],{"className":27383,"style":1015},[216],[23,27385,27387],{"className":27386},[1019,1020,1021,1022],[23,27388,6881],{"className":27389},[612,1022],[23,27391,297],{"className":27392},[296],[23,27394,27396],{"className":27395},[204],[23,27397,27399],{"className":27398,"style":19375},[208],[23,27400],{},[23,27402],{"className":27403,"style":149},[148],[23,27405,69],{"className":27406},[153],[23,27408],{"className":27409,"style":149},[148],[23,27411,27413,27417,27485,27488,27491,27494,27497],{"className":27412},[126],[23,27414],{"className":27415,"style":27416},[130],"height:1.1901em;vertical-align:-0.345em;",[23,27418,27420,27423,27482],{"className":27419},[135],[23,27421],{"className":27422},[140,192],[23,27424,27426],{"className":27425},[71],[23,27427,27429,27474],{"className":27428},[199,200],[23,27430,27432,27471],{"className":27431},[204],[23,27433,27435,27449,27457],{"className":27434,"style":10196},[208],[23,27436,27437,27440],{"style":10199},[23,27438],{"className":27439,"style":217},[216],[23,27441,27443],{"className":27442},[1019,1020,1021,1022],[23,27444,27446],{"className":27445},[135,1022],[23,27447,9952],{"className":27448},[135,1022],[23,27450,27451,27454],{"style":235},[23,27452],{"className":27453,"style":217},[216],[23,27455],{"className":27456,"style":243},[242],[23,27458,27459,27462],{"style":10222},[23,27460],{"className":27461,"style":217},[216],[23,27463,27465],{"className":27464},[1019,1020,1021,1022],[23,27466,27468],{"className":27467},[135,1022],[23,27469,4435],{"className":27470},[135,1022],[23,27472,297],{"className":27473},[296],[23,27475,27477],{"className":27476},[204],[23,27478,27480],{"className":27479,"style":10244},[208],[23,27481],{},[23,27483],{"className":27484},[169,192],[23,27486,54],{"className":27487},[140],[23,27489,4435],{"className":27490},[135],[23,27492],{"className":27493,"style":392},[148],[23,27495,4529],{"className":27496},[612],[23,27498],{"className":27499,"style":392},[148],[23,27501,27503,27506,27512],{"className":27502},[126],[23,27504],{"className":27505,"style":131},[130],[23,27507,27509],{"className":27508},[135],[23,27510,21310],{"className":27511,"style":7088},[135,5391],[23,27513,66],{"className":27514},[169]," 结合，得到",[23,27517,27519],{"className":27518,"translate":27},[26],[23,27520,27522,27602],{"className":27521,"translate":27},[31],[23,27523,27525],{"className":27524},[35],[37,27526,27527],{"xmlns":39,"display":40},[42,27528,27529,27599],{},[45,27530,27531,27539,27541,27543,27545,27551,27597],{},[19137,27532,27533,27535,27537],{},[48,27534,11],{},[48,27536,5342],{},[51,27538,6881],{},[88,27540,5360],{},[51,27542,7378],{},[88,27544,5360],{},[71,27546,27547,27549],{},[2581,27548,4435],{},[2581,27550,9952],{},[45,27552,27553,27555,27557,27559,27595],{},[51,27554,54],{"fence":122},[2581,27556,4435],{},[51,27558,4529],{},[25516,27560,27561],{},[71,27562,27563,27593],{},[45,27564,27565,27575,27577,27583,27585,27591],{},[898,27566,27567,27569],{},[48,27568,63],{},[45,27570,27571,27573],{},[48,27572,26918],{"mathvariant":111},[48,27574,555],{"mathvariant":111},[51,27576,54],{"stretchy":53},[898,27578,27579,27581],{},[48,27580,920],{},[2581,27582,1532],{},[48,27584,20680],{"mathvariant":111},[898,27586,27587,27589],{},[48,27588,920],{},[2581,27590,4435],{},[51,27592,66],{"stretchy":53},[2581,27594,9952],{},[51,27596,66],{"fence":122},[48,27598,112],{"mathvariant":111},[114,27600,27601],{"encoding":116},"p_e^*\\ \\ge\\ \\frac12\\left(1-\\sqrt{\\frac{D_{\\mathrm{KL}}(P_0\\|P_1)}{2}}\\right).",[23,27603,27605,27677],{"className":27604,"ariaHidden":122},[121],[23,27606,27608,27611,27662,27665,27668,27671,27674],{"className":27607},[126],[23,27609],{"className":27610,"style":20722},[130],[23,27612,27614,27617],{"className":27613},[135],[23,27615,11],{"className":27616},[135,136],[23,27618,27620],{"className":27619},[998],[23,27621,27623,27654],{"className":27622},[199,200],[23,27624,27626,27651],{"className":27625},[204],[23,27627,27629,27640],{"className":27628,"style":7045},[208],[23,27630,27631,27634],{"style":19345},[23,27632],{"className":27633,"style":1015},[216],[23,27635,27637],{"className":27636},[1019,1020,1021,1022],[23,27638,5342],{"className":27639},[135,136,1022],[23,27641,27642,27645],{"style":4989},[23,27643],{"className":27644,"style":1015},[216],[23,27646,27648],{"className":27647},[1019,1020,1021,1022],[23,27649,6881],{"className":27650},[612,1022],[23,27652,297],{"className":27653},[296],[23,27655,27657],{"className":27656},[204],[23,27658,27660],{"className":27659,"style":19375},[208],[23,27661],{},[23,27663,5360],{"className":27664},[148],[23,27666],{"className":27667,"style":149},[148],[23,27669,7378],{"className":27670},[153],[23,27672,5360],{"className":27673},[148],[23,27675],{"className":27676,"style":149},[148],[23,27678,27680,27684,27746,27749,28024,28027],{"className":27679},[126],[23,27681],{"className":27682,"style":27683},[130],"height:3em;vertical-align:-1.25em;",[23,27685,27687,27690,27743],{"className":27686},[135],[23,27688],{"className":27689},[140,192],[23,27691,27693],{"className":27692},[71],[23,27694,27696,27735],{"className":27695},[199,200],[23,27697,27699,27732],{"className":27698},[204],[23,27700,27702,27713,27721],{"className":27701,"style":12409},[208],[23,27703,27704,27707],{"style":212},[23,27705],{"className":27706,"style":217},[216],[23,27708,27710],{"className":27709},[135],[23,27711,9952],{"className":27712},[135],[23,27714,27715,27718],{"style":235},[23,27716],{"className":27717,"style":217},[216],[23,27719],{"className":27720,"style":243},[242],[23,27722,27723,27726],{"style":246},[23,27724],{"className":27725,"style":217},[216],[23,27727,27729],{"className":27728},[135],[23,27730,4435],{"className":27731},[135],[23,27733,297],{"className":27734},[296],[23,27736,27738],{"className":27737},[204],[23,27739,27741],{"className":27740,"style":6287},[208],[23,27742],{},[23,27744],{"className":27745},[169,192],[23,27747],{"className":27748,"style":280},[148],[23,27750,27752,27758,27761,27764,27767,27770,28018],{"className":27751},[1839],[23,27753,27755],{"className":27754,"style":1844},[140,1843],[23,27756,54],{"className":27757},[1066,10069],[23,27759,4435],{"className":27760},[135],[23,27762],{"className":27763,"style":392},[148],[23,27765,4529],{"className":27766},[612],[23,27768],{"className":27769,"style":392},[148],[23,27771,27773],{"className":27772},[135,25667],[23,27774,27776,28009],{"className":27775},[199,200],[23,27777,27779,28006],{"className":27778},[204],[23,27780,27783,27989],{"className":27781,"style":27782},[208],"height:1.7044em;",[23,27784,27787,27791],{"className":27785,"style":27786},[8672],"top:-4.4em;",[23,27788],{"className":27789,"style":27790},[216],"height:4.4em;",[23,27792,27795],{"className":27793,"style":27794},[135],"padding-left:1em;",[23,27796,27798,27801,27986],{"className":27797},[135],[23,27799],{"className":27800},[140,192],[23,27802,27804],{"className":27803},[71],[23,27805,27807,27978],{"className":27806},[199,200],[23,27808,27810,27975],{"className":27809},[204],[23,27811,27813,27824,27832],{"className":27812,"style":209},[208],[23,27814,27815,27818],{"style":212},[23,27816],{"className":27817,"style":217},[216],[23,27819,27821],{"className":27820},[135],[23,27822,9952],{"className":27823},[135],[23,27825,27826,27829],{"style":235},[23,27827],{"className":27828,"style":217},[216],[23,27830],{"className":27831,"style":243},[242],[23,27833,27834,27837],{"style":246},[23,27835],{"className":27836,"style":217},[216],[23,27838,27840,27886,27889,27929,27932,27972],{"className":27839},[135],[23,27841,27843,27846],{"className":27842},[135],[23,27844,63],{"className":27845,"style":144},[135,136],[23,27847,27849],{"className":27848},[998],[23,27850,27852,27878],{"className":27851},[199,200],[23,27853,27855,27875],{"className":27854},[204],[23,27856,27858],{"className":27857,"style":2026},[208],[23,27859,27860,27863],{"style":27171},[23,27861],{"className":27862,"style":1015},[216],[23,27864,27866],{"className":27865},[1019,1020,1021,1022],[23,27867,27869],{"className":27868},[135,1022],[23,27870,27872],{"className":27871},[135,1022],[23,27873,27187],{"className":27874},[135,5391,1022],[23,27876,297],{"className":27877},[296],[23,27879,27881],{"className":27880},[204],[23,27882,27884],{"className":27883,"style":2048},[208],[23,27885],{},[23,27887,54],{"className":27888},[140],[23,27890,27892,27895],{"className":27891},[135],[23,27893,920],{"className":27894,"style":1047},[135,136],[23,27896,27898],{"className":27897},[998],[23,27899,27901,27921],{"className":27900},[199,200],[23,27902,27904,27918],{"className":27903},[204],[23,27905,27907],{"className":27906,"style":18647},[208],[23,27908,27909,27912],{"style":21199},[23,27910],{"className":27911,"style":1015},[216],[23,27913,27915],{"className":27914},[1019,1020,1021,1022],[23,27916,1532],{"className":27917},[135,1022],[23,27919,297],{"className":27920},[296],[23,27922,27924],{"className":27923},[204],[23,27925,27927],{"className":27926,"style":2048},[208],[23,27928],{},[23,27930,20680],{"className":27931},[135],[23,27933,27935,27938],{"className":27934},[135],[23,27936,920],{"className":27937,"style":1047},[135,136],[23,27939,27941],{"className":27940},[998],[23,27942,27944,27964],{"className":27943},[199,200],[23,27945,27947,27961],{"className":27946},[204],[23,27948,27950],{"className":27949,"style":18647},[208],[23,27951,27952,27955],{"style":21199},[23,27953],{"className":27954,"style":1015},[216],[23,27956,27958],{"className":27957},[1019,1020,1021,1022],[23,27959,4435],{"className":27960},[135,1022],[23,27962,297],{"className":27963},[296],[23,27965,27967],{"className":27966},[204],[23,27968,27970],{"className":27969,"style":2048},[208],[23,27971],{},[23,27973,66],{"className":27974},[169],[23,27976,297],{"className":27977},[296],[23,27979,27981],{"className":27980},[204],[23,27982,27984],{"className":27983,"style":6287},[208],[23,27985],{},[23,27987],{"className":27988},[169,192],[23,27990,27992,27995],{"style":27991},"top:-3.6644em;",[23,27993],{"className":27994,"style":27790},[216],[23,27996,27999],{"className":27997,"style":27998},[25777],"min-width:1.02em;height:2.48em;",[8688,28000,28003],{"xmlns":8690,"width":8691,"height":28001,"viewBox":28002,"preserveAspectRatio":8694},"2.48em","0 0 400000 2592",[8696,28004],{"d":28005},"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",[23,28007,297],{"className":28008},[296],[23,28010,28012],{"className":28011},[204],[23,28013,28016],{"className":28014,"style":28015},[208],"height:0.7356em;",[23,28017],{},[23,28019,28021],{"className":28020,"style":1844},[169,1843],[23,28022,66],{"className":28023},[1066,10069],[23,28025],{"className":28026,"style":280},[148],[23,28028,112],{"className":28029},[135],[11,28031,28032,28033,28036],{},"这是 Le Cam 两点法的核心不等式：如果两个分布很难区分（KL 很小），那么任何分类器的错误率都不可能低。注意方向——",[18,28034,28035],{},"KL 给的是错误率的下界，不是上界","。KL 很大并不意味着问题容易，只意味着这个下界变得松甚至变得平凡。",[11,28038,28039],{},"上界则需要另一个量：",[23,28041,28043],{"className":28042,"translate":27},[26],[23,28044,28046,28102],{"className":28045,"translate":27},[31],[23,28047,28049],{"className":28048},[35],[37,28050,28051],{"xmlns":39,"display":40},[42,28052,28053,28099],{},[45,28054,28055,28063,28065,28071,28073,28079,28081,28087,28089,28095,28097],{},[19137,28056,28057,28059,28061],{},[48,28058,11],{},[48,28060,5342],{},[51,28062,6881],{},[51,28064,12268],{},[71,28066,28067,28069],{},[2581,28068,4435],{},[2581,28070,9952],{},[88,28072,90],{},[45,28074,28075,28077],{},[48,28076,2533],{"mathvariant":111},[48,28078,24979],{"mathvariant":111},[51,28080,54],{"stretchy":53},[898,28082,28083,28085],{},[48,28084,920],{},[2581,28086,1532],{},[51,28088,747],{"separator":122},[898,28090,28091,28093],{},[48,28092,920],{},[2581,28094,4435],{},[51,28096,66],{"stretchy":53},[48,28098,112],{"mathvariant":111},[114,28100,28101],{"encoding":116},"p_e^*\\le \\frac12\\,\\mathrm{BC}(P_0,P_1).",[23,28103,28105,28171],{"className":28104,"ariaHidden":122},[121],[23,28106,28108,28111,28162,28165,28168],{"className":28107},[126],[23,28109],{"className":28110,"style":20722},[130],[23,28112,28114,28117],{"className":28113},[135],[23,28115,11],{"className":28116},[135,136],[23,28118,28120],{"className":28119},[998],[23,28121,28123,28154],{"className":28122},[199,200],[23,28124,28126,28151],{"className":28125},[204],[23,28127,28129,28140],{"className":28128,"style":7045},[208],[23,28130,28131,28134],{"style":19345},[23,28132],{"className":28133,"style":1015},[216],[23,28135,28137],{"className":28136},[1019,1020,1021,1022],[23,28138,5342],{"className":28139},[135,136,1022],[23,28141,28142,28145],{"style":4989},[23,28143],{"className":28144,"style":1015},[216],[23,28146,28148],{"className":28147},[1019,1020,1021,1022],[23,28149,6881],{"className":28150},[612,1022],[23,28152,297],{"className":28153},[296],[23,28155,28157],{"className":28156},[204],[23,28158,28160],{"className":28159,"style":19375},[208],[23,28161],{},[23,28163],{"className":28164,"style":149},[148],[23,28166,12268],{"className":28167},[153],[23,28169],{"className":28170,"style":149},[148],[23,28172,28174,28177,28239,28242,28248,28251,28291,28294,28297,28337,28340],{"className":28173},[126],[23,28175],{"className":28176,"style":12390},[130],[23,28178,28180,28183,28236],{"className":28179},[135],[23,28181],{"className":28182},[140,192],[23,28184,28186],{"className":28185},[71],[23,28187,28189,28228],{"className":28188},[199,200],[23,28190,28192,28225],{"className":28191},[204],[23,28193,28195,28206,28214],{"className":28194,"style":12409},[208],[23,28196,28197,28200],{"style":212},[23,28198],{"className":28199,"style":217},[216],[23,28201,28203],{"className":28202},[135],[23,28204,9952],{"className":28205},[135],[23,28207,28208,28211],{"style":235},[23,28209],{"className":28210,"style":217},[216],[23,28212],{"className":28213,"style":243},[242],[23,28215,28216,28219],{"style":246},[23,28217],{"className":28218,"style":217},[216],[23,28220,28222],{"className":28221},[135],[23,28223,4435],{"className":28224},[135],[23,28226,297],{"className":28227},[296],[23,28229,28231],{"className":28230},[204],[23,28232,28234],{"className":28233,"style":6287},[208],[23,28235],{},[23,28237],{"className":28238},[169,192],[23,28240],{"className":28241,"style":280},[148],[23,28243,28245],{"className":28244},[135],[23,28246,25180],{"className":28247},[135,5391],[23,28249,54],{"className":28250},[140],[23,28252,28254,28257],{"className":28253},[135],[23,28255,920],{"className":28256,"style":1047},[135,136],[23,28258,28260],{"className":28259},[998],[23,28261,28263,28283],{"className":28262},[199,200],[23,28264,28266,28280],{"className":28265},[204],[23,28267,28269],{"className":28268,"style":18647},[208],[23,28270,28271,28274],{"style":21199},[23,28272],{"className":28273,"style":1015},[216],[23,28275,28277],{"className":28276},[1019,1020,1021,1022],[23,28278,1532],{"className":28279},[135,1022],[23,28281,297],{"className":28282},[296],[23,28284,28286],{"className":28285},[204],[23,28287,28289],{"className":28288,"style":2048},[208],[23,28290],{},[23,28292,747],{"className":28293},[824],[23,28295],{"className":28296,"style":280},[148],[23,28298,28300,28303],{"className":28299},[135],[23,28301,920],{"className":28302,"style":1047},[135,136],[23,28304,28306],{"className":28305},[998],[23,28307,28309,28329],{"className":28308},[199,200],[23,28310,28312,28326],{"className":28311},[204],[23,28313,28315],{"className":28314,"style":18647},[208],[23,28316,28317,28320],{"style":21199},[23,28318],{"className":28319,"style":1015},[216],[23,28321,28323],{"className":28322},[1019,1020,1021,1022],[23,28324,4435],{"className":28325},[135,1022],[23,28327,297],{"className":28328},[296],[23,28330,28332],{"className":28331},[204],[23,28333,28335],{"className":28334,"style":2048},[208],[23,28336],{},[23,28338,66],{"className":28339},[169],[23,28341,112],{"className":28342},[135],[11,28344,28345,28346,28660],{},"这是因为 ",[23,28347,28349,28405],{"className":28348,"translate":27},[31],[23,28350,28352],{"className":28351},[35],[37,28353,28354],{"xmlns":39},[42,28355,28356,28402],{},[45,28357,28358,28360,28362,28364,28366,28372,28374,28380,28382,28384,28386],{},[51,28359,1557],{},[48,28361,6967],{},[51,28363,6959],{},[51,28365,9010],{"stretchy":53},[898,28367,28368,28370],{},[48,28369,11],{},[2581,28371,1532],{},[51,28373,747],{"separator":122},[898,28375,28376,28378],{},[48,28377,11],{},[2581,28379,4435],{},[51,28381,9020],{"stretchy":53},[51,28383,12268],{},[51,28385,1557],{},[25516,28387,28388],{},[45,28389,28390,28396],{},[898,28391,28392,28394],{},[48,28393,11],{},[2581,28395,1532],{},[898,28397,28398,28400],{},[48,28399,11],{},[2581,28401,4435],{},[114,28403,28404],{"encoding":116},"\\int\\min\\{p_0,p_1\\}\\le\\int\\sqrt{p_0p_1}",[23,28406,28408,28524],{"className":28407,"ariaHidden":122},[121],[23,28409,28411,28414,28417,28420,28423,28426,28466,28469,28472,28512,28515,28518,28521],{"className":28410},[126],[23,28412],{"className":28413,"style":20450},[130],[23,28415,1557],{"className":28416,"style":20455},[1786,1790,20454],[23,28418],{"className":28419,"style":280},[148],[23,28421,6967],{"className":28422},[1786],[23,28424,9010],{"className":28425},[140],[23,28427,28429,28432],{"className":28428},[135],[23,28430,11],{"className":28431},[135,136],[23,28433,28435],{"className":28434},[998],[23,28436,28438,28458],{"className":28437},[199,200],[23,28439,28441,28455],{"className":28440},[204],[23,28442,28444],{"className":28443,"style":18647},[208],[23,28445,28446,28449],{"style":2029},[23,28447],{"className":28448,"style":1015},[216],[23,28450,28452],{"className":28451},[1019,1020,1021,1022],[23,28453,1532],{"className":28454},[135,1022],[23,28456,297],{"className":28457},[296],[23,28459,28461],{"className":28460},[204],[23,28462,28464],{"className":28463,"style":2048},[208],[23,28465],{},[23,28467,747],{"className":28468},[824],[23,28470],{"className":28471,"style":280},[148],[23,28473,28475,28478],{"className":28474},[135],[23,28476,11],{"className":28477},[135,136],[23,28479,28481],{"className":28480},[998],[23,28482,28484,28504],{"className":28483},[199,200],[23,28485,28487,28501],{"className":28486},[204],[23,28488,28490],{"className":28489,"style":18647},[208],[23,28491,28492,28495],{"style":2029},[23,28493],{"className":28494,"style":1015},[216],[23,28496,28498],{"className":28497},[1019,1020,1021,1022],[23,28499,4435],{"className":28500},[135,1022],[23,28502,297],{"className":28503},[296],[23,28505,28507],{"className":28506},[204],[23,28508,28510],{"className":28509,"style":2048},[208],[23,28511],{},[23,28513,9020],{"className":28514},[169],[23,28516],{"className":28517,"style":149},[148],[23,28519,12268],{"className":28520},[153],[23,28522],{"className":28523,"style":149},[148],[23,28525,28527,28530,28533,28536],{"className":28526},[126],[23,28528],{"className":28529,"style":25657},[130],[23,28531,1557],{"className":28532,"style":20455},[1786,1790,20454],[23,28534],{"className":28535,"style":280},[148],[23,28537,28539],{"className":28538},[135,25667],[23,28540,28542,28652],{"className":28541},[199,200],[23,28543,28545,28649],{"className":28544},[204],[23,28546,28548,28637],{"className":28547,"style":25677},[208],[23,28549,28551,28554],{"className":28550,"style":7130},[8672],[23,28552],{"className":28553,"style":217},[216],[23,28555,28557,28597],{"className":28556,"style":25687},[135],[23,28558,28560,28563],{"className":28559},[135],[23,28561,11],{"className":28562},[135,136],[23,28564,28566],{"className":28565},[998],[23,28567,28569,28589],{"className":28568},[199,200],[23,28570,28572,28586],{"className":28571},[204],[23,28573,28575],{"className":28574,"style":18647},[208],[23,28576,28577,28580],{"style":2029},[23,28578],{"className":28579,"style":1015},[216],[23,28581,28583],{"className":28582},[1019,1020,1021,1022],[23,28584,1532],{"className":28585},[135,1022],[23,28587,297],{"className":28588},[296],[23,28590,28592],{"className":28591},[204],[23,28593,28595],{"className":28594,"style":2048},[208],[23,28596],{},[23,28598,28600,28603],{"className":28599},[135],[23,28601,11],{"className":28602},[135,136],[23,28604,28606],{"className":28605},[998],[23,28607,28609,28629],{"className":28608},[199,200],[23,28610,28612,28626],{"className":28611},[204],[23,28613,28615],{"className":28614,"style":18647},[208],[23,28616,28617,28620],{"style":2029},[23,28618],{"className":28619,"style":1015},[216],[23,28621,28623],{"className":28622},[1019,1020,1021,1022],[23,28624,4435],{"className":28625},[135,1022],[23,28627,297],{"className":28628},[296],[23,28630,28632],{"className":28631},[204],[23,28633,28635],{"className":28634,"style":2048},[208],[23,28636],{},[23,28638,28639,28642],{"style":25770},[23,28640],{"className":28641,"style":217},[216],[23,28643,28645],{"className":28644,"style":25778},[25777],[8688,28646,28647],{"xmlns":8690,"width":8691,"height":25781,"viewBox":25782,"preserveAspectRatio":8694},[8696,28648],{"d":25785},[23,28650,297],{"className":28651},[296],[23,28653,28655],{"className":28654},[204],[23,28656,28658],{"className":28657,"style":25795},[208],[23,28659],{},"（算术-几何平均不等式）。Bhattacharyya 系数可以直接计算的情形（例如高斯），这个上界就能给出正确的指数衰减率。",[11,28662,28663,28664,28667,28668,28727,28728,28732],{},"把两个方向放在一起，可以看到散度在决策问题里的角色：它们不是「错误率」，而是错误率的",[18,28665,28666],{},"证书","。全变差是唯一精确的那个，但它通常最难估计；Bhattacharyya 系数给出上界，KL 给出下界，Hellinger 则通过 ",[23,28669,28671,28693],{"className":28670,"translate":27},[31],[23,28672,28674],{"className":28673},[35],[37,28675,28676],{"xmlns":39},[42,28677,28678,28690],{},[45,28679,28680,28686,28688],{},[45,28681,28682,28684],{},[48,28683,20703],{"mathvariant":111},[48,28685,10790],{"mathvariant":111},[51,28687,12268],{},[48,28689,26249],{},[114,28691,28692],{"encoding":116},"\\mathrm{TV}\\le H",[23,28694,28696,28718],{"className":28695,"ariaHidden":122},[121],[23,28697,28699,28703,28709,28712,28715],{"className":28698},[126],[23,28700],{"className":28701,"style":28702},[130],"height:0.8193em;vertical-align:-0.136em;",[23,28704,28706],{"className":28705},[135],[23,28707,21310],{"className":28708,"style":7088},[135,5391],[23,28710],{"className":28711,"style":149},[148],[23,28713,12268],{"className":28714},[153],[23,28716],{"className":28717,"style":149},[148],[23,28719,28721,28724],{"className":28720},[126],[23,28722],{"className":28723,"style":470},[130],[23,28725,26249],{"className":28726,"style":26503},[135,136]," 把两边连起来。",[495,28729,28731],{"href":28730},"\u002Fblog\u002F2026\u002F0011-about-the-f-divergence","关于 f-散度的那篇文章","，正是把这些证书放在同一个凸函数框架下；而贝叶斯错误率则告诉我们，这些证书最终约束的是什么。",[2501,28734,28736],{"id":28735},"_44-错误率住在哪里","4.4 错误率住在哪里",[11,28738,28739,28740,28884,28885,28959],{},"由 3.6 节的恒等式，只有满足 ",[23,28741,28743,28777],{"className":28742,"translate":27},[31],[23,28744,28746],{"className":28745},[35],[37,28747,28748],{"xmlns":39},[42,28749,28750,28774],{},[45,28751,28752,28754,28756,28758,28760,28762,28768,28770,28772],{},[48,28753,736],{},[51,28755,54],{"stretchy":53},[48,28757,434],{},[51,28759,66],{"stretchy":53},[51,28761,9015],{"mathvariant":111},[4518,28763,28764,28766],{},[48,28765,736],{},[51,28767,6881],{},[51,28769,54],{"stretchy":53},[48,28771,434],{},[51,28773,66],{"stretchy":53},[114,28775,28776],{"encoding":116},"\\delta(X)\\ne\\delta^*(X)",[23,28778,28780,28840],{"className":28779,"ariaHidden":122},[121],[23,28781,28783,28786,28789,28792,28795,28798,28801,28837],{"className":28782},[126],[23,28784],{"className":28785,"style":131},[130],[23,28787,736],{"className":28788,"style":783},[135,136],[23,28790,54],{"className":28791},[140],[23,28793,434],{"className":28794,"style":454},[135,136],[23,28796,66],{"className":28797},[169],[23,28799],{"className":28800,"style":149},[148],[23,28802,28804,28831,28834],{"className":28803},[153],[23,28805,28807],{"className":28806},[153],[23,28808,28810],{"className":28809},[135,9093],[23,28811,28813],{"className":28812},[9097],[23,28814,28816,28819,28828],{"className":28815},[9101],[23,28817],{"className":28818,"style":5414},[130],[23,28820,28822],{"className":28821},[9108],[23,28823,28825],{"className":28824},[135],[23,28826,9115],{"className":28827},[153],[23,28829],{"className":28830},[9119],[23,28832],{"className":28833},[148,9123],[23,28835,69],{"className":28836},[153],[23,28838],{"className":28839,"style":149},[148],[23,28841,28843,28846,28875,28878,28881],{"className":28842},[126],[23,28844],{"className":28845,"style":131},[130],[23,28847,28849,28852],{"className":28848},[135],[23,28850,736],{"className":28851,"style":783},[135,136],[23,28853,28855],{"className":28854},[998],[23,28856,28858],{"className":28857},[199],[23,28859,28861],{"className":28860},[204],[23,28862,28864],{"className":28863,"style":6912},[208],[23,28865,28866,28869],{"style":4696},[23,28867],{"className":28868,"style":1015},[216],[23,28870,28872],{"className":28871},[1019,1020,1021,1022],[23,28873,6881],{"className":28874},[612,1022],[23,28876,54],{"className":28877},[140],[23,28879,434],{"className":28880,"style":454},[135,136],[23,28882,66],{"className":28883},[169]," 的样本才贡献错误，且权重是 ",[23,28886,28888,28917],{"className":28887,"translate":27},[31],[23,28889,28891],{"className":28890},[35],[37,28892,28893],{"xmlns":39},[42,28894,28895,28915],{},[45,28896,28897,28899,28901,28903,28905,28907,28909,28911,28913],{},[48,28898,60],{"mathvariant":111},[2581,28900,9952],{},[48,28902,9765],{},[51,28904,54],{"stretchy":53},[48,28906,434],{},[51,28908,66],{"stretchy":53},[51,28910,4529],{},[2581,28912,4435],{},[48,28914,60],{"mathvariant":111},[114,28916,16468],{"encoding":116},[23,28918,28920,28950],{"className":28919,"ariaHidden":122},[121],[23,28921,28923,28926,28929,28932,28935,28938,28941,28944,28947],{"className":28922},[126],[23,28924],{"className":28925,"style":131},[130],[23,28927,15672],{"className":28928},[135],[23,28930,9765],{"className":28931,"style":676},[135,136],[23,28933,54],{"className":28934},[140],[23,28936,434],{"className":28937,"style":454},[135,136],[23,28939,66],{"className":28940},[169],[23,28942],{"className":28943,"style":392},[148],[23,28945,4529],{"className":28946},[612],[23,28948],{"className":28949,"style":392},[148],[23,28951,28953,28956],{"className":28952},[126],[23,28954],{"className":28955,"style":131},[130],[23,28957,15703],{"className":28958},[135],"。这意味着贝叶斯错误率不是由决策边界的「长度」或「复杂度」决定的，而是由边界附近的后验质量决定的。",[11,28961,28962,28963,28992,28993,16573,29021,29049,29050,29107],{},"高维空间里，决策边界可以非常复杂，但如果后验在边界两侧都很尖锐（",[23,28964,28966,28980],{"className":28965,"translate":27},[31],[23,28967,28969],{"className":28968},[35],[37,28970,28971],{"xmlns":39},[42,28972,28973,28977],{},[45,28974,28975],{},[48,28976,9765],{},[114,28978,28979],{"encoding":116},"\\eta",[23,28981,28983],{"className":28982,"ariaHidden":122},[121],[23,28984,28986,28989],{"className":28985},[126],[23,28987],{"className":28988,"style":672},[130],[23,28990,9765],{"className":28991,"style":676},[135,136]," 接近 ",[23,28994,28996,29009],{"className":28995,"translate":27},[31],[23,28997,28999],{"className":28998},[35],[37,29000,29001],{"xmlns":39},[42,29002,29003,29007],{},[45,29004,29005],{},[2581,29006,1532],{},[114,29008,1532],{"encoding":116},[23,29010,29012],{"className":29011,"ariaHidden":122},[121],[23,29013,29015,29018],{"className":29014},[126],[23,29016],{"className":29017,"style":2770},[130],[23,29019,1532],{"className":29020},[135],[23,29022,29024,29037],{"className":29023,"translate":27},[31],[23,29025,29027],{"className":29026},[35],[37,29028,29029],{"xmlns":39},[42,29030,29031,29035],{},[45,29032,29033],{},[2581,29034,4435],{},[114,29036,4435],{"encoding":116},[23,29038,29040],{"className":29039,"ariaHidden":122},[121],[23,29041,29043,29046],{"className":29042},[126],[23,29044],{"className":29045,"style":2770},[130],[23,29047,4435],{"className":29048},[135],"），错误率仍然可以很低。反过来，一个简单的线性边界，如果穿过一大片 ",[23,29051,29053,29076],{"className":29052,"translate":27},[31],[23,29054,29056],{"className":29055},[35],[37,29057,29058],{"xmlns":39},[42,29059,29060,29073],{},[45,29061,29062,29064,29067,29069,29071],{},[48,29063,9765],{},[51,29065,29066],{},"≈",[2581,29068,4435],{},[48,29070,2875],{"mathvariant":111},[2581,29072,9952],{},[114,29074,29075],{"encoding":116},"\\eta\\approx 1\u002F2",[23,29077,29079,29098],{"className":29078,"ariaHidden":122},[121],[23,29080,29082,29086,29089,29092,29095],{"className":29081},[126],[23,29083],{"className":29084,"style":29085},[130],"height:0.6776em;vertical-align:-0.1944em;",[23,29087,9765],{"className":29088,"style":676},[135,136],[23,29090],{"className":29091,"style":149},[148],[23,29093,29066],{"className":29094},[153],[23,29096],{"className":29097,"style":149},[148],[23,29099,29101,29104],{"className":29100},[126],[23,29102],{"className":29103,"style":131},[130],[23,29105,10469],{"className":29106},[135]," 的区域，错误率就会很高。这解释了为什么「边界复杂度」本身不是泛化误差的可靠指标：真正决定错误率的是后验在边界附近的分布，而不是边界的几何形状。",[2501,29109,29111],{"id":29110},"_45-回归的贝叶斯风险","4.5 回归的贝叶斯风险",[11,29113,29114],{},"平方损失下，贝叶斯风险为",[23,29116,29118],{"className":29117,"translate":27},[26],[23,29119,29121,29199],{"className":29120,"translate":27},[31],[23,29122,29124],{"className":29123},[35],[37,29125,29126],{"xmlns":39,"display":40},[42,29127,29128,29196],{},[45,29129,29130,29136,29138,29140,29142,29144,29146,29148,29154,29156,29158,29160,29166,29168,29170,29172,29174,29182,29184,29186,29188,29190,29192,29194],{},[4518,29131,29132,29134],{},[48,29133,572],{},[51,29135,6881],{},[51,29137,69],{},[48,29139,902],{"mathvariant":571},[51,29141,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,29143,54],{"stretchy":53},[48,29145,369],{},[51,29147,4529],{},[4518,29149,29150,29152],{},[48,29151,8336],{},[51,29153,6881],{},[51,29155,54],{"stretchy":53},[48,29157,434],{},[51,29159,66],{"stretchy":53},[4518,29161,29162,29164],{},[51,29163,66],{"stretchy":53},[2581,29165,9952],{},[51,29167,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,29169,69],{},[48,29171,902],{"mathvariant":571},[51,29173,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[45,29175,29176,29178,29180],{},[48,29177,10790],{"mathvariant":111},[48,29179,495],{"mathvariant":111},[48,29181,1653],{"mathvariant":111},[51,29183,54],{"stretchy":53},[48,29185,369],{},[51,29187,60],{},[48,29189,434],{},[51,29191,66],{"stretchy":53},[51,29193,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,29195,112],{"mathvariant":111},[114,29197,29198],{"encoding":116},"R^*= \\mathbb E\\big[(Y-f^*(X))^2\\big]=\\mathbb E\\big[\\mathrm{Var}(Y\\mid X)\\big].",[23,29200,29202,29246,29276,29364,29400],{"className":29201,"ariaHidden":122},[121],[23,29203,29205,29208,29237,29240,29243],{"className":29204},[126],[23,29206],{"className":29207,"style":7045},[130],[23,29209,29211,29214],{"className":29210},[135],[23,29212,572],{"className":29213,"style":963},[135,136],[23,29215,29217],{"className":29216},[998],[23,29218,29220],{"className":29219},[199],[23,29221,29223],{"className":29222},[204],[23,29224,29226],{"className":29225,"style":7045},[208],[23,29227,29228,29231],{"style":4989},[23,29229],{"className":29230,"style":1015},[216],[23,29232,29234],{"className":29233},[1019,1020,1021,1022],[23,29235,6881],{"className":29236},[612,1022],[23,29238],{"className":29239,"style":149},[148],[23,29241,69],{"className":29242},[153],[23,29244],{"className":29245,"style":149},[148],[23,29247,29249,29252,29255,29261,29264,29267,29270,29273],{"className":29248},[126],[23,29250],{"className":29251,"style":2213},[130],[23,29253,902],{"className":29254},[135,644],[23,29256,29258],{"className":29257},[135],[23,29259,924],{"className":29260},[1066,1067],[23,29262,54],{"className":29263},[140],[23,29265,369],{"className":29266,"style":392},[135,136],[23,29268],{"className":29269,"style":392},[148],[23,29271,4529],{"className":29272},[612],[23,29274],{"className":29275,"style":392},[148],[23,29277,29279,29282,29311,29314,29317,29320,29349,29355,29358,29361],{"className":29278},[126],[23,29280],{"className":29281,"style":17156},[130],[23,29283,29285,29288],{"className":29284},[135],[23,29286,8336],{"className":29287,"style":11398},[135,136],[23,29289,29291],{"className":29290},[998],[23,29292,29294],{"className":29293},[199],[23,29295,29297],{"className":29296},[204],[23,29298,29300],{"className":29299,"style":7045},[208],[23,29301,29302,29305],{"style":4989},[23,29303],{"className":29304,"style":1015},[216],[23,29306,29308],{"className":29307},[1019,1020,1021,1022],[23,29309,6881],{"className":29310},[612,1022],[23,29312,54],{"className":29313},[140],[23,29315,434],{"className":29316,"style":454},[135,136],[23,29318,66],{"className":29319},[169],[23,29321,29323,29326],{"className":29322},[169],[23,29324,66],{"className":29325},[169],[23,29327,29329],{"className":29328},[998],[23,29330,29332],{"className":29331},[199],[23,29333,29335],{"className":29334},[204],[23,29336,29338],{"className":29337,"style":11002},[208],[23,29339,29340,29343],{"style":4989},[23,29341],{"className":29342,"style":1015},[216],[23,29344,29346],{"className":29345},[1019,1020,1021,1022],[23,29347,9952],{"className":29348},[135,1022],[23,29350,29352],{"className":29351},[135],[23,29353,945],{"className":29354},[1066,1067],[23,29356],{"className":29357,"style":149},[148],[23,29359,69],{"className":29360},[153],[23,29362],{"className":29363,"style":149},[148],[23,29365,29367,29370,29373,29379,29385,29388,29391,29394,29397],{"className":29366},[126],[23,29368],{"className":29369,"style":2213},[130],[23,29371,902],{"className":29372},[135,644],[23,29374,29376],{"className":29375},[135],[23,29377,924],{"className":29378},[1066,1067],[23,29380,29382],{"className":29381},[135],[23,29383,11070],{"className":29384},[135,5391],[23,29386,54],{"className":29387},[140],[23,29389,369],{"className":29390,"style":392},[135,136],[23,29392],{"className":29393,"style":149},[148],[23,29395,60],{"className":29396},[153],[23,29398],{"className":29399,"style":149},[148],[23,29401,29403,29406,29409,29412,29418],{"className":29402},[126],[23,29404],{"className":29405,"style":2213},[130],[23,29407,434],{"className":29408,"style":454},[135,136],[23,29410,66],{"className":29411},[169],[23,29413,29415],{"className":29414},[135],[23,29416,945],{"className":29417},[1066,1067],[23,29419,112],{"className":29420},[135],[11,29422,29423,29424,29452,29453,29567],{},"这是数据生成过程本身的噪声，任何模型都消不掉。对一个在训练集 ",[23,29425,29427,29440],{"className":29426,"translate":27},[31],[23,29428,29430],{"className":29429},[35],[37,29431,29432],{"xmlns":39},[42,29433,29434,29438],{},[45,29435,29436],{},[48,29437,63],{},[114,29439,63],{"encoding":116},[23,29441,29443],{"className":29442,"ariaHidden":122},[121],[23,29444,29446,29449],{"className":29445},[126],[23,29447],{"className":29448,"style":470},[130],[23,29450,63],{"className":29451,"style":144},[135,136]," 上学到的回归器 ",[23,29454,29456,29478],{"className":29455,"translate":27},[31],[23,29457,29459],{"className":29458},[35],[37,29460,29461],{"xmlns":39},[42,29462,29463,29475],{},[45,29464,29465],{},[898,29466,29467,29473],{},[16807,29468,29469,29471],{"accent":122},[48,29470,8336],{},[51,29472,16813],{},[48,29474,63],{},[114,29476,29477],{"encoding":116},"\\hat f_D",[23,29479,29481],{"className":29480,"ariaHidden":122},[121],[23,29482,29484,29487],{"className":29483},[126],[23,29485],{"className":29486,"style":16826},[130],[23,29488,29490,29532],{"className":29489},[135],[23,29491,29493],{"className":29492},[135,16830],[23,29494,29496,29524],{"className":29495},[199,200],[23,29497,29499,29521],{"className":29498},[204],[23,29500,29502,29510],{"className":29501,"style":16840},[208],[23,29503,29504,29507],{"style":7130},[23,29505],{"className":29506,"style":217},[216],[23,29508,8336],{"className":29509,"style":11398},[135,136],[23,29511,29512,29515],{"style":16851},[23,29513],{"className":29514,"style":217},[216],[23,29516,29518],{"className":29517,"style":16859},[16858],[23,29519,16813],{"className":29520},[135],[23,29522,297],{"className":29523},[296],[23,29525,29527],{"className":29526},[204],[23,29528,29530],{"className":29529,"style":16872},[208],[23,29531],{},[23,29533,29535],{"className":29534},[998],[23,29536,29538,29559],{"className":29537},[199,200],[23,29539,29541,29556],{"className":29540},[204],[23,29542,29544],{"className":29543,"style":2026},[208],[23,29545,29547,29550],{"style":29546},"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;",[23,29548],{"className":29549,"style":1015},[216],[23,29551,29553],{"className":29552},[1019,1020,1021,1022],[23,29554,63],{"className":29555,"style":144},[135,136,1022],[23,29557,297],{"className":29558},[296],[23,29560,29562],{"className":29561},[204],[23,29563,29565],{"className":29564,"style":2048},[208],[23,29566],{},"，有标准的偏差-方差分解：",[23,29569,29571],{"className":29570,"translate":27},[26],[23,29572,29574,29755],{"className":29573,"translate":27},[31],[23,29575,29577],{"className":29576},[35],[37,29578,29579],{"xmlns":39,"display":40},[42,29580,29581,29752],{},[45,29582,29583,29589,29591,29593,29595,29597,29607,29609,29611,29613,29619,29621,29623,29626,29644,29646,29705,29707,29750],{},[898,29584,29585,29587],{},[48,29586,902],{"mathvariant":571},[48,29588,63],{},[51,29590,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,29592,54],{"stretchy":53},[48,29594,369],{},[51,29596,4529],{},[898,29598,29599,29605],{},[16807,29600,29601,29603],{"accent":122},[48,29602,8336],{},[51,29604,16813],{},[48,29606,63],{},[51,29608,54],{"stretchy":53},[48,29610,434],{},[51,29612,66],{"stretchy":53},[4518,29614,29615,29617],{},[51,29616,66],{"stretchy":53},[2581,29618,9952],{},[51,29620,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,29622,69],{},[148,29624],{"linebreak":29625},"newline",[6961,29627,29628,29642],{},[6961,29629,29630,29640],{},[19137,29631,29632,29635,29638],{},[48,29633,29634],{},"σ",[48,29636,29637],{},"ε",[2581,29639,9952],{},[51,29641,8477],{"stretchy":122},[88,29643,335],{},[51,29645,5106],{},[6961,29647,29648,29698],{},[6961,29649,29650,29696],{},[45,29651,29652,29654,29660,29662,29664,29666,29668,29674,29684,29686,29688,29690],{},[51,29653,54],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[4518,29655,29656,29658],{},[48,29657,8336],{},[51,29659,6881],{},[51,29661,54],{"stretchy":53},[48,29663,434],{},[51,29665,66],{"stretchy":53},[51,29667,4529],{},[898,29669,29670,29672],{},[48,29671,902],{"mathvariant":571},[48,29673,63],{},[898,29675,29676,29682],{},[16807,29677,29678,29680],{"accent":122},[48,29679,8336],{},[51,29681,16813],{},[48,29683,63],{},[51,29685,54],{"stretchy":53},[48,29687,434],{},[51,29689,66],{"stretchy":53},[4518,29691,29692,29694],{},[51,29693,66],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[2581,29695,9952],{},[51,29697,8477],{"stretchy":122},[4518,29699,29700,29703],{},[88,29701,29702],{},"偏差",[2581,29704,9952],{},[51,29706,5106],{},[6961,29708,29709,29747],{},[6961,29710,29711,29745],{},[45,29712,29713,29725,29727,29737,29739,29741,29743],{},[898,29714,29715,29723],{},[45,29716,29717,29719,29721],{},[48,29718,10790],{"mathvariant":111},[48,29720,495],{"mathvariant":111},[48,29722,1653],{"mathvariant":111},[48,29724,63],{},[51,29726,54],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[898,29728,29729,29735],{},[16807,29730,29731,29733],{"accent":122},[48,29732,8336],{},[51,29734,16813],{},[48,29736,63],{},[51,29738,54],{"stretchy":53},[48,29740,434],{},[51,29742,66],{"stretchy":53},[51,29744,66],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,29746,8477],{"stretchy":122},[88,29748,29749],{},"方差",[48,29751,112],{"mathvariant":111},[114,29753,29754],{"encoding":116},"\\mathbb E_D\\big[(Y-\\hat f_D(X))^2\\big]\n= \\\\ \\underbrace{\\sigma_\\varepsilon^2}_{\\text{贝叶斯风险}}\n+\\underbrace{\\big(f^*(X)-\\mathbb E_D\\hat f_D(X)\\big)^2}_{\\text{偏差}^2}\n+\\underbrace{\\mathrm{Var}_D\\big(\\hat 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贝叶斯决策如何长成算法",[2501,30782,30784],{"id":30783},"_51-生成式与判别式","5.1 生成式与判别式",[11,30786,30787,30788,30912,30913,30941,30942,31008,31009,31053,31054,31057,31058,31061],{},"贝叶斯规则只依赖后验概率 ",[23,30789,30791,30828],{"className":30790,"translate":27},[31],[23,30792,30794],{"className":30793},[35],[37,30795,30796],{"xmlns":39},[42,30797,30798,30826],{},[45,30799,30800,30802,30804,30806,30808,30810,30812,30814,30816,30818,30820,30822,30824],{},[48,30801,9765],{},[51,30803,54],{"stretchy":53},[48,30805,753],{},[51,30807,66],{"stretchy":53},[51,30809,69],{},[48,30811,11],{},[51,30813,54],{"stretchy":53},[48,30815,369],{},[51,30817,69],{},[2581,30819,4435],{},[51,30821,60],{},[48,30823,753],{},[51,30825,66],{"stretchy":53},[114,30827,9792],{"encoding":116},[23,30829,30831,30858,30882,30900],{"className":30830,"ariaHidden":122},[121],[23,30832,30834,30837,30840,30843,30846,30849,30852,30855],{"className":30833},[126],[23,30835],{"className":30836,"style":131},[130],[23,30838,9765],{"className":30839,"style":676},[135,136],[23,30841,54],{"className":30842},[140],[23,30844,753],{"className":30845},[135,136],[23,30847,66],{"className":30848},[169],[23,30850],{"className":30851,"style":149},[148],[23,30853,69],{"className":30854},[153],[23,30856],{"className":30857,"style":149},[148],[23,30859,30861,30864,30867,30870,30873,30876,30879],{"className":30860},[126],[23,30862],{"className":30863,"style":131},[130],[23,30865,11],{"className":30866},[135,136],[23,30868,54],{"className":30869},[140],[23,30871,369],{"className":30872,"style":392},[135,136],[23,30874],{"className":30875,"style":149},[148],[23,30877,69],{"className":30878},[153],[23,30880],{"className":30881,"style":149},[148],[23,30883,30885,30888,30891,30894,30897],{"className":30884},[126],[23,30886],{"className":30887,"style":131},[130],[23,30889,4435],{"className":30890},[135],[23,30892],{"className":30893,"style":149},[148],[23,30895,60],{"className":30896},[153],[23,30898],{"className":30899,"style":149},[148],[23,30901,30903,30906,30909],{"className":30902},[126],[23,30904],{"className":30905,"style":131},[130],[23,30907,753],{"className":30908},[135,136],[23,30910,66],{"className":30911},[169],"。因此有两条路可以走：直接建模 ",[23,30914,30916,30929],{"className":30915,"translate":27},[31],[23,30917,30919],{"className":30918},[35],[37,30920,30921],{"xmlns":39},[42,30922,30923,30927],{},[45,30924,30925],{},[48,30926,9765],{},[114,30928,28979],{"encoding":116},[23,30930,30932],{"className":30931,"ariaHidden":122},[121],[23,30933,30935,30938],{"className":30934},[126],[23,30936],{"className":30937,"style":672},[130],[23,30939,9765],{"className":30940,"style":676},[135,136],"，或者分别建模类条件密度 ",[23,30943,30945,30969],{"className":30944,"translate":27},[31],[23,30946,30948],{"className":30947},[35],[37,30949,30950],{"xmlns":39},[42,30951,30952,30966],{},[45,30953,30954,30956,30958,30960,30962,30964],{},[48,30955,11],{},[51,30957,54],{"stretchy":53},[48,30959,753],{},[51,30961,60],{},[48,30963,660],{},[51,30965,66],{"stretchy":53},[114,30967,30968],{"encoding":116},"p(x\\mid y)",[23,30970,30972,30996],{"className":30971,"ariaHidden":122},[121],[23,30973,30975,30978,30981,30984,30987,30990,30993],{"className":30974},[126],[23,30976],{"className":30977,"style":131},[130],[23,30979,11],{"className":30980},[135,136],[23,30982,54],{"className":30983},[140],[23,30985,753],{"className":30986},[135,136],[23,30988],{"className":30989,"style":149},[148],[23,30991,60],{"className":30992},[153],[23,30994],{"className":30995,"style":149},[148],[23,30997,30999,31002,31005],{"className":30998},[126],[23,31000],{"className":31001,"style":131},[130],[23,31003,660],{"className":31004,"style":676},[135,136],[23,31006,66],{"className":31007},[169]," 与先验 ",[23,31010,31012,31032],{"className":31011,"translate":27},[31],[23,31013,31015],{"className":31014},[35],[37,31016,31017],{"xmlns":39},[42,31018,31019,31029],{},[45,31020,31021,31023,31025,31027],{},[48,31022,11],{},[51,31024,54],{"stretchy":53},[48,31026,660],{},[51,31028,66],{"stretchy":53},[114,31030,31031],{"encoding":116},"p(y)",[23,31033,31035],{"className":31034,"ariaHidden":122},[121],[23,31036,31038,31041,31044,31047,31050],{"className":31037},[126],[23,31039],{"className":31040,"style":131},[130],[23,31042,11],{"className":31043},[135,136],[23,31045,54],{"className":31046},[140],[23,31048,660],{"className":31049,"style":676},[135,136],[23,31051,66],{"className":31052},[169],"，再用贝叶斯公式合成。前者叫",[18,31055,31056],{},"判别式","（discriminative），后者叫",[18,31059,31060],{},"生成式","（generative）。",[11,31063,31064],{},"生成式的合成公式是",[23,31066,31068],{"className":31067,"translate":27},[26],[23,31069,31071,31157],{"className":31070,"translate":27},[31],[23,31072,31074],{"className":31073},[35],[37,31075,31076],{"xmlns":39,"display":40},[42,31077,31078,31154],{},[45,31079,31080,31082,31084,31086,31088,31090,31152],{},[48,31081,9765],{},[51,31083,54],{"stretchy":53},[48,31085,753],{},[51,31087,66],{"stretchy":53},[51,31089,69],{},[71,31091,31092,31112],{},[45,31093,31094,31100,31106,31108,31110],{},[898,31095,31096,31098],{},[48,31097,21693],{},[2581,31099,4435],{},[898,31101,31102,31104],{},[48,31103,11],{},[2581,31105,4435],{},[51,31107,54],{"stretchy":53},[48,31109,753],{},[51,31111,66],{"stretchy":53},[45,31113,31114,31120,31126,31128,31130,31132,31134,31140,31146,31148,31150],{},[898,31115,31116,31118],{},[48,31117,21693],{},[2581,31119,1532],{},[898,31121,31122,31124],{},[48,31123,11],{},[2581,31125,1532],{},[51,31127,54],{"stretchy":53},[48,31129,753],{},[51,31131,66],{"stretchy":53},[51,31133,5106],{},[898,31135,31136,31138],{},[48,31137,21693],{},[2581,31139,4435],{},[898,31141,31142,31144],{},[48,31143,11],{},[2581,31145,4435],{},[51,31147,54],{"stretchy":53},[48,31149,753],{},[51,31151,66],{"stretchy":53},[48,31153,112],{"mathvariant":111},[114,31155,31156],{"encoding":116},"\\eta(x)=\\frac{\\pi_1p_1(x)}{\\pi_0p_0(x)+\\pi_1p_1(x)}.",[23,31158,31160,31187],{"className":31159,"ariaHidden":122},[121],[23,31161,31163,31166,31169,31172,31175,31178,31181,31184],{"className":31162},[126],[23,31164],{"className":31165,"style":131},[130],[23,31167,9765],{"className":31168,"style":676},[135,136],[23,31170,54],{"className":31171},[140],[23,31173,753],{"className":31174},[135,136],[23,31176,66],{"className":31177},[169],[23,31179],{"className":31180,"style":149},[148],[23,31182,69],{"className":31183},[153],[23,31185],{"className":31186,"style":149},[148],[23,31188,31190,31193,31525],{"className":31189},[126],[23,31191],{"className":31192,"style":185},[130],[23,31194,31196,31199,31522],{"className":31195},[135],[23,31197],{"className":31198},[140,192],[23,31200,31202],{"className":31201},[71],[23,31203,31205,31514],{"className":31204},[199,200],[23,31206,31208,31511],{"className":31207},[204],[23,31209,31211,31406,31414],{"className":31210,"style":209},[208],[23,31212,31213,31216],{"style":212},[23,31214],{"className":31215,"style":217},[216],[23,31217,31219,31259,31299,31302,31305,31308,31311,31314,31317,31357,31397,31400,31403],{"className":31218},[135],[23,31220,31222,31225],{"className":31221},[135],[23,31223,21693],{"className":31224,"style":676},[135,136],[23,31226,31228],{"className":31227},[998],[23,31229,31231,31251],{"className":31230},[199,200],[23,31232,31234,31248],{"className":31233},[204],[23,31235,31237],{"className":31236,"style":18647},[208],[23,31238,31239,31242],{"style":21850},[23,31240],{"className":31241,"style":1015},[216],[23,31243,31245],{"className":31244},[1019,1020,1021,1022],[23,31246,1532],{"className":31247},[135,1022],[23,31249,297],{"className":31250},[296],[23,31252,31254],{"className":31253},[204],[23,31255,31257],{"className":31256,"style":2048},[208],[23,31258],{},[23,31260,31262,31265],{"className":31261},[135],[23,31263,11],{"className":31264},[135,136],[23,31266,31268],{"className":31267},[998],[23,31269,31271,31291],{"className":31270},[199,200],[23,31272,31274,31288],{"className":31273},[204],[23,31275,31277],{"className":31276,"style":18647},[208],[23,31278,31279,31282],{"style":2029},[23,31280],{"className":31281,"style":1015},[216],[23,31283,31285],{"className":31284},[1019,1020,1021,1022],[23,31286,1532],{"className":31287},[135,1022],[23,31289,297],{"className":31290},[296],[23,31292,31294],{"className":31293},[204],[23,31295,31297],{"className":31296,"style":2048},[208],[23,31298],{},[23,31300,54],{"className":31301},[140],[23,31303,753],{"className":31304},[135,136],[23,31306,66],{"className":31307},[169],[23,31309],{"className":31310,"style":392},[148],[23,31312,5106],{"className":31313},[612],[23,31315],{"className":31316,"style":392},[148],[23,31318,31320,31323],{"className":31319},[135],[23,31321,21693],{"className":31322,"style":676},[135,136],[23,31324,31326],{"className":31325},[998],[23,31327,31329,31349],{"className":31328},[199,200],[23,31330,31332,31346],{"className":31331},[204],[23,31333,31335],{"className":31334,"style":18647},[208],[23,31336,31337,31340],{"style":21850},[23,31338],{"className":31339,"style":1015},[216],[23,31341,31343],{"className":31342},[1019,1020,1021,1022],[23,31344,4435],{"className":31345},[135,1022],[23,31347,297],{"className":31348},[296],[23,31350,31352],{"className":31351},[204],[23,31353,31355],{"className":31354,"style":2048},[208],[23,31356],{},[23,31358,31360,31363],{"className":31359},[135],[23,31361,11],{"className":31362},[135,136],[23,31364,31366],{"className":31365},[998],[23,31367,31369,31389],{"className":31368},[199,200],[23,31370,31372,31386],{"className":31371},[204],[23,31373,31375],{"className":31374,"style":18647},[208],[23,31376,31377,31380],{"style":2029},[23,31378],{"className":31379,"style":1015},[216],[23,31381,31383],{"className":31382},[1019,1020,1021,1022],[23,31384,4435],{"className":31385},[135,1022],[23,31387,297],{"className":31388},[296],[23,31390,31392],{"className":31391},[204],[23,31393,31395],{"className":31394,"style":2048},[208],[23,31396],{},[23,31398,54],{"className":31399},[140],[23,31401,753],{"className":31402},[135,136],[23,31404,66],{"className":31405},[169],[23,31407,31408,31411],{"style":235},[23,31409],{"className":31410,"style":217},[216],[23,31412],{"className":31413,"style":243},[242],[23,31415,31416,31419],{"style":246},[23,31417],{"className":31418,"style":217},[216],[23,31420,31422,31462,31502,31505,31508],{"className":31421},[135],[23,31423,31425,31428],{"className":31424},[135],[23,31426,21693],{"className":31427,"style":676},[135,136],[23,31429,31431],{"className":31430},[998],[23,31432,31434,31454],{"className":31433},[199,200],[23,31435,31437,31451],{"className":31436},[204],[23,31438,31440],{"className":31439,"style":18647},[208],[23,31441,31442,31445],{"style":21850},[23,31443],{"className":31444,"style":1015},[216],[23,31446,31448],{"className":31447},[1019,1020,1021,1022],[23,31449,4435],{"className":31450},[135,1022],[23,31452,297],{"className":31453},[296],[23,31455,31457],{"className":31456},[204],[23,31458,31460],{"className":31459,"style":2048},[208],[23,31461],{},[23,31463,31465,31468],{"className":31464},[135],[23,31466,11],{"className":31467},[135,136],[23,31469,31471],{"className":31470},[998],[23,31472,31474,31494],{"className":31473},[199,200],[23,31475,31477,31491],{"className":31476},[204],[23,31478,31480],{"className":31479,"style":18647},[208],[23,31481,31482,31485],{"style":2029},[23,31483],{"className":31484,"style":1015},[216],[23,31486,31488],{"className":31487},[1019,1020,1021,1022],[23,31489,4435],{"className":31490},[135,1022],[23,31492,297],{"className":31493},[296],[23,31495,31497],{"className":31496},[204],[23,31498,31500],{"className":31499,"style":2048},[208],[23,31501],{},[23,31503,54],{"className":31504},[140],[23,31506,753],{"className":31507},[135,136],[23,31509,66],{"className":31510},[169],[23,31512,297],{"className":31513},[296],[23,31515,31517],{"className":31516},[204],[23,31518,31520],{"className":31519,"style":304},[208],[23,31521],{},[23,31523],{"className":31524},[169,192],[23,31526,112],{"className":31527},[135],[11,31529,31530,31531,31560,31561,31589],{},"只要两边的密度估计足够准，这个式子就给出贝叶斯后验。线性判别分析（LDA）假设两类共享协方差 ",[23,31532,31534,31548],{"className":31533,"translate":27},[31],[23,31535,31537],{"className":31536},[35],[37,31538,31539],{"xmlns":39},[42,31540,31541,31545],{},[45,31542,31543],{},[48,31544,22112],{"mathvariant":111},[114,31546,31547],{"encoding":116},"\\Sigma",[23,31549,31551],{"className":31550,"ariaHidden":122},[121],[23,31552,31554,31557],{"className":31553},[126],[23,31555],{"className":31556,"style":470},[130],[23,31558,22112],{"className":31559},[135],"，此时对数似然比是 ",[23,31562,31564,31577],{"className":31563,"translate":27},[31],[23,31565,31567],{"className":31566},[35],[37,31568,31569],{"xmlns":39},[42,31570,31571,31575],{},[45,31572,31573],{},[48,31574,753],{},[114,31576,753],{"encoding":116},[23,31578,31580],{"className":31579,"ariaHidden":122},[121],[23,31581,31583,31586],{"className":31582},[126],[23,31584],{"className":31585,"style":703},[130],[23,31587,753],{"className":31588},[135,136]," 的线性函数：",[23,31591,31593],{"className":31592,"translate":27},[26],[23,31594,31596,31685],{"className":31595,"translate":27},[31],[23,31597,31599],{"className":31598},[35],[37,31600,31601],{"xmlns":39,"display":40},[42,31602,31603,31682],{},[45,31604,31605,31607,31609,31639,31641,31643,31649,31651,31657,31663,31673,31675,31677,31680],{},[48,31606,22418],{},[51,31608,6959],{},[71,31610,31611,31625],{},[45,31612,31613,31619,31621,31623],{},[898,31614,31615,31617],{},[48,31616,11],{},[2581,31618,4435],{},[51,31620,54],{"stretchy":53},[48,31622,753],{},[51,31624,66],{"stretchy":53},[45,31626,31627,31633,31635,31637],{},[898,31628,31629,31631],{},[48,31630,11],{},[2581,31632,1532],{},[51,31634,54],{"stretchy":53},[48,31636,753],{},[51,31638,66],{"stretchy":53},[51,31640,69],{},[51,31642,54],{"stretchy":53},[898,31644,31645,31647],{},[48,31646,22105],{},[2581,31648,4435],{},[51,31650,4529],{},[898,31652,31653,31655],{},[48,31654,22105],{},[2581,31656,1532],{},[4518,31658,31659,31661],{},[51,31660,66],{"stretchy":53},[48,31662,22471],{"mathvariant":111},[4518,31664,31665,31667],{},[48,31666,22112],{"mathvariant":111},[45,31668,31669,31671],{},[51,31670,4529],{},[2581,31672,4435],{},[48,31674,753],{},[51,31676,5106],{},[88,31678,31679],{},"常数",[51,31681,747],{"separator":122},[114,31683,31684],{"encoding":116},"\\log\\frac{p_1(x)}{p_0(x)}\n=(\\mu_1-\\mu_0)^\\top\\Sigma^{-1}x+\\text{常数},",[23,31686,31688,31865,31923,32045],{"className":31687,"ariaHidden":122},[121],[23,31689,31691,31694,31699,31702,31856,31859,31862],{"className":31690},[126],[23,31692],{"className":31693,"style":185},[130],[23,31695,22575,31697],{"className":31696},[1786],[23,31698,7089],{"style":7088},[23,31700],{"className":31701,"style":280},[148],[23,31703,31705,31708,31853],{"className":31704},[135],[23,31706],{"className":31707},[140,192],[23,31709,31711],{"className":31710},[71],[23,31712,31714,31845],{"className":31713},[199,200],[23,31715,31717,31842],{"className":31716},[204],[23,31718,31720,31777,31785],{"className":31719,"style":209},[208],[23,31721,31722,31725],{"style":212},[23,31723],{"className":31724,"style":217},[216],[23,31726,31728,31768,31771,31774],{"className":31727},[135],[23,31729,31731,31734],{"className":31730},[135],[23,31732,11],{"className":31733},[135,136],[23,31735,31737],{"className":31736},[998],[23,31738,31740,31760],{"className":31739},[199,200],[23,31741,31743,31757],{"className":31742},[204],[23,31744,31746],{"className":31745,"style":18647},[208],[23,31747,31748,31751],{"style":2029},[23,31749],{"className":31750,"style":1015},[216],[23,31752,31754],{"className":31753},[1019,1020,1021,1022],[23,31755,1532],{"className":31756},[135,1022],[23,31758,297],{"className":31759},[296],[23,31761,31763],{"className":31762},[204],[23,31764,31766],{"className":31765,"style":2048},[208],[23,31767],{},[23,31769,54],{"className":31770},[140],[23,31772,753],{"className":31773},[135,136],[23,31775,66],{"className":31776},[169],[23,31778,31779,31782],{"style":235},[23,31780],{"className":31781,"style":217},[216],[23,31783],{"className":31784,"style":243},[242],[23,31786,31787,31790],{"style":246},[23,31788],{"className":31789,"style":217},[216],[23,31791,31793,31833,31836,31839],{"className":31792},[135],[23,31794,31796,31799],{"className":31795},[135],[23,31797,11],{"className":31798},[135,136],[23,31800,31802],{"className":31801},[998],[23,31803,31805,31825],{"className":31804},[199,200],[23,31806,31808,31822],{"className":31807},[204],[23,31809,31811],{"className":31810,"style":18647},[208],[23,31812,31813,31816],{"style":2029},[23,31814],{"className":31815,"style":1015},[216],[23,31817,31819],{"className":31818},[1019,1020,1021,1022],[23,31820,4435],{"className":31821},[135,1022],[23,31823,297],{"className":31824},[296],[23,31826,31828],{"className":31827},[204],[23,31829,31831],{"className":31830,"style":2048},[208],[23,31832],{},[23,31834,54],{"className":31835},[140],[23,31837,753],{"className":31838},[135,136],[23,31840,66],{"className":31841},[169],[23,31843,297],{"className":31844},[296],[23,31846,31848],{"className":31847},[204],[23,31849,31851],{"className":31850,"style":304},[208],[23,31852],{},[23,31854],{"className":31855},[169,192],[23,31857],{"className":31858,"style":149},[148],[23,31860,69],{"className":31861},[153],[23,31863],{"className":31864,"style":149},[148],[23,31866,31868,31871,31874,31914,31917,31920],{"className":31867},[126],[23,31869],{"className":31870,"style":131},[130],[23,31872,54],{"className":31873},[140],[23,31875,31877,31880],{"className":31876},[135],[23,31878,22105],{"className":31879},[135,136],[23,31881,31883],{"className":31882},[998],[23,31884,31886,31906],{"className":31885},[199,200],[23,31887,31889,31903],{"className":31888},[204],[23,31890,31892],{"className":31891,"style":18647},[208],[23,31893,31894,31897],{"style":2029},[23,31895],{"className":31896,"style":1015},[216],[23,31898,31900],{"className":31899},[1019,1020,1021,1022],[23,31901,4435],{"className":31902},[135,1022],[23,3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",[23,32064,32066,32092],{"className":32065,"translate":27},[31],[23,32067,32069],{"className":32068},[35],[37,32070,32071],{"xmlns":39},[42,32072,32073,32089],{},[45,32074,32075,32081,32083],{},[898,32076,32077,32079],{},[48,32078,22112],{"mathvariant":111},[2581,32080,1532],{},[51,32082,747],{"separator":122},[898,32084,32085,32087],{},[48,32086,22112],{"mathvariant":111},[2581,32088,4435],{},[114,32090,32091],{"encoding":116},"\\Sigma_0,\\Sigma_1",[23,32093,32095],{"className":32094,"ariaHidden":122},[121],[23,32096,32098,32101,32141,32144,32147],{"className":32097},[126],[23,32099],{"className":32100,"style":817},[130],[23,32102,32104,32107],{"className":32103},[135],[23,32105,22112],{"className":32106},[135],[23,32108,32110],{"className":32109},[998],[23,32111,32113,32133],{"className":32112},[199,200],[23,32114,32116,32130],{"className":32115},[204],[23,32117,32119],{"className":32118,"style":18647},[208],[23,32120,32121,32124],{"style":2029},[23,32122],{"className":32123,"style":1015},[216],[23,32125,32127],{"className":32126},[1019,1020,1021,1022],[23,32128,1532],{"className":32129},[135,1022],[23,32131,297],{"className":32132},[296],[23,32134,32136],{"className":32135},[204],[23,32137,32139],{"className":32138,"style":2048},[208],[23,32140],{},[23,32142,747],{"className":32143},[824],[23,32145],{"className":32146,"style":280},[148],[23,32148,32150,32153],{"className":32149},[135],[23,32151,22112],{"className":32152},[135],[23,32154,32156],{"className":32155},[998],[23,32157,32159,32179],{"className":32158},[199,200],[23,32160,32162,32176],{"className":32161},[204],[23,32163,32165],{"className":32164,"style":18647},[208],[23,32166,32167,32170],{"style":2029},[23,32168],{"className":32169,"style":1015},[216],[23,32171,32173],{"className":32172},[1019,1020,1021,1022],[23,32174,4435],{"className":32175},[135,1022],[23,32177,297],{"className":32178},[296],[23,32180,32182],{"className":32181},[204],[23,32183,32185],{"className":32184,"style":2048},[208],[23,32186],{},"，对数似然比出现 ",[23,32189,32191,32241],{"className":32190,"translate":27},[31],[23,32192,32194],{"className":32193},[35],[37,32195,32196],{"xmlns":39},[42,32197,32198,32238],{},[45,32199,32200,32206,32208,32220,32222,32234,32236],{},[4518,32201,32202,32204],{},[48,32203,753],{},[48,32205,22471],{"mathvariant":111},[51,32207,54],{"stretchy":53},[19137,32209,32210,32212,32214],{},[48,32211,22112],{"mathvariant":111},[2581,32213,4435],{},[45,32215,32216,32218],{},[51,32217,4529],{},[2581,32219,4435],{},[51,32221,4529],{},[19137,32223,32224,32226,32228],{},[48,32225,22112],{"mathvariant":111},[2581,32227,1532],{},[45,32229,32230,32232],{},[51,32231,4529],{},[2581,32233,4435],{},[51,32235,66],{"stretchy":53},[48,32237,753],{},[114,32239,32240],{"encoding":116},"x^\\top(\\Sigma_1^{-1}-\\Sigma_0^{-1})x",[23,32242,32244,32353],{"className":32243,"ariaHidden":122},[121],[23,32245,32247,32251,32280,32283,32344,32347,32350],{"className":32246},[126],[23,32248],{"className":32249,"style":32250},[130],"height:1.1205em;vertical-align:-0.2663em;",[23,32252,32254,32257],{"className":32253},[135],[23,32255,753],{"className":32256},[135,136],[23,32258,32260],{"className":32259},[998],[23,32261,32263],{"className":32262},[199],[23,32264,32266],{"className":32265},[204],[23,32267,32269],{"className":32268,"style":4751},[208],[23,32270,32271,32274],{"style":4696},[23,32272],{"className":32273,"style":1015},[216],[23,32275,32277],{"className":32276},[1019,1020,1021,1022],[23,32278,22471],{"className":32279},[135,1022],[23,32281,54],{"className":32282},[140],[23,32284,32286,32289],{"className":32285},[135],[23,32287,22112],{"className":32288},[135],[23,32290,32292],{"className":32291},[998],[23,32293,32295,32335],{"className":32294},[199,200],[23,32296,32298,32332],{"className":32297},[204],[23,32299,32302,32314],{"className":32300,"style":32301},[208],"height:0.8542em;",[23,32303,32305,32308],{"style":32304},"top:-2.4337em;margin-left:0em;margin-right:0.05em;",[23,32306],{"className":32307,"style":1015},[216],[23,32309,32311],{"className":32310},[1019,1020,1021,1022],[23,32312,4435],{"className":32313},[135,1022],[23,32315,32317,32320],{"style":32316},"top:-3.1031em;margin-right:0.05em;",[23,32318],{"className":32319,"style":1015},[216],[23,32321,32323],{"className":32322},[1019,1020,1021,1022],[23,32324,32326,32329],{"className":32325},[135,1022],[23,32327,4529],{"className":32328},[135,1022],[23,32330,4435],{"className":32331},[135,1022],[23,32333,297],{"className":32334},[296],[23,32336,32338],{"className":32337},[204],[23,32339,32342],{"className":32340,"style":32341},[208],"height:0.2663em;",[23,32343],{},[23,32345],{"className":32346,"style":392},[148],[23,32348,4529],{"className":32349},[612],[23,32351],{"className":32352,"style":392},[148],[23,32354,32356,32359,32416,32419],{"className":32355},[126],[23,32357],{"className":32358,"style":32250},[130],[23,32360,32362,32365],{"className":32361},[135],[23,32363,22112],{"className":32364},[135],[23,32366,32368],{"className":32367},[998],[23,32369,32371,32408],{"className":32370},[199,200],[23,32372,32374,32405],{"className":32373},[204],[23,32375,32377,32388],{"className":32376,"style":32301},[208],[23,32378,32379,32382],{"style":32304},[23,32380],{"className":32381,"style":1015},[216],[23,32383,32385],{"className":32384},[1019,1020,1021,1022],[23,32386,1532],{"className":32387},[135,1022],[23,32389,32390,32393],{"style":32316},[23,32391],{"className":32392,"style":1015},[216],[23,32394,32396],{"className":32395},[1019,1020,1021,1022],[23,32397,32399,32402],{"className":32398},[135,1022],[23,32400,4529],{"className":32401},[135,1022],[23,32403,4435],{"className":32404},[135,1022],[23,32406,297],{"className":32407},[296],[23,32409,32411],{"className":32410},[204],[23,32412,32414],{"className":32413,"style":32341},[208],[23,32415],{},[23,32417,66],{"className":32418},[169],[23,32420,753],{"className":32421},[135,136]," 这样的二次项，边界变成二次曲面。朴素贝叶斯再加一层条件独立假设，把 ",[23,32424,32426,32449],{"className":32425,"translate":27},[31],[23,32427,32429],{"className":32428},[35],[37,32430,32431],{"xmlns":39},[42,32432,32433,32447],{},[45,32434,32435,32437,32439,32441,32443,32445],{},[48,32436,11],{},[51,32438,54],{"stretchy":53},[48,32440,753],{},[51,32442,60],{},[48,32444,660],{},[51,32446,66],{"stretchy":53},[114,32448,30968],{"encoding":116},[23,32450,32452,32476],{"className":32451,"ariaHidden":122},[121],[23,32453,32455,32458,32461,32464,32467,32470,32473],{"className":32454},[126],[23,32456],{"className":32457,"style":131},[130],[23,32459,11],{"className":32460},[135,136],[23,32462,54],{"className":32463},[140],[23,32465,753],{"className":32466},[135,136],[23,32468],{"className":32469,"style":149},[148],[23,32471,60],{"className":32472},[153],[23,32474],{"className":32475,"style":149},[148],[23,32477,32479,32482,32485],{"className":32478},[126],[23,32480],{"className":32481,"style":131},[130],[23,32483,660],{"className":32484,"style":676},[135,136],[23,32486,66],{"className":32487},[169]," 拆成各个特征的乘积，用很大的偏差换取很小的方差。",[11,32490,32491,32492,32520,32521,32586],{},"两条路的目标是同一个 ",[23,32493,32495,32508],{"className":32494,"translate":27},[31],[23,32496,32498],{"className":32497},[35],[37,32499,32500],{"xmlns":39},[42,32501,32502,32506],{},[45,32503,32504],{},[48,32505,9765],{},[114,32507,28979],{"encoding":116},[23,32509,32511],{"className":32510,"ariaHidden":122},[121],[23,32512,32514,32517],{"className":32513},[126],[23,32515],{"className":32516,"style":672},[130],[23,32518,9765],{"className":32519,"style":676},[135,136],"，但统计性质不同。生成式利用了 ",[23,32522,32524,32547],{"className":32523,"translate":27},[31],[23,32525,32527],{"className":32526},[35],[37,32528,32529],{"xmlns":39},[42,32530,32531,32545],{},[45,32532,32533,32535,32537,32539,32541,32543],{},[48,32534,11],{},[51,32536,54],{"stretchy":53},[48,32538,753],{},[51,32540,60],{},[48,32542,660],{},[51,32544,66],{"stretchy":53},[114,32546,30968],{"encoding":116},[23,32548,32550,32574],{"className":32549,"ariaHidden":122},[121],[23,32551,32553,32556,32559,32562,32565,32568,32571],{"className":32552},[126],[23,32554],{"className":32555,"style":131},[130],[23,32557,11],{"className":32558},[135,136],[23,32560,54],{"className":32561},[140],[23,32563,753],{"className":32564},[135,136],[23,32566],{"className":32567,"style":149},[148],[23,32569,60],{"className":32570},[153],[23,32572],{"className":32573,"style":149},[148],[23,32575,32577,32580,32583],{"className":32576},[126],[23,32578],{"className":32579,"style":131},[130],[23,32581,660],{"className":32582,"style":676},[135,136],[23,32584,66],{"className":32585},[169]," 的结构，在模型正确时样本效率更高；判别式直接优化决策边界，在模型误设时通常更稳健。这不是「谁更好」的问题，而是「你愿意把假设放在哪一侧」的问题。",[11,32588,32589,32590,32593,32594,32622,32623,32696,32697,32755],{},"这里说的是 ",[18,32591,32592],{},"oracle 贝叶斯规则","：它假设 ",[23,32595,32597,32610],{"className":32596,"translate":27},[31],[23,32598,32600],{"className":32599},[35],[37,32601,32602],{"xmlns":39},[42,32603,32604,32608],{},[45,32605,32606],{},[48,32607,9765],{},[114,32609,28979],{"encoding":116},[23,32611,32613],{"className":32612,"ariaHidden":122},[121],[23,32614,32616,32619],{"className":32615},[126],[23,32617],{"className":32618,"style":672},[130],[23,32620,9765],{"className":32621,"style":676},[135,136]," 已知。实践中我们把估计的 ",[23,32624,32626,32644],{"className":32625,"translate":27},[31],[23,32627,32629],{"className":32628},[35],[37,32630,32631],{"xmlns":39},[42,32632,32633,32641],{},[45,32634,32635],{},[16807,32636,32637,32639],{"accent":122},[48,32638,9765],{},[51,32640,16813],{},[114,32642,32643],{"encoding":116},"\\hat\\eta",[23,32645,32647],{"className":32646,"ariaHidden":122},[121],[23,32648,32650,32653],{"className":32649},[126],[23,32651],{"className":32652,"style":5414},[130],[23,32654,32656],{"className":32655},[135,16830],[23,32657,32659,32688],{"className":32658},[199,200],[23,32660,32662,32685],{"className":32661},[204],[23,32663,32665,32673],{"className":32664,"style":779},[208],[23,32666,32667,32670],{"style":7130},[23,32668],{"className":32669,"style":217},[216],[23,32671,9765],{"className":32672,"style":676},[135,136],[23,32674,32675,32678],{"style":7130},[23,32676],{"className":32677,"style":217},[216],[23,32679,32682],{"className":32680,"style":32681},[16858],"left:-0.1944em;",[23,32683,16813],{"className":32684},[135],[23,32686,297],{"className":32687},[296],[23,32689,32691],{"className":32690},[204],[23,32692,32694],{"className":32693,"style":16872},[208],[23,32695],{}," 代入，得到的只是 plug-in 分类器；文献里的「贝叶斯分类器」有时指这个 oracle 规则，有时指朴素贝叶斯之类的具体算法，读的时候要分清。plug-in 分类器的风险与 ",[23,32698,32700,32717],{"className":32699,"translate":27},[31],[23,32701,32703],{"className":32702},[35],[37,32704,32705],{"xmlns":39},[42,32706,32707,32715],{},[45,32708,32709],{},[4518,32710,32711,32713],{},[48,32712,572],{},[51,32714,6881],{},[114,32716,14962],{"encoding":116},[23,32718,32720],{"className":32719,"ariaHidden":122},[121],[23,32721,32723,32726],{"className":32722},[126],[23,32724],{"className":32725,"style":6912},[130],[23,32727,32729,32732],{"className":32728},[135],[23,32730,572],{"className":32731,"style":963},[135,136],[23,32733,32735],{"className":32734},[998],[23,32736,32738],{"className":32737},[199],[23,32739,32741],{"className":32740},[204],[23,32742,32744],{"className":32743,"style":6912},[208],[23,32745,32746,32749],{"style":4696},[23,32747],{"className":32748,"style":1015},[216],[23,32750,32752],{"className":32751},[1019,1020,1021,1022],[23,32753,6881],{"className":32754},[612,1022]," 之间的差距，正是第 7 节要分解的估计误差。",[11,32757,32758,32759,32803,32804,32836,32837,32865,32866,1471],{},"值得一提的是，朴素贝叶斯的条件独立假设几乎总是错的，但它的分类性能常常不差。原因在于 0-1 损失只关心 ",[23,32760,32762,32782],{"className":32761,"translate":27},[31],[23,32763,32765],{"className":32764},[35],[37,32766,32767],{"xmlns":39},[42,32768,32769,32779],{},[45,32770,32771,32773,32775,32777],{},[48,32772,9765],{},[51,32774,54],{"stretchy":53},[48,32776,753],{},[51,32778,66],{"stretchy":53},[114,32780,32781],{"encoding":116},"\\eta(x)",[23,32783,32785],{"className":32784,"ariaHidden":122},[121],[23,32786,32788,32791,32794,32797,32800],{"className":32787},[126],[23,32789],{"className":32790,"style":131},[130],[23,32792,9765],{"className":32793,"style":676},[135,136],[23,32795,54],{"className":32796},[140],[23,32798,753],{"className":32799},[135,136],[23,32801,66],{"className":32802},[169]," 是否越过 ",[23,32805,32807,32824],{"className":32806,"translate":27},[31],[23,32808,32810],{"className":32809},[35],[37,32811,32812],{"xmlns":39},[42,32813,32814,32822],{},[45,32815,32816,32818,32820],{},[2581,32817,4435],{},[48,32819,2875],{"mathvariant":111},[2581,32821,9952],{},[114,32823,10469],{"encoding":116},[23,32825,32827],{"className":32826,"ariaHidden":122},[121],[23,32828,32830,32833],{"className":32829},[126],[23,32831],{"className":32832,"style":131},[130],[23,32834,10469],{"className":32835},[135],"，而不要求 ",[23,32838,32840,32853],{"className":32839,"translate":27},[31],[23,32841,32843],{"className":32842},[35],[37,32844,32845],{"xmlns":39},[42,32846,32847,32851],{},[45,32848,32849],{},[48,32850,9765],{},[114,32852,28979],{"encoding":116},[23,32854,32856],{"className":32855,"ariaHidden":122},[121],[23,32857,32859,32862],{"className":32858},[126],[23,32860],{"className":32861,"style":672},[130],[23,32863,9765],{"className":32864,"style":676},[135,136]," 的数值完全准确；只要假设带来的偏差没有把决策边界推得太远，分类结果就可以是对的。这个现象也提醒我们：",[18,32867,32868],{},"概率估计的质量与决策的质量不是同一件事",[11,32870,32871,32872,32915,32916,32919,32920,32985],{},"这个区别在损失函数上有一个精确的对应。0-1 损失只关心 ",[23,32873,32875,32894],{"className":32874,"translate":27},[31],[23,32876,32878],{"className":32877},[35],[37,32879,32880],{"xmlns":39},[42,32881,32882,32892],{},[45,32883,32884,32886,32888,32890],{},[48,32885,9765],{},[51,32887,54],{"stretchy":53},[48,32889,753],{},[51,32891,66],{"stretchy":53},[114,32893,32781],{"encoding":116},[23,32895,32897],{"className":32896,"ariaHidden":122},[121],[23,32898,32900,32903,32906,32909,32912],{"className":32899},[126],[23,32901],{"className":32902,"style":131},[130],[23,32904,9765],{"className":32905,"style":676},[135,136],[23,32907,54],{"className":32908},[140],[23,32910,753],{"className":32911},[135,136],[23,32913,66],{"className":32914},[169]," 是否越过阈值，因此它对概率的微小误差不敏感；交叉熵（对数损失）则是一个",[18,32917,32918],{},"严格 proper scoring rule","，它的总体最优解恰好是真实的条件分布 ",[23,32921,32923,32946],{"className":32922,"translate":27},[31],[23,32924,32926],{"className":32925},[35],[37,32927,32928],{"xmlns":39},[42,32929,32930,32944],{},[45,32931,32932,32934,32936,32938,32940,32942],{},[48,32933,11],{},[51,32935,54],{"stretchy":53},[48,32937,660],{},[51,32939,60],{},[48,32941,753],{},[51,32943,66],{"stretchy":53},[114,32945,6815],{"encoding":116},[23,32947,32949,32973],{"className":32948,"ariaHidden":122},[121],[23,32950,32952,32955,32958,32961,32964,32967,32970],{"className":32951},[126],[23,32953],{"className":32954,"style":131},[130],[23,32956,11],{"className":32957},[135,136],[23,32959,54],{"className":32960},[140],[23,32962,660],{"className":32963,"style":676},[135,136],[23,32965],{"className":32966,"style":149},[148],[23,32968,60],{"className":32969},[153],[23,32971],{"className":32972,"style":149},[148],[23,32974,32976,32979,32982],{"className":32975},[126],[23,32977],{"className":32978,"style":131},[130],[23,32980,753],{"className":32981},[135,136],[23,32983,66],{"className":32984},[169],"：",[23,32987,32989],{"className":32988,"translate":27},[26],[23,32990,32992,33101],{"className":32991,"translate":27},[31],[23,32993,32995],{"className":32994},[35],[37,32996,32997],{"xmlns":39,"display":40},[42,32998,32999,33098],{},[45,33000,33001,33003,33005,33007,33009,33011,33014,33016,33018,33020,33022,33024,33026,33028,33030,33032,33034,33036,33038,33040,33042,33044,33046,33048,33050,33052,33054,33056,33058,33064,33066,33068,33070,33072,33074,33076,33078,33080,33082,33084,33086,33088,33090,33092,33094,33096],{},[48,33002,902],{"mathvariant":571},[51,33004,924],{"stretchy":53},[51,33006,4529],{},[48,33008,22418],{},[51,33010,6959],{},[48,33012,33013],{},"q",[51,33015,54],{"stretchy":53},[48,33017,369],{},[51,33019,60],{},[48,33021,434],{},[51,33023,66],{"stretchy":53},[51,33025,945],{"stretchy":53},[51,33027,69],{},[48,33029,902],{"mathvariant":571},[51,33031,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,33033,26249],{"mathvariant":571},[51,33035,54],{"stretchy":53},[48,33037,11],{},[51,33039,54],{"stretchy":53},[51,33041,5758],{},[51,33043,60],{},[48,33045,434],{},[51,33047,66],{"stretchy":53},[51,33049,66],{"stretchy":53},[51,33051,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,33053,5106],{},[48,33055,902],{"mathvariant":571},[51,33057,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[45,33059,33060,33062],{},[48,33061,26918],{"mathvariant":111},[48,33063,555],{"mathvariant":111},[51,33065,54],{"stretchy":53},[48,33067,11],{},[51,33069,54],{"stretchy":53},[51,33071,5758],{},[51,33073,60],{},[48,33075,434],{},[51,33077,66],{"stretchy":53},[48,33079,20680],{"mathvariant":111},[48,33081,33013],{},[51,33083,54],{"stretchy":53},[51,33085,5758],{},[51,33087,60],{},[48,33089,434],{},[51,33091,66],{"stretchy":53},[51,33093,66],{"stretchy":53},[51,33095,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,33097,747],{"separator":122},[114,33099,33100],{"encoding":116},"\\mathbb E[-\\log q(Y\\mid X)]\n=\\mathbb E\\big[\\mathbb H(p(\\cdot\\mid X))\\big]\n+\\mathbb E\\big[\\mathrm{KL}(p(\\cdot\\mid X)\\|q(\\cdot\\mid X))\\big],",[23,33102,33104,33148,33170,33209,33236,33278,33311],{"className":33103,"ariaHidden":122},[121],[23,33105,33107,33110,33113,33116,33119,33122,33127,33130,33133,33136,33139,33142,33145],{"className":33106},[126],[23,33108],{"className":33109,"style":131},[130],[23,33111,902],{"className":33112},[135,644],[23,33114,924],{"className":33115},[140],[23,33117,4529],{"className":33118},[135],[23,33120],{"className":33121,"style":280},[148],[23,33123,22575,33125],{"className":33124},[1786],[23,33126,7089],{"style":7088},[23,33128],{"className":33129,"style":280},[148],[23,33131,33013],{"className":33132,"style":676},[135,136],[23,33134,54],{"className":33135},[140],[23,33137,369],{"className":33138,"style":392},[135,136],[23,33140],{"className":33141,"style":149},[148],[23,33143,60],{"className":33144},[153],[23,33146],{"className":33147,"style":149},[148],[23,33149,33151,33154,33157,33161,33164,33167],{"className":33150},[126],[23,33152],{"className":33153,"style":131},[130],[23,33155,434],{"className":33156,"style":454},[135,136],[23,33158,33160],{"className":33159},[169],")]",[23,33162],{"className":33163,"style":149},[148],[23,33165,69],{"className":33166},[153],[23,33168],{"className":33169,"style":149},[148],[23,33171,33173,33176,33179,33185,33188,33191,33194,33197,33200,33203,33206],{"className":33172},[126],[23,33174],{"className":33175,"style":2213},[130],[23,33177,902],{"className":33178},[135,644],[23,33180,33182],{"className":33181},[135],[23,33183,924],{"className":33184},[1066,1067],[23,33186,26249],{"className":33187},[135,644],[23,33189,54],{"className":33190},[140],[23,33192,11],{"className":33193},[135,136],[23,33195,54],{"className":33196},[140],[23,33198,5758],{"className":33199},[135],[23,33201],{"className":33202,"style":149},[148],[23,33204,60],{"className":33205},[153],[23,33207],{"className":33208,"style":149},[148],[23,33210,33212,33215,33218,33221,33227,33230,33233],{"className":33211},[126],[23,33213],{"className":33214,"style":2213},[130],[23,33216,434],{"className":33217,"style":454},[135,136],[23,33219,1095],{"className":33220},[169],[23,33222,33224],{"className":33223},[135],[23,33225,945],{"className":33226},[1066,1067],[23,33228],{"className":33229,"style":392},[148],[23,33231,5106],{"className":33232},[612],[23,33234],{"className":33235,"style":392},[148],[23,33237,33239,33242,33245,33251,33257,33260,33263,33266,33269,33272,33275],{"className":33238},[126],[23,33240],{"className":33241,"style":2213},[130],[23,33243,902],{"className":33244},[135,644],[23,33246,33248],{"className":33247},[135],[23,33249,924],{"className":33250},[1066,1067],[23,33252,33254],{"className":33253},[135],[23,33255,27187],{"className":33256},[135,5391],[23,33258,54],{"className":33259},[140],[23,33261,11],{"className":33262},[135,136],[23,33264,54],{"className":33265},[140],[23,33267,5758],{"className":33268},[135],[23,33270],{"className":33271,"style":149},[148],[23,33273,60],{"className":33274},[153],[23,33276],{"className":33277,"style":149},[148],[23,33279,33281,33284,33287,33290,33293,33296,33299,33302,33305,33308],{"className":33280},[126],[23,33282],{"className":33283,"style":131},[130],[23,33285,434],{"className":33286,"style":454},[135,136],[23,33288,66],{"className":33289},[169],[23,33291,20680],{"className":33292},[135],[23,33294,33013],{"className":33295,"style":676},[135,136],[23,33297,54],{"className":33298},[140],[23,33300,5758],{"className":33301},[135],[23,33303],{"className":33304,"style":149},[148],[23,33306,60],{"className":33307},[153],[23,33309],{"className":33310,"style":149},[148],[23,33312,33314,33317,33320,33323,33329],{"className":33313},[126],[23,33315],{"className":33316,"style":2213},[130],[23,33318,434],{"className":33319,"style":454},[135,136],[23,33321,1095],{"className":33322},[169],[23,33324,33326],{"className":33325},[135],[23,33327,945],{"className":33328},[1066,1067],[23,33330,747],{"className":33331},[824],[11,33333,33334,33335,33363,33364,33452],{},"右边第一项与 ",[23,33336,33338,33351],{"className":33337,"translate":27},[31],[23,33339,33341],{"className":33340},[35],[37,33342,33343],{"xmlns":39},[42,33344,33345,33349],{},[45,33346,33347],{},[48,33348,33013],{},[114,33350,33013],{"encoding":116},[23,33352,33354],{"className":33353,"ariaHidden":122},[121],[23,33355,33357,33360],{"className":33356},[126],[23,33358],{"className":33359,"style":672},[130],[23,33361,33013],{"className":33362,"style":676},[135,136]," 无关，第二项在 ",[23,33365,33367,33395],{"className":33366,"translate":27},[31],[23,33368,33370],{"className":33369},[35],[37,33371,33372],{"xmlns":39},[42,33373,33374,33392],{},[45,33375,33376,33378,33380,33382,33384,33386,33388,33390],{},[48,33377,33013],{},[51,33379,69],{},[48,33381,11],{},[51,33383,54],{"stretchy":53},[51,33385,5758],{},[51,33387,60],{},[48,33389,434],{},[51,33391,66],{"stretchy":53},[114,33393,33394],{"encoding":116},"q=p(\\cdot\\mid X)",[23,33396,33398,33416,33440],{"className":33397,"ariaHidden":122},[121],[23,33399,33401,33404,33407,33410,33413],{"className":33400},[126],[23,33402],{"className":33403,"style":672},[130],[23,33405,33013],{"className":33406,"style":676},[135,136],[23,33408],{"className":33409,"style":149},[148],[23,33411,69],{"className":33412},[153],[23,33414],{"className":33415,"style":149},[148],[23,33417,33419,33422,33425,33428,33431,33434,33437],{"className":33418},[126],[23,33420],{"className":33421,"style":131},[130],[23,33423,11],{"className":33424},[135,136],[23,33426,54],{"className":33427},[140],[23,33429,5758],{"className":33430},[135],[23,33432],{"className":33433,"style":149},[148],[23,33435,60],{"className":33436},[153],[23,33438],{"className":33439,"style":149},[148],[23,33441,33443,33446,33449],{"className":33442},[126],[23,33444],{"className":33445,"style":131},[130],[23,33447,434],{"className":33448,"style":454},[135,136],[23,33450,66],{"className":33451},[169]," 时为零。所以，在总体意义上，用交叉熵训练模型是在要求它给出校准的概率；用 0-1 损失训练，只要求它给出正确的分类。两者目标不同，不能互相替代。",[2501,33454,33456],{"id":33455},"_52-正则化就是先验","5.2 正则化就是先验",[11,33458,33459,33460,33737,33738,33797],{},"设参数化模型 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",[23,33739,33741,33759],{"className":33740,"translate":27},[31],[23,33742,33744],{"className":33743},[35],[37,33745,33746],{"xmlns":39},[42,33747,33748,33756],{},[45,33749,33750],{},[4518,33751,33752,33754],{},[48,33753,29634],{},[2581,33755,9952],{},[114,33757,33758],{"encoding":116},"\\sigma^2",[23,33760,33762],{"className":33761,"ariaHidden":122},[121],[23,33763,33765,33768],{"className":33764},[126],[23,33766],{"className":33767,"style":4693},[130],[23,33769,33771,33774],{"className":33770},[135],[23,33772,29634],{"className":33773,"style":676},[135,136],[23,33775,33777],{"className":33776},[998],[23,33778,33780],{"className":33779},[199],[23,33781,33783],{"className":33782},[204],[23,33784,33786],{"className":33785,"style":4693},[208],[23,33787,33788,33791],{"style":4696},[23,33789],{"className":33790,"style":1015},[216],[23,33792,33794],{"className":33793},[1019,1020,1021,1022],[23,33795,9952],{"className":33796},[135,1022],"。则负对数似然正比于平方误差：",[23,33799,33801],{"className":33800,"translate":27},[26],[23,33802,33804,33895],{"className":33803,"translate":27},[31],[23,33805,33807],{"className":33806},[35],[37,33808,33809],{"xmlns":39,"display":40},[42,33810,33811,33892],{},[45,33812,33813,33815,33817,33819,33821,33823,33825,33827,33829,33831,33833,33847,33854,33856,33862,33864,33870,33872,33878,33880,33886,33888,33890],{},[51,33814,4529],{},[48,33816,22418],{},[51,33818,6959],{},[48,33820,11],{},[51,33822,54],{"stretchy":53},[48,33824,63],{},[51,33826,60],{},[48,33828,57],{},[51,33830,66],{"stretchy":53},[51,33832,69],{},[71,33834,33835,33837],{},[2581,33836,4435],{},[45,33838,33839,33841],{},[2581,33840,9952],{},[4518,33842,33843,33845],{},[48,33844,29634],{},[2581,33846,9952],{},[6961,33848,33849,33852],{},[51,33850,33851],{},"∑",[48,33853,33493],{},[51,33855,54],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[898,33857,33858,33860],{},[48,33859,660],{},[48,33861,33493],{},[51,33863,4529],{},[898,33865,33866,33868],{},[48,33867,8336],{},[48,33869,57],{},[51,33871,54],{"stretchy":53},[898,33873,33874,33876],{},[48,33875,753],{},[48,33877,33493],{},[51,33879,66],{"stretchy":53},[4518,33881,33882,33884],{},[51,33883,66],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[2581,33885,9952],{},[51,33887,5106],{},[88,33889,31679],{},[48,33891,112],{"mathvariant":111},[114,33893,33894],{"encoding":116},"-\\log p(D\\mid\\theta)\n=\\frac{1}{2\\sigma^2}\\sum_i\\big(y_i-f_\\theta(x_i)\\big)^2+\\text{常数}.",[23,33896,33898,33936,33957,34167,34302],{"className":33897,"ariaHidden":122},[121],[23,33899,33901,33904,33907,33910,33915,33918,33921,33924,33927,33930,33933],{"className":33900},[126],[23,33902],{"className":33903,"style":131},[130],[23,33905,4529],{"className":33906},[135],[23,33908],{"className":33909,"style":280},[148],[23,33911,22575,33913],{"className":33912},[1786],[23,33914,7089],{"style":7088},[23,33916],{"className":33917,"style":280},[148],[23,33919,11],{"className":33920},[135,136],[23,33922,54],{"className":33923},[140],[23,33925,63],{"className":33926,"style":144},[135,136],[23,33928],{"className":33929,"style":149},[148],[23,33931,60],{"className":33932},[153],[23,33934],{"className":33935,"style":149},[148],[23,33937,33939,33942,33945,33948,33951,33954],{"className":33938},[126],[23,33940],{"className":33941,"style":131},[130],[23,33943,57],{"className":33944,"style":144},[135,136],[23,33946,66],{"className":33947},[169],[23,33949],{"className":33950,"style":149},[148],[23,33952,69],{"className":33953},[153],[23,33955],{"className":33956,"style":149},[148],[23,33958,33960,33964,34057,34060,34109,34112,34118,34158,34161,34164],{"className":33959},[126],[23,33961],{"className":33962,"style":33963},[130],"height:2.5991em;vertical-align:-1.2777em;",[23,33965,33967,33970,34054],{"className":33966},[135],[23,33968],{"className":33969},[140,192],[23,33971,33973],{"className":33972},[71],[23,33974,33976,34046],{"className":33975},[199,200],[23,33977,33979,34043],{"className":33978},[204],[23,33980,33982,34024,34032],{"className":33981,"style":12409},[208],[23,33983,33984,33987],{"style":212},[23,33985],{"className":33986,"style":217},[216],[23,33988,33990,33993],{"className":33989},[135],[23,33991,9952],{"className":33992},[135],[23,33994,33996,33999],{"className":33995},[135],[23,33997,29634],{"className":33998,"style":676},[135,136],[23,34000,34002],{"className":34001},[998],[23,34003,34005],{"className":34004},[199],[23,34006,34008],{"className":34007},[204],[23,34009,34012],{"className":34010,"style":34011},[208],"height:0.7401em;",[23,34013,34015,34018],{"style":34014},"top:-2.989em;margin-right:0.05em;",[23,34016],{"className":34017,"style":1015},[216],[23,34019,34021],{"className":34020},[1019,1020,1021,1022],[23,34022,9952],{"className":34023},[135,1022],[23,34025,34026,34029],{"style":235},[23,34027],{"className":34028,"style":217},[216],[23,34030],{"className":34031,"style":243},[242],[23,34033,34034,34037],{"style":246},[23,34035],{"className":34036,"style":217},[216],[23,34038,34040],{"className":34039},[135],[23,34041,4435],{"className":34042},[135],[23,34044,297],{"className":34045},[296],[23,34047,34049],{"className":34048},[204],[23,34050,34052],{"className":34051,"style":6287},[208],[23,34053],{},[23,34055],{"className":34056},[169,192],[23,34058],{"className":34059,"style":280},[148],[23,34061,34063],{"className":34062},[1786,7096],[23,34064,34066,34100],{"className":34065},[199,200],[23,34067,34069,34097],{"className":34068},[204],[23,34070,34073,34086],{"className":34071,"style":34072},[208],"height:1.05em;",[23,34074,34076,34080],{"style":34075},"top:-1.8723em;margin-left:0em;",[23,34077],{"className":34078,"style":34079},[216],"height:3.05em;",[23,34081,34083],{"className":34082},[1019,1020,1021,1022],[23,34084,33493],{"className":34085},[135,136,1022],[23,34087,34089,34092],{"style":34088},"top:-3.05em;",[23,34090],{"className":34091,"style":34079},[216],[23,34093,34094],{},[23,34095,33851],{"className":34096},[1786,1790,1791],[23,34098,297],{"className":34099},[296],[23,34101,34103],{"className":34102},[204],[23,34104,34107],{"className":34105,"style":34106},[208],"height:1.2777em;",[23,34108],{},[23,34110],{"className":34111,"style":280},[148],[23,34113,34115],{"className":34114},[135],[23,34116,54],{"className":34117},[1066,1067],[23,34119,34121,34124],{"className":34120},[135],[23,34122,660],{"className":34123,"style":676},[135,136],[23,34125,34127],{"className":34126},[998],[23,34128,34130,34150],{"className":34129},[199,200],[23,34131,34133,34147],{"className":34132},[204],[23,34134,34136],{"className":34135,"style":33647},[208],[23,34137,34138,34141],{"style":21850},[23,34139],{"className":34140,"style":1015},[216],[23,34142,34144],{"className":34143},[1019,1020,1021,1022],[23,34145,33493],{"className":34146},[135,136,1022],[23,34148,297],{"className":34149},[296],[23,34151,34153],{"className":34152},[204],[23,34154,34156],{"className":34155,"style":2048},[208],[23,34157],{},[23,34159],{"className":34160,"style":392},[148],[23,34162,4529],{"className":34163},[612],[23,34165],{"className":34166,"style":392},[148],[23,34168,34170,34174,34215,34218,34258,34261,34293,34296,34299],{"className":34169},[126],[23,34171],{"className":34172,"style":34173},[130],"height:1.404em;vertical-align:-0.35em;",[23,34175,34177,34180],{"className":34176},[135],[23,34178,8336],{"className":34179,"s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",[23,35382,35384,35416],{"className":35383,"translate":27},[31],[23,35385,35387],{"className":35386},[35],[37,35388,35389],{"xmlns":39},[42,35390,35391,35413],{},[45,35392,35393,35399,35401,35403,35405,35411],{},[4518,35394,35395,35397],{},[48,35396,29634],{},[2581,35398,9952],{},[48,35400,2875],{"mathvariant":111},[51,35402,54],{"stretchy":53},[48,35404,4806],{},[4518,35406,35407,35409],{},[48,35408,34348],{},[2581,35410,9952],{},[51,35412,66],{"stretchy":53},[114,35414,35415],{"encoding":116},"\\sigma^2\u002F(n\\tau^2)",[23,35417,35419],{"className":35418,"ariaHidden":122},[121],[23,35420,35422,35425,35454,35457,35460,35463,35492],{"className":35421},[126],[23,35423],{"className":35424,"style":10661},[130],[23,35426,35428,35431],{"className":35427},[135],[23,35429,29634],{"className":35430,"style":676},[135,136],[23,35432,35434],{"className":35433},[998],[23,35435,35437],{"className":35436},[199],[23,35438,35440],{"className":35439},[204],[23,35441,35443],{"className":35442,"style":4693},[208],[23,35444,35445,35448],{"style":4696},[23,35446],{"className":35447,"style":1015},[216],[23,35449,35451],{"className":35450},[1019,1020,1021,1022],[23,35452,9952],{"className":35453},[135,1022],[23,35455,2875],{"className":35456},[135],[23,35458,54],{"className":35459},[140],[23,35461,4806],{"className":35462},[135,136],[23,35464,35466,35469],{"className":35465},[135],[23,35467,34348],{"className":35468,"style":34407},[135,136],[23,35470,35472],{"className":35471},[998],[23,35473,35475],{"className":35474},[199],[23,35476,35478],{"className":35477},[204],[23,35479,35481],{"className":35480,"style":4693},[208],[23,35482,35483,35486],{"style":4696},[23,35484],{"className":35485,"style":1015},[216],[23,35487,35489],{"className":35488},[1019,1020,1021,1022],[23,35490,9952],{"className":35491},[135,1022],[23,35493,66],{"className":35494},[169],"。若先验换成 Laplace 分布 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",[23,35793,35795,35817],{"className":35794,"translate":27},[31],[23,35796,35798],{"className":35797},[35],[37,35799,35800],{"xmlns":39},[42,35801,35802,35814],{},[45,35803,35804,35806,35808],{},[48,35805,20680],{"mathvariant":111},[48,35807,57],{},[898,35809,35810,35812],{},[48,35811,20680],{"mathvariant":111},[2581,35813,4435],{},[114,35815,35816],{"encoding":116},"\\|\\theta\\|_1",[23,35818,35820],{"className":35819,"ariaHidden":122},[121],[23,35821,35823,35826,35829,35832],{"className":35822},[126],[23,35824],{"className":35825,"style":131},[130],[23,35827,20680],{"className":35828},[135],[23,35830,57],{"className":35831,"style":144},[135,136],[23,35833,35835,35838],{"className":35834},[135],[23,35836,20680],{"className":35837},[135],[23,35839,35841],{"className":35840},[998],[23,35842,35844,35864],{"className":35843},[199,200],[23,35845,35847,35861],{"className":35846},[204],[23,35848,35850],{"className":35849,"style":18647},[208],[23,35851,35852,35855],{"style":2029},[23,35853],{"className":35854,"style":1015},[216],[23,35856,35858],{"className":35857},[1019,1020,1021,1022],[23,35859,4435],{"className":35860},[135,1022],[23,35862,297],{"className":35863},[296],[23,35865,35867],{"className":35866},[204],[23,35868,35870],{"className":35869,"style":2048},[208],[23,35871],{},"，MAP 就变成 Lasso。",[3764,35874,35875,35887],{},[3767,35876,35877],{},[3770,35878,35879,35881,35884],{},[3773,35880,3790],{},[3773,35882,35883],{},"负对数先验",[3773,35885,35886],{},"等价惩罚",[3783,35888,35889,35899,36120,36272],{},[3770,35890,35891,35894,35896],{},[3788,35892,35893],{},"均匀先验",[3788,35895,31679],{},[3788,35897,35898],{},"无（MLE）",[3770,35900,35901,35991,36117],{},[3788,35902,35903,35904],{},"高斯 ",[23,35905,35907,35935],{"className":35906,"translate":27},[31],[23,35908,35910],{"className":35909},[35],[37,35911,35912],{"xmlns":39},[42,35913,35914,35932],{},[45,35915,35916,35918,35920,35922,35924,35930],{},[48,35917,12801],{"mathvariant":375},[51,35919,54],{"stretchy":53},[2581,35921,1532],{},[51,35923,747],{"separator":122},[4518,35925,35926,35928],{},[48,35927,34348],{},[2581,35929,9952],{},[51,35931,66],{"stretchy":53},[114,35933,35934],{"encoding":116},"\\mathcal N(0,\\tau^2)",[23,35936,35938],{"className":35937,"ariaHidden":122},[121],[23,35939,35941,35944,35947,35950,35953,35956,35959,35988],{"className":35940},[126],[23,35942],{"className":35943,"style":10661},[130],[23,35945,12801],{"className":35946,"style":22218},[135,412],[23,35948,54],{"className":35949},[140],[23,35951,1532],{"className":35952},[135],[23,35954,747],{"className":35955},[824],[23,35957],{"className":35958,"style":280},[148],[23,35960,35962,35965],{"className":35961},[135],[23,35963,34348],{"className":35964,"style":34407},[135,136],[23,35966,35968],{"className":35967},[998],[23,35969,35971],{"className":35970},[199],[23,35972,35974],{"className":35973},[204],[23,35975,35977],{"className":35976,"style":4693},[208],[23,35978,35979,35982],{"style":4696},[23,35980],{"className":35981,"style":1015},[216],[23,35983,35985],{"className":35984},[1019,1020,1021,1022],[23,35986,9952],{"className":35987},[135,1022],[23,35989,66],{"className":35990},[169],[3788,35992,35993],{},[23,35994,35996,36032],{"className":35995,"translate":27},[31],[23,35997,35999],{"className":35998},[35],[37,36000,36001],{"xmlns":39},[42,36002,36003,36029],{},[45,36004,36005,36007,36009,36015,36017,36019,36021,36027],{},[48,36006,20680],{"mathvariant":111},[48,36008,57],{},[4518,36010,36011,36013],{},[48,36012,20680],{"mathvariant":111},[2581,36014,9952],{},[48,36016,2875],{"mathvariant":111},[51,36018,54],{"stretchy":53},[2581,36020,9952],{},[4518,36022,36023,36025],{},[48,36024,34348],{},[2581,36026,9952],{},[51,36028,66],{"stretchy":53},[114,36030,36031],{"encoding":116},"\\|\\theta\\|^2\u002F(2\\tau^2)",[23,36033,36035],{"className":36034,"ariaHidden":122},[121],[23,36036,36038,36041,36044,36047,36076,36079,36082,36085,36114],{"className":36037},[126],[23,36039],{"className":36040,"style":10661},[130],[23,36042,20680],{"className":36043},[135],[23,36045,57],{"className":36046,"style":144},[135,136],[23,36048,36050,36053],{"className":36049},[135],[23,36051,20680],{"className":36052},[135],[23,36054,36056],{"className":36055},[998],[23,36057,36059],{"className":36058},[199],[23,36060,36062],{"className":36061},[204],[23,36063,36065],{"className":36064,"style":4693},[208],[23,36066,36067,36070],{"style":4696},[23,36068],{"className":36069,"style":1015},[216],[23,36071,36073],{"className":36072},[1019,1020,1021,1022],[23,36074,9952],{"className":36075},[135,1022],[23,36077,2875],{"className":36078},[135],[23,36080,54],{"className":36081},[140],[23,36083,9952],{"className":36084},[135],[23,36086,36088,36091],{"className":36087},[135],[23,36089,34348],{"className":36090,"style":34407},[135,136],[23,36092,36094],{"className":36093},[998],[23,36095,36097],{"className":36096},[199],[23,36098,36100],{"className":36099},[204],[23,36101,36103],{"className":36102,"style":4693},[208],[23,36104,36105,36108],{"style":4696},[23,36106],{"className":36107,"style":1015},[216],[23,36109,36111],{"className":36110},[1019,1020,1021,1022],[23,36112,9952],{"className":36113},[135,1022],[23,36115,66],{"className":36116},[169],[3788,36118,36119],{},"岭回归、权重衰减",[3770,36121,36122,36177,36269],{},[3788,36123,36124,36125],{},"Laplace ",[23,36126,36128,36150],{"className":36127,"translate":27},[31],[23,36129,36131],{"className":36130},[35],[37,36132,36133],{"xmlns":39},[42,36134,36135,36147],{},[45,36136,36137,36139,36141,36143,36145],{},[51,36138,54],{"stretchy":53},[2581,36140,1532],{},[51,36142,747],{"separator":122},[48,36144,35533],{},[51,36146,66],{"stretchy":53},[114,36148,36149],{"encoding":116},"(0,b)",[23,36151,36153],{"className":36152,"ariaHidden":122},[121],[23,36154,36156,36159,36162,36165,36168,36171,36174],{"className":36155},[126],[23,36157],{"className":36158,"style":131},[130],[23,36160,54],{"className":36161},[140],[23,36163,1532],{"className":36164},[135],[23,36166,747],{"className":36167},[824],[23,36169],{"className":36170,"style":280},[148],[23,36172,35533],{"className":36173},[135,136],[23,36175,66],{"className":36176},[169],[3788,36178,36179],{},[23,36180,36182,36208],{"className":36181,"translate":27},[31],[23,36183,36185],{"className":36184},[35],[37,36186,36187],{"xmlns":39},[42,36188,36189,36205],{},[45,36190,36191,36193,36195,36201,36203],{},[48,36192,20680],{"mathvariant":111},[48,36194,57],{},[898,36196,36197,36199],{},[48,36198,20680],{"mathvariant":111},[2581,36200,4435],{},[48,36202,2875],{"mathvariant":111},[48,36204,35533],{},[114,36206,36207],{"encoding":116},"\\|\\theta\\|_1\u002Fb",[23,36209,36211],{"className":36210,"ariaHidden":122},[121],[23,36212,36214,36217,36220,36223,36263,36266],{"className":36213},[126],[23,36215],{"className":36216,"style":131},[130],[23,36218,20680],{"className":36219},[135],[23,36221,57],{"className":36222,"style":144},[135,136],[23,36224,36226,36229],{"className":36225},[135],[23,36227,20680],{"className":36228},[135],[23,36230,36232],{"className":36231},[998],[23,36233,36235,36255],{"className":36234},[199,200],[23,36236,36238,36252],{"className":36237},[204],[23,36239,36241],{"className":36240,"style":18647},[208],[23,36242,36243,36246],{"style":2029},[23,36244],{"className":36245,"style":1015},[216],[23,36247,36249],{"className":36248},[1019,1020,1021,1022],[23,36250,4435],{"className":36251},[135,1022],[23,36253,297],{"className":36254},[296],[23,36256,36258],{"className":36257},[204],[23,36259,36261],{"className":36260,"style":2048},[208],[23,36262],{},[23,36264,2875],{"className":36265},[135],[23,36267,35533],{"className":36268},[135,136],[3788,36270,36271],{},"Lasso",[3770,36273,36274,36277,36280],{},[3788,36275,36276],{},"稀疏诱导先验",[3788,36278,36279],{},"非凸惩罚",[3788,36281,36282],{},"特征选择",[11,36284,36285],{},"这张表解释了一个常见的困惑：为什么正则化看起来像是「人为加上去的技巧」，却总是有效。它其实是在补充一个先验：参数不应该太大，或者不应该同时有很多非零分量。正则项不是外来的约束，而是模型假设的一部分，只是以惩罚的形式出现。",[11,36287,36288],{},"需要谨慎的是，深度网络里的权重衰减与高斯先验的对应只在全批量 MAP 的意义上成立。SGD 的噪声、批量大小、参数化方式（例如 BatchNorm 的缩放不变性）都会改变有效的先验。把它当成一个有用的类比，而不是一个严格的等价。",[2501,36290,36292],{"id":36291},"_53-似然比roc-与代价","5.3 似然比、ROC 与代价",[11,36294,36295],{},"二分类的贝叶斯规则可以写成似然比检验：",[23,36297,36299],{"className":36298,"translate":27},[26],[23,36300,36302,36408],{"className":36301,"translate":27},[31],[23,36303,36305],{"className":36304},[35],[37,36306,36307],{"xmlns":39,"display":40},[42,36308,36309,36405],{},[45,36310,36311,36313,36315,36317,36319,36321,36323,36325,36327,36329,36359,36361,36403],{},[48,36312,736],{},[51,36314,54],{"stretchy":53},[48,36316,753],{},[51,36318,66],{"stretchy":53},[51,36320,69],{},[2581,36322,4435],{},[148,36324],{"width":9915},[51,36326,14380],{},[148,36328],{"width":9915},[71,36330,36331,36345],{},[45,36332,36333,36339,36341,36343],{},[898,36334,36335,36337],{},[48,36336,11],{},[2581,36338,4435],{},[51,36340,54],{"stretchy":53},[48,36342,753],{},[51,36344,66],{"stretchy":53},[45,36346,36347,36353,36355,36357],{},[898,36348,36349,36351],{},[48,36350,11],{},[2581,36352,1532],{},[51,36354,54],{"stretchy":53},[48,36356,753],{},[51,36358,66],{"stretchy":53},[51,36360,7378],{},[71,36362,36363,36383],{},[45,36364,36365,36371,36373],{},[898,36366,36367,36369],{},[48,36368,21693],{},[2581,36370,1532],{},[88,36372,90],{},[898,36374,36375,36377],{},[48,36376,12710],{},[45,36378,36379,36381],{},[48,36380,12715],{"mathvariant":111},[48,36382,920],{"mathvariant":111},[45,36384,36385,36391,36393],{},[898,36386,36387,36389],{},[48,36388,21693],{},[2581,36390,4435],{},[88,36392,90],{},[898,36394,36395,36397],{},[48,36396,12710],{},[45,36398,36399,36401],{},[48,36400,12715],{"mathvariant":111},[48,36402,12801],{"mathvariant":111},[48,36404,112],{"mathvariant":111},[114,36406,36407],{"encoding":116},"\\delta(x)=1\n\\quad\\Longleftrightarrow\\quad\n\\frac{p_1(x)}{p_0(x)}\n\\ge\n\\frac{\\pi_0\\,c_{\\mathrm{FP}}}{\\pi_1\\,c_{\\mathrm{FN}}}.",[23,36409,36411,36438,36463,36632],{"className":36410,"ariaHidden":122},[121],[23,36412,36414,36417,36420,36423,36426,36429,36432,36435],{"className":36413},[126],[23,36415],{"className":36416,"style":131},[130],[23,36418,736],{"className":36419,"style":783},[135,136],[23,36421,54],{"className":36422},[140],[23,36424,753],{"className":36425},[135,136],[23,36427,66],{"className":36428},[169],[23,36430],{"className":36431,"style":149},[148],[23,36433,69],{"className":36434},[153],[23,36436],{"className":36437,"style":149},[148],[23,36439,36441,36445,36448,36451,36454,36457,36460],{"className":36440},[126],[23,36442],{"className":36443,"style":36444},[130],"height:0.6684em;vertical-align:-0.024em;",[23,36446,4435],{"className":36447},[135],[23,36449],{"className":36450,"style":14437},[148],[23,36452],{"className":36453,"style":149},[148],[23,36455,14380],{"className":36456},[153],[23,36458],{"className":36459,"style":14437},[148],[23,36461],{"className":36462,"style":149},[148],[23,36464,36466,36469,36623,36626,36629],{"className":36465},[126],[23,36467],{"className":36468,"style":185},[130],[23,36470,36472,36475,36620],{"className":36471},[135],[23,36473],{"className":36474},[140,192],[23,36476,36478],{"className":36477},[71],[23,36479,36481,36612],{"className":36480},[199,200],[23,36482,36484,36609],{"className":36483},[204],[23,36485,36487,36544,36552],{"className":36486,"style":209},[208],[23,36488,36489,36492],{"style":212},[23,36490],{"className":36491,"style":217},[216],[23,36493,36495,36535,36538,36541],{"className":36494},[135],[23,36496,36498,36501],{"className":36497},[135],[23,36499,11],{"className":36500},[135,136],[23,36502,36504],{"className":36503},[998],[23,36505,36507,36527],{"className":36506},[199,200],[23,36508,36510,36524],{"className":36509},[204],[23,36511,36513],{"className":36512,"style":18647},[208],[23,36514,36515,36518],{"style":2029},[23,36516],{"className":36517,"style":1015},[216],[23,36519,36521],{"className":36520},[1019,1020,1021,1022],[23,36522,1532],{"className":36523},[135,1022],[23,36525,297],{"className":36526},[296],[23,36528,36530],{"className":36529},[204],[23,36531,36533],{"className":36532,"style":2048},[208],[23,36534],{},[23,36536,54],{"className":36537},[140],[23,36539,753],{"className":36540},[135,136],[23,36542,66],{"className":36543},[169],[23,36545,36546,36549],{"style":235},[23,36547],{"className":36548,"style":217},[216],[23,36550],{"className":36551,"style":243},[242],[23,36553,36554,36557],{"style":246},[23,36555],{"className":36556,"style":217},[216],[23,36558,36560,36600,36603,36606],{"className":36559},[135],[23,36561,36563,36566],{"className":36562},[135],[23,36564,11],{"className":36565},[135,136],[23,36567,36569],{"className":36568},[998],[23,36570,36572,36592],{"className":36571},[199,200],[23,36573,36575,36589],{"className":36574},[204],[23,36576,36578],{"className":36577,"style":18647},[208],[23,36579,36580,36583],{"style":2029},[23,36581],{"className":36582,"style":1015},[216],[23,36584,36586],{"className":36585},[1019,1020,1021,1022],[23,36587,4435],{"className":36588},[135,1022],[23,36590,297],{"className":36591},[296],[23,36593,36595],{"className":36594},[204],[23,36596,36598],{"className":36597,"style":2048},[208],[23,36599],{},[23,36601,54],{"className":36602},[140],[23,36604,753],{"className":36605},[135,136],[23,36607,66],{"className":36608},[169],[23,36610,297],{"className":36611},[296],[23,36613,36615],{"className":36614},[204],[23,36616,36618],{"className":36617,"style":304},[208],[23,36619],{},[23,36621],{"className":36622},[169,192],[23,36624],{"className":36625,"style":149},[148],[23,36627,7378],{"className":36628},[153],[23,36630],{"className":36631,"style":149},[148],[23,36633,36635,36638,36872],{"className":36634},[126],[23,36636],{"className":36637,"style":13486},[130],[23,36639,36641,36644,36869],{"className":36640},[135],[23,36642],{"className":36643},[140,192],[23,36645,36647],{"className":36646},[71],[23,36648,36650,36861],{"className":36649},[199,200],[23,36651,36653,36858],{"className":36652},[204],[23,36654,36656,36753,36761],{"className":36655,"style":6056},[208],[23,36657,36658,36661],{"style":212},[23,36659],{"className":36660,"style":217},[216],[23,36662,36664,36704,36707],{"className":36663},[135],[23,36665,36667,36670],{"className":36666},[135],[23,36668,21693],{"className":36669,"style":676},[135,136],[23,36671,36673],{"className":36672},[998],[23,36674,36676,36696],{"className":36675},[199,200],[23,36677,36679,36693],{"className":36678},[204],[23,36680,36682],{"className":36681,"style":18647},[208],[23,36683,36684,36687],{"style":21850},[23,36685],{"className":36686,"style":1015},[216],[23,36688,36690],{"className":36689},[1019,1020,1021,1022],[23,36691,4435],{"className":36692},[135,1022],[23,36694,297],{"className":36695},[296],[23,36697,36699],{"className":36698},[204],[23,36700,36702],{"className":36701,"style":2048},[208],[23,36703],{},[23,36705],{"className":36706,"style":280},[148],[23,36708,36710,36713],{"className":36709},[135],[23,36711,12710],{"className":36712},[135,136],[23,36714,36716],{"className":36715},[998],[23,36717,36719,36745],{"className":36718},[199,200],[23,36720,36722,36742],{"className":36721},[204],[23,36723,36725],{"className":36724,"style":2026},[208],[23,36726,36727,36730],{"style":2029},[23,36728],{"className":36729,"style":1015},[216],[23,36731,36733],{"className":36732},[1019,1020,1021,1022],[23,36734,36736],{"className":36735},[135,1022],[23,36737,36739],{"className":36738},[135,1022],[23,36740,12849],{"className":36741},[135,5391,1022],[23,36743,297],{"className":36744},[296],[23,36746,36748],{"className":36747},[204],[23,36749,36751],{"className":36750,"style":2048},[208],[23,36752],{},[23,36754,36755,36758],{"style":235},[23,36756],{"className":36757,"style":217},[216],[23,36759],{"className":36760,"style":243},[242],[23,36762,36763,36766],{"style":246},[23,36764],{"className":36765,"style":217},[216],[23,36767,36769,36809,36812],{"className":36768},[135],[23,36770,36772,36775],{"className":36771},[135],[23,36773,21693],{"className":36774,"style":676},[135,136],[23,36776,36778],{"className":36777},[998],[23,36779,36781,36801],{"className":36780},[199,200],[23,36782,36784,36798],{"className":36783},[204],[23,36785,36787],{"className":36786,"style":18647},[208],[23,36788,36789,36792],{"style":21850},[23,36790],{"className":36791,"style":1015},[216],[23,36793,36795],{"className":36794},[1019,1020,1021,1022],[23,36796,1532],{"className":36797},[135,1022],[23,36799,297],{"className":36800},[296],[23,36802,36804],{"className":36803},[204],[23,36805,36807],{"className":36806,"style":2048},[208],[23,36808],{},[23,36810],{"className":36811,"style":280},[148],[23,36813,36815,36818],{"className":36814},[135],[23,36816,12710],{"className":36817},[135,136],[23,36819,36821],{"className":36820},[998],[23,36822,36824,36850],{"className":36823},[199,200],[23,36825,36827,36847],{"className":36826},[204],[23,36828,36830],{"className":36829,"style":2026},[208],[23,36831,36832,36835],{"style":2029},[23,36833],{"className":36834,"style":1015},[216],[23,36836,36838],{"className":36837},[1019,1020,1021,1022],[23,36839,36841],{"className":36840},[135,1022],[23,36842,36844],{"className":36843},[135,1022],[23,36845,12766],{"className":36846},[135,5391,1022],[23,36848,297],{"className":36849},[296],[23,36851,36853],{"className":36852},[204],[23,36854,36856],{"className":36855,"style":2048},[208],[23,36857],{},[23,36859,297],{"className":36860},[296],[23,36862,36864],{"className":36863},[204],[23,36865,36867],{"className":36866,"style":13685},[208],[23,36868],{},[23,36870],{"className":36871},[169,192],[23,36873,112],{"className":36874},[135],[11,36876,36877],{},"左边是数据提供的证据，右边是先验与代价决定的阈值。先验偏向负类，阈值升高；漏诊更贵，阈值降低。这个分解把「数据说什么」与「我们更怕哪种错误」分开了。",[11,36879,36880,36881,36910,36911,36965,36966,37020],{},"把阈值从低到高扫一遍，就得到 ROC 曲线。设阈值为 ",[23,36882,36884,36897],{"className":36883,"translate":27},[31],[23,36885,36887],{"className":36886},[35],[37,36888,36889],{"xmlns":39},[42,36890,36891,36895],{},[45,36892,36893],{},[48,36894,5345],{},[114,36896,5345],{"encoding":116},[23,36898,36900],{"className":36899,"ariaHidden":122},[121],[23,36901,36903,36907],{"className":36902},[126],[23,36904],{"className":36905,"style":36906},[130],"height:0.6151em;",[23,36908,5345],{"className":36909},[135,136],"，记真正率为 ",[23,36912,36914,36940],{"className":36913,"translate":27},[31],[23,36915,36917],{"className":36916},[35],[37,36918,36919],{"xmlns":39},[42,36920,36921,36937],{},[45,36922,36923,36931,36933,36935],{},[45,36924,36925,36927,36929],{},[48,36926,20703],{"mathvariant":111},[48,36928,920],{"mathvariant":111},[48,36930,572],{"mathvariant":111},[51,36932,54],{"stretchy":53},[48,36934,5345],{},[51,36936,66],{"stretchy":53},[114,36938,36939],{"encoding":116},"\\mathrm{TPR}(t)",[23,36941,36943],{"className":36942,"ariaHidden":122},[121],[23,36944,36946,36949,36956,36959,36962],{"className":36945},[126],[23,36947],{"className":36948,"style":131},[130],[23,36950,36952],{"className":36951},[135],[23,36953,36955],{"className":36954},[135,5391],"TPR",[23,36957,54],{"className":36958},[140],[23,36960,5345],{"className":36961},[135,136],[23,36963,66],{"className":36964},[169],"、假正率为 ",[23,36967,36969,36995],{"className":36968,"translate":27},[31],[23,36970,36972],{"className":36971},[35],[37,36973,36974],{"xmlns":39},[42,36975,36976,36992],{},[45,36977,36978,36986,36988,36990],{},[45,36979,36980,36982,36984],{},[48,36981,12715],{"mathvariant":111},[48,36983,920],{"mathvariant":111},[48,36985,572],{"mathvariant":111},[51,36987,54],{"stretchy":53},[48,36989,5345],{},[51,36991,66],{"stretchy":53},[114,36993,36994],{"encoding":116},"\\mathrm{FPR}(t)",[23,36996,36998],{"className":36997,"ariaHidden":122},[121],[23,36999,37001,37004,37011,37014,37017],{"className":37000},[126],[23,37002],{"className":37003,"style":131},[130],[23,37005,37007],{"className":37006},[135],[23,37008,37010],{"className":37009},[135,5391],"FPR",[23,37012,54],{"className":37013},[140],[23,37015,5345],{"className":37016},[135,136],[23,37018,66],{"className":37019},[169],"，则期望代价为",[23,37022,37024],{"className":37023,"translate":27},[26],[23,37025,37027,37129],{"className":37026,"translate":27},[31],[23,37028,37030],{"className":37029},[35],[37,37031,37032],{"xmlns":39,"display":40},[42,37033,37034,37126],{},[45,37035,37036,37038,37040,37042,37044,37046,37052,37054,37064,37066,37074,37076,37078,37080,37082,37088,37090,37100,37102,37104,37106,37108,37116,37118,37120,37122,37124],{},[48,37037,24979],{},[51,37039,54],{"stretchy":53},[48,37041,5345],{},[51,37043,66],{"stretchy":53},[51,37045,69],{},[898,37047,37048,37050],{},[48,37049,21693],{},[2581,37051,1532],{},[88,37053,90],{},[898,37055,37056,37058],{},[48,37057,12710],{},[45,37059,37060,37062],{},[48,37061,12715],{"mathvariant":111},[48,37063,920],{"mathvariant":111},[88,37065,90],{},[45,37067,37068,37070,37072],{},[48,37069,12715],{"mathvariant":111},[48,37071,920],{"mathvariant":111},[48,37073,572],{"mathvariant":111},[51,37075,54],{"stretchy":53},[48,37077,5345],{},[51,37079,66],{"stretchy":53},[51,37081,5106],{},[898,37083,37084,37086],{},[48,37085,21693],{},[2581,37087,4435],{},[88,37089,90],{},[898,37091,37092,37094],{},[48,37093,12710],{},[45,37095,37096,37098],{},[48,37097,12715],{"mathvariant":111},[48,37099,12801],{"mathvariant":111},[88,37101,90],{},[51,37103,54],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[2581,37105,4435],{},[51,37107,4529],{},[45,37109,37110,37112,37114],{},[48,37111,20703],{"mathvariant":111},[48,37113,920],{"mathvariant":111},[48,37115,572],{"mathvariant":111},[51,37117,54],{"stretchy":53},[48,37119,5345],{},[51,37121,66],{"stretchy":53},[51,37123,66],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,37125,112],{"mathvariant":111},[114,37127,37128],{"encoding":116},"C(t)=\\pi_0\\,c_{\\mathrm{FP}}\\,\\mathrm{FPR}(t)\n+\\pi_1\\,c_{\\mathrm{FN}}\\,\\big(1-\\mathrm{TPR}(t)\\big).",[23,37130,37132,37160,37282,37398],{"className":37131,"ariaHidden":122},[121],[23,37133,37135,37138,37142,37145,37148,37151,37154,37157],{"className":37134},[126],[23,37136],{"className":37137,"style":131},[130],[23,37139,24979],{"className":37140,"style":37141},[135,136],"margin-right:0.0715em;",[23,37143,54],{"className":37144},[140],[23,37146,5345],{"className":37147},[135,136],[23,37149,66],{"className":37150},[169],[23,37152],{"className":37153,"style":149},[148],[23,37155,69],{"className":37156},[153],[23,37158],{"className":37159,"style":149},[148],[23,37161,37163,37166,37206,37209,37255,37258,37264,37267,37270,37273,37276,37279],{"className":37162},[126],[23,37164],{"className":37165,"style":131},[130],[23,37167,37169,37172],{"className":37168},[135],[23,37170,21693],{"className":37171,"style":676},[135,136],[23,37173,37175],{"className":37174},[998],[23,37176,37178,37198],{"className":37177},[199,200],[23,37179,37181,37195],{"className":37180},[204],[23,37182,37184],{"className":37183,"style":18647},[208],[23,37185,37186,37189],{"style":21850},[23,37187],{"className":37188,"style":1015},[216],[23,37190,37192],{"className":37191},[1019,1020,1021,1022],[23,37193,1532],{"className":37194},[135,1022],[23,37196,297],{"className":37197},[296],[23,37199,37201],{"className":37200},[204],[23,37202,37204],{"className":37203,"style":2048},[208],[23,37205],{},[23,37207],{"className":37208,"style":280},[148],[23,37210,37212,37215],{"className":37211},[135],[23,37213,12710],{"className":37214},[135,136],[23,37216,37218],{"className":37217},[998],[23,37219,37221,37247],{"className":37220},[199,200],[23,37222,37224,37244],{"className":37223},[204],[23,37225,37227],{"className":37226,"style":2026},[208],[23,37228,37229,37232],{"style":2029},[23,37230],{"className":37231,"style":1015},[216],[23,37233,37235],{"className":37234},[1019,1020,1021,1022],[23,37236,37238],{"className":37237},[135,1022],[23,37239,37241],{"className":37240},[135,1022],[23,37242,12766],{"className":37243},[135,5391,1022],[23,37245,297],{"className":37246},[296],[23,37248,37250],{"className":37249},[204],[23,37251,37253],{"className":37252,"style":2048},[208],[23,37254],{},[23,37256],{"className":37257,"style":280},[148],[23,37259,37261],{"className":37260},[135],[23,37262,37010],{"className":37263},[135,5391],[23,37265,54],{"className":37266},[140],[23,37268,5345],{"className":37269},[135,136],[23,37271,66],{"className":37272},[169],[23,37274],{"className":37275,"style":392},[148],[23,37277,5106],{"className":37278},[612],[23,37280],{"className":37281,"style":392},[148],[23,37283,37285,37288,37328,37331,37377,37380,37386,37389,37392,37395],{"className":37284},[126],[23,37286],{"className":37287,"style":2213},[130],[23,37289,37291,37294],{"className":37290},[135],[23,37292,21693],{"className":37293,"style":676},[135,136],[23,37295,37297],{"className":37296},[998],[23,37298,37300,37320],{"className":37299},[199,200],[23,37301,37303,37317],{"className":37302},[204],[23,37304,37306],{"className":37305,"style":18647},[208],[23,37307,37308,37311],{"style":21850},[23,37309],{"className":37310,"style":1015},[216],[23,37312,37314],{"className":37313},[1019,1020,1021,1022],[23,37315,4435],{"className":37316},[135,1022],[23,37318,297],{"className":37319},[296],[23,37321,37323],{"className":37322},[204],[23,37324,37326],{"className":37325,"style":2048},[208],[23,37327],{},[23,37329],{"className":37330,"style":280},[148],[23,37332,37334,37337],{"className":37333},[135],[23,37335,12710],{"className":37336},[135,136],[23,37338,37340],{"className":37339},[998],[23,37341,37343,37369],{"className":37342},[199,200],[23,37344,37346,37366],{"className":37345},[204],[23,37347,37349],{"className":37348,"style":2026},[208],[23,37350,37351,37354],{"style":2029},[23,37352],{"className":37353,"style":1015},[216],[23,37355,37357],{"className":37356},[1019,1020,1021,1022],[23,37358,37360],{"className":37359},[135,1022],[23,37361,37363],{"className":37362},[135,1022],[23,37364,12849],{"className":37365},[135,5391,1022],[23,37367,297],{"className":37368},[296],[23,37370,37372],{"className":37371},[204],[23,37373,37375],{"className":37374,"style":2048},[208],[23,37376],{},[23,37378],{"className":37379,"style":280},[148],[23,37381,37383],{"className":37382},[135],[23,37384,54],{"className":37385},[1066,1067],[23,37387,4435],{"className":37388},[135],[23,37390],{"className":37391,"style":392},[148],[23,37393,4529],{"className":37394},[612],[23,37396],{"className":37397,"style":392},[148],[23,37399,37401,37404,37410,37413,37416,37419,37425],{"className":37400},[126],[23,37402],{"className":37403,"style":2213},[130],[23,37405,37407],{"className":37406},[135],[23,37408,36955],{"className":37409},[135,5391],[23,37411,54],{"className":37412},[140],[23,37414,5345],{"className":37415},[135,136],[23,37417,66],{"className":37418},[169],[23,37420,37422],{"className":37421},[135],[23,37423,66],{"className":37424},[1066,1067],[23,37426,112],{"className":37427},[135],[11,37429,37430],{},"贝叶斯决策做的不是「选一个看起来最好的点」，而是根据先验与代价，在 ROC 上选一个使这个线性函数最小的点。等代价线的斜率是",[23,37432,37434],{"className":37433,"translate":27},[26],[23,37435,37437,37525],{"className":37436,"translate":27},[31],[23,37438,37440],{"className":37439},[35],[37,37441,37442],{"xmlns":39,"display":40},[42,37443,37444,37522],{},[45,37445,37446,37476,37478,37520],{},[71,37447,37448,37462],{},[45,37449,37450,37452,37454],{},[48,37451,1606],{},[88,37453,90],{},[45,37455,37456,37458,37460],{},[48,37457,20703],{"mathvariant":111},[48,37459,920],{"mathvariant":111},[48,37461,572],{"mathvariant":111},[45,37463,37464,37466,37468],{},[48,37465,1606],{},[88,37467,90],{},[45,37469,37470,37472,37474],{},[48,37471,12715],{"mathvariant":111},[48,37473,920],{"mathvariant":111},[48,37475,572],{"mathvariant":111},[51,37477,69],{},[71,37479,37480,37500],{},[45,37481,37482,37488,37490],{},[898,37483,37484,37486],{},[48,37485,21693],{},[2581,37487,1532],{},[88,37489,90],{},[898,37491,37492,37494],{},[48,37493,12710],{},[45,37495,37496,37498],{},[48,37497,12715],{"mathvariant":111},[48,37499,920],{"mathvariant":111},[45,37501,37502,37508,37510],{},[898,37503,37504,37506],{},[48,37505,21693],{},[2581,37507,4435],{},[88,37509,90],{},[898,37511,37512,37514],{},[48,37513,12710],{},[45,37515,37516,37518],{},[48,37517,12715],{"mathvariant":111},[48,37519,12801],{"mathvariant":111},[48,37521,112],{"mathvariant":111},[114,37523,37524],{"encoding":116},"\\frac{d\\,\\mathrm{TPR}}{d\\,\\mathrm{FPR}}\n=\\frac{\\pi_0\\,c_{\\mathrm{FP}}}{\\pi_1\\,c_{\\mathrm{FN}}}.",[23,37526,37528,37623],{"className":37527,"ariaHidden":122},[121],[23,37529,37531,37534,37614,37617,37620],{"className":37530},[126],[23,37532],{"className":37533,"style":6229},[130],[23,37535,37537,37540,37611],{"className":37536},[135],[23,37538],{"className":37539},[140,192],[23,37541,37543],{"className":37542},[71],[23,37544,37546,37603],{"className":37545},[199,200],[23,37547,37549,37600],{"className":37548},[204],[23,37550,37552,37572,37580],{"className":37551,"style":5607},[208],[23,37553,37554,37557],{"style":212},[23,37555],{"className":37556,"style":217},[216],[23,37558,37560,37563,37566],{"className":37559},[135],[23,37561,1606],{"className":37562},[135,136],[23,37564],{"className":37565,"style":280},[148],[23,37567,37569],{"className":37568},[135],[23,37570,37010],{"className":37571},[135,5391],[23,37573,37574,37577],{"style":235},[23,37575],{"className":37576,"style":217},[216],[23,37578],{"className":37579,"style":243},[242],[23,37581,37582,37585],{"style":246},[23,37583],{"className":37584,"style":217},[216],[23,37586,37588,37591,37594],{"className":37587},[135],[23,37589,1606],{"className":37590},[135,136],[23,37592],{"className":37593,"style":280},[148],[23,37595,37597],{"className":37596},[135],[23,37598,36955],{"className":37599},[135,5391],[23,37601,297],{"className":37602},[296],[23,37604,37606],{"className":37605},[204],[23,37607,37609],{"className":37608,"style":6287},[208],[23,37610],{},[23,37612],{"className":37613},[169,192],[23,37615],{"className":37616,"style":149},[148],[23,37618,69],{"className":37619},[153],[23,37621],{"className":37622,"style":149},[148],[23,37624,37626,37629,37863],{"className":37625},[126],[23,37627],{"className":37628,"style":13486},[130],[23,37630,37632,37635,37860],{"className":37631},[135],[23,37633],{"className":37634},[140,192],[23,37636,37638],{"className":37637},[71],[23,37639,37641,37852],{"className":37640},[199,200],[23,37642,37644,37849],{"className":37643},[204],[23,37645,37647,37744,37752],{"className":37646,"style":6056},[208],[23,37648,37649,37652],{"style":212},[23,37650],{"className":37651,"style":217},[216],[23,37653,37655,37695,37698],{"className":37654},[135],[23,37656,37658,37661],{"className":37657},[135],[23,37659,21693],{"className":37660,"style":676},[135,136],[23,37662,37664],{"className":37663},[998],[23,37665,37667,37687],{"className":37666},[199,200],[23,37668,37670,37684],{"className":37669},[204],[23,37671,37673],{"className":37672,"style":18647},[208],[23,37674,37675,37678],{"style":21850},[23,37676],{"className":37677,"style":1015},[216],[23,37679,37681],{"className":37680},[1019,1020,1021,1022],[23,37682,4435],{"className":37683},[135,1022],[23,37685,297],{"className":37686},[296],[23,37688,37690],{"className":37689},[204],[23,37691,37693],{"className":37692,"style":2048},[208],[23,37694],{},[23,37696],{"className":37697,"style":280},[148],[23,37699,37701,37704],{"className":37700},[135],[23,37702,12710],{"className":37703},[135,136],[23,37705,37707],{"className":37706},[998],[23,37708,37710,37736],{"className":37709},[199,200],[23,37711,37713,37733],{"className":37712},[204],[23,37714,37716],{"className":37715,"style":2026},[208],[23,37717,37718,37721],{"style":2029},[23,37719],{"className":37720,"style":1015},[216],[23,37722,37724],{"className":37723},[1019,1020,1021,1022],[23,37725,37727],{"className":37726},[135,1022],[23,37728,37730],{"className":37729},[135,1022],[23,37731,12849],{"className":37732},[135,5391,1022],[23,37734,297],{"className":37735},[296],[23,37737,37739],{"className":37738},[204],[23,37740,37742],{"className":37741,"style":2048},[208],[23,37743],{},[23,37745,37746,37749],{"style":235},[23,37747],{"className":37748,"style":217},[216],[23,37750],{"className":37751,"style":243},[242],[23,37753,37754,37757],{"style":246},[23,37755],{"className":37756,"style":217},[216],[23,37758,37760,37800,37803],{"className":37759},[135],[23,37761,37763,37766],{"className":37762},[135],[23,37764,21693],{"className":37765,"style":676},[135,136],[23,37767,37769],{"className":37768},[998],[23,37770,37772,37792],{"className":37771},[199,200],[23,37773,37775,37789],{"className":37774},[204],[23,37776,37778],{"className":37777,"style":18647},[208],[23,37779,37780,37783],{"style":21850},[23,37781],{"className":37782,"style":1015},[216],[23,37784,37786],{"className":37785},[1019,1020,1021,1022],[23,37787,1532],{"className":37788},[135,1022],[23,37790,297],{"className":37791},[296],[23,37793,37795],{"className":37794},[204],[23,37796,37798],{"className":37797,"style":2048},[208],[23,37799],{},[23,37801],{"className":37802,"style":280},[148],[23,37804,37806,37809],{"className":37805},[135],[23,37807,12710],{"className":37808},[135,136],[23,37810,37812],{"className":37811},[998],[23,37813,37815,37841],{"className":37814},[199,200],[23,37816,37818,37838],{"className":37817},[204],[23,37819,37821],{"className":37820,"style":2026},[208],[23,37822,37823,37826],{"style":2029},[23,37824],{"className":37825,"style":1015},[216],[23,37827,37829],{"className":37828},[1019,1020,1021,1022],[23,37830,37832],{"className":37831},[135,1022],[23,37833,37835],{"className":37834},[135,1022],[23,37836,12766],{"className":37837},[135,5391,1022],[23,37839,297],{"className":37840},[296],[23,37842,37844],{"className":37843},[204],[23,37845,37847],{"className":37846,"style":2048},[208],[23,37848],{},[23,37850,297],{"className":37851},[296],[23,37853,37855],{"className":37854},[204],[23,37856,37858],{"className":37857,"style":13685},[208],[23,37859],{},[23,37861],{"className":37862},[169,192],[23,37864,112],{"className":37865},[135],[11,37867,37868,37869,1471],{},"这也说明 AUC 只是一个与阈值无关的排序指标。两个模型的 AUC 可以一个高一个低，但在某个特定的代价结构下，AUC 低的模型反而更优。",[18,37870,37871],{},"脱离代价谈「最好的分类器」是没有意义的",[2501,37873,37875],{"id":37874},"_54-拒绝选项与选择性分类","5.4 拒绝选项与选择性分类",[11,37877,37878,37879,37929,37930,37933],{},"3.5 节已经导出了 Chow 规则：当最大后验低于 ",[23,37880,37882,37899],{"className":37881,"translate":27},[31],[23,37883,37885],{"className":37884},[35],[37,37886,37887],{"xmlns":39},[42,37888,37889,37897],{},[45,37890,37891,37893,37895],{},[2581,37892,4435],{},[51,37894,4529],{},[48,37896,12710],{},[114,37898,14071],{"encoding":116},[23,37900,37902,37920],{"className":37901,"ariaHidden":122},[121],[23,37903,37905,37908,37911,37914,37917],{"className":37904},[126],[23,37906],{"className":37907,"style":9400},[130],[23,37909,4435],{"className":37910},[135],[23,37912],{"className":37913,"style":392},[148],[23,37915,4529],{"className":37916},[612],[23,37918],{"className":37919,"style":392},[148],[23,37921,37923,37926],{"className":37922},[126],[23,37924],{"className":37925,"style":703},[130],[23,37927,12710],{"className":37928},[135,136]," 时拒绝。把这句话放到整个数据集上，就得到一条",[18,37931,37932],{},"风险-覆盖率曲线","：随着拒绝阈值升高，被拒绝的样本增多，覆盖率下降，但在被接受的样本上错误率也下降。",[11,37935,37936],{},"选择性分类的理论目标不是最大化准确率，而是最小化",[23,37938,37940],{"className":37939,"translate":27},[26],[23,37941,37943,37970],{"className":37942,"translate":27},[31],[23,37944,37946],{"className":37945},[35],[37,37947,37948],{"xmlns":39,"display":40},[42,37949,37950,37967],{},[45,37951,37952,37955,37957,37965],{},[88,37953,37954],{},"选择性风险",[51,37956,69],{},[71,37958,37959,37962],{},[88,37960,37961],{},"被接受样本上的错误数",[88,37963,37964],{},"被接受样本数",[51,37966,747],{"separator":122},[114,37968,37969],{"encoding":116},"\\text{选择性风险}=\\frac{\\text{被接受样本上的错误数}}{\\text{被接受样本数}},",[23,37971,37973,37994],{"className":37972,"ariaHidden":122},[121],[23,37974,37976,37979,37985,37988,37991],{"className":37975},[126],[23,37977],{"className":37978,"style":470},[130],[23,37980,37982],{"className":37981},[135,6514],[23,37983,37954],{"className":37984},[135,7238],[23,37986],{"className":37987,"style":149},[148],[23,37989,69],{"className":37990},[153],[23,37992],{"className":37993,"style":149},[148],[23,37995,37997,38001,38069],{"className":37996},[126],[23,37998],{"className":37999,"style":38000},[130],"height:2.0463em;vertical-align:-0.686em;",[23,38002,38004,38007,38066],{"className":38003},[135],[23,38005],{"className":38006},[140,192],[23,38008,38010],{"className":38009},[71],[23,38011,38013,38058],{"className":38012},[199,200],[23,38014,38016,38055],{"className":38015},[204],[23,38017,38019,38033,38041],{"className":38018,"style":24272},[208],[23,38020,38021,38024],{"style":212},[23,38022],{"className":38023,"style":217},[216],[23,38025,38027],{"className":38026},[135],[23,38028,38030],{"className":38029},[135,6514],[23,38031,37964],{"className":38032},[135,7238],[23,38034,38035,38038],{"style":235},[23,38036],{"className":38037,"style":217},[216],[23,38039],{"className":38040,"style":243},[242],[23,38042,38043,38046],{"style":246},[23,38044],{"className":38045,"style":217},[216],[23,38047,38049],{"className":38048},[135],[23,38050,38052],{"className":38051},[135,6514],[23,38053,37961],{"className":38054},[135,7238],[23,38056,297],{"className":38057},[296],[23,38059,38061],{"className":38060},[204],[23,38062,38064],{"className":38063,"style":6287},[208],[23,38065],{},[23,38067],{"className":38068},[169,192],[23,38070,747],{"className":38071},[824],[11,38073,38074,38075,38103,38104,38132],{},"同时满足覆盖率约束。这两个量之间的权衡，仍然由损失函数决定：拒绝的代价 ",[23,38076,38078,38091],{"className":38077,"translate":27},[31],[23,38079,38081],{"className":38080},[35],[37,38082,38083],{"xmlns":39},[42,38084,38085,38089],{},[45,38086,38087],{},[48,38088,12710],{},[114,38090,12710],{"encoding":116},[23,38092,38094],{"className":38093,"ariaHidden":122},[121],[23,38095,38097,38100],{"className":38096},[126],[23,38098],{"className":38099,"style":703},[130],[23,38101,12710],{"className":38102},[135,136]," 越小，模型越倾向于拒绝；",[23,38105,38107,38120],{"className":38106,"translate":27},[31],[23,38108,38110],{"className":38109},[35],[37,38111,38112],{"xmlns":39},[42,38113,38114,38118],{},[45,38115,38116],{},[48,38117,12710],{},[114,38119,12710],{"encoding":116},[23,38121,38123],{"className":38122,"ariaHidden":122},[121],[23,38124,38126,38129],{"className":38125},[126],[23,38127],{"className":38128,"style":703},[130],[23,38130,12710],{"className":38131},[135,136]," 越大，模型越倾向于冒险。",[11,38134,38135],{},"在医疗、自动驾驶、内容审核这些场景里，拒绝选项往往比「提高模型容量」更有效。一个知道自己不知道的模型，比一个在所有样本上都自信地犯错的模型更有用。",[2501,38137,38139],{"id":38138},"_55-类别不平衡改损失还是改数据","5.5 类别不平衡：改损失，还是改数据",[11,38141,38142],{},"类别不平衡经常被当成一个数据处理问题，但在决策论的框架里，它首先是一个先验与损失的问题。",[11,38144,38145,38146,38270,38271,38432,38433,38584],{},"设原分布的类先验为 ",[23,38147,38149,38175],{"className":38148,"translate":27},[31],[23,38150,38152],{"className":38151},[35],[37,38153,38154],{"xmlns":39},[42,38155,38156,38172],{},[45,38157,38158,38164,38166],{},[898,38159,38160,38162],{},[48,38161,21693],{},[2581,38163,1532],{},[51,38165,747],{"separator":122},[898,38167,38168,38170],{},[48,38169,21693],{},[2581,38171,4435],{},[114,38173,38174],{"encoding":116},"\\pi_0,\\pi_1",[23,38176,38178],{"className":38177,"ariaHidden":122},[121],[23,38179,38181,38184,38224,38227,38230],{"className":38180},[126],[23,38182],{"className":38183,"style":672},[130],[23,38185,38187,38190],{"className":38186},[135],[23,38188,21693],{"className":38189,"style":676},[135,136],[23,38191,38193],{"className":38192},[998],[23,38194,38196,38216],{"className":38195},[199,200],[23,38197,38199,38213],{"className":38198},[204],[23,38200,38202],{"className":38201,"style":18647},[208],[23,38203,38204,38207],{"style":21850},[23,38205],{"className":38206,"style":1015},[216],[23,38208,38210],{"className":38209},[1019,1020,1021,1022],[23,38211,1532],{"className":38212},[135,1022],[23,38214,297],{"className":38215},[296],[23,38217,38219],{"className":38218},[204],[23,38220,38222],{"className":38221,"style":2048},[208],[23,38223],{},[23,38225,747],{"className":38226},[824],[23,38228],{"className":38229,"style":280},[148],[23,38231,38233,38236],{"className":38232},[135],[23,38234,21693],{"className":38235,"style":676},[135,136],[23,38237,38239],{"className":38238},[998],[23,38240,38242,38262],{"className":38241},[199,200],[23,38243,38245,38259],{"className":38244},[204],[23,38246,38248],{"className":38247,"style":18647},[208],[23,38249,38250,38253],{"style":21850},[23,38251],{"className":38252,"style":1015},[216],[23,38254,38256],{"className":38255},[1019,1020,1021,1022],[23,38257,4435],{"className":38258},[135,1022],[23,38260,297],{"className":38261},[296],[23,38263,38265],{"className":38264},[204],[23,38266,38268],{"className":38267,"style":2048},[208],[23,38269],{},"，重采样后的先验为 ",[23,38272,38274,38305],{"className":38273,"translate":27},[31],[23,38275,38277],{"className":38276},[35],[37,38278,38279],{"xmlns":39},[42,38280,38281,38302],{},[45,38282,38283,38292,38294],{},[19137,38284,38285,38287,38289],{},[48,38286,21693],{},[2581,38288,1532],{},[51,38290,38291],{"mathvariant":111,"lspace":1522,"rspace":1522},"′",[51,38293,747],{"separator":122},[19137,38295,38296,38298,38300],{},[48,38297,21693],{},[2581,38299,4435],{},[51,38301,38291],{"mathvariant":111,"lspace":1522,"rspace":1522},[114,38303,38304],{"encoding":116},"\\pi'_0,\\pi'_1",[23,38306,38308],{"className":38307,"ariaHidden":122},[121],[23,38309,38311,38315,38372,38375,38378],{"className":38310},[126],[23,38312],{"className":38313,"style":38314},[130],"height:1em;vertical-align:-0.2481em;",[23,38316,38318,38321],{"className":38317},[135],[23,38319,21693],{"className":38320,"style":676},[135,136],[23,38322,38324],{"className":38323},[998],[23,38325,38327,38363],{"className":38326},[199,200],[23,38328,38330,38360],{"className":38329},[204],[23,38331,38334,38346],{"className":38332,"style":38333},[208],"height:0.7519em;",[23,38335,38337,38340],{"style":38336},"top:-2.4519em;margin-left:-0.0359em;margin-right:0.05em;",[23,38338],{"className":38339,"style":1015},[216],[23,38341,38343],{"className":38342},[1019,1020,1021,1022],[23,38344,1532],{"className":38345},[135,1022],[23,38347,38348,38351],{"style":4696},[23,38349],{"className":38350,"style":1015},[216],[23,38352,38354],{"className":38353},[1019,1020,1021,1022],[23,38355,38357],{"className":38356},[135,1022],[23,38358,38291],{"className":38359},[135,1022],[23,38361,297],{"className":38362},[296],[23,38364,38366],{"className":38365},[204],[23,38367,38370],{"className":38368,"style":38369},[208],"height:0.2481em;",[23,38371],{},[23,38373,747],{"className":38374},[824],[23,38376],{"className":38377,"style":280},[148],[23,38379,38381,38384],{"className":38380},[135],[23,38382,21693],{"className":38383,"style":676},[135,136],[23,38385,38387],{"className":38386},[998],[23,38388,38390,38424],{"className":38389},[199,200],[23,38391,38393,38421],{"className":38392},[204],[23,38394,38396,38407],{"className":38395,"style":38333},[208],[23,38397,38398,38401],{"style":38336},[23,38399],{"className":38400,"style":1015},[216],[23,38402,38404],{"className":38403},[1019,1020,1021,1022],[23,38405,4435],{"className":38406},[135,1022],[23,38408,38409,38412],{"style":4696},[23,38410],{"className":38411,"style":1015},[216],[23,38413,38415],{"className":38414},[1019,1020,1021,1022],[23,38416,38418],{"className":38417},[135,1022],[23,38419,38291],{"className":38420},[135,1022],[23,38422,297],{"className":38423},[296],[23,38425,38427],{"className":38426},[204],[23,38428,38430],{"className":38429,"style":38369},[208],[23,38431],{},"。同一个似然比 ",[23,38434,38436,38474],{"className":38435,"translate":27},[31],[23,38437,38439],{"className":38438},[35],[37,38440,38441],{"xmlns":39},[42,38442,38443,38471],{},[45,38444,38445,38451,38453,38455,38457,38459,38465,38467,38469],{},[898,38446,38447,38449],{},[48,38448,11],{},[2581,38450,4435],{},[51,38452,54],{"stretchy":53},[48,38454,753],{},[51,38456,66],{"stretchy":53},[48,38458,2875],{"mathvariant":111},[898,38460,38461,38463],{},[48,38462,11],{},[2581,38464,1532],{},[51,38466,54],{"stretchy":53},[48,38468,753],{},[51,38470,66],{"stretchy":53},[114,38472,38473],{"encoding":116},"p_1(x)\u002Fp_0(x)",[23,38475,38477],{"className":38476,"ariaHidden":122},[121],[23,38478,38480,38483,38523,38526,38529,38532,38535,38575,38578,38581],{"className":38479},[126],[23,38481],{"className":38482,"style":131},[130],[23,38484,38486,38489],{"className":38485},[135],[23,38487,11],{"className":38488},[135,136],[23,38490,38492],{"className":38491},[998],[23,38493,38495,38515],{"className":38494},[199,200],[23,38496,38498,38512],{"className":38497},[204],[23,38499,38501],{"className":38500,"style":18647},[208],[23,38502,38503,38506],{"style":2029},[23,38504],{"className":38505,"style":1015},[216],[23,38507,38509],{"className":38508},[1019,1020,1021,1022],[23,38510,4435],{"className":38511},[135,1022],[23,38513,297],{"className":38514},[296],[23,38516,38518],{"className":38517},[204],[23,38519,38521],{"className":38520,"style":2048},[208],[23,38522],{},[23,38524,54],{"className":38525},[140],[23,38527,753],{"className":38528},[135,136],[23,38530,66],{"className":38531},[169],[23,38533,2875],{"className":38534},[135],[23,38536,38538,38541],{"className":38537},[135],[23,38539,11],{"className":38540},[135,136],[23,38542,38544],{"className":38543},[998],[23,38545,38547,38567],{"className":38546},[199,200],[23,38548,38550,38564],{"className":38549},[204],[23,38551,38553],{"className":38552,"style":18647},[208],[23,38554,38555,38558],{"style":2029},[23,38556],{"className":38557,"style":1015},[216],[23,38559,38561],{"className":38560},[1019,1020,1021,1022],[23,38562,1532],{"className":38563},[135,1022],[23,38565,297],{"className":38566},[296],[23,38568,38570],{"className":38569},[204],[23,38571,38573],{"className":38572,"style":2048},[208],[23,38574],{},[23,38576,54],{"className":38577},[140],[23,38579,753],{"className":38580},[135,136],[23,38582,66],{"className":38583},[169]," 在两种分布下给出的后验几率不同：",[23,38586,38588],{"className":38587,"translate":27},[26],[23,38589,38591,38725],{"className":38590,"translate":27},[31],[23,38592,38594],{"className":38593},[35],[37,38595,38596],{"xmlns":39,"display":40},[42,38597,38598,38722],{},[45,38599,38600,38639,38641,38680,38682,38720],{},[71,38601,38602,38621],{},[45,38603,38604,38611,38613,38615,38617,38619],{},[898,38605,38606,38608],{},[48,38607,11],{},[88,38609,38610],{},"orig",[51,38612,54],{"stretchy":53},[2581,38614,4435],{},[51,38616,60],{},[48,38618,753],{},[51,38620,66],{"stretchy":53},[45,38622,38623,38629,38631,38633,38635,38637],{},[898,38624,38625,38627],{},[48,38626,11],{},[88,38628,38610],{},[51,38630,54],{"stretchy":53},[2581,38632,1532],{},[51,38634,60],{},[48,38636,753],{},[51,38638,66],{"stretchy":53},[51,38640,69],{},[71,38642,38643,38662],{},[45,38644,38645,38652,38654,38656,38658,38660],{},[898,38646,38647,38649],{},[48,38648,11],{},[88,38650,38651],{},"res",[51,38653,54],{"stretchy":53},[2581,38655,4435],{},[51,38657,60],{},[48,38659,753],{},[51,38661,66],{"stretchy":53},[45,38663,38664,38670,38672,38674,38676,38678],{},[898,38665,38666,38668],{},[48,38667,11],{},[88,38669,38651],{},[51,38671,54],{"stretchy":53},[2581,38673,1532],{},[51,38675,60],{},[48,38677,753],{},[51,38679,66],{"stretchy":53},[51,38681,5758],{},[71,38683,38684,38702],{},[45,38685,38686,38692,38694],{},[898,38687,38688,38690],{},[48,38689,21693],{},[2581,38691,4435],{},[88,38693,90],{},[19137,38695,38696,38698,38700],{},[48,38697,21693],{},[2581,38699,1532],{},[51,38701,38291],{"mathvariant":111,"lspace":1522,"rspace":1522},[45,38703,38704,38712,38714],{},[19137,38705,38706,38708,38710],{},[48,38707,21693],{},[2581,38709,4435],{},[51,38711,38291],{"mathvariant":111,"lspace":1522,"rspace":1522},[88,38713,90],{},[898,38715,38716,38718],{},[48,38717,21693],{},[2581,38719,1532],{},[48,38721,112],{"mathvariant":111},[114,38723,38724],{"encoding":116},"\\frac{p_{\\text{orig}}(1\\mid x)}{p_{\\text{orig}}(0\\mid x)}\n=\n\\frac{p_{\\text{res}}(1\\mid x)}{p_{\\text{res}}(0\\mid x)}\n\\cdot\n\\frac{\\pi_1\\,\\pi'_0}{\\pi'_1\\,\\pi_0}.",[23,38726,38728,38936,39141],{"className":38727,"ariaHidden":122},[121],[23,38729,38731,38735,38927,38930,38933],{"className":38730},[126],[23,38732],{"className":38733,"style":38734},[130],"height:2.3991em;vertical-align:-0.9721em;",[23,38736,38738,38741,38924],{"className":38737},[135],[23,38739],{"className":38740},[140,192],[23,38742,38744],{"className":38743},[71],[23,38745,38747,38915],{"className":38746},[199,200],[23,38748,38750,38912],{"className":38749},[204],[23,38751,38753,38829,38837],{"className":38752,"style":209},[208],[23,38754,38755,38758],{"style":212},[23,38756],{"className":38757,"style":217},[216],[23,38759,38761,38808,38811,38814,38817,38820,38823,38826],{"className":38760},[135],[23,38762,38764,38767],{"className":38763},[135],[23,38765,11],{"className":38766},[135,136],[23,38768,38770],{"className":38769},[998],[23,38771,38773,38800],{"className":38772},[199,200],[23,38774,38776,38797],{"className":38775},[204],[23,38777,38780],{"className":38778,"style":38779},[208],"height:0.3175em;",[23,38781,38782,38785],{"style":2029},[23,38783],{"className":38784,"style":1015},[216],[23,38786,38788],{"className":38787},[1019,1020,1021,1022],[23,38789,38791],{"className":38790},[135,1022],[23,38792,38794],{"className":38793},[135,6514,1022],[23,38795,38610],{"className":38796},[135,1022],[23,38798,297],{"className":38799},[296],[23,38801,38803],{"className":38802},[204],[23,38804,38806],{"className":38805,"style":14292},[208],[23,38807],{},[23,38809,54],{"className":38810},[140],[23,38812,1532],{"className":38813},[135],[23,38815],{"className":38816,"style":149},[148],[23,38818,60],{"className":38819},[153],[23,38821],{"className":38822,"style":149},[148],[23,38824,753],{"className":38825},[135,136],[23,38827,66],{"className":38828},[169],[23,38830,38831,38834],{"style":235},[23,38832],{"className":38833,"style":217},[216],[23,38835],{"className":38836,"style":243},[242],[23,38838,38839,38842],{"style":246},[23,38840],{"className":38841,"style":217},[216],[23,38843,38845,38891,38894,38897,38900,38903,38906,38909],{"className":38844},[135],[23,38846,38848,38851],{"className":38847},[135],[23,38849,11],{"className":38850},[135,136],[23,38852,38854],{"className":38853},[998],[23,38855,38857,38883],{"className":38856},[199,200],[23,38858,38860,38880],{"className":38859},[204],[23,38861,38863],{"className":38862,"style":38779},[208],[23,38864,38865,38868],{"style":2029},[23,38866],{"className":38867,"style":1015},[216],[23,38869,38871],{"className":38870},[1019,1020,1021,1022],[23,38872,38874],{"className":38873},[135,1022],[23,38875,38877],{"className":38876},[135,6514,1022],[23,38878,38610],{"className":38879},[135,1022],[23,38881,297],{"className":38882},[296],[23,38884,38886],{"className":38885},[204],[23,38887,38889],{"className":38888,"style":14292},[208],[23,38890],{},[23,38892,54],{"className":38893},[140],[23,38895,4435],{"className":38896},[135],[23,38898],{"className":38899,"style":149},[148],[23,38901,60],{"className":38902},[153],[23,38904],{"className":38905,"style":149},[148],[23,38907,753],{"className":38908},[135,136],[23,38910,66],{"className":38911},[169],[23,38913,297],{"className":38914},[296],[23,38916,38918],{"className":38917},[204],[23,38919,38922],{"className":38920,"style":38921},[208],"height:0.9721em;",[23,38923],{},[23,38925],{"className":38926},[169,192],[23,38928],{"className":38929,"style":149},[148],[23,38931,69],{"className":38932},[153],[23,38934],{"className":38935,"style":149},[148],[23,38937,38939,38942,39132,39135,39138],{"className":38938},[126],[23,38940],{"className":38941,"style":185},[130],[23,38943,38945,38948,39129],{"className":38944},[135],[23,38946],{"className":38947},[140,192],[23,38949,38951],{"className":38950},[71],[23,38952,38954,39121],{"className":38953},[199,200],[23,38955,38957,39118],{"className":38956},[204],[23,38958,38960,39035,39043],{"className":38959,"style":209},[208],[23,38961,38962,38965],{"style":212},[23,38963],{"className":38964,"style":217},[216],[23,38966,38968,39014,39017,39020,39023,39026,39029,39032],{"className":38967},[135],[23,38969,38971,38974],{"className":38970},[135],[23,38972,11],{"className":38973},[135,136],[23,38975,38977],{"className":38976},[998],[23,38978,38980,39006],{"className":38979},[199,200],[23,38981,38983,39003],{"className":38982},[204],[23,38984,38986],{"className":38985,"style":6499},[208],[23,38987,38988,38991],{"style":2029},[23,38989],{"className":38990,"style":1015},[216],[23,38992,38994],{"className":38993},[1019,1020,1021,1022],[23,38995,38997],{"className":38996},[135,1022],[23,38998,39000],{"className":38999},[135,6514,1022],[23,39001,38651],{"className":39002},[135,1022],[23,39004,297],{"className":39005},[296],[23,39007,39009],{"className":39008},[204],[23,39010,39012],{"className":39011,"style":2048},[208],[23,39013],{},[23,39015,54],{"className":39016},[140],[23,39018,1532],{"className":39019},[135],[23,39021],{"className":39022,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19},[135,1022],[23,39321,297],{"className":39322},[296],[23,39324,39326],{"className":39325},[204],[23,39327,39329],{"className":39328,"style":2048},[208],[23,39330],{},[23,39332],{"className":39333,"style":280},[148],[23,39335,39337,39340],{"className":39336},[135],[23,39338,21693],{"className":39339,"style":676},[135,136],[23,39341,39343],{"className":39342},[998],[23,39344,39346,39380],{"className":39345},[199,200],[23,39347,39349,39377],{"className":39348},[204],[23,39350,39352,39363],{"className":39351,"style":38333},[208],[23,39353,39354,39357],{"style":38336},[23,39355],{"className":39356,"style":1015},[216],[23,39358,39360],{"className":39359},[1019,1020,1021,1022],[23,39361,1532],{"className":39362},[135,1022],[23,39364,39365,39368],{"style":4696},[23,39366],{"className":39367,"style":1015},[216],[23,39369,39371],{"className":39370},[1019,1020,1021,1022],[23,39372,39374],{"className":39373},[135,1022],[23,39375,38291],{"className":39376},[135,1022],[23,39378,297],{"className":39379},[296],[23,39381,39383],{"className":39382},[204],[23,39384,39386],{"className":39385,"style":38369},[208],[23,39387],{},[23,39389,297],{"className":39390},[296],[23,39392,39394],{"className":39393},[204],[23,39395,39398],{"className":39396,"style":39397},[208],"height:0.9523em;",[23,39399],{},[23,39401],{"className":39402},[169,192],[23,39404,112],{"className":39405},[135],[11,39407,39408],{},"如果重采样之后不把这个因子校正回去，决策阈值就偏了。反过来，如果不想重采样，也可以直接在原分布上训练，然后按 3.5 节的公式移动阈值。两条路在理论上等价，差别只在实现难度与估计方差。",[11,39410,39411],{},"真正不能省掉的是损失函数。若两类错误的代价确实不同，那么无论怎么采样，都必须把代价写进决策规则；否则得到的只是一个在改动后的分布上「看起来平衡」的分类器，而不是原问题的最优决策。",[338,39413,39415],{"id":39414},"_6-从参数到分布","6. 从参数到分布",[2501,39417,39419],{"id":39418},"_61-三个预测器","6.1 三个预测器",[11,39421,39422,39423,39451,39452,39480],{},"给定训练集 ",[23,39424,39426,39439],{"className":39425,"translate":27},[31],[23,39427,39429],{"className":39428},[35],[37,39430,39431],{"xmlns":39},[42,39432,39433,39437],{},[45,39434,39435],{},[48,39436,63],{},[114,39438,63],{"encoding":116},[23,39440,39442],{"className":39441,"ariaHidden":122},[121],[23,39443,39445,39448],{"className":39444},[126],[23,39446],{"className":39447,"style":470},[130],[23,39449,63],{"className":39450,"style":144},[135,136],"，预测 ",[23,39453,39455,39468],{"className":39454,"translate":27},[31],[23,39456,39458],{"className":39457},[35],[37,39459,39460],{"xmlns":39},[42,39461,39462,39466],{},[45,39463,39464],{},[48,39465,369],{},[114,39467,369],{"encoding":116},[23,39469,39471],{"className":39470,"ariaHidden":122},[121],[23,39472,39474,39477],{"className":39473},[126],[23,39475],{"className":39476,"style":470},[130],[23,39478,369],{"className":39479,"style":392},[135,136]," 至少有三层做法：",[39482,39483,39484,39664,39932],"ol",{},[349,39485,39486,39489,39490,39551,39552,39663],{},[18,39487,39488],{},"代入式","：先估计一个参数 ",[23,39491,39493,39511],{"className":39492,"translate":27},[31],[23,39494,39496],{"className":39495},[35],[37,39497,39498],{"xmlns":39},[42,39499,39500,39508],{},[45,39501,39502],{},[16807,39503,39504,39506],{"accent":122},[48,39505,57],{},[51,39507,16813],{},[114,39509,39510],{"encoding":116},"\\hat\\theta",[23,39512,39514],{"className":39513,"ariaHidden":122},[121],[23,39515,39517,39520],{"className":39516},[126],[23,39518],{"className":39519,"style":16840},[130],[23,39521,39523],{"className":39522},[135,16830],[23,39524,39526],{"className":39525},[199],[23,39527,39529],{"className":39528},[204],[23,39530,39532,39540],{"className":39531,"style":16840},[208],[23,39533,39534,39537],{"style":7130},[23,39535],{"className":39536,"style":217},[216],[23,39538,57],{"className":39539,"style":144},[135,136],[23,39541,39542,39545],{"style":16851},[23,39543],{"className":39544,"style":217},[216],[23,39546,39548],{"className":39547,"style":34865},[16858],[23,39549,16813],{"className":39550},[135],"，再用 ",[23,39553,39555,39587],{"className":39554,"translate":27},[31],[23,39556,39558],{"className":39557},[35],[37,39559,39560],{"xmlns":39},[42,39561,39562,39584],{},[45,39563,39564,39566,39568,39570,39572,39574,39576,39582],{},[48,39565,11],{},[51,39567,54],{"stretchy":53},[48,39569,660],{},[51,39571,60],{},[48,39573,753],{},[51,39575,747],{"separator":122},[16807,39577,39578,39580],{"accent":122},[48,39579,57],{},[51,39581,16813],{},[51,39583,66],{"stretchy":53},[114,39585,39586],{"encoding":116},"p(y\\mid x,\\hat\\theta)",[23,39588,39590,39614],{"className":39589,"ariaHidden":122},[121],[23,39591,39593,39596,39599,39602,39605,39608,39611],{"className":39592},[126],[23,39594],{"className":39595,"style":131},[130],[23,39597,11],{"className":39598},[135,136],[23,39600,54],{"className":39601},[140],[23,39603,660],{"className":39604,"style":676},[135,136],[23,39606],{"className":39607,"style":149},[148],[23,39609,60],{"className":39610},[153],[23,39612],{"className":39613,"style":149},[148],[23,39615,39617,39620,39623,39626,39629,39660],{"className":39616},[126],[23,39618],{"className":39619,"style":16966},[130],[23,39621,753],{"className":39622},[135,136],[23,39624,747],{"className":39625},[824],[23,39627],{"className":39628,"style":280},[148],[23,39630,39632],{"className":39631},[135,16830],[23,39633,39635],{"className":39634},[199],[23,39636,39638],{"className":39637},[204],[23,39639,39641,39649],{"className":39640,"style":16840},[208],[23,39642,39643,39646],{"style":7130},[23,39644],{"className":39645,"style":217},[216],[23,39647,57],{"className":39648,"style":144},[135,136],[23,39650,39651,39654],{"style":16851},[23,39652],{"className":39653,"style":217},[216],[23,39655,39657],{"className":39656,"style":34865},[16858],[23,39658,16813],{"className":39659},[135],[23,39661,66],{"className":39662},[169]," 预测；",[349,39665,39666,39669,39670],{},[18,39667,39668],{},"后验预测","：对参数积分，\n",[23,39671,39673],{"className":39672,"translate":27},[26],[23,39674,39676,39748],{"className":39675,"translate":27},[31],[23,39677,39679],{"className":39678},[35],[37,39680,39681],{"xmlns":39,"display":40},[42,39682,39683,39745],{},[45,39684,39685,39692,39694,39696,39698,39700,39702,39704,39706,39708,39710,39712,39714,39716,39718,39720,39722,39724,39726,39728,39730,39732,39734,39736,39738,39740,39742],{},[16807,39686,39687,39689],{"accent":122},[48,39688,11],{},[51,39690,39691],{},"ˉ",[51,39693,54],{"stretchy":53},[48,39695,660],{},[51,39697,60],{},[48,39699,753],{},[51,39701,66],{"stretchy":53},[51,39703,69],{},[51,39705,1557],{},[48,39707,11],{},[51,39709,54],{"stretchy":53},[48,39711,660],{},[51,39713,60],{},[48,39715,753],{},[51,39717,747],{"separator":122},[48,39719,57],{},[51,39721,66],{"stretchy":53},[88,39723,90],{},[48,39725,11],{},[51,39727,54],{"stretchy":53},[48,39729,57],{},[51,39731,60],{},[48,39733,63],{},[51,39735,66],{"stretchy":53},[88,39737,90],{},[48,39739,1606],{},[48,39741,57],{},[51,39743,39744],{"separator":122},";",[114,39746,39747],{"encoding":116},"\\bar p(y\\mid x)=\\int p(y\\mid x,\\theta)\\,p(\\theta\\mid D)\\,d\\theta;",[23,39749,39751,39815,39836,39866,39908],{"className":39750,"ariaHidden":122},[121],[23,39752,39754,39757,39800,39803,39806,39809,39812],{"className":39753},[126],[23,39755],{"className":39756,"style":131},[130],[23,39758,39760],{"className":39759},[135,16830],[23,39761,39763,39792],{"className":39762},[199,200],[23,39764,39766,39789],{"className":39765},[204],[23,39767,39770,39778],{"className":39768,"style":39769},[208],"height:0.5678em;",[23,39771,39772,39775],{"style":7130},[23,39773],{"className":39774,"style":217},[216],[23,39776,11],{"className":39777},[135,136],[23,39779,39780,39783],{"style":7130},[23,39781],{"className":39782,"style":217},[216],[23,39784,39786],{"className":39785,"style":34865},[16858],[23,39787,39691],{"className":39788},[135],[23,39790,297],{"className":39791},[296],[23,39793,39795],{"className":39794},[204],[23,39796,39798],{"className":39797,"style":16872},[208],[23,39799],{},[23,39801,54],{"className":39802},[140],[23,39804,660],{"className":39805,"style":676},[135,136],[23,39807],{"className":39808,"style":149},[148],[23,39810,60],{"className":39811},[153],[23,39813],{"className":39814,"style":149},[148],[23,39816,39818,39821,39824,39827,39830,39833],{"className":39817},[126],[23,39819],{"className":39820,"style":131},[130],[23,39822,753],{"className":39823},[135,136],[23,39825,66],{"className":39826},[169],[23,39828],{"className":39829,"style":149},[148],[23,39831,69],{"className":39832},[153],[23,39834],{"className":39835,"style":149},[148],[23,39837,39839,39842,39845,39848,39851,39854,39857,39860,39863],{"className":39838},[126],[23,39840],{"className":39841,"style":3696},[130],[23,39843,1557],{"className":39844,"style":1792},[1786,1790,1791],[23,39846],{"className":39847,"style":280},[148],[23,39849,11],{"className":39850},[135,136],[23,39852,54],{"className":39853},[140],[23,39855,660],{"className":39856,"style":676},[135,136],[23,39858],{"className":39859,"style":149},[148],[23,39861,60],{"className":39862},[153],[23,39864],{"className":39865,"style":149},[148],[23,39867,39869,39872,39875,39878,39881,39884,39887,39890,39893,39896,39899,39902,39905],{"className":39868},[126],[23,39870],{"className":39871,"style":131},[130],[23,39873,753],{"className":39874},[135,136],[23,39876,747],{"className":39877},[824],[23,39879],{"className":39880,"style":280},[148],[23,39882,57],{"className":39883,"style":144},[135,136],[23,39885,66],{"className":39886},[169],[23,39888],{"className":39889,"style":280},[148],[23,39891,11],{"className":39892},[135,136],[23,39894,54],{"className":39895},[140],[23,39897,57],{"className":39898,"style":144},[135,136],[23,39900],{"className":39901,"style":149},[148],[23,39903,60],{"className":39904},[153],[23,39906],{"className":39907,"style":149},[148],[23,39909,39911,39914,39917,39920,39923,39926,39929],{"className":39910},[126],[23,39912],{"className":39913,"style":131},[130],[23,39915,63],{"className":39916,"style":144},[135,136],[23,39918,66],{"className":39919},[169],[23,39921],{"className":39922,"style":280},[148],[23,39924,1606],{"className":39925},[135,136],[23,39927,57],{"className":39928,"style":144},[135,136],[23,39930,39744],{"className":39931},[824],[349,39933,39934,39936,39937],{},[18,39935,13785],{},"：在预测分布上再做一次决策，\n",[23,39938,39940],{"className":39939,"translate":27},[26],[23,39941,39943,40023],{"className":39942,"translate":27},[31],[23,39944,39946],{"className":39945},[35],[37,39947,39948],{"xmlns":39,"display":40},[42,39949,39950,40020],{},[45,39951,39952,39958,39960,39962,39964,39966,39968,39970,39980,39982,39984,39986,39988,39990,39992,39994,39996,40002,40004,40006,40008,40010,40012,40014,40016,40018],{},[4518,39953,39954,39956],{},[48,39955,495],{},[51,39957,6881],{},[51,39959,54],{"stretchy":53},[48,39961,753],{},[51,39963,66],{"stretchy":53},[51,39965,69],{},[48,39967,6956],{},[51,39969,6959],{},[6961,39971,39972,39978],{},[45,39973,39974,39976],{},[48,39975,6967],{},[51,39977,6959],{},[48,39979,495],{},[51,39981,1557],{},[48,39983,555],{},[51,39985,54],{"stretchy":53},[48,39987,660],{},[51,39989,747],{"separator":122},[48,39991,495],{},[51,39993,66],{"stretchy":53},[88,39995,90],{},[16807,39997,39998,40000],{"accent":122},[48,39999,11],{},[51,40001,39691],{},[51,40003,54],{"stretchy":53},[48,40005,660],{},[51,40007,60],{},[48,40009,753],{},[51,40011,66],{"stretchy":53},[88,40013,90],{},[48,40015,1606],{},[48,40017,660],{},[48,40019,112],{"mathvariant":111},[114,40021,40022],{"encoding":116},"a^*(x)=\\arg\\min_a\\int L(y,a)\\,\\bar p(y\\mid x)\\,dy.",[23,40024,40026,40079,40227],{"className":40025,"ariaHidden":122},[121],[23,40027,40029,40032,40061,40064,40067,40070,40073,40076],{"className":40028},[126],[23,40030],{"className":40031,"style":131},[130],[23,40033,40035,40038],{"className":40034},[135],[23,40036,495],{"className":40037},[135,136],[23,40039,40041],{"className":40040},[998],[23,40042,40044],{"className":40043},[199],[23,40045,40047],{"className":40046},[204],[23,40048,40050],{"className":40049,"style":7045},[208],[23,40051,40052,40055],{"style":4989},[23,40053],{"className":40054,"style":1015},[216],[23,40056,40058],{"className":40057},[1019,1020,1021,1022],[23,40059,6881],{"className":40060},[612,1022],[23,40062,54],{"className":40063},[140],[23,40065,753],{"className":40066},[135,136],[23,40068,66],{"className":40069},[169],[23,40071],{"className":40072,"style":149},[148],[23,40074,69],{"className":40075},[153],[23,40077],{"className":40078,"style":149},[148],[23,40080,40082,40085,40090,40093,40137,40140,40143,40146,40149,40152,40155,40158,40161,40164,40167,40170,40212,40215,40218,40221,40224],{"className":40081},[126],[23,40083],{"className":40084,"style":3696},[130],[23,40086,7085,40088],{"className":40087},[1786],[23,40089,7089],{"style":7088},[23,40091],{"className":40092,"style":280},[148],[23,40094,40096],{"className":40095},[1786,7096],[23,40097,40099,40129],{"className":40098},[199,200],[23,40100,40102,40126],{"className":40101},[204],[23,40103,40105,40116],{"className":40104,"style":7106},[208],[23,40106,40107,40110],{"style":8780},[23,40108],{"className":40109,"style":217},[216],[23,40111,40113],{"className":40112},[1019,1020,1021,1022],[23,40114,495],{"className":40115},[135,136,1022],[23,40117,40118,40121],{"style":7130},[23,40119],{"className":40120,"style":217},[216],[23,40122,40123],{},[23,40124,6967],{"className":40125},[1786],[23,40127,297],{"className":40128},[296],[23,40130,40132],{"className":40131},[204],[23,40133,40135],{"className":40134,"style":8812},[208],[23,40136],{},[23,40138],{"className":40139,"style":280},[148],[23,40141,1557],{"className":40142,"style":1792},[1786,1790,1791],[23,40144],{"className":40145,"style":280},[148],[23,40147,555],{"className":40148},[135,136],[23,40150,54],{"className":40151},[140],[23,40153,660],{"className":40154,"style":676},[135,136],[23,40156,747],{"className":40157},[824],[23,40159],{"className":40160,"style":280},[148],[23,40162,495],{"className":40163},[135,136],[23,40165,66],{"className":40166},[169],[23,40168],{"className":40169,"style":280},[148],[23,40171,40173],{"className":40172},[135,16830],[23,40174,40176,40204],{"className":40175},[199,200],[23,40177,40179,40201],{"className":40178},[204],[23,40180,40182,40190],{"className":40181,"style":39769},[208],[23,40183,40184,40187],{"style":7130},[23,40185],{"className":40186,"style":217},[216],[23,40188,11],{"className":40189},[135,136],[23,40191,40192,40195],{"style":7130},[23,40193],{"className":40194,"style":217},[216],[23,40196,40198],{"className":40197,"style":34865},[16858],[23,40199,39691],{"className":40200},[135],[23,40202,297],{"className":40203},[296],[23,40205,40207],{"className":40206},[204],[23,40208,40210],{"className":40209,"style":16872},[208],[23,40211],{},[23,40213,54],{"className":40214},[140],[23,40216,660],{"className":40217,"style":676},[135,136],[23,40219],{"className":40220,"style":149},[148],[23,40222,60],{"className":40223},[153],[23,40225],{"className":40226,"style":149},[148],[23,40228,40230,40233,40236,40239,40242,40245,40248],{"className":40229},[126],[23,40231],{"className":40232,"style":131},[130],[23,40234,753],{"className":40235},[135,136],[23,40237,66],{"className":40238},[169],[23,40240],{"className":40241,"style":280},[148],[23,40243,1606],{"className":40244},[135,136],[23,40246,660],{"className":40247,"style":676},[135,136],[23,40249,112],{"className":40250},[135],[11,40252,40253],{},"第二层是贝叶斯学习的核心，第三层才是完整的贝叶斯决策。很多讨论停在第二层，把「后验预测」直接当成答案；但在有明确损失的问题里，真正的答案是第三层。",[11,40255,40256,40257,40336,40337,40365],{},"后验预测是一个混合分布，不是「把后验均值代进去」。由于 ",[23,40258,40260,40288],{"className":40259,"translate":27},[31],[23,40261,40263],{"className":40262},[35],[37,40264,40265],{"xmlns":39},[42,40266,40267,40285],{},[45,40268,40269,40271,40273,40275,40277,40279,40281,40283],{},[48,40270,11],{},[51,40272,54],{"stretchy":53},[48,40274,660],{},[51,40276,60],{},[48,40278,753],{},[51,40280,747],{"separator":122},[48,40282,57],{},[51,40284,66],{"stretchy":53},[114,40286,40287],{"encoding":116},"p(y\\mid x,\\theta)",[23,40289,40291,40315],{"className":40290,"ariaHidden":122},[121],[23,40292,40294,40297,40300,40303,40306,40309,40312],{"className":40293},[126],[23,40295],{"className":40296,"style":131},[130],[23,40298,11],{"className":40299},[135,136],[23,40301,54],{"className":40302},[140],[23,40304,660],{"className":40305,"style":676},[135,136],[23,40307],{"className":40308,"style":149},[148],[23,40310,60],{"className":40311},[153],[23,40313],{"className":40314,"style":149},[148],[23,40316,40318,40321,40324,40327,40330,40333],{"className":40317},[126],[23,40319],{"className":40320,"style":131},[130],[23,40322,753],{"className":40323},[135,136],[23,40325,747],{"className":40326},[824],[23,40328],{"className":40329,"style":280},[148],[23,40331,57],{"className":40332,"style":144},[135,136],[23,40334,66],{"className":40335},[169]," 通常对 ",[23,40338,40340,40353],{"className":40339,"translate":27},[31],[23,40341,40343],{"className":40342},[35],[37,40344,40345],{"xmlns":39},[42,40346,40347,40351],{},[45,40348,40349],{},[48,40350,57],{},[114,40352,3371],{"encoding":116},[23,40354,40356],{"className":40355,"ariaHidden":122},[121],[23,40357,40359,40362],{"className":40358},[126],[23,40360],{"className":40361,"style":779},[130],[23,40363,57],{"className":40364,"style":144},[135,136]," 非线性，",[23,40367,40369],{"className":40368,"translate":27},[26],[23,40370,40372,40452],{"className":40371,"translate":27},[31],[23,40373,40375],{"className":40374},[35],[37,40376,40377],{"xmlns":39,"display":40},[42,40378,40379,40449],{},[45,40380,40381,40383,40385,40387,40389,40391,40393,40395,40397,40399,40401,40403,40405,40407,40409,40411,40413,40415,40417,40419,40421,40423,40425,40427,40429,40431,40433,40435,40437,40439,40441,40443,40445,40447],{},[51,40382,1557],{},[48,40384,11],{},[51,40386,54],{"stretchy":53},[48,40388,660],{},[51,40390,60],{},[48,40392,753],{},[51,40394,747],{"separator":122},[48,40396,57],{},[51,40398,66],{"stretchy":53},[88,40400,90],{},[48,40402,11],{},[51,40404,54],{"stretchy":53},[48,40406,57],{},[51,40408,60],{},[48,40410,63],{},[51,40412,66],{"stretchy":53},[88,40414,90],{},[48,40416,1606],{},[48,40418,57],{},[51,40420,9015],{"mathvariant":111},[48,40422,11],{},[51,40424,54],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,40426,660],{},[51,40428,60],{},[48,40430,753],{},[51,40432,747],{"separator":122},[48,40434,902],{"mathvariant":571},[51,40436,924],{"stretchy":53},[48,40438,57],{},[51,40440,60],{},[48,40442,63],{},[51,40444,945],{"stretchy":53},[51,40446,66],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,40448,112],{"mathvariant":111},[114,40450,40451],{"encoding":116},"\\int p(y\\mid x,\\theta)\\,p(\\theta\\mid D)\\,d\\theta\n\\ne\np\\big(y\\mid x,\\mathbb E[\\theta\\mid D]\\big).",[23,40453,40455,40485,40527,40590,40617,40650],{"className":40454,"ariaHidden":122},[121],[23,40456,40458,40461,40464,40467,40470,40473,40476,40479,40482],{"className":40457},[126],[23,40459],{"className":40460,"style":3696},[130],[23,40462,1557],{"className":40463,"style":1792},[1786,1790,1791],[23,40465],{"className":40466,"style":280},[148],[23,40468,11],{"className":40469},[135,136],[23,40471,54],{"className":40472},[140],[23,40474,660],{"className":40475,"style":676},[135,136],[23,40477],{"className":40478,"style":149},[148],[23,40480,60],{"className":40481},[153],[23,40483],{"className":40484,"style":149},[148],[23,40486,40488,40491,40494,40497,40500,40503,40506,40509,40512,40515,40518,40521,40524],{"className":40487},[126],[23,40489],{"className":40490,"style":131},[130],[23,40492,753],{"className":40493},[135,136],[23,40495,747],{"className":40496},[824],[23,40498],{"className":40499,"style":280},[148],[23,40501,57],{"className":40502,"style":144},[135,136],[23,40504,66],{"className":40505},[169],[23,40507],{"className":40508,"style":280},[148],[23,40510,11],{"className":40511},[135,136],[23,40513,54],{"className":40514},[140],[23,40516,57],{"className":40517,"style":144},[135,136],[23,40519],{"className":40520,"style":149},[148],[23,40522,60],{"className":40523},[153],[23,40525],{"className":40526,"style":149},[148],[23,40528,40530,40533,40536,40539,40542,40545,40548,40551,40587],{"className":40529},[126],[23,40531],{"className":40532,"style":131},[130],[23,40534,63],{"className":40535,"style":144},[135,136],[23,40537,66],{"className":40538},[169],[23,40540],{"className":40541,"style":280},[148],[23,40543,1606],{"className":40544},[135,136],[23,40546,57],{"className":40547,"style":144},[135,136],[23,40549],{"className":40550,"style":149},[148],[23,40552,40554,40581,40584],{"className":40553},[153],[23,40555,40557],{"className":40556},[153],[23,40558,40560],{"className":40559},[135,9093],[23,40561,40563],{"className":40562},[9097],[23,40564,40566,40569,40578],{"className":40565},[9101],[23,40567],{"className":40568,"style":5414},[130],[23,40570,40572],{"className":40571},[9108],[23,40573,40575],{"className":40574},[135],[23,40576,9115],{"className":40577},[153],[23,40579],{"className":40580},[9119],[23,40582],{"className":40583},[148,9123],[23,40585,69],{"className":40586},[153],[23,40588],{"className":40589,"style":149},[148],[23,40591,40593,40596,40599,40605,40608,40611,40614],{"className":40592},[126],[23,40594],{"className":40595,"style":2213},[130],[23,40597,11],{"className":40598},[135,136],[23,40600,40602],{"className":40601},[135],[23,40603,54],{"className":40604},[1066,1067],[23,40606,660],{"className":40607,"style":676},[135,136],[23,40609],{"className":40610,"style":149},[148],[23,40612,60],{"className":40613},[153],[23,40615],{"className":40616,"style":149},[148],[23,40618,40620,40623,40626,40629,40632,40635,40638,40641,40644,40647],{"className":40619},[126],[23,40621],{"className":40622,"style":131},[130],[23,40624,753],{"className":40625},[135,136],[23,40627,747],{"className":40628},[824],[23,40630],{"className":40631,"style":280},[148],[23,40633,902],{"className":40634},[135,644],[23,40636,924],{"className":40637},[140],[23,40639,57],{"className":40640,"style":144},[135,136],[23,40642],{"className":40643,"style":149},[148],[23,40645,60],{"className":40646},[153],[23,40648],{"className":40649,"style":149},[148],[23,40651,40653,40656,40659,40662,40668],{"className":40652},[126],[23,40654],{"className":40655,"style":2213},[130],[23,40657,63],{"className":40658,"style":144},[135,136],[23,40660,945],{"className":40661},[169],[23,40663,40665],{"className":40664},[135],[23,40666,66],{"className":40667},[1066,1067],[23,40669,112],{"className":40670},[135],[11,40672,40673],{},"差别在不确定性大的区域尤其明显。设想后验在两个相距很远的参数值上各有一个峰，代入均值会得到一个位于两峰之间、几乎没有概率质量的预测；后验预测则会把两个峰的概率都保留下来。前者过度自信，后者诚实地表达了「我不知道是哪一个」。",[2501,40675,40677],{"id":40676},"_62-两种不确定性","6.2 两种不确定性",[11,40679,40680,40681,40684,40685,40713,40714,40742,40743,40746],{},"预测的不确定性可以分成两类。",[18,40682,40683],{},"偶然不确定性","（aleatoric uncertainty）来自数据生成过程本身的噪声：同样的 ",[23,40686,40688,40701],{"className":40687,"translate":27},[31],[23,40689,40691],{"className":40690},[35],[37,40692,40693],{"xmlns":39},[42,40694,40695,40699],{},[45,40696,40697],{},[48,40698,753],{},[114,40700,753],{"encoding":116},[23,40702,40704],{"className":40703,"ariaHidden":122},[121],[23,40705,40707,40710],{"className":40706},[126],[23,40708],{"className":40709,"style":703},[130],[23,40711,753],{"className":40712},[135,136]," 可能对应不同的 ",[23,40715,40717,40730],{"className":40716,"translate":27},[31],[23,40718,40720],{"className":40719},[35],[37,40721,40722],{"xmlns":39},[42,40723,40724,40728],{},[45,40725,40726],{},[48,40727,660],{},[114,40729,660],{"encoding":116},[23,40731,40733],{"className":40732,"ariaHidden":122},[121],[23,40734,40736,40739],{"className":40735},[126],[23,40737],{"className":40738,"style":672},[130],[23,40740,660],{"className":40741,"style":676},[135,136],"，即使知道真实参数也无法消除。",[18,40744,40745],{},"认知不确定性","（epistemic uncertainty）来自知识不足：我们不知道哪个参数是对的，但随着数据增多，它会收缩。",[11,40748,40749],{},"对平方损失，这两种不确定性由全方差公式精确分解：",[23,40751,40753],{"className":40752,"translate":27},[26],[23,40754,40756,40894],{"className":40755,"translate":27},[31],[23,40757,40759],{"className":40758},[35],[37,40760,40761],{"xmlns":39,"display":40},[42,40762,40763,40891],{},[45,40764,40765,40773,40775,40777,40779,40781,40783,40785,40787,40789,40838,40840,40889],{},[45,40766,40767,40769,40771],{},[48,40768,10790],{"mathvariant":111},[48,40770,495],{"mathvariant":111},[48,40772,1653],{"mathvariant":111},[51,40774,54],{"stretchy":53},[48,40776,369],{},[51,40778,60],{},[48,40780,753],{},[51,40782,747],{"separator":122},[48,40784,63],{},[51,40786,66],{"stretchy":53},[51,40788,69],{},[6961,40790,40791,40835],{},[6961,40792,40793,40833],{},[45,40794,40795,40807,40809,40817,40819,40821,40823,40825,40827,40829,40831],{},[898,40796,40797,40799],{},[48,40798,902],{"mathvariant":571},[45,40800,40801,40803,40805],{},[48,40802,57],{},[51,40804,60],{},[48,40806,63],{},[51,40808,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[45,40810,40811,40813,40815],{},[48,40812,10790],{"mathvariant":111},[48,40814,495],{"mathvariant":111},[48,40816,1653],{"mathvariant":111},[51,40818,54],{"stretchy":53},[48,40820,369],{},[51,40822,60],{},[48,40824,753],{},[51,40826,747],{"separator":122},[48,40828,57],{},[51,40830,66],{"stretchy":53},[51,40832,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,40834,8477],{"stretchy":122},[88,40836,40837],{},"偶然",[51,40839,5106],{},[6961,40841,40842,40886],{},[6961,40843,40844,40884],{},[45,40845,40846,40864,40866,40868,40870,40872,40874,40876,40878,40880,40882],{},[898,40847,40848,40856],{},[45,40849,40850,40852,40854],{},[48,40851,10790],{"mathvariant":111},[48,40853,495],{"mathvariant":111},[48,40855,1653],{"mathvariant":111},[45,40857,40858,40860,40862],{},[48,40859,57],{},[51,40861,60],{},[48,40863,63],{},[51,40865,54],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,40867,902],{"mathvariant":571},[51,40869,924],{"stretchy":53},[48,40871,369],{},[51,40873,60],{},[48,40875,753],{},[51,40877,747],{"separator":122},[48,40879,57],{},[51,40881,945],{"stretchy":53},[51,40883,66],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,40885,8477],{"stretchy":122},[88,40887,40888],{},"认知",[48,40890,112],{"mathvariant":111},[114,40892,40893],{"encoding":116},"\\mathrm{Var}(Y\\mid x,D)\n=\n\\underbrace{\\mathbb E_{\\theta\\mid D}\\big[\\mathrm{Var}(Y\\mid x,\\theta)\\big]}_{\\text{偶然}}\n+\n\\underbrace{\\mathrm{Var}_{\\theta\\mid D}\\big(\\mathbb E[Y\\mid x,\\theta]\\big)}_{\\text{认知}}.",[23,40895,40897,40924,40954,41178],{"className":40896,"ariaHidden":122},[121],[23,40898,40900,40903,40909,40912,40915,40918,40921],{"className":40899},[126],[23,40901],{"className":40902,"style":131},[130],[23,40904,40906],{"className":40905},[135],[23,40907,11070],{"className":40908},[135,5391],[23,40910,54],{"className":40911},[140],[23,40913,369],{"className":40914,"style":392},[135,136],[23,40916],{"className":40917,"style":149},[148],[23,40919,60],{"className":40920},[153],[23,40922],{"className":40923,"style":149},[148],[23,40925,40927,40930,40933,40936,40939,40942,40945,40948,40951],{"className":40926},[126],[23,40928],{"className":40929,"style":131},[130],[23,40931,753],{"className":40932},[135,136],[23,40934,747],{"className":40935},[824],[23,40937],{"className":40938,"style":280},[148],[23,40940,63],{"className":40941,"style":144},[135,136],[23,40943,66],{"className":40944},[169],[23,40946],{"className":40947,"style":149},[148],[23,40949,69],{"className":40950},[153],[23,40952],{"className":40953,"style":149},[148],[23,40955,40957,40961,41169,41172,41175],{"className":40956},[126],[23,40958],{"className":40959,"style":40960},[130],"height:2.5315em;vertical-align:-1.6815em;",[23,40962,40964],{"className":40963},[1839,6961],[23,40965,40967,41160],{"className":40966},[199,200],[23,40968,40970,41157],{"className":40969},[204],[23,40971,40974,40992],{"className":40972,"style":40973},[208],"height:0.85em;",[23,40975,40977,40980],{"style":40976},"top:-1.3185em;",[23,40978],{"className":40979,"style":217},[216],[23,40981,40983],{"className":40982},[1019,1020,1021,1022],[23,40984,40986],{"className":40985},[135,1022],[23,40987,40989],{"className":40988},[135,6514,1022],[23,40990,40837],{"className":40991},[135,7238,1022],[23,40993,40994,40997],{"style":7130},[23,40995],{"className":40996,"style":217},[216],[23,40998,41000],{"className":40999},[1839,6961],[23,41001,41003,41148],{"className":41002},[199,200],[23,41004,41006,41145],{"className":41005},[204],[23,41007,41009,41040],{"className":41008,"style":40973},[208],[23,41010,41013,41016],{"className":41011,"style":41012},[8672],"top:-1.9968em;",[23,41014],{"className":41015,"style":217},[216],[23,41017,41019,41026,41033],{"className":41018,"style":8681},[8680],[23,41020,41022],{"className":41021,"style":8686},[8685],[8688,41023,41024],{"xmlns":8690,"width":8691,"height":8692,"viewBox":8693,"preserveAspectRatio":8694},[8696,41025],{"d":8698},[23,41027,41029],{"className":41028,"style":8686},[8702],[8688,41030,41031],{"xmlns":8690,"width":8691,"height":8692,"viewBox":8693,"preserveAspectRatio":8705},[8696,41032],{"d":8708},[23,41034,41036],{"className":41035,"style":8686},[8712],[8688,41037,41038],{"xmlns":8690,"width":8691,"height":8692,"viewBox":8693,"preserveAspectRatio":8715},[8696,41039],{"d":8718},[23,41041,41042,41045],{"style":7130},[23,41043],{"className":41044,"style":217},[216],[23,41046,41048,41097,41103,41109,41112,41115,41118,41121,41124,41127,41130,41133,41136,41139],{"className":41047},[135],[23,41049,41051,41054],{"className":41050},[135],[23,41052,902],{"className":41053},[135,644],[23,41055,41057],{"className":41056},[998],[23,41058,41060,41089],{"className":41059},[199,200],[23,41061,41063,41086],{"className":41062},[204],[23,41064,41066],{"className":41065,"style":1008},[208],[23,41067,41068,41071],{"style":1011},[23,41069],{"className":41070,"style":1015},[216],[23,41072,41074],{"className":41073},[1019,1020,1021,1022],[23,41075,41077,41080,41083],{"className":41076},[135,1022],[23,41078,57],{"className":41079,"style":144},[135,136,1022],[23,41081,60],{"className":41082},[153,1022],[23,41084,63],{"className":41085,"style":144},[135,136,1022],[23,41087,297],{"className":41088},[296],[23,41090,41092],{"className":41091},[204],[23,41093,41095],{"className":41094,"style":1057},[208],[23,41096],{},[23,41098,41100],{"className":41099},[135],[23,41101,924],{"className":41102},[1066,1067],[23,41104,41106],{"className":41105},[135],[23,41107,11070],{"className":41108},[135,5391],[23,41110,54],{"className":41111},[140],[23,41113,369],{"className":41114,"style":392},[135,136],[23,41116],{"className":41117,"style":149},[148],[23,41119,60],{"className":41120},[153],[23,41122],{"className":41123,"style":149},[148],[23,41125,753],{"className":41126},[135,136],[23,41128,747],{"className":41129},[824],[23,41131],{"className":41132,"style":280},[148],[23,41134,57],{"className":41135,"style":144},[135,136],[23,41137,66],{"className":41138},[169],[23,41140,41142],{"className":41141},[135],[23,41143,945],{"className":41144},[1066,1067],[23,41146,297],{"className":41147},[296],[23,41149,41151],{"className":41150},[204],[23,41152,41155],{"className":41153,"style":41154},[208],"height:1.0032em;",[23,41156],{},[23,41158,297],{"className":41159},[296],[23,41161,41163],{"className":41162},[204],[23,41164,41167],{"className":41165,"style":41166},[208],"height:1.6815em;",[23,41168],{},[23,41170],{"className":41171,"style":392},[148],[23,41173,5106],{"className":41174},[612],[23,41176],{"className":41177,"style":392},[148],[23,41179,41181,41184,41388,41391],{"className":41180},[126],[23,41182],{"className":41183,"style":40960},[130],[23,41185,41187],{"className":41186},[1839,6961],[23,41188,41190,41380],{"className":41189},[199,200],[23,41191,41193,41377],{"className":41192},[204],[23,41194,41196,41213],{"className":41195,"style":40973},[208],[23,41197,41198,41201],{"style":40976},[23,41199],{"className":41200,"style":217},[216],[23,41202,41204],{"className":41203},[1019,1020,1021,1022],[23,41205,41207],{"className":41206},[135,1022],[23,41208,41210],{"className":41209},[135,6514,1022],[23,41211,40888],{"className":41212},[135,7238,1022],[23,41214,41215,41218],{"style":7130},[23,41216],{"className":41217,"style":217},[216],[23,41219,41221],{"className":41220},[1839,6961],[23,41222,41224,41369],{"className":41223},[199,200],[23,41225,41227,41366],{"className":41226},[204],[23,41228,41230,41260],{"className":41229,"style":40973},[208],[23,41231,41233,41236],{"className":41232,"style":41012},[8672],[23,41234],{"className":41235,"style":217},[216],[23,41237,41239,41246,41253],{"className":41238,"style":8681},[8680],[23,41240,41242],{"className":41241,"style":8686},[8685],[8688,41243,41244],{"xmlns":8690,"width":8691,"height":8692,"viewBox":8693,"preserveAspectRatio":8694},[8696,41245],{"d":8698},[23,41247,41249],{"className":41248,"style":8686},[8702],[8688,41250,41251],{"xmlns":8690,"width":8691,"height":8692,"viewBox":8693,"preserveAspectRatio":8705},[8696,41252],{"d":8708},[23,41254,41256],{"className":41255,"style":8686},[8712],[8688,41257,41258],{"xmlns":8690,"width":8691,"height":8692,"viewBox":8693,"preserveAspectRatio":8715},[8696,41259],{"d":8718},[23,41261,41262,41265],{"style":7130},[23,41263],{"className":41264,"style":217},[216],[23,41266,41268,41321,41327,41330,41333,41336,41339,41342,41345,41348,41351,41354,41357,41360],{"className":41267},[135],[23,41269,41271,41277],{"className":41270},[135],[23,41272,41274],{"className":41273},[135],[23,41275,11070],{"className":41276},[135,5391],[23,41278,41280],{"className":41279},[998],[23,41281,41283,41313],{"className":41282},[199,200],[23,41284,41286,41310],{"className":41285},[204],[23,41287,41289],{"className":41288,"style":1008},[208],[23,41290,41292,41295],{"style":41291},"top:-2.5198em;margin-right:0.05em;",[23,41293],{"className":41294,"style":1015},[216],[23,41296,41298],{"className":41297},[1019,1020,1021,1022],[23,41299,41301,41304,41307],{"className":41300},[135,1022],[23,41302,57],{"className":41303,"style":144},[135,136,1022],[23,41305,60],{"className":41306},[153,1022],[23,41308,63],{"className":41309,"style":144},[135,136,1022],[23,41311,297],{"className":41312},[296],[23,41314,41316],{"className":41315},[204],[23,41317,41319],{"className":41318,"style":1057},[208],[23,41320],{},[23,41322,41324],{"className":41323},[135],[23,41325,54],{"className":41326},[1066,1067],[23,41328,902],{"className":41329},[135,644],[23,41331,924],{"className":41332},[140],[23,41334,369],{"className":41335,"style":392},[135,136],[23,41337],{"className":41338,"style":149},[148],[23,41340,60],{"className":41341},[153],[23,41343],{"className":41344,"style":149},[148],[23,41346,753],{"className":41347},[135,136],[23,41349,747],{"className":41350},[824],[23,41352],{"className":41353,"style":280},[148],[23,41355,57],{"className":41356,"style":144},[135,136],[23,41358,945],{"className":41359},[169],[23,41361,41363],{"className":41362},[135],[23,41364,66],{"className":41365},[1066,1067],[23,41367,297],{"className":41368},[296],[23,41370,41372],{"className":41371},[204],[23,41373,41375],{"className":41374,"style":41154},[208],[23,41376],{},[23,41378,297],{"className":41379},[296],[23,41381,41383],{"className":41382},[204],[23,41384,41386],{"className":41385,"style":41166},[208],[23,41387],{},[23,41389],{"className":41390,"style":280},[148],[23,41392,112],{"className":41393},[135],[11,41395,41396],{},"第一项是「即使知道模型参数，预测仍然会波动的部分」；第二项是「不同参数给出的预测均值之间的差异」。在数据稀疏的区域，后验宽，第二项大；随着数据增加，后验收缩，第二项趋于零。第一项则不会消失。",[11,41398,41399],{},"这个分解也是主动学习的理论基础：我们想选的样本，是那些能最大程度降低认知不确定性的样本，而不是那些本身就带噪声的样本。",[2501,41401,41403],{"id":41402},"_63-推断本身也是决策","6.3 推断本身也是决策",[11,41405,41406],{},"在参数空间上定义一个损失，报一个点估计，本质上也是一次决策：",[3764,41408,41409,41419],{},[3767,41410,41411],{},[3770,41412,41413,41416],{},[3773,41414,41415],{},"参数空间上的损失",[3773,41417,41418],{},"最优估计",[3783,41420,41421,41427,41433],{},[3770,41422,41423,41425],{},[3788,41424,13800],{},[3788,41426,11331],{},[3770,41428,41429,41431],{},[3788,41430,13807],{},[3788,41432,13810],{},[3770,41434,41435,41437],{},[3788,41436,13792],{},[3788,41438,41439],{},"后验众数（MAP）",[11,41441,41442],{},"所以贝叶斯估计不是与决策论并列的东西，而是决策论在参数空间上的应用。后验均值与 MAP 会打架，是因为它们对应不同的损失，没有谁更「正确」。若你关心的是平方误差，均值是对的；若你关心的是「最可能的那个值」，众数是对的；若你关心的是「误差不超过某个范围」，分位数才是对的。",[11,41444,41445],{},"同样的逻辑适用于区间估计。一个可信区间是一个「集合值」的行动，它的损失函数衡量区间是否覆盖真值、以及区间有多宽。置信区间与可信区间的差异当然有概率哲学的成分；但从决策论看，它们也对应不同的损失：一个关心重复抽样下的覆盖率，一个关心给定数据下的后验覆盖概率。",[2501,41447,41449],{"id":41448},"_64-后验的集中","6.4 后验的集中",[11,41451,41452,41453,41456],{},"在正则条件下，后验会随着数据增多而集中到真实参数附近。",[18,41454,41455],{},"Bernstein–von Mises 定理","给出了更精确的结论：后验在适当的缩放与中心化之后，趋近于一个正态分布，",[23,41458,41460],{"className":41459,"translate":27},[26],[23,41461,41463,41544],{"className":41462,"translate":27},[31],[23,41464,41466],{"className":41465},[35],[37,41467,41468],{"xmlns":39,"display":40},[42,41469,41470,41541],{},[45,41471,41472,41476,41478,41480,41482,41484,41494,41496,41498,41500,41502,41505,41507,41509,41511,41513,41515,41517,41519,41521,41527,41537,41539],{},[25516,41473,41474],{},[48,41475,4806],{},[88,41477,90],{},[51,41479,54],{"stretchy":53},[48,41481,57],{},[51,41483,4529],{},[898,41485,41486,41492],{},[16807,41487,41488,41490],{"accent":122},[48,41489,57],{},[51,41491,16813],{},[48,41493,4806],{},[51,41495,66],{"stretchy":53},[51,41497,60],{},[48,41499,63],{},[88,41501,5360],{},[51,41503,41504],{},"⇒",[88,41506,5360],{},[48,41508,12801],{"mathvariant":375},[51,41510,54],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[2581,41512,1532],{},[51,41514,747],{"separator":122},[88,41516,5360],{},[48,41518,34353],{},[51,41520,54],{"stretchy":53},[898,41522,41523,41525],{},[48,41524,57],{},[2581,41526,1532],{},[4518,41528,41529,41531],{},[51,41530,66],{"stretchy":53},[45,41532,41533,41535],{},[51,41534,4529],{},[2581,41536,4435],{},[51,41538,66],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,41540,747],{"separator":122},[114,41542,41543],{"encoding":116},"\\sqrt n\\,(\\theta-\\hat\\theta_n)\\mid D\n\\ \\Rightarrow\\\n\\mathcal N\\big(0,\\ I(\\theta_0)^{-1}\\big),",[23,41545,41547,41619,41705,41729],{"className":41546,"ariaHidden":122},[121],[23,41548,41550,41554,41601,41604,41607,41610,41613,41616],{"className":41549},[126],[23,41551],{"className":41552,"style":41553},[130],"height:1.0992em;vertical-align:-0.25em;",[23,41555,41557],{"className":41556},[135,25667],[23,41558,41560,41592],{"className":41559},[199,200],[23,41561,41563,41589],{"className":41562},[204],[23,41564,41567,41576],{"className":41565,"style":41566},[208],"height:0.8492em;",[23,41568,41570,41573],{"className":41569,"style":7130},[8672],[23,41571],{"className":41572,"style":217},[216],[23,41574,4806],{"className":41575,"style":25687},[135,136],[23,41577,41579,41582],{"style":41578},"top:-2.8092em;",[23,41580],{"className":41581,"style":217},[216],[23,41583,41585],{"className":41584,"style":25778},[25777],[8688,41586,41587],{"xmlns":8690,"width":8691,"height":25781,"viewBox":25782,"preserveAspectRatio":8694},[8696,41588],{"d":25785},[23,41590,297],{"className":41591},[296],[23,41593,41595],{"className":41594},[204],[23,41596,41599],{"className":41597,"style":41598},[208],"height:0.1908em;",[23,41600],{},[23,41602],{"className":41603,"style":280},[148],[23,41605,54],{"className":41606},[140],[23,41608,57],{"className":41609,"style":144},[135,136],[23,41611],{"className":41612,"style":392},[148],[23,41614,4529],{"className":41615},[612],[23,41617],{"className":41618,"style":392},[148],[23,41620,41622,41625,41693,41696,41699,41702],{"className":41621},[126],[23,41623],{"className":41624,"style":16966},[130],[23,41626,41628,41659],{"className":41627},[135],[23,41629,41631],{"className":41630},[135,16830],[23,41632,41634],{"className":41633},[199],[23,41635,41637],{"className":41636},[204],[23,41638,41640,41648],{"className":41639,"style":16840},[208],[23,41641,41642,41645],{"style":7130},[23,41643],{"className":41644,"style":217},[216],[23,41646,57],{"className":41647,"style":144},[135,136],[23,41649,41650,41653],{"style":16851},[23,41651],{"className":41652,"style":217},[216],[23,41654,41656],{"className":41655,"style":34865},[16858],[23,41657,16813],{"className":41658},[135],[23,41660,41662],{"className":41661},[998],[23,41663,41665,41685],{"className":41664},[199,200],[23,41666,41668,41682],{"className":41667},[204],[23,41669,41671],{"className":41670,"style":6499},[208],[23,41672,41673,41676],{"style":27171},[23,41674],{"className":41675,"style":1015},[216],[23,41677,41679],{"className":41678},[1019,1020,1021,1022],[23,41680,4806],{"className":41681},[135,136,1022],[23,41683,297],{"className":41684},[296],[23,41686,41688],{"className":41687},[204],[23,41689,41691],{"className":41690,"style":2048},[208],[23,41692],{},[23,41694,66],{"className":41695},[169],[23,41697],{"className":41698,"style":149},[148],[23,41700,60],{"className":41701},[153],[23,41703],{"className":41704,"style":149},[148],[23,41706,41708,41711,41714,41717,41720,41723,41726],{"className":41707},[126],[23,41709],{"className":41710,"style":470},[130],[23,41712,63],{"className":41713,"style":144},[135,136],[23,41715,5360],{"className":41716},[148],[23,41718],{"className":41719,"style":149},[148],[23,41721,41504],{"className":41722},[153],[23,41724,5360],{"className":41725},[148],[23,41727],{"className":41728,"style":149},[148],[23,41730,41732,41735,41738,41744,41747,41750,41753,41756,41759,41762,41802,41837,41843],{"className":41731},[126],[23,41733],{"className":41734,"style":17156},[130],[23,41736,12801],{"className":41737,"style":22218},[135,412],[23,41739,41741],{"className":41740},[135],[23,41742,54],{"className":41743},[1066,1067],[23,41745,1532],{"className":41746},[135],[23,41748,747],{"className":41749},[824],[23,41751,5360],{"className":41752},[148],[23,41754],{"className":41755,"style":280},[148],[23,41757,34353],{"className":41758,"style":454},[135,136],[23,41760,54],{"className":41761},[140],[23,41763,41765,41768],{"className":41764},[135],[23,41766,57],{"className":41767,"style":144},[135,136],[23,41769,41771],{"className":41770},[998],[23,41772,41774,41794],{"className":41773},[199,200],[23,41775,41777,41791],{"className":41776},[204],[23,41778,41780],{"className":41779,"style":18647},[208],[23,41781,41782,41785],{"style":27171},[23,41783],{"className":41784,"style":1015},[216],[23,41786,41788],{"className":41787},[1019,1020,1021,1022],[23,41789,1532],{"className":41790},[135,1022],[23,41792,297],{"className":41793},[296],[23,41795,41797],{"className":41796},[204],[23,41798,41800],{"className":41799,"style":2048},[208],[23,41801],{},[23,41803,41805,41808],{"className":41804},[169],[23,41806,66],{"className":41807},[169],[23,41809,41811],{"className":41810},[998],[23,41812,41814],{"className":41813},[199],[23,41815,41817],{"className":41816},[204],[23,41818,41820],{"className":41819,"style":11002},[208],[23,41821,41822,41825],{"style":4989},[23,41823],{"className":41824,"style":1015},[216],[23,41826,41828],{"className":41827},[1019,1020,1021,1022],[23,41829,41831,41834],{"className":41830},[135,1022],[23,41832,4529],{"className":41833},[135,1022],[23,41835,4435],{"className":41836},[135,1022],[23,41838,41840],{"className":41839},[135],[23,41841,66],{"className":41842},[1066,1067],[23,41844,747],{"className":41845},[824],[11,41847,24429,41848,41933],{},[23,41849,41851,41875],{"className":41850,"translate":27},[31],[23,41852,41854],{"className":41853},[35],[37,41855,41856],{"xmlns":39},[42,41857,41858,41872],{},[45,41859,41860,41862,41864,41870],{},[48,41861,34353],{},[51,41863,54],{"stretchy":53},[898,41865,41866,41868],{},[48,41867,57],{},[2581,41869,1532],{},[51,41871,66],{"stretchy":53},[114,41873,41874],{"encoding":116},"I(\\theta_0)",[23,41876,41878],{"className":41877,"ariaHidden":122},[121],[23,41879,41881,41884,41887,41890,41930],{"className":41880},[126],[23,41882],{"className":41883,"style":131},[130],[23,41885,34353],{"className":41886,"style":454},[135,136],[23,41888,54],{"className":41889},[140],[23,41891,41893,41896],{"className":41892},[135],[23,41894,57],{"className":41895,"style":144},[135,136],[23,41897,41899],{"className":41898},[998],[23,41900,41902,41922],{"className":41901},[199,200],[23,41903,41905,41919],{"className":41904},[204],[23,41906,41908],{"className":41907,"style":18647},[208],[23,41909,41910,41913],{"style":27171},[23,41911],{"className":41912,"style":1015},[216],[23,41914,41916],{"className":41915},[1019,1020,1021,1022],[23,41917,1532],{"className":41918},[135,1022],[23,41920,297],{"className":41921},[296],[23,41923,41925],{"className":41924},[204],[23,41926,41928],{"className":41927,"style":2048},[208],[23,41929],{},[23,41931,66],{"className":41932},[169]," 是 Fisher 信息。等价地，后验近似为",[23,41935,41937],{"className":41936,"translate":27},[26],[23,41938,41940,42016],{"className":41939,"translate":27},[31],[23,41941,41943],{"className":41942},[35],[37,41944,41945],{"xmlns":39,"display":40},[42,41946,41947,42013],{},[45,41948,41949,41951,41953,41955,41957,41959,41961,41963,41965,42011],{},[48,41950,11],{},[51,41952,54],{"stretchy":53},[48,41954,57],{},[51,41956,60],{},[48,41958,63],{},[51,41960,66],{"stretchy":53},[51,41962,29066],{},[48,41964,12801],{"mathvariant":375},[45,41966,41967,41969,41979,41981,41983,41989,41991,41993,41999,42009],{},[51,41968,54],{"fence":122},[898,41970,41971,41977],{},[16807,41972,41973,41975],{"accent":122},[48,41974,57],{},[51,41976,16813],{},[48,41978,4806],{},[51,41980,747],{"separator":122},[88,41982,5360],{},[71,41984,41985,41987],{},[2581,41986,4435],{},[48,41988,4806],{},[48,41990,34353],{},[51,41992,54],{"stretchy":53},[898,41994,41995,41997],{},[48,41996,57],{},[2581,41998,1532],{},[4518,42000,42001,42003],{},[51,42002,66],{"stretchy":53},[45,42004,42005,42007],{},[51,42006,4529],{},[2581,42008,4435],{},[51,42010,66],{"fence":122},[48,42012,112],{"mathvariant":111},[114,42014,42015],{"encoding":116},"p(\\theta\\mid D)\\approx\n\\mathcal N\\left(\\hat\\theta_n,\\ \\frac{1}{n}I(\\theta_0)^{-1}\\right).",[23,42017,42019,42043,42064],{"className":42018,"ariaHidden":122},[121],[23,42020,42022,42025,42028,42031,42034,42037,42040],{"className":42021},[126],[23,42023],{"className":42024,"style":131},[130],[23,42026,11],{"className":42027},[135,136],[23,42029,54],{"className":42030},[140],[23,42032,57],{"className":42033,"style":144},[135,136],[23,42035],{"className":42036,"style":149},[148],[23,42038,60],{"className":42039},[153],[23,42041],{"className":42042,"style":149},[148],[23,42044,42046,42049,42052,42055,42058,42061],{"className":42045},[126],[23,42047],{"className":42048,"style":131},[130],[23,42050,63],{"className":42051,"style":144},[135,136],[23,42053,66],{"className":42054},[169],[23,42056],{"className":42057,"style":149},[148],[23,42059,29066],{"className":42060},[153],[23,42062],{"className":42063,"style":149},[148],[23,42065,42067,42070,42073,42076,42311,42314],{"className":42066},[126],[23,42068],{"className":42069,"style":20789},[130],[23,42071,12801],{"className":42072,"style":22218},[135,412],[23,42074],{"className":42075,"style":280},[148],[23,42077,42079,42085,42153,42156,42159,42162,42224,42227,42230,42270,42305],{"className":42078},[1839],[23,42080,42082],{"className":42081,"style":1844},[140,1843],[23,42083,54],{"className":42084},[1066,1021],[23,42086,42088,42119],{"className":42087},[135],[23,42089,42091],{"className":42090},[135,16830],[23,42092,42094],{"className":42093},[199],[23,42095,42097],{"className":42096},[204],[23,42098,42100,42108],{"className":42099,"style":16840},[208],[23,42101,42102,42105],{"style":7130},[23,42103],{"className":42104,"style":217},[216],[23,42106,57],{"className":42107,"style":144},[135,136],[23,42109,42110,42113],{"style":16851},[23,42111],{"className":42112,"style":217},[216],[23,42114,42116],{"className":42115,"style":34865},[16858],[23,42117,16813],{"className":42118},[135],[23,42120,42122],{"className":42121},[998],[23,42123,42125,42145],{"className":42124},[199,200],[23,42126,42128,42142],{"className":42127},[204],[23,42129,42131],{"className":42130,"style":6499},[208],[23,42132,42133,42136],{"style":27171},[23,42134],{"className":42135,"style":1015},[216],[23,42137,42139],{"className":42138},[1019,1020,1021,1022],[23,42140,4806],{"className":42141},[135,136,1022],[23,42143,297],{"className":42144},[296],[23,42146,42148],{"className":42147},[204],[23,42149,42151],{"className":42150,"style":2048},[208],[23,42152],{},[23,42154,747],{"className":42155},[824],[23,42157,5360],{"className":42158},[148],[23,42160],{"className":42161,"style":280},[148],[23,42163,42165,42168,42221],{"className":42164},[135],[23,42166],{"className":42167},[140,192],[23,42169,42171],{"className":42170},[71],[23,42172,42174,42213],{"className":42173},[199,200],[23,42175,42177,42210],{"className":42176},[204],[23,42178,42180,42191,42199],{"className":42179,"style":12409},[208],[23,42181,42182,42185],{"style":212},[23,42183],{"className":42184,"style":217},[216],[23,42186,42188],{"className":42187},[135],[23,42189,4806],{"className":42190},[135,136],[23,42192,42193,42196],{"style":235},[23,42194],{"className":42195,"style":217},[216],[23,42197],{"className":42198,"style":243},[242],[23,42200,42201,42204],{"style":246},[23,42202],{"className":42203,"style":217},[216],[23,42205,42207],{"className":42206},[135],[23,42208,4435],{"className":42209},[135],[23,42211,297],{"className":42212},[296],[23,42214,42216],{"className":42215},[204],[23,42217,42219],{"className":42218,"style":6287},[208],[23,42220],{},[23,42222],{"className":42223},[169,192],[23,42225,34353],{"className":42226,"style":454},[135,136],[23,42228,54],{"className":42229},[140],[23,42231,42233,42236],{"className":42232},[135],[23,42234,57],{"className":42235,"style":144},[135,136],[23,42237,42239],{"className":42238},[998],[23,42240,42242,42262],{"className":42241},[199,200],[23,42243,42245,42259],{"className":42244},[204],[23,42246,42248],{"className":42247,"style":18647},[208],[23,42249,42250,42253],{"style":27171},[23,42251],{"className":42252,"style":1015},[216],[23,42254,42256],{"className":42255},[1019,1020,1021,1022],[23,42257,1532],{"className":42258},[135,1022],[23,42260,297],{"className":42261},[296],[23,42263,42265],{"className":42264},[204],[23,42266,42268],{"className":42267,"style":2048},[208],[23,42269],{},[23,42271,42273,42276],{"className":42272},[169],[23,42274,66],{"className":42275},[169],[23,42277,42279],{"className":42278},[998],[23,42280,42282],{"className":42281},[199],[23,42283,42285],{"className":42284},[204],[23,42286,42288],{"className":42287,"style":11002},[208],[23,42289,42290,42293],{"style":4989},[23,42291],{"className":42292,"style":1015},[216],[23,42294,42296],{"className":42295},[1019,1020,1021,1022],[23,42297,42299,42302],{"className":42298},[135,1022],[23,42300,4529],{"className":42301},[135,1022],[23,42303,4435],{"className":42304},[135,1022],[23,42306,42308],{"className":42307,"style":1844},[169,1843],[23,42309,66],{"className":42310},[1066,1021],[23,42312],{"className":42313,"style":280},[148],[23,42315,112],{"className":42316},[135],[11,42318,42319,42322],{},[18,42320,42321],{},"Doob 定理","则从另一个角度保证一致性：对先验几乎所有的参数值，后验都会收敛到真值附近的任意邻域。注意「先验几乎处处」这个限定：先验可能把质量放在一个坏集合上，只是那个集合的概率为零。",[11,42324,42325,42326,42426],{},"这两个定理解释了为什么在大样本下，贝叶斯与频率派的答案往往一致。后验的宽度以 ",[23,42327,42329,42356],{"className":42328,"translate":27},[31],[23,42330,42332],{"className":42331},[35],[37,42333,42334],{"xmlns":39},[42,42335,42336,42353],{},[45,42337,42338,42341,42343,42345,42347,42351],{},[48,42339,42340],{},"O",[51,42342,54],{"stretchy":53},[2581,42344,4435],{},[48,42346,2875],{"mathvariant":111},[25516,42348,42349],{},[48,42350,4806],{},[51,42352,66],{"stretchy":53},[114,42354,42355],{"encoding":116},"O(1\u002F\\sqrt n)",[23,42357,42359],{"className":42358,"ariaHidden":122},[121],[23,42360,42362,42366,42369,42372,42376,42423],{"className":42361},[126],[23,42363],{"className":42364,"style":42365},[130],"height:1.0503em;vertical-align:-0.25em;",[23,42367,42340],{"className":42368,"style":144},[135,136],[23,42370,54],{"className":42371},[140],[23,42373,42375],{"className":42374},[135],"1\u002F",[23,42377,42379],{"className":42378},[135,25667],[23,42380,42382,42414],{"className":42381},[199,200],[23,42383,42385,42411],{"className":42384},[204],[23,42386,42389,42398],{"className":42387,"style":42388},[208],"height:0.8003em;",[23,42390,42392,42395],{"className":42391,"style":7130},[8672],[23,42393],{"className":42394,"style":217},[216],[23,42396,4806],{"className":42397,"style":25687},[135,136],[23,42399,42401,42404],{"style":42400},"top:-2.7603em;",[23,42402],{"className":42403,"style":217},[216],[23,42405,42407],{"className":42406,"style":25778},[25777],[8688,42408,42409],{"xmlns":8690,"width":8691,"height":25781,"viewBox":25782,"preserveAspectRatio":8694},[8696,42410],{"d":25785},[23,42412,297],{"className":42413},[296],[23,42415,42417],{"className":42416},[204],[23,42418,42421],{"className":42419,"style":42420},[208],"height:0.2397em;",[23,42422],{},[23,42424,66],{"className":42425},[169]," 收缩，先验对形状的影响随之减弱，后验主要由似然决定。但这也划出了边界：当模型奇异（参数不可识别、网络层数过多、对称性导致 Fisher 信息退化）时，Bernstein–von Mises 失效，后验的形状本身携带信息，不能用一个高斯替代。这正是奇异学习理论要处理的情形。",[2501,42428,42430],{"id":42429},"_65-边缘似然与奥卡姆剃刀","6.5 边缘似然与奥卡姆剃刀",[11,42432,42433,42434,42437],{},"贝叶斯模型比较的核心量是",[18,42435,42436],{},"边缘似然","（marginal likelihood），也就是贝叶斯公式的分母：",[23,42439,42441],{"className":42440,"translate":27},[26],[23,42442,42444,42510],{"className":42443,"translate":27},[31],[23,42445,42447],{"className":42446},[35],[37,42448,42449],{"xmlns":39,"display":40},[42,42450,42451,42507],{},[45,42452,42453,42455,42457,42459,42461,42463,42465,42467,42469,42471,42473,42475,42477,42479,42481,42483,42485,42487,42489,42491,42493,42495,42497,42499,42501,42503,42505],{},[48,42454,11],{},[51,42456,54],{"stretchy":53},[48,42458,63],{},[51,42460,60],{},[48,42462,34732],{},[51,42464,66],{"stretchy":53},[51,42466,69],{},[51,42468,1557],{},[48,42470,11],{},[51,42472,54],{"stretchy":53},[48,42474,63],{},[51,42476,60],{},[48,42478,57],{},[51,42480,747],{"separator":122},[48,42482,34732],{},[51,42484,66],{"stretchy":53},[88,42486,90],{},[48,42488,11],{},[51,42490,54],{"stretchy":53},[48,42492,57],{},[51,42494,60],{},[48,42496,34732],{},[51,42498,66],{"stretchy":53},[88,42500,90],{},[48,42502,1606],{},[48,42504,57],{},[48,42506,112],{"mathvariant":111},[114,42508,42509],{"encoding":116},"p(D\\mid M)=\\int p(D\\mid\\theta,M)\\,p(\\theta\\mid M)\\,d\\theta .",[23,42511,42513,42537,42559,42589,42631],{"className":42512,"ariaHidden":122},[121],[23,42514,42516,42519,42522,42525,42528,42531,42534],{"className":42515},[126],[23,42517],{"className":42518,"style":131},[130],[23,42520,11],{"className":42521},[135,136],[23,42523,54],{"className":42524},[140],[23,42526,63],{"className":42527,"style":144},[135,136],[23,42529],{"className":42530,"style":149},[148],[23,42532,60],{"className":42533},[153],[23,42535],{"className":42536,"style":149},[148],[23,42538,42540,42543,42547,42550,42553,42556],{"className":42539},[126],[23,42541],{"className":42542,"style":131},[130],[23,42544,34732],{"className":42545,"style":42546},[135,136],"margin-right:0.109em;",[23,42548,66],{"className":42549},[169],[23,42551],{"className":42552,"style":149},[148],[23,42554,69],{"className":42555},[153],[23,42557],{"className":42558,"style":149},[148],[23,42560,42562,42565,42568,42571,42574,42577,42580,42583,42586],{"className":42561},[126],[23,42563],{"className":42564,"style":3696},[130],[23,42566,1557],{"className":42567,"style":1792},[1786,1790,1791],[23,42569],{"className":42570,"style":280},[148],[23,42572,11],{"className":42573},[135,136],[23,42575,54],{"className":42576},[140],[23,42578,63],{"className":42579,"style":144},[135,136],[23,42581],{"className":42582,"style":149},[148],[23,42584,60],{"className":42585},[153],[23,42587],{"className":42588,"style":149},[148],[23,42590,42592,42595,42598,42601,42604,42607,42610,42613,42616,42619,42622,42625,42628],{"className":42591},[126],[23,42593],{"className":42594,"style":131},[130],[23,42596,57],{"className":42597,"style":144},[135,136],[23,42599,747],{"className":42600},[824],[23,42602],{"className":42603,"style":280},[148],[23,42605,34732],{"className":42606,"style":42546},[135,136],[23,42608,66],{"className":42609},[169],[23,42611],{"className":42612,"style":280},[148],[23,42614,11],{"className":42615},[135,136],[23,42617,54],{"className":42618},[140],[23,42620,57],{"className":42621,"style":144},[135,136],[23,42623],{"className":42624,"style":149},[148],[23,42626,60],{"className":42627},[153],[23,42629],{"className":42630,"style":149},[148],[23,42632,42634,42637,42640,42643,42646,42649,42652],{"className":42633},[126],[23,42635],{"className":42636,"style":131},[130],[23,42638,34732],{"className":42639,"style":42546},[135,136],[23,42641,66],{"className":42642},[169],[23,42644],{"className":42645,"style":280},[148],[23,42647,1606],{"className":42648},[135,136],[23,42650,57],{"className":42651,"style":144},[135,136],[23,42653,112],{"className":42654},[135],[11,42656,42657,42658,32985],{},"两个模型的比较用",[18,42659,42660],{},"贝叶斯因子",[23,42662,42664],{"className":42663,"translate":27},[26],[23,42665,42667,42818],{"className":42666,"translate":27},[31],[23,42668,42670],{"className":42669},[35],[37,42671,42672],{"xmlns":39,"display":40},[42,42673,42674,42815],{},[45,42675,42676,42687,42689,42727,42729,42731,42769,42771,42781,42783,42813],{},[898,42677,42678,42684],{},[45,42679,42680,42682],{},[48,42681,2533],{"mathvariant":111},[48,42683,12715],{"mathvariant":111},[2581,42685,42686],{},"12",[51,42688,69],{},[71,42690,42691,42709],{},[45,42692,42693,42695,42697,42699,42701,42707],{},[48,42694,11],{},[51,42696,54],{"stretchy":53},[48,42698,63],{},[51,42700,60],{},[898,42702,42703,42705],{},[48,42704,34732],{},[2581,42706,4435],{},[51,42708,66],{"stretchy":53},[45,42710,42711,42713,42715,42717,42719,42725],{},[48,42712,11],{},[51,42714,54],{"stretchy":53},[48,42716,63],{},[51,42718,60],{},[898,42720,42721,42723],{},[48,42722,34732],{},[2581,42724,9952],{},[51,42726,66],{"stretchy":53},[51,42728,747],{"separator":122},[148,42730],{"width":750},[71,42732,42733,42751],{},[45,42734,42735,42737,42739,42745,42747,42749],{},[48,42736,11],{},[51,42738,54],{"stretchy":53},[898,42740,42741,42743],{},[48,42742,34732],{},[2581,42744,4435],{},[51,42746,60],{},[48,42748,63],{},[51,42750,66],{"stretchy":53},[45,42752,42753,42755,42757,42763,42765,42767],{},[48,42754,11],{},[51,42756,54],{"stretchy":53},[898,42758,42759,42761],{},[48,42760,34732],{},[2581,42762,9952],{},[51,42764,60],{},[48,42766,63],{},[51,42768,66],{"stretchy":53},[51,42770,69],{},[898,42772,42773,42779],{},[45,42774,42775,42777],{},[48,42776,2533],{"mathvariant":111},[48,42778,12715],{"mathvariant":111},[2581,42780,42686],{},[51,42782,5758],{},[71,42784,42785,42799],{},[45,42786,42787,42789,42791,42797],{},[48,42788,11],{},[51,42790,54],{"stretchy":53},[898,42792,42793,42795],{},[48,42794,34732],{},[2581,42796,4435],{},[51,42798,66],{"stretchy":53},[45,42800,42801,42803,42805,42811],{},[48,42802,11],{},[51,42804,54],{"stretchy":53},[898,42806,42807,42809],{},[48,42808,34732],{},[2581,42810,9952],{},[51,42812,66],{"stretchy":53},[48,42814,112],{"mathvariant":111},[114,42816,42817],{"encoding":116},"\\mathrm{BF}_{12}=\\frac{p(D\\mid M_1)}{p(D\\mid M_2)},\n\\qquad\n\\frac{p(M_1\\mid D)}{p(M_2\\mid D)}\n=\\mathrm{BF}_{12}\\cdot\\frac{p(M_1)}{p(M_2)}.",[23,42819,42821,42884,43265,43326],{"className":42820,"ariaHidden":122},[121],[23,42822,42824,42828,42875,42878,42881],{"className":42823},[126],[23,42825],{"className":42826,"style":42827},[130],"height:0.8333em;vertical-align:-0.15em;",[23,42829,42831,42838],{"className":42830},[135],[23,42832,42834],{"className":42833},[135],[23,42835,42837],{"className":42836},[135,5391],"BF",[23,42839,42841],{"className":42840},[998],[23,42842,42844,42867],{"className":42843},[199,200],[23,42845,42847,42864],{"className":42846},[204],[23,42848,42850],{"className":42849,"style":18647},[208],[23,42851,42852,42855],{"style":8187},[23,42853],{"className":42854,"style":1015},[216],[23,42856,42858],{"className":42857},[1019,1020,1021,1022],[23,42859,42861],{"className":42860},[135,1022],[23,42862,42686],{"className":42863},[135,1022],[23,42865,297],{"className":42866},[296],[23,42868,42870],{"className":42869},[204],[23,42871,42873],{"className":42872,"style":2048},[208],[23,42874],{},[23,42876],{"className":42877,"style":149},[148],[23,42879,69],{"className":42880},[153],[23,42882],{"className":42883,"style":149},[148],[23,42885,42887,42890,43069,43072,43075,43078,43256,43259,43262],{"className":42886},[126],[23,42888],{"className":42889,"style":185},[130],[23,42891,42893,42896,43066],{"className":42892},[135],[23,42894],{"className":42895},[140,192],[23,42897,42899],{"className":42898},[71],[23,42900,42902,43058],{"className":42901},[199,200],[23,42903,42905,43055],{"className":42904},[204],[23,42906,42908,42978,42986],{"className":42907,"style":209},[208],[23,42909,42910,42913],{"style":212},[23,42911],{"className":42912,"style":217},[216],[23,42914,42916,42919,42922,42925,42928,42931,42934,42975],{"className":42915},[135],[23,42917,11],{"className":42918},[135,136],[23,42920,54],{"className":42921},[140],[23,42923,63],{"className":42924,"style":144},[135,136],[23,42926],{"className":42927,"style":149},[148],[23,42929,60],{"className":42930},[153],[23,42932],{"className":42933,"style":149},[148],[23,42935,42937,42940],{"className":42936},[135],[23,42938,34732],{"className":42939,"style":42546},[135,136],[23,42941,42943],{"className":42942},[998],[23,42944,42946,42967],{"className":42945},[199,200],[23,42947,42949,42964],{"className":42948},[204],[23,42950,42952],{"className":42951,"style":18647},[208],[23,42953,42955,42958],{"style":42954},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[23,42956],{"className":42957,"style":1015},[216],[23,42959,42961],{"className":42960},[1019,1020,1021,1022],[23,42962,9952],{"className":42963},[135,1022],[23,42965,297],{"className":42966},[296],[23,42968,42970],{"className":42969},[204],[23,42971,42973],{"className":42972,"style":2048},[208],[23,42974],{},[23,42976,66],{"className":42977},[169],[23,42979,42980,42983],{"style":235},[23,42981],{"className":42982,"style":217},[216],[23,42984],{"className":42985,"style":243},[242],[23,42987,42988,42991],{"style":246},[23,42989],{"className":42990,"style":217},[216],[23,42992,42994,42997,43000,43003,43006,43009,43012,43052],{"className":42993},[135],[23,42995,11],{"className":42996},[135,136],[23,42998,54],{"className":42999},[140],[23,43001,63],{"className":43002,"style":144},[135,136],[23,43004],{"className":43005,"style":149},[148],[23,43007,60],{"className":43008},[153],[23,43010],{"className":43011,"style":149},[148],[23,43013,43015,43018],{"className":43014},[135],[23,43016,34732],{"className":43017,"style":42546},[135,136],[23,43019,43021],{"className":43020},[998],[23,43022,43024,43044],{"className":43023},[199,200],[23,43025,43027,43041],{"className":43026},[204],[23,43028,43030],{"className":43029,"style":18647},[208],[23,43031,43032,43035],{"style":42954},[23,43033],{"className":43034,"style":1015},[216],[23,43036,43038],{"className":43037},[1019,1020,1021,1022],[23,43039,4435],{"className":43040},[135,1022],[23,43042,297],{"className":43043},[296],[23,43045,43047],{"className":43046},[204],[23,43048,43050],{"className":43049,"style":2048},[208],[23,43051],{},[23,43053,66],{"className":43054},[169],[23,43056,297],{"className":43057},[296],[23,43059,43061],{"className":43060},[204],[23,43062,43064],{"className":43063,"style":304},[208],[23,43065],{},[23,43067],{"className":43068},[169,192],[23,43070,747],{"className":43071},[824],[23,43073],{"className":43074,"style":828},[148],[23,43076],{"className":43077,"style":280},[148],[23,43079,43081,43084,43253],{"className":43080},[135],[23,43082],{"className":43083},[140,192],[23,43085,43087],{"className":43086},[71],[23,43088,43090,43245],{"className":43089},[199,200],[23,43091,43093,43242],{"className":43092},[204],[23,43094,43096,43165,43173],{"className":43095,"style":209},[208],[23,43097,43098,43101],{"style":212},[23,43099],{"className":43100,"style":217},[216],[23,43102,43104,43107,43110,43150,43153,43156,43159,43162],{"className":43103},[135],[23,43105,11],{"className":43106},[135,136],[23,43108,54],{"className":43109}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416,43417,43420],{"style":246},[23,43418],{"className":43419,"style":217},[216],[23,43421,43423,43426,43429,43469],{"className":43422},[135],[23,43424,11],{"className":43425},[135,136],[23,43427,54],{"className":43428},[140],[23,43430,43432,43435],{"className":43431},[135],[23,43433,34732],{"className":43434,"style":42546},[135,136],[23,43436,43438],{"className":43437},[998],[23,43439,43441,43461],{"className":43440},[199,200],[23,43442,43444,43458],{"className":43443},[204],[23,43445,43447],{"className":43446,"style":18647},[208],[23,43448,43449,43452],{"style":42954},[23,43450],{"className":43451,"style":1015},[216],[23,43453,43455],{"className":43454},[1019,1020,1021,1022],[23,43456,4435],{"className":43457},[135,1022],[23,43459,297],{"className":43460},[296],[23,43462,43464],{"className":43463},[204],[23,43465,43467],{"className":43466,"style":2048},[208],[23,43468],{},[23,43470,66],{"className":43471},[169],[23,43473,297],{"className":43474},[296],[23,43476,43478],{"className":43477},[204],[23,43479,43481],{"className":43480,"style":304},[208],[23,43482],{},[23,43484],{"className":43485},[169,192],[23,43487,112],{"className":43488},[135],[11,43490,43491,43492,43552,43553,34437],{},"边缘似然自动带有奥卡姆惩罚。用 Laplace 近似可以看出这一点。设 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M)\n\\approx\np(D\\mid\\hat\\theta)\\,p(\\hat\\theta\\mid M)\\,(2\\pi)^{d\u002F2}|H|^{-1\u002F2}.",[23,43979,43981,44005,44026,44050,44139],{"className":43980,"ariaHidden":122},[121],[23,43982,43984,43987,43990,43993,43996,43999,44002],{"className":43983},[126],[23,43985],{"className":43986,"style":131},[130],[23,43988,11],{"className":43989},[135,136],[23,43991,54],{"className":43992},[140],[23,43994,63],{"className":43995,"style":144},[135,136],[23,43997],{"className":43998,"style":149},[148],[23,44000,60],{"className":44001},[153],[23,44003],{"className":44004,"style":149},[148],[23,44006,44008,44011,44014,44017,44020,44023],{"className":44007},[126],[23,44009],{"className":44010,"style":131},[130],[23,44012,34732],{"className":44013,"style":42546},[135,136],[23,44015,66],{"className":44016},[169],[23,44018],{"className":44019,"style":149},[148],[23,44021,29066],{"className":44022},[153],[23,44024],{"className":44025,"style":149},[148],[23,44027,44029,44032,44035,44038,44041,44044,44047],{"className":44028},[126],[23,44030],{"className":44031,"style":131},[130],[23,44033,11],{"className":44034},[135,136],[23,44036,54],{"className":44037},[140],[23,44039,63],{"className":44040,"style":144},[135,136],[23,44042],{"className":44043,"style":149},[148],[23,44045,60],{"className":44046},[153],[23,44048],{"className":44049,"style":149},[148],[23,44051,44053,44056,44087,44090,44093,44096,44099,44130,44133,44136],{"className":44052},[126],[23,44054],{"className":44055,"style":16966},[130],[23,44057,44059],{"className":44058},[135,16830],[23,44060,44062],{"className":44061},[199],[23,44063,44065],{"className":44064},[204],[23,44066,44068,44076],{"className":44067,"style":16840},[208],[23,44069,44070,44073],{"style":7130},[23,44071],{"className":44072,"style":217},[216],[23,44074,57],{"className":44075,"style":144},[135,136],[23,44077,44078,44081],{"style":16851},[23,44079],{"className":44080,"style":217},[216],[23,44082,44084],{"className":44083,"style":34865},[16858],[23,44085,16813],{"className":44086},[135],[23,44088,66],{"className":44089},[169],[23,44091],{"className":44092,"style":280},[148],[23,44094,11],{"className":44095},[135,136],[23,44097,54],{"className":44098},[140],[23,44100,44102],{"className":44101},[135,16830],[23,44103,44105],{"className":44104},[199],[23,44106,44108],{"className":44107},[204],[23,44109,44111,44119],{"className":44110,"style":16840},[208],[23,44112,44113,44116],{"style":7130},[23,44114],{"className":44115,"style":217},[216],[23,44117,57],{"className":44118,"style":144},[135,136],[23,44120,44121,44124],{"style":16851},[23,44122],{"className":44123,"style":217},[216],[23,44125,44127],{"className":44126,"style":34865},[16858],[23,44128,16813],{"className":44129},[135],[23,44131],{"className":44132,"style":149},[148],[23,44134,60],{"className":44135},[153],[23,44137],{"className":44138,"style":149},[148],[23,44140,44142,44146,44149,44152,44155,44158,44161,44164,44200,44203,44206,44241],{"className":44141},[126],[23,44143],{"className":44144,"style":44145},[130],"height:1.188em;vertical-align:-0.25em;",[23,44147,34732],{"className":44148,"style":42546},[135,136],[23,44150,66],{"className":44151},[169],[23,44153],{"className":44154,"style":280},[148],[23,44156,54],{"className":44157},[140],[23,44159,9952],{"className":44160},[135],[23,44162,21693],{"className":44163,"style":676},[135,136],[23,44165,44167,44170],{"className":44166},[169],[23,44168,66],{"className":44169},[169],[23,44171,44173],{"className":44172},[998],[23,44174,44176],{"className":44175},[199],[23,44177,44179],{"className":44178},[204],[23,44180,44183],{"className":44181,"style":44182},[208],"height:0.938em;",[23,44184,44185,44188],{"style":4989},[23,44186],{"className":44187,"style":1015},[216],[23,44189,44191],{"className":44190},[1019,1020,1021,1022],[23,44192,44194,44197],{"className":44193},[135,1022],[23,44195,1606],{"className":44196},[135,136,1022],[23,44198,20560],{"className":44199},[135,1022],[23,44201,60],{"className":44202},[135],[23,44204,26249],{"className":44205,"style":26503},[135,136],[23,44207,44209,44212],{"className":44208},[135],[23,44210,60],{"className":44211},[135],[23,44213,44215],{"className":44214},[998],[23,44216,44218],{"className":44217},[199],[23,44219,44221],{"className":44220},[204],[23,44222,44224],{"className":44223,"style":44182},[208],[23,44225,44226,44229],{"style":4989},[23,44227],{"className":44228,"style":1015},[216],[23,44230,44232],{"className":44231},[1019,1020,1021,1022],[23,44233,44235,44238],{"className":44234},[135,1022],[23,44236,4529],{"className":44237},[135,1022],[23,44239,10469],{"className":44240},[135,1022],[23,44242,112],{"className":44243},[135],[11,44245,44246,44247,44350],{},"在正则模型里 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的主项。参数越多，惩罚越重；但更准确地说，惩罚的不是维度本身，而是参数空间中「能给出不同预测」的有效体积。一个参数如果对预测没有影响，它就不会贡献体积，也就不会被惩罚。",[11,44980,44981,44982,1471],{},"边缘似然也有它自己的问题：它对先验敏感。一个过分分散的先验会把概率质量摊到大量「几乎不预测数据」的区域，导致模型被过度惩罚。这就是 Lindley 悖论的一个侧面：当原假设是一个点、备择假设是一个很宽的分布时，贝叶斯因子可能强烈支持原假设，即使频率派的检验已经拒绝了它。",[18,44983,44984],{},"贝叶斯方法不会自动给出「正确」的模型选择，它只是把选择的标准写清楚了",[338,44986,44988],{"id":44987},"_7-机器学习理论中的位置","7. 机器学习理论中的位置",[2501,44990,44992],{"id":44991},"_71-风险分解","7.1 风险分解",[11,44994,4410,44995,45025,45026,8055,45208,45270],{},[23,44996,44998,45012],{"className":44997,"translate":27},[31],[23,44999,45001],{"className":45000},[35],[37,45002,45003],{"xmlns":39},[42,45004,45005,45009],{},[45,45006,45007],{},[48,45008,26249],{"mathvariant":375},[114,45010,45011],{"encoding":116},"\\mathcal H",[23,45013,45015],{"className":45014,"ariaHidden":122},[121],[23,45016,45018,45021],{"className":45017},[126],[23,45019],{"className":45020,"style":470},[130],[23,45022,26249],{"className":45023,"style":45024},[135,412],"margin-right:0.0097em;"," 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是从数据中学到的模型。则",[23,45272,45274],{"className":45273,"translate":27},[26],[23,45275,45277,45379],{"className":45276,"translate":27},[31],[23,45278,45280],{"className":45279},[35],[37,45281,45282],{"xmlns":39,"display":40},[42,45283,45284,45376],{},[45,45285,45286,45288,45290,45296,45298,45300,45306,45308,45337,45339,45374],{},[48,45287,572],{},[51,45289,54],{"stretchy":53},[16807,45291,45292,45294],{"accent":122},[48,45293,18368],{},[51,45295,16813],{},[51,45297,66],{"stretchy":53},[51,45299,4529],{},[4518,45301,45302,45304],{},[48,45303,572],{},[51,45305,6881],{},[51,45307,69],{},[6961,45309,45310,45334],{},[6961,45311,45312,45332],{},[45,45313,45314,45316,45322,45324,45330],{},[51,45315,54],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[898,45317,45318,45320],{},[48,45319,572],{},[48,45321,26249],{"mathvariant":375},[51,45323,4529],{},[4518,45325,45326,45328],{},[48,45327,572],{},[51,45329,6881],{},[51,45331,66],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,45333,8477],{"stretchy":122},[88,45335,45336],{},"近似误差",[51,45338,5106],{},[6961,45340,45341,45371],{},[6961,45342,45343,45369],{},[45,45344,45345,45347,45349,45351,45357,45359,45361,45367],{},[51,45346,54],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,45348,572],{},[51,45350,54],{"stretchy":53},[16807,45352,45353,45355],{"accent":122},[48,45354,18368],{},[51,45356,16813],{},[51,45358,66],{"stretchy":53},[51,45360,4529],{},[898,45362,45363,45365],{},[48,45364,572],{},[48,45366,26249],{"mathvariant":375},[51,45368,66],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,45370,8477],{"stretchy":122},[88,45372,45373],{},"估计误差",[48,45375,112],{"mathvariant":111},[114,45377,45378],{"encoding":116},"R(\\hat 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",[23,45925,45927,45944],{"className":45926,"translate":27},[31],[23,45928,45930],{"className":45929},[35],[37,45931,45932],{"xmlns":39},[42,45933,45934,45942],{},[45,45935,45936],{},[4518,45937,45938,45940],{},[48,45939,572],{},[51,45941,6881],{},[114,45943,14962],{"encoding":116},[23,45945,45947],{"className":45946,"ariaHidden":122},[121],[23,45948,45950,45953],{"className":45949},[126],[23,45951],{"className":45952,"style":6912},[130],[23,45954,45956,45959],{"className":45955},[135],[23,45957,572],{"className":45958,"style":963},[135,136],[23,45960,45962],{"className":45961},[998],[23,45963,45965],{"className":45964},[199],[23,45966,45968],{"className":45967},[204],[23,45969,45971],{"className":45970,"style":6912},[208],[23,45972,45973,45976],{"style":4696},[23,45974],{"className":45975,"style":1015},[216],[23,45977,45979],{"className":45978},[1019,1020,1021,1022],[23,45980,6881],{"className":45981},[612,1022]," 是这两项的共同起点。它既不属于模型，也不属于算法，而是问题本身的下限。学习理论的大部分内容，都可以理解为在给定 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的情况下，分别研究这两项能压到多低。",[11,46043,46044],{},"这也解释了为什么「更复杂的模型」有时有用、有时有害：它降低近似误差，却可能提高估计误差。深度学习成功的部分原因，是它在大规模数据上把估计误差压得足够低，同时用极宽的模型类把近似误差压得很小。这不是对偏差-方差权衡的否定，而是权衡曲线在数据量增大后的重新定位。",[2501,46046,46048],{"id":46047},"_72-一致性","7.2 一致性",[11,46050,46051,46052,15001,46124,46127,46128,8055],{},"一个决策规则序列 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\\xrightarrow{\\ P\\ }\\ R^*",[23,46215,46217,46357],{"className":46216,"ariaHidden":122},[121],[23,46218,46220,46224,46227,46230,46270,46273,46276,46279,46351,46354],{"className":46219},[126],[23,46221],{"className":46222,"style":46223},[130],"height:1.3503em;vertical-align:-0.25em;",[23,46225,572],{"className":46226,"style":963},[135,136],[23,46228,54],{"className":46229},[140],[23,46231,46233,46236],{"className":46232},[135],[23,46234,736],{"className":46235,"style":783},[135,136],[23,46237,46239],{"className":46238},[998],[23,46240,46242,46262],{"className":46241},[199,200],[23,46243,46245,46259],{"className":46244},[204],[23,46246,46248],{"className":46247,"style":6499},[208],[23,46249,46250,46253],{"style":46103},[23,46251],{"className":46252,"style":1015},[216],[23,46254,46256],{"className":46255},[1019,1020,1021,1022],[23,46257,4806],{"className":46258},[135,136,1022],[23,46260,297],{"className":46261},[296],[23,46263,46265],{"className":46264},[204],[23,46266,46268],{"className":46267,"style":2048},[208],[23,46269],{},[23,46271,66],{"className":46272},[169],[23,46274,5360],{"className":46275},[148],[23,46277],{"className":46278,"style":149},[148],[23,46280,46283],{"className":46281},[153,46282],"x-arrow",[23,46284,46286,46342],{"className":46285},[199,200],[23,46287,46289,46339],{"className":46288},[204],[23,46290,46293,46321],{"className":46291,"style":46292},[208],"height:1.1003em;",[23,46294,46296,46299],{"style":46295},"top:-3.322em;",[23,46297],{"className":46298,"style":1015},[216],[23,46300,46303],{"className":46301},[1019,1020,1021,1022,46302],"x-arrow-pad",[23,46304,46306,46312,46315],{"className":46305},[135,1022],[23,46307,46309],{"className":46308},[148,1022],[23,46310,5360],{"className":46311},[1022],[23,46313,920],{"className":46314,"style":1047},[135,136,1022],[23,46316,46318],{"className":46317},[148,1022],[23,46319,5360],{"className":46320},[1022],[23,46322,46325,46328],{"className":46323,"style":46324},[8672],"top:-2.689em;",[23,46326],{"className":46327,"style":1015},[216],[23,46329,46332],{"className":46330,"style":46331},[25777],"height:0.522em;min-width:1.469em;",[8688,46333,46336],{"xmlns":8690,"width":8691,"height":46334,"viewBox":46335,"preserveAspectRatio":8715},"0.522em","0 0 400000 522",[8696,46337],{"d":46338},"M0 241v40h399891c-47.3 35.3-84 78-110 128\n-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20\n 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7\n 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85\n-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5\n-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67\n 151.7 139 205zm0 0v40h399900v-40z",[23,46340,297],{"className":46341},[296],[23,46343,46345],{"className":46344},[204],[23,46346,46349],{"className":46347,"style":46348},[208],"height:0.011em;",[23,46350],{},[23,46352,5360],{"className":46353},[148],[23,46355],{"className":46356,"style":149},[148],[23,46358,46360,46363],{"className":46359},[126],[23,46361],{"className":46362,"style":7045},[130],[23,46364,46366,46369],{"className":46365},[135],[23,46367,572],{"className":46368,"style":963},[135,136],[23,46370,46372],{"className":46371},[998],[23,46373,46375],{"className":46374},[199],[23,46376,46378],{"className":46377},[204],[23,46379,46381],{"className":46380,"style":7045},[208],[23,46382,46383,46386],{"style":4989},[23,46384],{"className":46385,"style":1015},[216],[23,46387,46389],{"className":46388},[1019,1020,1021,1022],[23,46390,6881],{"className":46391},[612,1022],[11,46393,46394,46395,46397],{},"依概率成立。注意这里收敛的是",[18,46396,870],{},"，不是后验估计。一个模型可以是一致的，但给出的概率完全不校准；反过来，一个校准很好的模型也可能因为模型类太小而无法逼近贝叶斯风险。",[11,46399,46400,46428,46429,46481,46482,46543,46544,46572,46573,46601,46602,46666],{},[23,46401,46403,46416],{"className":46402,"translate":27},[31],[23,46404,46406],{"className":46405},[35],[37,46407,46408],{"xmlns":39},[42,46409,46410,46414],{},[45,46411,46412],{},[48,46413,4836],{},[114,46415,4836],{"encoding":116},[23,46417,46419],{"className":46418,"ariaHidden":122},[121],[23,46420,46422,46425],{"className":46421},[126],[23,46423],{"className":46424,"style":779},[130],[23,46426,4836],{"className":46427,"style":4851},[135,136]," 近邻是最经典的例子。Stone 定理说：若 ",[23,46430,46432,46451],{"className":46431,"translate":27},[31],[23,46433,46435],{"className":46434},[35],[37,46436,46437],{"xmlns":39},[42,46438,46439,46448],{},[45,46440,46441,46443,46445],{},[48,46442,4836],{},[51,46444,568],{},[48,46446,46447],{"mathvariant":111},"∞",[114,46449,46450],{"encoding":116},"k\\to\\infty",[23,46452,46454,46472],{"className":46453,"ariaHidden":122},[121],[23,46455,46457,46460,46463,46466,46469],{"className":46456},[126],[23,46458],{"className":46459,"style":779},[130],[23,46461,4836],{"className":46462,"style":4851},[135,136],[23,46464],{"className":46465,"style":149},[148],[23,46467,568],{"className":46468},[153],[23,46470],{"className":46471,"style":149},[148],[23,46473,46475,46478],{"className":46474},[126],[23,46476],{"className":46477,"style":703},[130],[23,46479,46447],{"className":46480},[135]," 且 ",[23,46483,46485,46507],{"className":46484,"translate":27},[31],[23,46486,46488],{"className":46487},[35],[37,46489,46490],{"xmlns":39},[42,46491,46492,46504],{},[45,46493,46494,46496,46498,46500,46502],{},[48,46495,4836],{},[48,46497,2875],{"mathvariant":111},[48,46499,4806],{},[51,46501,568],{},[2581,46503,1532],{},[114,46505,46506],{"encoding":116},"k\u002Fn\\to0",[23,46508,46510,46534],{"className":46509,"ariaHidden":122},[121],[23,46511,46513,46516,46519,46522,46525,46528,46531],{"className":46512},[126],[23,46514],{"className":46515,"style":131},[130],[23,46517,4836],{"className":46518,"style":4851},[135,136],[23,46520,2875],{"className":46521},[135],[23,46523,4806],{"className":46524},[135,136],[23,46526],{"className":46527,"style":149},[148],[23,46529,568],{"className":46530},[153],[23,46532],{"className":46533,"style":149},[148],[23,46535,46537,46540],{"className":46536},[126],[23,46538],{"className":46539,"style":2770},[130],[23,46541,1532],{"className":46542},[135],"，则 ",[23,46545,46547,46560],{"className":46546,"translate":27},[31],[23,46548,46550],{"className":46549},[35],[37,46551,46552],{"xmlns":39},[42,46553,46554,46558],{},[45,46555,46556],{},[48,46557,4836],{},[114,46559,4836],{"encoding":116},[23,46561,46563],{"className":46562,"ariaHidden":122},[121],[23,46564,46566,46569],{"className":46565},[126],[23,46567],{"className":46568,"style":779},[130],[23,46570,4836],{"className":46571,"style":4851},[135,136],"-NN 是普遍一致的。",[23,46574,46576,46589],{"className":46575,"translate":27},[31],[23,46577,46579],{"className":46578},[35],[37,46580,46581],{"xmlns":39},[42,46582,46583,46587],{},[45,46584,46585],{},[2581,46586,4435],{},[114,46588,4435],{"encoding":116},[23,46590,46592],{"className":46591,"ariaHidden":122},[121],[23,46593,46595,46598],{"className":46594},[126],[23,46596],{"className":46597,"style":2770},[130],[23,46599,4435],{"className":46600},[135],"-NN 更极端：它的渐近错误率不超过 ",[23,46603,46605,46625],{"className":46604,"translate":27},[31],[23,46606,46608],{"className":46607},[35],[37,46609,46610],{"xmlns":39},[42,46611,46612,46622],{},[45,46613,46614,46616],{},[2581,46615,9952],{},[4518,46617,46618,46620],{},[48,46619,572],{},[51,46621,6881],{},[114,46623,46624],{"encoding":116},"2R^*",[23,46626,46628],{"className":46627,"ariaHidden":122},[121],[23,46629,46631,46634,46637],{"className":46630},[126],[23,46632],{"className":46633,"style":6912},[130],[23,46635,9952],{"className":46636},[135],[23,46638,46640,46643],{"className":46639},[135],[23,46641,572],{"className":46642,"style":963},[135,136],[23,46644,46646],{"className":46645},[998],[23,46647,46649],{"className":46648},[199],[23,46650,46652],{"className":46651},[204],[23,46653,46655],{"className":46654,"style":6912},[208],[23,46656,46657,46660],{"style":4696},[23,46658],{"className":46659,"style":1015},[216],[23,46661,46663],{"className":46662},[1019,1020,1021,1022],[23,46664,6881],{"className":46665},[612,1022],"。这个「二倍」的常数很说明问题——即使是最简单的局部方法，也能在渐近意义下接近贝叶斯风险，代价是收敛速度很慢，而且在高维里被维数灾难拖垮。",[11,46668,46669,46670,46756],{},"一致性是一个渐近性质，它不告诉你有限样本下会发生什么。一个普遍一致的估计器可能在 ",[23,46671,46673,46697],{"className":46672,"translate":27},[31],[23,46674,46676],{"className":46675},[35],[37,46677,46678],{"xmlns":39},[42,46679,46680,46694],{},[45,46681,46682,46684,46686],{},[48,46683,4806],{},[51,46685,69],{},[4518,46687,46688,46691],{},[2581,46689,46690],{},"10",[2581,46692,46693],{},"6",[114,46695,46696],{"encoding":116},"n=10^6",[23,46698,46700,46718],{"className":46699,"ariaHidden":122},[121],[23,46701,46703,46706,46709,46712,46715],{"className":46702},[126],[23,46704],{"className":46705,"style":703},[130],[23,46707,4806],{"className":46708},[135,136],[23,46710],{"className":46711,"style":149},[148],[23,46713,69],{"className":46714},[153],[23,46716],{"className":46717,"style":149},[148],[23,46719,46721,46724,46727],{"className":46720},[126],[23,46722],{"className":46723,"style":4693},[130],[23,46725,4435],{"className":46726},[135],[23,46728,46730,46733],{"className":46729},[135],[23,46731,1532],{"className":46732},[135],[23,46734,46736],{"className":46735},[998],[23,46737,46739],{"className":46738},[199],[23,46740,46742],{"className":46741},[204],[23,46743,46745],{"className":46744,"style":4693},[208],[23,46746,46747,46750],{"style":4696},[23,46748],{"className":46749,"style":1015},[216],[23,46751,46753],{"className":46752},[1019,1020,1021,1022],[23,46754,46693],{"className":46755},[135,1022]," 时仍然表现很差；一个不一致的估计器可能在某个特定问题上表现得很好。贝叶斯风险给出的是终点，不是路线。",[2501,46758,46760],{"id":46759},"_73-pac-bayes把信念放进泛化界","7.3 PAC-Bayes：把「信念」放进泛化界",[11,46762,46763,46764,46806],{},"PAC-Bayes 把贝叶斯式的分布对象与频率式的泛化保证放在同一个不等式里。这里的 ",[23,46765,46767,46785],{"className":46766,"translate":27},[31],[23,46768,46770],{"className":46769},[35],[37,46771,46772],{"xmlns":39},[42,46773,46774,46782],{},[45,46775,46776,46778,46780],{},[48,46777,920],{},[51,46779,747],{"separator":122},[48,46781,22051],{},[114,46783,46784],{"encoding":116},"P,Q",[23,46786,46788],{"className":46787,"ariaHidden":122},[121],[23,46789,46791,46794,46797,46800,46803],{"className":46790},[126],[23,46792],{"className":46793,"style":817},[130],[23,46795,920],{"className":46796,"style":1047},[135,136],[23,46798,747],{"className":46799},[824],[23,46801],{"className":46802,"style":280},[148],[23,46804,22051],{"className":46805},[135,136]," 是假设空间上的分布，不要与数据分布混淆。",[6857,46808,46810,47051,47640],{"title":46809},"定理 7.1（McAllester 界）",[11,46811,46812,46813,46841,46842,46894,46895,46969,46970,47021,47022,47050],{},"设先验 ",[23,46814,46816,46829],{"className":46815,"translate":27},[31],[23,46817,46819],{"className":46818},[35],[37,46820,46821],{"xmlns":39},[42,46822,46823,46827],{},[45,46824,46825],{},[48,46826,920],{},[114,46828,920],{"encoding":116},[23,46830,46832],{"className":46831,"ariaHidden":122},[121],[23,46833,46835,46838],{"className":46834},[126],[23,46836],{"className":46837,"style":470},[130],[23,46839,920],{"className":46840,"style":1047},[135,136]," 在看到数据前固定，损失取值于 ",[23,46843,46845,46867],{"className":46844,"translate":27},[31],[23,46846,46848],{"className":46847},[35],[37,46849,46850],{"xmlns":39},[42,46851,46852,46864],{},[45,46853,46854,46856,46858,46860,46862],{},[51,46855,924],{"stretchy":53},[2581,46857,1532],{},[51,46859,747],{"separator":122},[2581,46861,4435],{},[51,46863,945],{"stretchy":53},[114,46865,46866],{"encoding":116},"[0,1]",[23,46868,46870],{"className":46869,"ariaHidden":122},[121],[23,46871,46873,46876,46879,46882,46885,46888,46891],{"className":46872},[126],[23,46874],{"className":46875,"style":131},[130],[23,46877,924],{"className":46878},[140],[23,46880,1532],{"className":46881},[135],[23,46883,747],{"className":46884},[824],[23,46886],{"className":46887,"style":280},[148],[23,46889,4435],{"className":46890},[135],[23,46892,945],{"className":46893},[169],"。对任意 ",[23,46896,46898,46924],{"className":46897,"translate":27},[31],[23,46899,46901],{"className":46900},[35],[37,46902,46903],{"xmlns":39},[42,46904,46905,46921],{},[45,46906,46907,46909,46911,46913,46915,46917,46919],{},[48,46908,736],{},[51,46910,372],{},[51,46912,54],{"stretchy":53},[2581,46914,1532],{},[51,46916,747],{"separator":122},[2581,46918,4435],{},[51,46920,66],{"stretchy":53},[114,46922,46923],{"encoding":116},"\\delta\\in(0,1)",[23,46925,46927,46945],{"className":46926,"ariaHidden":122},[121],[23,46928,46930,46933,46936,46939,46942],{"className":46929},[126],[23,46931],{"className":46932,"style":4450},[130],[23,46934,736],{"className":46935,"style":783},[135,136],[23,46937],{"className":46938,"style":149},[148],[23,46940,372],{"className":46941},[153],[23,46943],{"className":46944,"style":149},[148],[23,46946,46948,46951,46954,46957,46960,46963,46966],{"className":46947},[126],[23,46949],{"className":46950,"style":131},[130],[23,46952,54],{"className":46953},[140],[23,46955,1532],{"className":46956},[135],[23,46958,747],{"className":46959},[824],[23,46961],{"className":46962,"style":280},[148],[23,46964,4435],{"className":46965},[135],[23,46967,66],{"className":46968},[169],"，以至少 ",[23,46971,46973,46991],{"className":46972,"translate":27},[31],[23,46974,46976],{"className":46975},[35],[37,46977,46978],{"xmlns":39},[42,46979,46980,46988],{},[45,46981,46982,46984,46986],{},[2581,46983,4435],{},[51,46985,4529],{},[48,46987,736],{},[114,46989,46990],{"encoding":116},"1-\\delta",[23,46992,46994,47012],{"className":46993,"ariaHidden":122},[121],[23,46995,46997,47000,47003,47006,47009],{"className":46996},[126],[23,46998],{"className":46999,"style":9400},[130],[23,47001,4435],{"className":47002},[135],[23,47004],{"className":47005,"style":392},[148],[23,47007,4529],{"className":47008},[612],[23,47010],{"className":47011,"style":392},[148],[23,47013,47015,47018],{"className":47014},[126],[23,47016],{"className":47017,"style":779},[130],[23,47019,736],{"className":47020,"style":783},[135,136]," 的概率，对所有后验分布 ",[23,47023,47025,47038],{"className":47024,"translate":27},[31],[23,47026,47028],{"className":47027},[35],[37,47029,47030],{"xmlns":39},[42,47031,47032,47036],{},[45,47033,47034],{},[48,47035,22051],{},[114,47037,22051],{"encoding":116},[23,47039,47041],{"className":47040,"ariaHidden":122},[121],[23,47042,47044,47047],{"className":47043},[126],[23,47045],{"className":47046,"style":817},[130],[23,47048,22051],{"className":47049},[135,136]," 同时成立：",[23,47052,47054],{"className":47053,"translate":27},[26],[23,47055,47057,47179],{"className":47056,"translate":27},[31],[23,47058,47060],{"className":47059},[35],[37,47061,47062],{"xmlns":39,"display":40},[42,47063,47064,47176],{},[45,47065,47066,47078,47080,47082,47084,47086,47088,47090,47092,47104,47106,47116,47118,47120,47122,47124,47126,47174],{},[898,47067,47068,47070],{},[48,47069,902],{"mathvariant":571},[45,47071,47072,47074,47076],{},[48,47073,18368],{},[51,47075,917],{},[48,47077,22051],{},[51,47079,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,47081,572],{},[51,47083,54],{"stretchy":53},[48,47085,18368],{},[51,47087,66],{"stretchy":53},[51,47089,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,47091,12268],{},[898,47093,47094,47096],{},[48,47095,902],{"mathvariant":571},[45,47097,47098,47100,47102],{},[48,47099,18368],{},[51,47101,917],{},[48,47103,22051],{},[51,47105,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[898,47107,47108,47114],{},[16807,47109,47110,47112],{"accent":122},[48,47111,572],{},[51,47113,16813],{},[48,47115,4806],{},[51,47117,54],{"stretchy":53},[48,47119,18368],{},[51,47121,66],{"stretchy":53},[51,47123,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,47125,5106],{},[25516,47127,47128],{},[71,47129,47130,47168],{},[45,47131,47132,47138,47140,47142,47144,47146,47148,47150,47152,47154,47156,47158,47162,47164,47166],{},[45,47133,47134,47136],{},[48,47135,26918],{"mathvariant":111},[48,47137,555],{"mathvariant":111},[51,47139,54],{"stretchy":53},[48,47141,22051],{},[48,47143,20680],{"mathvariant":111},[48,47145,920],{},[51,47147,66],{"stretchy":53},[51,47149,5106],{},[48,47151,22418],{},[51,47153,6959],{},[51,47155,54],{"stretchy":53},[2581,47157,9952],{},[25516,47159,47160],{},[48,47161,4806],{},[48,47163,2875],{"mathvariant":111},[48,47165,736],{},[51,47167,66],{"stretchy":53},[45,47169,47170,47172],{},[2581,47171,9952],{},[48,47173,4806],{},[51,47175,747],{"separator":122},[114,47177,47178],{"encoding":116},"\\mathbb E_{h\\sim Q}\\big[R(h)\\big]\n\\le\n\\mathbb E_{h\\sim Q}\\big[\\hat R_n(h)\\big]\n+\n\\sqrt{\\frac{\\mathrm{KL}(Q\\|P)+\\log(2\\sqrt n\u002F\\delta)}{2n}},",[23,47180,47182,47270,47426],{"className":47181,"ariaHidden":122},[121],[23,47183,47185,47188,47237,47243,47246,47249,47252,47255,47261,47264,47267],{"className":47184},[126],[23,47186],{"className":47187,"style":2213},[130],[23,47189,47191,47194],{"className":47190},[135],[23,47192,902],{"className":47193},[135,644],[23,47195,47197],{"className":47196},[998],[23,47198,47200,47229],{"className":47199},[199,200],[23,47201,47203,47226],{"className":47202},[204],[23,47204,47206],{"className":47205,"style":34192},[208],[23,47207,47208,47211],{"style":2029},[23,47209],{"className":47210,"style":1015},[216],[23,47212,47214],{"className":47213},[1019,1020,1021,1022],[23,47215,47217,47220,47223],{"className":47216},[135,1022],[23,47218,18368],{"className":47219},[135,136,1022],[23,47221,917],{"className":47222},[153,1022],[23,47224,22051],{"className":47225},[135,136,1022],[23,47227,297],{"className":47228},[296],[23,47230,47232],{"className":47231},[204],[23,47233,47235],{"className":47234,"style":14292},[208],[23,47236],{},[23,47238,47240],{"className":47239},[135],[23,47241,924],{"className":47242},[1066,1067],[23,47244,572],{"className":47245,"style":963},[135,136],[23,47247,54],{"className":47248},[140],[23,47250,18368],{"className":47251},[135,136],[23,47253,66],{"className":47254},[169],[23,47256,47258],{"className":47257},[135],[23,47259,945],{"className":47260},[1066,1067],[23,47262],{"className":47263,"style":149},[148],[23,47265,12268],{"className":47266},[153],[23,47268],{"className":47269,"style":149},[148],[23,47271,47273,47277,47326,47332,47402,47405,47408,47411,47417,47420,47423],{"className":47272},[126],[23,47274],{"className":47275,"style":47276},[130],"height:1.2968em;vertical-align:-0.35em;",[23,47278,47280,47283],{"className":47279},[135],[23,47281,902],{"className":47282},[135,644],[23,47284,47286],{"className":47285},[998],[23,47287,47289,47318],{"className":47288},[199,200],[23,47290,47292,47315],{"className":47291},[204],[23,47293,47295],{"className":47294,"style":34192},[208],[23,47296,47297,47300],{"style":2029},[23,47298],{"className":47299,"style":1015},[216],[23,47301,47303],{"className":47302},[1019,1020,1021,1022],[23,47304,47306,47309,47312],{"className":47305},[135,1022],[23,47307,18368],{"className":47308},[135,136,1022],[23,47310,917],{"className":47311},[153,1022],[23,47313,22051],{"className":47314},[135,136,1022],[23,47316,297],{"className":47317},[296],[23,47319,47321],{"className":47320},[204],[23,47322,47324],{"className":47323,"style":14292},[208],[23,47325],{},[23,47327,47329],{"className":47328},[135],[23,47330,924],{"className":47331},[1066,1067],[23,47333,47335,47368],{"className":47334},[135],[23,47336,47338],{"className":47337},[135,16830],[23,47339,47341],{"className":47340},[199],[23,47342,47344],{"className":47343},[204],[23,47345,47348,47356],{"className":47346,"style":47347},[208],"height:0.9468em;",[23,47349,47350,47353],{"style":7130},[23,47351],{"className":47352,"style":217},[216],[23,47354,572],{"className":47355,"style":963},[135,136],[23,47357,47359,47362],{"style":47358},"top:-3.2523em;",[23,47360],{"className":47361,"style":217},[216],[23,47363,47365],{"className":47364,"style":34865},[16858],[23,47366,16813],{"className":47367},[135],[23,47369,47371],{"className":47370},[998],[23,47372,47374,47394],{"className":47373},[199,200],[23,47375,47377,47391],{"className":47376},[204],[23,47378,47380],{"className":47379,"style":6499},[208],[23,47381,47382,47385],{"style":45102},[23,47383],{"className":47384,"style":1015},[216],[23,47386,47388],{"className":47387},[1019,1020,1021,1022],[23,47389,4806],{"className":47390},[135,136,1022],[23,47392,297],{"className":47393},[296],[23,47395,47397],{"className":47396},[204],[23,47398,47400],{"className":47399,"style":2048},[208],[23,47401],{},[23,47403,54],{"className":47404},[140],[23,47406,18368],{"className":47407},[135,136],[23,47409,66],{"className":47410},[169],[23,47412,47414],{"className":47413},[135],[23,47415,945],{"className":47416},[1066,1067],[23,47418],{"className":47419,"style":392},[148],[23,47421,5106],{"className":47422},[612],[23,47424],{"className":47425,"style":392},[148],[23,47427,47429,47433,47637],{"className":47428},[126],[23,47430],{"className":47431,"style":47432},[130],"height:2.44em;vertical-align:-0.7105em;",[23,47434,47436],{"className":47435},[135,25667],[23,47437,47439,47628],{"className":47438},[199,200],[23,47440,47442,47625],{"className":47441},[204],[23,47443,47446,47612],{"className":47444,"style":47445},[208],"height:1.7295em;",[23,47447,47449,47452],{"className":47448,"style":27786},[8672],[23,47450],{"className":47451,"style":27790},[216],[23,47453,47455],{"className":47454,"style":27794},[135],[23,47456,47458,47461,47609],{"className":47457},[135],[23,47459],{"className":47460},[140,192],[23,47462,47464],{"className":47463},[71],[23,47465,47467,47601],{"className":47466},[199,200],[23,47468,47470,47598],{"className":47469},[204],[23,47471,47474,47488,4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R_n",[23,47668,47670],{"className":47669,"ariaHidden":122},[121],[23,47671,47673,47677],{"className":47672},[126],[23,47674],{"className":47675,"style":47676},[130],"height:1.0968em;vertical-align:-0.15em;",[23,47678,47680,47711],{"className":47679},[135],[23,47681,47683],{"className":47682},[135,16830],[23,47684,47686],{"className":47685},[199],[23,47687,47689],{"className":47688},[204],[23,47690,47692,47700],{"className":47691,"style":47347},[208],[23,47693,47694,47697],{"style":7130},[23,47695],{"className":47696,"style":217},[216],[23,47698,572],{"className":47699,"style":963},[135,136],[23,47701,47702,47705],{"style":47358},[23,47703],{"className":47704,"style":217},[216],[23,47706,47708],{"className":47707,"style":34865},[16858],[23,47709,16813],{"className":47710},[135],[23,47712,47714],{"className":47713},[998],[23,47715,47717,47737],{"className":47716},[199,200],[23,47718,47720,47734],{"className":47719},[204],[23,47721,47723],{"className":47722,"style":6499},[208],[23,47724,47725,47728],{"style":45102},[23,47726],{"className":47727,"style":1015},[216],[23,47729,47731],{"className":47730},[1019,1020,1021,1022],[23,47732,4806],{"className":47733},[135,136,1022],[23,47735,297],{"className":47736},[296],[23,47738,47740],{"className":47739},[204],[23,47741,47743],{"className":47742,"style":2048},[208],[23,47744],{}," 是经验风险。",[11,47747,47748,47749,47777,47778,47806],{},"这个界的美妙之处在于：",[23,47750,47752,47765],{"className":47751,"translate":27},[31],[23,47753,47755],{"className":47754},[35],[37,47756,47757],{"xmlns":39},[42,47758,47759,47763],{},[45,47760,47761],{},[48,47762,920],{},[114,47764,920],{"encoding":116},[23,47766,47768],{"className":47767,"ariaHidden":122},[121],[23,47769,47771,47774],{"className":47770},[126],[23,47772],{"className":47773,"style":470},[130],[23,47775,920],{"className":47776,"style":1047},[135,136]," 必须在看到数据前固定，",[23,47779,47781,47794],{"className":47780,"translate":27},[31],[23,47782,47784],{"className":47783},[35],[37,47785,47786],{"xmlns":39},[42,47787,47788,47792],{},[45,47789,47790],{},[48,47791,22051],{},[114,47793,22051],{"encoding":116},[23,47795,47797],{"className":47796,"ariaHidden":122},[121],[23,47798,47800,47803],{"className":47799},[126],[23,47801],{"className":47802,"style":817},[130],[23,47804,22051],{"className":47805},[135,136]," 可以是任何依赖于数据的分布。KL 项衡量「为了解释数据，你把信念从先验挪了多远」。先验越集中、后验越贴近先验，界就越紧；后验越远离先验，界就越松。",[11,47808,47809,47810,47813,47814,47842,47843,47871],{},"这正是奥卡姆剃刀的概率版本，只不过它作用在",[18,47811,47812],{},"分布","而不是单个假设上。一个在大量假设上分散的 ",[23,47815,47817,47830],{"className":47816,"translate":27},[31],[23,47818,47820],{"className":47819},[35],[37,47821,47822],{"xmlns":39},[42,47823,47824,47828],{},[45,47825,47826],{},[48,47827,22051],{},[114,47829,22051],{"encoding":116},[23,47831,47833],{"className":47832,"ariaHidden":122},[121],[23,47834,47836,47839],{"className":47835},[126],[23,47837],{"className":47838,"style":817},[130],[23,47840,22051],{"className":47841},[135,136]," 会付出很大的 KL，即使它的经验风险很低；一个集中在少数假设上的 ",[23,47844,47846,47859],{"className":47845,"translate":27},[31],[23,47847,47849],{"className":47848},[35],[37,47850,47851],{"xmlns":39},[42,47852,47853,47857],{},[45,47854,47855],{},[48,47856,22051],{},[114,47858,22051],{"encoding":116},[23,47860,47862],{"className":47861,"ariaHidden":122},[121],[23,47863,47865,47868],{"className":47864},[126],[23,47866],{"className":47867,"style":817},[130],[23,47869,22051],{"className":47870},[135,136]," 则可以用较小的 KL 换取较好的界。",[11,47873,47874],{},"PAC-Bayes 也有它的局限：界在深度网络里常常很松，常数项可能大到没有实际意义。但它提供了一个正确的概念框架：泛化不是「参数少」的同义词，而是「后验没有比先验复杂太多」。后来的 flatness、sharpness、信息论泛化界，都可以在这个框架里找到位置。",[2501,47876,47878],{"id":47877},"_74-变分推断当后验收敛不了","7.4 变分推断：当后验收敛不了",[11,47880,47881,47882,47925,47926,47955],{},"精确后验通常不可行，因为归一化常数 ",[23,47883,47885,47904],{"className":47884,"translate":27},[31],[23,47886,47888],{"className":47887},[35],[37,47889,47890],{"xmlns":39},[42,47891,47892,47902],{},[45,47893,47894,47896,47898,47900],{},[48,47895,11],{},[51,47897,54],{"stretchy":53},[48,47899,63],{},[51,47901,66],{"stretchy":53},[114,47903,3945],{"encoding":116},[23,47905,47907],{"className":47906,"ariaHidden":122},[121],[23,47908,47910,47913,47916,47919,47922],{"className":47909},[126],[23,47911],{"className":47912,"style":131},[130],[23,47914,11],{"className":47915},[135,136],[23,47917,54],{"className":47918},[140],[23,47920,63],{"className":47921,"style":144},[135,136],[23,47923,66],{"className":47924},[169]," 是一个高维积分。变分推断用一个受限制的分布族 ",[23,47927,47929,47943],{"className":47928,"translate":27},[31],[23,47930,47932],{"className":47931},[35],[37,47933,47934],{"xmlns":39},[42,47935,47936,47940],{},[45,47937,47938],{},[48,47939,22051],{"mathvariant":375},[114,47941,47942],{"encoding":116},"\\mathcal Q",[23,47944,47946],{"className":47945,"ariaHidden":122},[121],[23,47947,47949,47952],{"className":47948},[126],[23,47950],{"className":47951,"style":408},[130],[23,47953,22051],{"className":47954},[135,412]," 去逼近后验，目标是",[23,47957,47959],{"className":47958,"translate":27},[26],[23,47960,47962,48042],{"className":47961,"translate":27},[31],[23,47963,47965],{"className":47964},[35],[37,47966,47967],{"xmlns":39,"display":40},[42,47968,47969,48039],{},[45,47970,47971,47977,47979,47981,47983,47999,48001,48007,48009,48011,48013,48015,48017,48019,48021,48023,48025,48027,48029,48031,48033,48035,48037],{},[4518,47972,47973,47975],{},[48,47974,33013],{},[51,47976,6881],{},[51,47978,69],{},[48,47980,6956],{},[51,47982,6959],{},[6961,47984,47985,47991],{},[45,47986,47987,47989],{},[48,47988,6967],{},[51,47990,6959],{},[45,47992,47993,47995,47997],{},[48,47994,33013],{},[51,47996,372],{},[48,47998,22051],{"mathvariant":375},[88,48000,5360],{},[45,48002,48003,48005],{},[48,48004,26918],{"mathvariant":111},[48,48006,555],{"mathvariant":111},[51,48008,54],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,48010,33013],{},[51,48012,54],{"stretchy":53},[48,48014,57],{},[51,48016,66],{"stretchy":53},[88,48018,5360],{},[48,48020,20680],{"mathvariant":111},[88,48022,5360],{},[48,48024,11],{},[51,48026,54],{"stretchy":53},[48,48028,57],{},[51,48030,60],{},[48,48032,63],{},[51,48034,66],{"stretchy":53},[51,48036,66],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,48038,112],{"mathvariant":111},[114,48040,48041],{"encoding":116},"q^*=\\arg\\min_{q\\in\\mathcal Q}\\ \\mathrm{KL}\\big(q(\\theta)\\ \\|\\ p(\\theta\\mid D)\\big).",[23,48043,48045,48090,48215],{"className":48044,"ariaHidden":122},[121],[23,48046,48048,48052,48081,48084,48087],{"className":48047},[126],[23,48049],{"className":48050,"style":48051},[130],"height:0.9331em;vertical-align:-0.1944em;",[23,48053,48055,48058],{"className":48054},[135],[23,48056,33013],{"className":48057,"style":676},[135,136],[23,48059,48061],{"className":48060},[998],[23,48062,48064],{"className":48063},[199],[23,48065,48067],{"className":48066},[204],[23,48068,48070],{"className":48069,"style":7045},[208],[23,48071,48072,48075],{"style":4989},[23,48073],{"className":48074,"style":1015},[216],[23,48076,48078],{"className":48077},[1019,1020,1021,1022],[23,48079,6881],{"className":48080},[612,1022],[23,48082],{"className":48083,"style":149},[148],[23,48085,69],{"className":48086},[153],[23,48088],{"className":48089,"style":149},[148],[23,48091,48093,48097,48102,48105,48158,48161,48164,48170,48176,48179,48182,48185,48188,48191,48194,48197,48200,48203,48206,48209,48212],{"className":48092},[126],[23,48094],{"className":48095,"style":48096},[130],"height:1.7304em;vertical-align:-0.8804em;",[23,48098,7085,48100],{"className":48099},[1786],[23,48101,7089],{"style":7088},[23,48103],{"className":48104,"style":280},[148],[23,48106,48108],{"className":48107},[1786,7096],[23,48109,48111,48150],{"className":48110},[199,200],[23,48112,48114,48147],{"className":48113},[204],[23,48115,48117,48137],{"className":48116,"style":7106},[208],[23,48118,48119,48122],{"style":7109},[23,48120],{"className":48121,"style":217},[216],[23,48123,48125],{"className":48124},[1019,1020,1021,1022],[23,48126,48128,48131,48134],{"className":48127},[135,1022],[23,48129,33013],{"className":48130,"style":676},[135,136,1022],[23,48132,372],{"className":48133},[153,1022],[23,48135,22051],{"className":48136},[135,412,1022],[23,48138,48139,48142],{"style":7130},[23,48140],{"className":48141,"style":217},[216],[23,48143,48144],{},[23,48145,6967],{"className":48146},[1786],[23,48148,297],{"className":48149},[296],[23,48151,48153],{"className":48152},[204],[23,48154,48156],{"className":48155,"style":5683},[208],[23,48157],{},[23,48159,5360],{"className":48160},[148],[23,48162],{"className":48163,"style":280},[148],[23,48165,48167],{"className":48166},[135],[23,48168,27187],{"className":48169},[135,5391],[23,48171,48173],{"className":48172},[135],[23,48174,54],{"className":48175},[1066,1067],[23,48177,33013],{"className":48178,"style":676},[135,136],[23,48180,54],{"className":48181},[140],[23,48183,57],{"className":48184,"style":144},[135,136],[23,48186,66],{"className":48187},[169],[23,48189,5360],{"className":48190},[148],[23,48192,20680],{"className":48193},[135],[23,48195,5360],{"className":48196},[148],[23,48198,11],{"className":48199},[135,136],[23,48201,54],{"className":48202},[140],[23,48204,57],{"className":48205,"style":144},[135,136],[23,48207],{"className":48208,"style":149},[148],[23,48210,60],{"className":48211},[153],[23,48213],{"className":48214,"style":149},[148],[23,48216,48218,48221,48224,48227,48233],{"className":48217},[126],[23,48219],{"className":48220,"style":2213},[130],[23,48222,63],{"className":48223,"style":144},[135,136],[23,48225,66],{"className":48226},[169],[23,48228,48230],{"className":48229},[135],[23,48231,66],{"className":48232},[1066,1067],[23,48234,112],{"className":48235},[135],[11,48237,48238,48239,48282],{},"直接最小化这个 KL 仍然需要 ",[23,48240,48242,48261],{"className":48241,"translate":27},[31],[23,48243,48245],{"className":48244},[35],[37,48246,48247],{"xmlns":39},[42,48248,48249,48259],{},[45,48250,48251,48253,48255,48257],{},[48,48252,11],{},[51,48254,54],{"stretchy":53},[48,48256,63],{},[51,48258,66],{"stretchy":53},[114,48260,3945],{"encoding":116},[23,48262,48264],{"className":48263,"ariaHidden":122},[121],[23,48265,48267,48270,48273,48276,48279],{"className":48266},[126],[23,48268],{"className":48269,"style":131},[130],[23,48271,11],{"className":48272},[135,136],[23,48274,54],{"className":48275},[140],[23,48277,63],{"className":48278,"style":144},[135,136],[23,48280,66],{"className":48281},[169],"，于是利用恒等式",[23,48284,48286],{"className":48285,"translate":27},[26],[23,48287,48289,48423],{"className":48288,"translate":27},[31],[23,48290,48292],{"className":48291},[35],[37,48293,48294],{"xmlns":39,"display":40},[42,48295,48296,48420],{},[45,48297,48298,48300,48302,48304,48306,48308,48310,48312,48380,48382,48388,48390,48392,48394,48396,48398,48400,48402,48404,48406,48408,48410,48412,48414,48416,48418],{},[48,48299,22418],{},[51,48301,6959],{},[48,48303,11],{},[51,48305,54],{"stretchy":53},[48,48307,63],{},[51,48309,66],{"stretchy":53},[51,48311,69],{},[6961,48313,48314,48362],{},[6961,48315,48316,48360],{},[45,48317,48318,48324],{},[898,48319,48320,48322],{},[48,48321,902],{"mathvariant":571},[48,48323,33013],{},[45,48325,48326,48328,48330,48332,48358],{},[51,48327,924],{"fence":122},[48,48329,22418],{},[51,48331,6959],{},[71,48333,48334,48348],{},[45,48335,48336,48338,48340,48342,48344,48346],{},[48,48337,11],{},[51,48339,54],{"stretchy":53},[48,48341,63],{},[51,48343,747],{"separator":122},[48,48345,57],{},[51,48347,66],{"stretchy":53},[45,48349,48350,48352,48354,48356],{},[48,48351,33013],{},[51,48353,54],{"stretchy":53},[48,48355,57],{},[51,48357,66],{"stretchy":53},[51,48359,945],{"fence":122},[51,48361,8477],{"stretchy":122},[45,48363,48364,48374,48376,48378],{},[45,48365,48366,48368,48370,48372],{},[48,48367,902],{"mathvariant":111},[48,48369,555],{"mathvariant":111},[48,48371,2533],{"mathvariant":111},[48,48373,42340],{"mathvariant":111},[51,48375,54],{"stretchy":53},[48,48377,33013],{},[51,48379,66],{"stretchy":53},[51,48381,5106],{},[45,48383,48384,48386],{},[48,48385,26918],{"mathvariant":111},[48,48387,555],{"mathvariant":111},[51,48389,54],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,48391,33013],{},[51,48393,54],{"stretchy":53},[48,48395,57],{},[51,48397,66],{"stretchy":53},[88,48399,5360],{},[48,48401,20680],{"mathvariant":111},[88,48403,5360],{},[48,48405,11],{},[51,48407,54],{"stretchy":53},[48,48409,57],{},[51,48411,60],{},[48,48413,63],{},[51,48415,66],{"stretchy":53},[51,48417,66],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,48419,747],{"separator":122},[114,48421,48422],{"encoding":116},"\\log p(D)\n=\n\\underbrace{\\mathbb E_{q}\\left[\\log\\frac{p(D,\\theta)}{q(\\theta)}\\right]}_{\\mathrm{ELBO}(q)}\n+\n\\mathrm{KL}\\big(q(\\theta)\\ \\|\\ p(\\theta\\mid D)\\big),",[23,48424,48426,48461,48757,48814],{"className":48425,"ariaHidden":122},[121],[23,48427,48429,48432,48437,48440,48443,48446,48449,48452,48455,48458],{"className":48428},[126],[23,48430],{"className":48431,"style":131},[130],[23,48433,22575,48435],{"className":48434},[1786],[23,48436,7089],{"style":7088},[23,48438],{"className":48439,"style":280},[148],[23,48441,11],{"className":48442},[135,136],[23,48444,54],{"className":48445},[140],[23,48447,63],{"className":48448,"style":144},[135,136],[23,48450,66],{"className":48451},[169],[23,48453],{"className":48454,"style":149},[148],[23,48456,69],{"className":48457},[153],[23,48459],{"className":48460,"style":149},[148],[23,48462,48464,48468,48748,48751,48754],{"className":48463},[126],[23,48465],{"className":48466,"style":48467},[130],"height:3.948em;vertical-align:-2.498em;",[23,48469,48471],{"className":48470},[1839,6961],[23,48472,48474,48739],{"className":48473},[199,200],[23,48475,48477,48736],{"className":48476},[204],[23,48478,48481,48509],{"className":48479,"style":48480},[208],"height:1.45em;",[23,48482,48484,48487],{"style":48483},"top:-1.127em;",[23,48485],{"className":48486,"style":1713},[216],[23,48488,48490],{"className":48489},[1019,1020,1021,1022],[23,48491,48493,48500,48503,48506],{"className":48492},[135,1022],[23,48494,48496],{"className":48495},[135,1022],[23,48497,48499],{"className":48498},[135,5391,1022],"ELBO",[23,48501,54],{"className":48502},[140,1022],[23,48504,33013],{"className":48505,"style":676},[135,136,1022],[23,48507,66],{"className":48508},[169,1022],[23,48510,48512,48515],{"style":48511},"top:-3.45em;",[23,48513],{"className":48514,"style":1713},[216],[23,48516,48518],{"className":48517},[1839,6961],[23,48519,48521,48727],{"className":48520},[199,200],[23,48522,48524,48724],{"className":48523},[204],[23,48525,48527,48558],{"className":48526,"style":48480},[208],[23,48528,48531,48534],{"className":48529,"style":48530},[8672],"top:-1.852em;",[23,48532],{"className":48533,"style":1713},[216],[23,48535,48537,48544,48551],{"className":48536,"style":8681},[8680],[23,48538,48540],{"className":48539,"style":8686},[8685],[8688,48541,48542],{"xmlns":8690,"width":8691,"height":8692,"viewBox":8693,"preserveAspectRatio":8694},[8696,48543],{"d":8698},[23,48545,48547],{"className":48546,"style":8686},[8702],[8688,48548,48549],{"xmlns":8690,"width":8691,"height":8692,"viewBox":8693,"preserveAspectRatio":8705},[8696,48550],{"d":8708},[23,48552,48554],{"className":48553,"style":8686},[8712],[8688,48555,48556],{"xmlns":8690,"width":8691,"height":8692,"viewBox":8693,"preserveAspectRatio":8715},[8696,48557],{"d":8718},[23,48559,48560,48563],{"style":48511},[23,48561],{"className":48562,"style":1713},[216],[23,48564,48566,48609,48612],{"className":48565},[135],[23,48567,48569,48572],{"className":48568},[135],[23,48570,902],{"className":48571},[135,644],[23,48573,48575],{"className":48574},[998],[23,48576,48578,48601],{"className":48577},[199,200],[23,48579,48581,48598],{"className":48580},[204],[23,48582,48584],{"className":48583,"style":6499},[208],[23,48585,48586,48589],{"style":2029},[23,48587],{"className":48588,"style":1015},[216],[23,48590,48592],{"className":48591},[1019,1020,1021,1022],[23,48593,48595],{"className":48594},[135,1022],[23,48596,33013],{"className":48597,"style":676},[135,136,1022],[23,48599,297],{"className":48600},[296],[23,48602,48604],{"className":48603},[204],[23,48605,48607],{"className":48606,"style":14292},[208],[23,48608],{},[23,48610],{"className":48611,"style":280},[148],[23,48613,48615,48621,48626,48629,48718],{"className":48614},[1839],[23,48616,48618],{"className":48617,"style":1844},[140,1843],[23,48619,924],{"className":48620},[1066,1021],[23,48622,22575,48624],{"className":48623},[1786],[23,48625,7089],{"style":7088},[23,48627],{"className":48628,"style":280},[148],[23,48630,48632,48635,48715],{"className":48631},[135],[23,48633],{"className":48634},[140,192],[23,48636,48638],{"className":48637},[71],[23,48639,48641,48707],{"className":48640},[199,200],[23,48642,48644,48704],{"className":48643},[204],[23,48645,48647,48667,48675],{"className":48646,"style":209},[208],[23,48648,48649,48652],{"style":212},[23,48650],{"className":48651,"style":217},[216],[23,48653,48655,48658,48661,48664],{"className":48654},[135],[23,48656,33013],{"className":48657,"style":676},[135,136],[23,48659,54],{"className":48660},[140],[23,48662,57],{"className":48663,"style":144},[135,136],[23,48665,66],{"className":48666},[169],[23,48668,48669,48672],{"style":235},[23,48670],{"className":48671,"style":217},[216],[23,48673],{"className":48674,"style":243},[242],[23,48676,48677,48680],{"style":246},[23,48678],{"className":48679,"style":217},[216],[23,48681,48683,48686,48689,48692,48695,48698,48701],{"className":48682},[135],[23,48684,11],{"className":48685},[135,136],[23,48687,54],{"className":48688},[140],[23,48690,63],{"className":48691,"style":144},[135,136],[23,48693,747],{"className":48694},[824],[23,48696],{"className":48697,"style":280},[148],[23,48699,57],{"className":48700,"style":144},[135,136],[23,48702,66],{"className":48703},[169],[23,48705,297],{"className":48706},[296],[23,48708,48710],{"className":48709},[204],[23,48711,48713],{"className":48712,"style":304},[208],[23,48714],{},[23,48716],{"className":48717},[169,192],[23,48719,48721],{"className":48720,"style":1844},[169,1843],[23,48722,945],{"className":48723},[1066,1021],[23,48725,297],{"className":48726},[296],[23,48728,48730],{"className":48729},[204],[23,48731,48734],{"className":48732,"style":48733},[208],"height:1.598em;",[23,48735],{},[23,48737,297],{"className":48738},[296],[23,48740,48742],{"className":48741},[204],[23,48743,48746],{"className":48744,"style":48745},[208],"height:2.498em;",[23,48747],{},[23,48749],{"className":48750,"style":392},[148],[23,48752,5106],{"className":48753},[612],[23,48755],{"className":48756,"style":392},[148],[23,48758,48760,48763,48769,48775,48778,48781,48784,48787,48790,48793,48796,48799,48802,48805,48808,48811],{"className":48759},[126],[23,48761],{"className":48762,"style":2213},[130],[23,48764,48766],{"className":48765},[135],[23,48767,27187],{"className":48768},[135,5391],[23,48770,48772],{"className":48771},[135],[23,48773,54],{"className":48774},[1066,1067],[23,48776,33013],{"className":48777,"style":676},[135,136],[23,48779,54],{"className":48780},[140],[23,48782,57],{"className":48783,"style":144},[135,136],[23,48785,66],{"className":48786},[169],[23,48788,5360],{"className":48789},[148],[23,48791,20680],{"className":48792},[135],[23,48794,5360],{"className":48795},[148],[23,48797,11],{"className":48798},[135,136],[23,48800,54],{"className":48801},[140],[23,48803,57],{"className":48804,"style":144},[135,136],[23,48806],{"className":48807,"style":149},[148],[23,48809,60],{"className":48810},[153],[23,48812],{"className":48813,"style":149},[148],[23,48815,48817,48820,48823,48826,48832],{"className":48816},[126],[23,48818],{"className":48819,"style":2213},[130],[23,48821,63],{"className":48822,"style":144},[135,136],[23,48824,66],{"className":48825},[169],[23,48827,48829],{"className":48828},[135],[23,48830,66],{"className":48831},[1066,1067],[23,48833,747],{"className":48834},[824],[11,48836,48837,48838,48841],{},"把问题转化为最大化",[18,48839,48840],{},"证据下界","（ELBO）：",[23,48843,48845],{"className":48844,"translate":27},[26],[23,48846,48848,48940],{"className":48847,"translate":27},[31],[23,48849,48851],{"className":48850},[35],[37,48852,48853],{"xmlns":39,"display":40},[42,48854,48855,48937],{},[45,48856,48857,48867,48869,48871,48873,48875,48881,48883,48885,48887,48889,48891,48893,48895,48897,48899,48901,48903,48909,48911,48913,48915,48917,48919,48921,48923,48925,48927,48929,48931,48933,48935],{},[45,48858,48859,48861,48863,48865],{},[48,48860,902],{"mathvariant":111},[48,48862,555],{"mathvariant":111},[48,48864,2533],{"mathvariant":111},[48,48866,42340],{"mathvariant":111},[51,48868,54],{"stretchy":53},[48,48870,33013],{},[51,48872,66],{"stretchy":53},[51,48874,69],{},[898,48876,48877,48879],{},[48,48878,902],{"mathvariant":571},[48,48880,33013],{},[51,48882,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,48884,22418],{},[51,48886,6959],{},[48,48888,11],{},[51,48890,54],{"stretchy":53},[48,48892,63],{},[51,48894,60],{},[48,48896,57],{},[51,48898,66],{"stretchy":53},[51,48900,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,48902,4529],{},[45,48904,48905,48907],{},[48,48906,26918],{"mathvariant":111},[48,48908,555],{"mathvariant":111},[51,48910,54],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,48912,33013],{},[51,48914,54],{"stretchy":53},[48,48916,57],{},[51,48918,66],{"stretchy":53},[88,48920,5360],{},[48,48922,20680],{"mathvariant":111},[88,48924,5360],{},[48,48926,11],{},[51,48928,54],{"stretchy":53},[48,48930,57],{},[51,48932,66],{"stretchy":53},[51,48934,66],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,48936,112],{"mathvariant":111},[114,48938,48939],{"encoding":116},"\\mathrm{ELBO}(q)\n=\n\\mathbb E_q\\big[\\log p(D\\mid\\theta)\\big]\n-\n\\mathrm{KL}\\big(q(\\theta)\\ \\|\\ p(\\theta)\\big).",[23,48941,48943,48973,49054,49081],{"className":48942,"ariaHidden":122},[121],[23,48944,48946,48949,48955,48958,48961,48964,48967,48970],{"className":48945},[126],[23,48947],{"className":48948,"style":131},[130],[23,48950,48952],{"className":48951},[135],[23,48953,48499],{"className":48954},[135,5391],[23,48956,54],{"className":48957},[140],[23,48959,33013],{"className":48960,"style":676},[135,136],[23,48962,66],{"className":48963},[169],[23,48965],{"className":48966,"style":149},[148],[23,48968,69],{"className":48969},[153],[23,48971],{"className":48972,"style":149},[148],[23,48974,48976,48979,49019,49025,49028,49033,49036,49039,49042,49045,49048,49051],{"className":48975},[126],[23,48977],{"className":48978,"style":2213},[130],[23,48980,48982,48985],{"className":48981},[135],[23,48983,902],{"className":48984},[135,644],[23,48986,48988],{"className":48987},[998],[23,48989,48991,49011],{"className":48990},[199,200],[23,48992,48994,49008],{"className":48993},[204],[23,48995,48997],{"className":48996,"style":6499},[208],[23,48998,48999,49002],{"style":2029},[23,49000],{"className":49001,"style":1015},[216],[23,49003,49005],{"className":49004},[1019,1020,1021,1022],[23,49006,33013],{"className":49007,"style":676},[135,136,1022],[23,49009,297],{"className":49010},[296],[23,49012,49014],{"className":49013},[204],[23,49015,49017],{"className":49016,"style":14292},[208],[23,49018],{},[23,49020,49022],{"className":49021},[135],[23,49023,924],{"className":49024},[1066,1067],[23,49026],{"className":49027,"style":280},[148],[23,49029,22575,49031],{"className":49030},[1786],[23,49032,7089],{"style":7088},[23,49034],{"className":49035,"style":280},[148],[23,49037,11],{"className":49038},[135,136],[23,49040,54],{"className":49041},[140],[23,49043,63],{"className":49044,"style":144},[135,136],[23,49046],{"className":49047,"style":149},[148],[23,49049,60],{"className":49050},[153],[23,49052],{"className":49053,"style":149},[148],[23,49055,49057,49060,49063,49066,49072,49075,49078],{"className":49056},[126],[23,49058],{"className":49059,"style":2213},[130],[23,49061,57],{"className":49062,"style":144},[135,136],[23,49064,66],{"className":49065},[169],[23,49067,49069],{"className":49068},[135],[23,49070,945],{"className":49071},[1066,1067],[23,49073],{"className":49074,"style":392},[148],[23,49076,4529],{"className":49077},[612],[23,49079],{"className":49080,"style":392},[148],[23,49082,49084,49087,49093,49099,49102,49105,49108,49111,49114,49117,49120,49123,49126,49129,49132,49138],{"className":49083},[126],[23,49085],{"className":49086,"style":2213},[130],[23,49088,49090],{"className":49089},[135],[23,49091,27187],{"className":49092},[135,5391],[23,49094,49096],{"className":49095},[135],[23,49097,54],{"className":49098},[1066,1067],[23,49100,33013],{"className":49101,"style":676},[135,136],[23,49103,54],{"className":49104},[140],[23,49106,57],{"className":49107,"style":144},[135,136],[23,49109,66],{"className":49110},[169],[23,49112,5360],{"className":49113},[148],[23,49115,20680],{"className":49116},[135],[23,49118,5360],{"className":49119},[148],[23,49121,11],{"className":49122},[135,136],[23,49124,54],{"className":49125},[140],[23,49127,57],{"className":49128,"style":144},[135,136],[23,49130,66],{"className":49131},[169],[23,49133,49135],{"className":49134},[135],[23,49136,66],{"className":49137},[1066,1067],[23,49139,112],{"className":49140},[135],[11,49142,49143,49144,49172,49173,49201,49202,49297,49298,4136,49354,49382],{},"第一项鼓励 ",[23,49145,49147,49160],{"className":49146,"translate":27},[31],[23,49148,49150],{"className":49149},[35],[37,49151,49152],{"xmlns":39},[42,49153,49154,49158],{},[45,49155,49156],{},[48,49157,33013],{},[114,49159,33013],{"encoding":116},[23,49161,49163],{"className":49162,"ariaHidden":122},[121],[23,49164,49166,49169],{"className":49165},[126],[23,49167],{"className":49168,"style":672},[130],[23,49170,33013],{"className":49171,"style":676},[135,136]," 解释数据，第二项惩罚 ",[23,49174,49176,49189],{"className":49175,"translate":27},[31],[23,49177,49179],{"className":49178},[35],[37,49180,49181],{"xmlns":39},[42,49182,49183,49187],{},[45,49184,49185],{},[48,49186,33013],{},[114,49188,33013],{"encoding":116},[23,49190,49192],{"className":49191,"ariaHidden":122},[121],[23,49193,49195,49198],{"className":49194},[126],[23,49196],{"className":49197,"style":672},[130],[23,49199,33013],{"className":49200,"style":676},[135,136]," 偏离先验。最大化 ELBO 等价于最小化 ",[23,49203,49205,49243],{"className":49204,"translate":27},[31],[23,49206,49208],{"className":49207},[35],[37,49209,49210],{"xmlns":39},[42,49211,49212,49240],{},[45,49213,49214,49220,49222,49224,49226,49228,49230,49232,49234,49236,49238],{},[45,49215,49216,49218],{},[48,49217,26918],{"mathvariant":111},[48,49219,555],{"mathvariant":111},[51,49221,54],{"stretchy":53},[48,49223,33013],{},[48,49225,20680],{"mathvariant":111},[48,49227,11],{},[51,49229,54],{"stretchy":53},[48,49231,57],{},[51,49233,60],{},[48,49235,63],{},[51,49237,66],{"stretchy":53},[51,49239,66],{"stretchy":53},[114,49241,49242],{"encoding":116},"\\mathrm{KL}(q\\|p(\\theta\\mid D))",[23,49244,49246,49285],{"className":49245,"ariaHidden":122},[121],[23,49247,49249,49252,49258,49261,49264,49267,49270,49273,49276,49279,49282],{"className":49248},[126],[23,49250],{"className":49251,"style":131},[130],[23,49253,49255],{"className":49254},[135],[23,49256,27187],{"className":49257},[135,5391],[23,49259,54],{"className":49260},[140],[23,49262,33013],{"className":49263,"style":676},[135,136],[23,49265,20680],{"className":49266},[135],[23,49268,11],{"className":49269},[135,136],[23,49271,54],{"className":49272},[140],[23,49274,57],{"className":49275,"style":144},[135,136],[23,49277],{"className":49278,"style":149},[148],[23,49280,60],{"className":49281},[153],[23,49283],{"className":49284,"style":149},[148],[23,49286,49288,49291,49294],{"className":49287},[126],[23,49289],{"className":49290,"style":131},[130],[23,49292,63],{"className":49293,"style":144},[135,136],[23,49295,1095],{"className":49296},[169],"，因为 ",[23,49299,49301,49325],{"className":49300,"translate":27},[31],[23,49302,49304],{"className":49303},[35],[37,49305,49306],{"xmlns":39},[42,49307,49308,49322],{},[45,49309,49310,49312,49314,49316,49318,49320],{},[48,49311,22418],{},[51,49313,6959],{},[48,49315,11],{},[51,49317,54],{"stretchy":53},[48,49319,63],{},[51,49321,66],{"stretchy":53},[114,49323,49324],{"encoding":116},"\\log p(D)",[23,49326,49328],{"className":49327,"ariaHidden":122},[121],[23,49329,49331,49334,49339,49342,49345,49348,49351],{"className":49330},[126],[23,49332],{"className":49333,"style":131},[130],[23,49335,22575,49337],{"className":49336},[1786],[23,49338,7089],{"style":7088},[23,49340],{"className":49341,"style":280},[148],[23,49343,11],{"className":49344},[135,136],[23,49346,54],{"className":49347},[140],[23,49349,63],{"className":49350,"style":144},[135,136],[23,49352,66],{"className":49353},[169],[23,49355,49357,49370],{"className":49356,"translate":27},[31],[23,49358,49360],{"className":49359},[35],[37,49361,49362],{"xmlns":39},[42,49363,49364,49368],{},[45,49365,49366],{},[48,49367,33013],{},[114,49369,33013],{"encoding":116},[23,49371,49373],{"className":49372,"ariaHidden":122},[121],[23,49374,49376,49379],{"className":49375},[126],[23,49377],{"className":49378,"style":672},[130],[23,49380,33013],{"className":49381,"style":676},[135,136]," 无关。",[11,49384,49385,49386,49447,49448,49451,49452,49480,49481,49447,49542,49545,49546,49611],{},"KL 的方向很重要。",[23,49387,49389,49417],{"className":49388,"translate":27},[31],[23,49390,49392],{"className":49391},[35],[37,49393,49394],{"xmlns":39},[42,49395,49396,49414],{},[45,49397,49398,49404,49406,49408,49410,49412],{},[45,49399,49400,49402],{},[48,49401,26918],{"mathvariant":111},[48,49403,555],{"mathvariant":111},[51,49405,54],{"stretchy":53},[48,49407,33013],{},[48,49409,20680],{"mathvariant":111},[48,49411,11],{},[51,49413,66],{"stretchy":53},[114,49415,49416],{"encoding":116},"\\mathrm{KL}(q\\|p)",[23,49418,49420],{"className":49419,"ariaHidden":122},[121],[23,49421,49423,49426,49432,49435,49438,49441,49444],{"className":49422},[126],[23,49424],{"className":49425,"style":131},[130],[23,49427,49429],{"className":49428},[135],[23,49430,27187],{"className":49431},[135,5391],[23,49433,54],{"className":49434},[140],[23,49436,33013],{"className":49437,"style":676},[135,136],[23,49439,20680],{"className":49440},[135],[23,49442,11],{"className":49443},[135,136],[23,49445,66],{"className":49446},[169]," 倾向",[18,49449,49450],{},"寻模","（mode-seeking）：只要 ",[23,49453,49455,49468],{"className":49454,"translate":27},[31],[23,49456,49458],{"className":49457},[35],[37,49459,49460],{"xmlns":39},[42,49461,49462,49466],{},[45,49463,49464],{},[48,49465,33013],{},[114,49467,33013],{"encoding":116},[23,49469,49471],{"className":49470,"ariaHidden":122},[121],[23,49472,49474,49477],{"className":49473},[126],[23,49475],{"className":49476,"style":672},[130],[23,49478,33013],{"className":49479,"style":676},[135,136]," 覆盖了后验的一个峰，它就不太在意漏掉其他峰，因此常常低估方差。反方向 ",[23,49482,49484,49512],{"className":49483,"translate":27},[31],[23,49485,49487],{"className":49486},[35],[37,49488,49489],{"xmlns":39},[42,49490,49491,49509],{},[45,49492,49493,49499,49501,49503,49505,49507],{},[45,49494,49495,49497],{},[48,49496,26918],{"mathvariant":111},[48,49498,555],{"mathvariant":111},[51,49500,54],{"stretchy":53},[48,49502,11],{},[48,49504,20680],{"mathvariant":111},[48,49506,33013],{},[51,49508,66],{"stretchy":53},[114,49510,49511],{"encoding":116},"\\mathrm{KL}(p\\|q)",[23,49513,49515],{"className":49514,"ariaHidden":122},[121],[23,49516,49518,49521,49527,49530,49533,49536,49539],{"className":49517},[126],[23,49519],{"className":49520,"style":131},[130],[23,49522,49524],{"className":49523},[135],[23,49525,27187],{"className":49526},[135,5391],[23,49528,54],{"className":49529},[140],[23,49531,11],{"className":49532},[135,136],[23,49534,20680],{"className":49535},[135],[23,49537,33013],{"className":49538,"style":676},[135,136],[23,49540,66],{"className":49541},[169],[18,49543,49544],{},"覆盖质量","（mass-covering），但它的计算需要 ",[23,49547,49549,49572],{"className":49548,"translate":27},[31],[23,49550,49552],{"className":49551},[35],[37,49553,49554],{"xmlns":39},[42,49555,49556,49570],{},[45,49557,49558,49560,49562,49564,49566,49568],{},[48,49559,11],{},[51,49561,54],{"stretchy":53},[48,49563,57],{},[51,49565,60],{},[48,49567,63],{},[51,49569,66],{"stretchy":53},[114,49571,4032],{"encoding":116},[23,49573,49575,49599],{"className":49574,"ariaHidden":122},[121],[23,49576,49578,49581,49584,49587,49590,49593,49596],{"className":49577},[126],[23,49579],{"className":49580,"style":131},[130],[23,49582,11],{"className":49583},[135,136],[23,49585,54],{"className":49586},[140],[23,49588,57],{"className":49589,"style":144},[135,136],[23,49591],{"className":49592,"style":149},[148],[23,49594,60],{"className":49595},[153],[23,49597],{"className":49598,"style":149},[148],[23,49600,49602,49605,49608],{"className":49601},[126],[23,49603],{"className":49604,"style":131},[130],[23,49606,63],{"className":49607,"style":144},[135,136],[23,49609,66],{"className":49610},[169]," 的归一化常数。变分推断通常用前者，因为 ELBO 只需要未归一化的联合分布。",[11,49613,49614,49615,49644,49645,1471],{},"从决策论的角度看，变分推断本身也是一个决策问题：在受限的分布族里，选一个使 KL 最小的近似。这里的「损失」是 KL，行动是一个分布，而不是一个点。贝叶斯决策理论的框架在这里再次出现，只是行动空间从 ",[23,49616,49618,49632],{"className":49617,"translate":27},[31],[23,49619,49621],{"className":49620},[35],[37,49622,49623],{"xmlns":39},[42,49624,49625,49629],{},[45,49626,49627],{},[48,49628,500],{"mathvariant":375},[114,49630,49631],{"encoding":116},"\\mathcal A",[23,49633,49635],{"className":49634,"ariaHidden":122},[121],[23,49636,49638,49641],{"className":49637},[126],[23,49639],{"className":49640,"style":470},[130],[23,49642,500],{"className":49643},[135,412]," 变成了 ",[23,49646,49648,49661],{"className":49647,"translate":27},[31],[23,49649,49651],{"className":49650},[35],[37,49652,49653],{"xmlns":39},[42,49654,49655,49659],{},[45,49656,49657],{},[48,49658,22051],{"mathvariant":375},[114,49660,47942],{"encoding":116},[23,49662,49664],{"className":49663,"ariaHidden":122},[121],[23,49665,49667,49670],{"className":49666},[126],[23,49668],{"className":49669,"style":408},[130],[23,49671,22051],{"className":49672},[135,412],[2501,49674,49676],{"id":49675},"_75-贝叶斯神经网络与不确定性","7.5 贝叶斯神经网络与不确定性",[11,49678,49679,49680,49710,49711,49777],{},"把参数换成网络权重 ",[23,49681,49683,49697],{"className":49682,"translate":27},[31],[23,49684,49686],{"className":49685},[35],[37,49687,49688],{"xmlns":39},[42,49689,49690,49695],{},[45,49691,49692],{},[48,49693,49694],{},"w",[114,49696,49694],{"encoding":116},[23,49698,49700],{"className":49699,"ariaHidden":122},[121],[23,49701,49703,49706],{"className":49702},[126],[23,49704],{"className":49705,"style":703},[130],[23,49707,49694],{"className":49708,"style":49709},[135,136],"margin-right:0.0269em;","，后验 ",[23,49712,49714,49738],{"className":49713,"translate":27},[31],[23,49715,49717],{"className":49716},[35],[37,49718,49719],{"xmlns":39},[42,49720,49721,49735],{},[45,49722,49723,49725,49727,49729,49731,49733],{},[48,49724,11],{},[51,49726,54],{"stretchy":53},[48,49728,49694],{},[51,49730,60],{},[48,49732,63],{},[51,49734,66],{"stretchy":53},[114,49736,49737],{"encoding":116},"p(w\\mid D)",[23,49739,49741,49765],{"className":49740,"ariaHidden":122},[121],[23,49742,49744,49747,49750,49753,49756,49759,49762],{"className":49743},[126],[23,49745],{"className":49746,"style":131},[130],[23,49748,11],{"className":49749},[135,136],[23,49751,54],{"className":49752},[140],[23,49754,49694],{"className":49755,"style":49709},[135,136],[23,49757],{"className":49758,"style":149},[148],[23,49760,60],{"className":49761},[153],[23,49763],{"className":49764,"style":149},[148],[23,49766,49768,49771,49774],{"className":49767},[126],[23,49769],{"className":49770,"style":131},[130],[23,49772,63],{"className":49773,"style":144},[135,136],[23,49775,66],{"className":49776},[169]," 就是贝叶斯神经网络。精确后验不可行，实践中用 MC dropout、Laplace 近似、变分推断或深度集成来近似。",[11,49779,49780],{},"对分类问题，预测熵可以分解为",[23,49782,49784],{"className":49783,"translate":27},[26],[23,49785,49787,49873],{"className":49786,"translate":27},[31],[23,49788,49790],{"className":49789},[35],[37,49791,49792],{"xmlns":39,"display":40},[42,49793,49794,49870],{},[45,49795,49796,49798,49800,49802,49804,49806,49808,49810,49812,49814,49826,49828,49830,49832,49834,49836,49838,49840,49842,49844,49846,49848,49850,49852,49854,49856,49858,49860,49862,49864,49866,49868],{},[48,49797,26249],{"mathvariant":571},[51,49799,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,49801,369],{},[51,49803,60],{},[48,49805,753],{},[51,49807,747],{"separator":122},[48,49809,63],{},[51,49811,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,49813,69],{},[898,49815,49816,49818],{},[48,49817,902],{"mathvariant":571},[45,49819,49820,49822,49824],{},[48,49821,49694],{},[51,49823,60],{},[48,49825,63],{},[51,49827,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,49829,26249],{"mathvariant":571},[51,49831,924],{"stretchy":53},[48,49833,369],{},[51,49835,60],{},[48,49837,753],{},[51,49839,747],{"separator":122},[48,49841,49694],{},[51,49843,945],{"stretchy":53},[51,49845,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,49847,5106],{},[48,49849,34353],{"mathvariant":571},[51,49851,54],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,49853,369],{},[51,49855,39744],{"separator":122},[48,49857,49694],{},[51,49859,60],{},[48,49861,753],{},[51,49863,747],{"separator":122},[48,49865,63],{},[51,49867,66],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,49869,112],{"mathvariant":111},[114,49871,49872],{"encoding":116},"\\mathbb H\\big[Y\\mid x,D\\big]\n=\n\\mathbb E_{w\\mid D}\\big[\\mathbb H[Y\\mid x,w]\\big]\n+\n\\mathbb I\\big(Y;w\\mid x,D\\big).",[23,49874,49876,49903,49936,50015,50051,50087],{"className":49875,"ariaHidden":122},[121],[23,49877,49879,49882,49885,49891,49894,49897,49900],{"className":49878},[126],[23,49880],{"className":49881,"style":2213},[130],[23,49883,26249],{"className":49884},[135,644],[23,49886,49888],{"className":49887},[135],[23,49889,924],{"className":49890},[1066,1067],[23,49892,369],{"className":49893,"style":392},[135,136],[23,49895],{"className":49896,"style":149},[148],[23,49898,60],{"className":49899},[153],[23,49901],{"className":49902,"style":149},[148],[23,49904,49906,49909,49912,49915,49918,49921,49927,49930,49933],{"className":49905},[126],[23,49907],{"className":49908,"style":2213},[130],[23,49910,753],{"className":49911},[135,136],[23,49913,747],{"className":49914},[824],[23,49916],{"className":49917,"style":280},[148],[23,49919,63],{"className":49920,"style":144},[135,136],[23,49922,49924],{"className":49923},[135],[23,49925,945],{"className":49926},[1066,1067],[23,49928],{"className":49929,"style":149},[148],[23,49931,69],{"className":49932},[153],[23,49934],{"className":49935,"style":149},[148],[23,49937,49939,49942,49991,49997,50000,50003,50006,50009,50012],{"className":49938},[126],[23,49940],{"className":49941,"style":988},[130],[23,49943,49945,49948],{"className":49944},[135],[23,49946,902],{"className":49947},[135,644],[23,49949,49951],{"className":49950},[998],[23,49952,49954,49983],{"className":49953},[199,200],[23,49955,49957,49980],{"className":49956},[204],[23,49958,49960],{"className":49959,"style":1008},[208],[23,49961,49962,49965],{"style":1011},[23,49963],{"className":49964,"style":1015},[216],[23,49966,49968],{"className":49967},[1019,1020,1021,1022],[23,49969,49971,49974,49977],{"className":49970},[135,1022],[23,49972,49694],{"className":49973,"style":49709},[135,136,1022],[23,49975,60],{"className":49976},[153,1022],[23,49978,63],{"className":49979,"style":144},[135,136,1022],[23,49981,297],{"className":49982},[296],[23,49984,49986],{"className":49985},[204],[23,49987,49989],{"className":49988,"style":1057},[208],[23,49990],{},[23,49992,49994],{"className":49993},[135],[23,49995,924],{"className":49996},[1066,1067],[23,49998,26249],{"className":49999},[135,644],[23,50001,924],{"className":50002},[140],[23,50004,369],{"className":50005,"style":392},[135,136],[23,50007],{"className":50008,"style":149},[148],[23,50010,60],{"className":50011},[153],[23,50013],{"className":50014,"style":149},[148],[23,50016,50018,50021,50024,50027,50030,50033,50036,50042,50045,50048],{"className":50017},[126],[23,50019],{"className":50020,"style":2213},[130],[23,50022,753],{"className":50023},[135,136],[23,50025,747],{"className":50026},[824],[23,50028],{"className":50029,"style":280},[148],[23,50031,49694],{"className":50032,"style":49709},[135,136],[23,50034,945],{"className":50035},[169],[23,50037,50039],{"className":50038},[135],[23,50040,945],{"className":50041},[1066,1067],[23,50043],{"className":50044,"style":392},[148],[23,50046,5106],{"className":50047},[612],[23,50049],{"className":50050,"style":392},[148],[23,50052,50054,50057,50060,50066,50069,50072,50075,50078,50081,50084],{"className":50053},[126],[23,50055],{"className":50056,"style":2213},[130],[23,50058,34353],{"className":50059},[135,644],[23,50061,50063],{"className":50062},[135],[23,50064,54],{"className":50065},[1066,1067],[23,50067,369],{"className":50068,"style":392},[135,136],[23,50070,39744],{"className":50071},[824],[23,50073],{"className":50074,"style":280},[148],[23,50076,49694],{"className":50077,"style":49709},[135,136],[23,50079],{"className":50080,"style":149},[148],[23,50082,60],{"className":50083},[153],[23,50085],{"className":50086,"style":149},[148],[23,50088,50090,50093,50096,50099,50102,50105,50111],{"className":50089},[126],[23,50091],{"className":50092,"style":2213},[130],[23,50094,753],{"className":50095},[135,136],[23,50097,747],{"className":50098},[824],[23,50100],{"className":50101,"style":280},[148],[23,50103,63],{"className":50104,"style":144},[135,136],[23,50106,50108],{"className":50107},[135],[23,50109,66],{"className":50110},[1066,1067],[23,50112,112],{"className":50113},[135],[11,50115,50116],{},"第一项是偶然不确定性：即使固定权重，类别仍然有噪声。第二项是认知不确定性：不同权重给出的预测分布之间的差异。BALD 等主动学习准则，就是用它来挑选「最能减少认知不确定性」的样本。",[11,50118,50119],{},"但「贝叶斯」这个标签不保证校准。一个近似的后验可能给出过度自信的预测；深度集成不是严格的贝叶斯方法，却常常校准得更好。判断的标准应该是决策质量与校准，而不是方法的名字。一个模型如果能在高损失的区域拒绝预测，或者把不确定性传递给下游决策，它就已经在做贝叶斯决策了，哪怕它没有显式地计算后验。",[2501,50121,50123],{"id":50122},"_76-序贯决策后验采样与探索","7.6 序贯决策：后验采样与探索",[11,50125,50126,50127,50130,50131,50296],{},"在序贯决策里，后验不只是用来「估计」，还用来「探索」。",[18,50128,50129],{},"Thompson 采样","的规则很简单：每一步从后验中采样一个环境参数 ",[23,50132,50134,50171],{"className":50133,"translate":27},[31],[23,50135,50137],{"className":50136},[35],[37,50138,50139],{"xmlns":39},[42,50140,50141,50168],{},[45,50142,50143,50150,50152,50154,50156,50158,50160,50166],{},[16807,50144,50145,50147],{"accent":122},[48,50146,57],{},[51,50148,50149],{},"~",[51,50151,917],{},[48,50153,11],{},[51,50155,54],{"stretchy":53},[48,50157,57],{},[51,50159,60],{},[898,50161,50162,50164],{},[48,50163,63],{},[48,50165,5345],{},[51,50167,66],{"stretchy":53},[114,50169,50170],{"encoding":116},"\\tilde\\theta\\sim p(\\theta\\mid D_t)",[23,50172,50174,50222,50246],{"className":50173,"ariaHidden":122},[121],[23,50175,50177,50181,50213,50216,50219],{"className":50176},[126],[23,50178],{"className":50179,"style":50180},[130],"height:0.9313em;",[23,50182,50184],{"className":50183},[135,16830],[23,50185,50187],{"className":50186},[199],[23,50188,50190],{"className":50189},[204],[23,50191,50193,50201],{"className":50192,"style":50180},[208],[23,50194,50195,50198],{"style":7130},[23,50196],{"className":50197,"style":217},[216],[23,50199,57],{"className":50200,"style":144},[135,136],[23,50202,50204,50207],{"style":50203},"top:-3.6134em;",[23,50205],{"className":50206,"style":217},[216],[23,50208,50210],{"className":50209,"style":34865},[16858],[23,50211,50149],{"className":50212},[135],[23,50214],{"className":50215,"style":149},[148],[23,50217,917],{"className":50218},[153],[23,50220],{"className":50221,"style":149},[148],[23,50223,50225,50228,50231,50234,50237,50240,50243],{"className":50224},[126],[23,50226],{"className":50227,"style":131},[130],[23,50229,11],{"className":50230},[135,136],[23,50232,54],{"className":50233},[140],[23,50235,57],{"className":50236,"style":144},[135,136],[23,50238],{"className":50239,"style":149},[148],[23,50241,60],{"className":50242},[153],[23,50244],{"className":50245,"style":149},[148],[23,50247,50249,50252,50293],{"className":50248},[126],[23,50250],{"className":50251,"style":131},[130],[23,50253,50255,50258],{"className":50254},[135],[23,50256,63],{"className":50257,"style":144},[135,136],[23,50259,50261],{"className":50260},[998],[23,50262,50264,50285],{"className":50263},[199,200],[23,50265,50267,50282],{"className":50266},[204],[23,50268,50271],{"className":50269,"style":50270},[208],"height:0.2806em;",[23,50272,50273,50276],{"style":27171},[23,50274],{"className":50275,"style":1015},[216],[23,50277,50279],{"className":50278},[1019,1020,1021,1022],[23,50280,5345],{"className":50281},[135,136,1022],[23,50283,297],{"className":50284},[296],[23,50286,50288],{"className":50287},[204],[23,50289,50291],{"className":50290,"style":2048},[208],[23,50292],{},[23,50294,66],{"className":50295},[169],"，然后按这个采样出的环境做最优动作。",[11,50298,50299],{},"这个规则自动平衡了利用与探索。不确定性大的动作在采样中更可能显得最优，因此会被更频繁地尝试；随着数据增多，后验收窄，采样出的环境越来越接近真实环境，策略逐渐转向利用。它的贝叶斯遗憾有理论保证，而且实现起来只需要一个后验采样器。",[11,50301,50302],{},"从决策论的角度看，Thompson 采样把「信息」当成了决策的一部分：当前动作的收益不只是它带来的即时回报，还包括它揭示的信息。完整的贝叶斯解决方案是动态规划，在信念空间（belief space）上求解；Thompson 采样是它的一个可计算的近似。",[2501,50304,50306],{"id":50305},"_77-主动学习与贝叶斯优化","7.7 主动学习与贝叶斯优化",[11,50308,50309],{},"主动学习问的是：下一个标签应该买哪一个？从决策论看，这是一个两级决策：先选一个样本，再在拿到标签后做预测。理想的采集函数是「期望后验风险降低」：",[23,50311,50313],{"className":50312,"translate":27},[26],[23,50314,50316,50416],{"className":50315,"translate":27},[31],[23,50317,50319],{"className":50318},[35],[37,50320,50321],{"xmlns":39,"display":40},[42,50322,50323,50413],{},[45,50324,50325,50332,50334,50336,50338,50348,50350,50376,50411],{},[898,50326,50327,50329],{},[48,50328,753],{},[88,50330,50331],{},"next",[51,50333,69],{},[48,50335,6956],{},[51,50337,6959],{},[6961,50339,50340,50346],{},[45,50341,50342,50344],{},[48,50343,6967],{},[51,50345,6959],{},[48,50347,753],{},[88,50349,5360],{},[898,50351,50352,50354],{},[48,50353,902],{"mathvariant":571},[45,50355,50356,50358,50360,50362,50364,50366,50368,50370,50372,50374],{},[48,50357,660],{},[51,50359,917],{},[48,50361,11],{},[51,50363,54],{"stretchy":53},[48,50365,660],{},[51,50367,60],{},[48,50369,753],{},[51,50371,747],{"separator":122},[48,50373,63],{},[51,50375,66],{"stretchy":53},[45,50377,50378,50380,50386,50388,50390,50393,50395,50397,50399,50401,50403,50405,50407,50409],{},[51,50379,924],{"fence":122},[4518,50381,50382,50384],{},[48,50383,572],{},[51,50385,6881],{},[51,50387,54],{"stretchy":53},[48,50389,63],{},[51,50391,50392],{},"∪",[51,50394,9010],{"stretchy":53},[51,50396,54],{"stretchy":53},[48,50398,753],{},[51,50400,747],{"separator":122},[48,50402,660],{},[51,50404,66],{"stretchy":53},[51,50406,9020],{"stretchy":53},[51,50408,66],{"stretchy":53},[51,50410,945],{"fence":122},[51,50412,747],{"separator":122},[114,50414,50415],{"encoding":116},"x_{\\text{next}}\n=\n\\arg\\min_{x}\\ \n\\mathbb E_{y\\sim p(y\\mid x,D)}\n\\left[\nR^*(D\\cup\\{(x,y)\\})\n\\right],",[23,50417,50419,50480],{"className":50418,"ariaHidden":122},[121],[23,50420,50422,50425,50471,50474,50477],{"className":50421},[126],[23,50423],{"className":50424,"style":12730},[130],[23,50426,50428,50431],{"className":50427},[135],[23,50429,753],{"className":50430},[135,136],[23,50432,50434],{"className":50433},[998],[23,50435,50437,50463],{"className":50436},[199,200],[23,50438,50440,50460],{"className":50439},[204],[23,50441,50443],{"className":50442,"style":50270},[208],[23,50444,50445,50448],{"style":2029},[23,50446],{"className":50447,"style":1015},[216],[23,50449,50451],{"className":50450},[1019,1020,1021,1022],[23,50452,50454],{"className":50453},[135,1022],[23,50455,50457],{"className":50456},[135,6514,1022],[23,50458,50331],{"className":50459},[135,1022],[23,50461,297],{"className":50462},[296],[23,50464,50466],{"className":50465},[204],[23,50467,50469],{"className":50468,"style":2048},[208],[23,50470],{},[23,50472],{"className":50473,"style":149},[148],[23,50475,69],{"className":50476},[153],[23,50478],{"className":50479,"style":149},[148],[23,50481,50483,50487,50492,50495,50542,50545,50548,50618,50621,50694,50697],{"className":50482},[126],[23,50484],{"className":50485,"style":50486},[130],"height:1.45em;vertical-align:-0.7em;",[23,50488,7085,50490],{"className":50489},[1786],[23,50491,7089],{"style":7088},[23,50493],{"className":50494,"style":280},[148],[23,50496,50498],{"className":50497},[1786,7096],[23,50499,50501,50534],{"className":50500},[199,200],[23,50502,50504,50531],{"className":50503},[204],[23,50505,50507,50521],{"className":50506,"style":7106},[208],[23,50508,50509,50512],{"style":8780},[23,50510],{"className":50511,"style":217},[216],[23,50513,50515],{"className":50514},[1019,1020,1021,1022],[23,50516,50518],{"className":50517},[135,1022],[23,50519,753],{"className":50520},[135,136,1022],[23,50522,50523,50526],{"style":7130},[23,50524],{"className":50525,"style":217},[216],[23,50527,50528],{},[23,50529,6967],{"className":50530},[1786],[23,50532,297],{"className":50533},[296],[23,50535,50537],{"className":50536},[204],[23,50538,50540],{"className":50539,"style":8812},[208],[23,50541],{},[23,50543,5360],{"className":50544},[148],[23,50546],{"className":50547,"style":280},[148],[23,50549,50551,50554],{"className":50550},[135],[23,50552,902],{"className":50553},[135,644],[23,50555,50557],{"className":50556},[998],[23,50558,50560,50610],{"className":50559},[199,200],[23,50561,50563,50607],{"className":50562},[204],[23,50564,50566],{"className":50565,"style":1008},[208],[23,50567,50568,50571],{"style":1011},[23,50569],{"className":50570,"style":1015},[216],[23,50572,50574],{"className":50573},[1019,1020,1021,1022],[23,50575,50577,50580,50583,50586,50589,50592,50595,50598,50601,50604],{"className":50576},[135,1022],[23,50578,660],{"className":50579,"style":676},[135,136,1022],[23,50581,917],{"className":50582},[153,1022],[23,50584,11],{"className":50585},[135,136,1022],[23,50587,54],{"className":50588},[140,1022],[23,50590,660],{"className":50591,"style":676},[135,136,1022],[23,50593,60],{"className":50594},[153,1022],[23,50596,753],{"className":50597},[135,136,1022],[23,50599,747],{"className":50600},[824,1022],[23,50602,63],{"className":50603,"style":144},[135,136,1022],[23,50605,66],{"className":50606},[169,1022],[23,50608,297],{"className":50609},[296],[23,50611,50613],{"className":50612},[204],[23,50614,50616],{"className":50615,"style":1057},[208],[23,50617],{},[23,50619],{"className":50620,"style":280},[148],[23,50622,50624,50627,50656,50659,50662,50665,50668,50671,50675,50678,50681,50684,50687,50691],{"className":50623},[1839],[23,50625,924],{"className":50626,"style":1844},[140,1843],[23,50628,50630,50633],{"className":50629},[135],[23,50631,572],{"className":50632,"style":963},[135,136],[23,50634,50636],{"className":50635},[998],[23,50637,50639],{"className":50638},[199],[23,50640,50642],{"className":50641},[204],[23,50643,50645],{"className":50644,"style":7045},[208],[23,50646,50647,50650],{"style":4989},[23,50648],{"className":50649,"style":1015},[216],[23,50651,50653],{"className":50652},[1019,1020,1021,1022],[23,50654,6881],{"className":50655},[612,1022],[23,50657,54],{"className":50658},[140],[23,50660,63],{"className":50661,"style":144},[135,136],[23,50663],{"className":50664,"style":392},[148],[23,50666,50392],{"className":50667},[612],[23,50669],{"className":50670,"style":392},[148],[23,50672,50674],{"className":50673},[140],"{(",[23,50676,753],{"className":50677},[135,136],[23,50679,747],{"className":50680},[824],[23,50682],{"className":50683,"style":280},[148],[23,50685,660],{"className":50686,"style":676},[135,136],[23,50688,50690],{"className":50689},[169],")})",[23,50692,945],{"className":50693,"style":1844},[169,1843],[23,50695],{"className":50696,"style":280},[148],[23,50698,747],{"className":50699},[824],[11,50701,24429,50702,50776],{},[23,50703,50705,50729],{"className":50704,"translate":27},[31],[23,50706,50708],{"className":50707},[35],[37,50709,50710],{"xmlns":39},[42,50711,50712,50726],{},[45,50713,50714,50720,50722,50724],{},[4518,50715,50716,50718],{},[48,50717,572],{},[51,50719,6881],{},[51,50721,54],{"stretchy":53},[48,50723,63],{},[51,50725,66],{"stretchy":53},[114,50727,50728],{"encoding":116},"R^*(D)",[23,50730,50732],{"className":50731,"ariaHidden":122},[121],[23,50733,50735,50738,50767,50770,50773],{"className":50734},[126],[23,50736],{"className":50737,"style":131},[130],[23,50739,50741,50744],{"className":50740},[135],[23,50742,572],{"className":50743,"style":963},[135,136],[23,50745,50747],{"className":50746},[998],[23,50748,50750],{"className":50749},[199],[23,50751,50753],{"className":50752},[204],[23,50754,50756],{"className":50755,"style":6912},[208],[23,50757,50758,50761],{"style":4696},[23,50759],{"className":50760,"style":1015},[216],[23,50762,50764],{"className":50763},[1019,1020,1021,1022],[23,50765,6881],{"className":50766},[612,1022],[23,50768,54],{"className":50769},[140],[23,50771,63],{"className":50772,"style":144},[135,136],[23,50774,66],{"className":50775},[169]," 是当前数据下的贝叶斯风险。这个量通常算不出来，于是有了各种近似：熵、互信息、BALD、期望模型变化。",[11,50778,50779],{},"贝叶斯优化把目标函数本身当成随机变量，用后验选择下一个评估点。Expected Improvement 就是",[23,50781,50783],{"className":50782,"translate":27},[26],[23,50784,50786,50856],{"className":50785,"translate":27},[31],[23,50787,50789],{"className":50788},[35],[37,50790,50791],{"xmlns":39,"display":40},[42,50792,50793,50853],{},[45,50794,50795,50801,50803,50805,50807,50809,50811,50813,50815,50817,50819,50821,50823,50825,50833,50835,50837,50839,50841,50843,50845,50847,50849,50851],{},[45,50796,50797,50799],{},[48,50798,902],{"mathvariant":111},[48,50800,34353],{"mathvariant":111},[51,50802,54],{"stretchy":53},[48,50804,753],{},[51,50806,66],{"stretchy":53},[51,50808,69],{},[48,50810,902],{"mathvariant":571},[51,50812,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,50814,9551],{},[51,50816,6959],{},[51,50818,9010],{"stretchy":53},[2581,50820,1532],{},[51,50822,747],{"separator":122},[88,50824,5360],{},[898,50826,50827,50829,50831],{},[48,50828,8336],{},[48,50830,6967],{},[51,50832,6959],{},[51,50834,4529],{},[48,50836,8336],{},[51,50838,54],{"stretchy":53},[48,50840,753],{},[51,50842,66],{"stretchy":53},[51,50844,9020],{"stretchy":53},[51,50846,60],{},[48,50848,63],{},[51,50850,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,50852,747],{"separator":122},[114,50854,50855],{"encoding":116},"\\mathrm{EI}(x)=\\mathbb E\\big[\\max\\{0,\\ f_{\\min}-f(x)\\}\\mid D\\big],",[23,50857,50859,50890,50988,51015],{"className":50858,"ariaHidden":122},[121],[23,50860,50862,50865,50872,50875,50878,50881,50884,50887],{"className":50861},[126],[23,50863],{"className":50864,"style":131},[130],[23,50866,50868],{"className":50867},[135],[23,50869,50871],{"className":50870},[135,5391],"EI",[23,50873,54],{"className":50874},[140],[23,50876,753],{"className":50877},[135,136],[23,50879,66],{"className":50880},[169],[23,50882],{"className":50883,"style":149},[148],[23,50885,69],{"className":50886},[153],[23,50888],{"className":50889,"style":149},[148],[23,50891,50893,50896,50899,50905,50908,50911,50914,50917,50920,50923,50926,50979,50982,50985],{"className":50892},[126],[23,50894],{"className":50895,"style":2213},[130],[23,50897,902],{"className":50898},[135,644],[23,50900,50902],{"className":50901},[135],[23,50903,924],{"className":50904},[1066,1067],[23,50906],{"className":50907,"style":280},[148],[23,50909,9551],{"className":50910},[1786],[23,50912,9010],{"className":50913},[140],[23,50915,1532],{"className":50916},[135],[23,50918,747],{"className":50919},[824],[23,50921,5360],{"className":50922},[148],[23,50924],{"className":50925,"style":280},[148],[23,50927,50929,50932],{"className":50928},[135],[23,50930,8336],{"className":50931,"style":11398},[135,136],[23,50933,50935],{"className":50934},[998],[23,50936,50938,50971],{"className":50937},[199,200],[23,50939,50941,50968],{"className":50940},[204],[23,50942,50944],{"className":50943,"style":38779},[208],[23,50945,50946,50949],{"style":29546},[23,50947],{"className":50948,"style":1015},[216],[23,50950,50952],{"className":50951},[1019,1020,1021,1022],[23,50953,50955],{"className":50954},[135,1022],[23,50956,50958,50962,50965],{"className":50957},[1786,1022],[23,50959,50961],{"className":50960},[1022],"m",[23,50963,33493],{"className":50964},[1022],[23,50966,4806],{"className":50967},[1022],[23,50969,297],{"className":50970},[296],[23,50972,50974],{"className":50973},[204],[23,50975,50977],{"className":50976,"style":2048},[208],[23,50978],{},[23,50980],{"className":50981,"style":392},[148],[23,50983,4529],{"className":50984},[612],[23,50986],{"className":50987,"style":392},[148],[23,50989,50991,50994,50997,51000,51003,51006,51009,51012],{"className":50990},[126],[23,50992],{"className":50993,"style":131},[130],[23,50995,8336],{"className":50996,"style":11398},[135,136],[23,50998,54],{"className":50999},[140],[23,51001,753],{"className":51002},[135,136],[23,51004,15220],{"className":51005},[169],[23,51007],{"className":51008,"style":149},[148],[23,51010,60],{"className":51011},[153],[23,51013],{"className":51014,"style":149},[148],[23,51016,51018,51021,51024,51030],{"className":51017},[126],[23,51019],{"className":51020,"style":2213},[130],[23,51022,63],{"className":51023,"style":144},[135,136],[23,51025,51027],{"className":51026},[135],[23,51028,945],{"className":51029},[1066,1067],[23,51031,747],{"className":51032},[824],[11,51034,51035],{},"它把「可能变好多少」与「变好的概率」乘在一起。探索与利用的权衡，在这里被写成了一个一维积分的最大化问题。",[2501,51037,51039],{"id":51038},"_78-可容许性与完整类定理","7.8 可容许性与完整类定理",[11,51041,51042,51043,15001,51071,51074,51075,51137],{},"最后回到 2.4 节留下的那个说法。一个决策规则 ",[23,51044,51046,51059],{"className":51045,"translate":27},[31],[23,51047,51049],{"className":51048},[35],[37,51050,51051],{"xmlns":39},[42,51052,51053,51057],{},[45,51054,51055],{},[48,51056,736],{},[114,51058,7337],{"encoding":116},[23,51060,51062],{"className":51061,"ariaHidden":122},[121],[23,51063,51065,51068],{"className":51064},[126],[23,51066],{"className":51067,"style":779},[130],[23,51069,736],{"className":51070,"style":783},[135,136],[18,51072,51073],{},"可容许的","（admissible），若不存在另一个规则 ",[23,51076,51078,51096],{"className":51077,"translate":27},[31],[23,51079,51081],{"className":51080},[35],[37,51082,51083],{"xmlns":39},[42,51084,51085,51093],{},[45,51086,51087],{},[4518,51088,51089,51091],{},[48,51090,736],{},[51,51092,38291],{"mathvariant":111,"lspace":1522,"rspace":1522},[114,51094,51095],{"encoding":116},"\\delta'",[23,51097,51099],{"className":51098,"ariaHidden":122},[121],[23,51100,51102,51105],{"className":51101},[126],[23,51103],{"className":51104,"style":38333},[130],[23,51106,51108,51111],{"className":51107},[135],[23,51109,736],{"className":51110,"style":783},[135,136],[23,51112,51114],{"className":51113},[998],[23,51115,51117],{"className":51116},[199],[23,51118,51120],{"className":51119},[204],[23,51121,51123],{"className":51122,"style":38333},[208],[23,51124,51125,51128],{"style":4696},[23,51126],{"className":51127,"style":1015},[216],[23,51129,51131],{"className":51130},[1019,1020,1021,1022],[23,51132,51134],{"className":51133},[135,1022],[23,51135,38291],{"className":51136},[135,1022],"，使得",[23,51139,51141],{"className":51140,"translate":27},[26],[23,51142,51144,51187],{"className":51143,"translate":27},[31],[23,51145,51147],{"className":51146},[35],[37,51148,51149],{"xmlns":39,"display":40},[42,51150,51151,51184],{},[45,51152,51153,51155,51157,51163,51165,51167,51169,51171,51173,51175,51177,51180,51182],{},[48,51154,572],{},[51,51156,54],{"stretchy":53},[4518,51158,51159,51161],{},[48,51160,736],{},[51,51162,38291],{"mathvariant":111,"lspace":1522,"rspace":1522},[51,51164,66],{"stretchy":53},[51,51166,12268],{},[48,51168,572],{},[51,51170,54],{"stretchy":53},[48,51172,736],{},[51,51174,66],{"stretchy":53},[148,51176],{"width":9915},[88,51178,51179],{},"对所有分布 ",[48,51181,920],{},[51,51183,747],{"separator":122},[114,51185,51186],{"encoding":116},"R(\\delta')\\le R(\\delta)\\quad\\text{对所有分布 }P,",[23,51188,51190,51248],{"className":51189,"ariaHidden":122},[121],[23,51191,51193,51197,51200,51203,51236,51239,51242,51245],{"className":51192},[126],[23,51194],{"className":51195,"style":51196},[130],"height:1.0519em;vertical-align:-0.25em;",[23,51198,572],{"className":51199,"style":963},[135,136],[23,51201,54],{"className":51202},[140],[23,51204,51206,51209],{"className":51205},[135],[23,51207,736],{"className":51208,"style":783},[135,136],[23,51210,51212],{"className":51211},[998],[23,51213,51215],{"className":51214},[199],[23,51216,51218],{"className":51217},[204],[23,51219,51222],{"className":51220,"style":51221},[208],"height:0.8019em;",[23,51223,51224,51227],{"style":4989},[23,51225],{"className":51226,"style":1015},[216],[23,51228,51230],{"className":51229},[1019,1020,1021,1022],[23,51231,51233],{"className":51232},[135,1022],[23,51234,38291],{"className":51235},[135,1022],[23,51237,66],{"className":51238},[169],[23,51240],{"className":51241,"style":149},[148],[23,51243,12268],{"className":51244},[153],[23,51246],{"className":51247,"style":149},[148],[23,51249,51251,51254,51257,51260,51263,51266,51269,51279,51282],{"className":51250},[126],[23,51252],{"className":51253,"style":131},[130],[23,51255,572],{"className":51256,"style":963},[135,136],[23,51258,54],{"className":51259},[140],[23,51261,736],{"className":51262,"style":783},[135,136],[23,51264,66],{"className":51265},[169],[23,51267],{"className":51268,"style":14437},[148],[23,51270,51272,51276],{"className":51271},[135,6514],[23,51273,51275],{"className":51274},[135,7238],"对所有分布",[23,51277,5360],{"className":51278},[135],[23,51280,920],{"className":51281,"style":1047},[135,136],[23,51283,747],{"className":51284},[824],[11,51286,51287,51288,51316],{},"且至少对一个 ",[23,51289,51291,51304],{"className":51290,"translate":27},[31],[23,51292,51294],{"className":51293},[35],[37,51295,51296],{"xmlns":39},[42,51297,51298,51302],{},[45,51299,51300],{},[48,51301,920],{},[114,51303,920],{"encoding":116},[23,51305,51307],{"className":51306,"ariaHidden":122},[121],[23,51308,51310,51313],{"className":51309},[126],[23,51311],{"className":51312,"style":470},[130],[23,51314,920],{"className":51315,"style":1047},[135,136]," 严格小于。换句话说，可容许就是「没有被另一个规则全面支配」。一个处处比别的规则差的决策规则，理应被淘汰；可容许性把这个直觉形式化了。",[6857,51318,51320],{"title":51319},"定理 7.2（Wald 完整类定理）",[11,51321,51322,51323,51352],{},"在损失函数凸、行动空间紧、实验族满足适当正则条件（例如被某个 ",[23,51324,51326,51340],{"className":51325,"translate":27},[31],[23,51327,51329],{"className":51328},[35],[37,51330,51331],{"xmlns":39},[42,51332,51333,51337],{},[45,51334,51335],{},[48,51336,29634],{},[114,51338,51339],{"encoding":116},"\\sigma",[23,51341,51343],{"className":51342,"ariaHidden":122},[121],[23,51344,51346,51349],{"className":51345},[126],[23,51347],{"className":51348,"style":703},[130],[23,51350,29634],{"className":51351,"style":676},[135,136],"-有限测度支配）时，每一个可容许的决策规则都是某个先验（或先验序列的极限）下的贝叶斯规则。",[11,51354,51355],{},"这个定理的意义在于：在适当条件下，它把「频率派的极大极小规则」「看起来没有先验的规则」也纳入了贝叶斯框架。先验不再是一个可选的信仰，而是可容许性这个要求所对应的对偶变量。你可以拒绝某个具体的先验，但很难拒绝「存在某个先验」这件事。",[11,51357,51358,51359,1471],{},"定理的条件不能省。损失不凸、行动空间不紧，或者问题是非参数的，结论可能失效；此时需要广义先验、极限贝叶斯规则，或者放弃完整类定理的结论。这些技术细节不改变核心信息：",[18,51360,51361],{},"贝叶斯规则不是许多决策规则中的一种，而是所有不被支配的决策规则的一般形式",[11,51363,51364],{},"从实践的角度看，这个定理不告诉你该选哪个先验，只告诉你：如果你想要一个不被全面支配的规则，你总可以把它写成某个先验下的贝叶斯规则。先验的选择仍然是建模的一部分，但「要不要用先验」已经不再是一个真正的选项。",[338,51366,51368],{"id":51367},"_8-几个可以手算的例子","8. 几个可以手算的例子",[2501,51370,51372],{"id":51371},"_81-betabernoulli-的完整更新","8.1 Beta–Bernoulli 的完整更新",[11,51374,51375,51376,51444,51445,4821,51473,51502],{},"取先验 ",[23,51377,51379,51411],{"className":51378,"translate":27},[31],[23,51380,51382],{"className":51381},[35],[37,51383,51384],{"xmlns":39},[42,51385,51386,51408],{},[45,51387,51388,51398,51400,51402,51404,51406],{},[45,51389,51390,51392,51394,51396],{},[48,51391,2533],{"mathvariant":111},[48,51393,5342],{"mathvariant":111},[48,51395,5345],{"mathvariant":111},[48,51397,495],{"mathvariant":111},[51,51399,54],{"stretchy":53},[2581,51401,24919],{},[51,51403,747],{"separator":122},[2581,51405,24919],{},[51,51407,66],{"stretchy":53},[114,51409,51410],{"encoding":116},"\\mathrm{Beta}(3,3)",[23,51412,51414],{"className":51413,"ariaHidden":122},[121],[23,51415,51417,51420,51426,51429,51432,51435,51438,51441],{"className":51416},[126],[23,51418],{"className":51419,"style":131},[130],[23,51421,51423],{"className":51422},[135],[23,51424,5392],{"className":51425},[135,5391],[23,51427,54],{"className":51428},[140],[23,51430,24919],{"className":51431},[135],[23,51433,747],{"className":51434},[824],[23,51436],{"className":51437,"style":280},[148],[23,51439,24919],{"className":51440},[135],[23,51442,66],{"className":51443},[169],"，观测到 ",[23,51446,51448,51461],{"className":51447,"translate":27},[31],[23,51449,51451],{"className":51450},[35],[37,51452,51453],{"xmlns":39},[42,51454,51455,51459],{},[45,51456,51457],{},[2581,51458,46690],{},[114,51460,46690],{"encoding":116},[23,51462,51464],{"className":51463,"ariaHidden":122},[121],[23,51465,51467,51470],{"className":51466},[126],[23,51468],{"className":51469,"style":2770},[130],[23,51471,46690],{"className":51472},[135],[23,51474,51476,51490],{"className":51475,"translate":27},[31],[23,51477,51479],{"className":51478},[35],[37,51480,51481],{"xmlns":39},[42,51482,51483,51488],{},[45,51484,51485],{},[2581,51486,51487],{},"7",[114,51489,51487],{"encoding":116},[23,51491,51493],{"className":51492,"ariaHidden":122},[121],[23,51494,51496,51499],{"className":51495},[126],[23,51497],{"className":51498,"style":2770},[130],[23,51500,51487],{"className":51501},[135]," 次正面。后验为",[23,51504,51506],{"className":51505,"translate":27},[26],[23,51507,51509,51575],{"className":51508,"translate":27},[31],[23,51510,51512],{"className":51511},[35],[37,51513,51514],{"xmlns":39,"display":40},[42,51515,51516,51572],{},[45,51517,51518,51528,51530,51532,51534,51536,51538,51540,51542,51544,51546,51548,51550,51560,51562,51564,51566,51568,51570],{},[45,51519,51520,51522,51524,51526],{},[48,51521,2533],{"mathvariant":111},[48,51523,5342],{"mathvariant":111},[48,51525,5345],{"mathvariant":111},[48,51527,495],{"mathvariant":111},[51,51529,54],{"stretchy":53},[2581,51531,24919],{},[51,51533,5106],{},[2581,51535,51487],{},[51,51537,747],{"separator":122},[88,51539,5360],{},[2581,51541,24919],{},[51,51543,5106],{},[2581,51545,24919],{},[51,51547,66],{"stretchy":53},[51,51549,69],{},[45,51551,51552,51554,51556,51558],{},[48,51553,2533],{"mathvariant":111},[48,51555,5342],{"mathvariant":111},[48,51557,5345],{"mathvariant":111},[48,51559,495],{"mathvariant":111},[51,51561,54],{"stretchy":53},[2581,51563,46690],{},[51,51565,747],{"separator":122},[2581,51567,46693],{},[51,51569,66],{"stretchy":53},[48,51571,112],{"mathvariant":111},[114,51573,51574],{"encoding":116},"\\mathrm{Beta}(3+7,\\ 3+3)=\\mathrm{Beta}(10,6).",[23,51576,51578,51605,51636,51657],{"className":51577,"ariaHidden":122},[121],[23,51579,51581,51584,51590,51593,51596,51599,51602],{"className":51580},[126],[23,51582],{"className":51583,"style":131},[130],[23,51585,51587],{"className":51586},[135],[23,51588,5392],{"className":51589},[135,5391],[23,51591,54],{"className":51592},[140],[23,51594,24919],{"className":51595},[135],[23,51597],{"className":51598,"style":392},[148],[23,51600,5106],{"className":51601},[612],[23,51603],{"className":51604,"style":392},[148],[23,51606,51608,51612,51615,51618,51621,51624,51627,51630,51633],{"className":51607},[126],[23,51609],{"className":51610,"style":51611},[130],"height:0.8389em;vertical-align:-0.1944em;",[23,51613,51487],{"className":51614},[135],[23,51616,747],{"className":51617},[824],[23,51619,5360],{"className":51620},[148],[23,51622],{"className":51623,"style":280},[148],[23,51625,24919],{"className":51626},[135],[23,51628],{"className":51629,"style":392},[148],[23,51631,5106],{"className":51632},[612],[23,51634],{"className":51635,"style":392},[148],[23,51637,51639,51642,51645,51648,51651,51654],{"className":51638},[126],[23,51640],{"className":51641,"style":131},[130],[23,51643,24919],{"className":51644},[135],[23,51646,66],{"className":51647},[169],[23,51649],{"className":51650,"style":149},[148],[23,51652,69],{"className":51653},[153],[23,51655],{"className":51656,"style":149},[148],[23,51658,51660,51663,51669,51672,51675,51678,51681,51684,51687],{"className":51659},[126],[23,51661],{"className":51662,"style":131},[130],[23,51664,51666],{"className":51665},[135],[23,51667,5392],{"className":51668},[135,5391],[23,51670,54],{"className":51671},[140],[23,51673,46690],{"className":51674},[135],[23,51676,747],{"className":51677},[824],[23,51679],{"className":51680,"style":280},[148],[23,51682,46693],{"className":51683},[135],[23,51685,66],{"className":51686},[169],[23,51688,112],{"className":51689},[135],[11,51691,51692],{},"把几个估计量放在一起：",[3764,51694,51695,51705],{},[3767,51696,51697],{},[3770,51698,51699,51702],{},[3773,51700,51701],{},"量",[3773,51703,51704],{},"值",[3783,51706,51707,51771,51835,51899,51964],{},[3770,51708,51709,51712],{},[3788,51710,51711],{},"先验均值",[3788,51713,51714],{},[23,51715,51717,51740],{"className":51716,"translate":27},[31],[23,51718,51720],{"className":51719},[35],[37,51721,51722],{"xmlns":39},[42,51723,51724,51737],{},[45,51725,51726,51728,51730,51732,51734],{},[2581,51727,24919],{},[48,51729,2875],{"mathvariant":111},[2581,51731,46693],{},[51,51733,69],{},[2581,51735,51736],{},"0.5",[114,51738,51739],{"encoding":116},"3\u002F6=0.5",[23,51741,51743,51762],{"className":51742,"ariaHidden":122},[121],[23,51744,51746,51749,51753,51756,51759],{"className":51745},[126],[23,51747],{"className":51748,"style":131},[130],[23,51750,51752],{"className":51751},[135],"3\u002F6",[23,51754],{"className":51755,"style":149},[148],[23,51757,69],{"className":51758},[153],[23,51760],{"className":51761,"style":149},[148],[23,51763,51765,51768],{"className":51764},[126],[23,51766],{"className":51767,"style":2770},[130],[23,51769,51736],{"className":51770},[135],[3770,51772,51773,51776],{},[3788,51774,51775],{},"最大似然估计",[3788,51777,51778],{},[23,51779,51781,51804],{"className":51780,"translate":27},[31],[23,51782,51784],{"className":51783},[35],[37,51785,51786],{"xmlns":39},[42,51787,51788,51801],{},[45,51789,51790,51792,51794,51796,51798],{},[2581,51791,51487],{},[48,51793,2875],{"mathvariant":111},[2581,51795,46690],{},[51,51797,69],{},[2581,51799,51800],{},"0.7",[114,51802,51803],{"encoding":116},"7\u002F10=0.7",[23,51805,51807,51826],{"className":51806,"ariaHidden":122},[121],[23,51808,51810,51813,51817,51820,51823],{"className":51809},[126],[23,51811],{"className":51812,"style":131},[130],[23,51814,51816],{"className":51815},[135],"7\u002F10",[23,51818],{"className":51819,"style":149},[148],[23,51821,69],{"className":51822},[153],[23,51824],{"className":51825,"style":149},[148],[23,51827,51829,51832],{"className":51828},[126],[23,51830],{"className":51831,"style":2770},[130],[23,51833,51800],{"className":51834},[135],[3770,51836,51837,51839],{},[3788,51838,11331],{},[3788,51840,51841],{},[23,51842,51844,51868],{"className":51843,"translate":27},[31],[23,51845,51847],{"className":51846},[35],[37,51848,51849],{"xmlns":39},[42,51850,51851,51865],{},[45,51852,51853,51855,51857,51860,51862],{},[2581,51854,46690],{},[48,51856,2875],{"mathvariant":111},[2581,51858,51859],{},"16",[51,51861,69],{},[2581,51863,51864],{},"0.625",[114,51866,51867],{"encoding":116},"10\u002F16=0.625",[23,51869,51871,51890],{"className":51870,"ariaHidden":122},[121],[23,51872,51874,51877,51881,51884,51887],{"className":51873},[126],[23,51875],{"className":51876,"style":131},[130],[23,51878,51880],{"className":51879},[135],"10\u002F16",[23,51882],{"className":51883,"style":149},[148],[23,51885,69],{"className":51886},[153],[23,51888],{"className":51889,"style":149},[148],[23,51891,51893,51896],{"className":51892},[126],[23,51894],{"className":51895,"style":2770},[130],[23,51897,51864],{"className":51898},[135],[3770,51900,51901,51903],{},[3788,51902,13795],{},[3788,51904,51905],{},[23,51906,51908,51933],{"className":51907,"translate":27},[31],[23,51909,51911],{"className":51910},[35],[37,51912,51913],{"xmlns":39},[42,51914,51915,51930],{},[45,51916,51917,51920,51922,51925,51927],{},[2581,51918,51919],{},"9",[48,51921,2875],{"mathvariant":111},[2581,51923,51924],{},"14",[51,51926,29066],{},[2581,51928,51929],{},"0.643",[114,51931,51932],{"encoding":116},"9\u002F14\\approx0.643",[23,51934,51936,51955],{"className":51935,"ariaHidden":122},[121],[23,51937,51939,51942,51946,51949,51952],{"className":51938},[126],[23,51940],{"className":51941,"style":131},[130],[23,51943,51945],{"className":51944},[135],"9\u002F14",[23,51947],{"className":51948,"style":149},[148],[23,51950,29066],{"className":51951},[153],[23,51953],{"className":51954,"style":149},[148],[23,51956,51958,51961],{"className":51957},[126],[23,51959],{"className":51960,"style":2770},[130],[23,51962,51929],{"className":51963},[135],[3770,51965,51966,52104],{},[3788,51967,51968,51969],{},"后验预测 ",[23,51970,51972,52004],{"className":51971,"translate":27},[31],[23,51973,51975],{"className":51974},[35],[37,51976,51977],{"xmlns":39},[42,51978,51979,52001],{},[45,51980,51981,51983,51985,51991,51993,51995,51997,51999],{},[48,51982,11],{},[51,51984,54],{"stretchy":53},[898,51986,51987,51989],{},[48,51988,753],{},[88,51990,6412],{},[51,51992,69],{},[2581,51994,4435],{},[51,51996,60],{},[48,51998,63],{},[51,52000,66],{"stretchy":53},[114,52002,52003],{"encoding":116},"p(x_{\\text{new}}=1\\mid D)",[23,52005,52007,52074,52092],{"className":52006,"ariaHidden":122},[121],[23,52008,52010,52013,52016,52019,52065,52068,52071],{"className":52009},[126],[23,52011],{"className":52012,"style":131},[130],[23,52014,11],{"className":52015},[135,136],[23,52017,54],{"className":52018},[140],[23,52020,52022,52025],{"className":52021},[135],[23,52023,753],{"className":52024},[135,136],[23,52026,52028],{"className":52027},[998],[23,52029,52031,52057],{"className":52030},[199,200],[23,52032,52034,52054],{"className":52033},[204],[23,52035,52037],{"className":52036,"style":6499},[208],[23,52038,52039,52042],{"style":2029},[23,52040],{"className":52041,"style":1015},[216],[23,52043,52045],{"className":52044},[1019,1020,1021,1022],[23,52046,52048],{"className":52047},[135,1022],[23,52049,52051],{"className":52050},[135,6514,1022],[23,52052,6412],{"className":52053},[135,1022],[23,52055,297],{"className":52056},[296],[23,52058,52060],{"className":52059},[204],[23,52061,52063],{"className":52062,"style":2048},[208],[23,52064],{},[23,52066],{"className":52067,"style":149},[148],[23,52069,69],{"className":52070},[153],[23,52072],{"className":52073,"style":149},[148],[23,52075,52077,52080,52083,52086,52089],{"className":52076},[126],[23,52078],{"className":52079,"style":131},[130],[23,52081,4435],{"className":52082},[135],[23,52084],{"className":52085,"style":149},[148],[23,52087,60],{"className":52088},[153],[23,52090],{"className":52091,"style":149},[148],[23,52093,52095,52098,52101],{"className":52094},[126],[23,52096],{"className":52097,"style":131},[130],[23,52099,63],{"className":52100,"style":144},[135,136],[23,52102,66],{"className":52103},[169],[3788,52105,52106],{},[23,52107,52109,52122],{"className":52108,"translate":27},[31],[23,52110,52112],{"className":52111},[35],[37,52113,52114],{"xmlns":39},[42,52115,52116,52120],{},[45,52117,52118],{},[2581,52119,51864],{},[114,52121,51864],{"encoding":116},[23,52123,52125],{"className":52124,"ariaHidden":122},[121],[23,52126,52128,52131],{"className":52127},[126],[23,52129],{"className":52130,"style":2770},[130],[23,52132,51864],{"className":52133},[135],[11,52135,52136,52137,52165,52166,52194,52195,52224,52225,52254,52255,52314,52315,1471],{},"后验均值被先验从 ",[23,52138,52140,52153],{"className":52139,"translate":27},[31],[23,52141,52143],{"className":52142},[35],[37,52144,52145],{"xmlns":39},[42,52146,52147,52151],{},[45,52148,52149],{},[2581,52150,51800],{},[114,52152,51800],{"encoding":116},[23,52154,52156],{"className":52155,"ariaHidden":122},[121],[23,52157,52159,52162],{"className":52158},[126],[23,52160],{"className":52161,"style":2770},[130],[23,52163,51800],{"className":52164},[135]," 拉回到 ",[23,52167,52169,52182],{"className":52168,"translate":27},[31],[23,52170,52172],{"className":52171},[35],[37,52173,52174],{"xmlns":39},[42,52175,52176,52180],{},[45,52177,52178],{},[2581,52179,51864],{},[114,52181,51864],{"encoding":116},[23,52183,52185],{"className":52184,"ariaHidden":122},[121],[23,52186,52188,52191],{"className":52187},[126],[23,52189],{"className":52190,"style":2770},[130],[23,52192,51864],{"className":52193},[135],"，这就是收缩。数据越多，拉力越弱：若观测到 ",[23,52196,52198,52212],{"className":52197,"translate":27},[31],[23,52199,52201],{"className":52200},[35],[37,52202,52203],{"xmlns":39},[42,52204,52205,52210],{},[45,52206,52207],{},[2581,52208,52209],{},"100",[114,52211,52209],{"encoding":116},[23,52213,52215],{"className":52214,"ariaHidden":122},[121],[23,52216,52218,52221],{"className":52217},[126],[23,52219],{"className":52220,"style":2770},[130],[23,52222,52209],{"className":52223},[135]," 次中 ",[23,52226,52228,52242],{"className":52227,"translate":27},[31],[23,52229,52231],{"className":52230},[35],[37,52232,52233],{"xmlns":39},[42,52234,52235,52240],{},[45,52236,52237],{},[2581,52238,52239],{},"70",[114,52241,52239],{"encoding":116},[23,52243,52245],{"className":52244,"ariaHidden":122},[121],[23,52246,52248,52251],{"className":52247},[126],[23,52249],{"className":52250,"style":2770},[130],[23,52252,52239],{"className":52253},[135]," 次正面，后验均值变为 ",[23,52256,52258,52283],{"className":52257,"translate":27},[31],[23,52259,52261],{"className":52260},[35],[37,52262,52263],{"xmlns":39},[42,52264,52265,52280],{},[45,52266,52267,52270,52272,52275,52277],{},[2581,52268,52269],{},"73",[48,52271,2875],{"mathvariant":111},[2581,52273,52274],{},"106",[51,52276,29066],{},[2581,52278,52279],{},"0.689",[114,52281,52282],{"encoding":116},"73\u002F106\\approx0.689",[23,52284,52286,52305],{"className":52285,"ariaHidden":122},[121],[23,52287,52289,52292,52296,52299,52302],{"className":52288},[126],[23,52290],{"className":52291,"style":131},[130],[23,52293,52295],{"className":52294},[135],"73\u002F106",[23,52297],{"className":52298,"style":149},[148],[23,52300,29066],{"className":52301},[153],[23,52303],{"className":52304,"style":149},[148],[23,52306,52308,52311],{"className":52307},[126],[23,52309],{"className":52310,"style":2770},[130],[23,52312,52279],{"className":52313},[135],"，已经非常接近 ",[23,52316,52318,52331],{"className":52317,"translate":27},[31],[23,52319,52321],{"className":52320},[35],[37,52322,52323],{"xmlns":39},[42,52324,52325,52329],{},[45,52326,52327],{},[2581,52328,51800],{},[114,52330,51800],{"encoding":116},[23,52332,52334],{"className":52333,"ariaHidden":122},[121],[23,52335,52337,52340],{"className":52336},[126],[23,52338],{"className":52339,"style":2770},[130],[23,52341,51800],{"className":52342},[135],[11,52344,52345,52346,52378,52379,52407,52408,52436,52437,52465],{},"如果决策是「猜下一次是否为正面」，且两类错误的代价相同，那么只要后验预测大于 ",[23,52347,52349,52366],{"className":52348,"translate":27},[31],[23,52350,52352],{"className":52351},[35],[37,52353,52354],{"xmlns":39},[42,52355,52356,52364],{},[45,52357,52358,52360,52362],{},[2581,52359,4435],{},[48,52361,2875],{"mathvariant":111},[2581,52363,9952],{},[114,52365,10469],{"encoding":116},[23,52367,52369],{"className":52368,"ariaHidden":122},[121],[23,52370,52372,52375],{"className":52371},[126],[23,52373],{"className":52374,"style":131},[130],[23,52376,10469],{"className":52377},[135]," 就猜正面，",[23,52380,52382,52395],{"className":52381,"translate":27},[31],[23,52383,52385],{"className":52384},[35],[37,52386,52387],{"xmlns":39},[42,52388,52389,52393],{},[45,52390,52391],{},[2581,52392,51864],{},[114,52394,51864],{"encoding":116},[23,52396,52398],{"className":52397,"ariaHidden":122},[121],[23,52399,52401,52404],{"className":52400},[126],[23,52402],{"className":52403,"style":2770},[130],[23,52405,51864],{"className":52406},[135]," 满足条件。若两类错误的代价不同：把反面误判为正面的代价为 ",[23,52409,52411,52424],{"className":52410,"translate":27},[31],[23,52412,52414],{"className":52413},[35],[37,52415,52416],{"xmlns":39},[42,52417,52418,52422],{},[45,52419,52420],{},[2581,52421,9952],{},[114,52423,9952],{"encoding":116},[23,52425,52427],{"className":52426,"ariaHidden":122},[121],[23,52428,52430,52433],{"className":52429},[126],[23,52431],{"className":52432,"style":2770},[130],[23,52434,9952],{"className":52435},[135],"，把正面误判为反面的代价为 ",[23,52438,52440,52453],{"className":52439,"translate":27},[31],[23,52441,52443],{"className":52442},[35],[37,52444,52445],{"xmlns":39},[42,52446,52447,52451],{},[45,52448,52449],{},[2581,52450,4435],{},[114,52452,4435],{"encoding":116},[23,52454,52456],{"className":52455,"ariaHidden":122},[121],[23,52457,52459,52462],{"className":52458},[126],[23,52460],{"className":52461,"style":2770},[130],[23,52463,4435],{"className":52464},[135],"，则阈值变为",[23,52467,52469],{"className":52468,"translate":27},[26],[23,52470,52472,52544],{"className":52471,"translate":27},[31],[23,52473,52475],{"className":52474},[35],[37,52476,52477],{"xmlns":39,"display":40},[42,52478,52479,52541],{},[45,52480,52481,52517,52519,52531,52533,52539],{},[71,52482,52483,52493],{},[898,52484,52485,52487],{},[48,52486,12710],{},[45,52488,52489,52491],{},[48,52490,12715],{"mathvariant":111},[48,52492,920],{"mathvariant":111},[45,52494,52495,52505,52507],{},[898,52496,52497,52499],{},[48,52498,12710],{},[45,52500,52501,52503],{},[48,52502,12715],{"mathvariant":111},[48,52504,920],{"mathvariant":111},[51,52506,5106],{},[898,52508,52509,52511],{},[48,52510,12710],{},[45,52512,52513,52515],{},[48,52514,12715],{"mathvariant":111},[48,52516,12801],{"mathvariant":111},[51,52518,69],{},[71,52520,52521,52523],{},[2581,52522,9952],{},[45,52524,52525,52527,52529],{},[2581,52526,9952],{},[51,52528,5106],{},[2581,52530,4435],{},[51,52532,69],{},[71,52534,52535,52537],{},[2581,52536,9952],{},[2581,52538,24919],{},[51,52540,747],{"separator":122},[114,52542,52543],{"encoding":116},"\\frac{c_{\\mathrm{FP}}}{c_{\\mathrm{FP}}+c_{\\mathrm{FN}}}\n=\\frac{2}{2+1}=\\frac23,",[23,52545,52547,52765,52856],{"className":52546,"ariaHidden":122},[121],[23,52548,52550,52553,52756,52759,52762],{"className":52549},[126],[23,52551],{"className":52552,"style":13486},[130],[23,52554,52556,52559,52753],{"className":52555},[135],[23,52557],{"className":52558},[140,192],[23,52560,52562],{"className":52561},[71],[23,52563,52565,52745],{"className":52564},[199,200],[23,52566,52568,52742],{"className":52567},[204],[23,52569,52571,52680,52688],{"className":52570,"style":6056},[208],[23,52572,52573,52576],{"style":212},[23,52574],{"className":52575,"style":217},[216],[23,52577,52579,52625,52628,52631,52634],{"className":52578},[135],[23,52580,52582,52585],{"className":52581},[135],[23,52583,12710],{"className":52584},[135,136],[23,52586,52588],{"className":52587},[998],[23,52589,52591,52617],{"className":52590},[199,200],[23,52592,52594,52614],{"className":52593},[204],[23,52595,52597],{"className":52596,"style":2026},[208],[23,52598,52599,52602],{"style":2029},[23,52600],{"className":52601,"style":1015},[216],[23,52603,52605],{"className":52604},[1019,1020,1021,1022],[23,52606,52608],{"className":52607},[135,1022],[23,52609,52611],{"className":52610},[135,1022],[23,52612,12766],{"className":52613},[135,5391,1022],[23,52615,297],{"className":52616},[296],[23,52618,52620],{"className":52619},[204],[23,52621,52623],{"className":52622,"style":2048},[208],[23,52624],{},[23,52626],{"className":52627,"style":392},[148],[23,52629,5106],{"className":52630},[612],[23,52632],{"className":52633,"style":392},[148],[23,52635,52637,52640],{"className":52636},[135],[23,52638,12710],{"className":52639},[135,136],[23,52641,52643],{"className":52642},[998],[23,52644,52646,52672],{"className":52645},[199,200],[23,52647,52649,52669],{"className":52648},[204],[23,52650,52652],{"className":52651,"style":2026},[208],[23,52653,52654,52657],{"style":2029},[23,52655],{"className":52656,"style":1015},[216],[23,52658,52660],{"className":52659},[1019,1020,1021,1022],[23,52661,52663],{"className":52662},[135,1022],[23,52664,52666],{"className":52665},[135,1022],[23,52667,12849],{"className":52668},[135,5391,1022],[23,52670,297],{"className":52671},[296],[23,52673,52675],{"className":52674},[204],[23,52676,52678],{"className":52677,"style":2048},[208],[23,52679],{},[23,52681,52682,52685],{"style":235},[23,52683],{"className":52684,"style":217},[216],[23,52686],{"className":52687,"style":243},[242],[23,52689,52690,52693],{"style":246},[23,52691],{"className":52692,"style":217},[216],[23,52694,52696],{"className":52695},[135],[23,52697,52699,52702],{"className":52698},[135],[23,52700,12710],{"className":52701},[135,136],[23,52703,52705],{"className":52704},[998],[23,52706,52708,52734],{"className":52707},[199,200],[23,52709,52711,52731],{"className":52710},[204],[23,52712,52714],{"className":52713,"style":2026},[208],[23,52715,52716,52719],{"style":2029},[23,52717],{"className":52718,"style":1015},[216],[23,52720,52722],{"className":52721},[1019,1020,1021,1022],[23,52723,52725],{"className":52724},[135,1022],[23,52726,52728],{"className":52727},[135,1022],[23,52729,12766],{"className":52730},[135,5391,1022],[23,52732,297],{"className":52733},[296],[23,52735,52737],{"className":52736},[204],[23,52738,52740],{"className":52739,"style":2048},[208],[23,52741],{},[23,52743,297],{"className":52744},[296],[23,52746,52748],{"className":52747},[204],[23,52749,52751],{"className":52750,"style":13685},[208],[23,52752],{},[23,52754],{"className":52755},[169,192],[23,52757],{"className":52758,"style":149},[148],[23,52760,69],{"className":52761},[153],[23,52763],{"className":52764,"style":149},[148],[23,52766,52768,52772,52847,52850,52853],{"className":52767},[126],[23,52769],{"className":52770,"style":52771},[130],"height:2.0908em;vertical-align:-0.7693em;",[23,52773,52775,52778,52844],{"className":52774},[135],[23,52776],{"className":52777},[140,192],[23,52779,52781],{"className":52780},[71],[23,52782,52784,52835],{"className":52783},[199,200],[23,52785,52787,52832],{"className":52786},[204],[23,52788,52790,52813,52821],{"className":52789,"style":12409},[208],[23,52791,52792,52795],{"style":212},[23,52793],{"className":52794,"style":217},[216],[23,52796,52798,52801,52804,52807,52810],{"className":52797},[135],[23,52799,9952],{"className":52800},[135],[23,52802],{"className":52803,"style":392},[148],[23,52805,5106],{"className":52806},[612],[23,52808],{"className":52809,"style":392},[148],[23,52811,4435],{"className":52812},[135],[23,52814,52815,52818],{"style":235},[23,52816],{"className":52817,"style":217},[216],[23,52819],{"className":52820,"style":243},[242],[23,52822,52823,52826],{"style":246},[23,52824],{"className":52825,"style":217},[216],[23,52827,52829],{"className":52828},[135],[23,52830,9952],{"className":52831},[135],[23,52833,297],{"className":52834},[296],[23,52836,52838],{"className":52837},[204],[23,52839,52842],{"className":52840,"style":52841},[208],"height:0.7693em;",[23,52843],{},[23,52845],{"className":52846},[169,192],[23,52848],{"className":52849,"style":149},[148],[23,52851,69],{"className":52852},[153],[23,52854],{"className":52855,"style":149},[148],[23,52857,52859,52862,52924],{"className":52858},[126],[23,52860],{"className":52861,"style":12390},[130],[23,52863,52865,52868,52921],{"className":52864},[135],[23,52866],{"className":52867},[140,192],[23,52869,52871],{"className":52870},[71],[23,52872,52874,52913],{"className":52873},[199,200],[23,52875,52877,52910],{"className":52876},[204],[23,52878,52880,52891,52899],{"className":52879,"style":12409},[208],[23,52881,52882,52885],{"style":212},[23,52883],{"className":52884,"style":217},[216],[23,52886,52888],{"className":52887},[135],[23,52889,24919],{"className":52890},[135],[23,52892,52893,52896],{"style":235},[23,52894],{"className":52895,"style":217},[216],[23,52897],{"className":52898,"style":243},[242],[23,52900,52901,52904],{"style":246},[23,52902],{"className":52903,"style":217},[216],[23,52905,52907],{"className":52906},[135],[23,52908,9952],{"className":52909},[135],[23,52911,297],{"className":52912},[296],[23,52914,52916],{"className":52915},[204],[23,52917,52919],{"className":52918,"style":6287},[208],[23,52920],{},[23,52922],{"className":52923},[169,192],[23,52925,747],{"className":52926},[824],[11,52928,52929,52930,52987],{},"而 ",[23,52931,52933,52955],{"className":52932,"translate":27},[31],[23,52934,52936],{"className":52935},[35],[37,52937,52938],{"xmlns":39},[42,52939,52940,52952],{},[45,52941,52942,52944,52946,52948,52950],{},[2581,52943,51864],{},[51,52945,9983],{},[2581,52947,9952],{},[48,52949,2875],{"mathvariant":111},[2581,52951,24919],{},[114,52953,52954],{"encoding":116},"0.625\u003C2\u002F3",[23,52956,52958,52977],{"className":52957,"ariaHidden":122},[121],[23,52959,52961,52965,52968,52971,52974],{"className":52960},[126],[23,52962],{"className":52963,"style":52964},[130],"height:0.6835em;vertical-align:-0.0391em;",[23,52966,51864],{"className":52967},[135],[23,52969],{"className":52970,"style":149},[148],[23,52972,9983],{"className":52973},[153],[23,52975],{"className":52976,"style":149},[148],[23,52978,52980,52983],{"className":52979},[126],[23,52981],{"className":52982,"style":131},[130],[23,52984,52986],{"className":52985},[135],"2\u002F3","，最优行动变成猜反面。同一个后验，换一个损失，结论就变了。这正是贝叶斯决策理论与「最大后验」的分水岭：后验只有一个，行动却取决于代价。",[2501,52989,52991],{"id":52990},"_82-高斯先验与收缩","8.2 高斯先验与收缩",[11,52993,52994,52995,53145,53146,53296,53297,53359,53360,53388],{},"设观测模型 ",[23,52996,52998,53034],{"className":52997,"translate":27},[31],[23,52999,53001],{"className":53000},[35],[37,53002,53003],{"xmlns":39},[42,53004,53005,53031],{},[45,53006,53007,53013,53015,53017,53019,53021,53023,53029],{},[898,53008,53009,53011],{},[48,53010,660],{},[48,53012,33493],{},[51,53014,917],{},[48,53016,12801],{"mathvariant":375},[51,53018,54],{"stretchy":53},[48,53020,57],{},[51,53022,747],{"separator":122},[4518,53024,53025,53027],{},[48,53026,29634],{},[2581,53028,9952],{},[51,53030,66],{"stretchy":53},[114,53032,53033],{"encoding":116},"y_i\\sim\\mathcal N(\\theta,\\sigma^2)",[23,53035,53037,53092],{"className":53036,"ariaHidden":122},[121],[23,53038,53040,53043,53083,53086,53089],{"className":53039},[126],[23,53041],{"className":53042,"style":672},[130],[23,53044,53046,53049],{"className":53045},[135],[23,53047,660],{"className":53048,"style":676},[135,136],[23,53050,53052],{"className":53051},[998],[23,53053,53055,53075],{"className":53054},[199,200],[23,53056,53058,53072],{"className":53057},[204],[23,53059,53061],{"className":53060,"style":33647},[208],[23,53062,53063,53066],{"style":21850},[23,53064],{"className":53065,"style":1015},[216],[23,53067,53069],{"className":53068},[1019,1020,1021,1022],[23,53070,33493],{"className":53071},[135,136,1022],[23,53073,297],{"className":53074},[296],[23,53076,53078],{"className":53077},[204],[23,53079,53081],{"className":53080,"style":2048},[208],[23,53082],{},[23,53084],{"className":53085,"style":149},[148],[23,53087,917],{"className":53088},[153],[23,53090],{"className":53091,"style":149},[148],[23,53093,53095,53098,53101,53104,53107,53110,53113,53142],{"className":53094},[126],[23,53096],{"className":53097,"style":10661},[130],[23,53099,12801],{"className":53100,"style":22218},[135,412],[23,53102,54],{"className":53103},[140],[23,53105,57],{"className":53106,"style":144},[135,136],[23,53108,747],{"className":53109},[824],[23,53111],{"className":53112,"style":280},[148],[23,53114,53116,53119],{"className":53115},[135],[23,53117,29634],{"className":53118,"style":676},[135,136],[23,53120,53122],{"className":53121},[998],[23,53123,53125],{"className":53124},[199],[23,53126,53128],{"className":53127},[204],[23,53129,53131],{"className":53130,"style":4693},[208],[23,53132,53133,53136],{"style":4696},[23,53134],{"className":53135,"style":1015},[216],[23,53137,53139],{"className":53138},[1019,1020,1021,1022],[23,53140,9952],{"className":53141},[135,1022],[23,53143,66],{"className":53144},[169],"，先验 ",[23,53147,53149,53185],{"className":53148,"translate":27},[31],[23,53150,53152],{"className":53151},[35],[37,53153,53154],{"xmlns":39},[42,53155,53156,53182],{},[45,53157,53158,53160,53162,53164,53166,53172,53174,53180],{},[48,53159,57],{},[51,53161,917],{},[48,53163,12801],{"mathvariant":375},[51,53165,54],{"stretchy":53},[898,53167,53168,53170],{},[48,53169,22105],{},[2581,53171,1532],{},[51,53173,747],{"separator":122},[4518,53175,53176,53178],{},[48,53177,34348],{},[2581,53179,9952],{},[51,53181,66],{"stretchy":53},[114,53183,53184],{"encoding":116},"\\theta\\sim\\mathcal N(\\mu_0,\\tau^2)",[23,53186,53188,53206],{"className":53187,"ariaHidden":122},[121],[23,53189,53191,53194,53197,53200,53203],{"className":53190},[126],[23,53192],{"className":53193,"style":779},[130],[23,53195,57],{"className":53196,"style":144},[135,136],[23,53198],{"className":53199,"style":149},[148],[23,53201,917],{"className":53202},[153],[23,53204],{"className":53205,"style":149},[148],[23,53207,53209,53212,53215,53218,53258,53261,53264,53293],{"className":53208},[126],[23,53210],{"className":53211,"style":10661},[130],[23,53213,12801],{"className":53214,"style":22218},[135,412],[23,53216,54],{"className":53217},[140],[23,53219,53221,53224],{"className":53220},[135],[23,53222,22105],{"className":53223},[135,136],[23,53225,53227],{"className":53226},[998],[23,53228,53230,53250],{"className":53229},[199,200],[23,53231,53233,53247],{"className":53232},[204],[23,53234,53236],{"className":53235,"style":18647},[208],[23,53237,53238,53241],{"style":2029},[23,53239],{"className":53240,"style":1015},[216],[23,53242,53244],{"className":53243},[1019,1020,1021,1022],[23,53245,1532],{"className":53246},[135,1022],[23,53248,297],{"className":53249},[296],[23,53251,53253],{"className":53252},[204],[23,53254,53256],{"className":53255,"style":2048},[208],[23,53257],{},[23,53259,747],{"className":53260},[824],[23,53262],{"className":53263,"style":280},[148],[23,53265,53267,53270],{"className":53266},[135],[23,53268,34348],{"className":53269,"style":34407},[135,136],[23,53271,53273],{"className":53272},[998],[23,53274,53276],{"className":53275},[199],[23,53277,53279],{"className":53278},[204],[23,53280,53282],{"className":53281,"style":4693},[208],[23,53283,53284,53287],{"style":4696},[23,53285],{"className":53286,"style":1015},[216],[23,53288,53290],{"className":53289},[1019,1020,1021,1022],[23,53291,9952],{"className":53292},[135,1022],[23,53294,66],{"className":53295},[169],"，样本均值为 ",[23,53298,53300,53318],{"className":53299,"translate":27},[31],[23,53301,53303],{"className":53302},[35],[37,53304,53305],{"xmlns":39},[42,53306,53307,53315],{},[45,53308,53309],{},[16807,53310,53311,53313],{"accent":122},[48,53312,753],{},[51,53314,39691],{},[114,53316,53317],{"encoding":116},"\\bar x",[23,53319,53321],{"className":53320,"ariaHidden":122},[121],[23,53322,53324,53327],{"className":53323},[126],[23,53325],{"className":53326,"style":39769},[130],[23,53328,53330],{"className":53329},[135,16830],[23,53331,53333],{"className":53332},[199],[23,53334,53336],{"className":53335},[204],[23,53337,53339,53347],{"className":53338,"style":39769},[208],[23,53340,53341,53344],{"style":7130},[23,53342],{"className":53343,"style":217},[216],[23,53345,753],{"className":53346},[135,136],[23,53348,53349,53352],{"style":7130},[23,53350],{"className":53351,"style":217},[216],[23,53353,53356],{"className":53354,"style":53355},[16858],"left:-0.2222em;",[23,53357,39691],{"className":53358},[135],"，样本量为 ",[23,53361,53363,53376],{"className":53362,"translate":27},[31],[23,53364,53366],{"className":53365},[35],[37,53367,53368],{"xmlns":39},[42,53369,53370,53374],{},[45,53371,53372],{},[48,53373,4806],{},[114,53375,4806],{"encoding":116},[23,53377,53379],{"className":53378,"ariaHidden":122},[121],[23,53380,53382,53385],{"className":53381},[126],[23,53383],{"className":53384,"style":703},[130],[23,53386,4806],{"className":53387},[135,136],"。后验仍是高斯：",[23,53390,53392],{"className":53391,"translate":27},[26],[23,53393,53395,53439],{"className":53394,"translate":27},[31],[23,53396,53398],{"className":53397},[35],[37,53399,53400],{"xmlns":39,"display":40},[42,53401,53402,53436],{},[45,53403,53404,53406,53408,53410,53412,53414,53416,53422,53424,53432,53434],{},[48,53405,57],{},[51,53407,60],{},[48,53409,63],{},[51,53411,917],{},[48,53413,12801],{"mathvariant":375},[51,53415,54],{"stretchy":53},[898,53417,53418,53420],{},[48,53419,22105],{},[48,53421,4806],{},[51,53423,747],{"separator":122},[19137,53425,53426,53428,53430],{},[48,53427,34348],{},[48,53429,4806],{},[2581,53431,9952],{},[51,53433,66],{"stretchy":53},[51,53435,747],{"separator":122},[114,53437,53438],{"encoding":116},"\\theta\\mid D\\sim\\mathcal N(\\mu_n,\\tau_n^2),",[23,53440,53442,53460,53478],{"className":53441,"ariaHidden":122},[121],[23,53443,53445,53448,53451,53454,53457],{"className":53444},[126],[23,53446],{"className":53447,"style":131},[130],[23,53449,57],{"className":53450,"style":144},[135,136],[23,53452],{"className":53453,"style":149},[148],[23,53455,60],{"className":53456},[153],[23,53458],{"className":53459,"style":149},[148],[23,53461,53463,53466,53469,53472,53475],{"className":53462},[126],[23,53464],{"className":53465,"style":470},[130],[23,53467,63],{"className":53468,"style":144},[135,136],[23,53470],{"className":53471,"style":149},[148],[23,53473,917],{"className":53474},[153],[23,53476],{"className":53477,"style":149},[148],[23,53479,53481,53484,53487,53490,53530,53533,53536,53588,53591],{"className":53480},[126],[23,53482],{"className":53483,"style":26499},[130],[23,53485,12801],{"className":53486,"style":22218},[135,412],[23,53488,54],{"className":53489},[140],[23,53491,53493,53496],{"className":53492},[135],[23,53494,22105],{"className":53495},[135,136],[23,53497,53499],{"className":53498},[998],[23,53500,53502,53522],{"className":53501},[199,200],[23,53503,53505,53519],{"className":53504},[204],[23,53506,53508],{"className":53507,"style":6499},[208],[23,53509,53510,53513],{"style":2029},[23,53511],{"className":53512,"style":1015},[216],[23,53514,53516],{"className":53515},[1019,1020,1021,1022],[23,53517,4806],{"className":53518},[135,136,1022],[23,53520,297],{"className":53521},[296],[23,53523,53525],{"className":53524},[204],[23,53526,53528],{"className":53527,"style":2048},[208],[23,53529],{},[23,53531,747],{"className":53532},[824],[23,53534],{"className":53535,"style":280},[148],[23,53537,53539,53542],{"className":53538},[135],[23,53540,34348],{"className":53541,"style":34407},[135,136],[23,53543,53545],{"className":53544},[998],[23,53546,53548,53580],{"className":53547},[199,200],[23,53549,53551,53577],{"className":53550},[204],[23,53552,53554,53566],{"className":53553,"style":11002},[208],[23,53555,53557,53560],{"style":53556},"top:-2.453em;margin-left:-0.1132em;margin-right:0.05em;",[23,53558],{"className":53559,"style":1015},[216],[23,53561,53563],{"className":53562},[1019,1020,1021,1022],[23,53564,4806],{"className":53565},[135,136,1022],[23,53567,53568,53571],{"style":4989},[23,53569],{"className":53570,"style":1015},[216],[23,53572,53574],{"className":53573},[1019,1020,1021,1022],[23,53575,9952],{"className":53576},[135,1022],[23,53578,297],{"className":53579},[296],[23,53581,53583],{"className":53582},[204],[23,53584,53586],{"className":53585,"style":19375},[208],[23,53587],{},[23,53589,66],{"className":53590},[169],[23,53592,747],{"className":53593},[824],[11,53595,2109],{},[23,53597,53599],{"className":53598,"translate":27},[26],[23,53600,53602,53724],{"className":53601,"translate":27},[31],[23,53603,53605],{"className":53604},[35],[37,53606,53607],{"xmlns":39,"display":40},[42,53608,53609,53721],{},[45,53610,53611,53617,53619,53679,53681,53683,53695,53697,53707,53709,53719],{},[898,53612,53613,53615],{},[48,53614,22105],{},[48,53616,4806],{},[51,53618,69],{},[71,53620,53621,53655],{},[45,53622,53623,53629,53631,53637,53639,53641,53647,53649],{},[898,53624,53625,53627],{},[48,53626,22105],{},[2581,53628,1532],{},[48,53630,2875],{"mathvariant":111},[4518,53632,53633,53635],{},[48,53634,34348],{},[2581,53636,9952],{},[51,53638,5106],{},[48,53640,4806],{},[16807,53642,53643,53645],{"accent":122},[48,53644,753],{},[51,53646,39691],{},[48,53648,2875],{"mathvariant":111},[4518,53650,53651,53653],{},[48,53652,29634],{},[2581,53654,9952],{},[45,53656,53657,53659,53661,53667,53669,53671,53673],{},[2581,53658,4435],{},[48,53660,2875],{"mathvariant":111},[4518,53662,53663,53665],{},[48,53664,34348],{},[2581,53666,9952],{},[51,53668,5106],{},[48,53670,4806],{},[48,53672,2875],{"mathvariant":111},[4518,53674,53675,53677],{},[48,53676,29634],{},[2581,53678,9952],{},[51,53680,747],{"separator":122},[148,53682],{"width":750},[71,53684,53685,53687],{},[2581,53686,4435],{},[19137,53688,53689,53691,53693],{},[48,53690,34348],{},[48,53692,4806],{},[2581,53694,9952],{},[51,53696,69],{},[71,53698,53699,53701],{},[2581,53700,4435],{},[4518,53702,53703,53705],{},[48,53704,34348],{},[2581,53706,9952],{},[51,53708,5106],{},[71,53710,53711,53713],{},[48,53712,4806],{},[4518,53714,53715,53717],{},[48,53716,29634],{},[2581,53718,9952],{},[48,53720,112],{"mathvariant":111},[114,53722,53723],{"encoding":116},"\\mu_n=\n\\frac{\\mu_0\u002F\\tau^2+n\\bar 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5768,"style":149},[148],[23,55770,55772,55775],{"className":55771},[126],[23,55773],{"className":55774,"style":2770},[130],[23,55776,55743],{"className":55777},[135],[3770,55779,55780,55783,55843],{},[3788,55781,55782],{},"25",[3788,55784,55785],{},[23,55786,55788,55812],{"className":55787,"translate":27},[31],[23,55789,55791],{"className":55790},[35],[37,55792,55793],{"xmlns":39},[42,55794,55795,55809],{},[45,55796,55797,55799,55801,55804,55806],{},[2581,55798,55782],{},[48,55800,2875],{"mathvariant":111},[2581,55802,55803],{},"26",[51,55805,29066],{},[2581,55807,55808],{},"0.962",[114,55810,55811],{"encoding":116},"25\u002F26\\approx0.962",[23,55813,55815,55834],{"className":55814,"ariaHidden":122},[121],[23,55816,55818,55821,55825,55828,55831],{"className":55817},[126],[23,55819],{"className":55820,"style":131},[130],[23,55822,55824],{"className":55823},[135],"25\u002F26",[23,55826],{"className":55827,"style":149},[148],[23,55829,29066],{"className":55830},[153],[23,55832],{"className":55833,"style":149},[148],[23,55835,55837,55840],{"className":55836},[126],[23,55838],{"className":55839,"style":2770},[130],[23,55841,55808],{"className":55842},[135],[3788,55844,55845],{},[23,55846,55848,55871],{"className":55847,"translate":27},[31],[23,55849,55851],{"className":55850},[35],[37,55852,55853],{"xmlns":39},[42,55854,55855,55868],{},[45,55856,55857,55859,55861,55863,55865],{},[2581,55858,4435],{},[48,55860,2875],{"mathvariant":111},[2581,55862,55803],{},[51,55864,29066],{},[2581,55866,55867],{},"0.038",[114,55869,55870],{"encoding":116},"1\u002F26\\approx0.038",[23,55872,55874,55893],{"className":55873,"ariaHidden":122},[121],[23,55875,55877,55880,55884,55887,55890],{"className":55876},[126],[23,55878],{"className":55879,"style":131},[130],[23,55881,55883],{"className":55882},[135],"1\u002F26",[23,55885],{"className":55886,"style":149},[148],[23,55888,29066],{"className":55889},[153],[23,55891],{"className":55892,"style":149},[148],[23,55894,55896,55899],{"className":55895},[126],[23,55897],{"className":55898,"style":2770},[130],[23,55900,55867],{"className":55901},[135],[3770,55903,55904,55906,55966],{},[3788,55905,52209],{},[3788,55907,55908],{},[23,55909,55911,55935],{"className":55910,"translate":27},[31],[23,55912,55914],{"className":55913},[35],[37,55915,55916],{"xmlns":39},[42,55917,55918,55932],{},[45,55919,55920,55922,55924,55927,55929],{},[2581,55921,52209],{},[48,55923,2875],{"mathvariant":111},[2581,55925,55926],{},"101",[51,55928,29066],{},[2581,55930,55931],{},"0.990",[114,55933,55934],{"encoding":116},"100\u002F101\\approx0.990",[23,55936,55938,55957],{"className":55937,"ariaHidden":122},[121],[23,55939,55941,55944,55948,55951,55954],{"className":55940},[126],[23,55942],{"className":55943,"style":131},[130],[23,55945,55947],{"className":55946},[135],"100\u002F101",[23,55949],{"className":55950,"style":149},[148],[23,55952,29066],{"className":55953},[153],[23,55955],{"className":55956,"style":149},[148],[23,55958,55960,55963],{"className":55959},[126],[23,55961],{"className":55962,"style":2770},[130],[23,55964,55931],{"className":55965},[135],[3788,55967,55968],{},[23,55969,55971,55994],{"className":55970,"translate":27},[31],[23,55972,55974],{"className":55973},[35],[37,55975,55976],{"xmlns":39},[42,55977,55978,55991],{},[45,55979,55980,55982,55984,55986,55988],{},[2581,55981,4435],{},[48,55983,2875],{"mathvariant":111},[2581,55985,55926],{},[51,55987,29066],{},[2581,55989,55990],{},"0.0099",[114,55992,55993],{"encoding":116},"1\u002F101\\approx0.0099",[23,55995,55997,56016],{"className":55996,"ariaHidden":122},[121],[23,55998,56000,56003,56007,56010,56013],{"className":55999},[126],[23,56001],{"className":56002,"style":131},[130],[23,56004,56006],{"className":56005},[135],"1\u002F101",[23,56008],{"className":56009,"style":149},[148],[23,56011,29066],{"className":56012},[153],[23,56014],{"className":56015,"style":149},[148],[23,56017,56019,56022],{"className":56018},[126],[23,56020],{"className":56021,"style":2770},[130],[23,56023,55990],{"className":56024},[135],[11,56026,56027],{},"样本越多，后验越靠近样本均值，方差越小。先验的作用不是「永远拉住估计」，而是在小样本时提供一个稳定的起点，随着证据积累自动退场。",[2501,56029,56031],{"id":56030},"_83-医疗筛查先验似然与代价","8.3 医疗筛查：先验、似然与代价",[11,56033,56034,56035,56069,56070,56103,56104,56137],{},"设某疾病的患病率为 ",[23,56036,56038,56055],{"className":56037,"translate":27},[31],[23,56039,56041],{"className":56040},[35],[37,56042,56043],{"xmlns":39},[42,56044,56045,56052],{},[45,56046,56047,56049],{},[2581,56048,4435],{},[48,56050,56051],{"mathvariant":111},"%",[114,56053,56054],{"encoding":116},"1\\%",[23,56056,56058],{"className":56057,"ariaHidden":122},[121],[23,56059,56061,56065],{"className":56060},[126],[23,56062],{"className":56063,"style":56064},[130],"height:0.8056em;vertical-align:-0.0556em;",[23,56066,56068],{"className":56067},[135],"1%","，检测的灵敏度为 ",[23,56071,56073,56090],{"className":56072,"translate":27},[31],[23,56074,56076],{"className":56075},[35],[37,56077,56078],{"xmlns":39},[42,56079,56080,56087],{},[45,56081,56082,56085],{},[2581,56083,56084],{},"99",[48,56086,56051],{"mathvariant":111},[114,56088,56089],{"encoding":116},"99\\%",[23,56091,56093],{"className":56092,"ariaHidden":122},[121],[23,56094,56096,56099],{"className":56095},[126],[23,56097],{"className":56098,"style":56064},[130],[23,56100,56102],{"className":56101},[135],"99%","，特异度为 ",[23,56105,56107,56124],{"className":56106,"translate":27},[31],[23,56108,56110],{"className":56109},[35],[37,56111,56112],{"xmlns":39},[42,56113,56114,56121],{},[45,56115,56116,56119],{},[2581,56117,56118],{},"95",[48,56120,56051],{"mathvariant":111},[114,56122,56123],{"encoding":116},"95\\%",[23,56125,56127],{"className":56126,"ariaHidden":122},[121],[23,56128,56130,56133],{"className":56129},[126],[23,56131],{"className":56132,"style":56064},[130],[23,56134,56136],{"className":56135},[135],"95%","。一个人检测阳性，患病概率是多少？",[11,56139,56140],{},"用贝叶斯公式：",[23,56142,56144],{"className":56143,"translate":27},[26],[23,56145,56147,56345],{"className":56146,"translate":27},[31],[23,56148,56150],{"className":56149},[35],[37,56151,56152],{"xmlns":39,"display":40},[42,56153,56154,56342],{},[1518,56155,56156,56259,56306],{"rowspacing":1520,"columnalign":1521,"columnspacing":1522},[1524,56157,56158,56176],{},[1527,56159,56160],{},[1530,56161,56162],{"scriptlevel":1532,"displaystyle":122},[45,56163,56164,56166,56168,56170,56172,56174],{},[48,56165,920],{},[51,56167,54],{"stretchy":53},[48,56169,63],{},[51,56171,60],{},[51,56173,5106],{},[51,56175,66],{"stretchy":53},[1527,56177,56178],{},[1530,56179,56180],{"scriptlevel":1532,"displaystyle":122},[45,56181,56182,56184,56186],{},[45,56183],{},[51,56185,69],{},[71,56187,56188,56210],{},[45,56189,56190,56192,56194,56196,56198,56200,56202,56204,56206,56208],{},[48,56191,920],{},[51,56193,54],{"stretchy":53},[51,56195,5106],{},[51,56197,60],{},[48,56199,63],{},[51,56201,66],{"stretchy":53},[48,56203,920],{},[51,56205,54],{"stretchy":53},[48,56207,63],{},[51,56209,66],{"stretchy":53},[45,56211,56212,56214,56216,56218,56220,56222,56224,56226,56228,56230,56232,56234,56236,56238,56240,56242,56245,56247,56249,56251,56253,56255,56257],{},[48,56213,920],{},[51,56215,54],{"stretchy":53},[51,56217,5106],{},[51,56219,60],{},[48,56221,63],{},[51,56223,66],{"stretchy":53},[48,56225,920],{},[51,56227,54],{"stretchy":53},[48,56229,63],{},[51,56231,66],{"stretchy":53},[51,56233,5106],{},[48,56235,920],{},[51,56237,54],{"stretchy":53},[51,56239,5106],{},[51,56241,60],{},[48,56243,56244],{"mathvariant":111},"¬",[48,56246,63],{},[51,56248,66],{"stretchy":53},[48,56250,920],{},[51,56252,54],{"stretchy":53},[48,56254,56244],{"mathvariant":111},[48,56256,63],{},[51,56258,66],{"stretchy":53},[1524,56260,56261,56267],{},[1527,56262,56263],{},[1530,56264,56265],{"scriptlevel":1532,"displaystyle":122},[45,56266],{},[1527,56268,56269],{},[1530,56270,56271],{"scriptlevel":1532,"displaystyle":122},[45,56272,56273,56275,56277],{},[45,56274],{},[51,56276,69],{},[71,56278,56279,56289],{},[45,56280,56281,56284,56286],{},[2581,56282,56283],{},"0.99",[51,56285,563],{},[2581,56287,56288],{},"0.01",[45,56290,56291,56293,56295,56297,56299,56302,56304],{},[2581,56292,56283],{},[51,56294,563],{},[2581,56296,56288],{},[51,56298,5106],{},[2581,56300,56301],{},"0.05",[51,56303,563],{},[2581,56305,56283],{},[1524,56307,56308,56314],{},[1527,56309,56310],{},[1530,56311,56312],{"scriptlevel":1532,"displaystyle":122},[45,56313],{},[1527,56315,56316],{},[1530,56317,56318],{"scriptlevel":1532,"displaystyle":122},[45,56319,56320,56322,56324,56337,56339],{},[45,56321],{},[51,56323,69],{},[71,56325,56326,56328],{},[2581,56327,55990],{},[45,56329,56330,56332,56334],{},[2581,56331,55990],{},[51,56333,5106],{},[2581,56335,56336],{},"0.0495",[51,56338,29066],{},[2581,56340,56341],{},"0.167.",[114,56343,56344],{"encoding":116},"\\begin{aligned}\nP(D\\mid +)\n&=\\frac{P(+\\mid D)P(D)}{P(+\\mid D)P(D)+P(+\\mid \\neg D)P(\\neg D)}\\\\\n&=\\frac{0.99\\times0.01}{0.99\\times0.01+0.05\\times0.99}\\\\\n&=\\frac{0.0099}{0.0099+0.0495}\n\\approx0.167 .\n\\end{aligned}",[23,56346,56348],{"className":56347,"ariaHidden":122},[121],[23,56349,56351,56355],{"className":56350},[126],[23,56352],{"className":56353,"style":56354},[130],"height:7.1445em;vertical-align:-3.3223em;",[23,56356,56358],{"className":56357},[135],[23,56359,56361,56438],{"className":56360},[1518],[23,56362,56364],{"className":56363},[1696],[23,56365,56367,56429],{"className":56366},[199,200],[23,56368,56370,56426],{"className":56369},[204],[23,56371,56374,56408,56417],{"className":56372,"style":56373},[208],"height:3.8223em;",[23,56375,56377,56381],{"style":56376},"top:-5.8223em;",[23,56378],{"className":56379,"style":56380},[216],"height:3.427em;",[23,56382,56384,56387,56390,56393,56396,56399,56402,56405],{"className":56383},[135],[23,56385,920],{"className":56386,"style":1047},[135,136],[23,56388,54],{"className":56389},[140],[23,56391,63],{"className":56392,"style":144},[135,136],[23,56394],{"className":56395,"style":149},[148],[23,56397,60],{"className":56398},[153],[23,56400],{"className":56401,"style":149},[148],[23,56403,5106],{"className":56404},[135],[23,56406,66],{"className":56407},[169],[23,56409,56411,56414],{"style":56410},"top:-3.2648em;",[23,56412],{"className":56413,"style":56380},[216],[23,56415],{"className":56416},[135],[23,56418,56420,56423],{"style":56419},"top:-0.8741em;",[23,56421],{"className":56422,"style":56380},[216],[23,56424],{"className":56425},[135],[23,56427,297],{"className":56428},[296],[23,56430,56432],{"className":56431},[204],[23,56433,56436],{"className":56434,"style":56435},[208],"height:3.3223em;",[23,56437],{},[23,56439,56441],{"className":56440},[1753],[23,56442,56444,56888],{"className":56443},[199,200],[23,56445,56447,56885],{"className":56446},[204],[23,56448,56450,56649,56779],{"className":56449,"style":56373},[208],[23,56451,56452,56455],{"style":56376},[23,56453],{"className":56454,"style":56380},[216],[23,56456,56458,56461,56464,56467,56470],{"className":56457},[135],[23,56459],{"className":56460},[135],[23,56462],{"className":56463,"style":149},[148],[23,56465,69],{"className":56466},[153],[23,56468],{"className":56469,"style":149},[148],[23,56471,56473,56476,56646],{"className":56472},[135],[23,56474],{"className":56475},[140,192],[23,56477,56479],{"className":56478},[71],[23,56480,56482,56638],{"className":56481},[199,200],[23,56483,56485,56635],{"className":56484},[204],[23,56486,56488,56583,56591],{"className":56487,"style":209},[208],[23,56489,56490,56493],{"style":212},[23,56491],{"className":56492,"style":217},[216],[23,56494,56496,56499,56502,56505,56508,56511,56514,56517,56520,56523,56526,56529,56532,56535,56538,56541,56544,56547,56550,56553,56556,56559,56562,56565,56568,56571,56574,56577,56580],{"className":56495},[135],[23,56497,920],{"className":56498,"style":1047},[135,136],[23,56500,54],{"className":56501},[140],[23,56503,5106],{"className":56504},[135],[23,56506],{"className":56507,"style":149},[148],[23,56509,60],{"className":56510},[153],[23,56512],{"className":56513,"style":149},[148],[23,56515,63],{"className":56516,"style":144},[135,136],[23,56518,66],{"className":56519},[169],[23,56521,920],{"className":56522,"style":1047},[135,136],[23,56524,54],{"className":56525},[140],[23,56527,63],{"className":56528,"style":144},[135,136],[23,56530,66],{"className":56531},[169],[23,56533],{"className":56534,"style":392},[148],[23,56536,5106],{"className":56537},[612],[23,56539],{"className":56540,"style":392},[148],[23,56542,920],{"className":56543,"style":1047},[135,136],[23,56545,54],{"className":56546},[140],[23,56548,5106],{"className":56549},[135],[23,56551],{"className":56552,"style":149},[148],[23,56554,60],{"className":56555},[153],[23,56557],{"className":56558,"style":149},[148],[23,56560,56244],{"className":56561},[135],[23,56563,63],{"className":56564,"style":144},[135,136],[23,56566,66],{"className":56567},[169],[23,56569,920],{"className":56570,"style":1047},[135,136],[23,56572,54],{"className":56573},[140],[23,56575,56244],{"className":56576},[135],[23,56578,63],{"className":56579,"style":144},[135,136],[23,56581,66],{"className":56582},[169],[23,56584,56585,56588],{"style":235},[23,56586],{"className":56587,"style":217},[216],[23,56589],{"className":56590,"style":243},[242],[23,56592,56593,56596],{"style":246},[23,56594],{"className":56595,"style":217},[216],[23,56597,56599,56602,56605,56608,56611,56614,56617,56620,56623,56626,56629,56632],{"className":56598},[135],[23,56600,920],{"className":56601,"style":1047},[135,136],[23,56603,54],{"className":56604},[140],[23,56606,5106],{"className":56607},[135],[23,56609],{"className":56610,"style":149},[148],[23,56612,60],{"className":56613},[153],[23,56615],{"className":56616,"style":149},[148],[23,56618,63],{"className":56619,"style":144},[135,136],[23,56621,66],{"className":56622},[169],[23,56624,920],{"className":56625,"style":1047},[135,136],[23,56627,54],{"className":56628},[140],[23,56630,63],{"className":56631,"style":144},[135,136],[23,56633,66],{"className":56634},[169],[23,56636,297],{"className":56637},[296],[23,56639,56641],{"className":56640},[204],[23,56642,56644],{"className":56643,"style":304},[208],[23,56645],{},[23,56647],{"className":56648},[169,192],[23,56650,56651,56654],{"style":56410},[23,56652],{"className":56653,"style":56380},[216],[23,56655,56657,56660,56663,56666,56669],{"className":56656},[135],[23,56658],{"className":56659},[135],[23,56661],{"className":56662,"style":149},[148],[23,56664,69],{"className":56665},[153],[23,56667],{"className":56668,"style":149},[148],[23,56670,56672,56675,56776],{"className":56671},[135],[23,56673],{"className":56674},[140,192],[23,56676,56678],{"className":56677},[71],[23,56679,56681,56768],{"className":56680},[199,200],[23,56682,56684,56765],{"className":56683},[204],[23,56685,56687,56734,56742],{"className":56686,"style":12409},[208],[23,56688,56689,56692],{"style":212},[23,56690],{"className":56691,"style":217},[216],[23,56693,56695,56698,56701,56704,56707,56710,56713,56716,56719,56722,56725,56728,56731],{"className":56694},[135],[23,56696,56283],{"className":56697},[135],[23,56699],{"className":56700,"style":392},[148],[23,56702,563],{"className":56703},[612],[23,56705],{"className":56706,"style":392},[148],[23,56708,56288],{"className":56709},[135],[23,56711],{"className":56712,"style":392},[148],[23,56714,5106],{"className":56715},[612],[23,56717],{"className":56718,"style":392},[148],[23,56720,56301],{"className":56721},[135],[23,56723],{"className":56724,"style":392},[148],[23,56726,563],{"className":56727},[612],[23,56729],{"className":56730,"style":392},[148],[23,56732,56283],{"className":56733},[135],[23,56735,56736,56739],{"style":235},[23,56737],{"className":56738,"style":217},[216],[23,56740],{"className":56741,"style":243},[242],[23,56743,56744,56747],{"style":246},[23,56745],{"className":56746,"style":217},[216],[23,56748,56750,56753,56756,56759,56762],{"className":56749},[135],[23,56751,56283],{"className":56752},[135],[23,56754],{"className":56755,"style":392},[148],[23,56757,563],{"className":56758},[612],[23,56760],{"className":56761,"style":392},[148],[23,56763,56288],{"className":56764},[135],[23,56766,297],{"className":56767},[296],[23,56769,56771],{"className":56770},[204],[23,56772,56774],{"className":56773,"style":52841},[208],[23,56775],{},[23,56777],{"className":56778},[169,192],[23,56780,56781,56784],{"style":56419},[23,56782],{"className":56783,"style":56380},[216],[23,56785,56787,56790,56793,56796,56799,56873,56876,56879,56882],{"className":56786},[135],[23,56788],{"className":56789},[135],[23,56791],{"className":56792,"style":149},[148],[23,56794,69],{"className":56795},[153],[23,56797],{"className":56798,"style":149},[148],[23,56800,56802,56805,56870],{"className":56801},[135],[23,56803],{"className":56804},[140,192],[23,56806,56808],{"className":56807},[71],[23,56809,56811,56862],{"className":56810},[199,200],[23,56812,56814,56859],{"className":56813},[204],[23,56815,56817,56840,56848],{"className":56816,"style":12409},[208],[23,56818,56819,56822],{"style":212},[23,56820],{"className":56821,"style":217},[216],[23,56823,56825,56828,56831,56834,56837],{"className":56824},[135],[23,56826,55990],{"className":56827},[135],[23,56829],{"className":56830,"style":392},[148],[23,56832,5106],{"className":56833},[612],[23,56835],{"className":56836,"style":392},[148],[23,56838,56336],{"className":56839},[135],[23,56841,56842,56845],{"style":235},[23,56843],{"className":56844,"style":217},[216],[23,56846],{"className":56847,"style":243},[242],[23,56849,56850,56853],{"style":246},[23,56851],{"className":56852,"style":217},[216],[23,56854,56856],{"className":56855},[135],[23,56857,55990],{"className":56858},[135],[23,56860,297],{"className":56861},[296],[23,56863,56865],{"className":56864},[204],[23,56866,56868],{"className":56867,"style":52841},[208],[23,56869],{},[23,56871],{"className":56872},[169,192],[23,56874],{"className":56875,"style":149},[148],[23,56877,29066],{"className":56878},[153],[23,56880],{"className":56881,"style":149},[148],[23,56883,56341],{"className":56884},[135],[23,56886,297],{"className":56887},[296],[23,56889,56891],{"className":56890},[204],[23,56892,56894],{"className":56893,"style":56435},[208],[23,56895],{},[11,56897,56898,56899,56932,56933,56965,56966,56996,56997,4136,57063,57129],{},"尽管检测看起来很准，阳性者真正患病的概率只有 ",[23,56900,56902,56919],{"className":56901,"translate":27},[31],[23,56903,56905],{"className":56904},[35],[37,56906,56907],{"xmlns":39},[42,56908,56909,56916],{},[45,56910,56911,56914],{},[2581,56912,56913],{},"16.7",[48,56915,56051],{"mathvariant":111},[114,56917,56918],{"encoding":116},"16.7\\%",[23,56920,56922],{"className":56921,"ariaHidden":122},[121],[23,56923,56925,56928],{"className":56924},[126],[23,56926],{"className":56927,"style":56064},[130],[23,56929,56931],{"className":56930},[135],"16.7%","，因为健康人占了绝大多数，",[23,56934,56936,56952],{"className":56935,"translate":27},[31],[23,56937,56939],{"className":56938},[35],[37,56940,56941],{"xmlns":39},[42,56942,56943,56949],{},[45,56944,56945,56947],{},[2581,56946,55679],{},[48,56948,56051],{"mathvariant":111},[114,56950,56951],{"encoding":116},"5\\%",[23,56953,56955],{"className":56954,"ariaHidden":122},[121],[23,56956,56958,56961],{"className":56957},[126],[23,56959],{"className":56960,"style":56064},[130],[23,56962,56964],{"className":56963},[135],"5%"," 的假阳性率乘以 ",[23,56967,56969,56984],{"className":56968,"translate":27},[31],[23,56970,56972],{"className":56971},[35],[37,56973,56974],{"xmlns":39},[42,56975,56976,56982],{},[45,56977,56978,56980],{},[2581,56979,56084],{},[48,56981,56051],{"mathvariant":111},[114,56983,56089],{"encoding":116},[23,56985,56987],{"className":56986,"ariaHidden":122},[121],[23,56988,56990,56993],{"className":56989},[126],[23,56991],{"className":56992,"style":56064},[130],[23,56994,56102],{"className":56995},[135]," 的基础人数，产生了一个比真阳性更大的绝对数量。这就是基础率谬误：把 ",[23,56998,57000,57024],{"className":56999,"translate":27},[31],[23,57001,57003],{"className":57002},[35],[37,57004,57005],{"xmlns":39},[42,57006,57007,57021],{},[45,57008,57009,57011,57013,57015,57017,57019],{},[48,57010,920],{},[51,57012,54],{"stretchy":53},[51,57014,5106],{},[51,57016,60],{},[48,57018,63],{},[51,57020,66],{"stretchy":53},[114,57022,57023],{"encoding":116},"P(+\\mid D)",[23,57025,57027,57051],{"className":57026,"ariaHidden":122},[121],[23,57028,57030,57033,57036,57039,57042,57045,57048],{"className":57029},[126],[23,57031],{"className":57032,"style":131},[130],[23,57034,920],{"className":57035,"style":1047},[135,136],[23,57037,54],{"className":57038},[140],[23,57040,5106],{"className":57041},[135],[23,57043],{"className":57044,"style":149},[148],[23,57046,60],{"className":57047},[153],[23,57049],{"className":57050,"style":149},[148],[23,57052,57054,57057,57060],{"className":57053},[126],[23,57055],{"className":57056,"style":131},[130],[23,57058,63],{"className":57059,"style":144},[135,136],[23,57061,66],{"className":57062},[169],[23,57064,57066,57090],{"className":57065,"translate":27},[31],[23,57067,57069],{"className":57068},[35],[37,57070,57071],{"xmlns":39},[42,57072,57073,57087],{},[45,57074,57075,57077,57079,57081,57083,57085],{},[48,57076,920],{},[51,57078,54],{"stretchy":53},[48,57080,63],{},[51,57082,60],{},[51,57084,5106],{},[51,57086,66],{"stretchy":53},[114,57088,57089],{"encoding":116},"P(D\\mid +)",[23,57091,57093,57117],{"className":57092,"ariaHidden":122},[121],[23,57094,57096,57099,57102,57105,57108,57111,57114],{"className":57095},[126],[23,57097],{"className":57098,"style":131},[130],[23,57100,920],{"className":57101,"style":1047},[135,136],[23,57103,54],{"className":57104},[140],[23,57106,63],{"className":57107,"style":144},[135,136],[23,57109],{"className":57110,"style":149},[148],[23,57112,60],{"className":57113},[153],[23,57115],{"className":57116,"style":149},[148],[23,57118,57120,57123,57126],{"className":57119},[126],[23,57121],{"className":57122,"style":131},[130],[23,57124,5106],{"className":57125},[135],[23,57127,66],{"className":57128},[169]," 混为一谈。",[11,57131,57132],{},"患病率变化时，同一个检测给出的后验完全不同：",[3764,57134,57135,57209],{},[3767,57136,57137],{},[3770,57138,57139,57142],{},[3773,57140,57141],{},"患病率",[3773,57143,57144],{},[23,57145,57147,57170],{"className":57146,"translate":27},[31],[23,57148,57150],{"className":57149},[35],[37,57151,57152],{"xmlns":39},[42,57153,57154,57168],{},[45,57155,57156,57158,57160,57162,57164,57166],{},[48,57157,920],{},[51,57159,54],{"stretchy":53},[48,57161,63],{},[51,57163,60],{},[51,57165,5106],{},[51,57167,66],{"stretchy":53},[114,57169,57089],{"encoding":116},[23,57171,57173,57197],{"className":57172,"ariaHidden":122},[121],[23,57174,57176,57179,57182,57185,57188,57191,57194],{"className":57175},[126],[23,57177],{"className":57178,"style":131},[130],[23,57180,920],{"className":57181,"style":1047},[135,136],[23,57183,54],{"className":57184},[140],[23,57186,63],{"className":57187,"style":144},[135,136],[23,57189],{"className":57190,"style":149},[148],[23,57192,60],{"className":57193},[153],[23,57195],{"className":57196,"style":149},[148],[23,57198,57200,57203,57206],{"className":57199},[126],[23,57201],{"className":57202,"style":131},[130],[23,57204,5106],{"className":57205},[135],[23,57207,66],{"className":57208},[169],[3783,57210,57211,57278,57343,57408],{},[3770,57212,57213,57247],{},[3788,57214,57215],{},[23,57216,57218,57234],{"className":57217,"translate":27},[31],[23,57219,57221],{"className":57220},[35],[37,57222,57223],{"xmlns":39},[42,57224,57225,57231],{},[45,57226,57227,57229],{},[2581,57228,51736],{},[48,57230,56051],{"mathvariant":111},[114,57232,57233],{"encoding":116},"0.5\\%",[23,57235,57237],{"className":57236,"ariaHidden":122},[121],[23,57238,57240,57243],{"className":57239},[126],[23,57241],{"className":57242,"style":56064},[130],[23,57244,57246],{"className":57245},[135],"0.5%",[3788,57248,57249],{},[23,57250,57252,57266],{"className":57251,"translate":27},[31],[23,57253,57255],{"className":57254},[35],[37,57256,57257],{"xmlns":39},[42,57258,57259,57264],{},[45,57260,57261],{},[2581,57262,57263],{},"0.090",[114,57265,57263],{"encoding":116},[23,57267,57269],{"className":57268,"ariaHidden":122},[121],[23,57270,57272,57275],{"className":57271},[126],[23,57273],{"className":57274,"style":2770},[130],[23,57276,57263],{"className":57277},[135],[3770,57279,57280,57312],{},[3788,57281,57282],{},[23,57283,57285,57300],{"className":57284,"translate":27},[31],[23,57286,57288],{"className":57287},[35],[37,57289,57290],{"xmlns":39},[42,57291,57292,57298],{},[45,57293,57294,57296],{},[2581,57295,4435],{},[48,57297,56051],{"mathvariant":111},[114,57299,56054],{"encoding":116},[23,57301,57303],{"className":57302,"ariaHidden":122},[121],[23,57304,57306,57309],{"className":57305},[126],[23,57307],{"className":57308,"style":56064},[130],[23,57310,56068],{"className":57311},[135],[3788,57313,57314],{},[23,57315,57317,57331],{"className":57316,"translate":27},[31],[23,57318,57320],{"className":57319},[35],[37,57321,57322],{"xmlns":39},[42,57323,57324,57329],{},[45,57325,57326],{},[2581,57327,57328],{},"0.167",[114,57330,57328],{"encoding":116},[23,57332,57334],{"className":57333,"ariaHidden":122},[121],[23,57335,57337,57340],{"className":57336},[126],[23,57338],{"className":57339,"style":2770},[130],[23,57341,57328],{"className":57342},[135],[3770,57344,57345,57377],{},[3788,57346,57347],{},[23,57348,57350,57365],{"className":57349,"translate":27},[31],[23,57351,57353],{"className":57352},[35],[37,57354,57355],{"xmlns":39},[42,57356,57357,57363],{},[45,57358,57359,57361],{},[2581,57360,55679],{},[48,57362,56051],{"mathvariant":111},[114,57364,56951],{"encoding":116},[23,57366,57368],{"className":57367,"ariaHidden":122},[121],[23,57369,57371,57374],{"className":57370},[126],[23,57372],{"className":57373,"style":56064},[130],[23,57375,56964],{"className":57376},[135],[3788,57378,57379],{},[23,57380,57382,57396],{"className":57381,"translate":27},[31],[23,57383,57385],{"className":57384},[35],[37,57386,57387],{"xmlns":39},[42,57388,57389,57394],{},[45,57390,57391],{},[2581,57392,57393],{},"0.510",[114,57395,57393],{"encoding":116},[23,57397,57399],{"className":57398,"ariaHidden":122},[121],[23,57400,57402,57405],{"className":57401},[126],[23,57403],{"className":57404,"style":2770},[130],[23,57406,57393],{"className":57407},[135],[3770,57409,57410,57444],{},[3788,57411,57412],{},[23,57413,57415,57431],{"className":57414,"translate":27},[31],[23,57416,57418],{"className":57417},[35],[37,57419,57420],{"xmlns":39},[42,57421,57422,57428],{},[45,57423,57424,57426],{},[2581,57425,46690],{},[48,57427,56051],{"mathvariant":111},[114,57429,57430],{"encoding":116},"10\\%",[23,57432,57434],{"className":57433,"ariaHidden":122},[121],[23,57435,57437,57440],{"className":57436},[126],[23,57438],{"className":57439,"style":56064},[130],[23,57441,57443],{"className":57442},[135],"10%",[3788,57445,57446],{},[23,57447,57449,57463],{"className":57448,"translate":27},[31],[23,57450,57452],{"className":57451},[35],[37,57453,57454],{"xmlns":39},[42,57455,57456,57461],{},[45,57457,57458],{},[2581,57459,57460],{},"0.688",[114,57462,57460],{"encoding":116},[23,57464,57466],{"className":57465,"ariaHidden":122},[121],[23,57467,57469,57472],{"className":57468},[126],[23,57470],{"className":57471,"style":2770},[130],[23,57473,57460],{"className":57474},[135],[11,57476,57477,57478,57506,57507,55155,57609,57711],{},"现在加入代价。设漏诊的代价是误诊的 ",[23,57479,57481,57494],{"className":57480,"translate":27},[31],[23,57482,57484],{"className":57483},[35],[37,57485,57486],{"xmlns":39},[42,57487,57488,57492],{},[45,57489,57490],{},[2581,57491,46690],{},[114,57493,46690],{"encoding":116},[23,57495,57497],{"className":57496,"ariaHidden":122},[121],[23,57498,57500,57503],{"className":57499},[126],[23,57501],{"className":57502,"style":2770},[130],[23,57504,46690],{"className":57505},[135]," 倍，即 ",[23,57508,57510,57536],{"className":57509,"translate":27},[31],[23,57511,57513],{"className":57512},[35],[37,57514,57515],{"xmlns":39},[42,57516,57517,57533],{},[45,57518,57519,57529,57531],{},[898,57520,57521,57523],{},[48,57522,12710],{},[45,57524,57525,57527],{},[48,57526,12715],{"mathvariant":111},[48,57528,12801],{"mathvariant":111},[51,57530,69],{},[2581,57532,46690],{},[114,57534,57535],{"encoding":116},"c_{\\mathrm{FN}}=10",[23,57537,57539,57600],{"className":57538,"ariaHidden":122},[121],[23,57540,57542,57545,57591,57594,57597],{"className":57541},[126],[23,57543],{"className":57544,"style":12730},[130],[23,57546,57548,57551],{"className":57547},[135],[23,57549,12710],{"className":57550},[135,136],[23,57552,57554],{"className":57553},[998],[23,57555,57557,57583],{"className":57556},[199,200],[23,57558,57560,57580],{"className":57559},[204],[23,57561,57563],{"className":57562,"style":2026},[208],[23,57564,57565,57568],{"style":2029},[23,57566],{"className":57567,"style":1015},[216],[23,57569,57571],{"className":57570},[1019,1020,1021,1022],[23,57572,57574],{"className":57573},[135,1022],[23,57575,57577],{"className":57576},[135,1022],[23,57578,12849],{"className":57579},[135,5391,1022],[23,57581,297],{"className":57582},[296],[23,57584,57586],{"className":57585},[204],[23,57587,57589],{"className":57588,"style":2048},[208],[23,57590],{},[23,57592],{"className":57593,"style":149},[148],[23,57595,69],{"className":57596},[153],[23,57598],{"className":57599,"style":149},[148],[23,57601,57603,57606],{"className":57602},[126],[23,57604],{"className":57605,"style":2770},[130],[23,57607,46690],{"className":57608},[135],[23,57610,57612,57638],{"className":57611,"translate":27},[31],[23,57613,57615],{"className":57614},[35],[37,57616,57617],{"xmlns":39},[42,57618,57619,57635],{},[45,57620,57621,57631,57633],{},[898,57622,57623,57625],{},[48,57624,12710],{},[45,57626,57627,57629],{},[48,57628,12715],{"mathvariant":111},[48,57630,920],{"mathvariant":111},[51,57632,69],{},[2581,57634,4435],{},[114,57636,57637],{"encoding":116},"c_{\\mathrm{FP}}=1",[23,57639,57641,57702],{"className":57640,"ariaHidden":122},[121],[23,57642,57644,57647,57693,57696,57699],{"className":57643},[126],[23,57645],{"className":57646,"style":12730},[130],[23,57648,57650,57653],{"className":57649},[135],[23,57651,12710],{"className":57652},[135,136],[23,57654,57656],{"className":57655},[998],[23,57657,57659,57685],{"className":57658},[199,200],[23,57660,57662,57682],{"className":57661},[204],[23,57663,57665],{"className":57664,"style":2026},[208],[23,57666,57667,57670],{"style":2029},[23,57668],{"className":57669,"style":1015},[216],[23,57671,57673],{"className":57672},[1019,1020,1021,1022],[23,57674,57676],{"className":57675},[135,1022],[23,57677,57679],{"className":57678},[135,1022],[23,57680,12766],{"className":57681},[135,5391,1022],[23,57683,297],{"className":57684},[296],[23,57686,57688],{"className":57687},[204],[23,57689,57691],{"className":57690,"style":2048},[208],[23,57692],{},[23,57694],{"className":57695,"style":149},[148],[23,57697,69],{"className":57698},[153],[23,57700],{"className":57701,"style":149},[148],[23,57703,57705,57708],{"className":57704},[126],[23,57706],{"className":57707,"style":2770},[130],[23,57709,4435],{"className":57710},[135],"，则治疗阈值是",[23,57713,57715],{"className":57714,"translate":27},[26],[23,57716,57718,57779],{"className":57717,"translate":27},[31],[23,57719,57721],{"className":57720},[35],[37,57722,57723],{"xmlns":39,"display":40},[42,57724,57725,57776],{},[45,57726,57727,57763,57765,57771,57773],{},[71,57728,57729,57739],{},[898,57730,57731,57733],{},[48,57732,12710],{},[45,57734,57735,57737],{},[48,57736,12715],{"mathvariant":111},[48,57738,920],{"mathvariant":111},[45,57740,57741,57751,57753],{},[898,57742,57743,57745],{},[48,57744,12710],{},[45,57746,57747,57749],{},[48,57748,12715],{"mathvariant":111},[48,57750,920],{"mathvariant":111},[51,57752,5106],{},[898,57754,57755,57757],{},[48,57756,12710],{},[45,57758,57759,57761],{},[48,57760,12715],{"mathvariant":111},[48,57762,12801],{"mathvariant":111},[51,57764,69],{},[71,57766,57767,57769],{},[2581,57768,4435],{},[2581,57770,13748],{},[51,57772,29066],{},[2581,57774,57775],{},"0.091.",[114,57777,57778],{"encoding":116},"\\frac{c_{\\mathrm{FP}}}{c_{\\mathrm{FP}}+c_{\\mathrm{FN}}}\n=\\frac{1}{11}\\approx0.091.",[23,57780,57782,58000,58077],{"className":57781,"ariaHidden":122},[121],[23,57783,57785,57788,57991,57994,57997],{"className":57784},[126],[23,57786],{"className":57787,"style":13486},[130],[23,57789,57791,57794,57988],{"className":57790},[135],[23,57792],{"className":57793},[140,192],[23,57795,57797],{"className":57796},[71],[23,57798,57800,57980],{"className":57799},[199,200],[23,57801,57803,57977],{"className":57802},[204],[23,57804,57806,57915,57923],{"className":57805,"style":6056},[208],[23,57807,57808,57811],{"style":212},[23,57809],{"className":57810,"style":217},[216],[23,57812,57814,57860,57863,57866,57869],{"className":57813},[135],[23,57815,57817,57820],{"className":57816},[135],[23,57818,12710],{"className":57819},[135,136],[23,57821,57823],{"className":57822},[998],[23,57824,57826,57852],{"className":57825},[199,200],[23,57827,57829,57849],{"className":57828},[204],[23,57830,57832],{"className":57831,"style":2026},[208],[23,57833,57834,57837],{"style":2029},[23,57835],{"className":57836,"style":1015},[216],[23,57838,57840],{"className":57839},[1019,1020,1021,1022],[23,57841,57843],{"className":57842},[135,1022],[23,57844,57846],{"className":57845},[135,1022],[23,57847,12766],{"className":57848},[135,5391,1022],[23,57850,297],{"className":57851},[296],[23,57853,57855],{"className":57854},[204],[23,57856,57858],{"className":57857,"style":2048},[208],[23,57859],{},[23,57861],{"className":57862,"style":392},[148],[23,57864,5106],{"className":57865},[612],[23,57867],{"className":57868,"style":392},[148],[23,57870,57872,57875],{"className":57871},[135],[23,57873,12710],{"className":57874},[135,136],[23,57876,57878],{"className":57877},[998],[23,57879,57881,57907],{"className":57880},[199,200],[23,57882,57884,57904],{"className":57883},[204],[23,57885,57887],{"className":57886,"style":2026},[208],[23,57888,57889,57892],{"style":2029},[23,57890],{"className":57891,"style":1015},[216],[23,57893,57895],{"className":57894},[1019,1020,1021,1022],[23,57896,57898],{"className":57897},[135,1022],[23,57899,57901],{"className":57900},[135,1022],[23,57902,12849],{"className":57903},[135,5391,1022],[23,57905,297],{"className":57906},[296],[23,57908,57910],{"className":57909},[204],[23,57911,57913],{"className":57912,"style":2048},[208],[23,57914],{},[23,57916,57917,57920],{"style":235},[23,57918],{"className":57919,"style":217},[216],[23,57921],{"className":57922,"style":243},[242],[23,57924,57925,57928],{"style":246},[23,57926],{"className":57927,"style":217},[216],[23,57929,57931],{"className":57930},[135],[23,57932,57934,57937],{"className":57933},[135],[23,57935,12710],{"className":57936},[135,136],[23,57938,57940],{"className":57939},[998],[23,57941,57943,57969],{"className":57942},[199,200],[23,57944,57946,57966],{"className":57945},[204],[23,57947,57949],{"className":57948,"style":2026},[208],[23,57950,57951,57954],{"style":2029},[23,57952],{"className":57953,"style":1015},[216],[23,57955,57957],{"className":57956},[1019,1020,1021,1022],[23,57958,57960],{"className":57959},[135,1022],[23,57961,57963],{"className":57962},[135,1022],[23,57964,12766],{"className":57965},[135,5391,1022],[23,57967,297],{"className":57968},[296],[23,57970,57972],{"className":57971},[204],[23,57973,57975],{"className":57974,"style":2048},[208],[23,57976],{},[23,57978,297],{"className":57979},[296],[23,57981,57983],{"className":57982},[204],[23,57984,57986],{"className":57985,"style":13685},[208],[23,57987],{},[23,57989],{"className":57990},[169,192],[23,57992],{"className":57993,"style":149},[148],[23,57995,69],{"className":57996},[153],[23,57998],{"className":57999,"style":149},[148],[23,58001,58003,58006,58068,58071,58074],{"className":58002},[126],[23,58004],{"className":58005,"style":12390},[130],[23,58007,58009,58012,58065],{"className":58008},[135],[23,58010],{"className":58011},[140,192],[23,58013,58015],{"className":58014},[71],[23,58016,58018,58057],{"className":58017},[199,200],[23,58019,58021,58054],{"className":58020},[204],[23,58022,58024,58035,58043],{"className":58023,"style":12409},[208],[23,58025,58026,58029],{"style":212},[23,58027],{"className":58028,"style":217},[216],[23,58030,58032],{"className":58031},[135],[23,58033,13748],{"className":58034},[135],[23,58036,58037,58040],{"style":235},[23,58038],{"className":58039,"style":217},[216],[23,58041],{"className":58042,"style":243},[242],[23,58044,58045,58048],{"style":246},[23,58046],{"className":58047,"style":217},[216],[23,58049,58051],{"className":58050},[135],[23,58052,4435],{"className":58053},[135],[23,58055,297],{"className":58056},[296],[23,58058,58060],{"className":58059},[204],[23,58061,58063],{"className":58062,"style":6287},[208],[23,58064],{},[23,58066],{"className":58067},[169,192],[23,58069],{"className":58070,"style":149},[148],[23,58072,29066],{"className":58073},[153],[23,58075],{"className":58076,"style":149},[148],[23,58078,58080,58083],{"className":58079},[126],[23,58081],{"className":58082,"style":2770},[130],[23,58084,57775],{"className":58085},[135],[11,58087,58088,58089,58119,58120,58231,58232,58119,58262,58350],{},"在患病率 ",[23,58090,58092,58107],{"className":58091,"translate":27},[31],[23,58093,58095],{"className":58094},[35],[37,58096,58097],{"xmlns":39},[42,58098,58099,58105],{},[45,58100,58101,58103],{},[2581,58102,4435],{},[48,58104,56051],{"mathvariant":111},[114,58106,56054],{"encoding":116},[23,58108,58110],{"className":58109,"ariaHidden":122},[121],[23,58111,58113,58116],{"className":58112},[126],[23,58114],{"className":58115,"style":56064},[130],[23,58117,56068],{"className":58118},[135]," 时，",[23,58121,58123,58156],{"className":58122,"translate":27},[31],[23,58124,58126],{"className":58125},[35],[37,58127,58128],{"xmlns":39},[42,58129,58130,58153],{},[45,58131,58132,58134,58136,58138,58140,58142,58144,58146,58148,58150],{},[48,58133,920],{},[51,58135,54],{"stretchy":53},[48,58137,63],{},[51,58139,60],{},[51,58141,5106],{},[51,58143,66],{"stretchy":53},[51,58145,29066],{},[2581,58147,57328],{},[51,58149,2579],{},[2581,58151,58152],{},"0.091",[114,58154,58155],{"encoding":116},"P(D\\mid +)\\approx0.167>0.091",[23,58157,58159,58183,58204,58222],{"className":58158,"ariaHidden":122},[121],[23,58160,58162,58165,58168,58171,58174,58177,58180],{"className":58161},[126],[23,58163],{"className":58164,"style":131},[130],[23,58166,920],{"className":58167,"style":1047},[135,136],[23,58169,54],{"className":58170},[140],[23,58172,63],{"className":58173,"style":144},[135,136],[23,58175],{"className":58176,"style":149},[148],[23,58178,60],{"className":58179},[153],[23,58181],{"className":58182,"style":149},[148],[23,58184,58186,58189,58192,58195,58198,58201],{"className":58185},[126],[23,58187],{"className":58188,"style":131},[130],[23,58190,5106],{"className":58191},[135],[23,58193,66],{"className":58194},[169],[23,58196],{"className":58197,"style":149},[148],[23,58199,29066],{"className":58200},[153],[23,58202],{"className":58203,"style":149},[148],[23,58205,58207,58210,58213,58216,58219],{"className":58206},[126],[23,58208],{"className":58209,"style":52964},[130],[23,58211,57328],{"className":58212},[135],[23,58214],{"className":58215,"style":149},[148],[23,58217,2579],{"className":58218},[153],[23,58220],{"className":58221,"style":149},[148],[23,58223,58225,58228],{"className":58224},[126],[23,58226],{"className":58227,"style":2770},[130],[23,58229,58152],{"className":58230},[135],"，应当治疗；在患病率 ",[23,58233,58235,58250],{"className":58234,"translate":27},[31],[23,58236,58238],{"className":58237},[35],[37,58239,58240],{"xmlns":39},[42,58241,58242,58248],{},[45,58243,58244,58246],{},[2581,58245,51736],{},[48,58247,56051],{"mathvariant":111},[114,58249,57233],{"encoding":116},[23,58251,58253],{"className":58252,"ariaHidden":122},[121],[23,58254,58256,58259],{"className":58255},[126],[23,58257],{"className":58258,"style":56064},[130],[23,58260,57246],{"className":58261},[135],[23,58263,58265,58293],{"className":58264,"translate":27},[31],[23,58266,58268],{"className":58267},[35],[37,58269,58270],{"xmlns":39},[42,58271,58272,58290],{},[45,58273,58274,58276,58278,58280,58282,58284,58286,58288],{},[48,58275,920],{},[51,58277,54],{"stretchy":53},[48,58279,63],{},[51,58281,60],{},[51,58283,5106],{},[51,58285,66],{"stretchy":53},[51,58287,29066],{},[2581,58289,57263],{},[114,58291,58292],{"encoding":116},"P(D\\mid +)\\approx0.090",[23,58294,58296,58320,58341],{"className":58295,"ariaHidden":122},[121],[23,58297,58299,58302,58305,58308,58311,58314,58317],{"className":58298},[126],[23,58300],{"className":58301,"style":131},[130],[23,58303,920],{"className":58304,"style":1047},[135,136],[23,58306,54],{"className":58307},[140],[23,58309,63],{"className":58310,"style":144},[135,136],[23,58312],{"className":58313,"style":149},[148],[23,58315,60],{"className":58316},[153],[23,58318],{"className":58319,"style":149},[148],[23,58321,58323,58326,58329,58332,58335,58338],{"className":58322},[126],[23,58324],{"className":58325,"style":131},[130],[23,58327,5106],{"className":58328},[135],[23,58330,66],{"className":58331},[169],[23,58333],{"className":58334,"style":149},[148],[23,58336,29066],{"className":58337},[153],[23,58339],{"className":58340,"style":149},[148],[23,58342,58344,58347],{"className":58343},[126],[23,58345],{"className":58346,"style":2770},[130],[23,58348,57263],{"className":58349},[135],"，已经落到阈值以下，治疗与不治疗的期望代价几乎相同。同一个阳性结果，在不同的先验与代价下导向不同的行动，这不是检测本身变了，而是决策问题变了。",[11,58352,58353],{},"这个例子也说明了为什么「准确率」不是一个好的汇报指标。检测的准确率是",[23,58355,58357],{"className":58356,"translate":27},[26],[23,58358,58360,58394],{"className":58359,"translate":27},[31],[23,58361,58363],{"className":58362},[35],[37,58364,58365],{"xmlns":39,"display":40},[42,58366,58367,58391],{},[45,58368,58369,58371,58373,58375,58377,58380,58382,58384,58386,58389],{},[2581,58370,56283],{},[51,58372,563],{},[2581,58374,56288],{},[51,58376,5106],{},[2581,58378,58379],{},"0.95",[51,58381,563],{},[2581,58383,56283],{},[51,58385,29066],{},[2581,58387,58388],{},"0.9504",[51,58390,747],{"separator":122},[114,58392,58393],{"encoding":116},"0.99\\times0.01+0.95\\times0.99\\approx0.9504,",[23,58395,58397,58415,58433,58451,58469],{"className":58396,"ariaHidden":122},[121],[23,58398,58400,58403,58406,58409,58412],{"className":58399},[126],[23,58401],{"className":58402,"style":9400},[130],[23,58404,56283],{"className":58405},[135],[23,58407],{"className":58408,"style":392},[148],[23,58410,563],{"className":58411},[612],[23,58413],{"className":58414,"style":392},[148],[23,58416,58418,58421,58424,58427,58430],{"className":58417},[126],[23,58419],{"className":58420,"style":9400},[130],[23,58422,56288],{"className":58423},[135],[23,58425],{"className":58426,"style":392},[148],[23,58428,5106],{"className":58429},[612],[23,58431],{"className":58432,"style":392},[148],[23,58434,58436,58439,58442,58445,58448],{"className":58435},[126],[23,58437],{"className":58438,"style":9400},[130],[23,58440,58379],{"className":58441},[135],[23,58443],{"className":58444,"style":392},[148],[23,58446,563],{"className":58447},[612],[23,58449],{"className":58450,"style":392},[148],[23,58452,58454,58457,58460,58463,58466],{"className":58453},[126],[23,58455],{"className":58456,"style":2770},[130],[23,58458,56283],{"className":58459},[135],[23,58461],{"className":58462,"style":149},[148],[23,58464,29066],{"className":58465},[153],[23,58467],{"className":58468,"style":149},[148],[23,58470,58472,58475,58478],{"className":58471},[126],[23,58473],{"className":58474,"style":51611},[130],[23,58476,58388],{"className":58477},[135],[23,58479,747],{"className":58480},[824],[11,58482,58483],{},"看起来很高，但它几乎完全由健康人贡献；在真正需要决策的那一小部分人身上，后验概率才是关键。",[2501,58485,58487],{"id":58486},"_84-高斯两类问题","8.4 高斯两类问题",[11,58489,58490,58491,58572,58573,4136,58601,58629,58630,58681],{},"设两类共享方差 ",[23,58492,58494,58516],{"className":58493,"translate":27},[31],[23,58495,58497],{"className":58496},[35],[37,58498,58499],{"xmlns":39},[42,58500,58501,58513],{},[45,58502,58503,58509,58511],{},[4518,58504,58505,58507],{},[48,58506,29634],{},[2581,58508,9952],{},[51,58510,69],{},[2581,58512,4435],{},[114,58514,58515],{"encoding":116},"\\sigma^2=1",[23,58517,58519,58563],{"className":58518,"ariaHidden":122},[121],[23,58520,58522,58525,58554,58557,58560],{"className":58521},[126],[23,58523],{"className":58524,"style":4693},[130],[23,58526,58528,58531],{"className":58527},[135],[23,58529,29634],{"className":58530,"style":676},[135,136],[23,58532,58534],{"className":58533},[998],[23,58535,58537],{"className":58536},[199],[23,58538,58540],{"className":58539},[204],[23,58541,58543],{"className":58542,"style":4693},[208],[23,58544,58545,58548],{"style":4696},[23,58546],{"className":58547,"style":1015},[216],[23,58549,58551],{"className":58550},[1019,1020,1021,1022],[23,58552,9952],{"className":58553},[135,1022],[23,58555],{"className":58556,"style":149},[148],[23,58558,69],{"className":58559},[153],[23,58561],{"className":58562,"style":149},[148],[23,58564,58566,58569],{"className":58565},[126],[23,58567],{"className":58568,"style":2770},[130],[23,58570,4435],{"className":58571},[135],"，均值分别为 ",[23,58574,58576,58589],{"className":58575,"translate":27},[31],[23,58577,58579],{"className":58578},[35],[37,58580,58581],{"xmlns":39},[42,58582,58583,58587],{},[45,58584,58585],{},[2581,58586,1532],{},[114,58588,1532],{"encoding":116},[23,58590,58592],{"className":58591,"ariaHidden":122},[121],[23,58593,58595,58598],{"className":58594},[126],[23,58596],{"className":58597,"style":2770},[130],[23,58599,1532],{"className":58600},[135],[23,58602,58604,58617],{"className":58603,"translate":27},[31],[23,58605,58607],{"className":58606},[35],[37,58608,58609],{"xmlns":39},[42,58610,58611,58615],{},[45,58612,58613],{},[2581,58614,9952],{},[114,58616,9952],{"encoding":116},[23,58618,58620],{"className":58619,"ariaHidden":122},[121],[23,58621,58623,58626],{"className":58622},[126],[23,58624],{"className":58625,"style":2770},[130],[23,58627,9952],{"className":58628},[135],"，先验相等。则 ",[23,58631,58633,58651],{"className":58632,"translate":27},[31],[23,58634,58636],{"className":58635},[35],[37,58637,58638],{"xmlns":39},[42,58639,58640,58648],{},[45,58641,58642,58644,58646],{},[48,58643,23374],{"mathvariant":111},[51,58645,69],{},[2581,58647,9952],{},[114,58649,58650],{"encoding":116},"\\Delta=2",[23,58652,58654,58672],{"className":58653,"ariaHidden":122},[121],[23,58655,58657,58660,58663,58666,58669],{"className":58656},[126],[23,58658],{"className":58659,"style":470},[130],[23,58661,23374],{"className":58662},[135],[23,58664],{"className":58665,"style":149},[148],[23,58667,69],{"className":58668},[153],[23,58670],{"className":58671,"style":149},[148],[23,58673,58675,58678],{"className":58674},[126],[23,58676],{"className":58677,"style":2770},[130],[23,58679,9952],{"className":58680},[135],"，贝叶斯错误率为",[23,58683,58685],{"className":58684,"translate":27},[26],[23,58686,58688,58723],{"className":58687,"translate":27},[31],[23,58689,58691],{"className":58690},[35],[37,58692,58693],{"xmlns":39,"display":40},[42,58694,58695,58720],{},[45,58696,58697,58705,58707,58709,58711,58713,58715,58717],{},[19137,58698,58699,58701,58703],{},[48,58700,11],{},[48,58702,5342],{},[51,58704,6881],{},[51,58706,69],{},[48,58708,22051],{},[51,58710,54],{"stretchy":53},[2581,58712,4435],{},[51,58714,66],{"stretchy":53},[51,58716,29066],{},[2581,58718,58719],{},"0.1587.",[114,58721,58722],{"encoding":116},"p_e^*=Q(1)\\approx0.1587.",[23,58724,58726,58792,58819],{"className":58725,"ariaHidden":122},[121],[23,58727,58729,58732,58783,58786,58789],{"className":58728},[126],[23,58730],{"className":58731,"style":20722},[130],[23,58733,58735,58738],{"className":58734},[135],[23,58736,11],{"className":58737},[135,136],[23,58739,58741],{"className":58740},[998],[23,58742,58744,58775],{"className":58743},[199,200],[23,58745,58747,58772],{"className":58746},[204],[23,58748,58750,58761],{"className":58749,"style":7045},[208],[23,58751,58752,58755],{"style":19345},[23,58753],{"className":58754,"style":1015},[216],[23,58756,58758],{"className":58757},[1019,1020,1021,1022],[23,58759,5342],{"className":58760},[135,136,1022],[23,58762,58763,58766],{"style":4989},[23,58764],{"className":58765,"style":1015},[216],[23,58767,58769],{"className":58768},[1019,1020,1021,1022],[23,58770,6881],{"className":58771},[612,1022],[23,58773,297],{"className":58774},[296],[23,58776,58778],{"className":58777},[204],[23,58779,58781],{"className":58780,"style":19375},[208],[23,58782],{},[23,58784],{"className":58785,"style":149},[148],[23,58787,69],{"className":58788},[153],[23,58790],{"className":58791,"style":149},[148],[23,58793,58795,58798,58801,58804,58807,58810,58813,58816],{"className":58794},[126],[23,58796],{"className":58797,"style":131},[130],[23,58799,22051],{"className":58800},[135,136],[23,58802,54],{"className":58803},[140],[23,58805,4435],{"className":58806},[135],[23,58808,66],{"className":58809},[169],[23,58811],{"className":58812,"style":149},[148],[23,58814,29066],{"className":58815},[153],[23,58817],{"className":58818,"style":149},[148],[23,58820,58822,58825],{"className":58821},[126],[23,58823],{"className":58824,"style":2770},[130],[23,58826,58719],{"className":58827},[135],[11,58829,58830,58831,8055,58859,58910,58911,58977,58978,59084],{},"若把均值差扩大到 ",[23,58832,58834,58847],{"className":58833,"translate":27},[31],[23,58835,58837],{"className":58836},[35],[37,58838,58839],{"xmlns":39},[42,58840,58841,58845],{},[45,58842,58843],{},[2581,58844,24930],{},[114,58846,24930],{"encoding":116},[23,58848,58850],{"className":58849,"ariaHidden":122},[121],[23,58851,58853,58856],{"className":58852},[126],[23,58854],{"className":58855,"style":2770},[130],[23,58857,24930],{"className":58858},[135],[23,58860,58862,58880],{"className":58861,"translate":27},[31],[23,58863,58865],{"className":58864},[35],[37,58866,58867],{"xmlns":39},[42,58868,58869,58877],{},[45,58870,58871,58873,58875],{},[48,58872,23374],{"mathvariant":111},[51,58874,69],{},[2581,58876,24930],{},[114,58878,58879],{"encoding":116},"\\Delta=4",[23,58881,58883,58901],{"className":58882,"ariaHidden":122},[121],[23,58884,58886,58889,58892,58895,58898],{"className":58885},[126],[23,58887],{"className":58888,"style":470},[130],[23,58890,23374],{"className":58891},[135],[23,58893],{"className":58894,"style":149},[148],[23,58896,69],{"className":58897},[153],[23,58899],{"className":58900,"style":149},[148],[23,58902,58904,58907],{"className":58903},[126],[23,58905],{"className":58906,"style":2770},[130],[23,58908,24930],{"className":58909},[135],"，错误率降为 ",[23,58912,58914,58938],{"className":58913,"translate":27},[31],[23,58915,58917],{"className":58916},[35],[37,58918,58919],{"xmlns":39},[42,58920,58921,58935],{},[45,58922,58923,58925,58927,58929,58931,58933],{},[48,58924,22051],{},[51,58926,54],{"stretchy":53},[2581,58928,9952],{},[51,58930,66],{"stretchy":53},[51,58932,29066],{},[2581,58934,24933],{},[114,58936,58937],{"encoding":116},"Q(2)\\approx0.0228",[23,58939,58941,58968],{"className":58940,"ariaHidden":122},[121],[23,58942,58944,58947,58950,58953,58956,58959,58962,58965],{"className":58943},[126],[23,58945],{"className":58946,"style":131},[130],[23,58948,22051],{"className":58949},[135,136],[23,58951,54],{"className":58952},[140],[23,58954,9952],{"className":58955},[135],[23,58957,66],{"className":58958},[169],[23,58960],{"className":58961,"style":149},[148],[23,58963,29066],{"className":58964},[153],[23,58966],{"className":58967,"style":149},[148],[23,58969,58971,58974],{"className":58970},[126],[23,58972],{"className":58973,"style":2770},[130],[23,58975,24933],{"className":58976},[135],"。错误率随马氏距离以 ",[23,58979,58981,59011],{"className":58980,"translate":27},[31],[23,58982,58984],{"className":58983},[35],[37,58985,58986],{"xmlns":39},[42,58987,58988,59008],{},[45,58989,58990],{},[4518,58991,58992,58994],{},[48,58993,5342],{},[45,58995,58996,58998,59004,59006],{},[51,58997,4529],{},[4518,58999,59000,59002],{},[48,59001,23374],{"mathvariant":111},[2581,59003,9952],{},[48,59005,2875],{"mathvariant":111},[2581,59007,24741],{},[114,59009,59010],{"encoding":116},"e^{-\\Delta^2\u002F8}",[23,59012,59014],{"className":59013,"ariaHidden":122},[121],[23,59015,59017,59020],{"className":59016},[126],[23,59018],{"className":59019,"style":24841},[130],[23,59021,59023,59026],{"className":59022},[135],[23,59024,5342],{"className":59025},[135,136],[23,59027,59029],{"className":59028},[998],[23,59030,59032],{"className":59031},[199],[23,59033,59035],{"className":59034},[204],[23,59036,59038],{"className":59037,"style":24841},[208],[23,59039,59040,59043],{"style":4696},[23,59041],{"className":59042,"style":1015},[216],[23,59044,59046],{"className":59045},[1019,1020,1021,1022],[23,59047,59049,59052,59081],{"className":59048},[135,1022],[23,59050,4529],{"className":59051},[135,1022],[23,59053,59055,59058],{"className":59054},[135,1022],[23,59056,23374],{"className":59057},[135,1022],[23,59059,59061],{"className":59060},[998],[23,59062,59064],{"className":59063},[199],[23,59065,59067],{"className":59066},[204],[23,59068,59070],{"className":59069,"style":24874},[208],[23,59071,59072,59075],{"style":24877},[23,59073],{"className":59074,"style":24881},[216],[23,59076,59078],{"className":59077},[1019,24885,1067,1022],[23,59079,9952],{"className":59080},[135,1022],[23,59082,24892],{"className":59083},[135,1022]," 的速率衰减，但永远不为零，因为两个高斯的支撑集在整个实轴上重叠。",[11,59086,59087],{},"Bhattacharyya 上界给出",[23,59089,59091],{"className":59090,"translate":27},[26],[23,59092,59094,59142],{"className":59093,"translate":27},[31],[23,59095,59097],{"className":59096},[35],[37,59098,59099],{"xmlns":39,"display":40},[42,59100,59101,59139],{},[45,59102,59103,59111,59113,59119,59137],{},[19137,59104,59105,59107,59109],{},[48,59106,11],{},[48,59108,5342],{},[51,59110,6881],{},[51,59112,12268],{},[71,59114,59115,59117],{},[2581,59116,4435],{},[2581,59118,9952],{},[4518,59120,59121,59123],{},[48,59122,5342],{},[45,59124,59125,59127,59133,59135],{},[51,59126,4529],{},[4518,59128,59129,59131],{},[48,59130,23374],{"mathvariant":111},[2581,59132,9952],{},[48,59134,2875],{"mathvariant":111},[2581,59136,24741],{},[51,59138,747],{"separator":122},[114,59140,59141],{"encoding":116},"p_e^*\\le\\frac12e^{-\\Delta^2\u002F8},",[23,59143,59145,59211],{"className":59144,"ariaHidden":122},[121],[23,59146,59148,59151,59202,59205,59208],{"className":59147},[126],[23,59149],{"className":59150,"style":20722},[130],[23,59152,59154,59157],{"className":59153},[135],[23,59155,11],{"className":59156},[135,136],[23,59158,59160],{"className":59159},[998],[23,59161,59163,59194],{"className":59162},[199,200],[23,59164,59166,59191],{"className":59165},[204],[23,59167,59169,59180],{"className":59168,"style":7045},[208],[23,59170,59171,59174],{"style":19345},[23,59172],{"className":59173,"style":1015},[216],[23,59175,59177],{"className":59176},[1019,1020,1021,1022],[23,59178,5342],{"className":59179},[135,136,1022],[23,59181,59182,59185],{"style":4989},[23,59183],{"className":59184,"style":1015},[216],[23,59186,59188],{"className":59187},[1019,1020,1021,1022],[23,59189,6881],{"className":59190},[612,1022],[23,59192,297],{"className":59193},[296],[23,59195,59197],{"className":59196},[204],[23,59198,59200],{"className":59199,"style":19375},[208],[23,59201],{},[23,59203],{"className":59204,"style":149},[148],[23,59206,12268],{"className":59207},[153],[23,59209],{"className":59210,"style":149},[148],[23,59212,59214,59217,59279,59343],{"className":59213},[126],[23,59215],{"className":59216,"style":12390},[130],[23,59218,59220,59223,59276],{"className":59219},[135],[23,59221],{"className":59222},[140,192],[23,59224,59226],{"className":59225},[71],[23,59227,59229,59268],{"className":59228},[199,200],[23,59230,59232,59265],{"className":59231},[204],[23,59233,59235,59246,59254],{"className":59234,"style":12409},[208],[23,59236,59237,59240],{"style":212},[23,59238],{"className":59239,"style":217},[216],[23,59241,59243],{"className":59242},[135],[23,59244,9952],{"className":59245},[135],[23,59247,59248,59251],{"style":235},[23,59249],{"className":59250,"style":217},[216],[23,59252],{"className":59253,"style":243},[242],[23,59255,59256,59259],{"style":246},[23,59257],{"className":59258,"style":217},[216],[23,59260,59262],{"className":59261},[135],[23,59263,4435],{"className":59264},[135],[23,59266,297],{"className":59267},[296],[23,59269,59271],{"className":59270},[204],[23,59272,59274],{"className":59273,"style":6287},[208],[23,59275],{},[23,59277],{"className":59278},[169,192],[23,59280,59282,59285],{"className":59281},[135],[23,59283,5342],{"className":59284},[135,136],[23,59286,59288],{"className":59287},[998],[23,59289,59291],{"className":59290},[199],[23,59292,59294],{"className":59293},[204],[23,59295,59297],{"className":59296,"style":26141},[208],[23,59298,59299,59302],{"style":4989},[23,59300],{"className":59301,"style":1015},[216],[23,59303,59305],{"className":59304},[1019,1020,1021,1022],[23,59306,59308,59311,59340],{"className":59307},[135,1022],[23,59309,4529],{"className":59310},[135,1022],[23,59312,59314,59317],{"className":59313},[135,1022],[23,59315,23374],{"className":59316},[135,1022],[23,59318,59320],{"className":59319},[998],[23,59321,59323],{"className":59322},[199],[23,59324,59326],{"className":59325},[204],[23,59327,59329],{"className":59328,"style":24874},[208],[23,59330,59331,59334],{"style":24877},[23,59332],{"className":59333,"style":24881},[216],[23,59335,59337],{"className":59336},[1019,24885,1067,1022],[23,59338,9952],{"className":59339},[135,1022],[23,59341,24892],{"className":59342},[135,1022],[23,59344,747],{"className":59345},[824],[11,59347,59348,59349,59399,59400,59429,59430,59399,59480,59508],{},"在 ",[23,59350,59352,59369],{"className":59351,"translate":27},[31],[23,59353,59355],{"className":59354},[35],[37,59356,59357],{"xmlns":39},[42,59358,59359,59367],{},[45,59360,59361,59363,59365],{},[48,59362,23374],{"mathvariant":111},[51,59364,69],{},[2581,59366,9952],{},[114,59368,58650],{"encoding":116},[23,59370,59372,59390],{"className":59371,"ariaHidden":122},[121],[23,59373,59375,59378,59381,59384,59387],{"className":59374},[126],[23,59376],{"className":59377,"style":470},[130],[23,59379,23374],{"className":59380},[135],[23,59382],{"className":59383,"style":149},[148],[23,59385,69],{"className":59386},[153],[23,59388],{"className":59389,"style":149},[148],[23,59391,59393,59396],{"className":59392},[126],[23,59394],{"className":59395,"style":2770},[130],[23,59397,9952],{"className":59398},[135]," 时为 ",[23,59401,59403,59417],{"className":59402,"translate":27},[31],[23,59404,59406],{"className":59405},[35],[37,59407,59408],{"xmlns":39},[42,59409,59410,59415],{},[45,59411,59412],{},[2581,59413,59414],{},"0.303",[114,59416,59414],{"encoding":116},[23,59418,59420],{"className":59419,"ariaHidden":122},[121],[23,59421,59423,59426],{"className":59422},[126],[23,59424],{"className":59425,"style":2770},[130],[23,59427,59414],{"className":59428},[135],"，在 ",[23,59431,59433,59450],{"className":59432,"translate":27},[31],[23,59434,59436],{"className":59435},[35],[37,59437,59438],{"xmlns":39},[42,59439,59440,59448],{},[45,59441,59442,59444,59446],{},[48,59443,23374],{"mathvariant":111},[51,59445,69],{},[2581,59447,24930],{},[114,59449,58879],{"encoding":116},[23,59451,59453,59471],{"className":59452,"ariaHidden":122},[121],[23,59454,59456,59459,59462,59465,59468],{"className":59455},[126],[23,59457],{"className":59458,"style":470},[130],[23,59460,23374],{"className":59461},[135],[23,59463],{"className":59464,"style":149},[148],[23,59466,69],{"className":59467},[153],[23,59469],{"className":59470,"style":149},[148],[23,59472,59474,59477],{"className":59473},[126],[23,59475],{"className":59476,"style":2770},[130],[23,59478,24930],{"className":59479},[135],[23,59481,59483,59496],{"className":59482,"translate":27},[31],[23,59484,59486],{"className":59485},[35],[37,59487,59488],{"xmlns":39},[42,59489,59490,59494],{},[45,59491,59492],{},[2581,59493,24936],{},[114,59495,24936],{"encoding":116},[23,59497,59499],{"className":59498,"ariaHidden":122},[121],[23,59500,59502,59505],{"className":59501},[126],[23,59503],{"className":59504,"style":2770},[130],[23,59506,24936],{"className":59507},[135],"。上界不紧，但它的指数部分与真实错误率一致；这正是我们在 4.2 节强调的「速率」而非「常数」。",[338,59510,59511],{"id":59511},"结语",[11,59513,59514],{},"把全文的线索压缩成一行：",[23,59516,59518],{"className":59517,"translate":27},[26],[23,59519,59521,59687],{"className":59520,"translate":27},[31],[23,59522,59524],{"className":59523},[35],[37,59525,59526],{"xmlns":39,"display":40},[42,59527,59528,59684],{},[45,59529,59530,59548,59550,59572,59574,59576,59579,59581,59603,59605,59607,59609,59611,59659,59661,59663,59665,59667,59682],{},[6961,59531,59532,59546],{},[6961,59533,59534,59544],{},[45,59535,59536,59538,59540,59542],{},[48,59537,11],{},[51,59539,54],{"stretchy":53},[48,59541,57],{},[51,59543,66],{"stretchy":53},[51,59545,8477],{"stretchy":122},[88,59547,3790],{},[51,59549,563],{},[6961,59551,59552,59570],{},[6961,59553,59554,59568],{},[45,59555,59556,59558,59560,59562,59564,59566],{},[48,59557,11],{},[51,59559,54],{"stretchy":53},[48,59561,63],{},[51,59563,60],{},[48,59565,57],{},[51,59567,66],{"stretchy":53},[51,59569,8477],{"stretchy":122},[88,59571,3844],{},[88,59573,5360],{},[148,59575],{"linebreak":29625},[51,59577,59578],{},"⟶",[88,59580,5360],{},[6961,59582,59583,59601],{},[6961,59584,59585,59599],{},[45,59586,59587,59589,59591,59593,59595,59597],{},[48,59588,11],{},[51,59590,54],{"stretchy":53},[48,59592,57],{},[51,59594,60],{},[48,59596,63],{},[51,59598,66],{"stretchy":53},[51,59600,8477],{"stretchy":122},[88,59602,4003],{},[88,59604,5360],{},[148,59606],{"linebreak":29625},[51,59608,59578],{},[88,59610,5360],{},[6961,59612,59613,59657],{},[6961,59614,59615,59655],{},[45,59616,59617,59619,59621,59631,59633,59635,59637,59639,59641,59643,59645,59647,59649,59651,59653],{},[48,59618,6956],{},[51,59620,6959],{},[6961,59622,59623,59629],{},[45,59624,59625,59627],{},[48,59626,6967],{},[51,59628,6959],{},[48,59630,495],{},[88,59632,5360],{},[48,59634,902],{"mathvariant":571},[51,59636,924],{"stretchy":53},[48,59638,555],{},[51,59640,54],{"stretchy":53},[48,59642,369],{},[51,59644,747],{"separator":122},[48,59646,495],{},[51,59648,66],{"stretchy":53},[51,59650,60],{},[48,59652,63],{},[51,59654,945],{"stretchy":53},[51,59656,8477],{"stretchy":122},[88,59658,480],{},[88,59660,5360],{},[148,59662],{"linebreak":29625},[51,59664,59578],{},[88,59666,5360],{},[6961,59668,59669,59679],{},[6961,59670,59671,59677],{},[4518,59672,59673,59675],{},[48,59674,572],{},[51,59676,6881],{},[51,59678,8477],{"stretchy":122},[88,59680,59681],{},"下限",[48,59683,112],{"mathvariant":111},[114,59685,59686],{"encoding":116},"\\underbrace{p(\\theta)}_{\\text{先验}}\n\\times\n\\underbrace{p(D\\mid\\theta)}_{\\text{似然}}\n\\ \\\\ \\longrightarrow\\\n\\underbrace{p(\\theta\\mid D)}_{\\text{后验}}\n\\ \\\\ \\longrightarrow\\\n\\underbrace{\\arg\\min_a\\ \\mathbb E[L(Y,a)\\mid D]}_{\\text{行动}}\n\\ \\\\ \\longrightarrow\\\n\\underbrace{R^*}_{\\text{下限}}.",[23,59688,59690,59828,59967,59970,59986,60125,60128,60143,60361,60364,60379],{"className":59689,"ariaHidden":122},[121],[23,59691,59693,59697,59819,59822,59825],{"className":59692},[126],[23,59694],{"className":59695,"style":59696},[130],"height:2.3263em;vertical-align:-1.5763em;",[23,59698,59700],{"className":59699},[1839,6961],[23,59701,59703,59810],{"className":59702},[199,200],[23,59704,59706,59807],{"className":59705},[204],[23,59707,59709,59727],{"className":59708,"style":8633},[208],[23,59710,59712,59715],{"style":59711},"top:-1.4237em;",[23,59713],{"className":59714,"style":217},[216],[23,59716,59718],{"className":59717},[1019,1020,1021,1022],[23,59719,59721],{"className":59720},[135,1022],[23,59722,59724],{"className":59723},[135,6514,1022],[23,59725,3790],{"className":59726},[135,7238,1022],[23,59728,59729,59732],{"style":7130},[23,59730],{"className":59731,"style":217},[216],[23,59733,59735],{"className":59734},[1839,6961],[23,59736,59738,59798],{"className":59737},[199,200],[23,59739,59741,59795],{"className":59740},[204],[23,59742,59744,59775],{"className":59743,"style":8633},[208],[23,59745,59748,59751],{"className":59746,"style":59747},[8672],"top:-2.102em;",[23,59749],{"className":59750,"style":217},[216],[23,59752,59754,59761,59768],{"className":59753,"style":8681},[8680],[23,59755,59757],{"className":59756,"style":8686},[8685],[8688,59758,59759],{"xmlns":8690,"width":8691,"height":8692,"viewBox":8693,"preserveAspectRatio":8694},[8696,59760],{"d":8698},[23,59762,59764],{"className":59763,"style":8686},[8702],[8688,59765,59766],{"xmlns":8690,"width":8691,"height":8692,"viewBox":8693,"preserveAspectRatio":8705},[8696,59767],{"d":8708},[23,59769,59771],{"className":59770,"style":8686},[8712],[8688,59772,59773],{"xmlns":8690,"width":8691,"height":8692,"viewBox":8693,"preserveAspectRatio":8715},[8696,59774],{"d":8718},[23,59776,59777,59780],{"style":7130},[23,59778],{"className":59779,"style":217},[216],[23,59781,59783,59786,59789,59792],{"className":59782},[135],[23,59784,11],{"className":59785},[135,136],[23,59787,54],{"className":59788},[140],[23,59790,57],{"className":59791,"style":144},[135,136],[23,59793,66],{"className":59794},[169],[23,59796,297],{"className":59797},[296],[23,59799,59801],{"className":59800},[204],[23,59802,59805],{"className":59803,"style":59804},[208],"height:0.898em;",[23,59806],{},[23,59808,297],{"className":59809},[296],[23,59811,59813],{"className":59812},[204],[23,59814,59817],{"className":59815,"style":59816},[208],"height:1.5763em;",[23,59818],{},[23,59820],{"className":59821,"style":392},[148],[23,59823,563],{"className":59824},[612],[23,59826],{"className":59827,"style":392},[148],[23,59829,59831,59834,59964],{"className":59830},[126],[23,59832],{"className":59833,"style":59696},[130],[23,59835,59837],{"className":59836},[1839,6961],[23,59838,59840,59956],{"className":59839},[199,200],[23,59841,59843,59953],{"className":59842},[204],[23,59844,59846,59863],{"className":59845,"style":8633},[208],[23,59847,59848,59851],{"style":59711},[23,59849],{"className":59850,"style":217},[216],[23,59852,59854],{"className":59853},[1019,1020,1021,1022],[23,59855,59857],{"className":59856},[135,1022],[23,59858,59860],{"className":59859},[135,6514,1022],[23,59861,3844],{"className":59862},[135,7238,1022],[23,59864,59865,59868],{"style":7130},[23,59866],{"className":59867,"style":217},[216],[23,59869,59871],{"className":59870},[1839,6961],[23,59872,59874,59945],{"className":59873},[199,200],[23,59875,59877,59942],{"className":59876},[204],[23,59878,59880,59910],{"className":59879,"style":8633},[208],[23,59881,59883,59886],{"className":59882,"style":59747},[8672],[23,59884],{"className":59885,"style":217},[216],[23,59887,59889,59896,59903],{"className":59888,"style":8681},[8680],[23,59890,59892],{"className":59891,"style":8686},[8685],[8688,59893,59894],{"xmlns":8690,"width":8691,"height":8692,"viewBox":8693,"preserveAspectRatio":8694},[8696,59895],{"d":8698},[23,59897,59899],{"className":59898,"style":8686},[8702],[8688,59900,59901],{"xmlns":8690,"width":8691,"height":8692,"viewBox":8693,"preserveAspectRatio":8705},[8696,59902],{"d":8708},[23,59904,59906],{"className":59905,"style":8686},[8712],[8688,59907,59908],{"xmlns":8690,"width":8691,"height":8692,"viewBox":8693,"preserveAspectRatio":8715},[8696,59909],{"d":8718},[23,59911,59912,59915],{"style":7130},[23,59913],{"className":59914,"style":217},[216],[23,59916,59918,59921,59924,59927,59930,59933,59936,59939],{"className":59917},[135],[23,59919,11],{"className":59920},[135,136],[23,59922,54],{"className":59923},[140],[23,59925,63],{"className":59926,"style":144},[135,136],[23,59928],{"className":59929,"style":149},[148],[23,59931,60],{"className":59932},[153],[23,59934],{"className":59935,"style":149},[148],[23,59937,57],{"className":59938,"style":144},[135,136],[23,59940,66],{"className":59941},[169],[23,59943,297],{"className":59944},[296],[23,59946,59948],{"className":5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为什么能定义「范数正则化」；正交分解为什么迫使最优解落在训练点张成的子空间里；以及这个结论的边界——什么样的正则项会让它失效。",[338,60687,60689],{"id":60688},"一无穷维优化","一、无穷维优化",[11,60691,60692,60693,60794,60795,60823,60824,60852],{},"线性模型 ",[23,60694,60696,60726],{"className":60695,"translate":27},[31],[23,60697,60699],{"className":60698},[35],[37,60700,60701],{"xmlns":39},[42,60702,60703,60723],{},[45,60704,60705,60707,60709,60711,60713,60715,60721],{},[48,60706,8336],{},[51,60708,54],{"stretchy":53},[48,60710,753],{},[51,60712,66],{"stretchy":53},[51,60714,69],{},[4518,60716,60717,60719],{},[48,60718,49694],{},[48,60720,22471],{"mathvariant":111},[48,60722,753],{},[114,60724,60725],{"encoding":116},"f(x)=w^\\top x",[23,60727,60729,60756],{"className":60728,"ariaHidden":122},[121],[23,60730,60732,60735,60738,60741,60744,60747,60750,60753],{"className":60731},[126],[23,60733],{"className":60734,"style":131},[130],[23,60736,8336],{"className":60737,"style":11398},[135,136],[23,60739,54],{"className":60740},[140],[23,60742,753],{"className":60743},[135,136],[23,60745,66],{"className":60746},[169],[23,60748],{"className":60749,"style":149},[148],[23,60751,69],{"className":60752},[153],[23,60754],{"className":60755,"style":149},[148],[23,60757,60759,60762,60791],{"className":60758},[126],[23,60760],{"className":60761,"style":4751},[130],[23,60763,60765,60768],{"className":60764},[135],[23,60766,49694],{"className":60767,"style":49709},[135,136],[23,60769,60771],{"className":60770},[998],[23,60772,60774],{"className":60773},[199],[23,60775,60777],{"className":60776},[204],[23,60778,60780],{"className":60779,"style":4751},[208],[23,60781,60782,60785],{"style":4696},[23,60783],{"className":60784,"style":1015},[216],[23,60786,60788],{"className":60787},[1019,1020,1021,1022],[23,60789,22471],{"className":60790},[135,1022],[23,60792,753],{"className":60793},[135,136]," 的表达能力受限于特征 ",[23,60796,60798,60811],{"className":60797,"translate":27},[31],[23,60799,60801],{"className":60800},[35],[37,60802,60803],{"xmlns":39},[42,60804,60805,60809],{},[45,60806,60807],{},[48,60808,753],{},[114,60810,753],{"encoding":116},[23,60812,60814],{"className":60813,"ariaHidden":122},[121],[23,60815,60817,60820],{"className":60816},[126],[23,60818],{"className":60819,"style":703},[130],[23,60821,753],{"className":60822},[135,136]," 的坐标。想拟合非线性关系，一个标准做法是先把 ",[23,60825,60827,60840],{"className":60826,"translate":27},[31],[23,60828,60830],{"className":60829},[35],[37,60831,60832],{"xmlns":39},[42,60833,60834,60838],{},[45,60835,60836],{},[48,60837,753],{},[114,60839,753],{"encoding":116},[23,60841,60843],{"className":60842,"ariaHidden":122},[121],[23,60844,60846,60849],{"className":60845},[126],[23,60847],{"className":60848,"style":703},[130],[23,60850,753],{"className":60851},[135,136]," 映射到一个更大的特征空间：",[23,60854,60856],{"className":60855,"translate":27},[26],[23,60857,60859,60898],{"className":60858,"translate":27},[31],[23,60860,60862],{"className":60861},[35],[37,60863,60864],{"xmlns":39,"display":40},[42,60865,60866,60895],{},[45,60867,60868,60870,60872,60874,60876,60878,60884,60887,60889,60891,60893],{},[48,60869,8336],{},[51,60871,54],{"stretchy":53},[48,60873,753],{},[51,60875,66],{"stretchy":53},[51,60877,69],{},[4518,60879,60880,60882],{},[48,60881,49694],{},[48,60883,22471],{"mathvariant":111},[48,60885,60886],{},"φ",[51,60888,54],{"stretchy":53},[48,60890,753],{},[51,60892,66],{"stretchy":53},[51,60894,747],{"separator":122},[114,60896,60897],{"encoding":116},"f(x)=w^\\top \\varphi(x),",[23,60899,60901,60928],{"className":60900,"ariaHidden":122},[121],[23,60902,60904,60907,60910,60913,60916,60919,60922,60925],{"className":60903},[126],[23,60905],{"className":60906,"style":131},[130],[23,60908,8336],{"className":60909,"style":11398},[135,136],[23,60911,54],{"className":60912},[140],[23,60914,753],{"className":60915},[135,136],[23,60917,66],{"className":60918},[169],[23,60920],{"className":60921,"style":149},[148],[23,60923,69],{"className":60924},[153],[23,60926],{"className":60927,"style":149},[148],[23,60929,60931,60934,60963,60966,60969,60972,60975],{"className":60930},[126],[23,60932],{"className":60933,"style":4967},[130],[23,60935,60937,60940],{"className":60936},[135],[23,60938,49694],{"className":60939,"style":49709},[135,136],[23,60941,60943],{"className":60942},[998],[23,60944,60946],{"className":60945},[199],[23,60947,60949],{"className":60948},[204],[23,60950,60952],{"className":60951,"style":4986},[208],[23,60953,60954,60957],{"style":4989},[23,60955],{"className":60956,"style":1015},[216],[23,60958,60960],{"className":60959},[1019,1020,1021,1022],[23,60961,22471],{"className":60962},[135,1022],[23,60964,60886],{"className":60965},[135,136],[23,60967,54],{"className":60968},[140],[23,60970,753],{"className":60971},[135,136],[23,60973,66],{"className":60974},[169],[23,60976,747],{"className":60977},[824],[11,60979,24429,60980,61083,61084,61113,61114,61142],{},[23,60981,60983,61009],{"className":60982,"translate":27},[31],[23,60984,60986],{"className":60985},[35],[37,60987,60988],{"xmlns":39},[42,60989,60990,61006],{},[45,60991,60992,60994,60996,60998,61000],{},[48,60993,60886],{},[51,60995,558],{},[48,60997,434],{"mathvariant":375},[51,60999,568],{},[4518,61001,61002,61004],{},[48,61003,572],{"mathvariant":571},[48,61005,1606],{},[114,61007,61008],{"encoding":116},"\\varphi:\\mathcal X\\to\\mathbb R^d",[23,61010,61012,61030,61048],{"className":61011,"ariaHidden":122},[121],[23,61013,61015,61018,61021,61024,61027],{"className":61014},[126],[23,61016],{"className":61017,"style":672},[130],[23,61019,60886],{"className":61020},[135,136],[23,61022],{"className":61023,"style":149},[148],[23,61025,558],{"className":61026},[153],[23,61028],{"className":61029,"style":149},[148],[23,61031,61033,61036,61039,61042,61045],{"className":61032},[126],[23,61034],{"className":61035,"style":470},[130],[23,61037,434],{"className":61038,"style":474},[135,412],[23,61040],{"className":61041,"style":149},[148],[23,61043,568],{"className":61044},[153],[23,61046],{"className":61047,"style":149},[148],[23,61049,61051,61054],{"className":61050},[126],[23,61052],{"className":61053,"style":4751},[130],[23,61055,61057,61060],{"className":61056},[135],[23,61058,572],{"className":61059},[135,644],[23,61061,61063],{"className":61062},[998],[23,61064,61066],{"className":61065},[199],[23,61067,61069],{"className":61068},[204],[23,61070,61072],{"className":61071,"style":4751},[208],[23,61073,61074,61077],{"style":4696},[23,61075],{"className":61076,"style":1015},[216],[23,61078,61080],{"className":61079},[1019,1020,1021,1022],[23,61081,1606],{"className":61082},[135,136,1022]," 是特征映射。多项式回归、样条基、径向基函数都属于这个套路。问题在于，",[23,61085,61087,61101],{"className":61086,"translate":27},[31],[23,61088,61090],{"className":61089},[35],[37,61091,61092],{"xmlns":39},[42,61093,61094,61098],{},[45,61095,61096],{},[48,61097,60886],{},[114,61099,61100],{"encoding":116},"\\varphi",[23,61102,61104],{"className":61103,"ariaHidden":122},[121],[23,61105,61107,61110],{"className":61106},[126],[23,61108],{"className":61109,"style":672},[130],[23,61111,60886],{"className":61112},[135,136]," 的维数可以很大，甚至是无穷维。例如高斯核对应的是一个无穷维的特征空间；如果显式地写出 ",[23,61115,61117,61130],{"className":61116,"translate":27},[31],[23,61118,61120],{"className":61119},[35],[37,61121,61122],{"xmlns":39},[42,61123,61124,61128],{},[45,61125,61126],{},[48,61127,60886],{},[114,61129,61100],{"encoding":116},[23,61131,61133],{"className":61132,"ariaHidden":122},[121],[23,61134,61136,61139],{"className":61135},[126],[23,61137],{"className":61138,"style":672},[130],[23,61140,60886],{"className":61141},[135,136],"，优化根本无从下手。",[11,61144,61145],{},"核方法绕开了显式特征：它只通过内积",[23,61147,61149],{"className":61148,"translate":27},[26],[23,61150,61152,61210],{"className":61151,"translate":27},[31],[23,61153,61155],{"className":61154},[35],[37,61156,61157],{"xmlns":39,"display":40},[42,61158,61159,61207],{},[45,61160,61161,61163,61165,61167,61169,61175,61177,61179,61182,61184,61186,61188,61190,61192,61194,61196,61202,61204],{},[48,61162,4836],{},[51,61164,54],{"stretchy":53},[48,61166,753],{},[51,61168,747],{"separator":122},[4518,61170,61171,61173],{},[48,61172,753],{},[51,61174,38291],{"mathvariant":111,"lspace":1522,"rspace":1522},[51,61176,66],{"stretchy":53},[51,61178,69],{},[51,61180,61181],{"stretchy":53},"⟨",[48,61183,60886],{},[51,61185,54],{"stretchy":53},[48,61187,753],{},[51,61189,66],{"stretchy":53},[51,61191,747],{"separator":122},[48,61193,60886],{},[51,61195,54],{"stretchy":53},[4518,61197,61198,61200],{},[48,61199,753],{},[51,61201,38291],{"mathvariant":111,"lspace":1522,"rspace":1522},[51,61203,66],{"stretchy":53},[51,61205,61206],{"stretchy":53},"⟩",[114,61208,61209],{"encoding":116},"k(x,x')=\\langle \\varphi(x),\\varphi(x')\\rangle",[23,61211,61213,61278],{"className":61212,"ariaHidden":122},[121],[23,61214,61216,61219,61222,61225,61228,61231,61234,61266,61269,61272,61275],{"className":61215},[126],[23,61217],{"className":61218,"style":51196},[130],[23,61220,4836],{"className":61221,"style":4851},[135,136],[23,61223,54],{"className":61224},[140],[23,61226,753],{"className":61227},[135,136],[23,61229,747],{"className":61230},[824],[23,61232],{"className":61233,"style":280},[148],[23,61235,61237,61240],{"className":61236},[135],[23,61238,753],{"className":61239},[135,136],[23,61241,61243],{"className":61242},[998],[23,61244,61246],{"className":61245},[199],[23,61247,61249],{"className":61248},[204],[23,61250,61252],{"className":61251,"style":51221},[208],[23,61253,61254,61257],{"style":4989},[23,61255],{"className":61256,"style":1015},[216],[23,61258,61260],{"className":61259},[1019,1020,1021,1022],[23,61261,61263],{"className":61262},[135,1022],[23,61264,38291],{"className":61265},[135,1022],[23,61267,66],{"className":61268},[169],[23,61270],{"className":61271,"style":149},[148],[23,61273,69],{"className":61274},[153],[23,61276],{"className":61277,"style":149},[148],[23,61279,61281,61284,61287,61290,61293,61296,61299,61302,61305,61308,61311,61343],{"className":61280},[126],[23,61282],{"className":61283,"style":51196},[130],[23,61285,61181],{"className":61286},[140],[23,61288,60886],{"className":61289},[135,136],[23,61291,54],{"className":61292},[140],[23,61294,753],{"className":61295},[135,136],[23,61297,66],{"className":61298},[169],[23,61300,747],{"className":61301},[824],[23,61303],{"className":61304,"style":280},[148],[23,61306,60886],{"className":61307},[135,136],[23,61309,54],{"className":61310},[140],[23,61312,61314,61317],{"className":61313},[135],[23,61315,753],{"className":61316},[135,136],[23,61318,61320],{"className":61319},[998],[23,61321,61323],{"className":61322},[199],[23,61324,61326],{"className":61325},[204],[23,61327,61329],{"className":61328,"style":51221},[208],[23,61330,61331,61334],{"style":4989},[23,61332],{"className":61333,"style":1015},[216],[23,61335,61337],{"className":61336},[1019,1020,1021,1022],[23,61338,61340],{"className":61339},[135,1022],[23,61341,38291],{"className":61342},[135,1022],[23,61344,61346],{"className":61345},[169],")⟩",[11,61348,61349,61350,61378],{},"来访问特征空间。这个 ",[23,61351,61353,61366],{"className":61352,"translate":27},[31],[23,61354,61356],{"className":61355},[35],[37,61357,61358],{"xmlns":39},[42,61359,61360,61364],{},[45,61361,61362],{},[48,61363,4836],{},[114,61365,4836],{"encoding":116},[23,61367,61369],{"className":61368,"ariaHidden":122},[121],[23,61370,61372,61375],{"className":61371},[126],[23,61373],{"className":61374,"style":779},[130],[23,61376,4836],{"className":61377,"style":4851},[135,136]," 就是核函数。于是，一个在特征空间中的线性模型，可以等价地写成",[23,61380,61382],{"className":61381,"translate":27},[26],[23,61383,61385,61425],{"className":61384,"translate":27},[31],[23,61386,61388],{"className":61387},[35],[37,61389,61390],{"xmlns":39,"display":40},[42,61391,61392,61422],{},[45,61393,61394,61396,61398,61400,61402,61404,61406,61408,61410,61412,61414,61416,61418,61420],{},[48,61395,8336],{},[51,61397,54],{"stretchy":53},[48,61399,753],{},[51,61401,66],{"stretchy":53},[51,61403,69],{},[51,61405,61181],{"stretchy":53},[48,61407,49694],{},[51,61409,747],{"separator":122},[48,61411,60886],{},[51,61413,54],{"stretchy":53},[48,61415,753],{},[51,61417,66],{"stretchy":53},[51,61419,61206],{"stretchy":53},[51,61421,747],{"separator":122},[114,61423,61424],{"encoding":116},"f(x)=\\langle w,\\varphi(x)\\rangle,",[23,61426,61428,61455],{"className":61427,"ariaHidden":122},[121],[23,61429,61431,61434,61437,61440,61443,61446,61449,61452],{"className":61430},[126],[23,61432],{"className":61433,"style":131},[130],[23,61435,8336],{"className":61436,"style":11398},[135,136],[23,61438,54],{"className":61439},[140],[23,61441,753],{"className":61442},[135,136],[23,61444,66],{"className":61445},[169],[23,61447],{"className":61448,"style":149},[148],[23,61450,69],{"className":61451},[153],[23,61453],{"className":61454,"style":149},[148],[23,61456,61458,61461,61464,61467,61470,61473,61476,61479,61482,61485],{"className":61457},[126],[23,61459],{"className":61460,"style":131},[130],[23,61462,61181],{"className":61463},[140],[23,61465,49694],{"className":61466,"style":49709},[135,136],[23,61468,747],{"className":61469},[824],[23,61471],{"className":61472,"style":280},[148],[23,61474,60886],{"className":61475},[135,136],[23,61477,54],{"className":61478},[140],[23,61480,753],{"className":61481},[135,136],[23,61483,61346],{"className":61484},[169],[23,61486,747],{"className":61487},[824],[11,61489,52929,61490,61518,61519,61547,61548,61576,61577,1471],{},[23,61491,61493,61506],{"className":61492,"translate":27},[31],[23,61494,61496],{"className":61495},[35],[37,61497,61498],{"xmlns":39},[42,61499,61500,61504],{},[45,61501,61502],{},[48,61503,49694],{},[114,61505,49694],{"encoding":116},[23,61507,61509],{"className":61508,"ariaHidden":122},[121],[23,61510,61512,61515],{"className":61511},[126],[23,61513],{"className":61514,"style":703},[130],[23,61516,49694],{"className":61517,"style":49709},[135,136]," 所在的希尔伯特空间（更准确地说，是 ",[23,61520,61522,61535],{"className":61521,"translate":27},[31],[23,61523,61525],{"className":61524},[35],[37,61526,61527],{"xmlns":39},[42,61528,61529,61533],{},[45,61530,61531],{},[48,61532,60886],{},[114,61534,61100],{"encoding":116},[23,61536,61538],{"className":61537,"ariaHidden":122},[121],[23,61539,61541,61544],{"className":61540},[126],[23,61542],{"className":61543,"style":672},[130],[23,61545,60886],{"className":61546},[135,136]," 张成的闭包）就是核 ",[23,61549,61551,61564],{"className":61550,"translate":27},[31],[23,61552,61554],{"className":61553},[35],[37,61555,61556],{"xmlns":39},[42,61557,61558,61562],{},[45,61559,61560],{},[48,61561,4836],{},[114,61563,4836],{"encoding":116},[23,61565,61567],{"className":61566,"ariaHidden":122},[121],[23,61568,61570,61573],{"className":61569},[126],[23,61571],{"className":61572,"style":779},[130],[23,61574,4836],{"className":61575,"style":4851},[135,136]," 对应的再生核希尔伯特空间 ",[23,61578,61580,61598],{"className":61579,"translate":27},[31],[23,61581,61583],{"className":61582},[35],[37,61584,61585],{"xmlns":39},[42,61586,61587,61595],{},[45,61588,61589],{},[898,61590,61591,61593],{},[48,61592,26249],{"mathvariant":375},[48,61594,4836],{},[114,61596,61597],{"encoding":116},"\\mathcal H_k",[23,61599,61601],{"className":61600,"ariaHidden":122},[121],[23,61602,61604,61607],{"className":61603},[126],[23,61605],{"className":61606,"style":42827},[130],[23,61608,61610,61613],{"className":61609},[135],[23,61611,26249],{"className":61612,"style":45024},[135,412],[23,61614,61616],{"className":61615},[998],[23,61617,61619,61640],{"className":61618},[199,200],[23,61620,61622,61637],{"className":61621},[204],[23,61623,61625],{"className":61624,"style":34192},[208],[23,61626,61628,61631],{"style":61627},"top:-2.55em;margin-left:-0.0097em;margin-right:0.05em;",[23,61629],{"className":61630,"style":1015},[216],[23,61632,61634],{"className":61633},[1019,1020,1021,1022],[23,61635,4836],{"className":61636,"style":4851},[135,136,1022],[23,61638,297],{"className":61639},[296],[23,61641,61643],{"className":61642},[204],[23,61644,61646],{"className":61645,"style":2048},[208],[23,61647],{},[11,61649,61650],{},"正则化经验风险最小化（regularized empirical risk minimization）的目标是",[23,61652,61654],{"className":61653,"translate":27},[26],[23,61655,61657,61780],{"className":61656,"translate":27},[31],[23,61658,61660],{"className":61659},[35],[37,61661,61662],{"xmlns":39,"display":40},[42,61663,61664,61777],{},[1518,61665,61667],{"width":61666},"100%",[1524,61668,61669,61672,61770,61772],{},[1527,61670],{"width":61671},"50%",[1527,61673,61674],{},[45,61675,61676,61682,61684,61686,61688,61704,61768],{},[4518,61677,61678,61680],{},[48,61679,8336],{},[51,61681,6881],{},[51,61683,69],{},[48,61685,6956],{},[51,61687,6959],{},[6961,61689,61690,61696],{},[45,61691,61692,61694],{},[48,61693,6967],{},[51,61695,6959],{},[45,61697,61698,61700,61702],{},[48,61699,8336],{},[51,61701,372],{},[48,61703,26249],{"mathvariant":375},[45,61705,61706,61708,61723,61725,61727,61733,61735,61737,61739,61745,61747,61749,61751,61754,61756,61758,61766],{},[51,61707,924],{"fence":122},[61709,61710,61711,61713,61721],"munderover",{},[51,61712,33851],{},[45,61714,61715,61717,61719],{},[48,61716,33493],{},[51,61718,69],{},[2581,61720,4435],{},[48,61722,4806],{},[48,61724,555],{},[51,61726,54],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[898,61728,61729,61731],{},[48,61730,660],{},[48,61732,33493],{},[51,61734,747],{"separator":122},[48,61736,8336],{},[51,61738,54],{"stretchy":53},[898,61740,61741,61743],{},[48,61742,753],{},[48,61744,33493],{},[51,61746,66],{"stretchy":53},[51,61748,66],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[51,61750,5106],{},[48,61752,61753],{},"λ",[48,61755,20680],{"mathvariant":111},[48,61757,8336],{},[19137,61759,61760,61762,61764],{},[48,61761,20680],{"mathvariant":111},[48,61763,26249],{"mathvariant":375},[2581,61765,9952],{},[51,61767,945],{"fence":122},[51,61769,747],{"separator":122},[1527,61771],{"width":61671},[1527,61773,61774],{},[88,61775,61776],{},"(1)",[114,61778,61779],{"encoding":116},"f^*=\\arg\\min_{f\\in\\mathcal H}\\left[\\sum_{i=1}^n L\\bigl(y_i,f(x_i)\\bigr)+\\lambda\\|f\\|_{\\mathcal H}^2\\right],\n\\tag{1}",[23,61781,61783,61827,62171],{"className":61782,"ariaHidden":122},[121],[23,61784,61786,61789,61818,61821,61824],{"className":61785},[126],[23,61787],{"className":61788,"style":48051},[130],[23,61790,61792,61795],{"className":61791},[135],[23,61793,8336],{"className":61794,"style":11398},[135,136],[23,61796,61798],{"className":61797},[998],[23,61799,61801],{"className":61800},[199],[23,61802,61804],{"className":61803},[204],[23,61805,61807],{"className":61806,"style":7045},[208],[23,61808,61809,61812],{"style":4989},[23,61810],{"className":61811,"style":1015},[216],[23,61813,61815],{"className":61814},[1019,1020,1021,1022],[23,61816,6881],{"className":61817},[612,1022],[23,61819],{"className":61820,"style":149},[148],[23,61822,69],{"className":61823},[153],[23,61825],{"className":61826,"style":149},[148],[23,61828,61830,61834,61839,61842,61896,61899,62165,62168],{"className":61829},[126],[23,61831],{"className":61832,"style":61833},[130],"height:3.0277em;vertical-align:-1.2777em;",[23,61835,7085,61837],{"className":61836},[1786],[23,61838,7089],{"style":7088},[23,61840],{"className":61841,"style":280},[148],[23,61843,61845],{"className":61844},[1786,7096],[23,61846,61848,61887],{"className":61847},[199,200],[23,61849,61851,61884],{"className":61850},[204],[23,61852,61854,61874],{"className":61853,"style":7106},[208],[23,61855,61856,61859],{"style":14738},[23,61857],{"className":61858,"style":217},[216],[23,61860,61862],{"className":61861},[1019,1020,1021,1022],[23,61863,61865,61868,61871],{"className":61864},[135,1022],[23,61866,8336],{"className":61867,"style":11398},[135,136,1022],[23,61869,372],{"className":61870},[153,1022],[23,61872,26249],{"className":61873,"style":45024},[135,412,1022],[23,61875,61876,61879],{"style":7130},[23,61877],{"className":61878,"style":217},[216],[23,61880,61881],{},[23,61882,6967],{"className":61883},[1786],[23,61885,297],{"className":61886},[296],[23,61888,61890],{"className":61889},[204],[23,61891,61894],{"className":61892,"style":61893},[208],"height:0.8882em;",[23,61895],{},[23,61897],{"className":61898,"style":280},[148],[23,61900,61902,61908,61974,61977,61980,61986,62026,62029,62032,62035,62038,62078,62081,62087,62090,62093,62096,62099,62102,62105,62159],{"className":61901},[1839],[23,61903,61905],{"className":61904,"style":1844},[140,1843],[23,61906,924],{"className":61907},[1066,10069],[23,61909,61911],{"className":61910},[1786,7096],[23,61912,61914,61966],{"className":61913},[199,200],[23,61915,61917,61963],{"className":61916},[204],[23,61918,61921,61941,61951],{"className":61919,"style":61920},[208],"height:1.6514em;",[23,61922,61923,61926],{"style":34075},[23,61924],{"className":61925,"style":34079},[216],[23,61927,61929],{"className":61928},[1019,1020,1021,1022],[23,61930,61932,61935,61938],{"className":61931},[135,1022],[23,61933,33493],{"className":61934},[135,136,1022],[23,61936,69],{"className":61937},[153,1022],[23,61939,4435],{"className":61940},[135,1022],[23,61942,61943,61946],{"style":34088},[23,61944],{"className":61945,"style":34079},[216],[23,61947,61948],{},[23,61949,33851],{"className":61950},[1786,1790,1791],[23,61952,61954,61957],{"style":61953},"top:-4.3em;margin-left:0em;",[23,61955],{"className":61956,"style":34079},[216],[23,61958,61960],{"className":61959},[1019,1020,1021,1022],[23,61961,4806],{"className":61962},[135,136,1022],[23,61964,297],{"className":61965},[296],[23,61967,61969],{"className":61968},[204],[23,61970,61972],{"className":61971,"style":34106},[208],[23,61973],{},[23,61975],{"className":61976,"style":280},[148],[23,61978,555],{"className":61979},[135,136],[23,61981,61983],{"className":61982},[140],[23,61984,54],{"className":61985},[1066,1067],[23,61987,61989,61992],{"className":61988},[135],[23,61990,660],{"className":61991,"style":676},[135,136],[23,61993,61995],{"className":61994},[998],[23,61996,61998,62018],{"className":61997},[199,200],[23,61999,62001,62015],{"className":62000},[204],[23,62002,62004],{"className":62003,"style":33647},[208],[23,62005,62006,62009],{"style":21850},[23,62007],{"className":62008,"style":1015},[216],[23,62010,62012],{"className":62011},[1019,1020,1021,1022],[23,62013,33493],{"className":62014},[135,136,1022],[23,62016,297],{"className":62017},[296],[23,62019,62021],{"className":62020},[204],[23,62022,62024],{"className":62023,"style":2048},[208],[23,62025],{},[23,62027,747],{"className":62028},[824],[23,62030],{"className":62031,"style":280},[148],[23,62033,8336],{"className":62034,"style":11398},[135,136],[23,62036,54],{"className":62037},[140],[23,62039,62041,62044],{"className":62040},[135],[23,62042,753],{"className":62043},[135,136],[23,62045,62047],{"className":62046},[998],[23,62048,62050,62070],{"className":62049},[199,200],[23,62051,62053,62067],{"className":62052},[204],[23,62054,62056],{"className":62055,"style":33647},[208],[23,62057,62058,62061],{"style":2029},[23,62059],{"className":62060,"style":1015},[216],[23,62062,62064],{"className":62063},[1019,1020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是核 ",[23,62225,62227,62240],{"className":62226,"translate":27},[31],[23,62228,62230],{"className":62229},[35],[37,62231,62232],{"xmlns":39},[42,62233,62234,62238],{},[45,62235,62236],{},[48,62237,4836],{},[114,62239,4836],{"encoding":116},[23,62241,62243],{"className":62242,"ariaHidden":122},[121],[23,62244,62246,62249],{"className":62245},[126],[23,62247],{"className":62248,"style":779},[130],[23,62250,4836],{"className":62251,"style":4851},[135,136]," 对应的 RKHS，",[23,62254,62256,62274],{"className":62255,"translate":27},[31],[23,62257,62259],{"className":62258},[35],[37,62260,62261],{"xmlns":39},[42,62262,62263,62271],{},[45,62264,62265,62267,62269],{},[48,62266,61753],{},[51,62268,7378],{},[2581,62270,1532],{},[114,62272,62273],{"encoding":116},"\\lambda\\ge 0",[23,62275,62277,62296],{"className":62276,"ariaHidden":122},[121],[23,62278,62280,62284,62287,62290,62293],{"className":62279},[126],[23,62281],{"className":62282,"style":62283},[130],"height:0.8304em;vertical-align:-0.136em;",[23,62285,61753],{"className":62286},[135,136],[23,62288],{"className":62289,"style":149},[148],[23,62291,7378],{"className":62292},[153],[23,62294],{"className":62295,"style":149},[148],[23,62297,62299,62302],{"className":62298},[126],[23,62300],{"className":62301,"style":2770},[130],[23,62303,1532],{"className":62304},[135]," 是正则化参数。若 ",[23,62307,62309,62322],{"className":62308,"translate":27},[31],[23,62310,62312],{"className":62311},[35],[37,62313,62314],{"xmlns":39},[42,62315,62316,62320],{},[45,62317,62318],{},[48,62319,26249],{"mathvariant":375},[114,62321,45011],{"encoding":116},[23,62323,62325],{"className":62324,"ariaHidden":122},[121],[23,62326,62328,62331],{"className":62327},[126],[23,62329],{"className":62330,"style":470},[130],[23,62332,26249],{"className":62333,"style":45024},[135,412]," 是无穷维的，这看起来是一个无穷维优化问题。表示定理说：它其实是一个 ",[23,62336,62338,62351],{"className":62337,"translate":27},[31],[23,62339,62341],{"className":62340},[35],[37,62342,62343],{"xmlns":39},[42,62344,62345,62349],{},[45,62346,62347],{},[48,62348,4806],{},[114,62350,4806],{"encoding":116},[23,62352,62354],{"className":62353,"ariaHidden":122},[121],[23,62355,62357,62360],{"className":62356},[126],[23,62358],{"className":62359,"style":703},[130],[23,62361,4806],{"className":62362},[135,136]," 维问题。",[338,62365,62367],{"id":62366},"二rkhs-速览","二、RKHS 速览",[2501,62369,62371],{"id":62370},"_21-正定核与再生性","2.1 正定核与再生性",[11,62373,4410,62374,62403,62404,15001,62500,62503,62504,62532,62533,62697,62698,7350],{},[23,62375,62377,62391],{"className":62376,"translate":27},[31],[23,62378,62380],{"className":62379},[35],[37,62381,62382],{"xmlns":39},[42,62383,62384,62388],{},[45,62385,62386],{},[48,62387,434],{"mathvariant":375},[114,62389,62390],{"encoding":116},"\\mathcal X",[23,62392,62394],{"className":62393,"ariaHidden":122},[121],[23,62395,62397,62400],{"className":62396},[126],[23,62398],{"className":62399,"style":470},[130],[23,62401,434],{"className":62402,"style":474},[135,412]," 是一个非空集合。一个对称函数 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definite kernel），如果对任意 ",[23,62505,62507,62520],{"className":62506,"translate":27},[31],[23,62508,62510],{"className":62509},[35],[37,62511,62512],{"xmlns":39},[42,62513,62514,62518],{},[45,62515,62516],{},[48,62517,4806],{},[114,62519,4806],{"encoding":116},[23,62521,62523],{"className":62522,"ariaHidden":122},[121],[23,62524,62526,62529],{"className":62525},[126],[23,62527],{"className":62528,"style":703},[130],[23,62530,4806],{"className":62531},[135,136],"、任意 ",[23,62534,62536,62571],{"className":62535,"translate":27},[31],[23,62537,62539],{"className":62538},[35],[37,62540,62541],{"xmlns":39},[42,62542,62543,62568],{},[45,62544,62545,62551,62553,62556,62558,62564,62566],{},[898,62546,62547,62549],{},[48,62548,753],{},[2581,62550,4435],{},[51,62552,747],{"separator":122},[51,62554,62555],{},"…",[51,62557,747],{"separator":122},[898,62559,62560,62562],{},[48,62561,753],{},[48,62563,4806],{},[51,62565,372],{},[48,62567,434],{"mathvariant":375},[114,62569,62570],{"encoding":116},"x_1,\\dots,x_n\\in\\mathcal X",[23,62572,62574,62688],{"className":62573,"ariaHidden":122},[121],[23,62575,62577,62581,62621,62624,62627,62630,62633,62636,62639,62679,62682,62685],{"className":62576},[126],[23,62578],{"className":62579,"style":62580},[130],"height:0.7335em;vertical-align:-0.1944em;",[23,62582,62584,62587],{"className":62583},[135],[23,62585,753],{"className":62586},[135,136],[23,62588,62590],{"className":62589},[998],[23,62591,62593,62613],{"className":62592},[199,200],[23,62594,62596,62610],{"className":62595},[204],[23,62597,62599],{"className":62598,"style":18647},[208],[23,62600,62601,62604],{"style":2029},[23,62602],{"className":62603,"style":1015},[216],[23,62605,62607],{"className":62606},[1019,1020,1021,1022],[23,62608,4435],{"className":62609},[135,1022],[23,62611,297],{"className":62612},[296],[23,62614,62616],{"className":62615},[204],[23,62617,62619],{"className":62618,"style":2048},[208],[23,62620],{},[23,62622,747],{"className":62623},[824],[23,62625],{"className":62626,"style":280},[148],[23,62628,62555],{"className":62629},[1839],[23,62631],{"className":62632,"style":280},[148],[23,62634,747],{"className":62635},[824],[23,62637],{"className":62638,"style":280},[148],[23,62640,62642,62645],{"className":62641},[135],[23,62643,753],{"className":62644},[135,136],[23,62646,62648],{"className":62647},[998],[23,62649,62651,62671],{"className":62650},[199,200],[23,62652,62654,62668],{"className":62653},[204],[23,62655,62657],{"className":62656,"style":6499},[208],[23,62658,62659,62662],{"style":2029},[23,62660],{"className":62661,"style":1015},[216],[23,62663,62665],{"className":62664},[1019,1020,1021,1022],[23,62666,4806],{"className":62667},[135,136,1022],[23,62669,297],{"className":62670},[296],[23,62672,62674],{"className":62673},[204],[23,62675,62677],{"className":62676,"style":2048},[208],[23,62678],{},[23,62680],{"className":62681,"style":149},[148],[23,62683,372],{"className":62684},[153],[23,62686],{"className":62687,"style":149},[148],[23,62689,62691,62694],{"className":62690},[126],[23,62692],{"className":62693,"style":470},[130],[23,62695,434],{"className":62696,"style":474},[135,412]," 和任意 ",[23,62699,62701,62735],{"className":62700,"translate":27},[31],[23,62702,62704],{"className":62703},[35],[37,62705,62706],{"xmlns":39},[42,62707,62708,62732],{},[45,62709,62710,62716,62718,62720,62722,62728,62730],{},[898,62711,62712,62714],{},[48,62713,12710],{},[2581,62715,4435],{},[51,62717,747],{"separator":122},[51,62719,62555],{},[51,62721,747],{"separator":122},[898,62723,62724,62726],{},[48,62725,12710],{},[48,62727,4806],{},[51,62729,372],{},[48,62731,572],{"mathvariant":571},[114,62733,62734],{"encoding":116},"c_1,\\dots,c_n\\in\\mathbb R",[23,62736,62738,62851],{"className":62737,"ariaHidden":122},[121],[23,62739,62741,62744,62784,62787,62790,62793,62796,62799,62802,62842,62845,62848],{"className":62740},[126],[23,62742],{"className":62743,"style":62580},[130],[23,62745,62747,62750],{"className":62746},[135],[23,62748,12710],{"className":62749},[135,136],[23,62751,62753],{"className":62752},[998],[23,62754,62756,62776],{"className":62755},[199,200],[23,62757,62759,62773],{"className":62758},[204],[23,62760,62762],{"className":62761,"style":18647},[208],[23,62763,62764,62767],{"style":2029},[23,62765],{"className":62766,"style":1015},[216],[23,62768,62770],{"className":62769},[1019,1020,1021,1022],[23,62771,4435],{"className":62772},[135,1022],[23,62774,297],{"className":62775},[296],[23,62777,62779],{"className":62778},[204],[23,62780,62782],{"className":62781,"style":2048},[208],[23,62783],{},[23,62785,747],{"className":62786},[824],[23,62788],{"className":62789,"style":280},[148],[23,62791,62555],{"className":62792},[1839],[23,62794],{"className":62795,"style":280},[148],[23,62797,747],{"className":62798},[824],[23,62800],{"className":62801,"style":280},[148],[23,62803,62805,62808],{"className":62804},[135],[23,62806,12710],{"className":62807},[135,136],[23,62809,62811],{"className":62810},[998],[23,62812,62814,62834],{"className":62813},[199,200],[23,62815,62817,62831],{"className":62816},[204],[23,62818,62820],{"className":62819,"style":6499},[208],[23,62821,62822,62825],{"style":2029},[23,62823],{"className":62824,"style":1015},[216],[23,62826,62828],{"className":62827},[1019,1020,1021,1022],[23,62829,4806],{"className":62830},[135,136,1022],[23,62832,297],{"className":62833},[296],[23,62835,62837],{"className":62836},[204],[23,62838,62840],{"className":62839,"style":2048},[208],[23,62841],{},[23,62843],{"className":62844,"style":149},[148],[23,62846,372],{"className":62847},[153],[23,62849],{"className":62850,"style":149},[148],[23,62852,62854,62857],{"className":62853},[126],[23,62855],{"className":62856,"style":640},[130],[23,62858,572],{"className":62859},[135,644],[23,62861,62863],{"className":62862,"translate":27},[26],[23,62864,62866,62944],{"className":62865,"translate":27},[31],[23,62867,62869],{"className":62868},[35],[37,62870,62871],{"xmlns":39,"display":40},[42,62872,62873,62941],{},[45,62874,62875,62889,62903,62909,62915,62917,62919,62921,62927,62929,62935,62937,62939],{},[61709,62876,62877,62879,62887],{},[51,62878,33851],{},[45,62880,62881,62883,62885],{},[48,62882,33493],{},[51,62884,69],{},[2581,62886,4435],{},[48,62888,4806],{},[61709,62890,62891,62893,62901],{},[51,62892,33851],{},[45,62894,62895,62897,62899],{},[48,62896,35518],{},[51,62898,69],{},[2581,62900,4435],{},[48,62902,4806],{},[898,62904,62905,62907],{},[48,62906,12710],{},[48,62908,33493],{},[898,62910,62911,62913],{},[48,62912,12710],{},[48,62914,35518],{},[88,62916,90],{},[48,62918,4836],{},[51,62920,54],{"stretchy":53},[898,62922,62923,62925],{},[48,62924,753],{},[48,62926,33493],{},[51,62928,747],{"separator":122},[898,62930,62931,62933],{},[48,62932,753],{},[48,62934,35518],{},[51,62936,66],{"stretchy":53},[51,62938,7378],{},[2581,62940,2583],{},[114,62942,62943],{"encoding":116},"\\sum_{i=1}^n\\sum_{j=1}^n c_i c_j\\,k(x_i,x_j)\\ge 0.",[23,62945,62947,63276],{"className":62946,"ariaHidden":122},[121],[23,62948,62950,62954,63018,63021,63086,63089,63129,63169,63172,63175,63178,63218,63221,63224,63264,63267,63270,63273],{"className":62949},[126],[23,62951],{"className":62952,"style":62953},[130],"height:3.0652em;vertical-align:-1.4138em;",[23,62955,62957],{"className":62956},[1786,7096],[23,62958,62960,63010],{"className":62959},[199,200],[23,62961,62963,63007],{"className":62962},[204],[23,62964,62966,62986,62996],{"className":62965,"style":61920},[208],[23,62967,62968,62971],{"style":34075},[23,62969],{"className":62970,"style":34079},[216],[23,62972,62974],{"className":62973},[1019,1020,1021,1022],[23,62975,62977,62980,62983],{"className":62976},[135,1022],[23,62978,33493],{"className":62979},[135,136,1022],[23,62981,69],{"className":62982},[153,1022],[23,62984,4435],{"className":62985},[135,1022],[23,62987,62988,62991],{"style":34088},[23,62989],{"className":62990,"style":34079},[216],[23,62992,62993],{},[23,62994,33851],{"className":62995},[1786,1790,1791],[23,62997,62998,63001],{"style":61953},[23,62999],{"className":63000,"style":34079},[216],[23,63002,63004],{"className":63003},[1019,1020,1021,1022],[23,63005,4806],{"className":63006},[135,136,1022],[23,63008,297],{"className":63009},[296],[23,63011,63013],{"className":63012},[204],[23,63014,63016],{"className":63015,"style":34106},[208],[23,63017],{},[23,63019],{"className":63020,"style":280},[148],[23,63022,63024],{"className":63023},[1786,7096],[23,63025,63027,63077],{"className":63026},[199,200],[23,63028,63030,63074],{"className":63029},[204],[23,63031,63033,63053,63063],{"className":63032,"style":61920},[208],[23,63034,63035,63038],{"style":34075},[23,63036],{"className":63037,"style":34079},[216],[23,63039,63041],{"className":63040},[1019,1020,1021,1022],[23,63042,63044,63047,63050],{"className":63043},[135,1022],[23,63045,35518],{"className":63046,"style":35603},[135,136,1022],[23,63048,69],{"className":63049},[153,1022],[23,63051,4435],{"className":63052},[135,1022],[23,63054,63055,63058],{"style":34088},[23,63056],{"className":63057,"style":34079},[216],[23,63059,63060],{},[23,63061,33851],{"className":63062},[1786,1790,1791],[23,63064,63065,63068],{"style":61953},[23,63066],{"className":63067,"style":34079},[216],[23,63069,63071],{"className":63070},[1019,1020,1021,1022],[23,63072,4806],{"className":63073},[135,136,1022],[23,63075,297],{"className":63076},[296],[23,63078,63080],{"className":63079},[204],[23,63081,63084],{"className":63082,"style":63083},[208],"height:1.4138em;",[23,63085],{},[23,63087],{"className":63088,"style":280},[148],[23,63090,63092,63095],{"className":63091},[135],[23,63093,12710],{"className":63094},[135,136],[23,63096,63098],{"className":63097},[998],[23,63099,63101,63121],{"className":63100},[199,200],[23,63102,63104,63118],{"className":63103},[204],[23,63105,63107],{"className":63106,"style":33647},[208],[23,63108,63109,63112],{"style":2029},[23,63110],{"className":63111,"style":1015},[216],[23,63113,63115],{"className":63114},[1019,1020,1021,1022],[23,63116,33493],{"className":63117},[135,136,1022],[23,63119,297],{"className":63120},[296],[23,63122,63124],{"className":63123},[204],[23,63125,63127],{"className":63126,"style":2048},[208],[23,63128],{},[23,63130,63132,63135],{"className":63131},[135],[23,63133,12710],{"className":63134},[135,136],[23,63136,63138],{"className":63137},[998],[23,63139,63141,63161],{"className":63140},[199,200],[23,63142,63144,63158],{"className":63143},[204],[23,63145,63147],{"className":63146,"style":33647},[208],[23,63148,63149,63152],{"style":2029},[23,63150],{"className":63151,"style":1015},[216],[23,63153,63155],{"className":63154},[1019,1020,1021,1022],[23,63156,35518],{"className":63157,"style":35603},[135,136,1022],[23,63159,297],{"className":63160},[296],[23,63162,63164],{"className":63163},[204],[23,63165,63167],{"className":63166,"style":14292},[208],[23,63168],{},[23,63170],{"className":63171,"style":280},[148],[23,63173,4836],{"className":63174,"style":4851},[135,136],[23,63176,54],{"className":63177},[140],[23,63179,63181,63184],{"className":63180},[135],[23,63182,753],{"className":63183},[135,136],[23,63185,63187],{"className":63186},[998],[23,63188,63190,63210],{"className":63189},[199,200],[23,63191,63193,63207],{"className":63192},[204],[23,63194,63196],{"className":63195,"style":33647},[208],[23,63197,63198,63201],{"style":2029},[23,63199],{"className":63200,"style":1015},[216],[23,63202,63204],{"className":63203},[1019,1020,1021,1022],[23,63205,33493],{"className":63206},[135,136,1022],[23,63208,297],{"className":63209},[296],[23,63211,63213],{"className":63212},[204],[23,63214,63216],{"className":63215,"style":2048},[208],[23,63217],{},[23,63219,747],{"className":63220},[824],[23,63222],{"className":63223,"style":280},[148],[23,63225,63227,63230],{"className":63226},[135],[23,63228,753],{"className":63229},[135,136],[23,63231,63233],{"className":63232},[998],[23,63234,63236,63256],{"className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",[23,63289,63291,63335],{"className":63290,"translate":27},[31],[23,63292,63294],{"className":63293},[35],[37,63295,63296],{"xmlns":39},[42,63297,63298,63332],{},[45,63299,63300,63310,63312,63314,63316,63322,63324,63330],{},[898,63301,63302,63304],{},[48,63303,26918],{},[45,63305,63306,63308],{},[48,63307,33493],{},[48,63309,35518],{},[51,63311,69],{},[48,63313,4836],{},[51,63315,54],{"stretchy":53},[898,63317,63318,63320],{},[48,63319,753],{},[48,63321,33493],{},[51,63323,747],{"separator":122},[898,63325,63326,63328],{},[48,63327,753],{},[48,63329,35518],{},[51,63331,66],{"stretchy":53},[114,63333,63334],{"encoding":116},"K_{ij}=k(x_i,x_j)",[23,63336,63338,63399],{"className":63337,"ariaHidden":122},[121],[23,63339,63341,63345,63390,63393,63396],{"className":63340},[126],[23,63342],{"className":63343,"style":63344},[130],"height:0.9694em;vertical-align:-0.2861em;",[23,63346,63348,63351],{"className":63347},[135],[23,63349,26918],{"className":63350,"style":37141},[135,136],[23,63352,63354],{"className":63353},[998],[23,63355,63357,63382],{"className":63356},[199,200],[23,63358,63360,63379],{"className":63359},[204],[23,63361,63363],{"className":63362,"style":33647},[208],[23,63364,63366,63369],{"style":63365},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[23,63367],{"className":63368,"style":1015},[216],[23,63370,63372],{"className":63371},[1019,1020,1021,1022],[23,63373,63375],{"className":63374},[135,1022],[23,63376,63378],{"className":63377,"style":35603},[135,136,1022],"ij",[23,63380,297],{"className":63381},[296],[23,63383,63385],{"className":63384},[204],[23,63386,63388],{"className":63387,"style":14292},[208],[23,63389],{},[23,63391],{"className":63392,"style":149},[148],[23,63394,69],{"className":63395},[153],[23,63397],{"className":63398,"style":149},[148],[23,63400,63402,63405,63408,63411,63451,63454,63457,63497],{"className":63401},[126],[23,63403],{"className":63404,"style":14253},[130],[23,63406,4836],{"className":63407,"style":4851},[135,136],[23,63409,54],{"className":63410},[140],[23,63412,63414,63417],{"className":63413},[135],[23,63415,753],{"className":63416},[135,136],[23,63418,63420],{"className":63419},[998],[23,63421,63423,63443],{"className":63422},[199,200],[23,63424,63426,63440],{"className":63425},[204],[23,63427,63429],{"className":63428,"style":33647},[208],[23,63430,63431,63434],{"style":2029},[23,63432],{"className":63433,"style":1015},[216],[23,63435,63437],{"className":63436},[1019,1020,1021,1022],[23,63438,33493],{"className":63439},[135,136,1022],[23,63441,297],{"className":63442},[296],[23,63444,63446],{"className":63445},[204],[23,63447,63449],{"className":63448,"style":2048},[208],[23,63450],{},[23,63452,747],{"className":63453},[824],[23,63455],{"className":63456,"style":280},[148],[23,63458,63460,63463],{"className":63459},[135],[23,63461,753],{"className":63462},[135,136],[23,63464,63466],{"className":63465},[998],[23,63467,63469,63489],{"className":63468},[199,200],[23,63470,63472,63486],{"className":63471},[204],[23,63473,63475],{"className":63474,"style":33647},[208],[23,63476,63477,63480],{"style":2029},[23,63478],{"className":63479,"style":1015},[216],[23,63481,63483],{"className":63482},[1019,1020,1021,1022],[23,63484,35518],{"className":63485,"style":35603},[135,136,1022],[23,63487,297],{"className":63488},[296],[23,63490,63492],{"className":63491},[204],[23,63493,63495],{"className":63494,"style":14292},[208],[23,63496],{},[23,63498,66],{"className":63499},[169]," 总是半正定的。",[6857,63502,63504,63635,63956,64170],{"title":63503},"定理 2.1（Moore–Aronszajn）",[11,63505,63506,63507,63535,63536,63605,63606,63634],{},"对任意正定核 ",[23,63508,63510,63523],{"className":63509,"translate":27},[31],[23,63511,63513],{"className":63512},[35],[37,63514,63515],{"xmlns":39},[42,63516,63517,63521],{},[45,63518,63519],{},[48,63520,4836],{},[114,63522,4836],{"encoding":116},[23,63524,63526],{"className":63525,"ariaHidden":122},[121],[23,63527,63529,63532],{"className":63528},[126],[23,63530],{"className":63531,"style":779},[130],[23,63533,4836],{"className":63534,"style":4851},[135,136],"，存在唯一的希尔伯特空间 ",[23,63537,63539,63556],{"className":63538,"translate":27},[31],[23,63540,63542],{"className":63541},[35],[37,63543,63544],{"xmlns":39},[42,63545,63546,63554],{},[45,63547,63548],{},[898,63549,63550,63552],{},[48,63551,26249],{"mathvariant":375},[48,63553,4836],{},[114,63555,61597],{"encoding":116},[23,63557,63559],{"className":63558,"ariaHidden":122},[121],[23,63560,63562,63565],{"className":63561},[126],[23,63563],{"className":63564,"style":42827},[130],[23,63566,63568,63571],{"className":63567},[135],[23,63569,26249],{"className":63570,"style":45024},[135,412],[23,63572,63574],{"className":63573},[998],[23,63575,63577,63597],{"className":63576},[199,200],[23,63578,63580,63594],{"className":63579},[204],[23,63581,63583],{"className":63582,"style":34192},[208],[23,63584,63585,63588],{"style":61627},[23,63586],{"className":63587,"style":1015},[216],[23,63589,63591],{"className":63590},[1019,1020,1021,1022],[23,63592,4836],{"className":63593,"style":4851},[135,136,1022],[23,63595,297],{"className":63596},[296],[23,63598,63600],{"className":63599},[204],[23,63601,63603],{"className":63602,"style":2048},[208],[23,63604],{},"，由 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X",[23,63661,63663,63681],{"className":63662,"ariaHidden":122},[121],[23,63664,63666,63669,63672,63675,63678],{"className":63665},[126],[23,63667],{"className":63668,"style":513},[130],[23,63670,753],{"className":63671},[135,136],[23,63673],{"className":63674,"style":149},[148],[23,63676,372],{"className":63677},[153],[23,63679],{"className":63680,"style":149},[148],[23,63682,63684,63687],{"className":63683},[126],[23,63685],{"className":63686,"style":470},[130],[23,63688,434],{"className":63689,"style":474},[135,412],[23,63691,63693,63725],{"className":63692,"translate":27},[31],[23,63694,63696],{"className":63695},[35],[37,63697,63698],{"xmlns":39},[42,63699,63700,63722],{},[45,63701,63702,63704,63706,63708,63710,63712,63714,63716],{},[48,63703,4836],{},[51,63705,54],{"stretchy":53},[51,63707,5758],{},[51,63709,747],{"separator":122},[48,63711,753],{},[51,63713,66],{"stretchy":53},[51,63715,372],{},[898,63717,63718,63720],{},[48,63719,26249],{"mathvariant":375},[48,63721,4836],{},[114,63723,63724],{"encoding":116},"k(\\cdot,x)\\in\\mathcal 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和 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",[23,64174,64176,64189],{"className":64175,"translate":27},[31],[23,64177,64179],{"className":64178},[35],[37,64180,64181],{"xmlns":39},[42,64182,64183,64187],{},[45,64184,64185],{},[48,64186,4836],{},[114,64188,4836],{"encoding":116},[23,64190,64192],{"className":64191,"ariaHidden":122},[121],[23,64193,64195,64198],{"className":64194},[126],[23,64196],{"className":64197,"style":779},[130],[23,64199,4836],{"className":64200,"style":4851},[135,136],[18,64202,64203],{},"再生核希尔伯特空间",[11,64205,64206,64207,64210,64211,64239,64240,64297],{},"第二条就是",[18,64208,64209],{},"再生性","（reproducing property）。它把「函数在 ",[23,64212,64214,64227],{"className":64213,"translate":27},[31],[23,64215,64217],{"className":64216},[35],[37,64218,64219],{"xmlns":39},[42,64220,64221,64225],{},[45,64222,64223],{},[48,64224,753],{},[114,64226,753],{"encoding":116},[23,64228,64230],{"className":64229,"ariaHidden":122},[121],[23,64231,64233,64236],{"className":64232},[126],[23,64234],{"className":64235,"style":703},[130],[23,64237,753],{"className":64238},[135,136]," 点的取值」变成了「与 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的内积」。等价地说，取值泛函",[23,64299,64301],{"className":64300,"translate":27},[26],[23,64302,64304,64360],{"className":64303,"translate":27},[31],[23,64305,64307],{"className":64306},[35],[37,64308,64309],{"xmlns":39,"display":40},[42,64310,64311,64357],{},[45,64312,64313,64319,64321,64327,64329,64331,64333,64335,64341,64343,64345,64347,64349,64351,64353,64355],{},[898,64314,64315,64317],{},[48,64316,736],{},[48,64318,753],{},[51,64320,558],{},[898,64322,64323,64325],{},[48,64324,26249],{"mathvariant":375},[48,64326,4836],{},[51,64328,568],{},[48,64330,572],{"mathvariant":571},[51,64332,747],{"separator":122},[148,64334],{"width":750},[898,64336,64337,64339],{},[48,64338,736],{},[48,64340,753],{},[51,64342,54],{"stretchy":53},[48,64344,8336],{},[51,64346,66],{"stretchy":53},[51,64348,69],{},[48,64350,8336],{},[51,64352,54],{"stretchy":53},[48,64354,753],{},[51,64356,66],{"stretchy":53},[114,64358,64359],{"encoding":116},"\\delta_x:\\mathcal H_k\\to\\mathbb R,\\qquad 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里的函数都「足够好」：它们是由核函数线性组合逼近出来的，而核的选择决定了这个空间的光滑性。",[2501,65722,65724],{"id":65723},"_22-特征映射与核技巧","2.2 特征映射与核技巧",[11,65726,65727],{},"令",[23,65729,65731],{"className":65730,"translate":27},[26],[23,65732,65734,65778],{"className":65733,"translate":27},[31],[23,65735,65737],{"className":65736},[35],[37,65738,65739],{"xmlns":39,"display":40},[42,65740,65741,65775],{},[45,65742,65743,65745,65747,65749,65751,65753,65755,65757,65759,65761,65763,65765,65767,65773],{},[48,65744,60886],{},[51,65746,54],{"stretchy":53},[48,65748,753],{},[51,65750,66],{"stretchy":53},[51,65752,69],{},[48,65754,4836],{},[51,65756,54],{"stretchy":53},[51,65758,5758],{},[51,65760,747],{"separator":122},[48,65762,753],{},[51,65764,66],{"stretchy":53},[51,65766,372],{},[898,65768,65769,65771],{},[48,65770,26249],{"mathvariant":375},[48,65772,4836],{},[48,65774,112],{"mathvariant":111},[114,65776,65777],{"encoding":116},"\\varphi(x)=k(\\cdot,x)\\in\\mathcal H_k.",[23,65779,65781,65808,65844],{"className":65780,"ariaHidden":122},[121],[23,65782,65784,65787,65790,65793,65796,65799,65802,65805],{"className":65783},[126],[23,65785],{"className":65786,"style":131},[130],[23,65788,60886],{"className":65789},[135,136],[23,65791,54],{"className":65792},[140],[23,65794,753],{"className":65795},[135,136],[23,65797,66],{"className":65798},[169],[23,65800],{"className":65801,"style":149},[148],[23,65803,69],{"className":65804},[153],[23,65806],{"className":65807,"style":149},[148],[23,65809,65811,65814,65817,65820,65823,65826,65829,65832,65835,65838,65841],{"className":65810},[126],[23,65812],{"className":65813,"style":131},[130],[23,65815,4836],{"className":65816,"style":4851},[135,136],[23,65818,54],{"className":65819},[140],[23,65821,5758],{"className":65822},[135],[23,65824,747],{"className":65825},[824],[23,65827],{"className":65828,"style":280},[148],[23,65830,753],{"className":65831},[135,136],[23,65833,66],{"className":65834},[169],[23,65836],{"className":65837,"style":149},[148],[23,65839,372],{"className":65840},[153],[23,65842],{"className":65843,"style":149},[148],[23,65845,65847,65850,65890],{"className":65846},[126],[23,65848],{"className":65849,"style":42827},[130],[23,65851,65853,65856],{"className":65852},[135],[23,65854,26249],{"className":65855,"style":45024},[135,412],[23,65857,65859],{"className":65858},[998],[23,65860,65862,65882],{"className":65861},[199,200],[23,65863,65865,65879],{"className":65864},[204],[23,65866,65868],{"className":65867,"style":34192},[208],[23,65869,65870,65873],{"style":61627},[23,65871],{"className":65872,"style":1015},[216],[23,65874,65876],{"className":65875},[1019,1020,1021,1022],[23,65877,4836],{"className":65878,"style":4851},[135,136,1022],[23,65880,297],{"className":65881},[296],[23,65883,65885],{"className":65884},[204],[23,65886,65888],{"className":65887,"style":2048},[208],[23,65889],{},[23,65891,112],{"className":65892},[135],[11,65894,65895,65896,65924,65925,65953,65954,66023],{},"则 ",[23,65897,65899,65912],{"className":65898,"translate":27},[31],[23,65900,65902],{"className":65901},[35],[37,65903,65904],{"xmlns":39},[42,65905,65906,65910],{},[45,65907,65908],{},[48,65909,60886],{},[114,65911,61100],{"encoding":116},[23,65913,65915],{"className":65914,"ariaHidden":122},[121],[23,65916,65918,65921],{"className":65917},[126],[23,65919],{"className":65920,"style":672},[130],[23,65922,60886],{"className":65923},[135,136]," 就是一个从 ",[23,65926,65928,65941],{"className":65927,"translate":27},[31],[23,65929,65931],{"className":65930},[35],[37,65932,65933],{"xmlns":39},[42,65934,65935,65939],{},[45,65936,65937],{},[48,65938,434],{"mathvariant":375},[114,65940,62390],{"encoding":116},[23,65942,65944],{"className":65943,"ariaHidden":122},[121],[23,65945,65947,65950],{"className":65946},[126],[23,65948],{"className":65949,"style":470},[130],[23,65951,434],{"className":65952,"style":474},[135,412]," 到 ",[23,65955,65957,65974],{"className":65956,"translate":27},[31],[23,65958,65960],{"className":65959},[35],[37,65961,65962],{"xmlns":39},[42,65963,65964,65972],{},[45,65965,65966],{},[898,65967,65968,65970],{},[48,65969,26249],{"mathvariant":375},[48,65971,4836],{},[114,65973,61597],{"encoding":116},[23,65975,65977],{"className":65976,"ariaHidden":122},[121],[23,65978,65980,65983],{"className":65979},[126],[23,65981],{"className":65982,"style":42827},[130],[23,65984,65986,65989],{"className":65985},[135],[23,65987,26249],{"className":65988,"style":45024},[135,412],[23,65990,65992],{"className":65991},[998],[23,65993,65995,66015],{"className":65994},[199,200],[23,65996,65998,66012],{"className":65997},[204],[23,65999,66001],{"className":66000,"style":34192},[208],[23,66002,66003,66006],{"style":61627},[23,66004],{"className":66005,"style":1015},[216],[23,66007,66009],{"className":66008},[1019,1020,1021,1022],[23,66010,4836],{"className":66011,"style":4851},[135,136,1022],[23,66013,297],{"className":66014},[296],[23,66016,66018],{"className":66017},[204],[23,66019,66021],{"className":66020,"style":2048},[208],[23,66022],{}," 的特征映射，并且",[23,66025,66027],{"className":66026,"translate":27},[26],[23,66028,66030,66096],{"className":66029,"translate":27},[31],[23,66031,66033],{"className":66032},[35],[37,66034,66035],{"xmlns":39,"display":40},[42,66036,66037,66093],{},[45,66038,66039,66041,66043,66045,66047,66053,66055,66057,66059,66061,66063,66065,66067,66069,66071,66073,66079,66081,66091],{},[48,66040,4836],{},[51,66042,54],{"stretchy":53},[48,66044,753],{},[51,66046,747],{"separator":122},[4518,66048,66049,66051],{},[48,66050,753],{},[51,66052,38291],{"mathvariant":111,"lspace":1522,"rspace":1522},[51,66054,66],{"stretchy":53},[51,66056,69],{},[51,66058,61181],{"stretchy":53},[48,66060,60886],{},[51,66062,54],{"stretchy":53},[48,66064,753],{},[51,66066,66],{"stretchy":53},[51,66068,747],{"separator":122},[48,66070,60886],{},[51,66072,54],{"stretchy":53},[4518,66074,66075,66077],{},[48,66076,753],{},[51,66078,38291],{"mathvariant":111,"lspace":1522,"rspace":1522},[51,66080,66],{"stretchy":53},[898,66082,66083,66085],{},[51,66084,61206],{"stretchy":53},[898,66086,66087,66089],{},[48,66088,26249],{"mathvariant":375},[48,66090,4836],{},[48,66092,112],{"mathvariant":111},[114,66094,66095],{"encoding":116},"k(x,x')=\\langle \\varphi(x),\\varphi(x')\\rangle_{\\mathcal H_k}.",[23,66097,66099,66164],{"className":66098,"ariaHidden":122},[121],[23,66100,66102,66105,66108,66111,66114,66117,66120,66152,66155,66158,66161],{"className":66101},[126],[23,66103],{"className":66104,"style":51196},[130],[23,66106,4836],{"className":66107,"style":4851},[135,136],[23,66109,54],{"className":66110},[140],[23,66112,753],{"className":66113},[135,136],[23,66115,747],{"className":66116},[824],[23,66118],{"className":66119,"style":280},[148],[23,66121,66123,66126],{"className":66122},[135],[23,66124,753],{"className":66125},[135,136],[23,66127,66129],{"className":66128},[998],[23,66130,66132],{"className":66131},[199],[23,66133,66135],{"className":66134},[204],[23,66136,66138],{"className":66137,"style":51221},[208],[23,66139,66140,66143],{"style":4989},[23,66141],{"className":66142,"style":1015},[216],[23,66144,66146],{"className":66145},[1019,1020,1021,1022],[23,66147,66149],{"className":66148},[135,1022],[23,66150,38291],{"className":66151},[135,1022],[23,66153,66],{"className":66154},[169],[23,66156],{"className":66157,"style":149},[148],[23,66159,69],{"className":66160},[153],[23,66162],{"className":66163,"style":149},[148],[23,66165,66167,66171,66174,66177,66180,66183,66186,66189,66192,66195,66198,66230,66233,66313],{"className":66166},[126],[23,66168],{"className":66169,"style":66170},[130],"height:1.0578em;vertical-align:-0.2559em;",[23,66172,61181],{"className":66173},[140],[23,66175,60886],{"className":66176},[135,136],[23,66178,54],{"className":66179},[140],[23,66181,753],{"className":66182},[135,136],[23,66184,66],{"className":66185},[169],[23,66187,747],{"className":66188},[824],[23,66190],{"className":66191,"style":280},[148],[23,66193,60886],{"className":66194},[135,136],[23,66196,54],{"className":66197},[140],[23,66199,66201,66204],{"className":66200},[135],[23,66202,753],{"className":66203},[135,136],[23,66205,66207],{"className":66206},[998],[23,66208,66210],{"className":66209},[199],[23,66211,66213],{"className":66212},[204],[23,66214,66216],{"className":66215,"style":51221},[208],[23,66217,66218,66221],{"style":4989},[23,66219],{"className":66220,"style":1015},[216],[23,66222,66224],{"className":66223},[1019,1020,1021,1022],[23,66225,66227],{"className":66226},[135,1022],[23,66228,38291],{"className":66229},[135,1022],[23,66231,66],{"className":66232},[169],[23,66234,66236,66239],{"className":66235},[169],[23,66237,61206],{"className":66238},[169],[23,66240,66242],{"className":66241},[998],[23,66243,66245,66305],{"className":66244},[199,200],[23,66246,66248,66302],{"className":66247},[204],[23,66249,66251],{"className":66250,"style":2026},[208],[23,66252,66253,66256],{"style":2029},[23,66254],{"className":66255,"style":1015},[216],[23,66257,66259],{"className":66258},[1019,1020,1021,1022],[23,66260,66262],{"className":66261},[135,1022],[23,66263,66265,66268],{"className":66264},[135,1022],[23,66266,26249],{"className":66267,"style":45024},[135,412,1022],[23,66269,66271],{"className":66270},[998],[23,66272,66274,66294],{"className":66273},[199,200],[23,66275,66277,66291],{"className":66276},[204],[23,66278,66280],{"className":66279,"style":1008},[208],[23,66281,66282,66285],{"style":64106},[23,66283],{"className":66284,"style":24881},[216],[23,66286,66288],{"className":66287},[1019,24885,1067,1022],[23,66289,4836],{"className":66290,"style":4851},[135,136,1022],[23,66292,297],{"className":66293},[296],[23,66295,66297],{"className":66296},[204],[23,66298,66300],{"className":66299,"style":64125},[208],[23,66301],{},[23,66303,297],{"className":66304},[296],[23,66306,66308],{"className":66307},[204],[23,66309,66311],{"className":66310,"style":64137},[208],[23,66312],{},[23,66314,112],{"className":66315},[135],[11,66317,9746,66318,66321,66322,66435,66436,66526,66527,66555],{},[18,66319,66320],{},"核技巧","（kernel trick）的来源：所有只涉及特征内积的算法，都可以把 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替换成 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",[23,66528,66530,66543],{"className":66529,"translate":27},[31],[23,66531,66533],{"className":66532},[35],[37,66534,66535],{"xmlns":39},[42,66536,66537,66541],{},[45,66538,66539],{},[48,66540,60886],{},[114,66542,61100],{"encoding":116},[23,66544,66546],{"className":66545,"ariaHidden":122},[121],[23,66547,66549,66552],{"className":66548},[126],[23,66550],{"className":66551,"style":672},[130],[23,66553,60886],{"className":66554},[135,136]," 的情况下运行。SVM、核岭回归、核 PCA 都是这样。",[11,66557,66558],{},"需要强调的是，核技巧和表示定理是两件事。核技巧说的是「内积可以隐式计算」；表示定理说的是「最优解具有有限维形式」。前者是计算技巧，后者是结构定理。",[2501,66560,66562],{"id":66561},"_23-常见核与它们的-rkhs","2.3 常见核与它们的 RKHS",[3764,66564,66565,66578],{},[3767,66566,66567],{},[3770,66568,66569,66572,66575],{},[3773,66570,66571],{},"核",[3773,66573,66574],{},"形式",[3773,66576,66577],{},"特征维数 \u002F RKHS",[3783,66579,66580,66703,66976,67225,67413],{},[3770,66581,66582,66585,66672],{},[3788,66583,66584],{},"线性核",[3788,66586,66587],{},[23,66588,66590,66616],{"className":66589,"translate":27},[31],[23,66591,66593],{"className":66592},[35],[37,66594,66595],{"xmlns":39},[42,66596,66597,66613],{},[45,66598,66599,66601,66603,66605,66611],{},[51,66600,61181],{"stretchy":53},[48,66602,753],{},[51,66604,747],{"separator":122},[4518,66606,66607,66609],{},[48,66608,753],{},[51,66610,38291],{"mathvariant":111,"lspace":1522,"rspace":1522},[51,66612,61206],{"stretchy":53},[114,66614,66615],{"encoding":116},"\\langle 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维",[3770,66704,66705,66708,66856],{},[3788,66706,66707],{},"多项式核",[3788,66709,66710],{},[23,66711,66713,66751],{"className":66712,"translate":27},[31],[23,66714,66716],{"className":66715},[35],[37,66717,66718],{"xmlns":39},[42,66719,66720,66748],{},[45,66721,66722,66724,66726,66728,66730,66736,66738,66740,66742],{},[51,66723,54],{"stretchy":53},[51,66725,61181],{"stretchy":53},[48,66727,753],{},[51,66729,747],{"separator":122},[4518,66731,66732,66734],{},[48,66733,753],{},[51,66735,38291],{"mathvariant":111,"lspace":1522,"rspace":1522},[51,66737,61206],{"stretchy":53},[51,66739,5106],{},[48,66741,12710],{},[4518,66743,66744,66746],{},[51,66745,66],{"stretchy":53},[48,66747,11],{},[114,66749,66750],{"encoding":116},"(\\langle 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核",[3788,67231,67232],{},[23,67233,67235,67281],{"className":67234,"translate":27},[31],[23,67236,67238],{"className":67237},[35],[37,67239,67240],{"xmlns":39},[42,67241,67242,67278],{},[45,67243,67244,67246,67248,67250,67252,67254,67256,67258,67260,67266,67272,67274,67276],{},[48,67245,25008],{},[51,67247,6959],{},[88,67249,67001],{},[51,67251,54],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[51,67253,4529],{},[48,67255,20680],{"mathvariant":111},[48,67257,753],{},[51,67259,4529],{},[4518,67261,67262,67264],{},[48,67263,753],{},[51,67265,38291],{"mathvariant":111,"lspace":1522,"rspace":1522},[898,67267,67268,67270],{},[48,67269,20680],{"mathvariant":111},[2581,67271,4435],{},[48,67273,2875],{"mathvariant":111},[48,67275,29634],{},[51,67277,66],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[114,67279,67280],{"encoding":116},"\\exp\\!\\bigl(-\\|x-x'\\|_1\u002F\\sigma\\bigr)",[23,67282,67284,67320],{"className":67283,"ariaHidden":122},[121],[23,67285,67287,67290,67293,67296,67302,67305,67308,67311,67314,67317],{"className":67286},[126],[23,67288],{"className":67289,"style":2213},[130],[23,67291,25008],{"className":67292},[1786],[23,67294],{"className":67295,"style":67060},[148],[23,67297,67299],{"className":67298},[140],[23,67300,54],{"className":67301},[1066,1067],[23,67303,4529],{"className":67304},[135],[23,67306,20680],{"className":67307},[135],[23,67309,753],{"className":67310},[135,136],[23,67312],{"className":67313,"style":392},[148],[23,67315,4529],{"className":67316},[612],[23,67318],{"className":67319,"style":392},[148],[23,67321,67323,67326,67358,67398,67401,67404],{"className":67322},[126],[23,67324],{"className":67325,"style":2213},[130],[23,67327,67329,67332],{"className":67328},[135],[23,67330,753],{"className":67331},[135,136],[23,67333,67335],{"className":67334},[998],[23,67336,67338],{"className":67337},[199],[23,67339,67341],{"className":67340},[204],[23,67342,67344],{"className":67343,"style":38333},[208],[23,67345,67346,67349],{"style":4696},[23,67347],{"className":67348,"style":1015},[216],[23,67350,67352],{"className":67351},[1019,1020,1021,1022],[23,67353,67355],{"className":67354},[135,1022],[23,67356,38291],{"className":67357},[135,1022],[23,67359,67361,67364],{"className":67360},[135],[23,67362,20680],{"className":67363},[135],[23,67365,67367],{"className":67366},[998],[23,67368,67370,67390],{"className":67369},[199,200],[23,67371,67373,67387],{"className":67372},[204],[23,67374,67376],{"className":67375,"style":18647},[208],[23,67377,67378,67381],{"style":2029},[23,67379],{"className":67380,"style":1015},[216],[23,67382,67384],{"className":67383},[1019,1020,1021,1022],[23,67385,4435],{"className":67386},[135,1022],[23,67388,297],{"className":67389},[296],[23,67391,67393],{"className":67392},[204],[23,67394,67396],{"className":67395,"style":2048},[208],[23,67397],{},[23,67399,2875],{"className":67400},[135],[23,67402,29634],{"className":67403,"style":676},[135,136],[23,67405,67407],{"className":67406},[169],[23,67408,66],{"className":67409},[1066,1067],[3788,67411,67412],{},"无穷维，Sobolev 型",[3770,67414,67415,67418,67421],{},[3788,67416,67417],{},"Matérn 核",[3788,67419,67420],{},"由修正 Bessel 函数给出",[3788,67422,67423,67424],{},"Sobolev 空间 ",[23,67425,67427,67456],{"className":67426,"translate":27},[31],[23,67428,67430],{"className":67429},[35],[37,67431,67432],{"xmlns":39},[42,67433,67434,67453],{},[45,67435,67436],{},[4518,67437,67438,67440],{},[48,67439,26249],{},[45,67441,67442,67445,67447,67449,67451],{},[48,67443,67444],{},"ν",[51,67446,5106],{},[48,67448,1606],{},[48,67450,2875],{"mathvariant":111},[2581,67452,9952],{},[114,67454,67455],{"encoding":116},"H^{\\nu+d\u002F2}",[23,67457,67459],{"className":67458,"ariaHidden":122},[121],[23,67460,67462,67465],{"className":67461},[126],[23,67463],{"className":67464,"style":35723},[130],[23,67466,67468,67471],{"className":67467},[135],[23,67469,26249],{"className":67470,"style":26503},[135,136],[23,67472,67474],{"className":67473},[998],[23,67475,67477],{"className":67476},[199],[23,67478,67480],{"className":67479},[204],[23,67481,67483],{"className":67482,"style":35723},[208],[23,67484,67485,67488],{"style":4696},[23,67486],{"className":67487,"style":1015},[216],[23,67489,67491],{"className":67490},[1019,1020,1021,1022],[23,67492,67494,67498,67501,67504],{"className":67493},[135,1022],[23,67495,67444],{"className":67496,"style":67497},[135,136,1022],"margin-right:0.0637em;",[23,67499,5106],{"className":67500},[612,1022],[23,67502,1606],{"className":67503},[135,136,1022],[23,67505,20560],{"className":67506},[135,1022],[11,67508,67509,67510,67579,67580,1471],{},"核的选择等价于选择函数空间的先验：高斯核偏好非常光滑的函数，Laplacian 核允许更尖锐的变化，线性核则把模型限制在线性函数里。表示定理本身对核的选择没有偏好；它只负责把优化问题从 ",[23,67511,67513,67530],{"className":67512,"translate":27},[31],[23,67514,67516],{"className":67515},[35],[37,67517,67518],{"xmlns":39},[42,67519,67520,67528],{},[45,67521,67522],{},[898,67523,67524,67526],{},[48,67525,26249],{"mathvariant":375},[48,67527,4836],{},[114,67529,61597],{"encoding":116},[23,67531,67533],{"className":67532,"ariaHidden":122},[121],[23,67534,67536,67539],{"className":67535},[126],[23,67537],{"className":67538,"style":42827},[130],[23,67540,67542,67545],{"className":67541},[135],[23,67543,26249],{"className":67544,"style":45024},[135,412],[23,67546,67548],{"className":67547},[998],[23,67549,67551,67571],{"className":67550},[199,200],[23,67552,67554,67568],{"className":67553},[204],[23,67555,67557],{"className":67556,"style":34192},[208],[23,67558,67559,67562],{"style":61627},[23,67560],{"className":67561,"style":1015},[216],[23,67563,67565],{"className":67564},[1019,1020,1021,1022],[23,67566,4836],{"className":67567,"style":4851},[135,136,1022],[23,67569,297],{"className":67570},[296],[23,67572,67574],{"className":67573},[204],[23,67575,67577],{"className":67576,"style":2048},[208],[23,67578],{}," 降到 ",[23,67581,67583,67601],{"className":67582,"translate":27},[31],[23,67584,67586],{"className":67585},[35],[37,67587,67588],{"xmlns":39},[42,67589,67590,67598],{},[45,67591,67592],{},[4518,67593,67594,67596],{},[48,67595,572],{"mathvariant":571},[48,67597,4806],{},[114,67599,67600],{"encoding":116},"\\mathbb R^n",[23,67602,67604],{"className":67603,"ariaHidden":122},[121],[23,67605,67607,67610],{"className":67606},[126],[23,67608],{"className":67609,"style":640},[130],[23,67611,67613,67616],{"className":67612},[135],[23,67614,572],{"className":67615},[135,644],[23,67617,67619],{"className":67618},[998],[23,67620,67622],{"className":67621},[199],[23,67623,67625],{"className":67624},[204],[23,67626,67628],{"className":67627,"style":66844},[208],[23,67629,67630,67633],{"style":4696},[23,67631],{"className":67632,"style":1015},[216],[23,67634,67636],{"className":67635},[1019,1020,1021,1022],[23,67637,4806],{"className":67638},[135,136,1022],[338,67640,67642],{"id":67641},"三表示定理","三、表示定理",[2501,67644,67646],{"id":67645},"_31-定理","3.1 定理",[6857,67648,67650,68217,68717,68720,68726,69069],{"title":67649},"定理 3.1（表示定理）",[11,67651,4410,67652,62223,67680,67708,67709,67977,67978,68118,68119,68216],{},[23,67653,67655,67668],{"className":67654,"translate":27},[31],[23,67656,67658],{"className":67657},[35],[37,67659,67660],{"xmlns":39},[42,67661,67662,67666],{},[45,67663,67664],{},[48,67665,26249],{"mathvariant":375},[114,67667,45011],{"encoding":116},[23,67669,67671],{"className":67670,"ariaHidden":122},[121],[23,67672,67674,67677],{"className":67673},[126],[23,67675],{"className":67676,"style":470},[130],[23,67678,26249],{"className":67679,"style":45024},[135,412],[23,67681,67683,67696],{"className":67682,"translate":27},[31],[23,67684,67686],{"className":67685},[35],[37,67687,67688],{"xmlns":39},[42,67689,67690,67694],{},[45,67691,67692],{},[48,67693,4836],{},[114,67695,4836],{"encoding":116},[23,67697,67699],{"className":67698,"ariaHidden":122},[121],[23,67700,67702,67705],{"className":67701},[126],[23,67703],{"className":67704,"style":779},[130],[23,67706,4836],{"className":67707,"style":4851},[135,136]," 对应的 RKHS，给定训练数据 ",[23,67710,67712,67766],{"className":67711,"translate":27},[31],[23,67713,67715],{"className":67714},[35],[37,67716,67717],{"xmlns":39},[42,67718,67719,67763],{},[45,67720,67721,67723,67729,67731,67737,67739,67741,67743,67745,67747,67753,67755,67761],{},[51,67722,54],{"stretchy":53},[898,67724,67725,67727],{},[48,67726,753],{},[2581,67728,4435],{},[51,67730,747],{"separator":122},[898,67732,67733,67735],{},[48,67734,660],{},[2581,67736,4435],{},[51,67738,66],{"stretchy":53},[51,67740,747],{"separator":122},[51,67742,62555],{},[51,67744,747],{"separator":122},[51,67746,54],{"stretchy":53},[898,67748,67749,67751],{},[48,67750,753],{},[48,67752,4806],{},[51,67754,747],{"separator":122},[898,67756,67757,67759],{},[48,67758,660],{},[48,67760,4806],{},[51,67762,66],{"stretchy":53},[114,67764,67765],{"encoding":116},"(x_1,y_1),\\dots,(x_n,y_n)",[23,67767,67769],{"className":67768,"ariaHidden":122},[121],[23,67770,67772,67775,67778,67818,67821,67824,67864,67867,67870,67873,67876,67879,67882,67885,67888,67928,67931,67934,67974],{"className":67771},[126],[23,67773],{"className":67774,"style":131},[130],[23,67776,54],{"className":67777},[140],[23,67779,67781,67784],{"className":67780},[135],[23,67782,753],{"className":67783},[135,136],[23,67785,67787],{"className":67786},[998],[23,67788,67790,67810],{"className":67789},[199,200],[23,67791,67793,67807],{"className":67792},[204],[23,67794,67796],{"className":67795,"style":18647},[208],[23,67797,67798,67801],{"style":2029},[23,67799],{"className":67800,"style":1015},[216],[23,67802,67804],{"className":67803},[1019,1020,1021,1022],[23,67805,4435],{"className":67806},[135,1022],[23,67808,297],{"className":67809},[296],[23,67811,67813],{"className":67812},[204],[23,67814,67816],{"className":67815,"style":2048},[208],[23,67817],{},[23,67819,747],{"className":67820},[824],[23,67822],{"className":67823,"style":280},[148],[23,67825,67827,67830],{"className":67826},[135],[23,67828,660],{"className":67829,"style":676},[135,136],[23,67831,67833],{"className":67832},[998],[23,67834,67836,67856],{"className":67835},[199,200],[23,67837,67839,67853],{"className":67838},[204],[23,67840,67842],{"className":67841,"style":18647},[208],[23,67843,67844,67847],{"style":21850},[23,67845],{"className":67846,"style":1015},[216],[23,67848,67850],{"className":67849},[1019,1020,1021,1022],[23,67851,4435],{"className":67852},[135,1022],[23,67854,297],{"className":67855},[296],[23,67857,67859],{"className":67858},[204],[23,67860,67862],{"className":67861,"style":2048},[208],[23,67863],{},[23,67865,66],{"className":67866},[169],[23,67868,747],{"className":67869},[824],[23,67871],{"className":67872,"style":280},[148],[23,67874,62555],{"className":67875},[1839],[23,67877],{"className":67878,"style":280},[148],[23,67880,747],{"className":67881},[824],[23,67883],{"className":67884,"style":280},[148],[23,67886,54],{"className":67887},[140],[23,67889,67891,67894],{"className":67890},[135],[23,67892,753],{"className":67893},[135,136],[23,67895,67897],{"className":67896},[998],[23,67898,67900,67920],{"className":67899},[199,200],[23,67901,67903,67917],{"className":67902},[204],[23,67904,67906],{"className":67905,"style":6499},[208],[23,67907,67908,67911],{"style":2029},[23,67909],{"className":67910,"style":1015},[216],[23,67912,67914],{"className":67913},[1019,1020,1021,1022],[23,67915,4806],{"className":67916},[135,136,1022],[23,67918,297],{"className":67919},[296],[23,67921,67923],{"className":67922},[204],[23,67924,67926],{"className":67925,"style":2048},[208],[23,67927],{},[23,67929,747],{"className":67930},[824],[23,67932],{"className":67933,"style":280},[148],[23,67935,67937,67940],{"className":67936},[135],[23,67938,660],{"className":67939,"style":676},[135,136],[23,67941,67943],{"className":67942},[998],[23,67944,67946,67966],{"className":67945},[199,200],[23,67947,67949,67963],{"className":67948},[204],[23,67950,67952],{"className":67951,"style":6499},[208],[23,67953,67954,67957],{"style":21850},[23,67955],{"className":67956,"style":1015},[216],[23,67958,67960],{"className":67959},[1019,1020,1021,1022],[23,67961,4806],{"className":67962},[135,136,1022],[23,67964,297],{"className":67965},[296],[23,67967,67969],{"className":67968},[204],[23,67970,67972],{"className":67971,"style":2048},[208],[23,67973],{},[23,67975,66],{"className":67976},[169],"。设 ",[23,67979,67981,68017],{"className":67980,"translate":27},[31],[23,67982,67984],{"className":67983},[35],[37,67985,67986],{"xmlns":39},[42,67987,67988,68014],{},[45,67989,67990,67992,67994,68000,68002,68004,68006,68008,68010,68012],{},[48,67991,555],{},[51,67993,558],{},[4518,67995,67996,67998],{},[48,67997,572],{"mathvariant":571},[48,67999,4806],{},[51,68001,568],{},[48,68003,572],{"mathvariant":571},[51,68005,50392],{},[51,68007,9010],{"stretchy":53},[51,68009,5106],{},[48,68011,46447],{"mathvariant":111},[51,68013,9020],{"stretchy":53},[114,68015,68016],{"encoding":116},"L:\\mathbb R^n\\to\\mathbb R\\cup\\{+\\infty\\}",[23,68018,68020,68038,68082,68100],{"className":68019,"ariaHidden":122},[121],[23,68021,68023,68026,68029,68032,68035],{"className":68022},[126],[23,68024],{"className":68025,"style":470},[130],[23,68027,555],{"className":68028},[135,136],[23,68030],{"className":68031,"style":149},[148],[23,68033,558],{"className":68034},[153],[23,68036],{"className":68037,"style":149},[148],[23,68039,68041,68044,68073,68076,68079],{"className":68040},[126],[23,68042],{"className":68043,"style":640},[130],[23,68045,68047,68050],{"className":68046},[135],[23,68048,572],{"className":68049},[135,644],[23,68051,68053],{"className":68052},[998],[23,68054,68056],{"className":68055},[199],[23,68057,68059],{"className":68058},[204],[23,68060,68062],{"className":68061,"style":66844},[208],[23,68063,68064,68067],{"style":4696},[23,68065],{"className":68066,"style":1015},[216],[23,68068,68070],{"className":68069},[1019,1020,1021,1022],[23,68071,4806],{"className":68072},[135,136,1022],[23,68074],{"className":68075,"style":149},[148],[23,68077,568],{"className":68078},[153],[23,68080],{"className":68081,"style":149},[148],[23,68083,68085,68088,68091,68094,68097],{"className":68084},[126],[23,68086],{"className":68087,"style":640},[130],[23,68089,572],{"className":68090},[135,644],[23,68092],{"className":68093,"style":392},[148],[23,68095,50392],{"className":68096},[612],[23,68098],{"className":68099,"style":392},[148],[23,68101,68103,68106,68109,68112,68115],{"className":68102},[126],[23,68104],{"className":68105,"style":131},[130],[23,68107,9010],{"className":68108},[140],[23,68110,5106],{"className":68111},[135],[23,68113,46447],{"className":68114},[135],[23,68116,9020],{"className":68117},[169]," 是任意损失函数，",[23,68120,68122,68153],{"className":68121,"translate":27},[31],[23,68123,68125],{"className":68124},[35],[37,68126,68127],{"xmlns":39},[42,68128,68129,68150],{},[45,68130,68131,68134,68136,68138,68140,68142,68144,68146,68148],{},[48,68132,68133],{"mathvariant":111},"Ω",[51,68135,558],{},[51,68137,924],{"stretchy":53},[2581,68139,1532],{},[51,68141,747],{"separator":122},[48,68143,46447],{"mathvariant":111},[51,68145,66],{"stretchy":53},[51,68147,568],{},[48,68149,572],{"mathvariant":571},[114,68151,68152],{"encoding":116},"\\Omega:[0,\\infty)\\to\\mathbb R",[23,68154,68156,68174,68207],{"className":68155,"ariaHidden":122},[121],[23,68157,68159,68162,68165,68168,68171],{"className":68158},[126],[23,68160],{"className":68161,"style":470},[130],[23,68163,68133],{"className":68164},[135],[23,68166],{"className":68167,"style":149},[148],[23,68169,558],{"className":68170},[153],[23,68172],{"className":68173,"style":149},[148],[23,68175,68177,68180,68183,68186,68189,68192,68195,68198,68201,68204],{"className":68176},[126],[23,68178],{"className":68179,"style":131},[130],[23,68181,924],{"className":68182},[140],[23,68184,1532],{"className":68185},[135],[23,68187,747],{"className":68188},[824],[23,68190],{"className":68191,"style":280},[148],[23,68193,46447],{"className":68194},[135],[23,68196,66],{"className":68197},[169],[23,68199],{"className":68200,"style":149},[148],[23,68202,568],{"className":68203},[153],[23,68205],{"className":68206,"style":149},[148],[23,68208,68210,68213],{"className":68209},[126],[23,68211],{"className":68212,"style":640},[130],[23,68214,572],{"className":68215},[135,644]," 是非减函数。考虑",[23,68218,68220],{"className":68219,"translate":27},[26],[23,68221,68223,68342],{"className":68222,"translate":27},[31],[23,68224,68226],{"className":68225},[35],[37,68227,68228],{"xmlns":39,"display":40},[42,68229,68230,68339],{},[1518,68231,68232],{"width":61666},[1524,68233,68234,68236,68332,68334],{},[1527,68235],{"width":61671},[1527,68237,68238],{},[45,68239,68240,68246,68248,68250,68252,68268,68330],{},[4518,68241,68242,68244],{},[48,68243,8336],{},[51,68245,6881],{},[51,68247,372],{},[48,68249,6956],{},[51,68251,6959],{},[6961,68253,68254,68260],{},[45,68255,68256,68258],{},[48,68257,6967],{},[51,68259,6959],{},[45,68261,68262,68264,68266],{},[48,68263,8336],{},[51,68265,372],{},[48,68267,26249],{"mathvariant":375},[45,68269,68270,68272,68274,68276,68278,68280,68286,68288,68290,68292,68294,68296,68298,68304,68306,68308,68310,68312,68314,68316,68318,68326,68328],{},[51,68271,9010],{"fence":122},[48,68273,555],{},[51,68275,54],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[48,68277,8336],{},[51,68279,54],{"stretchy":53},[898,68281,68282,68284],{},[48,68283,753],{},[2581,68285,4435],{},[51,68287,66],{"stretchy":53},[51,68289,747],{"separator":122},[51,68291,62555],{},[51,68293,747],{"separator":122},[48,68295,8336],{},[51,68297,54],{"stretchy":53},[898,68299,68300,68302],{},[48,68301,753],{},[48,68303,4806],{},[51,68305,66],{"stretchy":53},[51,68307,66],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[51,68309,5106],{},[48,68311,68133],{"mathvariant":111},[51,68313,54],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[48,68315,20680],{"mathvariant":111},[48,68317,8336],{},[19137,68319,68320,68322,68324],{},[48,68321,20680],{"mathvariant":111},[48,68323,26249],{"mathvariant":375},[2581,68325,9952],{},[51,68327,66],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[51,68329,9020],{"fence":122},[48,68331,112],{"mathvariant":111},[1527,68333],{"width":61671},[1527,68335,68336],{},[88,68337,68338],{},"(2)",[114,68340,68341],{"encoding":116},"f^*\\in\\arg\\min_{f\\in\\mathcal H}\\left\\{L\\bigl(f(x_1),\\dots,f(x_n)\\bigr)+\\Omega\\bigl(\\|f\\|_{\\mathcal H}^2\\bigr)\\right\\}.\n\\tag{2}",[23,68343,68345,68389,68696],{"className":68344,"ariaHidden":122},[121],[23,68346,68348,68351,68380,68383,68386],{"className":68347},[126],[23,68349],{"className":68350,"style":48051},[130],[23,68352,68354,68357],{"className":68353},[135],[23,68355,8336],{"className":68356,"style":11398},[135,136],[23,68358,68360],{"className":68359},[998],[23,68361,68363],{"className":68362},[199],[23,68364,68366],{"className":68365},[204],[23,68367,68369],{"className":68368,"style":7045},[208],[23,68370,68371,68374],{"style":4989},[23,68372],{"className":68373,"style":1015},[216],[23,68375,68377],{"className":68376},[1019,1020,1021,1022],[23,68378,6881],{"className":68379},[612,1022],[23,68381],{"className":68382,"style":149},[148],[23,68384,372],{"className":68385},[153],[23,68387],{"className":68388,"style":149},[148],[23,68390,68392,68396,68401,68404,68457,68460,68690,68693],{"className":68391},[126],[23,68393],{"className":68394,"style":68395},[130],"height:1.7523em;vertical-align:-0.8882em;",[23,68397,7085,68399],{"className":68398},[1786],[23,68400,7089],{"style":7088},[23,68402],{"className":68403,"style":280},[148],[23,68405,68407],{"className":68406},[1786,7096],[23,68408,68410,68449],{"className":68409},[199,200],[23,68411,68413,68446],{"className":68412},[204],[23,68414,68416,68436],{"className":68415,"style":7106},[208],[23,68417,68418,68421],{"style":14738},[23,68419],{"className":68420,"style":217},[216],[23,68422,68424],{"className":68423},[1019,1020,1021,1022],[23,68425,68427,68430,68433],{"className":68426},[135,1022],[23,68428,8336],{"className":68429,"style":11398},[135,136,1022],[23,68431,372],{"className":68432},[153,1022],[23,68434,26249],{"className":68435,"style":45024},[135,412,1022],[23,68437,68438,68441],{"style":7130},[23,68439],{"className":68440,"style":217},[216],[23,68442,68443],{},[23,68444,6967],{"className":68445},[1786],[23,68447,297],{"className":68448},[296],[23,68450,68452],{"className":68451},[204],[23,68453,68455],{"className":68454,"style":61893},[208],[23,68456],{},[23,68458],{"className":68459,"style":280},[148],[23,68461,68463,68469,68472,68478,68481,68484,68524,68527,68530,68533,68536,68539,68542,68545,68548,68551,68591,68594,68600,68603,68606,68609,68612,68618,68621,68624,68678,68684],{"className":68462},[1839],[23,68464,68466],{"className":68465,"style":1844},[140,1843],[23,68467,9010],{"className":68468},[1066,1067],[23,68470,555],{"className":68471},[135,136],[23,68473,68475],{"className":68474},[140],[23,68476,54],{"className":68477},[1066,1067],[23,68479,8336],{"className":68480,"style":11398},[135,136],[23,68482,54],{"className":68483},[140],[23,68485,68487,68490],{"className":68486},[135],[23,68488,753],{"className":68489},[135,136],[23,68491,68493],{"className":68492},[998],[23,68494,68496,68516],{"className":68495},[199,200],[23,68497,68499,68513],{"className":68498},[204],[23,68500,68502],{"className":68501,"style":18647},[208],[23,68503,68504,68507],{"style":2029},[23,68505],{"className":68506,"style":1015},[216],[23,68508,68510],{"className":68509},[1019,1020,1021,1022],[23,68511,4435],{"className":68512},[135,1022],[23,68514,297],{"className":68515},[296],[23,68517,68519],{"className":68518},[204],[23,68520,68522],{"className":68521,"style":2048},[208],[23,68523],{},[23,68525,66],{"className":68526},[169],[23,68528,747],{"className":68529},[824],[23,68531],{"className":68532,"style":280},[148],[23,68534,62555],{"className":68535},[1839],[23,68537],{"className":68538,"style":280},[148],[23,68540,747],{"className":68541},[824],[23,68543],{"className":68544,"style":280},[148],[23,68546,8336],{"className":68547,"style":11398},[135,136],[23,68549,54],{"className":68550},[140],[23,68552,68554,68557],{"className":68553},[135],[23,68555,753],{"className":68556},[135,136],[23,68558,68560],{"className":68559},[998],[23,68561,68563,68583],{"className":68562},[199,200],[23,68564,68566,68580],{"className":68565},[204],[23,68567,68569],{"className":68568,"style":6499},[208],[23,68570,68571,68574],{"style":2029},[23,68572],{"className":68573,"style":1015},[216],[23,68575,68577],{"className":68576},[1019,1020,1021,1022],[23,68578,4806],{"className":68579},[135,136,1022],[23,68581,297],{"className":68582},[296],[23,68584,68586],{"className":68585},[204],[23,68587,68589],{"className":68588,"style":2048},[208],[23,68590],{},[23,68592,66],{"className":68593},[169],[23,68595,68597],{"className":68596},[169],[23,68598,66],{"className":68599},[1066,1067],[23,68601],{"className":68602,"style":392},[148],[23,68604,5106],{"className":68605},[612],[23,68607],{"className":68608,"style":392},[148],[23,68610,68133],{"className":68611},[135],[23,68613,68615],{"className":68614},[140],[23,68616,54],{"className":68617},[1066,1067],[23,68619,20680],{"className":68620},[135],[23,68622,8336],{"className":68623,"style":11398},[135,136],[23,68625,68627,68630],{"className":68626},[135],[23,68628,20680],{"className":68629},[135],[23,68631,68633],{"className":68632},[998],[23,68634,68636,68670],{"className":68635},[199,200],[23,68637,68639,68667],{"className":68638},[204],[23,68640,68642,68656],{"className":68641,"style":11002},[208],[23,68643,68644,68647],{"style":19345},[23,68645],{"className":68646,"style":1015},[216],[23,68648,68650],{"className":68649},[1019,1020,1021,1022],[23,68651,68653],{"className":68652},[135,1022],[23,68654,26249],{"className":68655,"style":45024},[135,412,1022],[23,68657,68658,68661],{"style":4989},[23,68659],{"className":68660,"style":1015},[216],[23,68662,68664],{"className":68663},[1019,1020,1021,1022],[23,68665,9952],{"className":68666},[135,1022],[23,68668,297],{"className":68669},[296],[23,68671,68673],{"className":68672},[204],[23,68674,68676],{"className":68675,"style":19375},[208],[23,68677],{},[23,68679,68681],{"className":68680},[169],[23,68682,66],{"className":68683},[1066,1067],[23,68685,68687],{"className":68686,"style":1844},[169,1843],[23,68688,9020],{"className":68689},[1066,1067],[23,68691],{"className":68692,"style":280},[148],[23,68694,112],{"className":68695},[135],[23,68697,68699,68702],{"className":68698},[62174],[23,68700],{"className":68701,"style":68395},[130],[23,68703,68705,68708,68714],{"className":68704},[135,6514],[23,68706,54],{"className":68707},[135],[23,68709,68711],{"className":68710},[135],[23,68712,9952],{"className":68713},[135],[23,68715,66],{"className":68716},[135],[11,68718,68719],{},"则：",[11,68721,68722,68725],{},[18,68723,68724],{},"(i)"," 存在一个最优解具有形式",[23,68727,68729],{"className":68728,"translate":27},[26],[23,68730,68732,68813],{"className":68731,"translate":27},[31],[23,68733,68735],{"className":68734},[35],[37,68736,68737],{"xmlns":39,"display":40},[42,68738,68739,68810],{},[1518,68740,68741],{"width":61666},[1524,68742,68743,68745,68803,68805],{},[1527,68744],{"width":61671},[1527,68746,68747],{},[45,68748,68749,68755,68757,68759,68761,68763,68777,68783,68785,68787,68789,68791,68793,68799,68801],{},[4518,68750,68751,68753],{},[48,68752,8336],{},[51,68754,6881],{},[51,68756,54],{"stretchy":53},[48,68758,753],{},[51,68760,66],{"stretchy":53},[51,68762,69],{},[61709,68764,68765,68767,68775],{},[51,68766,33851],{},[45,68768,68769,68771,68773],{},[48,68770,33493],{},[51,68772,69],{},[2581,68774,4435],{},[48,68776,4806],{},[898,68778,68779,68781],{},[48,68780,4526],{},[48,68782,33493],{},[88,68784,90],{},[48,68786,4836],{},[51,68788,54],{"stretchy":53},[48,68790,753],{},[51,68792,747],{"separator":122},[898,68794,68795,68797],{},[48,68796,753],{},[48,68798,33493],{},[51,68800,66],{"stretchy":53},[51,68802,39744],{"separator":122},[1527,68804],{"width":61671},[1527,68806,68807],{},[88,68808,68809],{},"(3)",[114,68811,68812],{"encoding":116},"f^*(x)=\\sum_{i=1}^n\\alpha_i\\,k(x,x_i);\n\\tag{3}",[23,68814,68816,68869,69048],{"className":68815,"ariaHidden":122},[121],[23,68817,68819,68822,68851,68854,68857,68860,68863,68866],{"className":68818},[126],[23,68820],{"className":68821,"style":131},[130],[23,68823,68825,68828],{"className":68824},[135],[23,68826,8336],{"className":68827,"style":11398},[135,136],[23,68829,68831],{"className":68830},[998],[23,68832,68834],{"className":68833},[199],[23,68835,68837],{"className":68836},[204],[23,68838,68840],{"className":68839,"style":7045},[208],[23,68841,68842,68845],{"style":4989},[23,68843],{"className":68844,"style":1015},[216],[23,68846,68848],{"className":68847},[1019,1020,1021,1022],[23,68849,6881],{"className":68850},[612,1022],[23,68852,54],{"className":68853},[140],[23,68855,753],{"className":68856},[135,136],[23,68858,66],{"className":68859},[169],[23,68861],{"className":68862,"style":149},[148],[23,68864,69],{"className":68865},[153],[23,68867],{"className":68868,"style":149},[148],[23,68870,68872,68876,68940,68943,68984,68987,68990,68993,68996,68999,69002,69042,69045],{"className":68871},[126],[23,68873],{"className":68874,"style":68875},[130],"height:2.9291em;vertical-align:-1.2777em;",[23,68877,68879],{"className":68878},[1786,7096],[23,68880,68882,68932],{"className":68881},[199,200],[23,68883,68885,68929],{"className":68884},[204],[23,68886,68888,68908,68918],{"className":68887,"style":61920},[208],[23,68889,68890,68893],{"style":34075},[23,68891],{"className":68892,"style":34079},[216],[23,68894,68896],{"className":68895},[1019,1020,1021,1022],[23,68897,68899,68902,68905],{"className":68898},[135,1022],[23,68900,33493],{"className":68901},[135,136,1022],[23,68903,69],{"className":68904},[153,1022],[23,68906,4435],{"className":68907},[135,1022],[23,68909,68910,68913],{"style":34088},[23,68911],{"className":68912,"style":34079},[216],[23,68914,68915],{},[23,68916,33851],{"className":68917},[1786,1790,1791],[23,68919,68920,68923],{"style":61953},[23,68921],{"className":68922,"style":34079},[216],[23,68924,68926],{"className":68925},[1019,1020,1021,1022],[23,68927,4806],{"className":68928},[135,136,1022],[23,68930,297],{"className":68931},[296],[23,68933,68935],{"className":68934},[204],[23,68936,68938],{"className":68937,"style":34106},[208],[23,68939],{},[23,68941],{"className":68942,"style":280},[148],[23,68944,68946,68949],{"className":68945},[135],[23,68947,4526],{"className":68948,"style":4645},[135,136],[23,68950,68952],{"className":68951},[998],[23,68953,68955,68976],{"className":68954},[199,200],[23,68956,68958,68973],{"className":68957},[204],[23,68959,68961],{"className":68960,"style":33647},[208],[23,68962,68964,68967],{"style":68963},"top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;",[23,68965],{"className":68966,"style":1015},[216],[23,68968,68970],{"className":68969},[1019,1020,1021,1022],[23,68971,33493],{"className":68972},[135,136,1022],[23,68974,297],{"className":68975},[296],[23,68977,68979],{"className":68978},[204],[23,68980,68982],{"className":68981,"style":2048},[208],[23,68983],{},[23,68985],{"className":68986,"style":280},[148],[23,68988,4836],{"className":68989,"style":4851},[135,136],[23,68991,54],{"className":68992},[140],[23,68994,753],{"className":68995},[135,136],[23,68997,747],{"className":68998},[824],[23,69000],{"className":69001,"style":280},[148],[23,69003,69005,69008],{"className":69004},[135],[23,69006,753],{"className":69007},[135,136],[23,69009,69011],{"className":69010},[998],[23,69012,69014,69034],{"className":69013},[199,200],[23,69015,69017,69031],{"className":69016},[204],[23,69018,69020],{"className":69019,"style":33647},[208],[23,69021,69022,69025],{"style":2029},[23,69023],{"className":69024,"style":1015},[216],[23,69026,69028],{"className":69027},[1019,1020,1021,1022],[23,69029,33493],{"className":69030},[135,136,1022],[23,69032,297],{"className":69033},[296],[23,69035,69037],{"className":69036},[204],[23,69038,69040],{"className":69039,"style":2048},[208],[23,69041],{},[23,69043,66],{"className":69044},[169],[23,69046,39744],{"className":69047},[824],[23,69049,69051,69054],{"className":69050},[62174],[23,69052],{"className":69053,"style":68875},[130],[23,69055,69057,69060,69066],{"className":69056},[135,6514],[23,69058,54],{"className":69059},[135],[23,69061,69063],{"className":69062},[135],[23,69064,24919],{"className":69065},[135],[23,69067,66],{"className":69068},[135],[11,69070,69071,69074,69075,69104,69105,69108],{},[18,69072,69073],{},"(ii)"," 若 ",[23,69076,69078,69092],{"className":69077,"translate":27},[31],[23,69079,69081],{"className":69080},[35],[37,69082,69083],{"xmlns":39},[42,69084,69085,69089],{},[45,69086,69087],{},[48,69088,68133],{"mathvariant":111},[114,69090,69091],{"encoding":116},"\\Omega",[23,69093,69095],{"className":69094,"ariaHidden":122},[121],[23,69096,69098,69101],{"className":69097},[126],[23,69099],{"className":69100,"style":470},[130],[23,69102,68133],{"className":69103},[135]," 严格递增，则",[18,69106,69107],{},"每个","最优解都具有这个形式。",[11,69110,69111],{},"这里有三个细节值得强调。",[11,69113,69114,69115,69143,69144,69314,69315,69343,69344,69372,69373,69443,69444,69472],{},"第一，",[23,69116,69118,69131],{"className":69117,"translate":27},[31],[23,69119,69121],{"className":69120},[35],[37,69122,69123],{"xmlns":39},[42,69124,69125,69129],{},[45,69126,69127],{},[48,69128,555],{},[114,69130,555],{"encoding":116},[23,69132,69134],{"className":69133,"ariaHidden":122},[121],[23,69135,69137,69140],{"className":69136},[126],[23,69138],{"className":69139,"style":470},[130],[23,69141,555],{"className":69142},[135,136]," 只通过 ",[23,69145,69147,69189],{"className":69146,"translate":27},[31],[23,69148,69150],{"className":69149},[35],[37,69151,69152],{"xmlns":39},[42,69153,69154,69186],{},[45,69155,69156,69158,69160,69166,69168,69170,69172,69174,69176,69178,69184],{},[48,69157,8336],{},[51,69159,54],{"stretchy":53},[898,69161,69162,69164],{},[48,69163,753],{},[2581,69165,4435],{},[51,69167,66],{"stretchy":53},[51,69169,747],{"separator":122},[51,69171,62555],{},[51,69173,747],{"separator":122},[48,69175,8336],{},[51,69177,54],{"stretchy":53},[898,69179,69180,69182],{},[48,69181,753],{},[48,69183,4806],{},[51,69185,66],{"stretchy":53},[114,69187,69188],{"encoding":116},"f(x_1),\\dots,f(x_n)",[23,69190,69192],{"className":69191,"ariaHidden":122},[121],[23,69193,69195,69198,69201,69204,69244,69247,69250,69253,69256,69259,69262,69265,69268,69271,69311],{"className":69194},[126],[23,69196],{"className":69197,"style":131},[130],[23,69199,8336],{"className":69200,"style":11398},[135,136],[23,69202,54],{"className":69203},[140],[23,69205,69207,69210],{"className":69206},[135],[23,69208,753],{"className":69209},[135,136],[23,69211,69213],{"className":69212},[998],[23,69214,69216,69236],{"className":69215},[199,200],[23,69217,69219,69233],{"className":69218},[204],[23,69220,69222],{"className":69221,"style":18647},[208],[23,69223,69224,69227],{"style":2029},[23,69225],{"className":69226,"style":1015},[216],[23,69228,69230],{"className":69229},[1019,1020,1021,1022],[23,69231,4435],{"className":69232},[135,1022],[23,69234,297],{"className":69235},[296],[23,69237,69239],{"className":69238},[204],[23,69240,69242],{"className":69241,"style":2048},[208],[23,69243],{},[23,69245,66],{"className":69246},[169],[23,69248,747],{"className":69249},[824],[23,69251],{"className":69252,"style":280},[148],[23,69254,62555],{"className":69255},[1839],[23,69257],{"className":69258,"style":280},[148],[23,69260,747],{"className":69261},[824],[23,69263],{"className":69264,"style":280},[148],[23,69266,8336],{"className":69267,"style":11398},[135,136],[23,69269,54],{"className":69270},[140],[23,69272,69274,69277],{"className":69273},[135],[23,69275,753],{"className":69276},[135,136],[23,69278,69280],{"className":69279},[998],[23,69281,69283,69303],{"className":69282},[199,200],[23,69284,69286,69300],{"className":69285},[204],[23,69287,69289],{"className":69288,"style":6499},[208],[23,69290,69291,69294],{"style":2029},[23,69292],{"className":69293,"style":1015},[216],[23,69295,69297],{"className":69296},[1019,1020,1021,1022],[23,69298,4806],{"className":69299},[135,136,1022],[23,69301,297],{"className":69302},[296],[23,69304,69306],{"className":69305},[204],[23,69307,69309],{"className":69308,"style":2048},[208],[23,69310],{},[23,69312,66],{"className":69313},[169]," 这 ",[23,69316,69318,69331],{"className":69317,"translate":27},[31],[23,69319,69321],{"className":69320},[35],[37,69322,69323],{"xmlns":39},[42,69324,69325,69329],{},[45,69326,69327],{},[48,69328,4806],{},[114,69330,4806],{"encoding":116},[23,69332,69334],{"className":69333,"ariaHidden":122},[121],[23,69335,69337,69340],{"className":69336},[126],[23,69338],{"className":69339,"style":703},[130],[23,69341,4806],{"className":69342},[135,136]," 个值依赖 ",[23,69345,69347,69360],{"className":69346,"translate":27},[31],[23,69348,69350],{"className":69349},[35],[37,69351,69352],{"xmlns":39},[42,69353,69354,69358],{},[45,69355,69356],{},[48,69357,8336],{},[114,69359,8336],{"encoding":116},[23,69361,69363],{"className":69362,"ariaHidden":122},[121],[23,69364,69366,69369],{"className":69365},[126],[23,69367],{"className":69368,"style":5414},[130],[23,69370,8336],{"className":69371,"style":11398},[135,136],"（标签 ",[23,69374,69376,69394],{"className":69375,"translate":27},[31],[23,69377,69379],{"className":69378},[35],[37,69380,69381],{"xmlns":39},[42,69382,69383,69391],{},[45,69384,69385],{},[898,69386,69387,69389],{},[48,69388,660],{},[48,69390,33493],{},[114,69392,69393],{"encoding":116},"y_i",[23,69395,69397],{"className":69396,"ariaHidden":122},[121],[23,69398,69400,69403],{"className":69399},[126],[23,69401],{"className":69402,"style":672},[130],[23,69404,69406,69409],{"className":69405},[135],[23,69407,660],{"className":69408,"style":676},[135,136],[23,69410,69412],{"className":69411},[998],[23,69413,69415,69435],{"className":69414},[199,200],[23,69416,69418,69432],{"className":69417},[204],[23,69419,69421],{"className":69420,"style":33647},[208],[23,69422,69423,69426],{"style":21850},[23,69424],{"className":69425,"style":1015},[216],[23,69427,69429],{"className":69428},[1019,1020,1021,1022],[23,69430,33493],{"className":69431},[135,136,1022],[23,69433,297],{"className":69434},[296],[23,69436,69438],{"className":69437},[204],[23,69439,69441],{"className":69440,"style":2048},[208],[23,69442],{}," 已吸收进 ",[23,69445,69447,69460],{"className":69446,"translate":27},[31],[23,69448,69450],{"className":69449},[35],[37,69451,69452],{"xmlns":39},[42,69453,69454,69458],{},[45,69455,69456],{},[48,69457,555],{},[114,69459,555],{"encoding":116},[23,69461,69463],{"className":69462,"ariaHidden":122},[121],[23,69464,69466,69469],{"className":69465},[126],[23,69467],{"className":69468,"style":470},[130],[23,69470,555],{"className":69471},[135,136],"）。如果损失还依赖导数、积分或在其他点上的取值，定理的形式会改变。表示定理的标准版本针对的正是这种「点值型」经验风险。",[11,69474,69475,69476,69504,69505,69579,69580,69651,69652,69703],{},"第二，",[23,69477,69479,69492],{"className":69478,"translate":27},[31],[23,69480,69482],{"className":69481},[35],[37,69483,69484],{"xmlns":39},[42,69485,69486,69490],{},[45,69487,69488],{},[48,69489,68133],{"mathvariant":111},[114,69491,69091],{"encoding":116},[23,69493,69495],{"className":69494,"ariaHidden":122},[121],[23,69496,69498,69501],{"className":69497},[126],[23,69499],{"className":69500,"style":470},[130],[23,69502,68133],{"className":69503},[135]," 只要求非减，不要求凸、不要求可微。式 (1) 中的 ",[23,69506,69508,69532],{"className":69507,"translate":27},[31],[23,69509,69511],{"className":69510},[35],[37,69512,69513],{"xmlns":39},[42,69514,69515,69529],{},[45,69516,69517,69519,69521,69523],{},[48,69518,61753],{},[48,69520,20680],{"mathvariant":111},[48,69522,8336],{},[4518,69524,69525,69527],{},[48,69526,20680],{"mathvariant":111},[2581,69528,9952],{},[114,69530,69531],{"encoding":116},"\\lambda\\|f\\|^2",[23,69533,69535],{"className":69534,"ariaHidden":122},[121],[23,69536,69538,69541,69544,69547,69550],{"className":69537},[126],[23,69539],{"className":69540,"style":10661},[130],[23,69542,61753],{"className":69543},[135,136],[23,69545,20680],{"className":69546},[135],[23,69548,8336],{"className":69549,"style":11398},[135,136],[23,69551,69553,69556],{"className":69552},[135],[23,69554,20680],{"className":69555},[135],[23,69557,69559],{"className":69558},[998],[23,69560,69562],{"className":69561},[199],[23,69563,69565],{"className":69564},[204],[23,69566,69568],{"className":69567,"style":4693},[208],[23,69569,69570,69573],{"style":4696},[23,69571],{"className":69572,"style":1015},[216],[23,69574,69576],{"className":69575},[1019,1020,1021,1022],[23,69577,9952],{"className":69578},[135,1022]," 对应 ",[23,69581,69583,69609],{"className":69582,"translate":27},[31],[23,69584,69586],{"className":69585},[35],[37,69587,69588],{"xmlns":39},[42,69589,69590,69606],{},[45,69591,69592,69594,69596,69598,69600,69602,69604],{},[48,69593,68133],{"mathvariant":111},[51,69595,54],{"stretchy":53},[48,69597,5345],{},[51,69599,66],{"stretchy":53},[51,69601,69],{},[48,69603,61753],{},[48,69605,5345],{},[114,69607,69608],{"encoding":116},"\\Omega(t)=\\lambda t",[23,69610,69612,69639],{"className":69611,"ariaHidden":122},[121],[23,69613,69615,69618,69621,69624,69627,69630,69633,69636],{"className":69614},[126],[23,69616],{"className":69617,"style":131},[130],[23,69619,68133],{"className":69620},[135],[23,69622,54],{"className":69623},[140],[23,69625,5345],{"className":69626},[135,136],[23,69628,66],{"className":69629},[169],[23,69631],{"className":69632,"style":149},[148],[23,69634,69],{"className":69635},[153],[23,69637],{"className":69638,"style":149},[148],[23,69640,69642,69645,69648],{"className":69641},[126],[23,69643],{"className":69644,"style":779},[130],[23,69646,61753],{"className":69647},[135,136],[23,69649,5345],{"className":69650},[135,136],"（",[23,69653,69655,69673],{"className":69654,"translate":27},[31],[23,69656,69658],{"className":69657},[35],[37,69659,69660],{"xmlns":39},[42,69661,69662,69670],{},[45,69663,69664,69666,69668],{},[48,69665,61753],{},[51,69667,2579],{},[2581,69669,1532],{},[114,69671,69672],{"encoding":116},"\\lambda>0",[23,69674,69676,69694],{"className":69675,"ariaHidden":122},[121],[23,69677,69679,69682,69685,69688,69691],{"className":69678},[126],[23,69680],{"className":69681,"style":4450},[130],[23,69683,61753],{"className":69684},[135,136],[23,69686],{"className":69687,"style":149},[148],[23,69689,2579],{"className":69690},[153],[23,69692],{"className":69693,"style":149},[148],[23,69695,69697,69700],{"className":69696},[126],[23,69698],{"className":69699,"style":2770},[130],[23,69701,1532],{"className":69702},[135]," 时严格递增），是最常见的特例。",[11,69705,69706,69707,69710,69711,69739],{},"第三，第 1 条是",[18,69708,69709],{},"存在性","，第 2 条才是「任意最优解」。如果 ",[23,69712,69714,69727],{"className":69713,"translate":27},[31],[23,69715,69717],{"className":69716},[35],[37,69718,69719],{"xmlns":39},[42,69720,69721,69725],{},[45,69722,69723],{},[48,69724,68133],{"mathvariant":111},[114,69726,69091],{"encoding":116},[23,69728,69730],{"className":69729,"ariaHidden":122},[121],[23,69731,69733,69736],{"className":69732},[126],[23,69734],{"className":69735,"style":470},[130],[23,69737,68133],{"className":69738},[135]," 只是非减而非严格递增，可能存在落在子空间之外的最优解；我们会在第四节看到这一点。",[2501,69741,69743],{"id":69742},"_32-证明正交分解","3.2 证明：正交分解",[11,69745,69746],{},"证明的核心是一个正交分解。令",[23,69748,69750],{"className":69749,"translate":27},[26],[23,69751,69753,69826],{"className":69752,"translate":27},[31],[23,69754,69756],{"className":69755},[35],[37,69757,69758],{"xmlns":39,"display":40},[42,69759,69760,69823],{},[45,69761,69762,69768,69770,69772,69774,69776,69778,69780,69782,69784,69790,69792,69794,69796,69798,69800,69802,69804,69806,69812,69814,69816,69819,69821],{},[898,69763,69764,69766],{},[48,69765,26249],{"mathvariant":375},[2581,69767,1532],{},[51,69769,69],{},[48,69771,23],{"mathvariant":111},[51,69773,6959],{},[51,69775,9010],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[48,69777,4836],{},[51,69779,54],{"stretchy":53},[51,69781,5758],{},[51,69783,747],{"separator":122},[898,69785,69786,69788],{},[48,69787,753],{},[2581,69789,4435],{},[51,69791,66],{"stretchy":53},[51,69793,747],{"separator":122},[51,69795,62555],{},[51,69797,747],{"separator":122},[48,69799,4836],{},[51,69801,54],{"stretchy":53},[51,69803,5758],{},[51,69805,747],{"separator":122},[898,69807,69808,69810],{},[48,69809,753],{},[48,69811,4806],{},[51,69813,66],{"stretchy":53},[51,69815,9020],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[51,69817,69818],{},"⊆",[48,69820,26249],{"mathvariant":375},[48,69822,112],{"mathvariant":111},[114,69824,69825],{"encoding":116},"\\mathcal H_0=\\operatorname{span}\\bigl\\{k(\\cdot,x_1),\\dots,k(\\cdot,x_n)\\bigr\\}\\subseteq\\mathcal H.",[23,69827,69829,69884,70051],{"className":69828,"ariaHidden":122},[121],[23,69830,69832,69835,69875,69878,69881],{"className":69831},[126],[23,69833],{"className":69834,"style":42827},[130],[23,69836,69838,69841],{"className":69837},[135],[23,69839,26249],{"className":69840,"style":45024},[135,412],[23,69842,69844],{"className":69843},[998],[23,69845,69847,69867],{"className":69846},[199,200],[23,69848,69850,69864],{"className":69849},[204],[23,69851,69853],{"className":69852,"style":18647},[208],[23,69854,69855,69858],{"style":61627},[23,69856],{"className":69857,"style":1015},[216],[23,69859,69861],{"className":69860},[1019,1020,1021,1022],[23,69862,1532],{"className":69863},[135,1022],[23,69865,297],{"className":69866},[296],[23,69868,69870],{"className":69869},[204],[23,69871,69873],{"className":69872,"style":2048},[208],[23,69874],{},[23,69876],{"className":69877,"style":149},[148],[23,69879,69],{"className":69880},[153],[23,69882],{"className":69883,"style":149},[148],[23,69885,69887,69890,69896,69902,69905,69908,69911,69914,69917,69957,69960,69963,69966,69969,69972,69975,69978,69981,69984,69987,69990,69993,70033,70036,70042,70045,70048],{"className":69886},[126],[23,69888],{"className":69889,"style":2213},[130],[23,69891,69893],{"className":69892},[1786],[23,69894,23],{"className":69895},[135,5391],[23,69897,69899],{"className":69898},[140],[23,69900,9010],{"className":69901},[1066,1067],[23,69903,4836],{"className":69904,"style":4851},[135,136],[23,69906,54],{"className":69907},[140],[23,69909,5758],{"className":69910},[135],[23,69912,747],{"className":69913},[824],[23,69915],{"className":69916,"style":280},[148],[23,69918,69920,69923],{"className":69919},[135],[23,69921,753],{"className":69922},[135,136],[23,69924,69926],{"className":69925},[998],[23,69927,69929,69949],{"className":69928},[199,200],[23,69930,69932,69946],{"className":69931},[204],[23,69933,69935],{"className":69934,"style":18647},[208],[23,69936,69937,69940],{"style":2029},[23,69938],{"className":69939,"style":1015},[216],[23,69941,69943],{"className":69942},[1019,1020,1021,1022],[23,69944,4435],{"className":69945},[135,1022],[23,69947,297],{"className":69948},[296],[23,69950,69952],{"className":69951},[204],[23,69953,69955],{"className":69954,"style":2048},[208],[23,69956],{},[23,69958,66],{"className":69959},[169],[23,69961,747],{"className":69962},[824],[23,69964],{"className":69965,"style":280},[148],[23,69967,62555],{"className":69968},[1839],[23,69970],{"className":69971,"style":280},[148],[23,69973,747],{"className":69974},[824],[23,69976],{"className":69977,"style":280},[148],[23,69979,4836],{"className":69980,"style":4851},[135,136],[23,69982,54],{"className":69983},[140],[23,69985,5758],{"className":69986},[135],[23,69988,747],{"className":69989},[824],[23,69991],{"className":69992,"style":280},[148],[23,69994,69996,69999],{"className":69995},[135],[23,69997,753],{"className":69998},[135,136],[23,70000,70002],{"className":70001},[998],[23,70003,70005,70025],{"className":70004},[199,200],[23,70006,70008,70022],{"className":70007},[204],[23,70009,70011],{"className":70010,"style":6499},[208],[23,70012,70013,70016],{"style":2029},[23,70014],{"className":70015,"style":1015},[216],[23,70017,70019],{"className":70018},[1019,1020,1021,1022],[23,70020,4806],{"className":70021},[135,136,1022],[23,70023,297],{"className":70024},[296],[23,70026,70028],{"className":70027},[204],[23,70029,70031],{"className":70030,"style":2048},[208],[23,70032],{},[23,70034,66],{"className":70035},[169],[23,70037,70039],{"className":70038},[169],[23,70040,9020],{"className":70041},[1066,1067],[23,70043],{"className":70044,"style":149},[148],[23,70046,69818],{"className":70047},[153],[23,70049],{"className":70050,"style":149},[148],[23,70052,70054,70057,70060],{"className":70053},[126],[23,70055],{"className":70056,"style":470},[130],[23,70058,26249],{"className":70059,"style":45024},[135,412],[23,70061,112],{"className":70062},[135],[11,70064,70065,70066,70117],{},"对任意 ",[23,70067,70069,70087],{"className":70068,"translate":27},[31],[23,70070,70072],{"className":70071},[35],[37,70073,70074],{"xmlns":39},[42,70075,70076,70084],{},[45,70077,70078,70080,70082],{},[48,70079,8336],{},[51,70081,372],{},[48,70083,26249],{"mathvariant":375},[114,70085,70086],{"encoding":116},"f\\in\\mathcal H",[23,70088,70090,70108],{"className":70089,"ariaHidden":122},[121],[23,70091,70093,70096,70099,70102,70105],{"className":70092},[126],[23,70094],{"className":70095,"style":5414},[130],[23,70097,8336],{"className":70098,"style":11398},[135,136],[23,70100],{"className":70101,"style":149},[148],[23,70103,372],{"className":70104},[153],[23,70106],{"className":70107,"style":149},[148],[23,70109,70111,70114],{"className":70110},[126],[23,70112],{"className":70113,"style":470},[130],[23,70115,26249],{"className":70116,"style":45024},[135,412],"，把它分解为",[23,70119,70121],{"className":70120,"translate":27},[26],[23,70122,70124,70195],{"className":70123,"translate":27},[31],[23,70125,70127],{"className":70126},[35],[37,70128,70129],{"xmlns":39,"display":40},[42,70130,70131,70192],{},[45,70132,70133,70135,70137,70143,70145,70152,70154,70156,70162,70164,70170,70172,70174,70180,70182,70190],{},[48,70134,8336],{},[51,70136,69],{},[898,70138,70139,70141],{},[48,70140,8336],{},[51,70142,20680],{},[51,70144,5106],{},[898,70146,70147,70149],{},[48,70148,8336],{},[51,70150,70151],{},"⊥",[51,70153,747],{"separator":122},[148,70155],{"width":750},[898,70157,70158,70160],{},[48,70159,8336],{},[51,70161,20680],{},[51,70163,372],{},[898,70165,70166,70168],{},[48,70167,26249],{"mathvariant":375},[2581,70169,1532],{},[51,70171,747],{"separator":122},[148,70173],{"width":9915},[898,70175,70176,70178],{},[48,70177,8336],{},[51,70179,70151],{},[51,70181,372],{},[19137,70183,70184,70186,70188],{},[48,70185,26249],{"mathvariant":375},[2581,70187,1532],{},[51,70189,70151],{},[48,70191,112],{"mathvariant":111},[114,70193,70194],{"encoding":116},"f=f_\\parallel+f_\\perp,\\qquad f_\\parallel\\in\\mathcal H_0,\\quad f_\\perp\\in\\mathcal H_0^\\perp.",[23,70196,70198,70216,70273,70377,70481],{"className":70197,"ariaHidden":122},[121],[23,70199,70201,70204,70207,70210,70213],{"className":70200},[126],[23,70202],{"className":70203,"style":5414},[130],[23,70205,8336],{"className":70206,"style":11398},[135,136],[23,70208],{"className":70209,"style":149},[148],[23,70211,69],{"className":70212},[153],[23,70214],{"className":70215,"style":149},[148],[23,70217,70219,70223,70264,70267,70270],{"className":70218},[126],[23,70220],{"className":70221,"style":70222},[130],"height:1.0496em;vertical-align:-0.3552em;",[23,70224,70226,70229],{"className":70225},[135],[23,70227,8336],{"className":70228,"style":11398},[135,136],[23,70230,70232],{"className":70231},[998],[23,70233,70235,70256],{"className":70234},[199,200],[23,70236,70238,70253],{"className":70237},[204],[23,70239,70241],{"className":70240,"style":1008},[208],[23,70242,70244,70247],{"style":70243},"top:-2.5198em;margin-left:-0.1076em;margin-right:0.05em;",[23,70245],{"className":70246,"style":1015},[216],[23,70248,70250],{"className":70249},[1019,1020,1021,1022],[23,70251,20680],{"className":70252},[153,1022],[23,70254,297],{"className":70255},[296],[23,70257,70259],{"className":70258},[204],[23,70260,70262],{"className":70261,"style":1057},[208],[23,70263],{},[23,70265],{"className":70266,"style":392},[148],[23,70268,5106],{"className":70269},[612],[23,70271],{"className":70272,"style":392},[148],[23,70274,70276,70279,70319,70322,70325,70328,70368,70371,70374],{"className":70275},[126],[23,70277],{"className":70278,"style":70222},[130],[23,70280,70282,70285],{"className":70281},[135],[23,70283,8336],{"className":70284,"style":11398},[135,136],[23,70286,70288],{"className":70287},[998],[23,70289,70291,70311],{"className":70290},[199,200],[23,70292,70294,70308],{"className":70293},[204],[23,70295,70297],{"className":70296,"style":34192},[208],[23,70298,70299,70302],{"style":29546},[23,70300],{"className":70301,"style":1015},[216],[23,70303,70305],{"className":70304},[1019,1020,1021,1022],[23,70306,70151],{"className":70307},[153,1022],[23,70309,297],{"className":70310},[296],[23,70312,70314],{"className":70313},[204],[23,70315,70317],{"className":70316,"style":2048},[208],[23,70318],{},[23,70320,747],{"className":70321},[824],[23,70323],{"className":70324,"style":828},[148],[23,70326],{"className":70327,"style":280},[148],[23,70329,70331,70334],{"className":70330},[135],[23,70332,8336],{"className":70333,"style":11398},[135,136],[23,70335,70337],{"className":70336},[998],[23,70338,70340,70360],{"className":70339},[199,200],[23,70341,70343,70357],{"className":70342},[204],[23,70344,70346],{"className":70345,"style":1008},[208],[23,70347,70348,70351],{"style":70243},[23,70349],{"className":70350,"style":1015},[216],[23,70352,70354],{"className":70353},[1019,1020,1021,1022],[23,70355,20680],{"className":70356},[153,1022],[23,70358,297],{"className":70359},[296],[23,70361,70363],{"className":70362},[204],[23,70364,70366],{"className":70365,"style":1057},[208],[23,70367],{},[23,70369],{"className":70370,"style":149},[148],[23,70372,372],{"className":70373},[153],[23,70375],{"className":70376,"style":149},[148],[23,70378,70380,70383,70423,70426,70429,70432,70472,70475,70478],{"className":70379},[126],[23,70381],{"className":70382,"style":5414},[130],[23,70384,70386,70389],{"className":70385},[135],[23,70387,26249],{"className":70388,"style":45024},[135,412],[23,70390,70392],{"className":70391},[998],[23,70393,70395,70415],{"className":70394},[199,200],[23,70396,70398,70412],{"className":70397},[204],[23,70399,70401],{"className":70400,"style":18647},[208],[23,70402,70403,70406],{"style":61627},[23,70404],{"className":70405,"style":1015},[216],[23,70407,70409],{"className":70408},[1019,1020,1021,1022],[23,70410,1532],{"className":70411},[135,1022],[23,70413,297],{"className":70414},[296],[23,70416,70418],{"className":70417},[204],[23,70419,70421],{"className":70420,"style":2048},[208],[23,70422],{},[23,70424,747],{"className":70425},[824],[23,70427],{"className":70428,"style":14437},[148],[23,70430],{"className":70431,"style":280},[148],[23,70433,70435,70438],{"className":70434},[135],[23,70436,8336],{"className":70437,"style":11398},[135,136],[23,70439,70441],{"className":70440},[998],[23,70442,70444,70464],{"className":70443},[199,200],[23,70445,70447,70461],{"className":70446},[204],[23,70448,70450],{"className":70449,"style":34192},[208],[23,70451,70452,70455],{"style":29546},[23,70453],{"className":70454,"style":1015},[216],[23,70456,70458],{"className":70457},[1019,1020,1021,1022],[23,70459,70151],{"className":70460},[153,1022],[23,70462,297],{"className":70463},[296],[23,70465,70467],{"className":70466},[204],[23,70468,70470],{"className":70469,"style":2048},[208],[23,70471],{},[23,70473],{"className":70474,"style":149},[148],[23,70476,372],{"className":70477},[153],[23,70479],{"className":70480,"style":149},[148],[23,70482,70484,70488,70540],{"className":70483},[126],[23,70485],{"className":70486,"style":70487},[130],"height:1.1461em;vertical-align:-0.247em;",[23,70489,70491,70494],{"className":70490},[135],[23,70492,26249],{"className":70493,"style":45024},[135,412],[23,70495,70497],{"className":70496},[998],[23,70498,70500,70532],{"className":70499},[199,200],[23,70501,70503,70529],{"className":70502},[204],[23,70504,70506,70518],{"className":70505,"style":4986},[208],[23,70507,70509,70512],{"style":70508},"top:-2.453em;margin-left:-0.0097em;margin-right:0.05em;",[23,70510],{"className":70511,"style":1015},[216],[23,70513,70515],{"className":70514},[1019,1020,1021,1022],[23,70516,1532],{"className":70517},[135,1022],[23,70519,70520,70523],{"style":4989},[23,70521],{"className":70522,"style":1015},[216],[23,70524,70526],{"className":70525},[1019,1020,1021,1022],[23,70527,70151],{"className":70528},[153,1022],[23,70530,297],{"className":70531},[296],[23,70533,70535],{"className":70534},[204],[23,70536,70538],{"className":70537,"style":19375},[208],[23,70539],{},[23,70541,112],{"className":70542},[135],[11,70544,70545,70546,49297,70616,70777],{},"对每个训练点 ",[23,70547,70549,70567],{"className":70548,"translate":27},[31],[23,70550,70552],{"className":70551},[35],[37,70553,70554],{"xmlns":39},[42,70555,70556,70564],{},[45,70557,70558],{},[898,70559,70560,70562],{},[48,70561,753],{},[48,70563,33493],{},[114,70565,70566],{"encoding":116},"x_i",[23,70568,70570],{"className":70569,"ariaHidden":122},[121],[23,70571,70573,70576],{"className":70572},[126],[23,70574],{"className":70575,"style":12730},[130],[23,70577,70579,70582],{"className":70578},[135],[23,70580,753],{"className":70581},[135,136],[23,70583,70585],{"className":70584},[998],[23,70586,70588,70608],{"className":70587},[199,200],[23,70589,70591,70605],{"className":70590},[204],[23,70592,70594],{"className":70593,"style":33647},[208],[23,70595,70596,70599],{"style":2029},[23,70597],{"className":70598,"style":1015},[216],[23,70600,70602],{"className":70601},[1019,1020,1021,1022],[23,70603,33493],{"className":70604},[135,136,1022],[23,70606,297],{"className":70607},[296],[23,70609,70611],{"className":70610},[204],[23,70612,70614],{"className":70613,"style":2048},[208],[23,70615],{},[23,70617,70619,70655],{"className":70618,"translate":27},[31],[23,70620,70622],{"className":70621},[35],[37,70623,70624],{"xmlns":39},[42,70625,70626,70652],{},[45,70627,70628,70630,70632,70634,70636,70642,70644,70646],{},[48,70629,4836],{},[51,70631,54],{"stretchy":53},[51,70633,5758],{},[51,70635,747],{"separator":122},[898,70637,70638,70640],{},[48,70639,753],{},[48,70641,33493],{},[51,70643,66],{"stretchy":53},[51,70645,372],{},[898,70647,70648,70650],{},[48,70649,26249],{"mathvariant":375},[2581,70651,1532],{},[114,70653,70654],{"encoding":116},"k(\\cdot,x_i)\\in\\mathcal H_0",[23,70656,70658,70731],{"className":70657,"ariaHidden":122},[121],[23,70659,70661,70664,70667,70670,70673,70676,70679,70719,70722,70725,70728],{"className":70660},[126],[23,70662],{"className":70663,"style":131},[130],[23,70665,4836],{"className":70666,"style":4851},[135,136],[23,70668,54],{"className":70669},[140],[23,70671,5758],{"className":70672},[135],[23,70674,747],{"className":70675},[824],[23,70677],{"className":70678,"style":280},[148],[23,70680,70682,70685],{"className":70681},[135],[23,70683,753],{"className":70684},[135,136],[23,70686,70688],{"className":70687},[998],[23,70689,70691,70711],{"className":70690},[199,200],[23,70692,70694,70708],{"className":70693},[204],[23,70695,70697],{"className":70696,"style":33647},[208],[23,70698,70699,70702],{"style":2029},[23,70700],{"className":70701,"style":1015},[216],[23,70703,70705],{"className":70704},[1019,1020,1021,1022],[23,70706,33493],{"className":70707},[135,136,1022],[23,70709,297],{"className":70710},[296],[23,70712,70714],{"className":70713},[204],[23,70715,70717],{"className":70716,"style":2048},[208],[23,70718],{},[23,70720,66],{"className":70721},[169],[23,70723],{"className":70724,"style":149},[148],[23,70726,372],{"className":70727},[153],[23,70729],{"className":70730,"style":149},[148],[23,70732,70734,70737],{"className":70733},[126],[23,70735],{"className":70736,"style":42827},[130],[23,70738,70740,70743],{"className":70739},[135],[23,70741,26249],{"className":70742,"style":45024},[135,412],[23,70744,70746],{"className":70745},[998],[23,70747,70749,70769],{"className":70748},[199,200],[23,70750,70752,70766],{"className":70751},[204],[23,70753,70755],{"className":70754,"style":18647},[208],[23,70756,70757,70760],{"style":61627},[23,70758],{"className":70759,"style":1015},[216],[23,70761,70763],{"className":70762},[1019,1020,1021,1022],[23,70764,1532],{"className":70765},[135,1022],[23,70767,297],{"className":70768},[296],[23,70770,70772],{"className":70771},[204],[23,70773,70775],{"className":70774,"style":2048},[208],[23,70776],{},"，所以由再生性",[23,70779,70781],{"className":70780,"translate":27},[26],[23,70782,70784,70916],{"className":70783,"translate":27},[31],[23,70785,70787],{"className":70786},[35],[37,70788,70789],{"xmlns":39,"display":40},[42,70790,70791,70913],{},[45,70792,70793,70795,70797,70803,70805,70807,70809,70811,70813,70815,70817,70819,70821,70827,70829,70831,70833,70835,70837,70843,70845,70847,70849,70851,70853,70859,70861,70863,70865,70867,70873,70875,70877,70879,70881,70883,70889,70891,70893,70895,70901,70903,70909,70911],{},[48,70794,8336],{},[51,70796,54],{"stretchy":53},[898,70798,70799,70801],{},[48,70800,753],{},[48,70802,33493],{},[51,70804,66],{"stretchy":53},[51,70806,69],{},[51,70808,61181],{"stretchy":53},[48,70810,8336],{},[51,70812,747],{"separator":122},[48,70814,4836],{},[51,70816,54],{"stretchy":53},[51,70818,5758],{},[51,70820,747],{"separator":122},[898,70822,70823,70825],{},[48,70824,753],{},[48,70826,33493],{},[51,70828,66],{"stretchy":53},[51,70830,61206],{"stretchy":53},[148,70832],{"linebreak":29625},[51,70834,69],{},[51,70836,61181],{"stretchy":53},[898,70838,70839,70841],{},[48,70840,8336],{},[51,70842,20680],{},[51,70844,747],{"separator":122},[48,70846,4836],{},[51,70848,54],{"stretchy":53},[51,70850,5758],{},[51,70852,747],{"separator":122},[898,70854,70855,70857],{},[48,70856,753],{},[48,70858,33493],{},[51,70860,66],{"stretchy":53},[51,70862,61206],{"stretchy":53},[51,70864,5106],{},[51,70866,61181],{"stretchy":53},[898,70868,70869,70871],{},[48,70870,8336],{},[51,70872,70151],{},[51,70874,747],{"separator":122},[48,70876,4836],{},[51,70878,54],{"stretchy":53},[51,70880,5758],{},[51,70882,747],{"separator":122},[898,70884,70885,70887],{},[48,70886,753],{},[48,70888,33493],{},[51,70890,66],{"stretchy":53},[51,70892,61206],{"stretchy":53},[51,70894,69],{},[898,70896,70897,70899],{},[48,70898,8336],{},[51,70900,20680],{},[51,70902,54],{"stretchy":53},[898,70904,70905,70907],{},[48,70906,753],{},[48,70908,33493],{},[51,70910,66],{"stretchy":53},[48,70912,112],{"mathvariant":111},[114,70914,70915],{"encoding":116},"f(x_i)=\\langle f,k(\\cdot,x_i)\\rangle \\\\\n=\\langle f_\\parallel,k(\\cdot,x_i)\\rangle+\\langle f_\\perp,k(\\cdot,x_i)\\rangle\n=f_\\parallel(x_i).",[23,70917,70919,70983,71059,71062,71075,71198,71320],{"className":70918,"ariaHidden":122},[121],[23,70920,70922,70925,70928,70931,70971,70974,70977,70980],{"className":70921},[126],[23,70923],{"className":70924,"style":131},[130],[23,70926,8336],{"className":70927,"style":11398},[135,136],[23,70929,54],{"className":70930},[140],[23,70932,70934,70937],{"className":70933},[135],[23,70935,753],{"className":70936},[135,136],[23,70938,70940],{"className":70939},[998],[23,70941,70943,70963],{"className":70942},[199,200],[23,70944,70946,70960],{"className":70945},[204],[23,70947,70949],{"className":70948,"style":33647},[208],[23,70950,70951,70954],{"style":2029},[23,70952],{"className":70953,"style":1015},[216],[23,70955,70957],{"className":70956},[1019,1020,1021,1022],[23,70958,33493],{"className":70959},[135,136,1022],[23,70961,297],{"className":70962},[296],[23,70964,70966],{"className":70965},[204],[23,70967,70969],{"className":70968,"style":2048},[208],[23,70970],{},[23,70972,66],{"className":70973},[169],[23,70975],{"className":70976,"style":149},[148],[23,70978,69],{"className":70979},[153],[23,70981],{"className":70982,"style":149},[148],[23,70984,70986,70989,70992,70995,70998,71001,71004,71007,71010,71013,71016,71056],{"className":70985},[126],[23,70987],{"className":70988,"style":131},[130],[23,70990,61181],{"className":70991},[140],[23,70993,8336],{"className":70994,"style":11398},[135,136],[23,70996,747],{"className":70997},[824],[23,70999],{"className":71000,"style":280},[148],[23,71002,4836],{"className":71003,"style":4851},[135,136],[23,71005,54],{"className":71006},[140],[23,71008,5758],{"className":71009},[135],[23,71011,747],{"className":71012},[824],[23,71014],{"className":71015,"style":280},[148],[23,71017,71019,71022],{"className":71018},[135],[23,71020,753],{"className":71021},[135,136],[23,71023,71025],{"className":71024},[998],[23,71026,71028,71048],{"className":71027},[199,200],[23,71029,71031,71045],{"className":71030},[204],[23,71032,71034],{"className":71033,"style":33647},[208],[23,71035,71036,71039],{"style":2029},[23,71037],{"className":71038,"style":1015},[216],[23,71040,71042],{"className":71041},[1019,1020,1021,1022],[23,71043,33493],{"className":71044},[135,136,1022],[23,71046,297],{"className":71047},[296],[23,71049,71051],{"className":71050},[204],[23,71052,71054],{"className":71053,"style":2048},[208],[23,71055],{},[23,71057,61346],{"className":71058},[169],[23,71060],{"className":71061},[148,29625],[23,71063,71065,71069,71072],{"className":71064},[126],[23,71066],{"className":71067,"style":71068},[130],"height:0.3669em;",[23,71070,69],{"className":71071},[153],[23,71073],{"className":71074,"style":149},[148],[23,71076,71078,71082,71085,71125,71128,71131,71134,71137,71140,71143,71146,71186,71189,71192,71195],{"className":71077},[126],[23,71079],{"className":71080,"style":71081},[130],"height:1.1052em;vertical-align:-0.3552em;",[23,71083,61181],{"className":71084},[140],[23,71086,71088,71091],{"className":71087},[135],[23,71089,8336],{"className":71090,"style":11398},[135,136],[23,71092,71094],{"className":71093},[998],[23,71095,71097,71117],{"className":71096},[199,200],[23,71098,71100,71114],{"className":71099},[204],[23,71101,71103],{"className":71102,"style":1008},[208],[23,71104,71105,71108],{"style":70243},[23,71106],{"className":71107,"style":1015},[216],[23,71109,71111],{"className":71110},[1019,1020,1021,1022],[23,71112,20680],{"className":71113},[153,1022],[23,71115,297],{"className":71116},[296],[23,71118,71120],{"className":71119},[204],[23,71121,71123],{"className":71122,"style":1057},[208],[23,71124],{},[23,71126,747],{"className":71127},[824],[23,71129],{"className":71130,"style":280},[148],[23,71132,4836],{"className":71133,"style":4851},[135,136],[23,71135,54],{"className":71136},[140],[23,71138,5758],{"className":71139},[135],[23,71141,747],{"className":71142},[824],[23,71144],{"className":71145,"style":280},[148],[23,71147,71149,71152],{"className":71148},[135],[23,71150,753],{"className":71151},[135,136],[23,71153,71155],{"className":71154},[998],[23,71156,71158,71178],{"className":71157},[199,200],[23,71159,71161,71175],{"className":71160},[204],[23,71162,71164],{"className":71163,"style":33647},[208],[23,71165,71166,71169],{"style":2029},[23,71167],{"className":71168,"style":1015},[216],[23,71170,71172],{"className":71171},[1019,1020,1021,1022],[23,71173,33493],{"className":71174},[135,136,1022],[23,71176,297],{"className":71177},[296],[23,71179,71181],{"className":71180},[204],[23,71182,71184],{"className":71183,"style":2048},[208],[23,71185],{},[23,71187,61346],{"className":71188},[169],[23,71190],{"className":71191,"style":392},[148],[23,71193,5106],{"className":71194},[612],[23,71196],{"className":71197,"style":392},[148],[23,71199,71201,71204,71207,71247,71250,71253,71256,71259,71262,71265,71268,71308,71311,71314,71317],{"className":71200},[126],[23,71202],{"className":71203,"style":131},[130],[23,71205,61181],{"className":71206},[140],[23,71208,71210,71213],{"className":71209},[135],[23,71211,8336],{"className":71212,"style":11398},[135,136],[23,71214,71216],{"className":71215},[998],[23,71217,71219,71239],{"className":71218},[199,200],[23,71220,71222,71236],{"className":71221},[204],[23,71223,71225],{"className":71224,"style":34192},[208],[23,71226,71227,71230],{"style":29546},[23,71228],{"className":71229,"style":1015},[216],[23,71231,71233],{"className":71232},[1019,1020,1021,1022],[23,71234,70151],{"className":71235},[153,1022],[23,71237,297],{"className":71238},[296],[23,71240,71242],{"className":71241},[204],[23,71243,71245],{"className":71244,"style":2048},[208],[23,71246],{},[23,71248,747],{"className":71249},[824],[23,71251],{"className":71252,"style":280},[148],[23,71254,4836],{"className":71255,"style":4851},[135,136],[23,71257,54],{"className":71258},[140],[23,71260,5758],{"className":71261},[135],[23,71263,747],{"className":71264},[824],[23,71266],{"className":71267,"style":280},[148],[23,71269,71271,71274],{"className":71270},[135],[23,71272,753],{"className":71273},[135,136],[23,71275,71277],{"className":71276},[998],[23,71278,71280,71300],{"className":71279},[199,200],[23,71281,71283,71297],{"className":71282},[204],[23,71284,71286],{"className":71285,"style":33647},[208],[23,71287,71288,71291],{"style":2029},[23,71289],{"className":71290,"style":1015},[216],[23,71292,71294],{"className":71293},[1019,1020,1021,1022],[23,71295,33493],{"className":71296},[135,136,1022],[23,71298,297],{"className":71299},[296],[23,71301,71303],{"className":71302},[204],[23,71304,71306],{"className":71305,"style":2048},[208],[23,71307],{},[23,71309,61346],{"className":71310},[169],[23,71312],{"className":71313,"style":149},[148],[23,71315,69],{"className":71316},[153],[23,71318],{"className":71319,"style":149},[148],[23,71321,71323,71326,71366,71369,71409,71412],{"className":71322},[126],[23,71324],{"className":71325,"style":71081},[130],[23,71327,71329,71332],{"className":71328},[135],[23,71330,8336],{"className":71331,"style":11398},[135,136],[23,71333,71335],{"className":71334},[998],[23,71336,71338,71358],{"className":71337},[199,200],[23,71339,71341,71355],{"className":71340},[204],[23,71342,71344],{"className":71343,"style":1008},[208],[23,71345,71346,71349],{"style":70243},[23,71347],{"className":71348,"style":1015},[216],[23,71350,71352],{"className":71351},[1019,1020,1021,1022],[23,71353,20680],{"className":71354},[153,1022],[23,71356,297],{"className":71357},[296],[23,71359,71361],{"className":71360},[204],[23,71362,71364],{"className":71363,"style":1057},[208],[23,71365],{},[23,71367,54],{"className":71368},[140],[23,71370,71372,71375],{"className":71371},[135],[23,71373,753],{"className":71374},[135,136],[23,71376,71378],{"className":71377},[998],[23,71379,71381,71401],{"className":71380},[199,200],[23,71382,71384,71398],{"className":71383},[204],[23,71385,71387],{"className":71386,"style":33647},[208],[23,71388,71389,71392],{"style":2029},[23,71390],{"className":71391,"style":1015},[216],[23,71393,71395],{"className":71394},[1019,1020,1021,1022],[23,71396,33493],{"className":71397},[135,136,1022],[23,71399,297],{"className":71400},[296],[23,71402,71404],{"className":71403},[204],[23,71405,71407],{"className":71406,"style":2048},[208],[23,71408],{},[23,71410,66],{"className":71411},[169],[23,71413,112],{"className":71414},[135],[11,71416,71417,71418,71491],{},"也就是说，",[18,71419,71420,71490],{},[23,71421,71423,71441],{"className":71422,"translate":27},[31],[23,71424,71426],{"className":71425},[35],[37,71427,71428],{"xmlns":39},[42,71429,71430,71438],{},[45,71431,71432],{},[898,71433,71434,71436],{},[48,71435,8336],{},[51,71437,70151],{},[114,71439,71440],{"encoding":116},"f_\\perp",[23,71442,71444],{"className":71443,"ariaHidden":122},[121],[23,71445,71447,71450],{"className":71446},[126],[23,71448],{"className":71449,"style":5414},[130],[23,71451,71453,71456],{"className":71452},[135],[23,71454,8336],{"className":71455,"style":11398},[135,136],[23,71457,71459],{"className":71458},[998],[23,71460,71462,71482],{"className":71461},[199,200],[23,71463,71465,71479],{"className":71464},[204],[23,71466,71468],{"className":71467,"style":34192},[208],[23,71469,71470,71473],{"style":29546},[23,71471],{"className":71472,"style":1015},[216],[23,71474,71476],{"className":71475},[1019,1020,1021,1022],[23,71477,70151],{"className":71478},[153,1022],[23,71480,297],{"className":71481},[296],[23,71483,71485],{"className":71484},[204],[23,71486,71488],{"className":71487,"style":2048},[208],[23,71489],{}," 在所有训练点上的取值都是零","。它对损失函数没有任何贡献。",[11,71493,71494],{},"再看正则项。由勾股定理，",[23,71496,71498],{"className":71497,"translate":27},[26],[23,71499,71501,71581],{"className":71500,"translate":27},[31],[23,71502,71504],{"className":71503},[35],[37,71505,71506],{"xmlns":39,"display":40},[42,71507,71508,71578],{},[45,71509,71510,71512,71514,71522,71524,71526,71532,71540,71542,71544,71550,71558,71560,71562,71568,71576],{},[48,71511,20680],{"mathvariant":111},[48,71513,8336],{},[19137,71515,71516,71518,71520],{},[48,71517,20680],{"mathvariant":111},[48,71519,26249],{"mathvariant":375},[2581,71521,9952],{},[51,71523,69],{},[48,71525,20680],{"mathvariant":111},[898,71527,71528,71530],{},[48,71529,8336],{},[51,71531,20680],{},[19137,71533,71534,71536,71538],{},[48,71535,20680],{"mathvariant":111},[48,71537,26249],{"mathvariant":375},[2581,71539,9952],{},[51,71541,5106],{},[48,71543,20680],{"mathvariant":111},[898,71545,71546,71548],{},[48,71547,8336],{},[51,71549,70151],{},[19137,71551,71552,71554,71556],{},[48,71553,20680],{"mathvariant":111},[48,71555,26249],{"mathvariant":375},[2581,71557,9952],{},[51,71559,7378],{},[48,71561,20680],{"mathvariant":111},[898,71563,71564,71566],{},[48,71565,8336],{},[51,71567,20680],{},[19137,71569,71570,71572,71574],{},[48,71571,20680],{"mathvariant":111},[48,71573,26249],{"mathvariant":375},[2581,71575,9952],{},[48,71577,112],{"mathvariant":111},[114,71579,71580],{"encoding":116},"\\|f\\|_{\\mathcal H}^2=\\|f_\\parallel\\|_{\\mathcal H}^2+\\|f_\\perp\\|_{\\mathcal H}^2\n\\ge \\|f_\\parallel\\|_{\\mathcal H}^2.",[23,71582,71584,71659,71772,71884],{"className":71583,"ariaHidden":122},[121],[23,71585,71587,71590,71593,71596,71650,71653,71656],{"className":71586},[126],[23,71588],{"className":71589,"style":26499},[130],[23,71591,20680],{"className":71592},[135],[23,71594,8336],{"className":71595,"style":11398},[135,136],[23,71597,71599,71602],{"className":71598},[135],[23,71600,20680],{"className":71601},[135],[23,71603,71605],{"className":71604},[998],[23,71606,71608,71642],{"className":71607},[199,200],[23,71609,71611,71639],{"className":71610},[204],[23,71612,71614,71628],{"className":71613,"style":11002},[208],[23,71615,71616,71619],{"style":19345},[23,71617],{"className":71618,"style":1015},[216],[23,71620,71622],{"className":71621},[1019,1020,1021,1022],[23,71623,71625],{"className":71624},[135,1022],[23,71626,26249],{"className":71627,"style":45024},[135,412,1022],[23,71629,71630,71633],{"style":4989},[23,71631],{"className":71632,"style":1015},[216],[23,71634,71636],{"className":71635},[1019,1020,1021,1022],[23,71637,9952],{"className":71638},[135,1022],[23,71640,297],{"className":71641},[296],[23,71643,71645],{"className":71644},[204],[23,71646,71648],{"className":71647,"style":19375},[208],[23,71649],{},[23,71651],{"className":71652,"style":149},[148],[23,71654,69],{"className":71655},[153],[23,71657],{"className":71658,"style":149},[148],[23,71660,71662,71666,71669,71709,71763,71766,71769],{"className":71661},[126],[23,71663],{"className":71664,"style":71665},[130],"height:1.2193em;vertical-align:-0.3552em;",[23,71667,20680],{"className":71668},[135],[23,71670,71672,71675],{"className":71671},[135],[23,71673,8336],{"className":71674,"style":11398},[135,136],[23,71676,71678],{"className":71677},[998],[23,71679,71681,71701],{"className":71680},[199,200],[23,71682,71684,71698],{"className":71683},[204],[23,71685,71687],{"className":71686,"style":1008},[208],[23,71688,71689,71692],{"style":70243},[23,71690],{"className":71691,"style":1015},[216],[23,71693,71695],{"className":71694},[1019,1020,1021,1022],[23,71696,20680],{"className":71697},[153,1022],[23,71699,297],{"className":71700},[296],[23,71702,71704],{"className":71703},[204],[23,71705,71707],{"className":71706,"style":1057},[208],[23,71708],{},[23,71710,71712,71715],{"className":71711},[135],[23,71713,20680],{"className":71714},[135],[23,71716,71718],{"className":71717},[998],[23,71719,71721,71755],{"className":71720},[199,200],[23,71722,71724,71752],{"className":71723},[204],[23,71725,71727,71741],{"className":71726,"style":11002},[208],[23,71728,71729,71732],{"style":19345},[23,71730],{"className":71731,"style":1015},[216],[23,71733,71735],{"className":71734},[1019,1020,1021,1022],[23,71736,71738],{"className":71737},[135,1022],[23,71739,26249],{"className":71740,"style":45024},[135,412,1022],[23,71742,71743,71746],{"style":4989},[23,71744],{"className":71745,"style":1015},[216],[23,71747,71749],{"className":71748},[1019,1020,1021,1022],[23,71750,9952],{"className":71751},[135,1022],[23,71753,297],{"className":71754},[296],[23,71756,71758],{"className":71757},[204],[23,71759,71761],{"className":71760,"style":19375},[208],[23,71762],{},[23,71764],{"className":71765,"style":392},[148],[23,71767,5106],{"className":71768},[612],[23,71770],{"className":71771,"style":392},[148],[23,71773,71775,71778,71781,71821,71875,71878,71881],{"className":71774},[126],[23,71776],{"className":71777,"style":26499},[130],[23,71779,20680],{"className":71780},[135],[23,71782,71784,71787],{"className":71783},[135],[23,71785,8336],{"className":71786,"style":11398},[135,136],[23,71788,71790],{"className":71789},[998],[23,71791,71793,71813],{"className":71792},[199,200],[23,71794,71796,71810],{"className":71795},[204],[23,71797,71799],{"className":71798,"style":34192},[208],[23,71800,71801,71804],{"style":29546},[23,71802],{"className":71803,"style":1015},[216],[23,71805,71807],{"className":71806},[1019,1020,1021,1022],[23,71808,70151],{"className":71809},[153,1022],[23,71811,297],{"className":71812},[296],[23,71814,71816],{"className":71815},[204],[23,71817,71819],{"className":71818,"style":2048},[208],[23,71820],{},[23,71822,71824,71827],{"className":71823},[135],[23,71825,20680],{"className":71826},[135],[23,71828,71830],{"className":71829},[998],[23,71831,71833,71867],{"className":71832},[199,200],[23,71834,71836,71864],{"className":71835},[204],[23,71837,71839,71853],{"className":71838,"style":11002},[208],[23,71840,71841,71844],{"style":19345},[23,71842],{"className":71843,"style":1015},[216],[23,71845,71847],{"className":71846},[1019,1020,1021,1022],[23,71848,71850],{"className":71849},[135,1022],[23,71851,26249],{"className":71852,"style":45024},[135,412,1022],[23,71854,71855,71858],{"style":4989},[23,71856],{"className":71857,"style":1015},[216],[23,71859,71861],{"className":71860},[1019,1020,1021,1022],[23,71862,9952],{"className":71863},[135,1022],[23,71865,297],{"className":71866},[296],[23,71868,71870],{"className":71869},[204],[23,71871,71873],{"className":71872,"style":19375},[208],[23,71874],{},[23,71876],{"className":71877,"style":149},[148],[23,71879,7378],{"className":71880},[153],[23,71882],{"className":71883,"style":149},[148],[23,71885,71887,71890,71893,71933,71987],{"className":71886},[126],[23,71888],{"className":71889,"style":71665},[130],[23,71891,20680],{"className":71892},[135],[23,71894,71896,71899],{"className":71895},[135],[23,71897,8336],{"className":71898,"style":11398},[135,136],[23,71900,71902],{"className":71901},[998],[23,71903,71905,71925],{"className":71904},[199,200],[23,71906,71908,71922],{"className":71907},[204],[23,71909,71911],{"className":71910,"style":1008},[208],[23,71912,71913,71916],{"style":70243},[23,71914],{"className":71915,"style":1015},[216],[23,71917,71919],{"className":71918},[1019,1020,1021,1022],[23,71920,20680],{"className":71921},[153,1022],[23,71923,297],{"className":71924},[296],[23,71926,71928],{"className":71927},[204],[23,71929,71931],{"className":71930,"style":1057},[208],[23,71932],{},[23,71934,71936,71939],{"className":71935},[135],[23,71937,20680],{"className":71938},[135],[23,71940,71942],{"className":71941},[998],[23,71943,71945,71979],{"className":71944},[199,200],[23,71946,71948,71976],{"className":71947},[204],[23,71949,71951,71965],{"className":71950,"style":11002},[208],[23,71952,71953,71956],{"style":19345},[23,71954],{"className":71955,"style":1015},[216],[23,71957,71959],{"className":71958},[1019,1020,1021,1022],[23,71960,71962],{"className":71961},[135,1022],[23,71963,26249],{"className":71964,"style":45024},[135,412,1022],[23,71966,71967,71970],{"style":4989},[23,71968],{"className":71969,"style":1015},[216],[23,71971,71973],{"className":71972},[1019,1020,1021,1022],[23,71974,9952],{"className":71975},[135,1022],[23,71977,297],{"className":71978},[296],[23,71980,71982],{"className":71981},[204],[23,71983,71985],{"className":71984,"style":19375},[208],[23,71986],{},[23,71988,112],{"className":71989},[135],[11,71991,71992,71993,72021],{},"因为 ",[23,71994,71996,72009],{"className":71995,"translate":27},[31],[23,71997,71999],{"className":71998},[35],[37,72000,72001],{"xmlns":39},[42,72002,72003,72007],{},[45,72004,72005],{},[48,72006,68133],{"mathvariant":111},[114,72008,69091],{"encoding":116},[23,72010,72012],{"className":72011,"ariaHidden":122},[121],[23,72013,72015,72018],{"className":72014},[126],[23,72016],{"className":72017,"style":470},[130],[23,72019,68133],{"className":72020},[135]," 非减，",[23,72023,72025],{"className":72024,"translate":27},[26],[23,72026,72028,72084],{"className":72027,"translate":27},[31],[23,72029,72031],{"className":72030},[35],[37,72032,72033],{"xmlns":39,"display":40},[42,72034,72035,72081],{},[45,72036,72037,72039,72041,72043,72045,72053,72055,72057,72059,72061,72063,72069,72077,72079],{},[48,72038,68133],{"mathvariant":111},[51,72040,54],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[48,72042,20680],{"mathvariant":111},[48,72044,8336],{},[19137,72046,72047,72049,72051],{},[48,72048,20680],{"mathvariant":111},[48,72050,26249],{"mathvariant":375},[2581,72052,9952],{},[51,72054,66],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[51,72056,7378],{},[48,72058,68133],{"mathvariant":111},[51,72060,54],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[48,72062,20680],{"mathvariant":111},[898,72064,72065,72067],{},[48,72066,8336],{},[51,72068,20680],{},[19137,72070,72071,72073,72075],{},[48,72072,20680],{"mathvariant":111},[48,72074,26249],{"mathvariant":375},[2581,72076,9952],{},[51,72078,66],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[48,72080,112],{"mathvariant":111},[114,72082,72083],{"encoding":116},"\\Omega\\bigl(\\|f\\|_{\\mathcal H}^2\\bigr)\\ge  \\Omega\\bigl(\\|f_\\parallel\\|_{\\mathcal H}^2\\bigr).",[23,72085,72087,72177],{"className":72086,"ariaHidden":122},[121],[23,72088,72090,72093,72096,72102,72105,72108,72162,72168,72171,72174],{"className":72089},[126],[23,72091],{"className":72092,"style":17156},[130],[23,72094,68133],{"className":72095},[135],[23,72097,72099],{"className":72098},[140],[23,72100,54],{"className":72101},[1066,1067],[23,72103,20680],{"className":72104},[135],[23,72106,8336],{"className":72107,"style":11398},[135,136],[23,72109,72111,72114],{"className":72110},[135],[23,72112,20680],{"className":72113},[135],[23,72115,72117],{"className":72116},[998],[23,72118,72120,72154],{"className":72119},[199,200],[23,72121,72123,72151],{"className":72122},[204],[23,72124,72126,72140],{"className":72125,"style":11002},[208],[23,72127,72128,72131],{"style":19345},[23,72129],{"className":72130,"style":1015},[216],[23,72132,72134],{"className":72133},[1019,1020,1021,1022],[23,72135,72137],{"className":72136},[135,1022],[23,72138,26249],{"className":72139,"style":45024},[135,412,1022],[23,72141,72142,72145],{"style":4989},[23,72143],{"className":72144,"style":1015},[216],[23,72146,72148],{"className":72147},[1019,1020,1021,1022],[23,72149,9952],{"className":72150},[135,1022],[23,72152,297],{"className":72153},[296],[23,72155,72157],{"className":72156},[204],[23,72158,72160],{"className":72159,"style":19375},[208],[23,72161],{},[23,72163,72165],{"className":72164},[169],[23,72166,66],{"className":72167},[1066,1067],[23,72169],{"className":72170,"style":149},[148],[23,72172,7378],{"className":72173},[153],[23,72175],{"className":72176,"style":149},[148],[23,72178,72180,72183,72186,72192,72195,72235,72289,72295],{"className":72179},[126],[23,72181],{"className":72182,"style":71665},[130],[23,72184,68133],{"className":72185},[135],[23,72187,72189],{"className":72188},[140],[23,72190,54],{"className":72191},[1066,1067],[23,72193,20680],{"className":72194},[135],[23,72196,72198,72201],{"className":72197},[135],[23,72199,8336],{"className":72200,"style":11398},[135,136],[23,72202,72204],{"className":72203},[998],[23,72205,72207,72227],{"className":72206},[199,200],[23,72208,72210,72224],{"className":72209},[204],[23,72211,72213],{"className":72212,"style":1008},[208],[23,72214,72215,72218],{"style":70243},[23,72216],{"className":72217,"style":1015},[216],[23,72219,72221],{"className":72220},[1019,1020,1021,1022],[23,72222,20680],{"className":72223},[153,1022],[23,72225,297],{"className":72226},[296],[23,72228,72230],{"className":72229},[204],[23,72231,72233],{"className":72232,"style":1057},[208],[23,72234],{},[23,72236,72238,72241],{"className":72237},[135],[23,72239,20680],{"className":72240},[135],[23,72242,72244],{"className":72243},[998],[23,72245,72247,72281],{"className":72246},[199,200],[23,72248,72250,72278],{"className":72249},[204],[23,72251,72253,72267],{"className":72252,"style":11002},[208],[23,72254,72255,72258],{"style":19345},[23,72256],{"className":72257,"style":1015},[216],[23,72259,72261],{"className":72260},[1019,1020,1021,1022],[23,72262,72264],{"className":72263},[135,1022],[23,72265,26249],{"className":72266,"style":45024},[135,412,1022],[23,72268,72269,72272],{"style":4989},[23,72270],{"className":72271,"style":1015},[216],[23,72273,72275],{"className":72274},[1019,1020,1021,1022],[23,72276,9952],{"className":72277},[135,1022],[23,72279,297],{"className":72280},[296],[23,72282,72284],{"className":72283},[204],[23,72285,72287],{"className":72286,"style":19375},[208],[23,72288],{},[23,72290,72292],{"className":72291},[169],[23,72293,66],{"className":72294},[1066,1067],[23,72296,112],{"className":72297},[135],[11,72299,72300],{},"把两件事合起来：",[23,72302,72304],{"className":72303,"translate":27},[26],[23,72305,72307,72445],{"className":72306,"translate":27},[31],[23,72308,72310],{"className":72309},[35],[37,72311,72312],{"xmlns":39,"display":40},[42,72313,72314,72442],{},[45,72315,72316,72318,72320,72322,72324,72330,72332,72334,72336,72338,72340,72342,72348,72350,72352,72354,72356,72358,72360,72362,72368,72370,72372,72374,72376,72378,72384,72386,72392,72394,72396,72398,72400,72406,72408,72414,72416,72418,72420,72422,72424,72426,72432,72438,72440],{},[48,72317,555],{},[51,72319,54],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[48,72321,8336],{},[51,72323,54],{"stretchy":53},[898,72325,72326,72328],{},[48,72327,753],{},[2581,72329,4435],{},[51,72331,66],{"stretchy":53},[51,72333,747],{"separator":122},[51,72335,62555],{},[51,72337,747],{"separator":122},[48,72339,8336],{},[51,72341,54],{"stretchy":53},[898,72343,72344,72346],{},[48,72345,753],{},[48,72347,4806],{},[51,72349,66],{"stretchy":53},[51,72351,66],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[51,72353,5106],{},[48,72355,68133],{"mathvariant":111},[51,72357,54],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[48,72359,20680],{"mathvariant":111},[48,72361,8336],{},[4518,72363,72364,72366],{},[48,72365,20680],{"mathvariant":111},[2581,72367,9952],{},[51,72369,66],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[51,72371,7378],{},[148,72373],{"linebreak":29625},[48,72375,555],{},[51,72377,54],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[898,72379,72380,72382],{},[48,72381,8336],{},[51,72383,20680],{},[51,72385,54],{"stretchy":53},[898,72387,72388,72390],{},[48,72389,753],{},[2581,72391,4435],{},[51,72393,66],{"stretchy":53},[51,72395,747],{"separator":122},[51,72397,62555],{},[51,72399,747],{"separator":122},[898,72401,72402,72404],{},[48,72403,8336],{},[51,72405,20680],{},[51,72407,54],{"stretchy":53},[898,72409,72410,72412],{},[48,72411,753],{},[48,72413,4806],{},[51,72415,66],{"stretchy":53},[51,72417,66],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[51,72419,5106],{},[48,72421,68133],{"mathvariant":111},[51,72423,54],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[48,72425,20680],{"mathvariant":111},[898,72427,72428,72430],{},[48,72429,8336],{},[51,72431,20680],{},[4518,72433,72434,72436],{},[48,72435,20680],{"mathvariant":111},[2581,72437,9952],{},[51,72439,66],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[48,72441,112],{"mathvariant":111},[114,72443,72444],{"encoding":116},"L\\bigl(f(x_1),\\dots,f(x_n)\\bigr)+\\Omega\\bigl(\\|f\\|^2\\bigr)\n\\ge \n \\\\ \nL\\bigl(f_\\parallel(x_1),\\dots,f_\\parallel(x_n)\\bigr)+\\Omega\\bigl(\\|f_\\parallel\\|^2\\bigr).",[23,72446,72448,72594,72656,72659,72879],{"className":72447,"ariaHidden":122},[121],[23,72449,72451,72454,72457,72463,72466,72469,72509,72512,72515,72518,72521,72524,72527,72530,72533,72536,72576,72579,72585,72588,72591],{"className":72450},[126],[23,72452],{"className":72453,"style":2213},[130],[23,72455,555],{"className":72456},[135,136],[23,72458,72460],{"className":72459},[140],[23,72461,54],{"className":72462},[1066,1067],[23,72464,8336],{"className":72465,"style":11398},[135,136],[23,72467,54],{"className":72468},[140],[23,72470,72472,72475],{"className":72471},[135],[23,72473,753],{"className":72474},[135,136],[23,72476,72478],{"className":72477},[998],[23,72479,72481,72501],{"className":72480},[199,200],[23,72482,72484,72498],{"className":72483},[204],[23,72485,72487],{"className":72486,"style":18647},[208],[23,72488,72489,72492],{"style":2029},[23,72490],{"className":72491,"style":1015},[216],[23,72493,72495],{"className":72494},[1019,1020,1021,1022],[23,72496,4435],{"className":72497},[135,1022],[23,72499,297],{"className":72500},[296],[23,72502,72504],{"className":72503},[204],[23,72505,72507],{"className":72506,"style":2048},[208],[23,72508],{},[23,72510,66],{"className":72511},[169],[23,72513,747],{"className":72514},[824],[23,72516],{"className":72517,"style":280},[148],[23,72519,62555],{"className":72520},[1839],[23,72522],{"className":72523,"style":280},[148],[23,72525,747],{"className":72526},[824],[23,72528],{"className":72529,"style":280},[148],[23,72531,8336],{"className":72532,"style":11398},[135,136],[23,72534,54],{"className":72535},[140],[23,72537,72539,72542],{"className":72538},[135],[23,72540,753],{"className":72541},[135,136],[23,72543,72545],{"className":72544},[998],[23,72546,72548,72568],{"className":72547},[199,200],[23,72549,72551,72565],{"className":72550},[204],[23,72552,72554],{"className":72553,"style":6499},[208],[23,72555,72556,72559],{"style":2029},[23,72557],{"className":72558,"style":1015},[216],[23,72560,72562],{"className":72561},[1019,1020,1021,1022],[23,72563,4806],{"className":72564},[135,136,1022],[23,72566,297],{"className":72567},[296],[23,72569,72571],{"className":72570},[204],[23,72572,72574],{"className":72573,"style":2048},[208],[23,72575],{},[23,72577,66],{"className":72578},[169],[23,72580,72582],{"className":72581},[169],[23,72583,66],{"className":72584},[1066,1067],[23,72586],{"className":72587,"style":392},[148],[23,72589,5106],{"className":72590},[612],[23,72592],{"className":72593,"style":392},[148],[23,72595,72597,72600,72603,72609,72612,72615,72644,72650,72653],{"className":72596},[126],[23,72598],{"className":72599,"style":17156},[130],[23,72601,68133],{"className":72602},[135],[23,72604,72606],{"className":72605},[140],[23,72607,54],{"className":72608},[1066,1067],[23,72610,20680],{"className":72611},[135],[23,72613,8336],{"className":72614,"style":11398},[135,136],[23,72616,72618,72621],{"className":72617},[135],[23,72619,20680],{"className":72620},[135],[23,72622,72624],{"className":72623},[998],[23,72625,72627],{"className":72626},[199],[23,72628,72630],{"className":72629},[204],[23,72631,72633],{"className":72632,"style":11002},[208],[23,72634,72635,72638],{"style":4989},[23,72636],{"className":72637,"style":1015},[216],[23,72639,72641],{"className":72640},[1019,1020,1021,1022],[23,72642,9952],{"className":72643},[135,1022],[23,72645,72647],{"className":72646},[169],[23,72648,66],{"className":72649},[1066,1067],[23,72651],{"className":72652,"style":149},[148],[23,72654,7378],{"className":72655},[153],[23,72657],{"className":72658},[148,29625],[23,72660,72662,72665,72668,72674,72714,72717,72757,72760,72763,72766,72769,72772,72775,72778,72818,72821,72861,72864,72870,72873,72876],{"className":72661},[126],[23,72663],{"className":72664,"style":988},[130],[23,72666,555],{"className":72667},[135,136],[23,72669,72671],{"className":72670},[140],[23,72672,54],{"className":72673},[1066,1067],[23,72675,72677,72680],{"className":72676},[135],[23,72678,8336],{"className":72679,"style":11398},[135,136],[23,72681,72683],{"className":72682},[998],[23,72684,72686,72706],{"className":72685},[199,200],[23,72687,72689,72703],{"className":72688},[204],[23,72690,72692],{"className":72691,"style":1008},[208],[23,72693,72694,72697],{"style":70243},[23,72695],{"className":72696,"style":1015},[216],[23,72698,72700],{"className":72699},[1019,1020,1021,1022],[23,72701,20680],{"className":72702},[153,1022],[23,72704,297],{"className":72705},[296],[23,72707,72709],{"className":72708},[204],[23,72710,72712],{"className":72711,"style":1057},[208],[23,72713],{},[23,72715,54],{"className":72716},[140],[23,72718,72720,72723],{"className":72719},[135],[23,72721,753],{"className":72722},[135,136],[23,72724,72726],{"className":72725},[998],[23,72727,72729,72749],{"className":72728},[199,200],[23,72730,72732,72746],{"className":72731},[204],[23,72733,72735],{"className":72734,"style":18647},[208],[23,72736,72737,72740],{"style":2029},[23,72738],{"className":72739,"style":1015},[216],[23,72741,72743],{"className":72742},[1019,1020,1021,1022],[23,72744,4435],{"className":72745},[135,1022],[23,72747,297],{"className":72748},[296],[23,72750,72752],{"className":72751},[204],[23,72753,72755],{"className":72754,"style":2048},[208],[23,72756],{},[23,72758,66],{"className":72759},[169],[23,72761,747],{"className":72762},[824],[23,72764],{"className":72765,"style":280},[148],[23,72767,62555],{"className":72768},[1839],[23,72770],{"className":72771,"style":280},[148],[23,72773,747],{"className":72774},[824],[23,72776],{"className":72777,"style":280},[148],[23,72779,72781,72784],{"className":72780},[135],[23,72782,8336],{"className":72783,"style":11398},[135,136],[23,72785,72787],{"className":72786},[998],[23,72788,72790,72810],{"className":72789},[199,200],[23,72791,72793,72807],{"className":72792},[204],[23,72794,72796],{"className":72795,"style":1008},[208],[23,72797,72798,72801],{"style":70243},[23,72799],{"className":72800,"style":1015},[216],[23,72802,72804],{"className":72803},[1019,1020,1021,1022],[23,72805,20680],{"className":72806},[153,1022],[23,72808,297],{"className":72809},[296],[23,72811,72813],{"className":72812},[204],[23,72814,72816],{"className":72815,"style":1057},[208],[23,72817],{},[23,72819,54],{"className":72820},[140],[23,72822,72824,72827],{"className":72823},[135],[23,72825,753],{"className":72826},[135,136],[23,72828,72830],{"className":72829},[998],[23,72831,72833,72853],{"className":72832},[199,200],[23,72834,72836,72850],{"className":72835},[204],[23,72837,72839],{"className":72838,"style":6499},[208],[23,72840,72841,72844],{"style":2029},[23,72842],{"className":72843,"style":1015},[216],[23,72845,72847],{"className":72846},[1019,1020,1021,1022],[23,72848,4806],{"className":72849},[135,136,1022],[23,72851,297],{"className":72852},[296],[23,72854,72856],{"className":72855},[204],[23,72857,72859],{"className":72858,"style":2048},[208],[23,72860],{},[23,72862,66],{"className":72863},[169],[23,72865,72867],{"className":72866},[169],[23,72868,66],{"className":72869},[1066,1067],[23,72871],{"className":72872,"style":392},[148],[23,72874,5106],{"className":72875},[612],[23,72877],{"className":72878,"style":392},[148],[23,72880,72882,72885,72888,72894,72897,72937,72966,72972],{"className":72881},[126],[23,72883],{"className":72884,"style":71665},[130],[23,72886,68133],{"className":72887},[135],[23,72889,72891],{"className":72890},[140],[23,72892,54],{"className":72893},[1066,1067],[23,72895,20680],{"className":72896},[135],[23,72898,72900,72903],{"className":72899},[135],[23,72901,8336],{"className":72902,"style":11398},[135,136],[23,72904,72906],{"className":72905},[998],[23,72907,72909,72929],{"className":72908},[199,200],[23,72910,72912,72926],{"className":72911},[204],[23,72913,72915],{"className":72914,"style":1008},[208],[23,72916,72917,72920],{"style":70243},[23,72918],{"className":72919,"style":1015},[216],[23,72921,72923],{"className":72922},[1019,1020,1021,1022],[23,72924,20680],{"className":72925},[153,1022],[23,72927,297],{"className":72928},[296],[23,72930,72932],{"className":72931},[204],[23,72933,72935],{"className":72934,"style":1057},[208],[23,72936],{},[23,72938,72940,72943],{"className":72939},[135],[23,72941,20680],{"className":72942},[135],[23,72944,72946],{"className":72945},[998],[23,72947,72949],{"className":72948},[199],[23,72950,72952],{"className":72951},[204],[23,72953,72955],{"className":72954,"style":11002},[208],[23,72956,72957,72960],{"style":4989},[23,72958],{"className":72959,"style":1015},[216],[23,72961,72963],{"className":72962},[1019,1020,1021,1022],[23,72964,9952],{"className":72965},[135,1022],[23,72967,72969],{"className":72968},[169],[23,72970,66],{"className":72971},[1066,1067],[23,72973,112],{"className":72974},[135],[11,72976,72977,72978,73006,73007,73077,73078,73106,73107,73135,73136,73205,73206,73339],{},"所以，对任意 ",[23,72979,72981,72994],{"className":72980,"translate":27},[31],[23,72982,72984],{"className":72983},[35],[37,72985,72986],{"xmlns":39},[42,72987,72988,72992],{},[45,72989,72990],{},[48,72991,8336],{},[114,72993,8336],{"encoding":116},[23,72995,72997],{"className":72996,"ariaHidden":122},[121],[23,72998,73000,73003],{"className":72999},[126],[23,73001],{"className":73002,"style":5414},[130],[23,73004,8336],{"className":73005,"style":11398},[135,136],"，它的投影 ",[23,73008,73010,73028],{"className":73009,"translate":27},[31],[23,73011,73013],{"className":73012},[35],[37,73014,73015],{"xmlns":39},[42,73016,73017,73025],{},[45,73018,73019],{},[898,73020,73021,73023],{},[48,73022,8336],{},[51,73024,20680],{},[114,73026,73027],{"encoding":116},"f_\\parallel",[23,73029,73031],{"className":73030,"ariaHidden":122},[121],[23,73032,73034,73037],{"className":73033},[126],[23,73035],{"className":73036,"style":70222},[130],[23,73038,73040,73043],{"className":73039},[135],[23,73041,8336],{"className":73042,"style":11398},[135,136],[23,73044,73046],{"className":73045},[998],[23,73047,73049,73069],{"className":73048},[199,200],[23,73050,73052,73066],{"className":73051},[204],[23,73053,73055],{"className":73054,"style":1008},[208],[23,73056,73057,73060],{"style":70243},[23,73058],{"className":73059,"style":1015},[216],[23,73061,73063],{"className":73062},[1019,1020,1021,1022],[23,73064,20680],{"className":73065},[153,1022],[23,73067,297],{"className":73068},[296],[23,73070,73072],{"className":73071},[204],[23,73073,73075],{"className":73074,"style":1057},[208],[23,73076],{}," 的目标值都不比 ",[23,73079,73081,73094],{"className":73080,"translate":27},[31],[23,73082,73084],{"className":73083},[35],[37,73085,73086],{"xmlns":39},[42,73087,73088,73092],{},[45,73089,73090],{},[48,73091,8336],{},[114,73093,8336],{"encoding":116},[23,73095,73097],{"className":73096,"ariaHidden":122},[121],[23,73098,73100,73103],{"className":73099},[126],[23,73101],{"className":73102,"style":5414},[130],[23,73104,8336],{"className":73105,"style":11398},[135,136]," 差。若 ",[23,73108,73110,73123],{"className":73109,"translate":27},[31],[23,73111,73113],{"className":73112},[35],[37,73114,73115],{"xmlns":39},[42,73116,73117,73121],{},[45,73118,73119],{},[48,73120,8336],{},[114,73122,8336],{"encoding":116},[23,73124,73126],{"className":73125,"ariaHidden":122},[121],[23,73127,73129,73132],{"className":73128},[126],[23,73130],{"className":73131,"style":5414},[130],[23,73133,8336],{"className":73134,"style":11398},[135,136]," 是最优解，则 ",[23,73137,73139,73156],{"className":73138,"translate":27},[31],[23,73140,73142],{"className":73141},[35],[37,73143,73144],{"xmlns":39},[42,73145,73146,73154],{},[45,73147,73148],{},[898,73149,73150,73152],{},[48,73151,8336],{},[51,73153,20680],{},[114,73155,73027],{"encoding":116},[23,73157,73159],{"className":73158,"ariaHidden":122},[121],[23,73160,73162,73165],{"className":73161},[126],[23,73163],{"className":73164,"style":70222},[130],[23,73166,73168,73171],{"className":73167},[135],[23,73169,8336],{"className":73170,"style":11398},[135,136],[23,73172,73174],{"className":73173},[998],[23,73175,73177,73197],{"className":73176},[199,200],[23,73178,73180,73194],{"className":73179},[204],[23,73181,73183],{"className":73182,"style":1008},[208],[23,73184,73185,73188],{"style":70243},[23,73186],{"className":73187,"style":1015},[216],[23,73189,73191],{"className":73190},[1019,1020,1021,1022],[23,73192,20680],{"className":73193},[153,1022],[23,73195,297],{"className":73196},[296],[23,73198,73200],{"className":73199},[204],[23,73201,73203],{"className":73202,"style":1057},[208],[23,73204],{}," 也是最优解，且 ",[23,73207,73209,73235],{"className":73208,"translate":27},[31],[23,73210,73212],{"className":73211},[35],[37,73213,73214],{"xmlns":39},[42,73215,73216,73232],{},[45,73217,73218,73224,73226],{},[898,73219,73220,73222],{},[48,73221,8336],{},[51,73223,20680],{},[51,73225,372],{},[898,73227,73228,73230],{},[48,73229,26249],{"mathvariant":375},[2581,73231,1532],{},[114,73233,73234],{"encoding":116},"f_\\parallel\\in\\mathcal H_0",[23,73236,73238,73293],{"className":73237,"ariaHidden":122},[121],[23,73239,73241,73244,73284,73287,73290],{"className":73240},[126],[23,73242],{"className":73243,"style":70222},[130],[23,73245,73247,73250],{"className":73246},[135],[23,73248,8336],{"className":73249,"style":11398},[135,136],[23,73251,73253],{"className":73252},[998],[23,73254,73256,73276],{"className":73255},[199,200],[23,73257,73259,73273],{"className":73258},[204],[23,73260,73262],{"className":73261,"style":1008},[208],[23,73263,73264,73267],{"style":70243},[23,73265],{"className":73266,"style":1015},[216],[23,73268,73270],{"className":73269},[1019,1020,1021,1022],[23,73271,20680],{"className":73272},[153,1022],[23,73274,297],{"className":73275},[296],[23,73277,73279],{"className":73278},[204],[23,73280,73282],{"className":73281,"style":1057},[208],[23,73283],{},[23,73285],{"className":73286,"style":149},[148],[23,73288,372],{"className":73289},[153],[23,73291],{"className":73292,"style":149},[148],[23,73294,73296,73299],{"className":73295},[126],[23,73297],{"className":73298,"style":42827},[130],[23,73300,73302,73305],{"className":73301},[135],[23,73303,26249],{"className":73304,"style":45024},[135,412],[23,73306,73308],{"className":73307},[998],[23,73309,73311,73331],{"className":73310},[199,200],[23,73312,73314,73328],{"className":73313},[204],[23,73315,73317],{"className":73316,"style":18647},[208],[23,73318,73319,73322],{"style":61627},[23,73320],{"className":73321,"style":1015},[216],[23,73323,73325],{"className":73324},[1019,1020,1021,1022],[23,73326,1532],{"className":73327},[135,1022],[23,73329,297],{"className":73330},[296],[23,73332,73334],{"className":73333},[204],[23,73335,73337],{"className":73336,"style":2048},[208],[23,73338],{},"。这就证明了第 1 条。",[11,73341,73342,73343,73371,73372,46543,73497,73670,73671,73699,73700,73792,73793,73863,73864],{},"若 ",[23,73344,73346,73359],{"className":73345,"translate":27},[31],[23,73347,73349],{"className":73348},[35],[37,73350,73351],{"xmlns":39},[42,73352,73353,73357],{},[45,73354,73355],{},[48,73356,68133],{"mathvariant":111},[114,73358,69091],{"encoding":116},[23,73360,73362],{"className":73361,"ariaHidden":122},[121],[23,73363,73365,73368],{"className":73364},[126],[23,73366],{"className":73367,"style":470},[130],[23,73369,68133],{"className":73370},[135]," 严格递增且 ",[23,73373,73375,73397],{"className":73374,"translate":27},[31],[23,73376,73378],{"className":73377},[35],[37,73379,73380],{"xmlns":39},[42,73381,73382,73394],{},[45,73383,73384,73390,73392],{},[898,73385,73386,73388],{},[48,73387,8336],{},[51,73389,70151],{},[51,73391,9015],{"mathvariant":111},[2581,73393,1532],{},[114,73395,73396],{"encoding":116},"f_\\perp\\neq 0",[23,73398,73400,73488],{"className":73399,"ariaHidden":122},[121],[23,73401,73403,73406,73446,73449,73485],{"className":73402},[126],[23,73404],{"className":73405,"style":5414},[130],[23,73407,73409,73412],{"className":73408},[135],[23,73410,8336],{"className":73411,"style":11398},[135,136],[23,73413,73415],{"className":73414},[998],[23,73416,73418,73438],{"className":73417},[199,200],[23,73419,73421,73435],{"className":73420},[204],[23,73422,73424],{"className":73423,"style":34192},[208],[23,73425,73426,73429],{"style":29546},[23,73427],{"className":73428,"style":1015},[216],[23,73430,73432],{"className":73431},[1019,1020,1021,1022],[23,73433,70151],{"className":73434},[153,1022],[23,73436,297],{"className":73437},[296],[23,73439,73441],{"className":73440},[204],[23,73442,73444],{"className":73443,"style":2048},[208],[23,73445],{},[23,73447],{"className":73448,"style":149},[148],[23,73450,73452,73479,73482],{"className":73451},[153],[23,73453,73455],{"className":73454},[153],[23,73456,73458],{"className":73457},[135,9093],[23,73459,73461],{"className":73460},[9097],[23,73462,73464,73467,73476],{"className":73463},[9101],[23,73465],{"className":73466,"style":5414},[130],[23,73468,73470],{"className":73469},[9108],[23,73471,73473],{"className":73472},[135],[23,73474,9115],{"className":73475},[153],[23,73477],{"className":73478},[9119],[23,73480],{"className":73481},[148,9123],[23,73483,69],{"className":73484},[153],[23,73486],{"className":73487,"style":149},[148],[23,73489,73491,73494],{"className":73490},[126],[23,73492],{"className":73493,"style":2770},[130],[23,73495,1532],{"className":73496},[135],[23,73498,73500,73538],{"className":73499,"translate":27},[31],[23,73501,73503],{"className":73502},[35],[37,73504,73505],{"xmlns":39},[42,73506,73507,73535],{},[45,73508,73509,73511,73513,73519,73521,73523,73529],{},[48,73510,20680],{"mathvariant":111},[48,73512,8336],{},[4518,73514,73515,73517],{},[48,73516,20680],{"mathvariant":111},[2581,73518,9952],{},[51,73520,2579],{},[48,73522,20680],{"mathvariant":111},[898,73524,73525,73527],{},[48,73526,8336],{},[51,73528,20680],{},[4518,73530,73531,73533],{},[48,73532,20680],{"mathvariant":111},[2581,73534,9952],{},[114,73536,73537],{"encoding":116},"\\|f\\|^2>\\|f_\\parallel\\|^2",[23,73539,73541,73591],{"className":73540,"ariaHidden":122},[121],[23,73542,73544,73547,73550,73553,73582,73585,73588],{"className":73543},[126],[23,73545],{"className":73546,"style":10661},[130],[23,73548,20680],{"className":73549},[135],[23,73551,8336],{"className":73552,"style":11398},[135,136],[23,73554,73556,73559],{"className":73555},[135],[23,73557,20680],{"className":73558},[135],[23,73560,73562],{"className":73561},[998],[23,73563,73565],{"className":73564},[199],[23,73566,73568],{"className":73567},[204],[23,73569,73571],{"className":73570,"style":4693},[208],[23,73572,73573,73576],{"style":4696},[23,73574],{"className":73575,"style":1015},[216],[23,73577,73579],{"className":73578},[1019,1020,1021,1022],[23,73580,9952],{"className":73581},[135,1022],[23,73583],{"className":73584,"style":149},[148],[23,73586,2579],{"className":73587},[153],[23,73589],{"className":73590,"style":149},[148],[23,73592,73594,73598,73601,73641],{"className":73593},[126],[23,73595],{"className":73596,"style":73597},[130],"height:1.1693em;vertical-align:-0.3552em;",[23,73599,20680],{"className":73600},[135],[23,73602,73604,73607],{"className":73603},[135],[23,73605,8336],{"className":73606,"style":11398},[135,136],[23,73608,73610],{"className":73609},[998],[23,73611,73613,73633],{"className":73612},[199,200],[23,73614,73616,73630],{"className":73615},[204],[23,73617,73619],{"className":73618,"style":1008},[208],[23,73620,73621,73624],{"style":70243},[23,73622],{"className":73623,"style":1015},[216],[23,73625,73627],{"className":73626},[1019,1020,1021,1022],[23,73628,20680],{"className":73629},[153,1022],[23,73631,297],{"className":73632},[296],[23,73634,73636],{"className":73635},[204],[23,73637,73639],{"className":73638,"style":1057},[208],[23,73640],{},[23,73642,73644,73647],{"className":73643},[135],[23,73645,20680],{"className":73646},[135],[23,73648,73650],{"className":73649},[998],[23,73651,73653],{"className":73652},[199],[23,73654,73656],{"className":73655},[204],[23,73657,73659],{"className":73658,"style":4693},[208],[23,73660,73661,73664],{"style":4696},[23,73662],{"className":73663,"style":1015},[216],[23,73665,73667],{"className":73666},[1019,1020,1021,1022],[23,73668,9952],{"className":73669},[135,1022],"，上面的不等式是严格的，",[23,73672,73674,73687],{"className":73673,"translate":27},[31],[23,73675,73677],{"className":73676},[35],[37,73678,73679],{"xmlns":39},[42,73680,73681,73685],{},[45,73682,73683],{},[48,73684,8336],{},[114,73686,8336],{"encoding":116},[23,73688,73690],{"className":73689,"ariaHidden":122},[121],[23,73691,73693,73696],{"className":73692},[126],[23,73694],{"className":73695,"style":5414},[130],[23,73697,8336],{"className":73698,"style":11398},[135,136]," 不可能是最优解。因此每个最优解都必须满足 ",[23,73701,73703,73725],{"className":73702,"translate":27},[31],[23,73704,73706],{"className":73705},[35],[37,73707,73708],{"xmlns":39},[42,73709,73710,73722],{},[45,73711,73712,73718,73720],{},[898,73713,73714,73716],{},[48,73715,8336],{},[51,73717,70151],{},[51,73719,69],{},[2581,73721,1532],{},[114,73723,73724],{"encoding":116},"f_\\perp=0",[23,73726,73728,73783],{"className":73727,"ariaHidden":122},[121],[23,73729,73731,73734,73774,73777,73780],{"className":73730},[126],[23,73732],{"className":73733,"style":5414},[130],[23,73735,73737,73740],{"className":73736},[135],[23,73738,8336],{"className":73739,"style":11398},[135,136],[23,73741,73743],{"className":73742},[998],[23,73744,73746,73766],{"className":73745},[199,200],[23,73747,73749,73763],{"className":73748},[204],[23,73750,73752],{"className":73751,"style":34192},[208],[23,73753,73754,73757],{"style":29546},[23,73755],{"className":73756,"style":1015},[216],[23,73758,73760],{"className":73759},[1019,1020,1021,1022],[23,73761,70151],{"className":73762},[153,1022],[23,73764,297],{"className":73765},[296],[23,73767,73769],{"className":73768},[204],[23,73770,73772],{"className":73771,"style":2048},[208],[23,73773],{},[23,73775],{"className":73776,"style":149},[148],[23,73778,69],{"className":73779},[153],[23,73781],{"className":73782,"style":149},[148],[23,73784,73786,73789],{"className":73785},[126],[23,73787],{"className":73788,"style":2770},[130],[23,73790,1532],{"className":73791},[135],"，即落在 ",[23,73794,73796,73814],{"className":73795,"translate":27},[31],[23,73797,73799],{"className":73798},[35],[37,73800,73801],{"xmlns":39},[42,73802,73803,73811],{},[45,73804,73805],{},[898,73806,73807,73809],{},[48,73808,26249],{"mathvariant":375},[2581,73810,1532],{},[114,73812,73813],{"encoding":116},"\\mathcal H_0",[23,73815,73817],{"className":73816,"ariaHidden":122},[121],[23,73818,73820,73823],{"className":73819},[126],[23,73821],{"className":73822,"style":42827},[130],[23,73824,73826,73829],{"className":73825},[135],[23,73827,26249],{"className":73828,"style":45024},[135,412],[23,73830,73832],{"className":73831},[998],[23,73833,73835,73855],{"className":73834},[199,200],[23,73836,73838,73852],{"className":73837},[204],[23,73839,73841],{"className":73840,"style":18647},[208],[23,73842,73843,73846],{"style":61627},[23,73844],{"className":73845,"style":1015},[216],[23,73847,73849],{"className":73848},[1019,1020,1021,1022],[23,73850,1532],{"className":73851},[135,1022],[23,73853,297],{"className":73854},[296],[23,73856,73858],{"className":73857},[204],[23,73859,73861],{"className":73860,"style":2048},[208],[23,73862],{}," 中。第 2 条得证。",[23,73865,73867,73880],{"className":73866,"translate":27},[31],[23,73868,73870],{"className":73869},[35],[37,73871,73872],{"xmlns":39},[42,73873,73874,73878],{},[45,73875,73876],{},[48,73877,8247],{"mathvariant":111},[114,73879,8250],{"encoding":116},[23,73881,73883],{"className":73882,"ariaHidden":122},[121],[23,73884,73886,73889],{"className":73885},[126],[23,73887],{"className":73888,"style":8260},[130],[23,73890,8247],{"className":73891},[135,8264],[4079,73893,73894],{},[4082,73895],{"height":73896,"width":73897,"style":4086,"data-d2-theme":4087,"alt":73898,"src":73899},277,760,"表示定理的证明结构：正交分解与有限维归约","data:image\u002Fsvg+xml;base64,<svg width="1415" height="277" xmlns="http://www.w3.org/2000/svg" data-d2-version="v0.9.0" viewBox="0 0 1415 277"><svg width="1415" height="277" class="d2-3061015132 d2-svg" viewBox="-1 -1 1415 277"><rect width="1415" height="277" x="-1" y="-1" stroke-width="0" rx="0" style="fill:#fff"/><style>.d2-3061015132 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.fill-N1{fill:#0a0f25}.d2-3061015132 .fill-B6{fill:#f7f8fe}.d2-3061015132 .stroke-B1{stroke:#0d32b2}</style><g class="QQ=="><g class="shape"><path fill="#f7f8fe" stroke="#0d32b2" d="M0 100h113v66H0z" class="stroke-B1 fill-B6" style="stroke-width:2"/></g><text x="56.5" y="138.5" fill="#0a0f25" class="text-bold fill-N1" style="text-anchor:middle;font-size:16px">f = f∥ + f⊥</text></g><g class="Qg=="><g class="shape"><path fill="#f7f8fe" stroke="#0d32b2" d="M213 173h100v66H213z" class="stroke-B1 fill-B6" style="stroke-width:2"/></g><text x="263" y="211.5" fill="#0a0f25" class="text-bold fill-N1" style="text-anchor:middle;font-size:16px">f∥ ∈ H₀</text></g><g class="Qw=="><g class="shape"><path fill="#f7f8fe" stroke="#0d32b2" d="M429 0h111v66H429z" class="stroke-B1 fill-B6" style="stroke-width:2"/></g><text x="484.5" y="38.5" fill="#0a0f25" class="text-bold fill-N1" style="text-anchor:middle;font-size:16px">f⊥ ∈ H₀⊥</text></g><g class="RA=="><g class="shape"><path fill="#f7f8fe" stroke="#0d32b2" d="M413 209h142v66H413z" class="stroke-B1 fill-B6" style="stroke-width:2"/></g><text x="484" y="247.5" fill="#0a0f25" class="text-bold fill-N1" style="text-anchor:middle;font-size:16px">f(xᵢ) = f∥(xᵢ)</text></g><g class="RQ=="><g class="shape"><path fill="#f7f8fe" stroke="#0d32b2" d="M655 37h197v66H655z" class="stroke-B1 fill-B6" style="stroke-width:2"/></g><text x="753.5" y="75.5" fill="#0a0f25" class="text-bold fill-N1" style="text-anchor:middle;font-size:16px">‖f‖² = ‖f∥‖² + ‖f⊥‖²</text></g><g class="Rg=="><g class="shape"><path fill="#f7f8fe" stroke="#0d32b2" d="M682 163h144v66H682z" class="stroke-B1 fill-B6" style="stroke-width:2"/></g><text x="754" y="201.5" fill="#0a0f25" class="text-bold fill-N1" style="text-anchor:middle;font-size:16px">损失只看 f∥</text></g><g class="Rw=="><g class="shape"><path fill="#f7f8fe" stroke="#0d32b2" d="M952 100h200v66H952z" class="stroke-B1 fill-B6" style="stroke-width:2"/></g><text x="1052" y="138.5" fill="#0a0f25" class="text-bold 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是核矩阵，",[23,74752,74754,74797],{"className":74753,"translate":27},[31],[23,74755,74757],{"className":74756},[35],[37,74758,74759],{"xmlns":39},[42,74760,74761,74795],{},[45,74762,74763,74773,74775,74777,74779,74785,74787,74793],{},[898,74764,74765,74767],{},[48,74766,26918],{},[45,74768,74769,74771],{},[48,74770,33493],{},[48,74772,35518],{},[51,74774,69],{},[48,74776,4836],{},[51,74778,54],{"stretchy":53},[898,74780,74781,74783],{},[48,74782,753],{},[48,74784,33493],{},[51,74786,747],{"separator":122},[898,74788,74789,74791],{},[48,74790,753],{},[48,74792,35518],{},[51,74794,66],{"stretchy":53},[114,74796,63334],{"encoding":116},[23,74798,74800,74858],{"className":74799,"ariaHidden":122},[121],[23,74801,74803,74806,74849,74852,74855],{"className":74802},[126],[23,74804],{"className":74805,"style":63344},[130],[23,74807,74809,74812],{"className":74808},[135],[23,74810,26918],{"className":74811,"style":37141},[135,136],[23,74813,74815],{"className":74814},[998],[23,74816,74818,74841],{"className":74817},[199,200],[23,74819,74821,74838],{"className":74820},[204],[23,74822,74824],{"className":74823,"style":33647},[208],[23,74825,74826,74829],{"style":63365},[23,74827],{"className":74828,"style":1015},[216],[23,74830,74832],{"className":74831},[1019,1020,1021,1022],[23,74833,74835],{"className":74834},[135,1022],[23,74836,63378],{"className":74837,"style":35603},[135,136,1022],[23,74839,297],{"className":74840},[296],[23,74842,74844],{"className":74843},[204],[23,74845,74847],{"className":74846,"style":14292},[208],[23,74848],{},[23,74850],{"className":74851,"style":149},[148],[23,74853,69],{"className":74854},[153],[23,74856],{"className":74857,"style":149},[148],[23,74859,74861,74864,74867,74870,74910,74913,74916,74956],{"className":74860},[126],[23,74862],{"className":74863,"style":14253},[130],[23,74865,4836],{"className":74866,"style":4851},[135,136],[23,74868,54],{"className":74869},[140],[23,74871,74873,74876],{"className":74872},[135],[23,74874,753],{"className":74875},[135,136],[23,74877,74879],{"className":74878},[998],[23,74880,74882,74902],{"className":74881},[199,200],[23,74883,74885,74899],{"className":74884},[204],[23,74886,74888],{"className":74887,"style":33647},[208],[23,74889,74890,74893],{"style":2029},[23,74891],{"className":74892,"style":1015},[216],[23,74894,74896],{"className":74895},[1019,1020,1021,1022],[23,74897,33493],{"className":74898},[135,136,1022],[23,74900,297],{"className":74901},[296],[23,74903,74905],{"className":74904},[204],[23,74906,74908],{"className":74907,"style":2048},[208],[23,74909],{},[23,74911,747],{"className":74912},[824],[23,74914],{"className":74915,"style":280},[148],[23,74917,74919,74922],{"className":74918},[135],[23,74920,753],{"className":74921},[135,136],[23,74923,74925],{"className":74924},[998],[23,74926,74928,74948],{"className":74927},[199,200],[23,74929,74931,74945],{"className":74930},[204],[23,74932,74934],{"className":74933,"style":33647},[208],[23,74935,74936,74939],{"style":2029},[23,74937],{"className":74938,"style":1015},[216],[23,74940,74942],{"className":74941},[1019,1020,1021,1022],[23,74943,35518],{"className":74944,"style":35603},[135,136,1022],[23,74946,297],{"className":74947},[296],[23,74949,74951],{"className":74950},[204],[23,74952,74954],{"className":74953,"style":14292},[208],[23,74955],{},[23,74957,66],{"className":74958},[169],"。正则项则是",[23,74961,74963],{"className":74962,"translate":27},[26],[23,74964,74966,75118],{"className":74965,"translate":27},[31],[23,74967,74969],{"className":74968},[35],[37,74970,74971],{"xmlns":39,"display":40},[42,74972,74973,75115],{},[45,74974,74975,74977,74979,74987,74989,74991,74997,75003,75005,75007,75009,75011,75017,75019,75021,75027,75033,75035,75037,75039,75041,75047,75049,75051,75053,75055,75067,75073,75079,75081,75083,75085,75091,75093,75099,75101,75103,75109,75111,75113],{},[48,74976,20680],{"mathvariant":111},[48,74978,8336],{},[19137,74980,74981,74983,74985],{},[48,74982,20680],{"mathvariant":111},[48,74984,26249],{"mathvariant":375},[2581,74986,9952],{},[51,74988,69],{},[51,74990,61181],{"fence":122,"stretchy":122,"minsize":7403,"maxsize":7403},[6961,74992,74993,74995],{},[51,74994,33851],{},[48,74996,33493],{},[898,74998,74999,75001],{},[48,75000,4526],{},[48,75002,33493],{},[48,75004,4836],{},[51,75006,54],{"stretchy":53},[51,75008,5758],{},[51,75010,747],{"separator":122},[898,75012,75013,75015],{},[48,75014,753],{},[48,75016,33493],{},[51,75018,66],{"stretchy":53},[51,75020,747],{"separator":122},[6961,75022,75023,75025],{},[51,75024,33851],{},[48,75026,35518],{},[898,75028,75029,75031],{},[48,75030,4526],{},[48,75032,35518],{},[48,75034,4836],{},[51,75036,54],{"stretchy":53},[51,75038,5758],{},[51,75040,747],{"separator":122},[898,75042,75043,75045],{},[48,75044,753],{},[48,75046,35518],{},[51,75048,66],{"stretchy":53},[51,75050,61206],{"fence":122,"stretchy":122,"minsize":7403,"maxsize":7403},[148,75052],{"linebreak":29625},[51,75054,69],{},[6961,75056,75057,75059],{},[51,75058,33851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R^n}\\left\\{L\\bigl(K\\alpha\\bigr)+\\Omega\\bigl(\\alpha^\\top K\\alpha\\bigr)\\right\\}.\n\\tag{4}",[23,75993,75995,76189],{"className":75994,"ariaHidden":122},[121],[23,75996,75998,76002,76085,76088,76183,76186],{"className":75997},[126],[23,75999],{"className":76000,"style":76001},[130],"height:1.6747em;vertical-align:-0.7756em;",[23,76003,76005],{"className":76004},[1786,7096],[23,76006,76008,76076],{"className":76007},[199,200],[23,76009,76011,76073],{"className":76010},[204],[23,76012,76014,76063],{"className":76013,"style":7106},[208],[23,76015,76017,76020],{"style":76016},"top:-2.3518em;margin-left:0em;",[23,76018],{"className":76019,"style":217},[216],[23,76021,76023],{"className":76022},[1019,1020,1021,1022],[23,76024,76026,76029,76032],{"className":76025},[135,1022],[23,76027,4526],{"className":76028,"style":4645},[135,136,1022],[23,76030,372],{"className":76031},[153,1022],[23,76033,76035,76038],{"className":76034},[135,1022],[23,76036,572],{"className":76037},[135,644,1022],[23,76039,76041],{"className":76040},[998],[23,76042,76044],{"className":76043},[199],[23,76045,76047],{"className":76046},[204],[23,76048,76051],{"className":76049,"style":76050},[208],"height:0.5935em;",[23,76052,76054,76057],{"style":76053},"top:-2.786em;margin-right:0.0714em;",[23,76055],{"className":76056,"style":24881},[216],[23,76058,76060],{"className":76059},[1019,24885,1067,1022],[23,76061,4806],{"className":76062},[135,136,1022],[23,76064,76065,76068],{"style":7130},[23,76066],{"className":76067,"style":217},[216],[23,76069,76070],{},[23,76071,6967],{"className":76072},[1786],[23,76074,297],{"className":76075},[296],[23,76077,76079],{"className":76078},[204],[23,76080,76083],{"className":76081,"style":76082},[208],"height:0.7756em;",[23,76084],{},[23,76086],{"className":76087,"style":280},[148],[23,76089,76091,76097,76100,76106,76109,76112,76118,76121,76124,76127,76130,76136,76165,76168,76171,76177],{"className":76090},[1839],[23,76092,76094],{"className":76093,"style":1844},[140,1843],[23,76095,9010],{"className":76096},[1066,1067],[23,76098,555],{"className":76099},[135,136],[23,76101,76103],{"className":76102},[140],[23,76104,54],{"className":76105},[1066,1067],[23,76107,26918],{"className":76108,"style":37141},[135,136],[23,76110,4526],{"className":76111,"style":4645},[135,136],[23,76113,76115],{"className":76114},[169],[23,76116,66],{"className":76117},[1066,1067],[23,76119],{"className":76120,"style":392},[148],[23,76122,5106],{"className":76123},[612],[23,76125],{"className":76126,"style":392},[148],[23,76128,68133],{"className":76129},[135],[23,76131,76133],{"className":76132},[140],[23,76134,54],{"className":76135},[1066,1067],[23,76137,76139,76142],{"className":76138},[135],[23,76140,4526],{"className":76141,"style":4645},[135,136],[23,76143,76145],{"className":76144},[998],[23,76146,76148],{"className":76147},[199],[23,76149,76151],{"className":76150},[204],[23,76152,76154],{"className":76153,"style":4986},[208],[23,76155,76156,76159],{"style":4989},[23,76157],{"className":76158,"style":1015},[216],[23,76160,76162],{"className":76161},[1019,1020,1021,1022],[23,76163,22471],{"className":76164},[135,1022],[23,76166,26918],{"className":76167,"style":37141},[135,136],[23,76169,4526],{"className":76170,"style":4645},[135,136],[23,76172,76174],{"className":76173},[169],[23,76175,66],{"className":76176},[1066,1067],[23,76178,76180],{"className":76179,"style":1844},[169,1843],[23,76181,9020],{"className":76182},[1066,1067],[23,76184],{"className":76185,"style":280},[148],[23,76187,112],{"className":76188},[135],[23,76190,76192,76195],{"className":76191},[62174],[23,76193],{"className":76194,"style":76001},[130],[23,76196,76198,76201,76207],{"className":76197},[135,6514],[23,76199,54],{"className":76200},[135],[23,76202,76204],{"className":76203},[135],[23,76205,24930],{"className":76206},[135],[23,76208,66],{"className":76209},[135],[11,76211,76212,76213,76241,76242,76270,76271,76299,76300,76328,76329,76380,76381,76409,76410,76439,76440,76468],{},"这就是表示定理的全部实用价值：原本在无穷维空间里找 ",[23,76214,76216,76229],{"className":76215,"translate":27},[31],[23,76217,76219],{"className":76218},[35],[37,76220,76221],{"xmlns":39},[42,76222,76223,76227],{},[45,76224,76225],{},[48,76226,8336],{},[114,76228,8336],{"encoding":116},[23,76230,76232],{"className":76231,"ariaHidden":122},[121],[23,76233,76235,76238],{"className":76234},[126],[23,76236],{"className":76237,"style":5414},[130],[23,76239,8336],{"className":76240,"style":11398},[135,136],"，现在只需找 ",[23,76243,76245,76258],{"className":76244,"translate":27},[31],[23,76246,76248],{"className":76247},[35],[37,76249,76250],{"xmlns":39},[42,76251,76252,76256],{},[45,76253,76254],{},[48,76255,4806],{},[114,76257,4806],{"encoding":116},[23,76259,76261],{"className":76260,"ariaHidden":122},[121],[23,76262,76264,76267],{"className":76263},[126],[23,76265],{"className":76266,"style":703},[130],[23,76268,4806],{"className":76269},[135,136]," 个系数。核矩阵 ",[23,76272,76274,76287],{"className":76273,"translate":27},[31],[23,76275,76277],{"className":76276},[35],[37,76278,76279],{"xmlns":39},[42,76280,76281,76285],{},[45,76282,76283],{},[48,76284,26918],{},[114,76286,26918],{"encoding":116},[23,76288,76290],{"className":76289,"ariaHidden":122},[121],[23,76291,76293,76296],{"className":76292},[126],[23,76294],{"className":76295,"style":470},[130],[23,76297,26918],{"className":76298,"style":37141},[135,136]," 是半正定的；若 ",[23,76301,76303,76316],{"className":76302,"translate":27},[31],[23,76304,76306],{"className":76305},[35],[37,76307,76308],{"xmlns":39},[42,76309,76310,76314],{},[45,76311,76312],{},[48,76313,26918],{},[114,76315,26918],{"encoding":116},[23,76317,76319],{"className":76318,"ariaHidden":122},[121],[23,76320,76322,76325],{"className":76321},[126],[23,76323],{"className":76324,"style":470},[130],[23,76326,26918],{"className":76327,"style":37141},[135,136]," 可逆，则 ",[23,76330,76332,76350],{"className":76331,"translate":27},[31],[23,76333,76335],{"className":76334},[35],[37,76336,76337],{"xmlns":39},[42,76338,76339,76347],{},[45,76340,76341,76343,76345],{},[48,76342,4526],{},[51,76344,756],{},[48,76346,8336],{},[114,76348,76349],{"encoding":116},"\\alpha\\mapsto f",[23,76351,76353,76371],{"className":76352,"ariaHidden":122},[121],[23,76354,76356,76359,76362,76365,76368],{"className":76355},[126],[23,76357],{"className":76358,"style":59976},[130],[23,76360,4526],{"className":76361,"style":4645},[135,136],[23,76363],{"className":76364,"style":149},[148],[23,76366,756],{"className":76367},[153],[23,76369],{"className":76370,"style":149},[148],[23,76372,76374,76377],{"className":76373},[126],[23,76375],{"className":76376,"style":5414},[130],[23,76378,8336],{"className":76379,"style":11398},[135,136]," 是一一对应；若 ",[23,76382,76384,76397],{"className":76383,"translate":27},[31],[23,76385,76387],{"className":76386},[35],[37,76388,76389],{"xmlns":39},[42,76390,76391,76395],{},[45,76392,76393],{},[48,76394,26918],{},[114,76396,26918],{"encoding":116},[23,76398,76400],{"className":76399,"ariaHidden":122},[121],[23,76401,76403,76406],{"className":76402},[126],[23,76404],{"className":76405,"style":470},[130],[23,76407,26918],{"className":76408,"style":37141},[135,136]," 奇异，不同的 ",[23,76411,76413,76427],{"className":76412,"translate":27},[31],[23,76414,76416],{"className":76415},[35],[37,76417,76418],{"xmlns":39},[42,76419,76420,76424],{},[45,76421,76422],{},[48,76423,4526],{},[114,76425,76426],{"encoding":116},"\\alpha",[23,76428,76430],{"className":76429,"ariaHidden":122},[121],[23,76431,76433,76436],{"className":76432},[126],[23,76434],{"className":76435,"style":703},[130],[23,76437,4526],{"className":76438,"style":4645},[135,136]," 可能对应同一个函数（相差一个 ",[23,76441,76443,76456],{"className":76442,"translate":27},[31],[23,76444,76446],{"className":76445},[35],[37,76447,76448],{"xmlns":39},[42,76449,76450,76454],{},[45,76451,76452],{},[48,76453,26918],{},[114,76455,26918],{"encoding":116},[23,76457,76459],{"className":76458,"ariaHidden":122},[121],[23,76460,76462,76465],{"className":76461},[126],[23,76463],{"className":76464,"style":470},[130],[23,76466,26918],{"className":76467,"style":37141},[135,136]," 的零空间向量，其范数为零），但函数本身仍然是良定义的。",[2501,76470,76472],{"id":76471},"_34-唯一性系数与偏置项","3.4 唯一性、系数与偏置项",[11,76474,76475,76476,76479,76480,76483],{},"表示定理给出的是",[18,76477,76478],{},"解的形式","，不是",[18,76481,76482],{},"解的唯一性","。以下几点容易混淆：",[346,76485,76486,76911,76976],{},[349,76487,76488,69074,76491,76519,76520,76548,76549,76608,76609,76659,76660,76688,76689,76747,76748,76881,76882,76910],{},[18,76489,76490],{},"函数唯一 vs 系数唯一。",[23,76492,76494,76507],{"className":76493,"translate":27},[31],[23,76495,76497],{"className":76496},[35],[37,76498,76499],{"xmlns":39},[42,76500,76501,76505],{},[45,76502,76503],{},[48,76504,26918],{},[114,76506,26918],{"encoding":116},[23,76508,76510],{"className":76509,"ariaHidden":122},[121],[23,76511,76513,76516],{"className":76512},[126],[23,76514],{"className":76515,"style":470},[130],[23,76517,26918],{"className":76518,"style":37141},[135,136]," 奇异，",[23,76521,76523,76536],{"className":76522,"translate":27},[31],[23,76524,76526],{"className":76525},[35],[37,76527,76528],{"xmlns":39},[42,76529,76530,76534],{},[45,76531,76532],{},[48,76533,4526],{},[114,76535,76426],{"encoding":116},[23,76537,76539],{"className":76538,"ariaHidden":122},[121],[23,76540,76542,76545],{"className":76541},[126],[23,76543],{"className":76544,"style":703},[130],[23,76546,4526],{"className":76547,"style":4645},[135,136]," 不唯一，但 ",[23,76550,76552,76570],{"className":76551,"translate":27},[31],[23,76553,76555],{"className":76554},[35],[37,76556,76557],{"xmlns":39},[42,76558,76559,76567],{},[45,76560,76561],{},[4518,76562,76563,76565],{},[48,76564,8336],{},[51,76566,6881],{},[114,76568,76569],{"encoding":116},"f^*",[23,76571,76573],{"className":76572,"ariaHidden":122},[121],[23,76574,76576,76579],{"className":76575},[126],[23,76577],{"className":76578,"style":5414},[130],[23,76580,76582,76585],{"className":76581},[135],[23,76583,8336],{"className":76584,"style":11398},[135,136],[23,76586,76588],{"className":76587},[998],[23,76589,76591],{"className":76590},[199],[23,76592,76594],{"className":76593},[204],[23,76595,76597],{"className":76596,"style":6912},[208],[23,76598,76599,76602],{"style":4696},[23,76600],{"className":76601,"style":1015},[216],[23,76603,76605],{"className":76604},[1019,1020,1021,1022],[23,76606,6881],{"className":76607},[612,1022]," 可能唯一。核岭回归在 ",[23,76610,76612,76629],{"className":76611,"translate":27},[31],[23,76613,76615],{"className":76614},[35],[37,76616,76617],{"xmlns":39},[42,76618,76619,76627],{},[45,76620,76621,76623,76625],{},[48,76622,61753],{},[51,76624,2579],{},[2581,76626,1532],{},[114,76628,69672],{"encoding":116},[23,76630,76632,76650],{"className":76631,"ariaHidden":122},[121],[23,76633,76635,76638,76641,76644,76647],{"className":76634},[126],[23,76636],{"className":76637,"style":4450},[130],[23,76639,61753],{"className":76640},[135,136],[23,76642],{"className":76643,"style":149},[148],[23,76645,2579],{"className":76646},[153],[23,76648],{"className":76649,"style":149},[148],[23,76651,76653,76656],{"className":76652},[126],[23,76654],{"className":76655,"style":2770},[130],[23,76657,1532],{"className":76658},[135]," 时就是如此：目标函数关于 ",[23,76661,76663,76676],{"className":76662,"translate":27},[31],[23,76664,76666],{"className":76665},[35],[37,76667,76668],{"xmlns":39},[42,76669,76670,76674],{},[45,76671,76672],{},[48,76673,8336],{},[114,76675,8336],{"encoding":116},[23,76677,76679],{"className":76678,"ariaHidden":122},[121],[23,76680,76682,76685],{"className":76681},[126],[23,76683],{"className":76684,"style":5414},[130],[23,76686,8336],{"className":76687,"style":11398},[135,136]," 严格凸，",[23,76690,76692,76709],{"className":76691,"translate":27},[31],[23,76693,76695],{"className":76694},[35],[37,76696,76697],{"xmlns":39},[42,76698,76699,76707],{},[45,76700,76701],{},[4518,76702,76703,76705],{},[48,76704,8336],{},[51,76706,6881],{},[114,76708,76569],{"encoding":116},[23,76710,76712],{"className":76711,"ariaHidden":122},[121],[23,76713,76715,76718],{"className":76714},[126],[23,76716],{"className":76717,"style":5414},[130],[23,76719,76721,76724],{"className":76720},[135],[23,76722,8336],{"className":76723,"style":11398},[135,136],[23,76725,76727],{"className":76726},[998],[23,76728,76730],{"className":76729},[199],[23,76731,76733],{"className":76732},[204],[23,76734,76736],{"className":76735,"style":6912},[208],[23,76737,76738,76741],{"style":4696},[23,76739],{"className":76740,"style":1015},[216],[23,76742,76744],{"className":76743},[1019,1020,1021,1022],[23,76745,6881],{"className":76746},[612,1022]," 唯一；而 ",[23,76749,76751,76789],{"className":76750,"translate":27},[31],[23,76752,76754],{"className":76753},[35],[37,76755,76756],{"xmlns":39},[42,76757,76758,76786],{},[45,76759,76760,76762,76764,76766,76768,76770,76772,76774,76784],{},[48,76761,4526],{},[51,76763,69],{},[51,76765,54],{"stretchy":53},[48,76767,26918],{},[51,76769,5106],{},[48,76771,61753],{},[48,76773,34353],{},[4518,76775,76776,76778],{},[51,76777,66],{"stretchy":53},[45,76779,76780,76782],{},[51,76781,4529],{},[2581,76783,4435],{},[48,76785,660],{},[114,76787,76788],{"encoding":116},"\\alpha=(K+\\lambda I)^{-1}y",[23,76790,76792,76810,76831],{"className":76791,"ariaHidden":122},[121],[23,76793,76795,76798,76801,76804,76807],{"className":76794},[126],[23,76796],{"className":76797,"style":703},[130],[23,76799,4526],{"className":76800,"style":4645},[135,136],[23,76802],{"className":76803,"style":149},[148],[23,76805,69],{"className":76806},[153],[23,76808],{"className":76809,"style":149},[148],[23,76811,76813,76816,76819,76822,76825,76828],{"className":76812},[126],[23,76814],{"className":76815,"style":131},[130],[23,76817,54],{"className":76818},[140],[23,76820,26918],{"className":76821,"style":37141},[135,136],[23,76823],{"className":76824,"style":392},[148],[23,76826,5106],{"className":76827},[612],[23,76829],{"className":76830,"style":392},[148],[23,76832,76834,76837,76840,76843,76878],{"className":76833},[126],[23,76835],{"className":76836,"style":10661},[130],[23,76838,61753],{"className":76839},[135,136],[23,76841,34353],{"className":76842,"style":454},[135,136],[23,76844,76846,76849],{"className":76845},[169],[23,76847,66],{"className":76848},[169],[23,76850,76852],{"className":76851},[998],[23,76853,76855],{"className":76854},[199],[23,76856,76858],{"className":76857},[204],[23,76859,76861],{"className":76860,"style":4693},[208],[23,76862,76863,76866],{"style":4696},[23,76864],{"className":76865,"style":1015},[216],[23,76867,76869],{"className":76868},[1019,1020,1021,1022],[23,76870,76872,76875],{"className":76871},[135,1022],[23,76873,4529],{"className":76874},[135,1022],[23,76876,4435],{"className":76877},[135,1022],[23,76879,660],{"className":76880,"style":676},[135,136]," 只是其中一个系数表示，所有相差 ",[23,76883,76885,76898],{"className":76884,"translate":27},[31],[23,76886,76888],{"className":76887},[35],[37,76889,76890],{"xmlns":39},[42,76891,76892,76896],{},[45,76893,76894],{},[48,76895,26918],{},[114,76897,26918],{"encoding":116},[23,76899,76901],{"className":76900,"ariaHidden":122},[121],[23,76902,76904,76907],{"className":76903},[126],[23,76905],{"className":76906,"style":470},[130],[23,76908,26918],{"className":76909,"style":37141},[135,136]," 零空间的系数给出同一个函数。",[349,76912,76913,76975],{},[18,76914,76915,76916,76974],{},"损失不严格凸时，",[23,76917,76919,76936],{"className":76918,"translate":27},[31],[23,76920,76922],{"className":76921},[35],[37,76923,76924],{"xmlns":39},[42,76925,76926,76934],{},[45,76927,76928],{},[4518,76929,76930,76932],{},[48,76931,8336],{},[51,76933,6881],{},[114,76935,76569],{"encoding":116},[23,76937,76939],{"className":76938,"ariaHidden":122},[121],[23,76940,76942,76945],{"className":76941},[126],[23,76943],{"className":76944,"style":5414},[130],[23,76946,76948,76951],{"className":76947},[135],[23,76949,8336],{"className":76950,"style":11398},[135,136],[23,76952,76954],{"className":76953},[998],[23,76955,76957],{"className":76956},[199],[23,76958,76960],{"className":76959},[204],[23,76961,76963],{"className":76962,"style":6912},[208],[23,76964,76965,76968],{"style":4696},[23,76966],{"className":76967,"style":1015},[216],[23,76969,76971],{"className":76970},[1019,1020,1021,1022],[23,76972,6881],{"className":76973},[612,1022]," 也可能不唯一。"," 例如 hinge loss 在某些数据上会有平坦区域；表示定理仍然成立，但最优解可能是一个集合。",[349,76977,76978,76981,76982,77048,77049,77077,77078,77106,77107,77352,77353,77381,77382,77385],{},[18,76979,76980],{},"偏置项要单独处理。"," 如果模型是 ",[23,76983,76985,77009],{"className":76984,"translate":27},[31],[23,76986,76988],{"className":76987},[35],[37,76989,76990],{"xmlns":39},[42,76991,76992,77006],{},[45,76993,76994,76996,76998,77000,77002,77004],{},[48,76995,8336],{},[51,76997,54],{"stretchy":53},[48,76999,753],{},[51,77001,66],{"stretchy":53},[51,77003,5106],{},[48,77005,35533],{},[114,77007,77008],{"encoding":116},"f(x)+b",[23,77010,77012,77039],{"className":77011,"ariaHidden":122},[121],[23,77013,77015,77018,77021,77024,77027,77030,77033,77036],{"className":77014},[126],[23,77016],{"className":77017,"style":131},[130],[23,77019,8336],{"className":77020,"style":11398},[135,136],[23,77022,54],{"className":77023},[140],[23,77025,753],{"className":77026},[135,136],[23,77028,66],{"className":77029},[169],[23,77031],{"className":77032,"style":392},[148],[23,77034,5106],{"className":77035},[612],[23,77037],{"className":77038,"style":392},[148],[23,77040,77042,77045],{"className":77041},[126],[23,77043],{"className":77044,"style":779},[130],[23,77046,35533],{"className":77047},[135,136],"，而 ",[23,77050,77052,77065],{"className":77051,"translate":27},[31],[23,77053,77055],{"className":77054},[35],[37,77056,77057],{"xmlns":39},[42,77058,77059,77063],{},[45,77060,77061],{},[48,77062,35533],{},[114,77064,35533],{"encoding":116},[23,77066,77068],{"className":77067,"ariaHidden":122},[121],[23,77069,77071,77074],{"className":77070},[126],[23,77072],{"className":77073,"style":779},[130],[23,77075,35533],{"className":77076},[135,136]," 不受正则项约束，那么表示定理直接作用于 ",[23,77079,77081,77094],{"className":77080,"translate":27},[31],[23,77082,77084],{"className":77083},[35],[37,77085,77086],{"xmlns":39},[42,77087,77088,77092],{},[45,77089,77090],{},[48,77091,8336],{},[114,77093,8336],{"encoding":116},[23,77095,77097],{"className":77096,"ariaHidden":122},[121],[23,77098,77100,77103],{"className":77099},[126],[23,77101],{"className":77102,"style":5414},[130],[23,77104,8336],{"className":77105,"style":11398},[135,136]," 的部分仍然给出 ",[23,77108,77110,77158],{"className":77109,"translate":27},[31],[23,77111,77113],{"className":77112},[35],[37,77114,77115],{"xmlns":39},[42,77116,77117,77155],{},[45,77118,77119,77125,77127,77133,77139,77141,77143,77145,77147,77153],{},[4518,77120,77121,77123],{},[48,77122,8336],{},[51,77124,6881],{},[51,77126,69],{},[898,77128,77129,77131],{},[51,77130,33851],{},[48,77132,33493],{},[898,77134,77135,77137],{},[48,77136,4526],{},[48,77138,33493],{},[48,77140,4836],{},[51,77142,54],{"stretchy":53},[51,77144,5758],{},[51,77146,747],{"separator":122},[898,77148,77149,77151],{},[48,77150,753],{},[48,77152,33493],{},[51,77154,66],{"stretchy":53},[114,77156,77157],{"encoding":116},"f^*=\\sum_i\\alpha_i k(\\cdot,x_i)",[23,77159,77161,77205],{"className":77160,"ariaHidden":122},[121],[23,77162,77164,77167,77196,77199,77202],{"className":77163},[126],[23,77165],{"className":77166,"style":5414},[130],[23,77168,77170,77173],{"className":77169},[135],[23,77171,8336],{"className":77172,"style":11398},[135,136],[23,77174,77176],{"className":77175},[998],[23,77177,77179],{"className":77178},[199],[23,77180,77182],{"className":77181},[204],[23,77183,77185],{"className":77184,"style":6912},[208],[23,77186,77187,77190],{"style":4696},[23,77188],{"className":77189,"style":1015},[216],[23,77191,77193],{"className":77192},[1019,1020,1021,1022],[23,77194,6881],{"className":77195},[612,1022],[23,77197],{"className":77198,"style":149},[148],[23,77200,69],{"className":77201},[153],[23,77203],{"className":77204,"style":149},[148],[23,77206,77208,77211,77251,77254,77294,77297,77300,77303,77306,77309,77349],{"className":77207},[126],[23,77209],{"className":77210,"style":33575},[130],[23,77212,77214,77217],{"className":77213},[1786],[23,77215,33851],{"className":77216,"style":33582},[1786,1790,20454],[23,77218,77220],{"className":77219},[998],[23,77221,77223,77243],{"className":77222},[199,200],[23,77224,77226,77240],{"className":77225},[204],[23,77227,77229],{"className":77228,"style":33595},[208],[23,77230,77231,77234],{"style":33598},[23,77232],{"className":77233,"style":1015},[216],[23,77235,77237],{"className":77236},[1019,1020,1021,1022],[23,77238,33493],{"className":77239},[135,136,1022],[23,77241,297],{"className":77242},[296],[23,77244,77246],{"className":77245},[204],[23,77247,77249],{"className":77248,"style":33617},[208],[23,77250],{},[23,77252],{"className":77253,"style":280},[148],[23,77255,77257,77260],{"className":77256},[135],[23,77258,4526],{"className":77259,"style":4645},[135,136],[23,77261,77263],{"className":77262},[998],[23,77264,77266,77286],{"className":77265},[199,200],[23,77267,77269,77283],{"className":77268},[204],[23,77270,77272],{"className":77271,"style":33647},[208],[23,77273,77274,77277],{"style":68963},[23,77275],{"className":77276,"style":1015},[216],[23,77278,77280],{"className":77279},[1019,1020,1021,1022],[23,77281,33493],{"className":77282},[135,136,1022],[23,77284,297],{"className":77285},[296],[23,77287,77289],{"className":77288},[204],[23,77290,77292],{"className":77291,"style":2048},[208],[23,77293],{},[23,77295,4836],{"className":77296,"style":4851},[135,136],[23,77298,54],{"className":77299},[140],[23,77301,5758],{"className":77302},[135],[23,77304,747],{"className":77305},[824],[23,77307],{"className":77308,"style":280},[148],[23,77310,77312,77315],{"className":77311},[135],[23,77313,753],{"className":77314},[135,136],[23,77316,77318],{"className":77317},[998],[23,77319,77321,77341],{"className":77320},[199,200],[23,77322,77324,77338],{"className":77323},[204],[23,77325,77327],{"className":77326,"style":33647},[208],[23,77328,77329,77332],{"style":2029},[23,77330],{"className":77331,"style":1015},[216],[23,77333,77335],{"className":77334},[1019,1020,1021,1022],[23,77336,33493],{"className":77337},[135,136,1022],[23,77339,297],{"className":77340},[296],[23,77342,77344],{"className":77343},[204],[23,77345,77347],{"className":77346,"style":2048},[208],[23,77348],{},[23,77350,66],{"className":77351},[169],"，但 ",[23,77354,77356,77369],{"className":77355,"translate":27},[31],[23,77357,77359],{"className":77358},[35],[37,77360,77361],{"xmlns":39},[42,77362,77363,77367],{},[45,77364,77365],{},[48,77366,35533],{},[114,77368,35533],{"encoding":116},[23,77370,77372],{"className":77371,"ariaHidden":122},[121],[23,77373,77375,77378],{"className":77374},[126],[23,77376],{"className":77377,"style":779},[130],[23,77379,35533],{"className":77380},[135,136]," 需要通过有限维优化或 KKT 条件单独求出。更系统的处理是",[18,77383,77384],{},"半参数表示定理","（semi-parametric representer theorem）：若模型为",[23,77387,77389],{"className":77388,"translate":27},[26],[23,77390,77392,77468],{"className":77391,"translate":27},[31],[23,77393,77395],{"className":77394},[35],[37,77396,77397],{"xmlns":39,"display":40},[42,77398,77399,77465],{},[45,77400,77401,77403,77405,77407,77409,77411,77425,77431,77437,77439,77441,77443,77445,77447,77449,77451,77453,77455,77457,77459,77461,77463],{},[48,77402,8336],{},[51,77404,54],{"stretchy":53},[48,77406,753],{},[51,77408,66],{"stretchy":53},[51,77410,69],{},[61709,77412,77413,77415,77423],{},[51,77414,33851],{},[45,77416,77417,77419,77421],{},[48,77418,35518],{},[51,77420,69],{},[2581,77422,4435],{},[48,77424,11],{},[898,77426,77427,77429],{},[48,77428,4548],{},[48,77430,35518],{},[898,77432,77433,77435],{},[48,77434,18363],{},[48,77436,35518],{},[51,77438,54],{"stretchy":53},[48,77440,753],{},[51,77442,66],{"stretchy":53},[51,77444,5106],{},[48,77446,7089],{},[51,77448,54],{"stretchy":53},[48,77450,753],{},[51,77452,66],{"stretchy":53},[51,77454,747],{"separator":122},[148,77456],{"width":750},[48,77458,7089],{},[51,77460,372],{},[48,77462,26249],{"mathvariant":375},[51,77464,747],{"separator":122},[114,77466,77467],{"encoding":116},"f(x)=\\sum_{j=1}^p\\beta_j\\phi_j(x)+g(x),\\qquad g\\in\\mathcal H,",[23,77469,77471,77498,77673,77712],{"className":77470,"ariaHidden":122},[121],[23,77472,77474,77477,77480,77483,77486,77489,77492,77495],{"className":77473},[126],[23,77475],{"className":77476,"style":131},[130],[23,77478,8336],{"className":77479,"style":11398},[135,136],[23,77481,54],{"className":77482},[140],[23,77484,753],{"className":77485},[135,136],[23,77487,66],{"className":77488},[169],[23,77490],{"className":77491,"style":149},[148],[23,77493,69],{"className":77494},[153],[23,77496],{"className":77497,"style":149},[148],[23,77499,77501,77505,77571,77574,77615,77655,77658,77661,77664,77667,77670],{"className":77500},[126],[23,77502],{"className":77503,"style":77504},[130],"height:3.1123em;vertical-align:-1.4138em;",[23,77506,77508],{"className":77507},[1786,7096],[23,77509,77511,77563],{"className":77510},[199,200],[23,77512,77514,77560],{"className":77513},[204],[23,77515,77518,77538,77548],{"className":77516,"style":77517},[208],"height:1.6985em;",[23,77519,77520,77523],{"style":34075},[23,77521],{"className":77522,"style":34079},[216],[23,77524,77526],{"className":77525},[1019,1020,1021,1022],[23,77527,77529,77532,77535],{"className":77528},[135,1022],[23,77530,35518],{"className":77531,"style":35603},[135,136,1022],[23,77533,69],{"className":77534},[153,1022],[23,77536,4435],{"className":77537},[135,1022],[23,77539,77540,77543],{"style":34088},[23,77541],{"className":77542,"style":34079},[216],[23,77544,77545],{},[23,77546,33851],{"className":77547},[1786,1790,1791],[23,77549,77551,77554],{"style":77550},"top:-4.3471em;margin-left:0em;",[23,77552],{"className":77553,"style":34079},[216],[23,77555,77557],{"className":77556},[1019,1020,1021,1022],[23,77558,11],{"className":77559},[135,136,1022],[23,77561,297],{"className":77562},[296],[23,77564,77566],{"className":77565},[204],[23,77567,77569],{"className":77568,"style":63083},[208],[23,77570],{},[23,77572],{"className":77573,"style":280},[148],[23,77575,77577,77580],{"className":77576},[135],[23,77578,4548],{"className":77579,"style":4655},[135,136],[23,77581,77583],{"className":77582},[998],[23,77584,77586,77607],{"className":77585},[199,200],[23,77587,77589,77604],{"className":77588},[204],[23,77590,77592],{"className":77591,"style":33647},[208],[23,77593,77595,77598],{"style":77594},"top:-2.55em;margin-left:-0.0528em;margin-right:0.05em;",[23,77596],{"className":77597,"style":1015},[216],[23,77599,77601],{"className":77600},[1019,1020,1021,1022],[23,77602,35518],{"className":77603,"style":35603},[135,136,1022],[23,77605,297],{"className":77606},[296],[23,77608,77610],{"className":77609},[204],[23,77611,77613],{"className":77612,"style":14292},[208],[23,77614],{},[23,77616,77618,77621],{"className":77617},[135],[23,77619,18363],{"className":77620},[135,136],[23,77622,77624],{"className":77623},[998],[23,77625,77627,77647],{"className":77626},[199,200],[23,77628,77630,77644],{"className":77629},[204],[23,77631,77633],{"className":77632,"style":33647},[208],[23,77634,77635,77638],{"style":2029},[23,77636],{"className":77637,"style":1015},[216],[23,77639,77641],{"className":77640},[1019,1020,1021,1022],[23,77642,35518],{"className":77643,"style":35603},[135,136,1022],[23,77645,297],{"className":77646},[296],[23,77648,77650],{"className":77649},[204],[23,77651,77653],{"className":77652,"style":14292},[208],[23,77654],{},[23,77656,54],{"className":77657},[140],[23,77659,753],{"className":77660},[135,136],[23,77662,66],{"className":77663},[169],[23,77665],{"className":77666,"style":392},[148],[23,77668,5106],{"className":77669},[612],[23,77671],{"className":77672,"style":392},[148],[23,77674,77676,77679,77682,77685,77688,77691,77694,77697,77700,77703,77706,77709],{"className":77675},[126],[23,77677],{"className":77678,"style":131},[130],[23,77680,7089],{"className":77681,"style":676},[135,136],[23,77683,54],{"className":77684},[140],[23,77686,753],{"className":77687},[135,136],[23,77689,66],{"className":77690},[169],[23,77692,747],{"className":77693},[824],[23,77695],{"className":77696,"style":828},[148],[23,77698],{"className":77699,"style":280},[148],[23,77701,7089],{"className":77702,"style":676},[135,136],[23,77704],{"className":77705,"style":149},[148],[23,77707,372],{"className":77708},[153],[23,77710],{"className":77711,"style":149},[148],[23,77713,77715,77718,77721],{"className":77714},[126],[23,77716],{"className":77717,"style":817},[130],[23,77719,26249],{"className":77720,"style":45024},[135,412],[23,77722,747],{"className":77723},[824],[11,77725,24429,77726,77867,77868,77967,77968,77996,77997,78026],{},[23,77727,77729,77759],{"className":77728,"translate":27},[31],[23,77730,77732],{"className":77731},[35],[37,77733,77734],{"xmlns":39},[42,77735,77736,77756],{},[45,77737,77738,77744,77746,77748,77750],{},[898,77739,77740,77742],{},[48,77741,18363],{},[2581,77743,4435],{},[51,77745,747],{"separator":122},[51,77747,62555],{},[51,77749,747],{"separator":122},[898,77751,77752,77754],{},[48,77753,18363],{},[48,77755,11],{},[114,77757,77758],{"encoding":116},"\\phi_1,\\dots,\\phi_p",[23,77760,77762],{"className":77761,"ariaHidden":122},[121],[23,77763,77765,77769,77809,77812,77815,77818,77821,77824,77827],{"className":77764},[126],[23,77766],{"className":77767,"style":77768},[130],"height:0.9805em;vertical-align:-0.2861em;",[23,77770,77772,77775],{"className":77771},[135],[23,77773,18363],{"className":77774},[135,136],[23,77776,77778],{"className":77777},[998],[23,77779,77781,77801],{"className":77780},[199,200],[23,77782,77784,77798],{"className":77783},[204],[23,77785,77787],{"className":77786,"style":18647},[208],[23,77788,77789,77792],{"style":2029},[23,77790],{"className":77791,"style":1015},[216],[23,77793,77795],{"className":77794},[1019,1020,1021,1022],[23,77796,4435],{"className":77797},[135,1022],[23,77799,297],{"className":77800},[296],[23,77802,77804],{"className":77803},[204],[23,77805,77807],{"className":77806,"style":2048},[208],[23,77808],{},[23,77810,747],{"className":77811},[824],[23,77813],{"className":77814,"style":280},[148],[23,77816,62555],{"className":77817},[1839],[23,77819],{"className":77820,"style":280},[148],[23,77822,747],{"className":77823},[824],[23,77825],{"className":77826,"style":280},[148],[23,77828,77830,77833],{"className":77829},[135],[23,77831,18363],{"className":77832},[135,136],[23,77834,77836],{"className":77835},[998],[23,77837,77839,77859],{"className":77838},[199,200],[23,77840,77842,77856],{"className":77841},[204],[23,77843,77845],{"className":77844,"style":6499},[208],[23,77846,77847,77850],{"style":2029},[23,77848],{"className":77849,"style":1015},[216],[23,77851,77853],{"className":77852},[1019,1020,1021,1022],[23,77854,11],{"className":77855},[135,136,1022],[23,77857,297],{"className":77858},[296],[23,77860,77862],{"className":77861},[204],[23,77863,77865],{"className":77864,"style":14292},[208],[23,77866],{}," 张成一个有限维空间（例如常数、线性函数），正则项只惩罚 ",[23,77869,77871,77895],{"className":77870,"translate":27},[31],[23,77872,77874],{"className":77873},[35],[37,77875,77876],{"xmlns":39},[42,77877,77878,77892],{},[45,77879,77880,77882,77884],{},[48,77881,20680],{"mathvariant":111},[48,77883,7089],{},[19137,77885,77886,77888,77890],{},[48,77887,20680],{"mathvariant":111},[48,77889,26249],{"mathvariant":375},[2581,77891,9952],{},[114,77893,77894],{"encoding":116},"\\|g\\|_{\\mathcal H}^2",[23,77896,77898],{"className":77897,"ariaHidden":122},[121],[23,77899,77901,77905,77908,77911],{"className":77900},[126],[23,77902],{"className":77903,"style":77904},[130],"height:1.0894em;vertical-align:-0.2753em;",[23,77906,20680],{"className":77907},[135],[23,77909,7089],{"className":77910,"style":676},[135,136],[23,77912,77914,77917],{"className":77913},[135],[23,77915,20680],{"className":77916},[135],[23,77918,77920],{"className":77919},[998],[23,77921,77923,77958],{"className":77922},[199,200],[23,77924,77926,77955],{"className":77925},[204],[23,77927,77929,77944],{"className":77928,"style":4693},[208],[23,77930,77932,77935],{"style":77931},"top:-2.4247em;margin-left:0em;margin-right:0.05em;",[23,77933],{"className":77934,"style":1015},[216],[23,77936,77938],{"className":77937},[1019,1020,1021,1022],[23,77939,77941],{"className":77940},[135,1022],[23,77942,26249],{"className":77943,"style":45024},[135,412,1022],[23,77945,77946,77949],{"style":4696},[23,77947],{"className":77948,"style":1015},[216],[23,77950,77952],{"className":77951},[1019,1020,1021,1022],[23,77953,9952],{"className":77954},[135,1022],[23,77956,297],{"className":77957},[296],[23,77959,77961],{"className":77960},[204],[23,77962,77965],{"className":77963,"style":77964},[208],"height:0.2753em;",[23,77966],{},"，则最优的 ",[23,77969,77971,77984],{"className":77970,"translate":27},[31],[23,77972,77974],{"className":77973},[35],[37,77975,77976],{"xmlns":39},[42,77977,77978,77982],{},[45,77979,77980],{},[48,77981,7089],{},[114,77983,7089],{"encoding":116},[23,77985,77987],{"className":77986,"ariaHidden":122},[121],[23,77988,77990,77993],{"className":77989},[126],[23,77991],{"className":77992,"style":672},[130],[23,77994,7089],{"className":77995,"style":676},[135,136]," 仍在训练点的张成空间中，而 ",[23,77998,78000,78014],{"className":77999,"translate":27},[31],[23,78001,78003],{"className":78002},[35],[37,78004,78005],{"xmlns":39},[42,78006,78007,78011],{},[45,78008,78009],{},[48,78010,4548],{},[114,78012,78013],{"encoding":116},"\\beta",[23,78015,78017],{"className":78016,"ariaHidden":122},[121],[23,78018,78020,78023],{"className":78019},[126],[23,78021],{"className":78022,"style":5414},[130],[23,78024,4548],{"className":78025,"style":4655},[135,136]," 通过一个有限维系统求解。SVM 的偏置、平滑样条的多项式部分都属于这种情况。",[338,78028,78030],{"id":78029},"四为什么是范数的非减函数","四、为什么是「范数的非减函数」",[2501,78032,78034],{"id":78033},"_41-充分性","4.1 充分性",[11,78036,78037,78038,78066,78067,78136,78137,78206,78207,63905,78379,78407],{},"第三节的证明已经说明：只要 ",[23,78039,78041,78054],{"className":78040,"translate":27},[31],[23,78042,78044],{"className":78043},[35],[37,78045,78046],{"xmlns":39},[42,78047,78048,78052],{},[45,78049,78050],{},[48,78051,68133],{"mathvariant":111},[114,78053,69091],{"encoding":116},[23,78055,78057],{"className":78056,"ariaHidden":122},[121],[23,78058,78060,78063],{"className":78059},[126],[23,78061],{"className":78062,"style":470},[130],[23,78064,68133],{"className":78065},[135]," 非减，投影 ",[23,78068,78070,78087],{"className":78069,"translate":27},[31],[23,78071,78073],{"className":78072},[35],[37,78074,78075],{"xmlns":39},[42,78076,78077,78085],{},[45,78078,78079],{},[898,78080,78081,78083],{},[48,78082,8336],{},[51,78084,20680],{},[114,78086,73027],{"encoding":116},[23,78088,78090],{"className":78089,"ariaHidden":122},[121],[23,78091,78093,78096],{"className":78092},[126],[23,78094],{"className":78095,"style":70222},[130],[23,78097,78099,78102],{"className":78098},[135],[23,78100,8336],{"className":78101,"style":11398},[135,136],[23,78103,78105],{"className":78104},[998],[23,78106,78108,78128],{"className":78107},[199,200],[23,78109,78111,78125],{"className":78110},[204],[23,78112,78114],{"className":78113,"style":1008},[208],[23,78115,78116,78119],{"style":70243},[23,78117],{"className":78118,"style":1015},[216],[23,78120,78122],{"className":78121},[1019,1020,1021,1022],[23,78123,20680],{"className":78124},[153,1022],[23,78126,297],{"className":78127},[296],[23,78129,78131],{"className":78130},[204],[23,78132,78134],{"className":78133,"style":1057},[208],[23,78135],{}," 就不会让目标变差，因此总有一个最优解落在 ",[23,78138,78140,78157],{"className":78139,"translate":27},[31],[23,78141,78143],{"className":78142},[35],[37,78144,78145],{"xmlns":39},[42,78146,78147,78155],{},[45,78148,78149],{},[898,78150,78151,78153],{},[48,78152,26249],{"mathvariant":375},[2581,78154,1532],{},[114,78156,73813],{"encoding":116},[23,78158,78160],{"className":78159,"ariaHidden":122},[121],[23,78161,78163,78166],{"className":78162},[126],[23,78164],{"className":78165,"style":42827},[130],[23,78167,78169,78172],{"className":78168},[135],[23,78170,26249],{"className":78171,"style":45024},[135,412],[23,78173,78175],{"className":78174},[998],[23,78176,78178,78198],{"className":78177},[199,200],[23,78179,78181,78195],{"className":78180},[204],[23,78182,78184],{"className":78183,"style":18647},[208],[23,78185,78186,78189],{"style":61627},[23,78187],{"className":78188,"style":1015},[216],[23,78190,78192],{"className":78191},[1019,1020,1021,1022],[23,78193,1532],{"className":78194},[135,1022],[23,78196,297],{"className":78197},[296],[23,78199,78201],{"className":78200},[204],[23,78202,78204],{"className":78203,"style":2048},[208],[23,78205],{}," 中。这一步只用了 ",[23,78208,78210,78248],{"className":78209,"translate":27},[31],[23,78211,78213],{"className":78212},[35],[37,78214,78215],{"xmlns":39},[42,78216,78217,78245],{},[45,78218,78219,78221,78223,78229,78231,78233,78239],{},[48,78220,20680],{"mathvariant":111},[48,78222,8336],{},[4518,78224,78225,78227],{},[48,78226,20680],{"mathvariant":111},[2581,78228,9952],{},[51,78230,7378],{},[48,78232,20680],{"mathvariant":111},[898,78234,78235,78237],{},[48,78236,8336],{},[51,78238,20680],{},[4518,78240,78241,78243],{},[48,78242,20680],{"mathvariant":111},[2581,78244,9952],{},[114,78246,78247],{"encoding":116},"\\|f\\|^2\\ge\\|f_\\parallel\\|^2",[23,78249,78251,78301],{"className":78250,"ariaHidden":122},[121],[23,78252,78254,78257,78260,78263,78292,78295,78298],{"className":78253},[126],[23,78255],{"className":78256,"style":10661},[130],[23,78258,20680],{"className":78259},[135],[23,78261,8336],{"className":78262,"style":11398},[135,136],[23,78264,78266,78269],{"className":78265},[135],[23,78267,20680],{"className":78268},[135],[23,78270,78272],{"className":78271},[998],[23,78273,78275],{"className":78274},[199],[23,78276,78278],{"className":78277},[204],[23,78279,78281],{"className":78280,"style":4693},[208],[23,78282,78283,78286],{"style":4696},[23,78284],{"className":78285,"style":1015},[216],[23,78287,78289],{"className":78288},[1019,1020,1021,1022],[23,78290,9952],{"className":78291},[135,1022],[23,78293],{"className":78294,"style":149},[148],[23,78296,7378],{"className":78297},[153],[23,78299],{"className":78300,"style":149},[148],[23,78302,78304,78307,78310,78350],{"className":78303},[126],[23,78305],{"className":78306,"style":73597},[130],[23,78308,20680],{"className":78309},[135],[23,78311,78313,78316],{"className":78312},[135],[23,78314,8336],{"className":78315,"style":11398},[135,136],[23,78317,78319],{"className":78318},[998],[23,78320,78322,78342],{"className":78321},[199,200],[23,78323,78325,78339],{"className":78324},[204],[23,78326,78328],{"className":78327,"style":1008},[208],[23,78329,78330,78333],{"style":70243},[23,78331],{"className":78332,"style":1015},[216],[23,78334,78336],{"className":78335},[1019,1020,1021,1022],[23,78337,20680],{"className":78338},[153,1022],[23,78340,297],{"className":78341},[296],[23,78343,78345],{"className":78344},[204],[23,78346,78348],{"className":78347,"style":1057},[208],[23,78349],{},[23,78351,78353,78356],{"className":78352},[135],[23,78354,20680],{"className":78355},[135],[23,78357,78359],{"className":78358},[998],[23,78360,78362],{"className":78361},[199],[23,78363,78365],{"className":78364},[204],[23,78366,78368],{"className":78367,"style":4693},[208],[23,78369,78370,78373],{"style":4696},[23,78371],{"className":78372,"style":1015},[216],[23,78374,78376],{"className":78375},[1019,1020,1021,1022],[23,78377,9952],{"className":78378},[135,1022],[23,78380,78382,78395],{"className":78381,"translate":27},[31],[23,78383,78385],{"className":78384},[35],[37,78386,78387],{"xmlns":39},[42,78388,78389,78393],{},[45,78390,78391],{},[48,78392,68133],{"mathvariant":111},[114,78394,69091],{"encoding":116},[23,78396,78398],{"className":78397,"ariaHidden":122},[121],[23,78399,78401,78404],{"className":78400},[126],[23,78402],{"className":78403,"style":470},[130],[23,78405,68133],{"className":78406},[135]," 的单调性，没有别的东西。",[2501,78409,78411],{"id":78410},"_42-反例非单调正则项","4.2 反例：非单调正则项",[11,78413,78414,78415,78443],{},"反过来，如果 ",[23,78416,78418,78431],{"className":78417,"translate":27},[31],[23,78419,78421],{"className":78420},[35],[37,78422,78423],{"xmlns":39},[42,78424,78425,78429],{},[45,78426,78427],{},[48,78428,68133],{"mathvariant":111},[114,78430,69091],{"encoding":116},[23,78432,78434],{"className":78433,"ariaHidden":122},[121],[23,78435,78437,78440],{"className":78436},[126],[23,78438],{"className":78439,"style":470},[130],[23,78441,68133],{"className":78442},[135]," 不是非减的，表示定理就可能失效。考虑一个最小的例子。",[11,78445,78446,78447,78528,78529,78697,78698,78779,78780,78895,78896,78988],{},"取 ",[23,78448,78450,78472],{"className":78449,"translate":27},[31],[23,78451,78453],{"className":78452},[35],[37,78454,78455],{"xmlns":39},[42,78456,78457,78469],{},[45,78458,78459,78461,78463],{},[48,78460,434],{"mathvariant":375},[51,78462,69],{},[4518,78464,78465,78467],{},[48,78466,572],{"mathvariant":571},[2581,78468,9952],{},[114,78470,78471],{"encoding":116},"\\mathcal X=\\mathbb R^2",[23,78473,78475,78493],{"className":78474,"ariaHidden":122},[121],[23,78476,78478,78481,78484,78487,78490],{"className":78477},[126],[23,78479],{"className":78480,"style":470},[130],[23,78482,434],{"className":78483,"style":474},[135,412],[23,78485],{"className":78486,"style":149},[148],[23,78488,69],{"className":78489},[153],[23,78491],{"className":78492,"style":149},[148],[23,78494,78496,78499],{"className":78495},[126],[23,78497],{"className":78498,"style":4693},[130],[23,78500,78502,78505],{"className":78501},[135],[23,78503,572],{"className":78504},[135,644],[23,78506,78508],{"className":78507},[998],[23,78509,78511],{"className":78510},[199],[23,78512,78514],{"className":78513},[204],[23,78515,78517],{"className":78516,"style":4693},[208],[23,78518,78519,78522],{"style":4696},[23,78520],{"className":78521,"style":1015},[216],[23,78523,78525],{"className":78524},[1019,1020,1021,1022],[23,78526,9952],{"className":78527},[135,1022],"，线性核 ",[23,78530,78532,78576],{"className":78531,"translate":27},[31],[23,78533,78535],{"className":78534},[35],[37,78536,78537],{"xmlns":39},[42,78538,78539,78573],{},[45,78540,78541,78543,78545,78547,78549,78555,78557,78559,78561,78563,78565,78571],{},[48,78542,4836],{},[51,78544,54],{"stretchy":53},[48,78546,753],{},[51,78548,747],{"separator":122},[4518,78550,78551,78553],{},[48,78552,753],{},[51,78554,38291],{"mathvariant":111,"lspace":1522,"rspace":1522},[51,78556,66],{"stretchy":53},[51,78558,69],{},[51,78560,61181],{"stretchy":53},[48,78562,753],{},[51,78564,747],{"separator":122},[4518,78566,78567,78569],{},[48,78568,753],{},[51,78570,38291],{"mathvariant":111,"lspace":1522,"rspace":1522},[51,78572,61206],{"stretchy":53},[114,78574,78575],{"encoding":116},"k(x,x')=\\langle x,x'\\rangle",[23,78577,78579,78644],{"className":78578,"ariaHidden":122},[121],[23,78580,78582,78585,78588,78591,78594,78597,78600,78632,78635,78638,78641],{"className":78581},[126],[23,78583],{"className":78584,"style":66372},[130],[23,78586,4836],{"className":78587,"style":4851},[135,136],[23,78589,54],{"className":78590},[140],[23,78592,753],{"className":78593},[135,136],[23,78595,747],{"className":78596},[824],[23,78598],{"className":78599,"style":280},[148],[23,78601,78603,78606],{"className":78602},[135],[23,78604,753],{"className":78605},[135,136],[23,78607,78609],{"className":78608},[998],[23,78610,78612],{"className":78611},[199],[23,78613,78615],{"className":78614},[204],[23,78616,78618],{"className":78617,"style":38333},[208],[23,78619,78620,78623],{"style":4696},[23,78621],{"className":78622,"style":1015},[216],[23,78624,78626],{"className":78625},[1019,1020,1021,1022],[23,78627,78629],{"className":78628},[135,1022],[23,78630,38291],{"className":78631},[135,1022],[23,78633,66],{"className":78634},[169],[23,78636],{"className":78637,"style":149},[148],[23,78639,69],{"className":78640},[153],[23,78642],{"className":78643,"style":149},[148],[23,78645,78647,78650,78653,78656,78659,78662,78694],{"className":78646},[126],[23,78648],{"className":78649,"style":66372},[130],[23,78651,61181],{"className":78652},[140],[23,78654,753],{"className":78655},[135,136],[23,78657,747],{"className":78658},[824],[23,78660],{"className":78661,"style":280},[148],[23,78663,78665,78668],{"className":78664},[135],[23,78666,753],{"className":78667},[135,136],[23,78669,78671],{"className":78670},[998],[23,78672,78674],{"className":78673},[199],[23,78675,78677],{"className":78676},[204],[23,78678,78680],{"className":78679,"style":38333},[208],[23,78681,78682,78685],{"style":4696},[23,78683],{"className":78684,"style":1015},[216],[23,78686,78688],{"className":78687},[1019,1020,1021,1022],[23,78689,78691],{"className":78690},[135,1022],[23,78692,38291],{"className":78693},[135,1022],[23,78695,61206],{"className":78696},[169],"，此时 ",[23,78699,78701,78723],{"className":78700,"translate":27},[31],[23,78702,78704],{"className":78703},[35],[37,78705,78706],{"xmlns":39},[42,78707,78708,78720],{},[45,78709,78710,78712,78714],{},[48,78711,26249],{"mathvariant":375},[51,78713,69],{},[4518,78715,78716,78718],{},[48,78717,572],{"mathvariant":571},[2581,78719,9952],{},[114,78721,78722],{"encoding":116},"\\mathcal H=\\mathbb R^2",[23,78724,78726,78744],{"className":78725,"ariaHidden":122},[121],[23,78727,78729,78732,78735,78738,78741],{"className":78728},[126],[23,78730],{"className":78731,"style":470},[130],[23,78733,26249],{"className":78734,"style":45024},[135,412],[23,78736],{"className":78737,"style":149},[148],[23,78739,69],{"className":78740},[153],[23,78742],{"className":78743,"style":149},[148],[23,78745,78747,78750],{"className":78746},[126],[23,78748],{"className":78749,"style":4693},[130],[23,78751,78753,78756],{"className":78752},[135],[23,78754,572],{"className":78755},[135,644],[23,78757,78759],{"className":78758},[998],[23,78760,78762],{"className":78761},[199],[23,78763,78765],{"className":78764},[204],[23,78766,78768],{"className":78767,"style":4693},[208],[23,78769,78770,78773],{"style":4696},[23,78771],{"className":78772,"style":1015},[216],[23,78774,78776],{"className":78775},[1019,1020,1021,1022],[23,78777,9952],{"className":78778},[135,1022],"。只有一个训练点 ",[23,78781,78783,78813],{"className":78782,"translate":27},[31],[23,78784,78786],{"className":78785},[35],[37,78787,78788],{"xmlns":39},[42,78789,78790,78810],{},[45,78791,78792,78798,78800,78802,78804,78806,78808],{},[898,78793,78794,78796],{},[48,78795,753],{},[2581,78797,4435],{},[51,78799,69],{},[51,78801,54],{"stretchy":53},[2581,78803,4435],{},[51,78805,747],{"separator":122},[2581,78807,1532],{},[51,78809,66],{"stretchy":53},[114,78811,78812],{"encoding":116},"x_1=(1,0)",[23,78814,78816,78871],{"className":78815,"ariaHidden":122},[121],[23,78817,78819,78822,78862,78865,78868],{"className":78818},[126],[23,78820],{"className":78821,"style":12730},[130],[23,78823,78825,78828],{"className":78824},[135],[23,78826,753],{"className":78827},[135,136],[23,78829,78831],{"className":78830},[998],[23,78832,78834,78854],{"className":78833},[199,200],[23,78835,78837,78851],{"className":78836},[204],[23,78838,78840],{"className":78839,"style":18647},[208],[23,78841,78842,78845],{"style":2029},[23,78843],{"className":78844,"style":1015},[216],[23,78846,78848],{"className":78847},[1019,1020,1021,1022],[23,78849,4435],{"className":78850},[135,1022],[23,78852,297],{"className":78853},[296],[23,78855,78857],{"className":78856},[204],[23,78858,78860],{"className":78859,"style":2048},[208],[23,78861],{},[23,78863],{"className":78864,"style":149},[148],[23,78866,69],{"className":78867},[153],[23,78869],{"className":78870,"style":149},[148],[23,78872,78874,78877,78880,78883,78886,78889,78892],{"className":78873},[126],[23,78875],{"className":78876,"style":131},[130],[23,78878,54],{"className":78879},[140],[23,78881,4435],{"className":78882},[135],[23,78884,747],{"className":78885},[824],[23,78887],{"className":78888,"style":280},[148],[23,78890,1532],{"className":78891},[135],[23,78893,66],{"className":78894},[169],"，标签 ",[23,78897,78899,78921],{"className":78898,"translate":27},[31],[23,78900,78902],{"className":78901},[35],[37,78903,78904],{"xmlns":39},[42,78905,78906,78918],{},[45,78907,78908,78914,78916],{},[898,78909,78910,78912],{},[48,78911,660],{},[2581,78913,4435],{},[51,78915,69],{},[2581,78917,1532],{},[114,78919,78920],{"encoding":116},"y_1=0",[23,78922,78924,78979],{"className":78923,"ariaHidden":122},[121],[23,78925,78927,78930,78970,78973,78976],{"className":78926},[126],[23,78928],{"className":78929,"style":672},[130],[23,78931,78933,78936],{"className":78932},[135],[23,78934,660],{"className":78935,"style":676},[135,136],[23,78937,78939],{"className":78938},[998],[23,78940,78942,78962],{"className":78941},[199,200],[23,78943,78945,78959],{"className":78944},[204],[23,78946,78948],{"className":78947,"style":18647},[208],[23,78949,78950,78953],{"style":21850},[23,78951],{"className":78952,"style":1015},[216],[23,78954,78956],{"className":78955},[1019,1020,1021,1022],[23,78957,4435],{"className":78958},[135,1022],[23,78960,297],{"className":78961},[296],[23,78963,78965],{"className":78964},[204],[23,78966,78968],{"className":78967,"style":2048},[208],[23,78969],{},[23,78971],{"className":78972,"style":149},[148],[23,78974,69],{"className":78975},[153],[23,78977],{"className":78978,"style":149},[148],[23,78980,78982,78985],{"className":78981},[126],[23,78983],{"className":78984,"style":2770},[130],[23,78986,1532],{"className":78987},[135],"。损失取平方损失",[23,78990,78992],{"className":78991,"translate":27},[26],[23,78993,78995,79045],{"className":78994,"translate":27},[31],[23,78996,78998],{"className":78997},[35],[37,78999,79000],{"xmlns":39,"display":40},[42,79001,79002,79042],{},[45,79003,79004,79006,79008,79010,79012,79018,79020,79022,79024,79026,79028,79034,79040],{},[48,79005,555],{},[51,79007,54],{"stretchy":53},[48,79009,8336],{},[51,79011,54],{"stretchy":53},[898,79013,79014,79016],{},[48,79015,753],{},[2581,79017,4435],{},[51,79019,66],{"stretchy":53},[51,79021,66],{"stretchy":53},[51,79023,69],{},[48,79025,8336],{},[51,79027,54],{"stretchy":53},[898,79029,79030,79032],{},[48,79031,753],{},[2581,79033,4435],{},[4518,79035,79036,79038],{},[51,79037,66],{"stretchy":53},[2581,79039,9952],{},[51,79041,747],{"separator":122},[114,79043,79044],{"encoding":116},"L(f(x_1))=f(x_1)^2,",[23,79046,79048,79118],{"className":79047,"ariaHidden":122},[121],[23,79049,79051,79054,79057,79060,79063,79066,79106,79109,79112,79115],{"className":79050},[126],[23,79052],{"className":79053,"style":131},[130],[23,79055,555],{"className":79056},[135,136],[23,79058,54],{"className":79059},[140],[23,79061,8336],{"className":79062,"style":11398},[135,136],[23,79064,54],{"className":79065},[140],[23,79067,79069,79072],{"className":79068},[135],[23,79070,753],{"className":79071},[135,136],[23,79073,79075],{"className":79074},[998],[23,79076,79078,79098],{"className":79077},[199,200],[23,79079,79081,79095],{"className":79080},[204],[23,79082,79084],{"className":79083,"style":18647},[208],[23,79085,79086,79089],{"style":2029},[23,79087],{"className":79088,"style":1015},[216],[23,79090,79092],{"className":79091},[1019,1020,1021,1022],[23,79093,4435],{"className":79094},[135,1022],[23,79096,297],{"className":79097},[296],[23,79099,79101],{"className":79100},[204],[23,79102,79104],{"className":79103,"style":2048},[208],[23,79105],{},[23,79107,1095],{"className":79108},[169],[23,79110],{"className":79111,"style":149},[148],[23,79113,69],{"className":79114},[153],[23,79116],{"className":79117,"style":149},[148],[23,79119,79121,79124,79127,79130,79170,79199],{"className":79120},[126],[23,79122],{"className":79123,"style":26499},[130],[23,79125,8336],{"className":79126,"style":11398},[135,136],[23,79128,54],{"className":79129},[140],[23,79131,79133,79136],{"className":79132},[135],[23,79134,753],{"className":79135},[135,136],[23,79137,79139],{"className":79138},[998],[23,79140,79142,79162],{"className":79141},[199,200],[23,79143,79145,79159],{"className":79144},[204],[23,79146,79148],{"className":79147,"style":18647},[208],[23,79149,79150,79153],{"style":2029},[23,79151],{"className":79152,"style":1015},[216],[23,79154,79156],{"className":79155},[1019,1020,1021,1022],[23,79157,4435],{"className":79158},[135,1022],[23,79160,297],{"className":79161},[296],[23,79163,79165],{"className":79164},[204],[23,79166,79168],{"className":79167,"style":2048},[208],[23,79169],{},[23,79171,79173,79176],{"className":79172},[169],[23,79174,66],{"className":79175},[169],[23,79177,79179],{"className":79178},[998],[23,79180,79182],{"className":79181},[199],[23,79183,79185],{"className":79184},[204],[23,79186,79188],{"className":79187,"style":11002},[208],[23,79189,79190,79193],{"style":4989},[23,79191],{"className":79192,"style":1015},[216],[23,79194,79196],{"className":79195},[1019,1020,1021,1022],[23,79197,9952],{"className":79198},[135,1022],[23,79200,747],{"className":79201},[824],[11,79203,79204],{},"正则项取",[23,79206,79208],{"className":79207,"translate":27},[26]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",[23,79355,79357,79378],{"className":79356,"translate":27},[31],[23,79358,79360],{"className":79359},[35],[37,79361,79362],{"xmlns":39},[42,79363,79364,79376],{},[45,79365,79366,79368,79370,79372,79374],{},[51,79367,924],{"stretchy":53},[2581,79369,1532],{},[51,79371,747],{"separator":122},[2581,79373,4435],{},[51,79375,945],{"stretchy":53},[114,79377,46866],{"encoding":116},[23,79379,79381],{"className":79380,"ariaHidden":122},[121],[23,79382,79384,79387,79390,79393,79396,79399,79402],{"className":79383},[126],[23,79385],{"className":79386,"style":131},[130],[23,79388,924],{"className":79389},[140],[23,79391,1532],{"className":79392},[135],[23,79394,747],{"className":79395},[824],[23,79397],{"className":79398,"style":280},[148],[23,79400,4435],{"className":79401},[135],[23,79403,945],{"className":79404},[169]," 上递减，不是非减函数。",[11,79407,79408],{},"训练点张成的子空间是",[23,79410,79412],{"className":79411,"translate":27},[26],[23,79413,79415,79485],{"className":79414,"translate":27},[31],[23,79416,79418],{"className":79417},[35],[37,79419,79420],{"xmlns":39,"display":40},[42,79421,79422,79482],{},[45,79423,79424,79430,79432,79434,79436,79438,79440,79442,79444,79446,79452,79454,79456,79458,79460,79462,79464,79466,79468,79470,79472,79474,79476,79478,79480],{},[898,79425,79426,79428],{},[48,79427,26249],{"mathvariant":375},[2581,79429,1532],{},[51,79431,69],{},[48,79433,23],{"mathvariant":111},[51,79435,6959],{},[51,79437,9010],{"stretchy":53},[48,79439,4836],{},[51,79441,54],{"stretchy":53},[51,79443,5758],{},[51,79445,747],{"separator":122},[898,79447,79448,79450],{},[48,79449,753],{},[2581,79451,4435],{},[51,79453,66],{"stretchy":53},[51,79455,9020],{"stretchy":53},[51,79457,69],{},[51,79459,9010],{"stretchy":53},[51,79461,54],{"stretchy":53},[48,79463,495],{},[51,79465,747],{"separator":122},[2581,79467,1532],{},[51,79469,66],{"stretchy":53},[51,79471,558],{},[48,79473,495],{},[51,79475,372],{},[48,79477,572],{"mathvariant":571},[51,79479,9020],{"stretchy":53},[48,79481,112],{"mathvariant":111},[114,79483,79484],{"encoding":116},"\\mathcal H_0=\\operatorname{span}\\{k(\\cdot,x_1)\\}=\\{(a,0):a\\in\\mathbb R\\}.",[23,79486,79488,79543,79625,79658,79676],{"className":79487,"ariaHidden":122},[121],[23,79489,79491,79494,79534,79537,79540],{"className":79490},[126],[23,79492],{"className":79493,"style":42827},[130],[23,79495,79497,79500],{"className":79496},[135],[23,79498,26249],{"className":79499,"style":45024},[135,412],[23,79501,79503],{"className":79502},[998],[23,79504,79506,79526],{"className":79505},[199,200],[23,79507,79509,79523],{"className":79508},[204],[23,79510,79512],{"className":79511,"style":18647},[208],[23,79513,79514,79517],{"style":61627},[23,79515],{"className":79516,"style":1015},[216],[23,79518,79520],{"className":79519},[1019,1020,1021,1022],[23,79521,1532],{"className":79522},[135,1022],[23,79524,297],{"className":79525},[296],[23,79527,79529],{"className":79528},[204],[23,79530,79532],{"className":79531,"style":2048},[208],[23,79533],{},[23,79535],{"className":79536,"style":149},[148],[23,79538,69],{"className":79539},[153],[23,79541],{"className":79542,"style":149},[148],[23,79544,79546,79549,79555,79558,79561,79564,79567,79570,79573,79613,79616,79619,79622],{"className":79545},[126],[23,79547],{"className":79548,"style":131},[130],[23,79550,79552],{"className":79551},[1786],[23,79553,23],{"className":79554},[135,5391],[23,79556,9010],{"className":79557},[140],[23,79559,4836],{"className":79560,"style":4851},[135,136],[23,79562,54],{"className":79563},[140],[23,79565,5758],{"className":79566},[135],[23,79568,747],{"className":79569},[824],[23,79571],{"className":79572,"style":280},[148],[23,79574,79576,79579],{"className":79575},[135],[23,79577,753],{"className":79578},[135,136],[23,79580,79582],{"className":79581},[998],[23,79583,79585,79605],{"className":79584},[199,200],[23,79586,79588,79602],{"className":79587},[204],[23,79589,79591],{"className":79590,"style":18647},[208],[23,79592,79593,79596],{"style":2029},[23,79594],{"className":79595,"style":1015},[216],[23,79597,79599],{"className":79598},[1019,1020,1021,1022],[23,79600,4435],{"className":79601},[135,1022],[23,79603,297],{"className":79604},[296],[23,79606,79608],{"className":79607},[204],[23,79609,79611],{"className":79610,"style":2048},[208],[23,79612],{},[23,79614,15220],{"className":79615},[169],[23,79617],{"className":79618,"style":149},[148],[23,79620,69],{"className":79621},[153],[23,79623],{"className":79624,"style":149},[148],[23,79626,79628,79631,79634,79637,79640,79643,79646,79649,79652,79655],{"className":79627},[126],[23,79629],{"className":79630,"style":131},[130],[23,79632,50674],{"className":79633},[140],[23,79635,495],{"className":79636},[135,136],[23,79638,747],{"className":79639},[824],[23,79641],{"className":79642,"style":280},[148],[23,79644,1532],{"className":79645},[135],[23,79647,66],{"className":79648},[169],[23,79650],{"className":79651,"style":149},[148],[23,79653,558],{"className":79654},[153],[23,79656],{"className":79657,"style":149},[148],[23,79659,79661,79664,79667,79670,79673],{"className":79660},[126],[23,79662],{"className":79663,"style":513},[130],[23,79665,495],{"className":79666},[135,136],[23,79668],{"className":79669,"style":149},[148],[23,79671,372],{"className":79672},[153],[23,79674],{"className":79675,"style":149},[148],[23,79677,79679,79682,79685,79688],{"className":79678},[126],[23,79680],{"className":79681,"style":131},[130],[23,79683,572],{"className":79684},[135,644],[23,79686,9020],{"className":79687},[169],[23,79689,112],{"className":79690},[135],[11,79692,59348,79693,79767,79768,79875,79876,77352,79904,79965,79966,79994,79995,80023,80024,79767,80098,80204,80205,77048,80233,79965,80294,79994,80322,80350,80351,80403,80404,80473],{},[23,79694,79696,79722],{"className":79695,"translate":27},[31],[23,79697,79699],{"className":79698},[35],[37,79700,79701],{"xmlns":39},[42,79702,79703,79719],{},[45,79704,79705,79707,79709,79711,79713,79715,79717],{},[48,79706,8336],{},[51,79708,69],{},[51,79710,54],{"stretchy":53},[2581,79712,1532],{},[51,79714,747],{"separator":122},[2581,79716,1532],{},[51,79718,66],{"stretchy":53},[114,79720,79721],{"encoding":116},"f=(0,0)",[23,79723,79725,79743],{"className":79724,"ariaHidden":122},[121],[23,79726,79728,79731,79734,79737,79740],{"className":79727},[126],[23,79729],{"className":79730,"style":5414},[130],[23,79732,8336],{"className":79733,"style":11398},[135,136],[23,79735],{"className":79736,"style":149},[148],[23,79738,69],{"className":79739},[153],[23,79741],{"className":79742,"style":149},[148],[23,79744,79746,79749,79752,79755,79758,79761,79764],{"className":79745},[126],[23,79747],{"className":79748,"style":131},[130],[23,79750,54],{"className":79751},[140],[23,79753,1532],{"className":79754},[135],[23,79756,747],{"className":79757},[824],[23,79759],{"className":79760,"style":280},[148],[23,79762,1532],{"className":79763},[135],[23,79765,66],{"className":79766},[169]," 处，",[23,79769,79771,79799],{"className":79770,"translate":27},[31],[23,79772,79774],{"className":79773},[35],[37,79775,79776],{"xmlns":39},[42,79777,79778,79796],{},[45,79779,79780,79782,79784,79790,79792,79794],{},[48,79781,8336],{},[51,79783,54],{"stretchy":53},[898,79785,79786,79788],{},[48,79787,753],{},[2581,79789,4435],{},[51,79791,66],{"stretchy":53},[51,79793,69],{},[2581,79795,1532],{},[114,79797,79798],{"encoding":116},"f(x_1)=0",[23,79800,79802,79866],{"className":79801,"ariaHidden":122},[121],[23,79803,79805,79808,79811,79814,79854,79857,79860,79863],{"className":79804},[126],[23,79806],{"className":79807,"style":131},[130],[23,79809,8336],{"className":79810,"style":11398},[135,136],[23,79812,54],{"className":79813},[140],[23,79815,79817,79820],{"className":79816},[135],[23,79818,753],{"className":79819},[135,136],[23,79821,79823],{"className":79822},[998],[23,79824,79826,79846],{"className":79825},[199,200],[23,79827,79829,79843],{"className":79828},[204],[23,79830,79832],{"className":79831,"style":18647},[208],[23,79833,79834,79837],{"style":2029},[23,79835],{"className":79836,"style":1015},[216],[23,79838,79840],{"className":79839},[1019,1020,1021,1022],[23,79841,4435],{"className":79842},[135,1022],[23,79844,297],{"className":79845},[296],[23,79847,79849],{"className":79848},[204],[23,79850,79852],{"className":79851,"style":2048},[208],[23,79853],{},[23,79855,66],{"className":79856},[169],[23,79858],{"className":79859,"style":149},[148],[23,79861,69],{"className":79862},[153],[23,79864],{"className":79865,"style":149},[148],[23,79867,79869,79872],{"className":79868},[126],[23,79870],{"className":79871,"style":2770},[130],[23,79873,1532],{"className":79874},[135],"，损失为 ",[23,79877,79879,79892],{"className":79878,"translate":27},[31],[23,79880,79882],{"className":79881},[35],[37,79883,79884],{"xmlns":39},[42,79885,79886,79890],{},[45,79887,79888],{},[2581,79889,1532],{},[114,79891,1532],{"encoding":116},[23,79893,79895],{"className":79894,"ariaHidden":122},[121],[23,79896,79898,79901],{"className":79897},[126],[23,79899],{"className":79900,"style":2770},[130],[23,79902,1532],{"className":79903},[135],[23,79905,79907,79929],{"className":79906,"translate":27},[31],[23,79908,79910],{"className":79909},[35],[37,79911,79912],{"xmlns":39},[42,79913,79914,79926],{},[45,79915,79916,79918,79920,79922,79924],{},[48,79917,20680],{"mathvariant":111},[48,79919,8336],{},[48,79921,20680],{"mathvariant":111},[51,79923,69],{},[2581,79925,1532],{},[114,79927,79928],{"encoding":116},"\\|f\\|=0",[23,79930,79932,79956],{"className":79931,"ariaHidden":122},[121],[23,79933,79935,79938,79941,79944,79947,79950,79953],{"className":79934},[126],[23,79936],{"className":79937,"style":131},[130],[23,79939,20680],{"className":79940},[135],[23,79942,8336],{"className":79943,"style":11398},[135,136],[23,79945,20680],{"className":79946},[135],[23,79948],{"className":79949,"style":149},[148],[23,79951,69],{"className":79952},[153],[23,79954],{"className":79955,"style":149},[148],[23,79957,79959,79962],{"className":79958},[126],[23,79960],{"className":79961,"style":2770},[130],[23,79963,1532],{"className":79964},[135],"，正则项为 ",[23,79967,79969,79982],{"className":79968,"translate":27},[31],[23,79970,79972],{"className":79971},[35],[37,79973,79974],{"xmlns":39},[42,79975,79976,79980],{},[45,79977,79978],{},[2581,79979,4435],{},[114,79981,4435],{"encoding":116},[23,79983,79985],{"className":79984,"ariaHidden":122},[121],[23,79986,79988,79991],{"className":79987},[126],[23,79989],{"className":79990,"style":2770},[130],[23,79992,4435],{"className":79993},[135],"，目标值为 ",[23,79996,79998,80011],{"className":79997,"translate":27},[31],[23,79999,80001],{"className":80000},[35],[37,80002,80003],{"xmlns":39},[42,80004,80005,80009],{},[45,80006,80007],{},[2581,80008,4435],{},[114,80010,4435],{"encoding":116},[23,80012,80014],{"className":80013,"ariaHidden":122},[121],[23,80015,80017,80020],{"className":80016},[126],[23,80018],{"className":80019,"style":2770},[130],[23,80021,4435],{"className":80022},[135],"。在 ",[23,80025,80027,80053],{"className":80026,"translate":27},[31],[23,80028,80030],{"className":80029},[35],[37,80031,80032],{"xmlns":39},[42,80033,80034,80050],{},[45,80035,80036,80038,80040,80042,80044,80046,80048],{},[48,80037,8336],{},[51,80039,69],{},[51,80041,54],{"stretchy":53},[2581,80043,1532],{},[51,80045,747],{"separator":122},[2581,80047,4435],{},[51,80049,66],{"stretchy":53},[114,80051,80052],{"encoding":116},"f=(0,1)",[23,80054,80056,80074],{"className":80055,"ariaHidden":122},[121],[23,80057,80059,80062,80065,80068,80071],{"className":80058},[126],[23,80060],{"className":80061,"style":5414},[130],[23,80063,8336],{"className":80064,"style":11398},[135,136],[23,80066],{"className":80067,"style":149},[148],[23,80069,69],{"className":80070},[153],[23,80072],{"className":80073,"style":149},[148],[23,80075,80077,80080,80083,80086,80089,80092,80095],{"className":80076},[126],[23,80078],{"className":80079,"style":131},[130],[23,80081,54],{"className":80082},[140],[23,80084,1532],{"className":80085},[135],[23,80087,747],{"className":80088},[824],[23,80090],{"className":80091,"style":280},[148],[23,80093,4435],{"className":80094},[135],[23,80096,66],{"className":80097},[169],[23,80099,80101,80128],{"className":80100,"translate":27},[31],[23,80102,80104],{"className":80103},[35],[37,80105,80106],{"xmlns":39},[42,80107,80108,80126],{},[45,80109,80110,80112,80114,80120,80122,80124],{},[48,80111,8336],{},[51,80113,54],{"stretchy":53},[898,80115,80116,80118],{},[48,80117,753],{},[2581,80119,4435],{},[51,80121,66],{"stretchy":53},[51,80123,69],{},[2581,80125,1532],{},[114,80127,79798],{"encoding":116},[23,80129,80131,80195],{"className":80130,"ariaHidden":122},[121],[23,80132,80134,80137,80140,80143,80183,80186,80189,80192],{"className":80133},[126],[23,80135],{"className":80136,"style":131},[130],[23,80138,8336],{"className":80139,"style":11398},[135,136],[23,80141,54],{"className":80142},[140],[23,80144,80146,80149],{"className":80145},[135],[23,80147,753],{"className":80148},[135,136],[23,80150,80152],{"className":80151},[998],[23,80153,80155,80175],{"className":80154},[199,200],[23,80156,80158,80172],{"className":80157},[204],[23,80159,80161],{"className":80160,"style":18647},[208],[23,80162,80163,80166],{"style":2029},[23,80164],{"className":80165,"style":1015},[216],[23,80167,80169],{"className":80168},[1019,1020,1021,1022],[23,80170,4435],{"className":80171},[135,1022],[23,80173,297],{"className":80174},[296],[23,80176,80178],{"className":80177},[204],[23,80179,80181],{"className":80180,"style":2048},[208],[23,80182],{},[23,80184,66],{"className":80185},[169],[23,80187],{"className":80188,"style":149},[148],[23,80190,69],{"className":80191},[153],[23,80193],{"className":80194,"style":149},[148],[23,80196,80198,80201],{"className":80197},[126],[23,80199],{"className":80200,"style":2770},[130],[23,80202,1532],{"className":80203},[135],"，损失仍为 ",[23,80206,80208,80221],{"className":80207,"translate":27},[31],[23,80209,80211],{"className":80210},[35],[37,80212,80213],{"xmlns":39},[42,80214,80215,80219],{},[45,80216,80217],{},[2581,80218,1532],{},[114,80220,1532],{"encoding":116},[23,80222,80224],{"className":80223,"ariaHidden":122},[121],[23,80225,80227,80230],{"className":80226},[126],[23,80228],{"className":80229,"style":2770},[130],[23,80231,1532],{"className":80232},[135],[23,80234,80236,80258],{"className":80235,"translate":27},[31],[23,80237,80239],{"className":80238},[35],[37,80240,80241],{"xmlns":39},[42,80242,80243,80255],{},[45,80244,80245,80247,80249,80251,80253],{},[48,80246,20680],{"mathvariant":111},[48,80248,8336],{},[48,80250,20680],{"mathvariant":111},[51,80252,69],{},[2581,80254,4435],{},[114,80256,80257],{"encoding":116},"\\|f\\|=1",[23,80259,80261,80285],{"className":80260,"ariaHidden":122},[121],[23,80262,80264,80267,80270,80273,80276,80279,80282],{"className":80263},[126],[23,80265],{"className":80266,"style":131},[130],[23,80268,20680],{"className":80269},[135],[23,80271,8336],{"className":80272,"style":11398},[135,136],[23,80274,20680],{"className":80275},[135],[23,80277],{"className":80278,"style":149},[148],[23,80280,69],{"className":80281},[153],[23,80283],{"className":80284,"style":149},[148],[23,80286,80288,80291],{"className":80287},[126],[23,80289],{"className":80290,"style":2770},[130],[23,80292,4435],{"className":80293},[135],[23,80295,80297,80310],{"className":80296,"translate":27},[31],[23,80298,80300],{"className":80299},[35],[37,80301,80302],{"xmlns":39},[42,80303,80304,80308],{},[45,80305,80306],{},[2581,80307,1532],{},[114,80309,1532],{"encoding":116},[23,80311,80313],{"className":80312,"ariaHidden":122},[121],[23,80314,80316,80319],{"className":80315},[126],[23,80317],{"className":80318,"style":2770},[130],[23,80320,1532],{"className":80321},[135],[23,80323,80325,80338],{"className":80324,"translate":27},[31],[23,80326,80328],{"className":80327},[35],[37,80329,80330],{"xmlns":39},[42,80331,80332,80336],{},[45,80333,80334],{},[2581,80335,1532],{},[114,80337,1532],{"encoding":116},[23,80339,80341],{"className":80340,"ariaHidden":122},[121],[23,80342,80344,80347],{"className":80343},[126],[23,80345],{"className":80346,"style":2770},[130],[23,80348,1532],{"className":80349},[135],"。所以最优解是 ",[23,80352,80354,80376],{"className":80353,"translate":27},[31],[23,80355,80357],{"className":80356},[35],[37,80358,80359],{"xmlns":39},[42,80360,80361,80373],{},[45,80362,80363,80365,80367,80369,80371],{},[51,80364,54],{"stretchy":53},[2581,80366,1532],{},[51,80368,747],{"separator":122},[2581,80370,4435],{},[51,80372,66],{"stretchy":53},[114,80374,80375],{"encoding":116},"(0,1)",[23,80377,80379],{"className":80378,"ariaHidden":122},[121],[23,80380,80382,80385,80388,80391,80394,80397,80400],{"className":80381},[126],[23,80383],{"className":80384,"style":131},[130],[23,80386,54],{"className":80387},[140],[23,80389,1532],{"className":80390},[135],[23,80392,747],{"className":80393},[824],[23,80395],{"className":80396,"style":280},[148],[23,80398,4435],{"className":80399},[135],[23,80401,66],{"className":80402},[169],"，它落在 ",[23,80405,80407,80424],{"className":80406,"translate":27},[31],[23,80408,80410],{"className":80409},[35],[37,80411,80412],{"xmlns":39},[42,80413,80414,80422],{},[45,80415,80416],{},[898,80417,80418,80420],{},[48,80419,26249],{"mathvariant":375},[2581,80421,1532],{},[114,80423,73813],{"encoding":116},[23,80425,80427],{"className":80426,"ariaHidden":122},[121],[23,80428,80430,80433],{"className":80429},[126],[23,80431],{"className":80432,"style":42827},[130],[23,80434,80436,80439],{"className":80435},[135],[23,80437,26249],{"className":80438,"style":45024},[135,412],[23,80440,80442],{"className":80441},[998],[23,80443,80445,80465],{"className":80444},[199,200],[23,80446,80448,80462],{"className":80447},[204],[23,80449,80451],{"className":80450,"style":18647},[208],[23,80452,80453,80456],{"style":61627},[23,80454],{"className":80455,"style":1015},[216],[23,80457,80459],{"className":80458},[1019,1020,1021,1022],[23,80460,1532],{"className":80461},[135,1022],[23,80463,297],{"className":80464},[296],[23,80466,80468],{"className":80467},[204],[23,80469,80471],{"className":80470,"style":2048},[208],[23,80472],{}," 的正交补里，而不是训练点的张成空间中。",[11,80475,80476,80477,80480],{},"这个反例说明：",[18,80478,80479],{},"非减性不是证明技巧，而是定理的本质条件。"," 正则项如果「喜欢」某个非零范数，它就会把解推离训练点子空间。",[2501,80482,80484],{"id":80483},"_43-充要条件","4.3 充要条件",[11,80486,80487],{},"上述反例可以推广。Dinuzzo 与 Schölkopf 在 2012 年给出了完整的刻画：在相当一般的条件下，表示定理对所有损失函数和所有数据集成立，当且仅当正则项是范数的非减函数。也就是说，",[23,80489,80491],{"className":80490,"translate":27},[26],[23,80492,80494,80532],{"className":80493,"translate":27},[31],[23,80495,80497],{"className":80496},[35],[37,80498,80499],{"xmlns":39,"display":40},[42,80500,80501,80529],{},[45,80502,80503,80505,80507,80509,80511,80513,80515,80517,80519,80521,80527],{},[48,80504,68133],{"mathvariant":111},[51,80506,54],{"stretchy":53},[48,80508,8336],{},[51,80510,66],{"stretchy":53},[51,80512,69],{},[48,80514,18368],{},[51,80516,54],{"stretchy":53},[48,80518,20680],{"mathvariant":111},[48,80520,8336],{},[898,80522,80523,80525],{},[48,80524,20680],{"mathvariant":111},[48,80526,26249],{"mathvariant":375},[51,80528,66],{"stretchy":53},[114,80530,80531],{"encoding":116},"\\Omega(f)=h(\\|f\\|_{\\mathcal H})",[23,80533,80535,80562],{"className":80534,"ariaHidden":122},[121],[23,80536,80538,80541,80544,80547,80550,80553,80556,80559],{"className":80537},[126],[23,80539],{"className":80540,"style":131},[130],[23,80542,68133],{"className":80543},[135],[23,80545,54],{"className":80546},[140],[23,80548,8336],{"className":80549,"style":11398},[135,136],[23,80551,66],{"className":80552},[169],[23,80554],{"className":80555,"style":149},[148],[23,80557,69],{"className":80558},[153],[23,80560],{"className":80561,"style":149},[148],[23,80563,80565,80568,80571,80574,80577,80580,80623],{"className":80564},[126],[23,80566],{"className":80567,"style":131},[130],[23,80569,18368],{"className":80570},[135,136],[23,80572,54],{"className":80573},[140],[23,80575,20680],{"className":80576},[135],[23,80578,8336],{"className":80579,"style":11398},[135,136],[23,80581,80583,80586],{"className":80582},[135],[23,80584,20680],{"className":80585},[135],[23,80587,80589],{"className":80588},[998],[23,80590,80592,80615],{"className":80591},[199,200],[23,80593,80595,80612],{"className":80594},[204],[23,80596,80598],{"className":80597,"style":2026},[208],[23,80599,80600,80603],{"style":2029},[23,80601],{"className":80602,"style":1015},[216],[23,80604,80606],{"className":80605},[1019,1020,1021,1022],[23,80607,80609],{"className":80608},[135,1022],[23,80610,26249],{"className":80611,"style":45024},[135,412,1022],[23,80613,297],{"className":80614},[296],[23,80616,80618],{"className":80617},[204],[23,80619,80621],{"className":80620,"style":2048},[208],[23,80622],{},[23,80624,66],{"className":80625},[169],[11,80627,80628,80629,80725],{},"对某个非减的 ",[23,80630,80632,80662],{"className":80631,"translate":27},[31],[23,80633,80635],{"className":80634},[35],[37,80636,80637],{"xmlns":39},[42,80638,80639,80659],{},[45,80640,80641,80643,80645,80647,80649,80651,80653,80655,80657],{},[48,80642,18368],{},[51,80644,558],{},[51,80646,924],{"stretchy":53},[2581,80648,1532],{},[51,80650,747],{"separator":122},[48,80652,46447],{"mathvariant":111},[51,80654,66],{"stretchy":53},[51,80656,568],{},[48,80658,572],{"mathvariant":571},[114,80660,80661],{"encoding":116},"h:[0,\\infty)\\to\\mathbb R",[23,80663,80665,80683,80716],{"className":80664,"ariaHidden":122},[121],[23,80666,80668,80671,80674,80677,80680],{"className":80667},[126],[23,80669],{"className":80670,"style":779},[130],[23,80672,18368],{"className":80673},[135,136],[23,80675],{"className":80676,"style":149},[148],[23,80678,558],{"className":80679},[153],[23,80681],{"className":80682,"style":149},[148],[23,80684,80686,80689,80692,80695,80698,80701,80704,80707,80710,80713],{"className":80685},[126],[23,80687],{"className":80688,"style":131},[130],[23,80690,924],{"className":80691},[140],[23,80693,1532],{"className":80694},[135],[23,80696,747],{"className":80697},[824],[23,80699],{"className":80700,"style":280},[148],[23,80702,46447],{"className":80703},[135],[23,80705,66],{"className":80706},[169],[23,80708],{"className":80709,"style":149},[148],[23,80711,568],{"className":80712},[153],[23,80714],{"className":80715,"style":149},[148],[23,80717,80719,80722],{"className":80718},[126],[23,80720],{"className":80721,"style":640},[130],[23,80723,572],{"className":80724},[135,644]," 成立，是表示定理的充要条件。",[11,80727,80728],{},"这里还要区分两个层次：",[346,80730,80731,80807],{},[349,80732,80733,80736,80737,80806],{},[18,80734,80735],{},"非减","：保证存在一个落在 ",[23,80738,80740,80757],{"className":80739,"translate":27},[31],[23,80741,80743],{"className":80742},[35],[37,80744,80745],{"xmlns":39},[42,80746,80747,80755],{},[45,80748,80749],{},[898,80750,80751,80753],{},[48,80752,26249],{"mathvariant":375},[2581,80754,1532],{},[114,80756,73813],{"encoding":116},[23,80758,80760],{"className":80759,"ariaHidden":122},[121],[23,80761,80763,80766],{"className":80762},[126],[23,80764],{"className":80765,"style":42827},[130],[23,80767,80769,80772],{"className":80768},[135],[23,80770,26249],{"className":80771,"style":45024},[135,412],[23,80773,80775],{"className":80774},[998],[23,80776,80778,80798],{"className":80777},[199,200],[23,80779,80781,80795],{"className":80780},[204],[23,80782,80784],{"className":80783,"style":18647},[208],[23,80785,80786,80789],{"style":61627},[23,80787],{"className":80788,"style":1015},[216],[23,80790,80792],{"className":80791},[1019,1020,1021,1022],[23,80793,1532],{"className":80794},[135,1022],[23,80796,297],{"className":80797},[296],[23,80799,80801],{"className":80800},[204],[23,80802,80804],{"className":80803,"style":2048},[208],[23,80805],{}," 中的最优解；",[349,80808,80809,80812,80813,80882],{},[18,80810,80811],{},"严格递增","：保证每个最优解都落在 ",[23,80814,80816,80833],{"className":80815,"translate":27},[31],[23,80817,80819],{"className":80818},[35],[37,80820,80821],{"xmlns":39},[42,80822,80823,80831],{},[45,80824,80825],{},[898,80826,80827,80829],{},[48,80828,26249],{"mathvariant":375},[2581,80830,1532],{},[114,80832,73813],{"encoding":116},[23,80834,80836],{"className":80835,"ariaHidden":122},[121],[23,80837,80839,80842],{"className":80838},[126],[23,80840],{"className":80841,"style":42827},[130],[23,80843,80845,80848],{"className":80844},[135],[23,80846,26249],{"className":80847,"style":45024},[135,412],[23,80849,80851],{"className":80850},[998],[23,80852,80854,80874],{"className":80853},[199,200],[23,80855,80857,80871],{"className":80856},[204],[23,80858,80860],{"className":80859,"style":18647},[208],[23,80861,80862,80865],{"style":61627},[23,80863],{"className":80864,"style":1015},[216],[23,80866,80868],{"className":80867},[1019,1020,1021,1022],[23,80869,1532],{"className":80870},[135,1022],[23,80872,297],{"className":80873},[296],[23,80875,80877],{"className":80876},[204],[23,80878,80880],{"className":80879,"style":2048},[208],[23,80881],{}," 中。",[11,80884,80885,80886,80938,80939,80967,80968,81037,81038,81107],{},"如果 ",[23,80887,80889,80908],{"className":80888,"translate":27},[31],[23,80890,80892],{"className":80891},[35],[37,80893,80894],{"xmlns":39},[42,80895,80896,80905],{},[45,80897,80898,80900,80903],{},[48,80899,68133],{"mathvariant":111},[51,80901,80902],{},"≡",[2581,80904,1532],{},[114,80906,80907],{"encoding":116},"\\Omega\\equiv 0",[23,80909,80911,80929],{"className":80910,"ariaHidden":122},[121],[23,80912,80914,80917,80920,80923,80926],{"className":80913},[126],[23,80915],{"className":80916,"style":470},[130],[23,80918,68133],{"className":80919},[135],[23,80921],{"className":80922,"style":149},[148],[23,80924,80902],{"className":80925},[153],[23,80927],{"className":80928,"style":149},[148],[23,80930,80932,80935],{"className":80931},[126],[23,80933],{"className":80934,"style":2770},[130],[23,80936,1532],{"className":80937},[135],"，目标函数只依赖 ",[23,80940,80942,80955],{"className":80941,"translate":27},[31],[23,80943,80945],{"className":80944},[35],[37,80946,80947],{"xmlns":39},[42,80948,80949,80953],{},[45,80950,80951],{},[48,80952,8336],{},[114,80954,8336],{"encoding":116},[23,80956,80958],{"className":80957,"ariaHidden":122},[121],[23,80959,80961,80964],{"className":80960},[126],[23,80962],{"className":80963,"style":5414},[130],[23,80965,8336],{"className":80966,"style":11398},[135,136]," 在训练点上的取值。任何在这些点上取值相同的函数都是最优解，其中很多并不在 ",[23,80969,80971,80988],{"className":80970,"translate":27},[31],[23,80972,80974],{"className":80973},[35],[37,80975,80976],{"xmlns":39},[42,80977,80978,80986],{},[45,80979,80980],{},[898,80981,80982,80984],{},[48,80983,26249],{"mathvariant":375},[2581,80985,1532],{},[114,80987,73813],{"encoding":116},[23,80989,80991],{"className":80990,"ariaHidden":122},[121],[23,80992,80994,80997],{"className":80993},[126],[23,80995],{"className":80996,"style":42827},[130],[23,80998,81000,81003],{"className":80999},[135],[23,81001,26249],{"className":81002,"style":45024},[135,412],[23,81004,81006],{"className":81005},[998],[23,81007,81009,81029],{"className":81008},[199,200],[23,81010,81012,81026],{"className":81011},[204],[23,81013,81015],{"className":81014,"style":18647},[208],[23,81016,81017,81020],{"style":61627},[23,81018],{"className":81019,"style":1015},[216],[23,81021,81023],{"className":81022},[1019,1020,1021,1022],[23,81024,1532],{"className":81025},[135,1022],[23,81027,297],{"className":81028},[296],[23,81030,81032],{"className":81031},[204],[23,81033,81035],{"className":81034,"style":2048},[208],[23,81036],{}," 里。表示定理的「存在」版本仍然成立（取 ",[23,81039,81041,81058],{"className":81040,"translate":27},[31],[23,81042,81044],{"className":81043},[35],[37,81045,81046],{"xmlns":39},[42,81047,81048,81056],{},[45,81049,81050],{},[898,81051,81052,81054],{},[48,81053,8336],{},[51,81055,20680],{},[114,81057,73027],{"encoding":116},[23,81059,81061],{"className":81060,"ariaHidden":122},[121],[23,81062,81064,81067],{"className":81063},[126],[23,81065],{"className":81066,"style":70222},[130],[23,81068,81070,81073],{"className":81069},[135],[23,81071,8336],{"className":81072,"style":11398},[135,136],[23,81074,81076],{"className":81075},[998],[23,81077,81079,81099],{"className":81078},[199,200],[23,81080,81082,81096],{"className":81081},[204],[23,81083,81085],{"className":81084,"style":1008},[208],[23,81086,81087,81090],{"style":70243},[23,81088],{"className":81089,"style":1015},[216],[23,81091,81093],{"className":81092},[1019,1020,1021,1022],[23,81094,20680],{"className":81095},[153,1022],[23,81097,297],{"className":81098},[296],[23,81100,81102],{"className":81101},[204],[23,81103,81105],{"className":81104,"style":1057},[208],[23,81106],{}," 即可），但「任意最优解」的版本不成立。因此，「任意最优解都满足」这一说法需要加上严格递增这个条件。",[338,81109,81111],{"id":81110},"五五个经典例子","五、五个经典例子",[2501,81113,81115],{"id":81114},"_51-核岭回归","5.1 核岭回归",[11,81117,81118,81119,69651,81189,81239],{},"取平方损失和 ",[23,81120,81122,81147],{"className":81121,"translate":27},[31],[23,81123,81125],{"className":81124},[35],[37,81126,81127],{"xmlns":39},[42,81128,81129,81145],{},[45,81130,81131,81133,81135,81137,81139,81141,81143],{},[48,81132,68133],{"mathvariant":111},[51,81134,54],{"stretchy":53},[48,81136,5345],{},[51,81138,66],{"stretchy":53},[51,81140,69],{},[48,81142,61753],{},[48,81144,5345],{},[114,81146,69608],{"encoding":116},[23,81148,81150,81177],{"className":81149,"ariaHidden":122},[121],[23,81151,81153,81156,81159,81162,81165,81168,81171,81174],{"className":81152},[126],[23,81154],{"className":81155,"style":131},[130],[23,81157,68133],{"className":81158},[135],[23,81160,54],{"className":81161},[140],[23,81163,5345],{"className":81164},[135,136],[23,81166,66],{"className":81167},[169],[23,81169],{"className":81170,"style":149},[148],[23,81172,69],{"className":81173},[153],[23,81175],{"className":81176,"style":149},[148],[23,81178,81180,81183,81186],{"className":81179},[126],[23,81181],{"className":81182,"style":779},[130],[23,81184,61753],{"className":81185},[135,136],[23,81187,5345],{"className":81188},[135,136],[23,81190,81192,81209],{"className":81191,"translate":27},[31],[23,81193,81195],{"className":81194},[35],[37,81196,81197],{"xmlns":39},[42,81198,81199,81207],{},[45,81200,81201,81203,81205],{},[48,81202,61753],{},[51,81204,2579],{},[2581,81206,1532],{},[114,81208,69672],{"encoding":116},[23,81210,81212,81230],{"className":81211,"ariaHidden":122},[121],[23,81213,81215,81218,81221,81224,81227],{"className":81214},[126],[23,81216],{"className":81217,"style":4450},[130],[23,81219,61753],{"className":81220},[135,136],[23,81222],{"className":81223,"style":149},[148],[23,81225,2579],{"className":81226},[153],[23,81228],{"className":81229,"style":149},[148],[23,81231,81233,81236],{"className":81232},[126],[23,81234],{"className":81235,"style":2770},[130],[23,81237,1532],{"className":81238},[135],"）：",[23,81241,81243],{"className":81242,"translate":27},[26],[23,81244,81246,81340],{"className":81245,"translate":27},[31],[23,81247,81249],{"className":81248},[35],[37,81250,81251],{"xmlns":39,"display":40},[42,81252,81253,81337],{},[45,81254,81255,81271,81335],{},[6961,81256,81257,81263],{},[45,81258,81259,81261],{},[48,81260,6967],{},[51,81262,6959],{},[45,81264,81265,81267,81269],{},[48,81266,8336],{},[51,81268,372],{},[48,81270,26249],{"mathvariant":375},[45,81272,81273,81275,81289,81291,81297,81299,81301,81303,81309,81311,81317,81319,81321,81323,81325,81333],{},[51,81274,9010],{"fence":122},[61709,81276,81277,81279,81287],{},[51,81278,33851],{},[45,81280,81281,81283,81285],{},[48,81282,33493],{},[51,81284,69],{},[2581,81286,4435],{},[48,81288,4806],{},[51,81290,54],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[898,81292,81293,81295],{},[48,81294,660],{},[48,81296,33493],{},[51,81298,4529],{},[48,81300,8336],{},[51,81302,54],{"stretchy":53},[898,81304,81305,81307],{},[48,81306,753],{},[48,81308,33493],{},[51,81310,66],{"stretchy":53},[4518,81312,81313,81315],{},[51,81314,66],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[2581,81316,9952],{},[51,81318,5106],{},[48,81320,61753],{},[48,81322,20680],{"mathvariant":111},[48,81324,8336],{},[19137,81326,81327,81329,81331],{},[48,81328,20680],{"mathvariant":111},[48,81330,26249],{"mathvariant":375},[2581,81332,9952],{},[51,81334,9020],{"fence":122},[48,81336,112],{"mathvariant":111},[114,81338,81339],{"encoding":116},"\\min_{f\\in\\mathcal H}\\left\\{\\sum_{i=1}^n\\bigl(y_i-f(x_i)\\bigr)^2+\\lambda\\|f\\|_{\\mathcal H}^2\\right\\}.",[23,81341,81343],{"className":81342,"ariaHidden":122},[121],[23,81344,81346,81349,81402,81405,81692,81695],{"className":81345},[126],[23,81347],{"className":81348,"style":61833},[130],[23,81350,81352],{"className":81351},[1786,7096],[23,81353,81355,81394],{"className":81354},[199,200],[23,81356,81358,81391],{"className":81357},[204],[23,81359,81361,81381],{"className":81360,"style":7106},[208],[23,81362,81363,81366],{"style":14738},[23,81364],{"className":81365,"style":217},[216],[23,81367,81369],{"className":81368},[1019,1020,1021,1022],[23,81370,81372,81375,81378],{"className":81371},[135,1022],[23,81373,8336],{"className":81374,"style":11398},[135,136,1022],[23,81376,372],{"className":81377},[153,1022],[23,81379,26249],{"className":81380,"style":45024},[135,412,1022],[23,81382,81383,81386],{"style":7130},[23,81384],{"className":81385,"style":217},[216],[23,81387,81388],{},[23,81389,6967],{"className":81390},[1786],[23,81392,297],{"className":81393},[296],[23,81395,81397],{"className":81396},[204],[23,81398,81400],{"className":81399,"style":61893},[208],[23,81401],{},[23,81403],{"className":81404,"style":280},[148],[23,81406,81408,81414,81478,81484,81524,81527,81530,81533,81536,81539,81579,81582,81614,81617,81620,81623,81626,81629,81632,81686],{"className":81407},[1839],[23,81409,81411],{"className":81410,"style":1844},[140,1843],[23,81412,9010],{"className":81413},[1066,10069],[23,81415,81417],{"className":81416},[1786,7096],[23,81418,81420,81470],{"className":81419},[199,200],[23,81421,81423,81467],{"className":81422},[204],[23,81424,81426,81446,81456],{"className":81425,"style":61920},[208],[23,81427,81428,81431],{"style":34075},[23,81429],{"className":81430,"style":34079},[216],[23,81432,81434],{"className":81433},[1019,1020,1021,1022],[23,81435,81437,81440,81443],{"className":81436},[135,1022],[23,81438,33493],{"className":81439},[135,136,1022],[23,81441,69],{"className":81442},[153,1022],[23,81444,4435],{"className":81445},[135,1022],[23,81447,81448,81451],{"style":34088},[23,81449],{"className":81450,"style":34079},[216],[23,81452,81453],{},[23,81454,33851],{"className":81455},[1786,1790,1791],[23,81457,81458,81461],{"style":61953},[23,81459],{"className":81460,"style":34079},[216],[23,81462,81464],{"className":81463},[1019,1020,1021,1022],[23,81465,4806],{"className":81466},[135,136,1022],[23,81468,297],{"className":81469},[296],[23,81471,81473],{"className":81472},[204],[23,81474,81476],{"className":81475,"style":34106},[208],[23,81477],{},[23,81479,81481],{"className":81480},[140],[23,81482,54],{"className":81483},[1066,1067],[23,81485,81487,81490],{"className":81486},[135],[23,81488,660],{"className":81489,"style":676},[135,136],[23,81491,81493],{"className":81492},[998],[23,81494,81496,81516],{"className":81495},[199,200],[23,81497,81499,81513],{"className":81498},[204],[23,81500,81502],{"className":81501,"style":33647},[208],[23,81503,81504,81507],{"style":21850},[23,81505],{"className":81506,"style":1015},[216],[23,81508,81510],{"className":81509},[1019,1020,1021,1022],[23,81511,33493],{"className":81512},[135,136,1022],[23,81514,297],{"className":81515},[296],[23,81517,81519],{"className":81518},[204],[23,81520,81522],{"className":81521,"style":2048},[208],[23,81523],{},[23,81525],{"className":81526,"style":392},[148],[23,81528,4529],{"className":81529},[612],[23,81531],{"className":81532,"style":392},[148],[23,81534,8336],{"className":81535,"style":11398},[135,136],[23,81537,54],{"className":81538},[140],[23,81540,81542,81545],{"className":81541},[135],[23,81543,753],{"className":81544},[135,136],[23,81546,81548],{"className":81547},[998],[23,81549,81551,81571],{"className":81550},[199,200],[23,81552,81554,81568],{"className":81553},[204],[23,81555,81557],{"className":81556,"style":33647},[208],[23,81558,81559,81562],{"style":2029},[23,81560],{"className":81561,"style":1015},[216],[23,81563,81565],{"className":81564},[1019,1020,1021,1022],[23,81566,33493],{"className":81567},[135,136,1022],[23,81569,297],{"className":81570},[296],[23,81572,81574],{"className":81573},[204],[23,81575,81577],{"className":81576,"style":2048},[208],[23,81578],{},[23,81580,66],{"className":81581},[169],[23,81583,81585,81591],{"className":81584},[169],[23,81586,81588],{"className":81587},[169],[23,81589,66],{"className":81590},[1066,1067],[23,81592,81594],{"className":81593},[998],[23,81595,81597],{"className":81596},[199],[23,81598,81600],{"className":81599},[204],[23,81601,81603],{"className":81602,"style":11188},[208],[23,81604,81605,81608],{"style":11191},[23,81606],{"className":81607,"style":1015},[216],[23,81609,81611],{"className":81610},[1019,1020,1021,1022],[23,81612,9952],{"className":81613},[135,1022],[23,81615],{"className":81616,"style":392},[148],[23,81618,5106],{"className":81619},[612],[23,81621],{"className":81622,"style":392},[148],[23,81624,61753],{"className":81625},[135,136],[23,81627,20680],{"className":81628},[135],[23,81630,8336],{"className":81631,"style":11398},[135,136],[23,81633,81635,81638],{"className":81634},[135],[23,81636,20680],{"className":81637},[135],[23,81639,81641],{"className":81640},[998],[23,81642,81644,81678],{"className":81643},[199,200],[23,81645,81647,81675],{"className":81646},[204],[23,81648,81650,81664],{"className":81649,"style":11002},[208],[23,81651,81652,81655],{"style":19345},[23,81653],{"className":81654,"style":1015},[216],[23,81656,81658],{"className":81657},[1019,1020,1021,1022],[23,81659,81661],{"className":81660},[135,1022],[23,81662,26249],{"className":81663,"style":45024},[135,412,1022],[23,81665,81666,81669],{"style":4989},[23,81667],{"className":81668,"style":1015},[216],[23,81670,81672],{"className":81671},[1019,1020,1021,1022],[23,81673,9952],{"className":81674},[135,1022],[23,81676,297],{"className":81677},[296],[23,81679,81681],{"className":81680},[204],[23,81682,81684],{"className":81683,"style":19375},[208],[23,81685],{},[23,81687,81689],{"className":81688,"style":1844},[169,1843],[23,81690,9020],{"className":81691},[1066,10069],[23,81693],{"className":81694,"style":280},[148],[23,81696,112],{"className":81697},[135],[11,81699,81700,81701,81945],{},"由表示定理，",[23,81702,81704,81751],{"className":81703,"translate":27},[31],[23,81705,81707],{"className":81706},[35],[37,81708,81709],{"xmlns":39},[42,81710,81711,81749],{},[45,81712,81713,81719,81721,81727,81733,81735,81737,81739,81741,81747],{},[4518,81714,81715,81717],{},[48,81716,8336],{},[51,81718,6881],{},[51,81720,69],{},[898,81722,81723,81725],{},[51,81724,33851],{},[48,81726,33493],{},[898,81728,81729,81731],{},[48,81730,4526],{},[48,81732,33493],{},[48,81734,4836],{},[51,81736,54],{"stretchy":53},[51,81738,5758],{},[51,81740,747],{"separator":122},[898,81742,81743,81745],{},[48,81744,753],{},[48,81746,33493],{},[51,81748,66],{"stretchy":53},[114,81750,77157],{"encoding":116},[23,81752,81754,81798],{"className":81753,"ariaHidden":122},[121],[23,81755,81757,81760,81789,81792,81795],{"className":81756},[126],[23,81758],{"className":81759,"style":5414},[130],[23,81761,81763,81766],{"className":81762},[135],[23,81764,8336],{"className":81765,"style":11398},[135,136],[23,81767,81769],{"className":81768},[998],[23,81770,81772],{"className":81771},[199],[23,81773,81775],{"className":81774},[204],[23,81776,81778],{"className":81777,"style":6912},[208],[23,81779,81780,81783],{"style":4696},[23,81781],{"className":81782,"style":1015},[216],[23,81784,81786],{"className":81785},[1019,1020,1021,1022],[23,81787,6881],{"className":81788},[612,1022],[23,81790],{"className":81791,"style":149},[148],[23,81793,69],{"className":81794},[153],[23,81796],{"className":81797,"style":149},[148],[23,81799,81801,81804,81844,81847,81887,81890,81893,81896,81899,81902,81942],{"className":81800},[126],[23,81802],{"className":81803,"style":33575},[130],[23,81805,81807,81810],{"className":81806},[1786],[23,81808,33851],{"className":81809,"style":33582},[1786,1790,20454],[23,81811,81813],{"className":81812},[998],[23,81814,81816,81836],{"className":81815},[199,200],[23,81817,81819,81833],{"className":81818},[204],[23,81820,81822],{"className":81821,"style":33595},[208],[23,81823,81824,81827],{"style":33598},[23,81825],{"className":81826,"style":1015},[216],[23,81828,81830],{"className":81829},[1019,1020,1021,1022],[23,81831,33493],{"className":81832},[135,136,1022],[23,81834,297],{"className":81835},[296],[23,81837,81839],{"className":81838},[204],[23,81840,81842],{"className":81841,"style":33617},[208],[23,81843],{},[23,81845],{"className":81846,"style":280},[148],[23,81848,81850,81853],{"className":81849},[135],[23,81851,4526],{"className":81852,"style":4645},[135,136],[23,81854,81856],{"className":81855},[998],[23,81857,81859,81879],{"className":81858},[199,200],[23,81860,81862,81876],{"className":81861},[204],[23,81863,81865],{"className":81864,"style":33647},[208],[23,81866,81867,81870],{"style":68963},[23,81868],{"className":81869,"style":1015},[216],[23,81871,81873],{"className":81872},[1019,1020,1021,1022],[23,81874,33493],{"className":81875},[135,136,1022],[23,81877,297],{"className":81878},[296],[23,81880,81882],{"className":81881},[204],[23,81883,81885],{"className":81884,"style":2048},[208],[23,81886],{},[23,81888,4836],{"className":81889,"style":4851},[135,136],[23,81891,54],{"className":81892},[140],[23,81894,5758],{"className":81895},[135],[23,81897,747],{"className":81898},[824],[23,81900],{"className":81901,"style":280},[148],[23,81903,81905,81908],{"className":81904},[135],[23,81906,753],{"className":81907},[135,136],[23,81909,81911],{"className":81910},[998],[23,81912,81914,81934],{"className":81913},[199,200],[23,81915,81917,81931],{"className":81916},[204],[23,81918,81920],{"className":81919,"style":33647},[208],[23,81921,81922,81925],{"style":2029},[23,81923],{"className":81924,"style":1015},[216],[23,81926,81928],{"className":81927},[1019,1020,1021,1022],[23,81929,33493],{"className":81930},[135,136,1022],[23,81932,297],{"className":81933},[296],[23,81935,81937],{"className":81936},[204],[23,81938,81940],{"className":81939,"style":2048},[208],[23,81941],{},[23,81943,66],{"className":81944},[169],"。代入式 (4)：",[23,81947,81949],{"className":81948,"translate":27},[26],[23,81950,81952,81998],{"className":81951,"translate":27},[31],[23,81953,81955],{"className":81954},[35],[37,81956,81957],{"xmlns":39,"display":40},[42,81958,81959,81995],{},[45,81960,81961,81963,81965,81967,81969,81971,81977,81979,81981,81983,81989,81991,81993],{},[48,81962,20680],{"mathvariant":111},[48,81964,660],{},[51,81966,4529],{},[48,81968,26918],{},[48,81970,4526],{},[4518,81972,81973,81975],{},[48,81974,20680],{"mathvariant":111},[2581,81976,9952],{},[51,81978,5106],{},[48,81980,61753],{},[88,81982,90],{},[4518,81984,81985,81987],{},[48,81986,4526],{},[48,81988,22471],{"mathvariant":111},[48,81990,26918],{},[48,81992,4526],{},[48,81994,112],{"mathvariant":111},[114,81996,81997],{"encoding":116},"\\|y-K\\alpha\\|^2+\\lambda\\,\\alpha^\\top K\\alpha.",[23,81999,82001,82022,82072],{"className":82000,"ariaHidden":122},[121],[23,82002,82004,82007,82010,82013,82016,82019],{"className":82003},[126],[23,82005],{"className":82006,"style":131},[130],[23,82008,20680],{"className":82009},[135],[23,82011,660],{"className":82012,"style":676},[135,136],[23,82014],{"className":82015,"style":392},[148],[23,82017,4529],{"className":82018},[612],[23,82020],{"className":82021,"style":392},[148],[23,82023,82025,82028,82031,82034,82063,82066,82069],{"className":82024},[126],[23,82026],{"className":82027,"style":26499},[130],[23,82029,26918],{"className":82030,"style":37141},[135,136],[23,82032,4526],{"className":82033,"style":4645},[135,136],[23,82035,82037,82040],{"className":82036},[135],[23,82038,20680],{"className":82039},[135],[23,82041,82043],{"className":82042},[998],[23,82044,82046],{"className":82045},[199],[23,82047,82049],{"className":82048},[204],[23,82050,82052],{"className":82051,"style":11002},[208],[23,82053,82054,82057],{"style":4989},[23,82055],{"className":82056,"style":1015},[216],[23,82058,82060],{"className":82059},[1019,1020,1021,1022],[23,82061,9952],{"className":82062},[135,1022],[23,82064],{"className":82065,"style":392},[148],[23,82067,5106],{"className":82068},[612],[23,82070],{"className":82071,"style":392},[148],[23,82073,82075,82078,82081,82084,82113,82116,82119],{"className":82074},[126],[23,82076],{"className":82077,"style":4986},[130],[23,82079,61753],{"className":82080},[135,136],[23,82082],{"className":82083,"style":280},[148],[23,82085,82087,82090],{"className":82086},[135],[23,82088,4526],{"className":82089,"style":4645},[135,136],[23,82091,82093],{"className":82092},[998],[23,82094,82096],{"className":82095},[199],[23,82097,82099],{"className":82098},[204],[23,82100,82102],{"className":82101,"style":4986},[208],[23,82103,82104,82107],{"style":4989},[23,82105],{"className":82106,"style":1015},[216],[23,82108,82110],{"className":82109},[1019,1020,1021,1022],[23,82111,22471],{"className":82112},[135,1022],[23,82114,26918],{"className":82115,"style":37141},[135,136],[23,82117,4526],{"className":82118,"style":4645},[135,136],[23,82120,112],{"className":82121},[135],[11,82123,16375,82124,82152],{},[23,82125,82127,82140],{"className":82126,"translate":27},[31],[23,82128,82130],{"className":82129},[35],[37,82131,82132],{"xmlns":39},[42,82133,82134,82138],{},[45,82135,82136],{},[48,82137,4526],{},[114,82139,76426],{"encoding":116},[23,82141,82143],{"className":82142,"ariaHidden":122},[121],[23,82144,82146,82149],{"className":82145},[126],[23,82147],{"className":82148,"style":703},[130],[23,82150,4526],{"className":82151,"style":4645},[135,136]," 求梯度：",[23,82154,82156],{"className":82155,"translate":27},[26],[23,82157,82159,82229],{"className":82158,"translate":27},[31],[23,82160,82162],{"className":82161},[35],[37,82163,82164],{"xmlns":39,"display":40},[42,82165,82166,82226],{},[45,82167,82168,82170,82172,82174,82176,82178,82180,82182,82184,82186,82188,82190,82192,82194,82196,82198,82200,82202,82204,82206,82208,82210,82212,82214,82216,82218,82220,82222,82224],{},[51,82169,4529],{},[2581,82171,9952],{},[48,82173,26918],{},[51,82175,54],{"stretchy":53},[48,82177,660],{},[51,82179,4529],{},[48,82181,26918],{},[48,82183,4526],{},[51,82185,66],{"stretchy":53},[51,82187,5106],{},[2581,82189,9952],{},[48,82191,61753],{},[48,82193,26918],{},[48,82195,4526],{},[51,82197,69],{},[2581,82199,9952],{},[48,82201,26918],{},[51,82203,54],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[51,82205,54],{"stretchy":53},[48,82207,26918],{},[51,82209,5106],{},[48,82211,61753],{},[48,82213,34353],{},[51,82215,66],{"stretchy":53},[48,82217,4526],{},[51,82219,4529],{},[48,82221,660],{},[51,82223,66],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[48,82225,112],{"mathvariant":111},[114,82227,82228],{"encoding":116},"-2K(y-K\\alpha)+2\\lambda K\\alpha\n=2K\\bigl((K+\\lambda I)\\alpha-y\\bigr).",[23,82230,82232,82262,82286,82313,82346,82373],{"className":82231,"ariaHidden":122},[121],[23,82233,82235,82238,82241,82244,82247,82250,82253,82256,82259],{"className":82234},[126],[23,82236],{"className":82237,"style":131},[130],[23,82239,4529],{"className":82240},[135],[23,82242,9952],{"className":82243},[135],[23,82245,26918],{"className":82246,"style":37141},[135,136],[23,82248,54],{"className":82249},[140],[23,82251,660],{"className":82252,"style":676},[135,136],[23,82254],{"className":82255,"style":392},[148],[23,82257,4529],{"className":82258},[612],[23,82260],{"className":82261,"style":392},[148],[23,82263,82265,82268,82271,82274,82277,82280,82283],{"className":82264},[126],[23,82266],{"className":82267,"style":131},[130],[23,82269,26918],{"className":82270,"style":37141},[135,136],[23,82272,4526],{"className":82273,"style":4645},[135,136],[23,82275,66],{"className":82276},[169],[23,82278],{"className":82279,"style":392},[148],[23,82281,5106],{"className":82282},[612],[23,82284],{"className":82285,"style":392},[148],[23,82287,82289,82292,82295,82298,82301,82304,82307,82310],{"className":82288},[126],[23,82290],{"className":82291,"style":779},[130],[23,82293,9952],{"className":82294},[135],[23,82296,61753],{"className":82297},[135,136],[23,82299,26918],{"className":82300,"style":37141},[135,136],[23,82302,4526],{"className":82303,"style":4645},[135,136],[23,82305],{"className":82306,"style":149},[148],[23,82308,69],{"className":82309},[153],[23,82311],{"className":82312,"style":149},[148],[23,82314,82316,82319,82322,82325,82331,82334,82337,82340,82343],{"className":82315},[126],[23,82317],{"className":82318,"style":2213},[130],[23,82320,9952],{"className":82321},[135],[23,82323,26918],{"className":82324,"style":37141},[135,136],[23,82326,82328],{"className":82327},[140],[23,82329,54],{"className":82330},[1066,1067],[23,82332,54],{"className":82333},[140],[23,82335,26918],{"className":82336,"style":37141},[135,136],[23,82338],{"className":82339,"style":392},[148],[23,82341,5106],{"className":82342},[612],[23,82344],{"className":82345,"style":392},[148],[23,82347,82349,82352,82355,82358,82361,82364,82367,82370],{"className":82348},[126],[23,82350],{"className":82351,"style":131},[130],[23,82353,61753],{"className":82354},[135,136],[23,82356,34353],{"className":82357,"style":454},[135,136],[23,82359,66],{"className":82360},[169],[23,82362,4526],{"className":82363,"style":4645},[135,136],[23,82365],{"className":82366,"style":392},[148],[23,82368,4529],{"className":82369},[612],[23,82371],{"className":82372,"style":392},[148],[23,82374,82376,82379,82382,82388],{"className":82375},[126],[23,82377],{"className":82378,"style":2213},[130],[23,82380,660],{"className":82381,"style":676},[135,136],[23,82383,82385],{"className":82384},[169],[23,82386,66],{"className":82387},[1066,1067],[23,82389,112],{"className":82390},[135],[11,82392,82393,82394,82497,82498,82548,82549,82605],{},"令梯度为零，得到正规方程 ",[23,82395,82397,82431],{"className":82396,"translate":27},[31],[23,82398,82400],{"className":82399},[35],[37,82401,82402],{"xmlns":39},[42,82403,82404,82428],{},[45,82405,82406,82408,82410,82412,82414,82416,82418,82420,82422,82424,82426],{},[48,82407,26918],{},[51,82409,54],{"stretchy":53},[48,82411,26918],{},[51,82413,5106],{},[48,82415,61753],{},[48,82417,34353],{},[51,82419,66],{"stretchy":53},[48,82421,4526],{},[51,82423,69],{},[48,82425,26918],{},[48,82427,660],{},[114,82429,82430],{"encoding":116},"K(K+\\lambda I)\\alpha=Ky",[23,82432,82434,82458,82485],{"className":82433,"ariaHidden":122},[121],[23,82435,82437,82440,82443,82446,82449,82452,82455],{"className":82436},[126],[23,82438],{"className":82439,"style":131},[130],[23,82441,26918],{"className":82442,"style":37141},[135,136],[23,82444,54],{"className":82445},[140],[23,82447,26918],{"className":82448,"style":37141},[135,136],[23,82450],{"className":82451,"style":392},[148],[23,82453,5106],{"className":82454},[612],[23,82456],{"className":82457,"style":392},[148],[23,82459,82461,82464,82467,82470,82473,82476,82479,82482],{"className":82460},[126],[23,82462],{"className":82463,"style":131},[130],[23,82465,61753],{"className":82466},[135,136],[23,82468,34353],{"className":82469,"style":454},[135,136],[23,82471,66],{"className":82472},[169],[23,82474,4526],{"className":82475,"style":4645},[135,136],[23,82477],{"className":82478,"style":149},[148],[23,82480,69],{"className":82481},[153],[23,82483],{"className":82484,"style":149},[148],[23,82486,82488,82491,82494],{"className":82487},[126],[23,82489],{"className":82490,"style":817},[130],[23,82492,26918],{"className":82493,"style":37141},[135,136],[23,82495,660],{"className":82496,"style":676},[135,136],"。由于 ",[23,82499,82501,82518],{"className":82500,"translate":27},[31],[23,82502,82504],{"className":82503},[35],[37,82505,82506],{"xmlns":39},[42,82507,82508,82516],{},[45,82509,82510,82512,82514],{},[48,82511,61753],{},[51,82513,2579],{},[2581,82515,1532],{},[114,82517,69672],{"encoding":116},[23,82519,82521,82539],{"className":82520,"ariaHidden":122},[121],[23,82522,82524,82527,82530,82533,82536],{"className":82523},[126],[23,82525],{"className":82526,"style":4450},[130],[23,82528,61753],{"className":82529},[135,136],[23,82531],{"className":82532,"style":149},[148],[23,82534,2579],{"className":82535},[153],[23,82537],{"className":82538,"style":149},[148],[23,82540,82542,82545],{"className":82541},[126],[23,82543],{"className":82544,"style":2770},[130],[23,82546,1532],{"className":82547},[135]," 时 ",[23,82550,82552,82572],{"className":82551,"translate":27},[31],[23,82553,82555],{"className":82554},[35],[37,82556,82557],{"xmlns":39},[42,82558,82559,82569],{},[45,82560,82561,82563,82565,82567],{},[48,82562,26918],{},[51,82564,5106],{},[48,82566,61753],{},[48,82568,34353],{},[114,82570,82571],{"encoding":116},"K+\\lambda I",[23,82573,82575,82593],{"className":82574,"ariaHidden":122},[121],[23,82576,82578,82581,82584,82587,82590],{"className":82577},[126],[23,82579],{"className":82580,"style":62460},[130],[23,82582,26918],{"className":82583,"style":37141},[135,136],[23,82585],{"className":82586,"style":392},[148],[23,82588,5106],{"className":82589},[612],[23,82591],{"className":82592,"style":392},[148],[23,82594,82596,82599,82602],{"className":82595},[126],[23,82597],{"className":82598,"style":779},[130],[23,82600,61753],{"className":82601},[135,136],[23,82603,34353],{"className":82604,"style":454},[135,136]," 可逆，",[23,82607,82609],{"className":82608,"translate":27},[26],[23,82610,82612,82649],{"className":82611,"translate":27},[31],[23,82613,82615],{"className":82614},[35],[37,82616,82617],{"xmlns":39,"display":40},[42,82618,82619,82647],{},[45,82620,82621,82623,82625,82627,82629,82631,82633,82635,82645],{},[48,82622,4526],{},[51,82624,69],{},[51,82626,54],{"stretchy":53},[48,82628,26918],{},[51,82630,5106],{},[48,82632,61753],{},[48,82634,34353],{},[4518,82636,82637,82639],{},[51,82638,66],{"stretchy":53},[45,82640,82641,82643],{},[51,82642,4529],{},[2581,82644,4435],{},[48,82646,660],{},[114,82648,76788],{"encoding":116},[23,82650,82652,82670,82691],{"className":82651,"ariaHidden":122},[121],[23,82653,82655,82658,82661,82664,82667],{"className":82654},[126],[23,82656],{"className":82657,"style":703},[130],[23,82659,4526],{"className":82660,"style":4645},[135,136],[23,82662],{"className":82663,"style":149},[148],[23,82665,69],{"className":82666},[153],[23,82668],{"className":82669,"style":149},[148],[23,82671,82673,82676,82679,82682,82685,82688],{"className":82672},[126],[23,82674],{"className":82675,"style":131},[130],[23,82677,54],{"className":82678},[140],[23,82680,26918],{"className":82681,"style":37141},[135,136],[23,82683],{"className":82684,"style":392},[148],[23,82686,5106],{"className":82687},[612],[23,82689],{"className":82690,"style":392},[148],[23,82692,82694,82697,82700,82703,82738],{"className":82693},[126],[23,82695],{"className":82696,"style":26499},[130],[23,82698,61753],{"className":82699},[135,136],[23,82701,34353],{"className":82702,"style":454},[135,136],[23,82704,82706,82709],{"className":82705},[169],[23,82707,66],{"className":82708},[169],[23,82710,82712],{"className":82711},[998],[23,82713,82715],{"className":82714},[199],[23,82716,82718],{"className":82717},[204],[23,82719,82721],{"className":82720,"style":11002},[208],[23,82722,82723,82726],{"style":4989},[23,82724],{"className":82725,"style":1015},[216],[23,82727,82729],{"className":82728},[1019,1020,1021,1022],[23,82730,82732,82735],{"className":82731},[135,1022],[23,82733,4529],{"className":82734},[135,1022],[23,82736,4435],{"className":82737},[135,1022],[23,82739,660],{"className":82740,"style":676},[135,136],[11,82742,82743,82744,82772,82773,82801],{},"是一个解（即使 ",[23,82745,82747,82760],{"className":82746,"translate":27},[31],[23,82748,82750],{"className":82749},[35],[37,82751,82752],{"xmlns":39},[42,82753,82754,82758],{},[45,82755,82756],{},[48,82757,26918],{},[114,82759,26918],{"encoding":116},[23,82761,82763],{"className":82762,"ariaHidden":122},[121],[23,82764,82766,82769],{"className":82765},[126],[23,82767],{"className":82768,"style":470},[130],[23,82770,26918],{"className":82771,"style":37141},[135,136]," 奇异，这个 ",[23,82774,82776,82789],{"className":82775,"translate":27},[31],[23,82777,82779],{"className":82778},[35],[37,82780,82781],{"xmlns":39},[42,82782,82783,82787],{},[45,82784,82785],{},[48,82786,4526],{},[114,82788,76426],{"encoding":116},[23,82790,82792],{"className":82791,"ariaHidden":122},[121],[23,82793,82795,82798],{"className":82794},[126],[23,82796],{"className":82797,"style":703},[130],[23,82799,4526],{"className":82800,"style":4645},[135,136]," 也满足正规方程；不同的解给出同一个函数）。预测为",[23,82803,82805],{"className":82804,"translate":27},[26],[23,82806,82808,82930],{"className":82807,"translate":27},[31],[23,82809,82811],{"className":82810},[35],[37,82812,82813],{"xmlns":39,"display":40},[42,82814,82815,82927],{},[45,82816,82817,82823,82825,82827,82829,82831,82833,82835,82837,82843,82845,82847,82849,82851,82853,82863,82865,82867,82869,82871,82873,82875,82877,82879,82881,82883,82885,82887,82889,82895,82897,82899,82901,82903,82905,82907,82909,82911,82917,82919,82925],{},[4518,82818,82819,82821],{},[48,82820,8336],{},[51,82822,6881],{},[51,82824,54],{"stretchy":53},[48,82826,753],{},[51,82828,66],{"stretchy":53},[51,82830,69],{},[48,82832,4836],{},[51,82834,54],{"stretchy":53},[48,82836,753],{},[4518,82838,82839,82841],{},[51,82840,66],{"stretchy":53},[48,82842,22471],{"mathvariant":111},[51,82844,54],{"stretchy":53},[48,82846,26918],{},[51,82848,5106],{},[48,82850,61753],{},[48,82852,34353],{},[4518,82854,82855,82857],{},[51,82856,66],{"stretchy":53},[45,82858,82859,82861],{},[51,82860,4529],{},[2581,82862,4435],{},[48,82864,660],{},[51,82866,747],{"separator":122},[148,82868],{"linebreak":29625},[48,82870,4836],{},[51,82872,54],{"stretchy":53},[48,82874,753],{},[51,82876,66],{"stretchy":53},[51,82878,69],{},[51,82880,54],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[48,82882,4836],{},[51,82884,54],{"stretchy":53},[48,82886,753],{},[51,82888,747],{"separator":122},[898,82890,82891,82893],{},[48,82892,753],{},[2581,82894,4435],{},[51,82896,66],{"stretchy":53},[51,82898,747],{"separator":122},[51,82900,62555],{},[51,82902,747],{"separator":122},[48,82904,4836],{},[51,82906,54],{"stretchy":53},[48,82908,753],{},[51,82910,747],{"separator":122},[898,82912,82913,82915],{},[48,82914,753],{},[48,82916,4806],{},[51,82918,66],{"stretchy":53},[4518,82920,82921,82923],{},[51,82922,66],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[48,82924,22471],{"mathvariant":111},[48,82926,112],{"mathvariant":111},[114,82928,82929],{"encoding":116},"f^*(x)=k(x)^\\top(K+\\lambda I)^{-1}y,\n\\\\\nk(x)=\\bigl(k(x,x_1),\\dots,k(x,x_n)\\bigr)^\\top.",[23,82931,82933,82986,83045,83098,83101,83128],{"className":82932,"ariaHidden":122},[121],[23,82934,82936,82939,82968,82971,82974,82977,82980,82983],{"className":82935},[126],[23,82937],{"className":82938,"style":131},[130],[23,82940,82942,82945],{"className":82941},[135],[23,82943,8336],{"className":82944,"style":11398},[135,136],[23,82946,82948],{"className":82947},[998],[23,82949,82951],{"className":82950},[199],[23,82952,82954],{"className":82953},[204],[23,82955,82957],{"className":82956,"style":7045},[208],[23,82958,82959,82962],{"style":4989},[23,82960],{"className":82961,"style":1015},[216],[23,82963,82965],{"className":82964},[1019,1020,1021,1022],[23,82966,6881],{"className":82967},[612,1022],[23,82969,54],{"className":82970},[140],[23,82972,753],{"className":82973},[135,136],[23,82975,66],{"className":82976},[169],[23,82978],{"className":82979,"style":149},[148],[23,82981,69],{"className":82982},[153],[23,82984],{"className":82985,"style":149},[148],[23,82987,82989,82992,82995,82998,83001,83030,83033,83036,83039,83042],{"className":82988},[126],[23,82990],{"className":82991,"style":4967},[130],[23,82993,4836],{"className":82994,"style":4851},[135,136],[23,82996,54],{"className":82997},[140],[23,82999,753],{"className":83000},[135,136],[23,83002,83004,83007],{"className":83003},[169],[23,83005,66],{"className":83006},[169],[23,83008,83010],{"className":83009},[998],[23,83011,83013],{"className":83012},[199],[23,83014,83016],{"className":83015},[204],[23,83017,83019],{"className":83018,"style":4986},[208],[23,83020,83021,83024],{"style":4989},[23,83022],{"className":83023,"style":1015},[216],[23,83025,83027],{"className":83026},[1019,1020,1021,1022],[23,83028,22471],{"className":83029},[135,1022],[23,83031,54],{"className":83032},[140],[23,83034,26918],{"className":83035,"style":37141},[135,136],[23,83037],{"className":83038,"style":392},[148],[23,83040,5106],{"className":83041},[612],[23,83043],{"className":83044,"style":392},[148],[23,83046,83048,83051,83054,83057,83092,83095],{"className":83047},[126],[23,83049],{"className":83050,"style":26499},[130],[23,83052,61753],{"className":83053},[135,136],[23,83055,34353],{"className":83056,"style":454},[135,136],[23,83058,83060,83063],{"className":83059},[169],[23,83061,66],{"className":83062},[169],[23,83064,83066],{"className":83065},[998],[23,83067,83069],{"className":83068},[199],[23,83070,83072],{"className":83071},[204],[23,83073,83075],{"className":83074,"style":11002},[208],[23,83076,83077,83080],{"style":4989},[23,83078],{"className":83079,"style":1015},[216],[23,83081,83083],{"className":83082},[1019,1020,1021,1022],[23,83084,83086,83089],{"className":83085},[135,1022],[23,83087,4529],{"className":83088},[135,1022],[23,83090,4435],{"className":83091},[135,1022],[23,83093,660],{"className":83094,"style":676},[135,136],[23,83096,747],{"className":83097},[824],[23,83099],{"className":83100},[148,29625],[23,83102,83104,83107,83110,83113,83116,83119,83122,83125],{"className":83103},[126],[23,83105],{"className":83106,"style":131},[130],[23,83108,4836],{"className":83109,"style":4851},[135,136],[23,83111,54],{"className":83112},[140],[23,83114,753],{"className":83115},[135,136],[23,83117,66],{"className":83118},[169],[23,83120],{"className":83121,"style":149},[148],[23,83123,69],{"className":83124},[153],[23,83126],{"className":83127,"style":149},[148],[23,83129,83131,83135,83141,83144,83147,83150,83153,83156,83196,83199,83202,83205,83208,83211,83214,83217,83220,83223,83226,83229,83232,83272,83275,83308],{"className":83130},[126],[23,83132],{"className":83133,"style":83134},[130],"height:1.439em;vertical-align:-0.35em;",[23,83136,83138],{"className":83137},[140],[23,83139,54],{"className":83140},[1066,1067],[23,83142,4836],{"className":83143,"style":4851},[135,136],[23,83145,54],{"className":83146},[140],[23,83148,753],{"className":83149},[135,136],[23,83151,747],{"className":83152},[824],[23,83154],{"className":83155,"style":280},[148],[23,83157,83159,83162],{"className":83158},[135],[23,83160,753],{"className":83161},[135,136],[23,83163,83165],{"className":83164},[998],[23,83166,83168,83188],{"className":83167},[199,200],[23,83169,83171,83185],{"className":83170},[204],[23,83172,83174],{"className":83173,"style":18647},[208],[23,83175,83176,83179],{"style":2029},[23,83177],{"className":83178,"style":1015},[216],[23,83180,83182],{"className":83181},[1019,1020,1021,1022],[23,83183,4435],{"className":83184},[135,1022],[23,83186,297],{"className":83187},[296],[23,83189,83191],{"className":83190},[204],[23,83192,83194],{"className":83193,"style":2048},[208],[23,83195],{},[23,83197,66],{"className":83198},[169],[23,83200,747],{"className":83201},[824],[23,83203],{"className":83204,"style":280},[148],[23,83206,62555],{"className":83207},[1839],[23,83209],{"className":83210,"style":280},[148],[23,83212,747],{"className":83213},[824],[23,83215],{"className":83216,"style":280},[148],[23,8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",[23,83315,83317,83341],{"className":83316,"translate":27},[31],[23,83318,83320],{"className":83319},[35],[37,83321,83322],{"xmlns":39},[42,83323,83324,83338],{},[45,83325,83326,83328,83330,83336],{},[48,83327,42340],{},[51,83329,54],{"stretchy":53},[4518,83331,83332,83334],{},[48,83333,4806],{},[2581,83335,24919],{},[51,83337,66],{"stretchy":53},[114,83339,83340],{"encoding":116},"O(n^3)",[23,83342,83344],{"className":83343,"ariaHidden":122},[121],[23,83345,83347,83350,83353,83356,83385],{"className":83346},[126],[23,83348],{"className":83349,"style":10661},[130],[23,83351,42340],{"className":83352,"style":144},[135,136],[23,83354,54],{"className":83355},[140],[23,83357,83359,83362],{"className":83358},[135],[23,83360,4806],{"className":83361},[135,136],[23,83363,83365],{"className":83364},[998],[23,83366,83368],{"className":83367},[199],[23,83369,83371],{"className":83370},[204],[23,83372,83374],{"className":83373,"style":4693},[208],[23,83375,83376,83379],{"style":4696},[23,83377],{"className":83378,"style":1015},[216],[23,83380,83382],{"className":83381},[1019,1020,1021,1022],[23,83383,24919],{"className":83384},[135,1022],[23,83386,66],{"className":83387},[169]," 时间、",[23,83390,83392,83416],{"className":83391,"translate":27},[31],[23,83393,83395],{"className":83394},[35],[37,83396,83397],{"xmlns":39},[42,83398,83399,83413],{},[45,83400,83401,83403,83405,83411],{},[48,83402,42340],{},[51,83404,54],{"stretchy":53},[4518,83406,83407,83409],{},[48,83408,4806],{},[2581,83410,9952],{},[51,83412,66],{"stretchy":53},[114,83414,83415],{"encoding":116},"O(n^2)",[23,83417,83419],{"className":83418,"ariaHidden":122},[121],[23,83420,83422,83425,83428,83431,83460],{"className":83421},[126],[23,83423],{"className":83424,"style":10661},[130],[23,83426,42340],{"className":83427,"style":144},[135,136],[23,83429,54],{"className":83430},[140],[23,83432,83434,83437],{"className":83433},[135],[23,83435,4806],{"className":83436},[135,136],[23,83438,83440],{"className":83439},[998],[23,83441,83443],{"className":83442},[199],[23,83444,83446],{"className":83445},[204],[23,83447,83449],{"className":83448,"style":4693},[208],[23,83450,83451,83454],{"style":4696},[23,83452],{"className":83453,"style":1015},[216],[23,83455,83457],{"className":83456},[1019,1020,1021,1022],[23,83458,9952],{"className":83459},[135,1022],[23,83461,66],{"className":83462},[169]," 内存；预测一个点需要 ",[23,83465,83467,83487],{"className":83466,"translate":27},[31],[23,83468,83470],{"className":83469},[35],[37,83471,83472],{"xmlns":39},[42,83473,83474,83484],{},[45,83475,83476,83478,83480,83482],{},[48,83477,42340],{},[51,83479,54],{"stretchy":53},[48,83481,4806],{},[51,83483,66],{"stretchy":53},[114,83485,83486],{"encoding":116},"O(n)",[23,83488,83490],{"className":83489,"ariaHidden":122},[121],[23,83491,83493,83496,83499,83502,83505],{"className":83492},[126],[23,83494],{"className":83495,"style":131},[130],[23,83497,42340],{"className":83498,"style":144},[135,136],[23,83500,54],{"className":83501},[140],[23,83503,4806],{"className":83504},[135,136],[23,83506,66],{"className":83507},[169]," 次核函数求值。",[2501,83510,83512],{"id":83511},"_52-支持向量机","5.2 支持向量机",[11,83514,83515],{},"软间隔 SVM 在 RKHS 中的 primal 问题是",[23,83517,83519],{"className":83518,"translate":27},[26],[23,83520,83522,83646],{"className":83521,"translate":27},[31],[23,83523,83525],{"className":83524},[35],[37,83526,83527],{"xmlns":39,"display":40},[42,83528,83529,83643],{},[45,83530,83531,83557,83641],{},[6961,83532,83533,83539],{},[45,83534,83535,83537],{},[48,83536,6967],{},[51,83538,6959],{},[45,83540,83541,83543,83545,83547,83549,83551,83553,83555],{},[48,83542,8336],{},[51,83544,372],{},[48,83546,26249],{"mathvariant":375},[51,83548,747],{"separator":122},[88,83550,90],{},[48,83552,35533],{},[51,83554,372],{},[48,83556,572],{"mathvariant":571},[45,83558,83559,83561,83567,83569,83571,83579,83581,83583,83597,83599,83601,83603,83605,83607,83609,83611,83617,83619,83621,83623,83629,83631,83633,83635,83637,83639],{},[51,83560,9010],{"fence":122},[71,83562,83563,83565],{},[2581,83564,4435],{},[2581,83566,9952],{},[48,83568,20680],{"mathvariant":111},[48,83570,8336],{},[19137,83572,83573,83575,83577],{},[48,83574,20680],{"mathvariant":111},[48,83576,26249],{"mathvariant":375},[2581,83578,9952],{},[51,83580,5106],{},[48,83582,24979],{},[61709,83584,83585,83587,83595],{},[51,83586,33851],{},[45,83588,83589,83591,83593],{},[48,83590,33493],{},[51,83592,69],{},[2581,83594,4435],{},[48,83596,4806],{},[48,83598,9551],{},[51,83600,6959],{},[51,83602,54],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[2581,83604,1532],{},[51,83606,747],{"separator":122},[2581,83608,4435],{},[51,83610,4529],{},[898,83612,83613,83615],{},[48,83614,660],{},[48,83616,33493],{},[51,83618,54],{"stretchy":53},[48,83620,8336],{},[51,83622,54],{"stretchy":53},[898,83624,83625,83627],{},[48,83626,753],{},[48,83628,33493],{},[51,83630,66],{"stretchy":53},[51,83632,5106],{},[48,83634,35533],{},[51,83636,66],{"stretchy":53},[51,83638,66],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[51,83640,9020],{"fence":122},[48,83642,112],{"mathvariant":111},[114,83644,83645],{"encoding":116},"\\min_{f\\in\\mathcal 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条件给出：",[346,85763,85764,86043,86322],{},[349,85765,73342,85766,46543,85951,63810],{},[23,85767,85769,85811],{"className":85768,"translate":27},[31],[23,85770,85772],{"className":85771},[35],[37,85773,85774],{"xmlns":39},[42,85775,85776,85808],{},[45,85777,85778,85784,85786,85788,85790,85796,85798,85800,85802,85804,85806],{},[898,85779,85780,85782],{},[48,85781,660],{},[48,85783,33493],{},[51,85785,54],{"stretchy":53},[48,85787,8336],{},[51,85789,54],{"stretchy":53},[898,85791,85792,85794],{},[48,85793,753],{},[48,85795,33493],{},[51,85797,66],{"stretchy":53},[51,85799,5106],{},[48,85801,35533],{},[51,85803,66],{"stretchy":53},[51,85805,2579],{},[2581,85807,4435],{},[114,85809,85810],{"encoding":116},"y_i(f(x_i)+b)>1",[23,85812,85814,85921,85942],{"className":85813,"ariaHidden":122},[121],[23,85815,85817,85820,85860,85863,85866,85869,85909,85912,85915,85918],{"className":85816},[126],[23,85818],{"className":85819,"style":131},[130],[23,85821,85823,85826],{"className":85822},[135],[23,85824,660],{"className":85825,"style":676},[135,136],[23,85827,85829],{"className":85828},[998],[23,85830,85832,85852],{"className":85831},[199,200],[23,85833,85835,85849],{"className":85834},[204],[23,85836,85838],{"className":85837,"style":33647},[208],[23,85839,85840,85843],{"style":21850},[23,85841],{"className":85842,"style":1015},[216],[23,85844,85846],{"className":85845},[1019,1020,1021,1022],[23,85847,33493],{"className":85848},[135,136,1022],[23,85850,297],{"className":85851},[296],[23,85853,85855],{"className":85854},[204],[23,85856,85858],{"className":85857,"style":2048},[208],[23,85859],{},[23,85861,54],{"className":85862},[140],[23,85864,8336],{"className":85865,"style":11398},[135,136],[23,85867,54],{"className":85868},[140],[23,85870,85872,85875],{"className":85871},[135],[23,85873,753],{"className":85874},[135,136],[23,85876,85878],{"className":85877},[998],[23,85879,85881,85901],{"className":85880},[199,200],[23,85882,85884,85898],{"className":85883},[204],[23,85885,85887],{"className":85886,"style":33647},[208],[23,85888,85889,85892],{"style":2029},[23,85890],{"className":85891,"style":1015},[216],[23,85893,85895],{"className":85894},[1019,1020,1021,1022],[23,85896,33493],{"className":85897},[135,136,1022],[23,85899,297],{"className":85900},[296],[23,85902,85904],{"className":85903},[204],[23,85905,85907],{"className":85906,"style":2048},[208],[23,85908],{},[23,85910,66],{"className":85911},[169],[23,85913],{"className":85914,"style":392},[148],[23,85916,5106],{"className":85917},[612],[23,85919],{"className":85920,"style":392},[148],[23,85922,85924,85927,85930,85933,85936,85939],{"className":85923},[126],[23,85925],{"className":85926,"style":131},[130],[23,85928,35533],{"className":85929},[135,136],[23,85931,66],{"className":85932},[169],[23,85934],{"className":85935,"style":149},[148],[23,85937,2579],{"className":85938},[153],[23,85940],{"className":85941,"style":149},[148],[23,85943,85945,85948],{"className":85944},[126],[23,85946],{"className":85947,"style":2770},[130],[23,85949,4435],{"className":85950},[135],[23,85952,85954,85976],{"className":85953,"translate":27},[31],[23,85955,85957],{"className":85956},[35],[37,85958,85959],{"xmlns":39},[42,85960,85961,85973],{},[45,85962,85963,85969,85971],{},[898,85964,85965,85967],{},[48,85966,4526],{},[48,85968,33493],{},[51,85970,69],{},[2581,85972,1532],{},[114,85974,85975],{"encoding":116},"\\alpha_i=0",[23,85977,85979,86034],{"className":85978,"ariaHidden":122},[121],[23,85980,85982,85985,86025,86028,86031],{"className":85981},[126],[23,85983],{"className":85984,"style":12730},[130],[23,85986,85988,85991],{"className":85987},[135],[23,85989,4526],{"className":85990,"style":4645},[135,136],[23,85992,85994],{"className":85993},[998],[23,85995,85997,86017],{"className":85996},[199,200],[23,85998,86000,86014],{"className":85999},[204],[23,86001,86003],{"className":86002,"style":33647},[208],[23,86004,86005,86008],{"style":68963},[23,86006],{"className":86007,"style":1015},[216],[23,86009,86011],{"className":86010},[1019,1020,1021,1022],[23,86012,33493],{"className":86013},[135,136,1022],[23,86015,297],{"className":86016},[296],[23,86018,86020],{"className":86019},[204],[23,86021,86023],{"className":86022,"style":2048},[208],[23,86024],{},[23,86026],{"className":86027,"style":149},[148],[23,86029,69],{"className":86030},[153],[23,86032],{"className":86033,"style":149},[148],[23,86035,86037,86040],{"className":86036},[126],[23,86038],{"className":86039,"style":2770},[130],[23,86041,1532],{"className":86042},[135],[349,86044,73342,86045,46543,86230,63810],{},[23,86046,86048,86090],{"className":86047,"translate":27},[31],[23,86049,86051],{"className":86050},[35],[37,86052,86053],{"xmlns":39},[42,86054,86055,86087],{},[45,86056,86057,86063,86065,86067,86069,86075,86077,86079,86081,86083,86085],{},[898,86058,86059,86061],{},[48,86060,660],{},[48,86062,33493],{},[51,86064,54],{"stretchy":53},[48,86066,8336],{},[51,86068,54],{"stretchy":53},[898,86070,86071,86073],{},[48,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C",[23,86539,86541,86560,86616],{"className":86540,"ariaHidden":122},[121],[23,86542,86544,86548,86551,86554,86557],{"className":86543},[126],[23,86545],{"className":86546,"style":86547},[130],"height:0.7804em;vertical-align:-0.136em;",[23,86549,1532],{"className":86550},[135],[23,86552],{"className":86553,"style":149},[148],[23,86555,12268],{"className":86556},[153],[23,86558],{"className":86559,"style":149},[148],[23,86561,86563,86567,86607,86610,86613],{"className":86562},[126],[23,86564],{"className":86565,"style":86566},[130],"height:0.786em;vertical-align:-0.15em;",[23,86568,86570,86573],{"className":86569},[135],[23,86571,4526],{"className":86572,"style":4645},[135,136],[23,86574,86576],{"className":86575},[998],[23,86577,86579,86599],{"className":86578},[199,200],[23,86580,86582,86596],{"className":86581},[204],[23,86583,86585],{"className":86584,"style":33647},[208],[23,86586,86587,86590],{"style":68963},[23,86588],{"className":86589,"style":1015},[216],[23,86591,86593],{"className":86592},[1019,1020,1021,1022],[23,86594,33493],{"className":86595},[135,136,1022],[23,86597,297],{"className":86598},[296],[23,86600,86602],{"className":86601},[204],[23,86603,86605],{"className":86604,"style":2048},[208],[23,86606],{},[23,86608],{"className":86609,"style":149},[148],[23,86611,12268],{"className":86612},[153],[23,86614],{"className":86615,"style":149},[148],[23,86617,86619,86622],{"className":86618},[126],[23,86620],{"className":86621,"style":470},[130],[23,86623,24979],{"className":86624,"style":37141},[135,136],[11,86626,86627,86720,86721,86724],{},[23,86628,86630,86652],{"className":86629,"translate":27},[31],[23,86631,86633],{"className":86632},[35],[37,86634,86635],{"xmlns":39},[42,86636,86637,86649],{},[45,86638,86639,86645,86647],{},[898,86640,86641,86643],{},[48,86642,4526],{},[48,86644,33493],{},[51,86646,2579],{},[2581,86648,1532],{},[114,86650,86651],{"encoding":116},"\\alpha_i>0",[23,86653,86655,86711],{"className":86654,"ariaHidden":122},[121],[23,86656,86658,86662,86702,86705,86708],{"className":86657},[126],[23,86659],{"className":86660,"style":86661},[130],"height:0.6891em;vertical-align:-0.15em;",[23,86663,86665,86668],{"className":86664},[135],[23,86666,4526],{"className":86667,"style":4645},[135,136],[23,86669,86671],{"className":86670},[998],[23,86672,86674,86694],{"className":86673},[199,200],[23,86675,86677,86691],{"className":86676},[204],[23,86678,86680],{"className":86679,"style":33647},[208],[23,86681,86682,86685],{"style":68963},[23,86683],{"className":86684,"style":1015},[216],[23,86686,86688],{"className":86687},[1019,1020,1021,1022],[23,86689,33493],{"className":86690},[135,136,1022],[23,86692,297],{"className":86693},[296],[23,86695,86697],{"className":86696},[204],[23,86698,86700],{"className":86699,"style":2048},[208],[23,86701],{},[23,86703],{"className":86704,"style":149},[148],[23,86706,2579],{"className":86707},[153],[23,86709],{"className":86710,"style":149},[148],[23,86712,86714,86717],{"className":86713},[126],[23,86715],{"className":86716,"style":2770},[130],[23,86718,1532],{"className":86719},[135]," 的点就是",[18,86722,86723],{},"支持向量","。决策函数为",[23,86726,86728],{"className":86727,"translate":27},[26],[23,86729,86731,86804],{"className":86730,"translate":27},[31],[23,86732,86734],{"className":86733},[35],[37,86735,86736],{"xmlns":39,"display":40},[42,86737,86738,86801],{},[45,86739,86740,86743,86745,86799],{},[48,86741,86742],{"mathvariant":111},"sign",[51,86744,6959],{},[45,86746,86747,86749,86763,86769,86775,86777,86779,86781,86783,86785,86791,86793,86795,86797],{},[51,86748,54],{"fence":122},[61709,86750,86751,86753,86761],{},[51,86752,33851],{},[45,86754,86755,86757,86759],{},[48,86756,33493],{},[51,86758,69],{},[2581,86760,4435],{},[48,86762,4806],{},[898,86764,86765,86767],{},[48,86766,4526],{},[48,86768,33493],{},[898,86770,86771,86773],{},[48,86772,660],{},[48,86774,33493],{},[88,86776,90],{},[48,86778,4836],{},[51,86780,54],{"stretchy":53},[48,86782,753],{},[51,86784,747],{"separator":122},[898,86786,86787,86789],{},[48,86788,753],{},[48,86790,33493],{},[51,86792,66],{"stretchy":53},[51,86794,5106],{},[48,86796,35533],{},[51,86798,66],{"fence":122},[48,86800,112],{"mathvariant":111},[114,86802,86803],{"encoding":116},"\\operatorname{sign}\\left(\\sum_{i=1}^n\\alpha_i y_i\\,k(x,x_i)+b\\right).",[23,86805,86807],{"className":86806,"ariaHidden":122},[121],[23,86808,86810,86813,86819,86822,87057,87060],{"className":86809},[126],[23,86811],{"className":86812,"style":61833},[130],[23,86814,86816],{"className":86815},[1786],[23,86817,86742],{"className":86818},[135,5391],[23,86820],{"className":86821,"style":280},[148],[23,86823,86825,86831,86895,86898,86938,86978,86981,86984,86987,86990,86993,86996,87036,87039,87042,87045,87048,87051],{"className":86824},[1839],[23,86826,86828],{"className":86827,"style":1844},[140,1843],[23,86829,54],{"className":86830},[1066,10069],[23,86832,86834],{"className":86833},[1786,7096],[23,86835,86837,86887],{"className":86836},[199,200],[23,86838,86840,86884],{"className":86839},[204],[23,86841,86843,86863,86873],{"className":86842,"style":61920},[208],[23,86844,86845,86848],{"style":34075},[23,86846],{"className":86847,"style":34079},[216],[23,86849,86851],{"className":86850},[1019,1020,1021,1022],[23,86852,86854,86857,86860],{"className":86853},[135,1022],[23,86855,33493],{"className":86856},[135,136,1022],[23,86858,69],{"className":86859},[153,1022],[23,86861,4435],{"className":86862},[135,1022],[23,86864,86865,86868],{"style":34088},[23,86866],{"className":86867,"style":34079},[216],[23,86869,86870],{},[23,86871,33851],{"className":86872},[1786,1790,1791],[23,86874,86875,86878],{"style":61953},[23,86876],{"className":86877,"style":34079},[216],[23,86879,86881],{"className":86880},[1019,1020,1021,1022],[23,86882,4806],{"className":86883},[135,136,1022],[23,86885,297],{"className":86886},[296],[23,86888,86890],{"className":86889},[204],[23,86891,86893],{"className":86892,"style":34106},[208],[23,86894],{},[23,86896],{"className":86897,"style":280},[148],[23,86899,86901,86904],{"className":86900},[135],[23,86902,4526],{"className":86903,"style":4645},[135,136],[23,86905,86907],{"className":86906},[998],[23,86908,86910,86930],{"className":86909},[199,200],[23,86911,86913,86927],{"className":86912},[204],[23,86914,86916],{"className":86915,"style":33647},[208],[23,86917,86918,86921],{"style":68963},[23,86919],{"className":86920,"style":1015},[216],[23,86922,86924],{"className":86923},[1019,1020,1021,1022],[23,86925,33493],{"className":86926},[135,136,1022],[23,86928,297],{"className":86929},[296],[23,86931,86933],{"className":86932},[204],[23,86934,86936],{"className":86935,"style":2048},[208],[23,86937],{},[23,86939,86941,86944],{"className":86940},[135],[23,86942,660],{"className":86943,"style":676},[135,136],[23,86945,86947],{"className":86946},[998],[23,86948,86950,86970],{"className":86949},[199,200],[23,86951,86953,86967],{"className":86952},[204],[23,86954,86956],{"className":86955,"style":33647},[208],[23,86957,86958,86961],{"style":21850},[23,86959],{"className":86960,"style":1015},[216],[23,86962,86964],{"className":86963},[1019,1020,1021,1022],[23,86965,33493],{"className":86966},[135,136,1022],[23,86968,297],{"className":86969},[296],[23,86971,86973],{"className":86972},[204],[23,86974,86976],{"className":86975,"style":2048},[208],[23,86977],{},[23,86979],{"className":86980,"style":280},[148],[23,86982,4836],{"className":86983,"style":4851},[135,136],[23,86985,54],{"className":86986},[140],[23,86988,753],{"className":86989},[135,136],[23,86991,747],{"className":86992},[824],[23,86994],{"className":86995,"style":280},[148],[23,86997,86999,87002],{"className":86998},[135],[23,87000,753],{"className":87001},[135,136],[23,87003,87005],{"className":87004},[998],[23,87006,87008,87028],{"className":87007},[199,200],[23,87009,87011,87025],{"className":87010},[204],[23,87012,87014],{"className":87013,"style":33647},[208],[23,87015,87016,87019],{"style":2029},[23,87017],{"className":87018,"style":1015},[216],[23,87020,87022],{"className":87021},[1019,1020,1021,1022],[23,87023,33493],{"className":87024},[135,136,1022],[23,87026,297],{"className":87027},[296],[23,87029,87031],{"className":87030},[204],[23,87032,87034],{"className":87033,"style":2048},[208],[23,87035],{},[23,87037,66],{"className":87038},[169],[23,87040],{"className":87041,"style":392},[148],[23,87043,5106],{"className":87044},[612],[23,87046],{"className":87047,"style":392},[148],[23,87049,35533],{"className":87050},[135,136],[23,87052,87054],{"className":87053,"style":1844},[169,1843],[23,87055,66],{"className":87056},[1066,10069],[23,87058],{"className":87059,"style":280},[148],[23,87061,112],{"className":87062},[135],[11,87064,87065,87066,87069,87070,87128,87129,87157,87158,87227,87228,87297,87298,87326],{},"这里要特别澄清一个常见误解：",[18,87067,87068],{},"SVM 的稀疏性不是表示定理给的。"," 表示定理只说 ",[23,87071,87073,87090],{"className":87072,"translate":27},[31],[23,87074,87076],{"className":87075},[35],[37,87077,87078],{"xmlns":39},[42,87079,87080,87088],{},[45,87081,87082],{},[4518,87083,87084,87086],{},[48,87085,8336],{},[51,87087,6881],{},[114,87089,76569],{"encoding":116},[23,87091,87093],{"className":87092,"ariaHidden":122},[121],[23,87094,87096,87099],{"className":87095},[126],[23,87097],{"className":87098,"style":5414},[130],[23,87100,87102,87105],{"className":87101},[135],[23,87103,8336],{"className":87104,"style":11398},[135,136],[23,87106,87108],{"className":87107},[998],[23,87109,87111],{"className":87110},[199],[23,87112,87114],{"className":87113},[204],[23,87115,87117],{"className":87116,"style":6912},[208],[23,87118,87119,87122],{"style":4696},[23,87120],{"className":87121,"style":1015},[216],[23,87123,87125],{"className":87124},[1019,1020,1021,1022],[23,87126,6881],{"className":87127},[612,1022]," 是 ",[23,87130,87132,87145],{"className":87131,"translate":27},[31],[23,87133,87135],{"className":87134},[35],[37,87136,87137],{"xmlns":39},[42,87138,87139,87143],{},[45,87140,87141],{},[48,87142,4806],{},[114,87144,4806],{"encoding":116},[23,87146,87148],{"className":87147,"ariaHidden":122},[121],[23,87149,87151,87154],{"className":87150},[126],[23,87152],{"className":87153,"style":703},[130],[23,87155,4806],{"className":87156},[135,136]," 个核函数的线性组合，",[23,87159,87161,87178],{"className":87160,"translate":27},[31],[23,87162,87164],{"className":87163},[35],[37,87165,87166],{"xmlns":39},[42,87167,87168,87176],{},[45,87169,87170],{},[898,87171,87172,87174],{},[48,87173,4526],{},[48,87175,33493],{},[114,87177,85529],{"encoding":116},[23,87179,87181],{"className":87180,"ariaHidden":122},[121],[23,87182,87184,87187],{"className":87183},[126],[23,87185],{"className":87186,"style":12730},[130],[23,87188,87190,87193],{"className":87189},[135],[23,87191,4526],{"className":87192,"style":4645},[135,136],[23,87194,87196],{"className":87195},[998],[23,87197,87199,87219],{"className":87198},[199,200],[23,87200,87202,87216],{"className":87201},[204],[23,87203,87205],{"className":87204,"style":33647},[208],[23,87206,87207,87210],{"style":68963},[23,87208],{"className":87209,"style":1015},[216],[23,87211,87213],{"className":87212},[1019,1020,1021,1022],[23,87214,33493],{"className":87215},[135,136,1022],[23,87217,297],{"className":87218},[296],[23,87220,87222],{"className":87221},[204],[23,87223,87225],{"className":87224,"style":2048},[208],[23,87226],{}," 完全可以全部非零。真正让大部分 ",[23,87229,87231,87248],{"className":87230,"translate":27},[31],[23,87232,87234],{"className":87233},[35],[37,87235,87236],{"xmlns":39},[42,87237,87238,87246],{},[45,87239,87240],{},[898,87241,87242,87244],{},[48,87243,4526],{},[48,87245,33493],{},[114,87247,85529],{"encoding":116},[23,87249,87251],{"className":87250,"ariaHidden":122},[121],[23,87252,87254,87257],{"className":87253},[126],[23,87255],{"className":87256,"style":12730},[130],[23,87258,87260,87263],{"className":87259},[135],[23,87261,4526],{"className":87262,"style":4645},[135,136],[23,87264,87266],{"className":87265},[998],[23,87267,87269,87289],{"className":87268},[199,200],[23,87270,87272,87286],{"className":87271},[204],[23,87273,87275],{"className":87274,"style":33647},[208],[23,87276,87277,87280],{"style":68963},[23,87278],{"className":87279,"style":1015},[216],[23,87281,87283],{"className":87282},[1019,1020,1021,1022],[23,87284,33493],{"className":87285},[135,136,1022],[23,87287,297],{"className":87288},[296],[23,87290,87292],{"className":87291},[204],[23,87293,87295],{"className":87294,"style":2048},[208],[23,87296],{}," 变成零的，是 hinge loss 的 KKT 条件：间隔大于 ",[23,87299,87301,87314],{"className":87300,"translate":27},[31],[23,87302,87304],{"className":87303},[35],[37,87305,87306],{"xmlns":39},[42,87307,87308,87312],{},[45,87309,87310],{},[2581,87311,4435],{},[114,87313,4435],{"encoding":116},[23,87315,87317],{"className":87316,"ariaHidden":122},[121],[23,87318,87320,87323],{"className":87319},[126],[23,87321],{"className":87322,"style":2770},[130],[23,87324,4435],{"className":87325},[135]," 的点对损失没有贡献，对应的乘子为零。",[2501,87328,87330],{"id":87329},"_53-核逻辑回归","5.3 核逻辑回归",[11,87332,87333],{},"取逻辑损失：",[23,87335,87337],{"className":87336,"translate":27},[26],[23,87338,87340,87450],{"className":87339,"translate":27},[31],[23,87341,87343],{"className":87342},[35],[37,87344,87345],{"xmlns":39,"display":40},[42,87346,87347,87447],{},[45,87348,87349,87365,87445],{},[6961,87350,87351,87357],{},[45,87352,87353,87355],{},[48,87354,6967],{},[51,87356,6959],{},[45,87358,87359,87361,87363],{},[48,87360,8336],{},[51,87362,372],{},[48,87364,26249],{"mathvariant":375},[45,87366,87367,87369,87383,87385,87387,87389,87391,87393,87395,87397,87399,87401,87407,87409,87411,87417,87419,87421,87423,87425,87431,87433,87435,87443],{},[51,87368,9010],{"fence":122},[61709,87370,87371,87373,87381],{},[51,87372,33851],{},[45,87374,87375,87377,87379],{},[48,87376,33493],{},[51,87378,69],{},[2581,87380,4435],{},[48,87382,4806],{},[48,87384,22418],{},[51,87386,6959],{},[51,87388,54],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[2581,87390,4435],{},[51,87392,5106],{},[48,87394,25008],{},[51,87396,6959],{},[51,87398,54],{"stretchy":53},[51,87400,4529],{},[898,87402,87403,87405],{},[48,87404,660],{},[48,87406,33493],{},[48,87408,8336],{},[51,87410,54],{"stretchy":53},[898,87412,87413,87415],{},[48,87414,753],{},[48,87416,33493],{},[51,87418,66],{"stretchy":53},[51,87420,66],{"stretchy":53},[51,87422,66],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[51,87424,5106],{},[71,87426,87427,87429],{},[48,87428,61753],{},[2581,87430,9952],{},[48,87432,20680],{"mathvariant":111},[48,87434,8336],{},[19137,87436,87437,87439,87441],{},[48,87438,20680],{"mathvariant":111},[48,87440,26249],{"mathvariant":375},[2581,87442,9952],{},[51,87444,9020],{"fence":122},[48,87446,112],{"mathvariant":111},[114,87448,87449],{"encoding":116},"\\min_{f\\in\\mathcal H}\\left\\{\n\\sum_{i=1}^n\\log\\bigl(1+\\exp(-y_i f(x_i))\\bigr)\n+\\frac{\\lambda}{2}\\|f\\|_{\\mathcal H}^2\n\\right\\}.",[23,87451,87453],{"className":87452,"ariaHidden":122},[121],[23,87454,87456,87459,87512,87515,87855,87858],{"className":87455},[126],[23,87457],{"className":87458,"style":61833},[130],[23,87460,87462],{"className":87461},[1786,7096],[23,87463,87465,87504],{"className":87464},[199,200],[23,87466,87468,87501],{"className":87467},[204],[23,87469,87471,87491],{"className":87470,"style":7106},[208],[23,87472,87473,87476],{"style":14738},[23,87474],{"className":87475,"style":217},[216],[23,87477,87479],{"className":87478},[1019,1020,1021,1022],[23,87480,87482,87485,87488],{"className":87481},[135,1022],[23,87483,8336],{"className":87484,"style":11398},[135,136,1022],[23,87486,372],{"className":87487},[153,1022],[23,87489,26249],{"className":87490,"style":45024},[135,412,1022],[23,87492,87493,87496],{"style":7130},[23,87494],{"className":87495,"style":217},[216],[23,87497,87498],{},[23,87499,6967],{"className":87500},[1786],[23,87502,297],{"className":87503},[296],[23,87505,87507],{"className":87506},[204],[23,87508,87510],{"className":87509,"style":61893},[208],[23,87511],{},[23,87513],{"className":87514,"style":280},[148],[23,87516,87518,87524,87588,87591,87596,87602,87605,87608,87611,87614,87617,87620,87623,87663,87666,87669,87709,87712,87718,87721,87724,87727,87789,87792,87795,87849],{"className":87517},[1839],[23,87519,87521],{"className":87520,"style":1844},[140,1843],[23,87522,9010],{"className":87523},[1066,10069],[23,87525,87527],{"className":87526},[1786,7096],[23,87528,87530,87580],{"className":87529},[199,200],[23,87531,87533,87577],{"className":87532},[204],[23,87534,87536,87556,87566],{"className":87535,"style":61920},[208],[23,87537,87538,87541],{"style":34075},[23,87539],{"className":87540,"style":34079},[216],[23,87542,87544],{"className":87543},[1019,1020,1021,1022],[23,87545,87547,87550,87553],{"className":87546},[135,1022],[23,87548,33493],{"className":87549},[135,136,1022],[23,87551,69],{"className":87552},[153,1022],[23,87554,4435],{"className":87555},[135,1022],[23,87557,87558,87561],{"style":34088},[23,87559],{"className":87560,"style":34079},[216],[23,87562,87563],{},[23,87564,33851],{"className":87565},[1786,1790,1791],[23,87567,87568,87571],{"style":61953},[23,87569],{"className":87570,"style":34079},[216],[23,87572,87574],{"className":87573},[1019,1020,1021,1022],[23,87575,4806],{"className":87576},[135,136,1022],[23,87578,297],{"className":87579},[296],[23,87581,87583],{"className":87582},[204],[23,87584,87586],{"className":87585,"style":34106},[208],[23,87587],{},[23,87589],{"className":87590,"style":280},[148],[23,87592,22575,87594],{"className":87593},[1786],[23,87595,7089],{"style":7088},[23,87597,87599],{"className":87598},[140],[23,87600,54],{"className":87601},[1066,1067],[23,87603,4435],{"className":87604},[135],[23,87606],{"className":87607,"style":392},[148],[23,87609,5106],{"className":87610},[612],[23,87612],{"className":87613,"style":392},[148],[23,87615,25008],{"className":87616},[1786],[23,87618,54],{"className":87619},[140],[23,87621,4529],{"className":87622},[135],[23,87624,87626,87629],{"className":87625},[135],[23,87627,660],{"className":87628,"style":676},[135,136],[23,87630,87632],{"className":87631},[998],[23,87633,87635,87655],{"className":87634},[199,200],[23,87636,87638,87652],{"className":87637},[204],[23,87639,87641],{"className":87640,"style":33647},[208],[23,87642,87643,87646],{"style":21850},[23,87644],{"className":87645,"style":1015},[216],[23,87647,87649],{"className":87648},[1019,1020,1021,1022],[23,87650,33493],{"className":87651},[135,136,1022],[23,87653,297],{"className":87654},[296],[23,87656,87658],{"className":87657},[204],[23,87659,87661],{"className":87660,"style":2048},[208],[23,87662],{},[23,87664,8336],{"className":87665,"style":11398},[135,136],[23,87667,54],{"className":87668},[140],[23,87670,87672,87675],{"className":87671},[135],[23,87673,753],{"className":87674},[135,136],[23,87676,87678],{"className":87677},[998],[23,87679,87681,87701],{"className":87680},[199,200],[23,87682,87684,87698],{"className":87683},[204],[23,87685,87687],{"className":87686,"style":33647},[208],[23,87688,87689,87692],{"style":2029},[23,87690],{"className":87691,"style":1015},[216],[23,87693,87695],{"className":87694},[1019,1020,1021,1022],[23,87696,33493],{"className":87697},[135,136,1022],[23,87699,297],{"className":87700},[296],[23,87702,87704],{"className":87703},[204],[23,87705,87707],{"className":87706,"style":2048},[208],[23,87708],{},[23,87710,1095],{"className":87711},[169],[23,87713,87715],{"className":87714},[169],[23,87716,66],{"className":87717},[1066,1067],[23,87719],{"className":87720,"style":392},[148],[23,87722,5106],{"className":87723},[612],[23,87725],{"className":87726,"style":392},[148],[23,87728,87730,87733,87786],{"className":87729},[135],[23,87731],{"className":87732},[140,192],[23,87734,87736],{"className":87735},[71],[23,87737,87739,87778],{"className":87738},[199,200],[23,87740,87742,87775],{"className":87741},[204],[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K\\alpha\n\\right\\}.",[23,88228,88230],{"className":88229,"ariaHidden":122},[121],[23,88231,88233,88236,88315,88318,88636,88639],{"className":88232},[126],[23,88234],{"className":88235,"style":61833},[130],[23,88237,88239],{"className":88238},[1786,7096],[23,88240,88242,88307],{"className":88241},[199,200],[23,88243,88245,88304],{"className":88244},[204],[23,88246,88248,88294],{"className":88247,"style":7106},[208],[23,88249,88250,88253],{"style":76016},[23,88251],{"className":88252,"style":217},[216],[23,88254,88256],{"className":88255},[1019,1020,1021,1022],[23,88257,88259,88262,88265],{"className":88258},[135,1022],[23,88260,4526],{"className":88261,"style":4645},[135,136,1022],[23,88263,372],{"className":88264},[153,1022],[23,88266,88268,88271],{"className":88267},[135,1022],[23,88269,572],{"className":88270},[135,644,1022],[23,88272,88274],{"className":88273},[998],[23,88275,88277],{"className":88276},[199],[23,88278,88280],{"className":88279},[204],[23,88281,88283],{"className":88282,"style":76050},[208],[23,88284,88285,88288],{"style":76053},[23,88286],{"className":88287,"style":24881},[216],[23,88289,88291],{"className":88290},[1019,24885,1067,1022],[23,88292,4806],{"className":88293},[135,136,1022],[23,88295,88296,88299],{"style":7130},[23,88297],{"className":88298,"style":217},[216],[23,88300,88301],{},[23,88302,6967],{"className":88303},[1786],[23,88305,297],{"className":88306},[296],[23,88308,88310],{"className":88309},[204],[23,88311,88313],{"className":88312,"style":76082},[208],[23,88314],{},[23,88316],{"className":88317,"style":280},[148],[23,88319,88321,88327,88391,88394,88399,88405,88408,88411,88414,88417,88420,88423,88426,88466,88469,88472,88475,88515,88518,88524,88527,88530,88533,88595,88624,88627,88630],{"className":88320},[1839],[23,88322,88324],{"className":88323,"style":1844},[140,1843],[23,88325,9010],{"className":88326},[1066,10069],[23,88328,88330],{"className":88329},[1786,7096],[23,88331,88333,88383],{"className":88332},[199,200],[23,88334,88336,88380],{"className":88335},[204],[23,88337,88339,88359,88369],{"className":88338,"style":61920},[208],[23,88340,88341,88344],{"style":34075},[23,88342],{"className":88343,"style":34079},[216],[23,88345,88347],{"className":88346},[1019,1020,1021,1022],[23,88348,88350,88353,88356],{"className":88349},[135,1022],[23,88351,33493],{"className":88352},[135,136,1022],[23,88354,69],{"className":88355},[153,1022],[23,88357,4435],{"className":88358},[135,1022],[23,88360,88361,88364],{"style":34088},[23,88362],{"className":88363,"style":34079},[216],[23,88365,88366],{},[23,88367,33851],{"className":88368},[1786,1790,1791],[23,88370,88371,88374],{"style":61953},[23,88372],{"className":88373,"style":34079},[216],[23,88375,88377],{"className":88376},[1019,1020,1021,1022],[23,88378,4806],{"className":88379},[135,136,1022],[23,88381,297],{"className":88382},[296],[23,88384,88386],{"className":88385},[204],[23,88387,88389],{"className":88388,"style":34106},[208],[23,88390],{},[23,88392],{"className":88393,"style":280},[148],[23,88395,22575,88397],{"className":88396},[1786],[23,88398,7089],{"style":7088},[23,88400,88402],{"className":88401},[140],[23,88403,54],{"className":88404},[1066,1067],[23,88406,4435],{"className":88407},[135],[23,88409],{"className":88410,"style":392},[148],[23,88412,5106],{"className":88413},[612],[23,88415],{"className":88416,"style":392},[148],[23,88418,25008],{"className":88419},[1786],[23,88421,54],{"className":88422},[140],[23,88424,4529],{"className":88425},[135],[23,88427,88429,88432],{"className":88428},[135],[23,88430,660],{"className":88431,"style":676},[135,136],[23,88433,88435],{"className":88434},[998],[23,88436,88438,88458],{"className":88437},[199,200],[23,88439,88441,88455],{"className":88440},[204],[23,88442,88444],{"className":88443,"style":33647},[208],[23,88445,88446,88449],{"style":21850},[23,88447],{"className":88448,"style":1015},[216],[23,88450,88452],{"className":88451},[1019,1020,1021,1022],[23,88453,33493],{"className":88454},[135,136,1022],[23,88456,297],{"className":88457},[296],[23,88459,88461],{"className":88460},[204],[23,88462,88464],{"className":88463,"style":2048},[208],[23,88465],{},[23,88467,54],{"className":88468},[140],[23,88470,26918],{"className":88471,"style":37141},[135,136],[23,88473,4526],{"className":88474,"style":4645},[135,136],[23,88476,88478,88481],{"className":88477},[169],[23,88479,66],{"className":88480},[169],[23,88482,88484],{"className":88483},[998],[23,88485,88487,88507],{"className":88486},[199,200],[23,88488,88490,88504],{"className":88489},[204],[23,88491,88493],{"className":88492,"style":33647},[208],[23,88494,88495,88498],{"style":2029},[23,88496],{"className":88497,"style":1015},[216],[23,88499,88501],{"className":88500},[1019,1020,1021,1022],[23,88502,33493],{"className":88503},[135,136,1022],[23,88505,297],{"className":88506},[296],[23,88508,88510],{"className":88509},[204],[23,88511,88513],{"className":88512,"style":2048},[208],[23,88514],{},[23,88516,66],{"classNa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Newton 法、IRLS 或梯度类方法求解。与 SVM 不同，逻辑损失不会产生精确的零系数，",[23,88646,88648,88661],{"className":88647,"translate":27},[31],[23,88649,88651],{"className":88650},[35],[37,88652,88653],{"xmlns":39},[42,88654,88655,88659],{},[45,88656,88657],{},[48,88658,4526],{},[114,88660,76426],{"encoding":116},[23,88662,88664],{"className":88663,"ariaHidden":122},[121],[23,88665,88667,88670],{"className":88666},[126],[23,88668],{"className":88669,"style":703},[130],[23,88671,4526],{"className":88672,"style":4645},[135,136]," 一般是稠密的。",[2501,88675,88677],{"id":88676},"_54-高斯过程","5.4 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的核岭回归解，也就是表示定理给出的形式。区别在于，GP 还给出了不确定度 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平滑样条",[11,89880,89881,89882,8055,89956,87128,89984,90012],{},"表示定理最早的重要应用之一是平滑样条。取 ",[23,89883,89885,89911],{"className":89884,"translate":27},[31],[23,89886,89888],{"className":89887},[35],[37,89889,89890],{"xmlns":39},[42,89891,89892,89908],{},[45,89893,89894,89896,89898,89900,89902,89904,89906],{},[48,89895,434],{"mathvariant":375},[51,89897,69],{},[51,89899,924],{"stretchy":53},[2581,89901,1532],{},[51,89903,747],{"separator":122},[2581,89905,4435],{},[51,89907,945],{"stretchy":53},[114,89909,89910],{"encoding":116},"\\mathcal X=[0,1]",[23,89912,89914,89932],{"className":89913,"ariaHidden":122},[121],[23,89915,89917,89920,89923,89926,89929],{"className":89916},[126],[23,89918],{"className":89919,"style":470},[130],[23,89921,434],{"className":89922,"style":474},[135,412],[23,89924],{"className":89925,"style":149},[148],[23,89927,69],{"className":89928},[153],[23,89930],{"className":89931,"style":149},[148],[23,89933,89935,89938,89941,89944,89947,89950,89953],{"className":89934},[126],[23,89936],{"className":89937,"style":131},[130],[23,89939,924],{"className":89940},[140],[23,89942,1532],{"className":89943},[135],[23,89945,747],{"className":89946},[824],[23,89948],{"className":89949,"style":280},[148],[23,89951,4435],{"className":89952},[135],[23,89954,945],{"className":89955},[169],[23,89957,89959,89972],{"className":89958,"translate":27},[31],[23,89960,89962],{"className":89961},[35],[37,89963,89964],{"xmlns":39},[42,89965,89966,89970],{},[45,89967,89968],{},[48,89969,26249],{"mathvariant":375},[114,89971,45011],{"encoding":116},[23,89973,89975],{"className":89974,"ariaHidden":122},[121],[23,89976,89978,89981],{"className":89977},[126],[23,89979],{"className":89980,"style":470},[130],[23,89982,26249],{"className":89983,"style":45024},[135,412],[23,89985,89987,90000],{"className":89986,"translate":27},[31],[23,89988,89990],{"className":89989},[35],[37,89991,89992],{"xmlns":39},[42,89993,89994,89998],{},[45,89995,89996],{},[48,89997,50961],{},[114,89999,50961],{"encoding":116},[23,90001,90003],{"className":90002,"ariaHidden":122},[121],[23,90004,90006,90009],{"className":90005},[126],[23,90007],{"className":90008,"style":703},[130],[23,90010,50961],{"className":90011},[135,136]," 阶 Sobolev 空间，正则项为",[23,90014,90016],{"className":90015,"translate":27},[26],[23,90017,90019,90075],{"className":90018,"translate":27},[31],[23,90020,90022],{"className":90021},[35],[37,90023,90024],{"xmlns":39,"display":40},[42,90025,90026,90072],{},[45,90027,90028,90030,90038,90040,90052,90054,90056,90058,90064,90066,90068,90070],{},[48,90029,61753],{},[19137,90031,90032,90034,90036],{},[51,90033,1557],{},[2581,90035,1532],{},[2581,90037,4435],{},[51,90039,54],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[4518,90041,90042,90044],{},[48,90043,8336],{},[45,90045,90046,90048,90050],{},[51,90047,54],{"stretchy":53},[48,90049,50961],{},[51,90051,66],{"stretchy":53},[51,90053,54],{"stretchy":53},[48,90055,753],{},[51,90057,66],{"stretchy":53},[4518,90059,90060,90062],{},[51,90061,66],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[2581,90063,9952],{},[88,90065,90],{},[48,90067,1606],{},[48,90069,753],{},[48,90071,112],{"mathvariant":111},[114,90073,90074],{"encoding":116},"\\lambda\\int_0^1\\bigl(f^{(m)}(x)\\bigr)^2\\,dx.",[23,90076,90078],{"className":90077,"ariaHidden":122},[121],[23,90079,90081,90085,90088,90091,90144,90150,90188,90191,90194,90197,90229,90232,90235,90238],{"className":90080},[126],[23,90082],{"className":90083,"style":90084},[130],"height:2.476em;vertical-align:-0.9119em;",[23,90086,61753],{"className":90087},[135,136],[23,90089],{"className":90090,"style":280},[148],[23,90092,90094,90097],{"className":90093},[1786],[23,90095,1557],{"className":90096,"style":1792},[1786,1790,1791],[23,90098,90100],{"className":90099},[998],[23,90101,90103,90136],{"className":90102},[199,200],[23,90104,90106,90133],{"className":90105},[204],[23,90107,90110,90121],{"className":90108,"style":90109},[208],"height:1.564em;",[23,90111,90112,90115],{"style":1808},[23,90113],{"className":90114,"style":1015},[216],[23,90116,90118],{"className":90117},[1019,1020,1021,1022],[23,90119,1532],{"className":90120},[135,1022],[23,90122,90124,90127],{"style":90123},"top:-3.8129em;margin-right:0.05em;",[23,90125],{"className":90126,"style":1015},[216],[23,90128,90130],{"className":90129},[1019,1020,1021,1022],[23,90131,4435],{"className":90132},[135,1022],[23,90134,297],{"className":90135},[296],[23,90137,90139],{"className":90138},[204],[23,90140,90142],{"className":90141,"style":1830},[208],[23,90143],{},[23,90145,90147],{"className":90146},[140],[23,90148,54],{"className":90149},[1066,1067],[23,90151,90153,90156],{"className":90152},[135],[23,90154,8336],{"className":90155,"style":11398},[135,136],[23,90157,90159],{"className":90158},[998],[23,90160,90162],{"className":90161},[199],[23,90163,90165],{"className":90164},[204],[23,90166,90168],{"className":90167,"style":44182},[208],[23,90169,90170,90173],{"style":4989},[23,90171],{"className":90172,"style":1015},[216],[23,90174,90176],{"className":90175},[1019,1020,1021,1022],[23,90177,90179,90182,90185],{"className":90178},[135,1022],[23,90180,54],{"className":90181},[140,1022],[23,90183,50961],{"className":90184},[135,136,1022],[23,90186,66],{"className":90187},[169,1022],[23,90189,54],{"className":90190},[140],[23,90192,753],{"className":90193},[135,136],[23,90195,66],{"className":90196},[169],[23,90198,90200,90206],{"className":90199},[169],[23,90201,90203],{"className":90202},[169],[23,90204,66],{"className":90205},[1066,1067],[23,90207,90209],{"className":90208},[998],[23,90210,90212],{"className":90211},[199],[23,90213,90215],{"className":90214},[204],[23,90216,90218],{"className":90217,"style":11188},[208],[23,90219,90220,90223],{"style":11191},[23,90221],{"className":90222,"style":1015},[216],[23,90224,90226],{"className":90225},[1019,1020,1021,1022],[23,90227,9952],{"className":90228},[135,1022],[23,90230],{"className":90231,"style":280},[148],[23,90233,1606],{"className":90234},[135,136],[23,90236,753],{"className":90237},[135,136],[23,90239,112],{"className":90240},[135],[11,90242,90243],{},"最小化",[23,90245,90247],{"className":90246,"translate":27},[26],[23,90248,90250,90348],{"className":90249,"translate":27},[31],[23,90251,90253],{"className":90252},[35],[37,90254,90255],{"xmlns":39,"display":40},[42,90256,90257,90345],{},[45,90258,90259,90273,90275,90281,90283,90285,90287,90293,90295,90301,90303,90305,90313,90315,90327,90329,90331,90333,90339,90341,90343],{},[61709,90260,90261,90263,90271],{},[51,90262,33851],{},[45,90264,90265,90267,90269],{},[48,90266,33493],{},[51,90268,69],{},[2581,90270,4435],{},[48,90272,4806],{},[51,90274,54],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[898,90276,90277,90279],{},[48,90278,660],{},[48,90280,33493],{},[51,90282,4529],{},[48,90284,8336],{},[51,90286,54],{"stretchy":53},[898,90288,90289,90291],{},[48,90290,753],{},[48,90292,33493],{},[51,90294,66],{"stretchy":53},[4518,90296,90297,90299],{},[51,90298,66],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[2581,90300,9952],{},[51,90302,5106],{},[48,90304,61753],{},[19137,90306,90307,90309,90311],{},[51,90308,1557],{},[2581,90310,1532],{},[2581,903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",[23,90733,90735,90755],{"className":90734,"translate":27},[31],[23,90736,90738],{"className":90737},[35],[37,90739,90740],{"xmlns":39},[42,90741,90742,90752],{},[45,90743,90744,90746,90748,90750],{},[2581,90745,9952],{},[48,90747,50961],{},[51,90749,4529],{},[2581,90751,4435],{},[114,90753,90754],{"encoding":116},"2m-1",[23,90756,90758,90779],{"className":90757,"ariaHidden":122},[121],[23,90759,90761,90764,90767,90770,90773,90776],{"className":90760},[126],[23,90762],{"className":90763,"style":9400},[130],[23,90765,9952],{"className":90766},[135],[23,90768,50961],{"className":90769},[135,136],[23,90771],{"className":90772,"style":392},[148],[23,90774,4529],{"className":90775},[612],[23,90777],{"className":90778,"style":392},[148],[23,90780,90782,90785],{"className":90781},[126],[23,90783],{"className":90784,"style":2770},[130],[23,90786,4435],{"className":90787},[135]," 次自然多项式样条。表示定理给出了「解在训练点核函数张成空间中」的结构；样条的多项式零空间（",[23,90790,90792,90805],{"className":90791,"translate":27},[31],[23,90793,90795],{"className":90794},[35],[37,90796,90797],{"xmlns":39},[42,90798,90799,90803],{},[45,90800,90801],{},[48,90802,50961],{},[114,90804,50961],{"encoding":116},[23,90806,90808],{"className":90807,"ariaHidden":122},[121],[23,90809,90811,90814],{"className":90810},[126],[23,90812],{"className":90813,"style":703},[130],[23,90815,50961],{"className":90816},[135,136]," 维）则由半参数版本处理。",[23,90819,90821,90839],{"className":90820,"translate":27},[31],[23,90822,90824],{"className":90823},[35],[37,90825,90826],{"xmlns":39},[42,90827,90828,90836],{},[45,90829,90830,90832,90834],{},[48,90831,50961],{},[51,90833,69],{},[2581,90835,9952],{},[114,90837,90838],{"encoding":116},"m=2",[23,90840,90842,90860],{"className":90841,"ariaHidden":122},[121],[23,90843,90845,90848,90851,90854,90857],{"className":90844},[126],[23,90846],{"className":90847,"style":703},[130],[23,90849,50961],{"className":90850},[135,136],[23,90852],{"className":90853,"style":149},[148],[23,90855,69],{"className":90856},[153],[23,90858],{"className":90859,"style":149},[148],[23,90861,90863,90866],{"className":90862},[126],[23,90864],{"className":90865,"style":2770},[130],[23,90867,9952],{"className":90868},[135]," 时就是常见的自然三次样条。",[11,90871,90872],{},"这也是 Kimeldorf–Wahba 定理的内容。它说明表示定理并不是核方法时代才出现的技巧，而是从样条估计中生长出来的结构结论。",[2501,90874,90876],{"id":90875},"_56-一张表","5.6 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I)^{-1}y",[23,91022,91024,91045],{"className":91023,"ariaHidden":122},[121],[23,91025,91027,91030,91033,91036,91039,91042],{"className":91026},[126],[23,91028],{"className":91029,"style":131},[130],[23,91031,54],{"className":91032},[140],[23,91034,26918],{"className":91035,"style":37141},[135,136],[23,91037],{"className":91038,"style":392},[148],[23,91040,5106],{"className":91041},[612],[23,91043],{"className":91044,"style":392},[148],[23,91046,91048,91051,91054,91057,91092],{"className":91047},[126],[23,91049],{"className":91050,"style":10661},[130],[23,91052,61753],{"className":91053},[135,136],[23,91055,34353],{"className":91056,"style":454},[135,136],[23,91058,91060,91063],{"className":91059},[169],[23,91061,66],{"className":91062},[169],[23,91064,91066],{"className":91065},[998],[23,91067,91069],{"className":91068},[199],[23,91070,91072],{"className":91071},[204],[23,91073,91075],{"className":91074,"style":4693},[208],[23,91076,91077,91080],{"style":4696},[23,91078],{"className":91079,"style":1015},[216],[23,91081,91083],{"className":91082},[1019,1020,1021,1022],[23,91084,91086,91089],{"className":91085},[135,1022],[23,91087,4529],{"className":91088},[135,1022],[23,91090,4435],{"className":91091},[135,1022],[23,91093,660],{"className":91094,"style":676},[135,136],[3788,91096,91097],{},"一般稠密",[3770,91099,91100,91103,91106,91251,91254],{},[3788,91101,91102],{},"SVM",[3788,91104,91105],{},"hinge",[3788,91107,91108],{},[23,91109,91111,91139],{"className":91110,"translate":27},[31],[23,91112,91114],{"className":91113},[35],[37,91115,91116],{"xmlns":39},[42,91117,91118,91136],{},[45,91119,91120,91126,91128,91130],{},[71,91121,91122,91124],{},[2581,91123,4435],{},[2581,91125,9952],{},[48,91127,20680],{"mathvariant":111},[48,91129,8336],{},[4518,91131,91132,91134],{},[48,91133,20680],{"mathvariant":111},[2581,91135,9952],{},[114,91137,91138],{"encoding":116},"\\frac12\\|f\\|^2",[23,91140,91142],{"className":91141,"ariaHidden":122},[121],[23,91143,91145,91148,91216,91219,91222],{"className":91144},[126],[23,91146],{"className":91147,"style":27416},[130],[23,91149,91151,91154,91213],{"className":91150},[135],[23,91152],{"className":91153},[140,192],[23,91155,91157],{"className":91156},[71],[23,91158,91160,91205],{"className":91159},[199,200],[23,91161,91163,91202],{"className":91162},[204],[23,91164,91166,91180,91188],{"className":91165,"style":10196},[208],[23,91167,91168,91171],{"style":10199},[23,91169],{"className":91170,"style":217},[216],[23,91172,91174],{"className":91173},[1019,1020,1021,1022],[23,91175,91177],{"className":91176},[135,1022],[23,91178,9952],{"className":91179},[135,1022],[23,91181,91182,91185],{"style":235},[23,91183],{"className":91184,"style":217},[216],[23,91186],{"className":91187,"style":243},[242],[23,91189,91190,91193],{"style":10222},[23,91191],{"className":91192,"style":217},[216],[23,91194,91196],{"className":91195},[1019,1020,1021,1022],[23,91197,91199],{"className":91198},[135,1022],[23,91200,4435],{"className":91201},[135,1022],[23,91203,297],{"className":91204},[296],[23,91206,91208],{"className":91207},[204],[23,91209,91211],{"className":91210,"style":10244},[208],[23,91212],{},[23,91214],{"className":91215},[169,192],[23,91217,20680],{"className":91218},[135],[23,91220,8336],{"className":91221,"style":11398},[135,136],[23,91223,91225,91228],{"className":91224},[135],[23,91226,20680],{"className":91227},[135],[23,91229,91231],{"className":91230},[998],[23,91232,91234],{"className":91233},[199],[23,91235,91237],{"className":91236},[204],[23,91238,91240],{"className":91239,"style":4693},[208],[23,91241,91242,91245],{"style":4696},[23,91243],{"className":91244,"style":1015},[216],[23,91246,91248],{"className":91247},[1019,1020,1021,1022],[23,91249,9952],{"className":91250},[135,1022],[3788,91252,91253],{},"对偶二次规划",[3788,91255,91256],{},"支持向量稀疏",[3770,91258,91259,91262,91265,91412,91415],{},[3788,91260,91261],{},"核逻辑回归",[3788,91263,91264],{},"logistic",[3788,91266,91267],{},[23,91268,91270,91298],{"className":91269,"translate":27},[31],[23,91271,91273],{"className":91272},[35],[37,91274,91275],{"xmlns":39},[42,91276,91277,91295],{},[45,91278,91279,91285,91287,91289],{},[71,91280,91281,91283],{},[48,91282,61753],{},[2581,91284,9952],{},[48,91286,20680],{"mathvariant":111},[48,91288,8336],{},[4518,91290,91291,91293],{},[48,91292,20680],{"mathvariant":111},[2581,91294,9952],{},[114,91296,91297],{"encoding":116},"\\frac{\\lambda}{2}\\|f\\|^2",[23,91299,91301],{"className":91300,"ariaHidden":122},[121],[23,91302,91304,91308,91377,91380,91383],{"className":91303},[126],[23,91305],{"className":91306,"style":91307},[130],"height:1.2251em;vertical-align:-0.345em;",[23,91309,91311,91314,91374],{"className":91310},[135],[23,91312],{"className":91313},[140,192],[23,91315,91317],{"className":91316},[71],[23,91318,91320,91366],{"className":91319},[199,200],[23,91321,91323,91363],{"className":91322},[204],[23,91324,91327,91341,91349],{"className":91325,"style":91326},[208],"height:0.8801em;",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先验",[3788,91429,91430],{},[23,91431,91433,91471],{"className":91432,"translate":27},[31],[23,91434,91436],{"className":91435},[35],[37,91437,91438],{"xmlns":39},[42,91439,91440,91468],{},[45,91441,91442,91444,91446,91448,91454,91456,91466],{},[51,91443,54],{"stretchy":53},[48,91445,26918],{},[51,91447,5106],{},[4518,91449,91450,91452],{},[48,91451,29634],{},[2581,91453,9952],{},[48,91455,34353],{},[4518,91457,91458,91460],{},[51,91459,66],{"stretchy":53},[45,91461,91462,91464],{},[51,91463,4529],{},[2581,91465,4435],{},[48,91467,660],{},[114,91469,91470],{"encoding":116},"(K+\\sigma^2 I)^{-1}y",[23,91472,91474,91495],{"className":91473,"ariaHidden":122},[121],[23,91475,91477,91480,91483,91486,91489,91492],{"className":91476},[126],[23,91478],{"className":91479,"style":131},[130],[23,91481,54],{"className":91482},[140],[23,91484,26918],{"className":91485,"style":37141},[135,136],[23,91487],{"className":91488,"style":392},[148],[23,91490,5106],{"className":91491},[612],[23,91493],{"className":91494,"style":392},[148],[23,91496,91498,91501,91530,91533,91568],{"className":91497},[126],[23,91499],{"className":91500,"style":10661},[130],[23,91502,91504,91507],{"className":91503},[135],[23,91505,29634],{"className":91506,"style":676},[135,136],[23,91508,91510],{"className":91509},[998],[23,91511,91513],{"className":91512},[199],[23,91514,91516],{"className":91515},[204],[23,91517,91519],{"className":91518,"style":4693},[208],[23,91520,91521,91524],{"style":4696},[23,91522],{"className":91523,"style":1015},[216],[23,91525,91527],{"className":91526},[1019,1020,1021,1022],[23,91528,9952],{"className":91529},[135,1022],[23,91531,34353],{"className":91532,"style":454},[135,136],[23,91534,91536,91539],{"className":91535},[169],[23,91537,66],{"className":91538},[169],[23,91540,91542],{"className":91541},[998],[23,91543,91545],{"className":91544},[199],[23,91546,91548],{"className":91547},[204],[23,91549,91551],{"className":91550,"style":4693},[208],[23,91552,91553,91556],{"style":4696},[23,91554],{"className":91555,"style":1015},[216],[23,91557,91559],{"className":91558},[1019,1020,1021,1022],[23,91560,91562,91565],{"className":91561},[135,1022],[23,91563,4529],{"className":91564},[135,1022],[23,91566,4435],{"className":91567},[135,1022],[23,91569,660],{"className":91570,"style":676},[135,136],[3788,91572,91097],{},[3770,91574,91575,91578,91580,91702,91705],{},[3788,91576,91577],{},"平滑样条",[3788,91579,90906],{},[3788,91581,91582],{},[23,91583,91585,91619],{"className":91584,"translate":27},[31],[23,91586,91588],{"className":91587},[35],[37,91589,91590],{"xmlns":39},[42,91591,91592,91616],{},[45,91593,91594,91596,91598,91610],{},[51,91595,1557],{},[51,91597,54],{"stretchy":53},[4518,91599,91600,91602],{},[48,91601,8336],{},[45,91603,91604,91606,91608],{},[51,91605,54],{"stretchy":53},[48,91607,50961],{},[51,91609,66],{"stretchy":53},[4518,91611,91612,91614],{},[51,91613,66],{"stretchy":53},[2581,91615,9952],{},[114,91617,91618],{"encoding":116},"\\int(f^{(m)})^2",[23,91620,91622],{"className":91621,"ariaHidden":122},[121],[23,91623,91625,91629,91632,91635,91673],{"className":91624},[126],[23,91626],{"className":91627,"style":91628},[130],"height:1.1941em;vertical-align:-0.3061em;",[23,91630,1557],{"className":91631,"style":20455},[1786,1790,20454],[23,91633,54],{"className":91634},[140],[23,91636,91638,91641],{"className":91637},[135],[23,91639,8336],{"className":91640,"style":11398},[135,136],[23,91642,91644],{"className":91643},[998],[23,91645,91647],{"className":91646},[199],[23,91648,91650],{"className":91649},[204],[23,91651,91653],{"className":91652,"style":35723},[208],[23,91654,91655,91658],{"style":4696},[23,91656],{"className":91657,"style":1015},[216],[23,91659,91661],{"className":91660},[1019,1020,1021,1022],[23,91662,91664,91667,91670],{"className":91663},[135,1022],[23,91665,54],{"className":91666},[140,1022],[23,91668,50961],{"className":91669},[135,136,1022],[23,91671,66],{"className":91672},[169,1022],[23,91674,91676,91679],{"className":91675},[169],[23,91677,66],{"className":91678},[169],[23,91680,91682],{"className":91681},[998],[23,91683,91685],{"className":91684},[199],[23,91686,91688],{"className":91687},[204],[23,91689,91691],{"className":91690,"style":4693},[208],[23,91692,91693,91696],{"style":4696},[23,91694],{"className":91695,"style":1015},[216],[23,91697,91699],{"className":91698},[1019,1020,1021,1022],[23,91700,9952],{"className":91701},[135,1022],[3788,91703,91704],{},"样条系数",[3788,91706,91097],{},[338,91708,91710],{"id":91709},"六计算代价与大规模近似","六、计算代价与大规模近似",[11,91712,91713,91714,91742,91743,91771,91772,91800,91801,91874,91875,91948,91949,91992,91993,8055,92090,92170],{},"表示定理把无穷维问题降到了 ",[23,91715,91717,91730],{"className":91716,"translate":27},[31],[23,91718,91720],{"className":91719},[35],[37,91721,91722],{"xmlns":39},[42,91723,91724,91728],{},[45,91725,91726],{},[48,91727,4806],{},[114,91729,4806],{"encoding":116},[23,91731,91733],{"className":91732,"ariaHidden":122},[121],[23,91734,91736,91739],{"className":91735},[126],[23,91737],{"className":91738,"style":703},[130],[23,91740,4806],{"className":91741},[135,136]," 维，但 ",[23,91744,91746,91759],{"className":91745,"translate":27},[31],[23,91747,91749],{"className":91748},[35],[37,91750,91751],{"xmlns":39},[42,91752,91753,91757],{},[45,91754,91755],{},[48,91756,4806],{},[114,91758,4806],{"encoding":116},[23,91760,91762],{"className":91761,"ariaHidden":122},[121],[23,91763,91765,91768],{"className":91764},[126],[23,91766],{"className":91767,"style":703},[130],[23,91769,4806],{"className":91770},[135,136]," 是训练样本数，可能很大。核矩阵 ",[23,91773,91775,91788],{"className":91774,"translate":27},[31],[23,91776,91778],{"className":91777},[35],[37,91779,91780],{"xmlns":39},[42,91781,91782,91786],{},[45,91783,91784],{},[48,91785,26918],{},[114,91787,26918],{"encoding":116},[23,91789,91791],{"className":91790,"ariaHidden":122},[121],[23,91792,91794,91797],{"className":91793},[126],[23,91795],{"className":91796,"style":470},[130],[23,91798,26918],{"className":91799,"style":37141},[135,136]," 需要 ",[23,91802,91804,91827],{"className":91803,"translate":27},[31],[23,91805,91807],{"className":91806},[35],[37,91808,91809],{"xmlns":39},[42,91810,91811,91825],{},[45,91812,91813,91815,91817,91823],{},[48,91814,42340],{},[51,91816,54],{"stretchy":53},[4518,91818,91819,91821],{},[48,91820,4806],{},[2581,91822,9952],{},[51,91824,66],{"stretchy":53},[114,91826,83415],{"encoding":116},[23,91828,91830],{"className":91829,"ariaHidden":122},[121],[23,91831,91833,91836,91839,91842,91871],{"className":91832},[126],[23,91834],{"className":91835,"style":10661},[130],[23,91837,42340],{"className":91838,"style":144},[135,136],[23,91840,54],{"className":91841},[140],[23,91843,91845,91848],{"className":91844},[135],[23,91846,4806],{"className":91847},[135,136],[23,91849,91851],{"className":91850},[998],[23,91852,91854],{"className":91853},[199],[23,91855,91857],{"className":91856},[204],[23,91858,91860],{"className":91859,"style":4693},[208],[23,91861,91862,91865],{"style":4696},[23,91863],{"className":91864,"style":1015},[216],[23,91866,91868],{"className":91867},[1019,1020,1021,1022],[23,91869,9952],{"className":91870},[135,1022],[23,91872,66],{"className":91873},[169]," 内存；核岭回归的 Cholesky 分解需要 ",[23,91876,91878,91901],{"className":91877,"translate":27},[31],[23,91879,91881],{"className":91880},[35],[37,91882,91883],{"xmlns":39},[42,91884,91885,91899],{},[45,91886,91887,91889,91891,91897],{},[48,91888,42340],{},[51,91890,54],{"stretchy":53},[4518,91892,91893,91895],{},[48,91894,4806],{},[2581,91896,24919],{},[51,91898,66],{"stretchy":53},[114,91900,83340],{"encoding":116},[23,91902,91904],{"className":91903,"ariaHidden":122},[121],[23,91905,91907,91910,91913,91916,91945],{"className":91906},[126],[23,91908],{"className":91909,"style":10661},[130],[23,91911,42340],{"className":91912,"style":144},[135,136],[23,91914,54],{"className":91915},[140],[23,91917,91919,91922],{"className":91918},[135],[23,91920,4806],{"className":91921},[135,136],[23,91923,91925],{"className":91924},[998],[23,91926,91928],{"className":91927},[199],[23,91929,91931],{"className":91930},[204],[23,91932,91934],{"className":91933,"style":4693},[208],[23,91935,91936,91939],{"style":4696},[23,91937],{"className":91938,"style":1015},[216],[23,91940,91942],{"className":91941},[1019,1020,1021,1022],[23,91943,24919],{"className":91944},[135,1022],[23,91946,66],{"className":91947},[169]," 时间；SVM 的通用二次规划更慢。预测一个新点需要计算它与所有训练点的核函数值，代价是 ",[23,91950,91952,91971],{"className":91951,"translate":27},[31],[23,91953,91955],{"className":91954},[35],[37,91956,91957],{"xmlns":39},[42,91958,91959,91969],{},[45,91960,91961,91963,91965,91967],{},[48,91962,42340],{},[51,91964,54],{"stretchy":53},[48,91966,4806],{},[51,91968,66],{"stretchy":53},[114,91970,83486],{"encoding":116},[23,91972,91974],{"className":91973,"ariaHidden":122},[121],[23,91975,91977,91980,91983,91986,91989],{"className":91976},[126],[23,91978],{"className":91979,"style":131},[130],[23,91981,42340],{"className":91982,"style":144},[135,136],[23,91984,54],{"className":91985},[140],[23,91987,4806],{"className":91988},[135,136],[23,91990,66],{"className":91991},[169],"（SVM 中是 ",[23,91994,91996,92025],{"className":91995,"translate":27},[31],[23,91997,91999],{"className":91998},[35],[37,92000,92001],{"xmlns":39},[42,92002,92003,92022],{},[45,92004,92005,92007,92009,92020],{},[48,92006,42340],{},[51,92008,54],{"stretchy":53},[898,92010,92011,92013],{},[48,92012,4806],{},[45,92014,92015,92018],{},[48,92016,92017],{"mathvariant":111},"S",[48,92019,10790],{"mathvariant":111},[51,92021,66],{"stretchy":53},[114,92023,92024],{"encoding":116},"O(n_{\\mathrm{SV}})",[23,92026,92028],{"className":92027,"ariaHidden":122},[121],[23,92029,92031,92034,92037,92040,92087],{"className":92030},[126],[23,92032],{"className":92033,"style":131},[130],[23,92035,42340],{"className":92036,"style":144},[135,136],[23,92038,54],{"className":92039},[140],[23,92041,92043,92046],{"className":92042},[135],[23,92044,4806],{"className":92045},[135,136],[23,92047,92049],{"className":92048},[998],[23,92050,92052,92079],{"className":92051},[199,200],[23,92053,92055,92076],{"className":92054},[204],[23,92056,92058],{"className":92057,"style":2026},[208],[23,92059,92060,92063],{"style":2029},[23,92061],{"className":92062,"style":1015},[216],[23,92064,92066],{"className":92065},[1019,1020,1021,1022],[23,92067,92069],{"className":92068},[135,1022],[23,92070,92072],{"className":92071},[135,1022],[23,92073,92075],{"className":92074,"style":7088},[135,5391,1022],"SV",[23,92077,297],{"className":92078},[296],[23,92080,92082],{"className":92081},[204],[23,92083,92085],{"className":92084,"style":2048},[208],[23,92086],{},[23,92088,66],{"className":92089},[169],[23,92091,92093,92115],{"className":92092,"translate":27},[31],[23,92094,92096],{"className":92095},[35],[37,92097,92098],{"xmlns":39},[42,92099,92100,92112],{},[45,92101,92102],{},[898,92103,92104,92106],{},[48,92105,4806],{},[45,92107,92108,92110],{},[48,92109,92017],{"mathvariant":111},[48,92111,10790],{"mathvariant":111},[114,92113,92114],{"encoding":116},"n_{\\mathrm{SV}}",[23,92116,92118],{"className":92117,"ariaHidden":122},[121],[23,92119,92121,92124],{"className":92120},[126],[23,92122],{"className":92123,"style":12730},[130],[23,92125,92127,92130],{"className":92126},[135],[23,92128,4806],{"className":92129},[135,136],[23,92131,92133],{"className":92132},[998],[23,92134,92136,92162],{"className":92135},[199,200],[23,92137,92139,92159],{"className":92138},[204],[23,92140,92142],{"className":92141,"style":2026},[208],[23,92143,92144,92147],{"style":2029},[23,92145],{"className":92146,"style":1015},[216],[23,92148,92150],{"className":92149},[1019,1020,1021,1022],[23,92151,92153],{"className":92152},[135,1022],[23,92154,92156],{"className":92155},[135,1022],[23,92157,92075],{"className":92158,"style":7088},[135,5391,1022],[23,92160,297],{"className":92161},[296],[23,92163,92165],{"className":92164},[204],[23,92166,92168],{"className":92167,"style":2048},[208],[23,92169],{}," 为支持向量数）。",[11,92172,92173,92174,92202],{},"当 ",[23,92175,92177,92190],{"className":92176,"translate":27},[31],[23,92178,92180],{"className":92179},[35],[37,92181,92182],{"xmlns":39},[42,92183,92184,92188],{},[45,92185,92186],{},[48,92187,4806],{},[114,92189,4806],{"encoding":116},[23,92191,92193],{"className":92192,"ariaHidden":122},[121],[23,92194,92196,92199],{"className":92195},[126],[23,92197],{"className":92198,"style":703},[130],[23,92200,4806],{"className":92201},[135,136]," 很大时，常用的近似方法有：",[346,92204,92205,92724,92919,92976],{},[349,92206,92207,92210,92211,92263,92264,92519,92520,67579,92593,63810],{},[18,92208,92209],{},"Nyström 近似","：选 ",[23,92212,92214,92233],{"className":92213,"translate":27},[31],[23,92215,92217],{"className":92216},[35],[37,92218,92219],{"xmlns":39},[42,92220,92221,92230],{},[45,92222,92223,92225,92228],{},[48,92224,50961],{},[51,92226,92227],{},"≪",[48,92229,4806],{},[114,92231,92232],{"encoding":116},"m\\ll n",[23,92234,92236,92254],{"className":92235,"ariaHidden":122},[121],[23,92237,92239,92242,92245,92248,92251],{"className":92238},[126],[23,92240],{"className":92241,"style":513},[130],[23,92243,50961],{"className":92244},[135,136],[23,92246],{"className":92247,"style":149},[148],[23,92249,92227],{"className":92250},[153],[23,92252],{"className":92253,"style":149},[148],[23,92255,92257,92260],{"className":92256},[126],[23,92258],{"className":92259,"style":703},[130],[23,92261,4806],{"className":92262},[135,136]," 个「地标点」，用 ",[23,92265,92267,92325],{"className":92266,"translate":27},[31],[23,92268,92270],{"className":92269},[35],[37,92271,92272],{"xmlns":39},[42,92273,92274,92322],{},[45,92275,92276,92278,92280,92292,92310],{},[48,92277,26918],{},[51,92279,29066],{},[898,92281,92282,92284],{},[48,92283,26918],{},[45,92285,92286,92288,92290],{},[48,92287,4806],{},[51,92289,747],{"separator":122},[48,92291,50961],{},[19137,92293,92294,92296,92304],{},[48,92295,26918],{},[45,92297,92298,92300,92302],{},[48,92299,50961],{},[51,92301,747],{"separator":122},[48,92303,50961],{},[45,92305,92306,92308],{},[51,92307,4529],{},[2581,92309,4435],{},[898,92311,92312,92314],{},[48,92313,26918],{},[45,92315,92316,92318,92320],{},[48,92317,50961],{},[51,92319,747],{"separator":122},[48,92321,4806],{},[114,92323,92324],{"encoding":116},"K\\approx K_{n,m}K_{m,m}^{-1}K_{m,n}",[23,92326,92328,92346],{"className":92327,"ariaHidden":122},[121],[23,92329,92331,92334,92337,92340,92343],{"className":92330},[126],[23,92332],{"className":92333,"style":470},[130],[23,92335,26918],{"className":92336,"style":37141},[135,136],[23,92338],{"className":92339,"style":149},[148],[23,92341,29066],{"className":92342},[153],[23,92344],{"className":92345,"style":149},[148],[23,92347,92349,92353,92402,92470],{"className":92348},[126],[23,92350],{"className":92351,"style":92352},[130],"height:1.1972em;vertical-align:-0.3831em;",[23,92354,92356,92359],{"className":92355},[135],[23,92357,26918],{"className":92358,"style":37141},[135,136],[23,92360,92362],{"className":92361},[998],[23,92363,92365,92394],{"className":92364},[199,200],[23,92366,92368,92391],{"className":92367},[204],[23,92369,92371],{"className":92370,"style":6499},[208],[23,92372,92373,92376],{"style":63365},[23,92374],{"className":92375,"style":1015},[216],[23,92377,92379],{"className":92378},[1019,1020,1021,1022],[23,92380,92382,92385,92388],{"className":92381},[135,1022],[23,92383,4806],{"className":92384},[135,136,1022],[23,92386,747],{"className":92387},[824,1022],[23,92389,50961],{"className":92390},[135,136,1022],[23,92392,297],{"className":92393},[296],[23,92395,92397],{"className":92396},[204],[23,92398,92400],{"className":92399,"style":14292},[208],[23,92401],{},[23,92403,92405,92408],{"className":92404},[135],[23,92406,26918],{"className":92407,"style":37141},[135,136],[23,92409,92411],{"className":92410},[998],[23,92412,92414,92461],{"className":92413},[199,200],[23,92415,92417,92458],{"className":92416},[204],[23,92418,92420,92441],{"className":92419,"style":4693},[208],[23,92421,92423,92426],{"style":92422},"top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;",[23,92424],{"className":92425,"style":1015},[216],[23,92427,92429],{"className":92428},[1019,1020,1021,1022],[23,92430,92432,92435,92438],{"className":92431},[135,1022],[23,92433,50961],{"className":92434},[135,136,1022],[23,92436,747],{"className":92437},[824,1022],[23,92439,50961],{"className":92440},[135,136,1022],[23,92442,92443,92446],{"style":4696},[23,92444],{"className":92445,"style":1015},[216],[23,92447,92449],{"className":92448},[1019,1020,1021,1022],[23,92450,92452,92455],{"className":92451},[135,1022],[23,92453,4529],{"className":92454},[135,1022],[23,92456,4435],{"className":92457},[135,1022],[23,92459,297],{"className":92460},[296],[23,92462,92464],{"className":92463},[204],[23,92465,92468],{"className":92466,"style":92467},[208],"height:0.3831em;",[23,92469],{},[23,92471,92473,92476],{"className":92472},[135],[23,92474,26918],{"className":92475,"style":37141},[135,136],[23,92477,92479],{"className":92478},[998],[23,92480,92482,92511],{"className":92481},[199,200],[23,92483,92485,92508],{"className":92484},[204],[23,92486,92488],{"className":92487,"style":6499},[208],[23,92489,92490,92493],{"style":63365},[23,92491],{"className":92492,"style":1015},[216],[23,92494,92496],{"className":92495},[1019,1020,1021,1022],[23,92497,92499,92502,92505],{"className":92498},[135,1022],[23,92500,50961],{"className":92501},[135,136,1022],[23,92503,747],{"className":92504},[824,1022],[23,92506,4806],{"className":92507},[135,136,1022],[23,92509,297],{"className":92510},[296],[23,92512,92514],{"className":92513},[204],[23,92515,92517],{"className":92516,"style":14292},[208],[23,92518],{}," 逼近核矩阵，代价从 ",[23,92521,92523,92546],{"className":92522,"translate":27},[31],[23,92524,92526],{"className":92525},[35],[37,92527,92528],{"xmlns":39},[42,92529,92530,92544],{},[45,92531,92532,92534,92536,92542],{},[48,92533,42340],{},[51,92535,54],{"stretchy":53},[4518,92537,92538,92540],{},[48,92539,4806],{},[2581,92541,24919],{},[51,92543,66],{"stretchy":53},[114,92545,83340],{"encoding":116},[23,92547,92549],{"className":92548,"ariaHidden":122},[121],[23,92550,92552,92555,92558,92561,92590],{"className":92551},[126],[23,92553],{"className":92554,"style":10661},[130],[23,92556,42340],{"className":92557,"style":144},[135,136],[23,92559,54],{"className":92560},[140],[23,92562,92564,92567],{"className":92563},[135],[23,92565,4806],{"className":92566},[135,136],[23,92568,92570],{"className":92569},[998],[23,92571,92573],{"className":92572},[199],[23,92574,92576],{"className":92575},[204],[23,92577,92579],{"className":92578,"style":4693},[208],[23,92580,92581,92584],{"style":4696},[23,92582],{"className":92583,"style":1015},[216],[23,92585,92587],{"className":92586},[1019,1020,1021,1022],[23,92588,24919],{"className":92589},[135,1022],[23,92591,66],{"className":92592},[169],[23,92594,92596,92630],{"className":92595,"translate":27},[31],[23,92597,92599],{"className":92598},[35],[37,92600,92601],{"xmlns":39},[42,92602,92603,92627],{},[45,92604,92605,92607,92609,92611,92617,92619,92625],{},[48,92606,42340],{},[51,92608,54],{"stretchy":53},[48,92610,4806],{},[4518,92612,92613,92615],{},[48,92614,50961],{},[2581,92616,9952],{},[51,92618,5106],{},[4518,92620,92621,92623],{},[48,92622,50961],{},[2581,92624,24919],{},[51,92626,66],{"stretchy":53},[114,92628,92629],{"encoding":116},"O(nm^2+m^3)",[23,92631,92633,92686],{"className":92632,"ariaHidden":122},[121],[23,92634,92636,92639,92642,92645,92648,92677,92680,92683],{"className":92635},[126],[23,92637],{"className":92638,"style":10661},[130],[23,92640,42340],{"className":92641,"style":144},[135,136],[23,92643,54],{"className":92644},[140],[23,92646,4806],{"className":92647},[135,136],[23,92649,92651,92654],{"className":92650},[135],[23,92652,50961],{"className":92653},[135,136],[23,92655,92657],{"className":92656},[998],[23,92658,92660],{"className":92659},[199],[23,92661,92663],{"className":92662},[204],[23,92664,92666],{"className":92665,"style":4693},[208],[23,92667,92668,92671],{"style":4696},[23,92669],{"className":92670,"style":1015},[216],[23,92672,92674],{"className":92673},[1019,1020,1021,1022],[23,92675,9952],{"className":92676},[135,1022],[23,92678],{"className":92679,"style":392},[148],[23,92681,5106],{"className":92682},[612],[23,92684],{"className":92685,"style":392},[148],[23,92687,92689,92692,92721],{"className":92688},[126],[23,92690],{"className":92691,"style":10661},[130],[23,92693,92695,92698],{"className":92694},[135],[23,92696,50961],{"className":92697},[135,136],[23,92699,92701],{"className":92700},[998],[23,92702,92704],{"className":92703},[199],[23,92705,92707],{"className":92706},[204],[23,92708,92710],{"className":92709,"style":4693},[208],[23,92711,92712,92715],{"style":4696},[23,92713],{"className":92714,"style":1015},[216],[23,92716,92718],{"className":92717},[1019,1020,1021,1022],[23,92719,24919],{"className":92720},[135,1022],[23,92722,66],{"className":92723},[169],[349,92725,92726,92729,92730,92758,92759,92787,92788,63810],{},[18,92727,92728],{},"随机傅里叶特征","：用 ",[23,92731,92733,92746],{"className":92732,"translate":27},[31],[23,92734,92736],{"className":92735},[35],[37,92737,92738],{"xmlns":39},[42,92739,92740,92744],{},[45,92741,92742],{},[48,92743,63],{},[114,92745,63],{"encoding":116},[23,92747,92749],{"className":92748,"ariaHidden":122},[121],[23,92750,92752,92755],{"className":92751},[126],[23,92753],{"className":92754,"style":470},[130],[23,92756,63],{"className":92757,"style":144},[135,136]," 个随机特征近似平移不变核，把核方法转成 ",[23,92760,92762,92775],{"className":92761,"translate":27},[31],[23,92763,92765],{"className":92764},[35],[37,92766,92767],{"xmlns":39},[42,92768,92769,92773],{},[45,92770,92771],{},[48,92772,63],{},[114,92774,63],{"encoding":116},[23,92776,92778],{"className":92777,"ariaHidden":122},[121],[23,92779,92781,92784],{"className":92780},[126],[23,92782],{"className":92783,"style":470},[130],[23,92785,63],{"className":92786,"style":144},[135,136]," 维的线性方法，代价约为 ",[23,92789,92791,92825],{"className":92790,"translate":27},[31],[23,92792,92794],{"className":92793},[35],[37,92795,92796],{"xmlns":39},[42,92797,92798,92822],{},[45,92799,92800,92802,92804,92806,92812,92814,92820],{},[48,92801,42340],{},[51,92803,54],{"stretchy":53},[48,92805,4806],{},[4518,92807,92808,92810],{},[48,92809,63],{},[2581,92811,9952],{},[51,92813,5106],{},[4518,92815,92816,92818],{},[48,92817,63],{},[2581,92819,24919],{},[51,92821,66],{"stretchy":53},[114,92823,92824],{"encoding":116},"O(nD^2+D^3)",[23,92826,92828,92881],{"className":92827,"ariaHidden":122},[121],[23,92829,92831,92834,92837,92840,92843,92872,92875,92878],{"className":92830},[126],[23,92832],{"className":92833,"style":10661},[130],[23,92835,42340],{"className":92836,"style":144},[135,136],[23,92838,54],{"className":92839},[140],[23,92841,4806],{"className":92842},[135,136],[23,92844,92846,92849],{"className":92845},[135],[23,92847,63],{"className":92848,"style":144},[135,136],[23,92850,92852],{"className":92851},[998],[23,92853,92855],{"className":92854},[199],[23,92856,92858],{"className":92857},[204],[23,92859,92861],{"className":92860,"style":4693},[208],[23,92862,92863,92866],{"style":4696},[23,92864],{"className":92865,"style":1015},[216],[23,92867,92869],{"className":92868},[1019,1020,1021,1022],[23,92870,9952],{"className":92871},[135,1022],[23,92873],{"className":92874,"style":392},[148],[23,92876,5106],{"className":92877},[612],[23,92879],{"className":92880,"style":392},[148],[23,92882,92884,92887,92916],{"className":92883},[126],[23,92885],{"className":92886,"style":10661},[130],[23,92888,92890,92893],{"className":92889},[135],[23,92891,63],{"className":92892,"style":144},[135,136],[23,92894,92896],{"className":92895},[998],[23,92897,92899],{"className":92898},[199],[23,92900,92902],{"className":92901},[204],[23,92903,92905],{"className":92904,"style":4693},[208],[23,92906,92907,92910],{"style":4696},[23,92908],{"className":92909,"style":1015},[216],[23,92911,92913],{"className":92912},[1019,1020,1021,1022],[23,92914,24919],{"className":92915},[135,1022],[23,92917,66],{"className":92918},[169],[349,92920,92921,92924,92925,92975],{},[18,92922,92923],{},"原问题方法","：对线性核或可显式低维特征化的核，直接在原空间优化，避免 ",[23,92926,92928,92945],{"className":92927,"translate":27},[31],[23,92929,92931],{"className":92930},[35],[37,92932,92933],{"xmlns":39},[42,92934,92935,92943],{},[45,92936,92937,92939,92941],{},[48,92938,4806],{},[51,92940,563],{},[48,92942,4806],{},[114,92944,60651],{"encoding":116},[23,92946,92948,92966],{"className":92947,"ariaHidden":122},[121],[23,92949,92951,92954,92957,92960,92963],{"className":92950},[126],[23,92952],{"className":92953,"style":5445},[130],[23,92955,4806],{"className":92956},[135,136],[23,92958],{"className":92959,"style":392},[148],[23,92961,563],{"className":92962},[612],[23,92964],{"className":92965,"style":392},[148],[23,92967,92969,92972],{"className":92968},[126],[23,92970],{"className":92971,"style":703},[130],[23,92973,4806],{"className":92974},[135,136]," 矩阵；",[349,92977,92978,92981],{},[18,92979,92980],{},"子集选择 \u002F 预算方法","：只保留一部分训练点作为核中心，控制模型规模。",[11,92983,92984,92985,93013],{},"这些方法改变的是「用多少个核函数」或「如何近似核矩阵」，而不是表示定理本身。表示定理保证精确解在 ",[23,92986,92988,93001],{"className":92987,"translate":27},[31],[23,92989,92991],{"className":92990},[35],[37,92992,92993],{"xmlns":39},[42,92994,92995,92999],{},[45,92996,92997],{},[48,92998,4806],{},[114,93000,4806],{"encoding":116},[23,93002,93004],{"className":93003,"ariaHidden":122},[121],[23,93005,93007,93010],{"className":93006},[126],[23,93008],{"className":93009,"style":703},[130],[23,93011,4806],{"className":93012},[135,136]," 维子空间里；近似方法则用低秩矩阵或低维特征来替代这个子空间的计算。",[338,93015,93017],{"id":93016},"七表示定理与深度学习","七、表示定理与深度学习",[11,93019,93020,93021,93024],{},"表示定理并没有随着深度学习的兴起而过时。相反，它在",[18,93022,93023],{},"神经正切核","（Neural Tangent Kernel, NTK）的框架里重新出现。",[11,93026,93027],{},"考虑一个无限宽的神经网络，用梯度下降在平方损失上训练。在适当的参数化和训练体制下，网络函数会收敛到核回归的解：",[23,93029,93031],{"className":93030,"translate":27},[26],[23,93032,93034,93112],{"className":93033,"translate":27},[31],[23,93035,93037],{"className":93036},[35],[37,93038,93039],{"xmlns":39,"display":40},[42,93040,93041,93109],{},[45,93042,93043,93045,93047,93049,93051,93053,93065,93067,93069,93071,93073,93075,93077,93089,93091,93093,93095,93105,93107],{},[48,93044,8336],{},[51,93046,54],{"stretchy":53},[48,93048,753],{},[51,93050,66],{"stretchy":53},[51,93052,69],{},[898,93054,93055,93057],{},[48,93056,4836],{},[45,93058,93059,93061,93063],{},[48,93060,12801],{"mathvariant":111},[48,93062,20703],{"mathvariant":111},[48,93064,26918],{"mathvariant":111},[51,93066,54],{"stretchy":53},[48,93068,753],{},[51,93070,747],{"separator":122},[48,93072,434],{},[51,93074,66],{"stretchy":53},[51,93076,54],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[898,93078,93079,93081],{},[48,93080,26918],{},[45,93082,93083,93085,93087],{},[48,93084,12801],{"mathvariant":111},[48,93086,20703],{"mathvariant":111},[48,93088,26918],{"mathvariant":111},[51,93090,5106],{},[48,93092,61753],{},[48,93094,34353],{},[4518,93096,93097,93099],{},[51,93098,66],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[45,93100,93101,93103],{},[51,93102,4529],{},[2581,93104,4435],{},[48,93106,660],{},[51,93108,747],{"separator":122},[114,93110,93111],{"encoding":116},"f(x)=k_{\\mathrm{NTK}}(x,X)\\bigl(K_{\\mathrm{NTK}}+\\lambda I\\bigr)^{-1}y,",[23,93113,93115,93142,93275],{"className":93114,"ariaHidden":122},[121],[23,93116,93118,93121,93124,93127,93130,93133,93136,93139],{"className":93117},[126],[23,93119],{"className":93120,"style":131},[130],[23,93122,8336],{"className":93123,"style":11398},[135,136],[23,93125,54],{"className":93126},[140],[23,93128,753],{"className":93129},[135,136],[23,93131,66],{"className":93132},[169],[23,93134],{"className":93135,"style":149},[148],[23,93137,69],{"className":93138},[153],[23,93140],{"className":93141,"style":149},[148],[23,93143,93145,93148,93196,93199,93202,93205,93208,93211,93214,93220,93266,93269,93272],{"className":93144},[126],[23,93146],{"className":93147,"style":2213},[130],[23,93149,93151,93154],{"className":93150},[135],[23,93152,4836],{"className":93153,"style":4851},[135,136],[23,93155,93157],{"className":93156},[998],[23,93158,93160,93188],{"className":93159},[199,200],[23,93161,93163,93185],{"className":93162},[204],[23,93164,93166],{"className":93165,"style":2026},[208],[23,93167,93169,93172],{"style":93168},"top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;",[23,93170],{"className":93171,"style":1015},[216],[23,93173,93175],{"className":93174},[1019,1020,1021,1022],[23,93176,93178],{"className":93177},[135,1022],[23,93179,93181],{"className":93180},[135,1022],[23,93182,93184],{"className":93183},[135,5391,1022],"NTK",[23,93186,297],{"className":93187},[296],[23,93189,93191],{"className":93190},[204],[23,93192,93194],{"className":93193,"style":2048},[208],[23,93195],{},[23,93197,54],{"className":93198},[140],[23,93200,753],{"className":93201},[135,136],[23,93203,747],{"className":93204},[824],[23,93206],{"className":93207,"style":280},[148],[23,93209,434],{"className":93210,"style":454},[135,136],[23,93212,66],{"className":93213},[169],[23,93215,93217],{"className":93216},[140],[23,93218,54],{"className":93219},[1066,1067],[23,93221,93223,93226],{"className":93222},[135],[23,93224,26918],{"className":93225,"style":37141},[135,136],[23,93227,93229],{"className":93228},[998],[23,93230,93232,93258],{"className":93231},[199,200],[23,93233,93235,93255],{"className":93234},[204],[23,93236,93238],{"className":93237,"style":2026},[208],[23,93239,93240,93243],{"style":63365},[23,93241],{"className":93242,"style":1015},[216],[23,93244,93246],{"className":93245},[1019,1020,1021,1022],[23,93247,93249],{"className":93248},[135,1022],[23,93250,93252],{"className":93251},[135,1022],[23,93253,93184],{"className":93254},[135,5391,1022],[23,93256,297],{"className":93257},[296],[23,93259,93261],{"className":93260},[204],[23,93262,93264],{"className":93263,"style":2048},[208],[23,93265],{},[23,93267],{"className":93268,"style":392},[148],[23,93270,5106],{"className":93271},[612],[23,93273],{"className":93274,"style":392},[148],[23,93276,93278,93281,93284,93287,93325,93328],{"className":93277},[126],[23,93279],{"className":93280,"style":34173},[130],[23,93282,61753],{"className":93283},[135,136],[23,93285,34353],{"className":93286,"style":454},[135,136],[23,93288,93290,93296],{"className":93289},[169],[23,93291,93293],{"className":93292},[169],[23,93294,66],{"className":93295},[1066,1067],[23,93297,93299],{"className":93298},[998],[23,93300,93302],{"className":93301},[199],[23,93303,93305],{"className":93304},[204],[23,93306,93308],{"className":93307,"style":11188},[208],[23,93309,93310,93313],{"style":11191},[23,93311],{"className":93312,"style":1015},[216],[23,93314,93316],{"className":93315},[1019,1020,1021,1022],[23,93317,93319,93322],{"className":93318},[135,1022],[23,93320,4529],{"className":93321},[135,1022],[23,93323,4435],{"className":93324},[135,1022],[23,93326,660],{"className":93327,"style":676},[135,136],[23,93329,747],{"className":93330},[824],[11,93332,24429,93333,93415,93416,93498],{},[23,93334,93336,93360],{"className":93335,"translate":27},[31],[23,93337,93339],{"className":93338},[35],[37,93340,93341],{"xmlns":39},[42,93342,93343,93357],{},[45,93344,93345],{},[898,93346,93347,93349],{},[48,93348,4836],{},[45,93350,93351,93353,93355],{},[48,93352,12801],{"mathvariant":111},[48,93354,20703],{"mathvariant":111},[48,93356,26918],{"mathvariant":111},[114,93358,93359],{"encoding":116},"k_{\\mathrm{NTK}}",[23,93361,93363],{"className":93362,"ariaHidden":122},[121],[23,93364,93366,93369],{"className":93365},[126],[23,93367],{"className":93368,"style":46082},[130],[23,93370,93372,93375],{"className":93371},[135],[23,93373,4836],{"className":93374,"style":4851},[135,136],[23,93376,93378],{"className":93377},[998],[23,93379,93381,93407],{"className":93380},[199,200],[23,93382,93384,93404],{"className":93383},[204],[23,93385,93387],{"className":93386,"style":2026},[208],[23,93388,93389,93392],{"style":93168},[23,93390],{"className":93391,"style":1015},[216],[23,93393,93395],{"className":93394},[1019,1020,1021,1022],[23,93396,93398],{"className":93397},[135,1022],[23,93399,93401],{"className":93400},[135,1022],[23,93402,93184],{"className":93403},[135,5391,1022],[23,93405,297],{"className":93406},[296],[23,93408,93410],{"className":93409},[204],[23,93411,93413],{"className":93412,"style":2048},[208],[23,93414],{}," 是网络对应的神经正切核，",[23,93417,93419,93443],{"className":93418,"translate":27},[31],[23,93420,93422],{"className":93421},[35],[37,93423,93424],{"xmlns":39},[42,93425,93426,93440],{},[45,93427,93428],{},[898,93429,93430,93432],{},[48,93431,26918],{},[45,93433,93434,93436,93438],{},[48,93435,12801],{"mathvariant":111},[48,93437,20703],{"mathvariant":111},[48,93439,26918],{"mathvariant":111},[114,93441,93442],{"encoding":116},"K_{\\mathrm{NTK}}",[23,93444,93446],{"className":93445,"ariaHidden":122},[121],[23,93447,93449,93452],{"className":93448},[126],[23,93450],{"className":93451,"style":42827},[130],[23,93453,93455,93458],{"className":93454},[135],[23,93456,26918],{"className":93457,"style":37141},[135,136],[23,93459,93461],{"className":93460},[998],[23,93462,93464,93490],{"className":93463},[199,200],[23,93465,93467,93487],{"className":93466},[204],[23,93468,93470],{"className":93469,"style":2026},[208],[23,93471,93472,93475],{"style":63365},[23,93473],{"className":93474,"style":1015},[216],[23,93476,93478],{"className":93477},[1019,1020,1021,1022],[23,93479,93481],{"className":93480},[135,1022],[23,93482,93484],{"className":93483},[135,1022],[23,93485,93184],{"className":93486},[135,5391,1022],[23,93488,297],{"className":93489},[296],[23,93491,93493],{"className":93492},[204],[23,93494,93496],{"className":93495,"style":2048},[208],[23,93497],{}," 是它在训练点上的核矩阵。这个式子与式 (3) 完全同构：预测值是训练点上核函数值的线性组合。",[11,93500,93501],{},"这解释了为什么在 NTK 机制里，深度网络的预测可以写成「训练目标的核加权和」。但也要注意它的边界：NTK 描述的是无限宽、线性化、平方损失、梯度流（或小学习率）的极限；它并不是对一般深度网络特征学习的完整解释。表示定理在这里的作用，是给这个极限提供一个清晰的结构语言。",[338,93503,93505],{"id":93504},"九结语","九、结语",[11,93507,93508,93509,93512],{},"表示定理的核心是一句几何话：",[18,93510,93511],{},"损失函数看不见训练点之外的正交分量，而正则项会惩罚它。"," 因此，最优解没有理由保留这个分量；在严格递增的正则项下，它必然落在训练点核函数张成的子空间里。",[11,93514,93515,93516,93630],{},"这个论证只用了两件事：再生性 ",[23,93517,93519,93561],{"className":93518,"translate":27},[31],[23,93520,93522],{"className":93521},[35],[37,93523,93524],{"xmlns":39},[42,93525,93526,93558],{},[45,93527,93528,93530,93532,93534,93536,93538,93540,93542,93544,93546,93548,93550,93552,93554,93556],{},[48,93529,8336],{},[51,93531,54],{"stretchy":53},[48,93533,753],{},[51,93535,66],{"stretchy":53},[51,93537,69],{},[51,93539,61181],{"stretchy":53},[48,93541,8336],{},[51,93543,747],{"separator":122},[48,93545,4836],{},[51,93547,54],{"stretchy":53},[51,93549,5758],{},[51,93551,747],{"separator":122},[48,93553,753],{},[51,93555,66],{"stretchy":53},[51,93557,61206],{"stretchy":53},[114,93559,93560],{"encoding":116},"f(x)=\\langle f,k(\\cdot,x)\\rangle",[23,93562,93564,93591],{"className":93563,"ariaHidden":122},[121],[23,93565,93567,93570,93573,93576,93579,93582,93585,93588],{"className":93566},[126],[23,93568],{"className":93569,"style":131},[130],[23,93571,8336],{"className":93572,"style":11398},[135,136],[23,93574,54],{"className":93575},[140],[23,93577,753],{"className":93578},[135,136],[23,93580,66],{"className":93581},[169],[23,93583],{"className":93584,"style":149},[148],[23,93586,69],{"className":93587},[153],[23,93589],{"className":93590,"style":149},[148],[23,93592,93594,93597,93600,93603,93606,93609,93612,93615,93618,93621,93624,93627],{"className":93593},[126],[23,93595],{"className":93596,"style":131},[130],[23,93598,61181],{"className":93599},[140],[23,93601,8336],{"className":93602,"style":11398},[135,136],[23,93604,747],{"className":93605},[824],[23,93607],{"className":93608,"style":280},[148],[23,93610,4836],{"className":93611,"style":4851},[135,136],[23,93613,54],{"className":93614},[140],[23,93616,5758],{"className":93617},[135],[23,93619,747],{"className":93620},[824],[23,93622],{"className":93623,"style":280},[148],[23,93625,753],{"className":93626},[135,136],[23,93628,61346],{"className":93629},[169],"，以及范数的勾股分解。它不需要损失凸、不需要可微、不需要核矩阵可逆。也正因如此，表示定理能同时覆盖核岭回归、SVM、核逻辑回归、高斯过程和平滑样条。",[11,93632,93633,93634],{},"但它的边界同样清楚：正则项必须是范数的非减函数；若只是非减而非严格递增，定理给出的是「存在一个」这样的解，而不是「每一个」解都如此。非单调的正则项可以让解离开训练点子空间，4.2 节的反例说明这不是技术细节，而是定理的本质。",[18,93635,93636,93637,93665,93666,93694],{},"无穷维并没有消失，它只是被 ",[23,93638,93640,93653],{"className":93639,"translate":27},[31],[23,93641,93643],{"className":93642},[35],[37,93644,93645],{"xmlns":39},[42,93646,93647,93651],{},[45,93648,93649],{},[48,93650,4806],{},[114,93652,4806],{"encoding":116},[23,93654,93656],{"className":93655,"ariaHidden":122},[121],[23,93657,93659,93662],{"className":93658},[126],[23,93660],{"className":93661,"style":703},[130],[23,93663,4806],{"className":93664},[135,136]," 个训练点压成了 ",[23,93667,93669,93682],{"className":93668,"translate":27},[31],[23,93670,93672],{"className":93671},[35],[37,93673,93674],{"xmlns":39},[42,93675,93676,93680],{},[45,93677,93678],{},[48,93679,4806],{},[114,93681,4806],{"encoding":116},[23,93683,93685],{"className":93684,"ariaHidden":122},[121],[23,93686,93688,93691],{"className":93687},[126],[23,93689],{"className":93690,"style":703},[130],[23,93692,4806],{"className":93693},[135,136]," 维。",{"title":60549,"searchDepth":60550,"depth":60550,"links":93696},[93697,93698,93703,93709,93714,93722,93723,93724],{"id":60688,"depth":60550,"text":60689},{"id":62366,"depth":60550,"text":62367,"children":93699},[93700,93701,93702],{"id":62370,"depth":60556,"text":62371},{"id":65723,"depth":60556,"text":65724},{"id":66561,"depth":60556,"text":66562},{"id":67641,"depth":60550,"text":67642,"children":93704},[93705,93706,93707,93708],{"id":67645,"depth":60556,"text":67646},{"id":69742,"depth":60556,"text":69743},{"id":73963,"depth":60556,"text":73964},{"id":76471,"depth":60556,"text":76472},{"id":78029,"depth":60550,"text":78030,"children":93710},[93711,93712,93713],{"id":78033,"depth":60556,"text":78034},{"id":78410,"depth":60556,"text":78411},{"id":80483,"depth":60556,"text":80484},{"id":81110,"depth":60550,"text":81111,"children":93715},[93716,93717,93718,93719,93720,93721],{"id":81114,"depth":60556,"text":81115},{"id":83511,"depth":60556,"text":83512},{"id":87329,"depth":60556,"text":87330},{"id":88676,"depth":60556,"text":88677},{"id":89877,"depth":60556,"text":89878},{"id":90875,"depth":60556,"text":90876},{"id":91709,"depth":60550,"text":91710},{"id":93016,"depth":60550,"text":93017},{"id":93504,"depth":60550,"text":93505},"核方法最反直觉的一点是：你明明在一个可能是无限维的函数空间里做优化，最后却只需要解一个 n×nn\\times nn×n 的线性系统或二次规划。把这件事说清楚的，就是表示定理。",{},"\u002Fblog\u002F2025\u002F2025-09-11-representer-theorem",{"title":60625,"description":93725},"blog\u002F2025\u002F2025-09-11-representer-theorem","表示定理是核方法的一个基石性结论。它指出：对于一个在再生核希尔伯特空间（RKHS）中带有范数正则化的经验风险最小化问题，其最优解一定可以写成核函数在训练样本点上的线性组合。",[60619],"2025-09-11T21:00:12+08:00","mTnCagDDEj7M4wOzjAllewJKiowioNxQIPyKgUsW0PI",{"id":93735,"title":93736,"body":93737,"description":60549,"draft":60609,"enableCodeHighlight":60609,"enableCodeHightlight":60609,"enableComment":60610,"enableD2":60609,"enableKatex":60609,"extension":60611,"image":60612,"important":60609,"location":60612,"meta":93815,"navigation":60610,"ogImage":60612,"path":93816,"seo":93817,"stem":93818,"summary":93819,"tags":93820,"time":93822,"__hash__":93823},"blog\u002Fblog\u002F2025\u002F2025-08-17-the-lamentable-programmers.md","可叹的程序员们",{"type":8,"value":93738,"toc":93813},[93739,93745,93748,93751,93754,93757,93760,93763,93766,93769,93772,93775,93778,93781,93784,93787,93790,93793,93796,93799],[93740,93741,93742],"blockquote",{},[11,93743,93744],{},"个人奋斗固然重要，但也必须顺应历史的进程。",[11,93746,93747],{},"我见过许多从事软件开发行业的程序员，有时也会加入网络社区的日常讨论，尤其是中文推特社区。那里聚集了大量来自腾讯、字节跳动、百度等大厂的国内互联网从业者。我一度也参与过他们关于程序员生活的争论。",[11,93749,93750],{},"中国的互联网行业可以算作一个站在「风口」上的特殊职业领域，也就是十年（2012—2022）的投资热点。这是许多互联网创业者高速扩张业务的时期，阿里巴巴、京东等皆在其列。",[11,93752,93753],{},"它对计算机人才的需求日益增长。许多计算机相关专业的毕业生赶上了自己的「时代红利」，随后他们取代了土木工程的地位，而后者曾预示大基建时代走向终结。一时间，计算机相关专业成了众多专业中的香饽饽。",[11,93755,93756],{},"毋庸置疑，计算机科学与软件开发属于高薪职业。这些毕业生身为程序员，比同龄人挣得更多，日子过得更体面。用不了多久，他们大体上就能迈入春风得意的中等收入阶层。",[11,93758,93759],{},"2012 到 2022 这十年，可以说是中国互联网的黄金时代。随之而来的高速扩张与高薪让他们沉溺于一种幻觉之中。当然，我可以理解：作为既得利益者，他们倾向于夸大前景、持乐观态度，因而在面对与自身利益相关的事情时很难冷静下来。",[11,93761,93762],{},"历史终将证明，繁荣终究是短暂的。",[11,93764,93765],{},"与美国——这个早已走完工业化并成为金融强国道路的老牌资本主义国家——不同，中国仍是一个以传统人力资源为基础的国家。经历新冠疫情之后，产业升级仍遥遥无期。这意味着中国的商业发展依然依赖最重要的因素：人力资源，而非资本增值。这也就是所谓的「人口红利」。因此，不难理解为什么 996 在中国如此泛滥，而政府对违反劳动法的行为总是睁一只眼闭一只眼。我们都知道生产关系决定生产力。中国的互联网行业本质上依附于实体经济，也就是说，互联网是服务于实体经济的。迄今为止，第一产业与第二产业分别占 40% 与 53%。换言之，互联网行业与第二、第三产业休戚与共。如果后者遭遇衰退，互联网行业必然首当其冲，且受损最为严重。",[11,93767,93768],{},"我也对中国互联网企业是否属于「新质生产力」持怀疑态度。查理·芒格（Charlie Munger）曾这样评价阿里巴巴：「我一直以为阿里巴巴是一家高科技公司，结果发现它不过是一家该死的零售商。」互联网巨头本质上就是「血汗工厂」。据我所知，大多数互联网业务在形式上都不过是 CRUD 的堆叠。人们津津乐道的「三高」——高并发、高性能、高可用，本质上都是在争夺存量市场。于是我们看到，程序员面试正朝着一种诡异而不正常的方向发展。正所谓「面试造火箭，工作拧螺丝」。程序员要过关斩将、通过层层严苛的考核，问题越来越多地脱离实际工作的需要，变成纯粹的智力游戏。如果你对这一现象提出异议，他们总会说：「你不干，有的是人干。」",[11,93770,93771],{},"的确，如果你观察产业结构就会发现，中国的各个产业如同一条条管道，它们的职责就是源源不断地泵出产品。",[11,93773,93774],{},"我曾面试过 Java 后端开发岗位。面试官问了我一些关于 Java 开发的问题。起初他只是问些基础知识，但随后便抛出了一些刁钻的难题，例如：如何优化 JVM 的底层架构？",[11,93776,93777],{},"中国程序员可能是全世界最勤奋、最辛苦的群体。他们中的大多数人不由自主地钻研苦刷海量的 LeetCode 算法题，只为在面试中答好那些模式化的问题，拿个最高分，拼命争得一份 offer。即便如此，在这场内卷的竞赛中，他们也逃不脱 35 岁被裁的宿命。",[11,93779,93780],{},"试想，即便你足够优秀，你又能否完全保证自己将来不会被弃用，尤其是在当下这个经济衰退的时期？",[11,93782,93783],{},"毕竟，人脑的学习能力是有限的，难以适应高速迭代的知识与经验。软件开发行业以技术的高速演进而闻名，一些开发框架、库的版本迭代快到每三天就发布一个新版本，塞进大量新特性与改动。此外，这些技术栈之间也彼此竞争。某个突出的技术栈取代其他技术栈而风靡一时，是常有的事。",[11,93785,93786],{},"在资本主义与自由主义的市场体系下，每个劳动者都应当有作为「工具人」的觉悟，程序员也不例外。但最糟糕的是，在被裁之后，程序员会陷入除写代码外别无技能的困境。人们常开玩笑地把程序员称作「码农」。若果真如此，那么农民工与程序员之间唯一的区别，不过是前者从事体力劳动，后者从事脑力劳动。而他们同样会随着时间推移、因磨损而贬值，就像一件被用旧的工具。这并非玩笑，而是严肃的事实。程序员如今实际上就是被物化为劳动工具的「工具」。你必须记住，除了你所谓的编程知识之外，你其实一无所有。",[11,93788,93789],{},"当我在推特上提出这一看法时，遭遇了不少反对。这无所谓。历史很快就会证明其正确性。",[11,93791,93792],{},"我经常在推特的骂战中看到程序员之间的争论，例如「React 还是 Vue」「要不要加大括号」这类毫无意义的琐碎问题。为此，他们总是分化出许多派别，互相争斗。",[11,93794,93795],{},"属于程序员的时代如今已渐近尾声，我们该何去何从？😔",[11,93797,93798],{},"最后，以下是我分享的一点心得：",[346,93800,93801,93804,93807,93810],{},[349,93802,93803],{},"趁年轻去谋取权力或资本，而不只是微薄的金钱；并且要及时意识到权力与资本的重要性。",[349,93805,93806],{},"不要把自己埋进代码和技术里，不要与社会脱节。要记得你是作为一个「人」而活着，而不是技术的化身。",[349,93808,93809],{},"学好英语。它不仅为出国移民和海外业务提供了基本技能，也足以开启与世界相连的信息之门。拥有熟练的英语能力，你能读懂世界上 90% 以上的信息。",[349,93811,93812],{},"探索一些副业或第二职业。要清醒地认识到：主业用来谋生，副业用来提升生活质量。",{"title":60549,"searchDepth":60550,"depth":60550,"links":93814},[],{},"\u002Fblog\u002F2025\u002F2025-08-17-the-lamentable-programmers",{"title":93736,"description":60549},"blog\u002F2025\u002F2025-08-17-the-lamentable-programmers","许多软件工程师和程序员常常陷入一种困境：他们对自己所掌握的技术细节沾沾自喜，却很少意识到生活中几乎没有什么事情与代码有关。",[93821],"thoughts","2025-08-17T00:00:00+08:00","LzgvRCx-_r2IhjfAgOtA5AVwS8dCHGrgF3-y9Kx1_1U",1794483700654]