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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。",[6857,6858,6859,6927,7251],"blockquote",{},[11,6860,6861,6864,6865,6926],{},[18,6862,6863],{},"定义 3.1（贝叶斯决策规则）"," 若决策规则 ",[23,6866,6868,6887],{"className":6867,"translate":27},[31],[23,6869,6871],{"className":6870},[35],[37,6872,6873],{"xmlns":39},[42,6874,6875,6884],{},[45,6876,6877],{},[4518,6878,6879,6881],{},[48,6880,736],{},[51,6882,6883],{},"∗",[114,6885,6886],{"encoding":116},"\\delta^*",[23,6888,6890],{"className":6889,"ariaHidden":122},[121],[23,6891,6893,6896],{"className":6892},[126],[23,6894],{"className":6895,"style":779},[130],[23,6897,6899,6902],{"className":6898},[135],[23,6900,736],{"className":6901,"style":783},[135,136],[23,6903,6905],{"className":6904},[998],[23,6906,6908],{"className":6907},[199],[23,6909,6911],{"className":6910},[204],[23,6912,6915],{"className":6913,"style":6914},[208],"height:0.6887em;",[23,6916,6917,6920],{"style":4696},[23,6918],{"className":6919,"style":1015},[216],[23,6921,6923],{"className":6922},[1019,1020,1021,1022],[23,6924,6883],{"className":6925},[612,1022]," 满足",[23,6928,6930],{"className":6929,"translate":27},[26],[23,6931,6933,7020],{"className":6932,"translate":27},[31],[23,6934,6936],{"className":6935},[35],[37,6937,6938],{"xmlns":39,"display":40},[42,6939,6940,7017],{},[45,6941,6942,6948,6950,6952,6954,6956,6959,6962,6980,6982,6984,6986,6988,6990,6992,6994,6996,6998,7000,7002,7004,7006,7008,7010,7013,7015],{},[4518,6943,6944,6946],{},[48,6945,736],{},[51,6947,6883],{},[51,6949,54],{"stretchy":53},[48,6951,753],{},[51,6953,66],{"stretchy":53},[51,6955,372],{},[48,6957,6958],{},"arg",[51,6960,6961],{},"⁡",[6963,6964,6965,6972],"munder",{},[45,6966,6967,6970],{},[48,6968,6969],{},"min",[51,6971,6961],{},[45,6973,6974,6976,6978],{},[48,6975,495],{},[51,6977,372],{},[48,6979,500],{"mathvariant":375},[88,6981,5360],{},[48,6983,902],{"mathvariant":571},[51,6985,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,6987,555],{},[51,6989,54],{"stretchy":53},[48,6991,369],{},[51,6993,747],{"separator":122},[48,6995,495],{},[51,6997,66],{"stretchy":53},[51,6999,60],{},[48,7001,434],{},[51,7003,69],{},[48,7005,753],{},[51,7007,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[148,7009],{"width":750},[88,7011,7012],{},"对几乎所有的 ",[48,7014,753],{},[51,7016,747],{"separator":122},[114,7018,7019],{"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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rule）。",[6857,7318,7319,7355],{},[11,7320,7321,7324,7325,7354],{},[18,7322,7323],{},"定理 3.1（逐点最优即全局最优）"," 对任意决策规则 ",[23,7326,7328,7342],{"className":7327,"translate":27},[31],[23,7329,7331],{"className":7330},[35],[37,7332,7333],{"xmlns":39},[42,7334,7335,7339],{},[45,7336,7337],{},[48,7338,736],{},[114,7340,7341],{"encoding":116},"\\delta",[23,7343,7345],{"className":7344,"ariaHidden":122},[121],[23,7346,7348,7351],{"className":7347},[126],[23,7349],{"className":7350,"style":779},[130],[23,7352,736],{"className":7353,"style":783},[135,136],"，都有",[23,7356,7358],{"className":7357,"translate":27},[26],[23,7359,7361,7443],{"className":7360,"translate":27},[31],[23,7362,7364],{"className":7363},[35],[37,7365,7366],{"xmlns":39,"display":40},[42,7367,7368,7440],{},[45,7369,7370,7372,7374,7376,7378,7380,7383,7385,7387,7389,7395,7397,7399,7405,7408,7424,7426,7428,7430,7432,7434,7436,7438],{},[48,7371,572],{},[51,7373,54],{"stretchy":53},[48,7375,736],{},[51,7377,66],{"stretchy":53},[88,7379,5360],{},[51,7381,7382],{},"≥",[88,7384,5360],{},[48,7386,572],{},[51,7388,54],{"stretchy":53},[4518,7390,7391,7393],{},[48,7392,736],{},[51,7394,6883],{},[51,7396,66],{"stretchy":53},[51,7398,69],{},[898,7400,7401,7403],{},[48,7402,902],{"mathvariant":571},[48,7404,434],{},[51,7406,924],{"fence":53,"stretchy":122,"minsize":7407,"maxsize":7407},"1.8em",[6963,7409,7410,7416],{},[45,7411,7412,7414],{},[48,7413,6969],{},[51,7415,6961],{},[45,7417,7418,7420,7422],{},[48,7419,495],{},[51,7421,372],{},[48,7423,500],{"mathvariant":375},[48,7425,1653],{},[51,7427,54],{"stretchy":53},[48,7429,495],{},[51,7431,747],{"separator":122},[48,7433,434],{},[51,7435,66],{"stretchy":53},[51,7437,945],{"fence":53,"stretchy":122,"minsize":7407,"maxsize":7407},[48,7439,112],{"mathvariant":111},[114,7441,7442],{"encoding":116},"R(\\delta)\\ \\ge\\ R(\\delta^*)=\\mathbb E_X\\Big[\\min_{a\\in\\mathcal A}r(a,X)\\Big].",[23,7444,7446,7479,7532],{"className":7445,"ariaHidden":122},[121],[23,7447,7449,7452,7455,7458,7461,7464,7467,7470,7473,7476],{"className":7448},[126],[23,7450],{"className":7451,"style":131},[130],[23,7453,572],{"className":7454,"style":963},[135,136],[23,7456,54],{"className":7457},[140],[23,7459,736],{"className":7460,"style":783},[135,136],[23,7462,66],{"className":7463},[169],[23,7465,5360],{"className":7466},[148],[23,7468],{"className":7469,"style":149},[148],[23,7471,7382],{"className":7472},[153],[23,7474,5360],{"className":7475},[148],[23,7477],{"className":7478,"style":149},[148],[23,7480,7482,7485,7488,7491,7520,7523,7526,7529],{"className":7481},[126],[23,7483],{"className":7484,"style":131},[130],[23,7486,572],{"className":7487,"style":963},[135,136],[23,7489,54],{"className":7490},[140],[23,7492,7494,7497],{"className":7493},[135],[23,7495,736],{"className":7496,"style":783},[135,136],[23,7498,7500],{"className":7499},[998],[23,7501,7503],{"className":7502},[199],[23,7504,7506],{"className":7505},[204],[23,7507,7509],{"className":7508,"style":7047},[208],[23,7510,7511,7514],{"style":4989},[23,7512],{"className":7513,"style":1015},[216],[23,7515,7517],{"className":7516},[1019,1020,1021,1022],[23,7518,6883],{"className":7519},[612,1022],[23,7521,66],{"className":7522},[169],[23,7524],{"className":7525,"style":149},[148],[23,7527,69],{"className":7528},[153],[23,7530],{"className":7531,"style":149},[148],[23,7533,7535,7539,7579,7586,7589,7642,7645,7648,7651,7654,7657,7660,7663,7666,7672],{"className":7534},[126],[23,7536],{"className":7537,"style":7538},[130],"height:1.9217em;vertical-align:-0.7717em;",[23,7540,7542,7545],{"className":7541},[135],[23,7543,902],{"className":7544},[135,644],[23,7546,7548],{"className":7547},[998],[23,7549,7551,7571],{"className":7550},[199,200],[23,7552,7554,7568],{"className":7553},[204],[23,7555,7557],{"className":7556,"style":2026},[208],[23,7558,7559,7562],{"style":2029},[23,7560],{"className":7561,"style":1015},[216],[23,7563,7565],{"className":7564},[1019,1020,1021,1022],[23,7566,434],{"className":7567,"style":454},[135,136,1022],[23,7569,297],{"className":7570},[296],[23,7572,7574],{"className":7573},[204],[23,7575,7577],{"className":7576,"style":2048},[208],[23,7578],{},[23,7580,7582],{"className":7581},[135],[23,7583,924],{"className":7584},[1066,7585],"size2",[23,7587],{"className":7588,"style":280},[148],[23,7590,7592],{"className":7591},[1786,7098],[23,7593,7595,7634],{"className":7594},[199,200],[23,7596,7598,7631],{"className":7597},[204],[23,7599,7601,7621],{"className":7600,"style":7108},[208],[23,7602,7603,7606],{"style":7111},[23,7604],{"className":7605,"style":217},[216],[23,7607,7609],{"className":7608},[1019,1020,1021,1022],[23,7610,7612,7615,7618],{"className":7611},[135,1022],[23,7613,495],{"className":7614},[135,136,1022],[23,7616,372],{"className":7617},[153,1022],[23,7619,500],{"className":7620},[135,412,1022],[23,7622,7623,7626],{"style":7132},[23,7624],{"className":7625,"style":217},[216],[23,7627,7628],{},[23,7629,6969],{"className":7630},[1786],[23,7632,297],{"className":7633},[296],[23,7635,7637],{"className":7636},[204],[23,7638,7640],{"className":7639,"style":7150},[208],[23,7641],{},[23,7643],{"className":7644,"style":280},[148],[23,7646,1653],{"className":7647,"style":144},[135,136],[23,7649,54],{"className":7650},[140],[23,7652,495],{"className":7653},[135,136],[23,7655,747],{"className":7656},[824],[23,7658],{"className":7659,"style":280},[148],[23,7661,434],{"className":7662,"style":454},[135,136],[23,7664,66],{"className":7665},[169],[23,7667,7669],{"className":7668},[135],[23,7670,945],{"className":7671},[1066,7585],[23,7673,112],{"className":7674},[135],[11,7676,7677,7680],{},[18,7678,7679],{},"证明。"," 由塔性质，",[23,7682,7684],{"className":7683,"translate":27},[26],[23,7685,7687,7779],{"className":7686,"translate":27},[31],[23,7688,7690],{"className":7689},[35],[37,7691,7692],{"xmlns":39,"display":40},[42,7693,7694,7776],{},[45,7695,7696,7698,7700,7702,7704,7706,7712,7714,7716,7718,7720,7722,7724,7726,7728,7730,7732,7734,7736,7738,7740,7742,7744,7746,7752,7754,7756,7758,7760,7762,7764,7766,7768,7770,7772,7774],{},[48,7697,572],{},[51,7699,54],{"stretchy":53},[48,7701,736],{},[51,7703,66],{"stretchy":53},[51,7705,69],{},[898,7707,7708,7710],{},[48,7709,902],{"mathvariant":571},[48,7711,434],{},[51,7713,924],{"fence":53,"stretchy":122,"minsize":7407,"maxsize":7407},[48,7715,902],{"mathvariant":571},[51,7717,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,7719,555],{},[51,7721,54],{"stretchy":53},[48,7723,369],{},[51,7725,747],{"separator":122},[48,7727,736],{},[51,7729,54],{"stretchy":53},[48,7731,434],{},[51,7733,66],{"stretchy":53},[51,7735,66],{"stretchy":53},[51,7737,60],{},[48,7739,434],{},[51,7741,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,7743,945],{"fence":53,"stretchy":122,"minsize":7407,"maxsize":7407},[51,7745,69],{},[898,7747,7748,7750],{},[48,7749,902],{"mathvariant":571},[48,7751,434],{},[51,7753,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,7755,1653],{},[51,7757,54],{"stretchy":53},[48,7759,736],{},[51,7761,54],{"stretchy":53},[48,7763,434],{},[51,7765,66],{"stretchy":53},[51,7767,747],{"separator":122},[48,7769,434],{},[51,7771,66],{"stretchy":53},[51,7773,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,7775,112],{"mathvariant":111},[114,7777,7778],{"encoding":116},"R(\\delta)=\\mathbb E_X\\Big[\\mathbb E\\big[L(Y,\\delta(X))\\mid X\\big]\\Big]\n=\\mathbb E_X\\big[r(\\delta(X),X)\\big].",[23,7780,7782,7809,7907,7937],{"className":7781,"ariaHidden":122},[121],[23,7783,7785,7788,7791,7794,7797,7800,7803,7806],{"className":7784},[126],[23,7786],{"className":7787,"style":131},[130],[23,7789,572],{"className":7790,"style":963},[135,136],[23,7792,54],{"className":7793},[140],[23,7795,736],{"className":7796,"style":783},[135,136],[23,7798,66],{"className":7799},[169],[23,7801],{"className":7802,"style":149},[148],[23,7804,69],{"className":7805},[153],[23,7807],{"className":7808,"style":149},[148],[23,7810,7812,7816,7856,7862,7865,7871,7874,7877,7880,7883,7886,7889,7892,7895,7898,7901,7904],{"className":7811},[126],[23,7813],{"className":7814,"style":7815},[130],"height:1.8em;vertical-align:-0.65em;",[23,7817,7819,7822],{"className":7818},[135],[23,7820,902],{"className":7821},[135,644],[23,7823,7825],{"className":7824},[998],[23,7826,7828,7848],{"className":7827},[199,200],[23,7829,7831,7845],{"className":7830},[204],[23,7832,7834],{"className":7833,"style":2026},[208],[23,7835,7836,7839],{"style":2029},[23,7837],{"className":7838,"style":1015},[216],[23,7840,7842],{"className":7841},[1019,1020,1021,1022],[23,7843,434],{"className":7844,"style":454},[135,136,1022],[23,7846,297],{"className":7847},[296],[23,7849,7851],{"className":7850},[204],[23,7852,7854],{"className":7853,"style":2048},[208],[23,7855],{},[23,7857,7859],{"className":7858},[135],[23,7860,924],{"className":7861},[1066,7585],[23,7863,902],{"className":7864},[135,644],[23,7866,7868],{"className":7867},[135],[23,7869,924],{"className":7870},[1066,1067],[23,7872,555],{"className":7873},[135,136],[23,7875,54],{"className":7876},[140],[23,7878,369],{"className":7879,"style":392},[135,136],[23,7881,747],{"className":7882},[824],[23,7884],{"className":7885,"style":280},[148],[23,7887,736],{"className":7888,"style":783},[135,136],[23,7890,54],{"className":7891},[140],[23,7893,434],{"className":7894,"style":454},[135,136],[23,7896,1095],{"className":7897},[169],[23,7899],{"className":7900,"style":149},[148],[23,7902,60],{"className":7903},[153],[23,7905],{"className":7906,"style":149},[148],[23,7908,7910,7913,7916,7922,7928,7931,7934],{"className":7909},[126],[23,7911],{"className":7912,"style":7815},[130],[23,7914,434],{"className":7915,"style":454},[135,136],[23,7917,7919],{"className":7918},[135],[23,7920,945],{"className":7921},[1066,1067],[23,7923,7925],{"className":7924},[135],[23,7926,945],{"className":7927},[1066,7585],[23,7929],{"className":7930,"style":149},[148],[23,7932,69],{"className":7933},[153],[23,7935],{"className":7936,"style":149},[148],[23,7938,7940,7943,7983,7989,7992,7995,7998,8001,8004,8007,8010,8013,8016,8019,8025],{"className":7939},[126],[23,7941],{"className":7942,"style":2213},[130],[23,7944,7946,7949],{"className":7945},[135],[23,7947,902],{"className":7948},[135,644],[23,7950,7952],{"className":7951},[998],[23,7953,7955,7975],{"className":7954},[199,200],[23,7956,7958,7972],{"className":7957},[204],[23,7959,7961],{"className":7960,"style":2026},[208],[23,7962,7963,7966],{"style":2029},[23,7964],{"className":7965,"style":1015},[216],[23,7967,7969],{"className":7968},[1019,1020,1021,1022],[23,7970,434],{"className":7971,"style":454},[135,136,1022],[23,7973,297],{"className":7974},[296],[23,7976,7978],{"className":7977},[204],[23,7979,7981],{"className":7980,"style":2048},[208],[23,7982],{},[23,7984,7986],{"className":7985},[135],[23,7987,924],{"className":7988},[1066,1067],[23,7990,1653],{"className":7991,"style":144},[135,136],[23,7993,54],{"className":7994},[140],[23,7996,736],{"className":7997,"style":783},[135,136],[23,7999,54],{"className":8000},[140],[23,8002,434],{"className":8003,"style":454},[135,136],[23,8005,66],{"className":8006},[169],[23,8008,747],{"className":8009},[824],[23,8011],{"className":8012,"style":280},[148],[23,8014,434],{"className":8015,"style":454},[135,136],[23,8017,66],{"className":8018},[169],[23,8020,8022],{"className":8021},[135],[23,8023,945],{"className":8024},[1066,1067],[23,8026,112],{"className":8027},[135],[11,8029,8030,8031,8059,8060,8236,8237],{},"对每个 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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 ",[23,8273,8275,8291],{"className":8274,"translate":27},[31],[23,8276,8278],{"className":8277},[35],[37,8279,8280],{"xmlns":39},[42,8281,8282,8288],{},[45,8283,8284,8286],{},[48,8285,6969],{},[51,8287,6961],{},[114,8289,8290],{"encoding":116},"\\min",[23,8292,8294],{"className":8293,"ariaHidden":122},[121],[23,8295,8297,8300],{"className":8296},[126],[23,8298],{"className":8299,"style":7108},[130],[23,8301,6969],{"className":8302},[1786]," 换成 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换成「取一列逼近的行动」即可，证明不变。",[11,8388,8389],{},"把上式改写成差值，可以得到一个更有解释力的恒等式：",[23,8391,8393],{"className":8392,"translate":27},[26],[23,8394,8396,8495],{"className":8395,"translate":27},[31],[23,8397,8399],{"className":8398},[35],[37,8400,8401],{"xmlns":39,"display":40},[42,8402,8403,8492],{},[45,8404,8405,8407,8409,8411,8413,8415,8421,8423,8429,8431,8488,8490],{},[48,8406,572],{},[51,8408,54],{"stretchy":53},[48,8410,736],{},[51,8412,66],{"stretchy":53},[51,8414,4529],{},[4518,8416,8417,8419],{},[48,8418,572],{},[51,8420,6883],{},[51,8422,69],{},[898,8424,8425,8427],{},[48,8426,902],{"mathvariant":571},[48,8428,434],{},[51,8430,924],{"fence":53,"stretchy":122,"minsize":7407,"maxsize":7407},[6963,8432,8433,8482],{},[6963,8434,8435,8479],{},[45,8436,8437,8439,8441,8443,8445,8447,8449,8451,8453,8455,8457,8467,8469,8471,8473,8475,8477],{},[48,8438,1653],{},[51,8440,54],{"stretchy":53},[48,8442,736],{},[51,8444,54],{"stretchy":53},[48,8446,434],{},[51,8448,66],{"stretchy":53},[51,8450,747],{"separator":122},[48,8452,434],{},[51,8454,66],{"stretchy":53},[51,8456,4529],{},[6963,8458,8459,8465],{},[45,8460,8461,8463],{},[48,8462,6969],{},[51,8464,6961],{},[48,8466,495],{},[48,8468,1653],{},[51,8470,54],{"stretchy":53},[48,8472,495],{},[51,8474,747],{"separator":122},[48,8476,434],{},[51,8478,66],{"stretchy":53},[51,8480,8481],{"stretchy":122},"⏟",[45,8483,8484,8486],{},[51,8485,7382],{},[2581,8487,1532],{},[51,8489,945],{"fence":53,"stretchy":122,"minsize":7407,"maxsize":7407},[48,8491,112],{"mathvariant":111},[114,8493,8494],{"encoding":116},"R(\\delta)-R^*\n=\\mathbb 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20-1h17z",[23,8724,8725,8728],{"style":7132},[23,8726],{"className":8727,"style":217},[216],[23,8729,8731,8734,8737,8740,8743,8746,8749,8752,8755,8758,8761,8764,8767,8770,8819,8822,8825,8828,8831,8834,8837,8840],{"className":8730},[135],[23,8732,1653],{"className":8733,"style":144},[135,136],[23,8735,54],{"className":8736},[140],[23,8738,736],{"className":8739,"style":783},[135,136],[23,8741,54],{"className":8742},[140],[23,8744,434],{"className":8745,"style":454},[135,136],[23,8747,66],{"className":8748},[169],[23,8750,747],{"className":8751},[824],[23,8753],{"className":8754,"style":280},[148],[23,8756,434],{"className":8757,"style":454},[135,136],[23,8759,66],{"className":8760},[169],[23,8762],{"className":8763,"style":392},[148],[23,8765,4529],{"className":8766},[612],[23,8768],{"className":8769,"style":392},[148],[23,8771,8773],{"className":8772},[1786,7098],[23,8774,8776,8810],{"className":8775},[199,200],[23,8777,8779,8807],{"className":8778},[204],[23,8780,8782,8797],{"className":8781,"style":7108},[208],[23,8783,8785,8788],{"style":8784},"top:-2.4em;margin-left:0em;",[23,8786],{"className":8787,"style":217},[216],[23,8789,8791],{"className":8790},[1019,1020,1021,1022],[23,8792,8794],{"className":8793},[135,1022],[23,8795,495],{"className":8796},[135,136,1022],[23,8798,8799,8802],{"style":7132},[23,8800],{"className":8801,"style":217},[216],[23,8803,8804],{},[23,8805,6969],{"className":8806},[1786],[23,8808,297],{"className":8809},[296],[23,8811,8813],{"className":8812},[204],[23,8814,8817],{"className":8815,"style":8816},[208],"height:0.7em;",[23,8818],{},[23,8820],{"className":8821,"style":280},[148],[23,8823,1653],{"className":8824,"style":144},[135,136],[23,8826,54],{"className":8827},[140],[23,8829,495],{"className":8830},[135,136],[23,8832,747],{"className":8833},[824],[23,8835],{"className":8836,"style":280},[148],[23,8838,434],{"className":8839,"style":454},[135,136],[23,8841,66],{"className":8842},[169],[23,8844,297],{"className":8845},[296],[23,8847,8849],{"className":8848},[204],[23,8850,8853],{"className":8851,"style":8852},[208],"height:1.348em;",[23,8854],{},[23,8856,297],{"className":8857},[296],[23,8859,8861],{"className":8860},[204],[23,8862,8865],{"className":8863,"style":8864},[208],"height:2.0943em;",[23,8866],{},[23,8868],{"className":8869,"style":280},[148],[23,8871,8873],{"className":8872},[135],[23,8874,945],{"className":8875},[1066,7585],[23,8877,112],{"className":8878},[135],[11,8880,8881,8882,8910,8911,8914,8915,8943,8944,8972],{},"右边每一项都是非负的，称为在 ",[23,8883,8885,8898],{"className":8884,"translate":27},[31],[23,8886,8888],{"className":8887},[35],[37,8889,8890],{"xmlns":39},[42,8891,8892,8896],{},[45,8893,8894],{},[48,8895,753],{},[114,8897,753],{"encoding":116},[23,8899,8901],{"className":8900,"ariaHidden":122},[121],[23,8902,8904,8907],{"className":8903},[126],[23,8905],{"className":8906,"style":703},[130],[23,8908,753],{"className":8909},[135,136]," 处的",[18,8912,8913],{},"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 ",[23,8916,8918,8931],{"className":8917,"translate":27},[31],[23,8919,8921],{"className":8920},[35],[37,8922,8923],{"xmlns":39},[42,8924,8925,8929],{},[45,8926,8927],{},[48,8928,753],{},[114,8930,753],{"encoding":116},[23,8932,8934],{"className":8933,"ariaHidden":122},[121],[23,8935,8937,8940],{"className":8936},[126],[23,8938],{"className":8939,"style":703},[130],[23,8941,753],{"className":8942},[135,136]," 上没有选到局部最优行动。不同 ",[23,8945,8947,8960],{"className":8946,"translate":27},[31],[23,8948,8950],{"className":8949},[35],[37,8951,8952],{"xmlns":39},[42,8953,8954,8958],{},[45,8955,8956],{},[48,8957,753],{},[114,8959,753],{"encoding":116},[23,8961,8963],{"className":8962,"ariaHidden":122},[121],[23,8964,8966,8969],{"className":8965},[126],[23,8967],{"className":8968,"style":703},[130],[23,8970,753],{"className":8971},[135,136]," 之间的错误不会相互抵消，也不会相互放大。",[2501,8974,8976],{"id":8975},"_32-0-1-损失最大后验","3.2 0-1 损失：最大后验",[11,8978,8979],{},"分类问题最常用的损失是 0-1 损失：",[23,8981,8983],{"className":8982,"translate":27},[26],[23,8984,8986,9030],{"className":8985,"translate":27},[31],[23,8987,8989],{"className":8988},[35],[37,8990,8991],{"xmlns":39,"display":40},[42,8992,8993,9027],{},[45,8994,8995,8997,8999,9001,9003,9005,9007,9009,9012,9015,9017,9020,9022,9025],{},[48,8996,555],{},[51,8998,54],{"stretchy":53},[48,9000,660],{},[51,9002,747],{"separator":122},[48,9004,495],{},[51,9006,66],{"stretchy":53},[51,9008,69],{},[2581,9010,4435],{"mathvariant":9011},"bold",[51,9013,9014],{"stretchy":53},"{",[48,9016,660],{},[51,9018,9019],{"mathvariant":111},"≠",[48,9021,495],{},[51,9023,9024],{"stretchy":53},"}",[48,9026,112],{"mathvariant":111},[114,9028,9029],{"encoding":116},"L(y,a)=\\mathbf 1\\{y\\ne a\\}.",[23,9031,9033,9069,9134],{"className":9032,"ariaHidden":122},[121],[23,9034,9036,9039,9042,9045,9048,9051,9054,9057,9060,9063,9066],{"className":9035},[126],[23,9037],{"className":9038,"style":131},[130],[23,9040,555],{"className":9041},[135,136],[23,9043,54],{"className":9044},[140],[23,9046,660],{"className":9047,"style":676},[135,136],[23,9049,747],{"className":9050},[824],[23,9052],{"className":9053,"style":280},[148],[23,9055,495],{"className":9056},[135,136],[23,9058,66],{"className":9059},[169],[23,9061],{"className":9062,"style":149},[148],[23,9064,69],{"className":9065},[153],[23,9067],{"className":9068,"style":149},[148],[23,9070,9072,9075,9079,9082,9085,9088,9131],{"className":9071},[126],[23,9073],{"className":9074,"style":131},[130],[23,9076,4435],{"className":9077},[135,9078],"mathbf",[23,9080,9014],{"className":9081},[140],[23,9083,660],{"className":9084,"style":676},[135,136],[23,9086],{"className":9087,"style":149},[148],[23,9089,9091,9124,9128],{"className":9090},[153],[23,9092,9094],{"className":9093},[153],[23,9095,9098],{"className":9096},[135,9097],"vbox",[23,9099,9102],{"className":9100},[9101],"thinbox",[23,9103,9106,9109,9120],{"className":9104},[9105],"rlap",[23,9107],{"className":9108,"style":5414},[130],[23,9110,9113],{"className":9111},[9112],"inner",[23,9114,9116],{"className":9115},[135],[23,9117,9119],{"className":9118},[153],"",[23,9121],{"className":9122},[9123],"fix",[23,9125],{"className":9126},[148,9127],"nobreak",[23,9129,69],{"className":9130},[153],[23,9132],{"className":9133,"style":149},[148],[23,9135,9137,9140,9143,9146],{"className":9136},[126],[23,9138],{"className":9139,"style":131},[130],[23,9141,495],{"className":9142},[135,136],[23,9144,9024],{"className":9145},[169],[23,9147,112],{"className":9148},[135],[11,9150,9151],{},"此时条件风险为",[23,9153,9155],{"className":9154,"translate":27},[26],[23,9156,9158,9230],{"className":9157,"translate":27},[31],[23,9159,9161],{"className":9160},[35],[37,9162,9163],{"xmlns":39,"display":40},[42,9164,9165,9227],{},[45,9166,9167,9169,9171,9173,9175,9177,9179,9181,9183,9185,9187,9189,9191,9193,9195,9197,9199,9201,9203,9205,9207,9209,9211,9213,9215,9217,9219,9221,9223,9225],{},[48,9168,1653],{},[51,9170,54],{"stretchy":53},[48,9172,495],{},[51,9174,747],{"separator":122},[48,9176,753],{},[51,9178,66],{"stretchy":53},[51,9180,69],{},[48,9182,902],{"mathvariant":571},[51,9184,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[2581,9186,4435],{"mathvariant":9011},[51,9188,9014],{"stretchy":53},[48,9190,369],{},[51,9192,9019],{"mathvariant":111},[48,9194,495],{},[51,9196,9024],{"stretchy":53},[51,9198,60],{},[48,9200,434],{},[51,9202,69],{},[48,9204,753],{},[51,9206,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,9208,69],{},[2581,9210,4435],{},[51,9212,4529],{},[48,9214,11],{},[51,9216,54],{"stretchy":53},[48,9218,495],{},[51,9220,60],{},[48,9222,753],{},[51,9224,66],{"stretchy":53},[51,9226,747],{"separator":122},[114,9228,9229],{"encoding":116},"r(a,x)=\\mathbb E\\big[\\mathbf 1\\{Y\\ne a\\}\\mid X=x\\big]=1-p(a\\mid x),",[23,9231,9233,9269,9335,9356,9374,9398,9417,9441],{"className":9232,"ariaHidden":122},[121],[23,9234,9236,9239,9242,9245,9248,9251,9254,9257,9260,9263,9266],{"className":9235},[126],[23,9237],{"className":9238,"style":131},[130],[23,9240,1653],{"className":9241,"style":144},[135,136],[23,9243,54],{"className":9244},[140],[23,9246,495],{"className":9247},[135,136],[23,9249,747],{"className":9250},[824],[23,9252],{"className":9253,"style":280},[148],[23,9255,753],{"className":9256},[135,136],[23,9258,66],{"className":9259},[169],[23,9261],{"className":9262,"style":149},[148],[23,9264,69],{"className":9265},[153],[23,9267],{"className":9268,"style":149},[148],[23,9270,9272,9275,9278,9284,9287,9290,9293,9296,9332],{"className":9271},[126],[23,9273],{"className":9274,"style":2213},[130],[23,9276,902],{"className":9277},[135,644],[23,9279,9281],{"className":9280},[135],[23,9282,924],{"className":9283},[1066,1067],[23,9285,4435],{"className":9286},[135,9078],[23,9288,9014],{"className":9289},[140],[23,9291,369],{"className":9292,"style":392},[135,136],[23,9294],{"className":9295,"style":149},[148],[23,9297,9299,9326,9329],{"className":9298},[153],[23,9300,9302],{"className":9301},[153],[23,9303,9305],{"className":9304},[135,9097],[23,9306,9308],{"className":9307},[9101],[23,9309,9311,9314,9323],{"className":9310},[9105],[23,9312],{"className":9313,"style":5414},[130],[23,9315,9317],{"className":9316},[9112],[23,9318,9320],{"className":9319},[135],[23,9321,9119],{"className":9322},[153],[23,9324],{"className":9325},[9123],[23,9327],{"className":9328},[148,9127],[23,9330,69],{"className":9331},[153],[23,9333],{"className":9334,"style":149},[148],[23,9336,9338,9341,9344,9347,9350,9353],{"className":9337},[126],[23,9339],{"className":9340,"style":131},[130],[23,9342,495],{"className":9343},[135,136],[23,9345,9024],{"className":9346},[169],[23,9348],{"className":9349,"style":149},[148],[23,9351,60],{"className":9352},[153],[23,9354],{"className":9355,"style":149},[148],[23,9357,9359,9362,9365,9368,9371],{"className":9358},[126],[23,9360],{"className":9361,"style":470},[130],[23,9363,434],{"className":9364,"style":454},[135,136],[23,9366],{"className":9367,"style":149},[148],[23,9369,69],{"className":9370},[153],[23,9372],{"className":9373,"style":149},[148],[23,9375,9377,9380,9383,9389,9392,9395],{"className":9376},[126],[23,9378],{"className":9379,"style":2213},[130],[23,9381,753],{"className":9382},[135,136],[23,9384,9386],{"className":9385},[135],[23,9387,945],{"className":9388},[1066,1067],[23,9390],{"className":9391,"style":149},[148],[23,9393,69],{"className":9394},[153],[23,9396],{"className":9397,"style":149},[148],[23,9399,9401,9405,9408,9411,9414],{"className":9400},[126],[23,9402],{"className":9403,"style":9404},[130],"height:0.7278em;vertical-align:-0.0833em;",[23,9406,4435],{"className":9407},[135],[23,9409],{"className":9410,"style":392},[148],[23,9412,4529],{"className":9413},[612],[23,9415],{"className":9416,"style":392},[148],[23,9418,9420,9423,9426,9429,9432,9435,9438],{"className":9419},[126],[23,9421],{"className":9422,"style":131},[130],[23,9424,11],{"className":9425},[135,136],[23,9427,54],{"className":9428},[140],[23,9430,495],{"className":9431},[135,136],[23,9433],{"className":9434,"style":149},[148],[23,9436,60],{"className":9437},[153],[23,9439],{"className":9440,"style":149},[148],[23,9442,9444,9447,9450,9453],{"className":9443},[126],[23,9445],{"className":9446,"style":131},[130],[23,9448,753],{"className":9449},[135,136],[23,9451,66],{"className":9452},[169],[23,9454,747],{"className":9455},[824],[11,9457,9458,9459,9515],{},"最小化 ",[23,9460,9462,9485],{"className":9461,"translate":27},[31],[23,9463,9465],{"className":9464},[35],[37,9466,9467],{"xmlns":39},[42,9468,9469,9483],{},[45,9470,9471,9473,9475,9477,9479,9481],{},[48,9472,1653],{},[51,9474,54],{"stretchy":53},[48,9476,495],{},[51,9478,747],{"separator":122},[48,9480,753],{},[51,9482,66],{"stretchy":53},[114,9484,2435],{"encoding":116},[23,9486,9488],{"className":9487,"ariaHidden":122},[121],[23,9489,9491,9494,9497,9500,9503,9506,9509,9512],{"className":9490},[126],[23,9492],{"className":9493,"style":131},[130],[23,9495,1653],{"className":9496,"style":144},[135,136],[23,9498,54],{"className":9499},[140],[23,9501,495],{"className":9502},[135,136],[23,9504,747],{"className":9505},[824],[23,9507],{"className":9508,"style":280},[148],[23,9510,753],{"className":9511},[135,136],[23,9513,66],{"className":9514},[169]," 等价于最大化后验概率：",[23,9517,9519],{"className":9518,"translate":27},[26],[23,9520,9522,9585],{"className":9521,"translate":27},[31],[23,9523,9525],{"className":9524},[35],[37,9526,9527],{"xmlns":39,"display":40},[42,9528,9529,9582],{},[45,9530,9531,9537,9539,9541,9543,9545,9547,9549,9566,9568,9570,9572,9574,9576,9578,9580],{},[4518,9532,9533,9535],{},[48,9534,736],{},[51,9536,6883],{},[51,9538,54],{"stretchy":53},[48,9540,753],{},[51,9542,66],{"stretchy":53},[51,9544,69],{},[48,9546,6958],{},[51,9548,6961],{},[6963,9550,9551,9558],{},[45,9552,9553,9556],{},[48,9554,9555],{},"max",[51,9557,6961],{},[45,9559,9560,9562,9564],{},[48,9561,660],{},[51,9563,372],{},[48,9565,369],{"mathvariant":375},[88,9567,5360],{},[48,9569,11],{},[51,9571,54],{"stretchy":53},[48,9573,660],{},[51,9575,60],{},[48,9577,753],{},[51,9579,66],{"stretchy":53},[48,9581,112],{"mathvariant":111},[114,9583,9584],{"encoding":116},"\\delta^*(x)=\\arg\\max_{y\\in\\mathcal Y}\\ p(y\\mid x).",[23,9586,9588,9641,9733],{"className":9587,"ariaHidden":122},[121],[23,9589,9591,9594,9623,9626,9629,9632,9635,9638],{"className":9590},[126],[23,9592],{"className":9593,"style":131},[130],[23,9595,9597,9600],{"className":9596},[135],[23,9598,736],{"className":9599,"style":783},[135,136],[23,9601,9603],{"className":9602},[998],[23,9604,9606],{"className":9605},[199],[23,9607,9609],{"className":9608},[204],[23,9610,9612],{"className":9611,"style":7047},[208],[23,9613,9614,9617],{"style":4989},[23,9615],{"className":9616,"style":1015},[216],[23,9618,9620],{"className":9619},[1019,1020,1021,1022],[23,9621,6883],{"className":9622},[612,1022],[23,9624,54],{"className":9625},[140],[23,9627,753],{"className":9628},[135,136],[23,9630,66],{"className":9631},[169],[23,9633],{"className":9634,"style":149},[148],[23,9636,69],{"className":9637},[153],[23,9639],{"className":9640,"style":149},[148],[23,9642,9644,9648,9653,9656,9709,9712,9715,9718,9721,9724,9727,9730],{"className":9643},[126],[23,9645],{"className":9646,"style":9647},[130],"height:1.6304em;vertical-align:-0.8804em;",[23,9649,7087,9651],{"className":9650},[1786],[23,9652,7091],{"style":7090},[23,9654],{"className":9655,"style":280},[148],[23,9657,9659],{"className":9658},[1786,7098],[23,9660,9662,9701],{"className":9661},[199,200],[23,9663,9665,9698],{"className":9664},[204],[23,9666,9668,9688],{"className":9667,"style":703},[208],[23,9669,9670,9673],{"style":7111},[23,9671],{"className":9672,"style":217},[216],[23,9674,9676],{"className":9675},[1019,1020,1021,1022],[23,9677,9679,9682,9685],{"className":9678},[135,1022],[23,9680,660],{"className":9681,"style":676},[135,136,1022],[23,9683,372],{"className":9684},[153,1022],[23,9686,369],{"className":9687,"style":413},[135,412,1022],[23,9689,9690,9693],{"style":7132},[23,9691],{"className":9692,"style":217},[216],[23,9694,9695],{},[23,9696,9555],{"className":9697},[1786],[23,9699,297],{"className":9700},[296],[23,9702,9704],{"className":9703},[204],[23,9705,9707],{"className":9706,"style":5683},[208],[23,9708],{},[23,9710,5360],{"className":9711},[148],[23,9713],{"className":9714,"style":280},[148],[23,9716,11],{"className":9717},[135,136],[23,9719,54],{"className":9720},[140],[23,9722,660],{"className":9723,"style":676},[135,136],[23,9725],{"className":9726,"style":149},[148],[23,9728,60],{"className":9729},[153],[23,9731],{"className":9732,"style":149},[148],[23,9734,9736,9739,9742,9745],{"className":9735},[126],[23,9737],{"className":9738,"style":131},[130],[23,9740,753],{"className":9741},[135,136],[23,9743,66],{"className":9744},[169],[23,9746,112],{"className":9747},[135],[11,9749,9750,9751,9754,9755,9881],{},"这就是",[18,9752,9753],{},"最大后验决策","（maximum 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& \\eta(x)\\ge \\tfrac12,\\\\[2pt]\n0, & 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",[23,10375,10377,10413],{"className":10376,"translate":27},[31],[23,10378,10380],{"className":10379},[35],[37,10381,10382],{"xmlns":39},[42,10383,10384,10410],{},[45,10385,10386,10388,10390,10392,10394,10396,10398,10400,10402,10404,10406,10408],{},[51,10387,9014],{"stretchy":53},[48,10389,753],{},[51,10391,558],{},[48,10393,9769],{},[51,10395,54],{"stretchy":53},[48,10397,753],{},[51,10399,66],{"stretchy":53},[51,10401,69],{},[2581,10403,4435],{},[48,10405,2875],{"mathvariant":111},[2581,10407,9956],{},[51,10409,9024],{"stretchy":53},[114,10411,10412],{"encoding":116},"\\{x:\\eta(x)=1\u002F2\\}",[23,10414,10416,10437,10464],{"className":10415,"ariaHidden":122},[121],[23,10417,10419,10422,10425,10428,10431,10434],{"className":10418},[126],[23,10420],{"className":10421,"style":131},[130],[23,10423,9014],{"className":10424},[140],[23,10426,753],{"className":10427},[135,136],[23,10429],{"className":10430,"style":149},[148],[23,10432,558],{"className":10433},[153],[23,10435],{"className":10436,"style":149},[148],[23,10438,10440,10443,10446,10449,10452,10455,10458,10461],{"className":10439},[126],[23,10441],{"className":10442,"style":131},[130],[23,10444,9769],{"className":10445,"style":676},[135,136],[23,10447,54],{"className":10448},[140],[23,10450,753],{"className":10451},[135,136],[23,10453,66],{"className":10454},[169],[23,10456],{"className":10457,"style":149},[148],[23,10459,69],{"className":10460},[153],[23,10462],{"className":10463,"style":149},[148],[23,10465,10467,10470,10474],{"className":10466},[126],[23,10468],{"className":10469,"style":131},[130],[23,10471,10473],{"className":10472},[135],"1\u002F2",[23,10475,9024],{"className":10476},[169],"。当 ",[23,10479,10481,10509],{"className":10480,"translate":27},[31],[23,10482,10484],{"className":10483},[35],[37,10485,10486],{"xmlns":39},[42,10487,10488,10506],{},[45,10489,10490,10492,10494,10496,10498,10500,10502,10504],{},[48,10491,9769],{},[51,10493,54],{"stretchy":53},[48,10495,753],{},[51,10497,66],{"stretchy":53},[51,10499,69],{},[2581,10501,4435],{},[48,10503,2875],{"mathvariant":111},[2581,10505,9956],{},[114,10507,10508],{"encoding":116},"\\eta(x)=1\u002F2",[23,10510,10512,10539],{"className":10511,"ariaHidden":122},[121],[23,10513,10515,10518,10521,10524,10527,10530,10533,10536],{"className":10514},[126],[23,10516],{"className":10517,"style":131},[130],[23,10519,9769],{"className":10520,"style":676},[135,136],[23,10522,54],{"className":10523},[140],[23,10525,753],{"className":10526},[135,136],[23,10528,66],{"className":10529},[169],[23,10531],{"className":10532,"style":149},[148],[23,10534,69],{"className":10535},[153],[23,10537],{"className":10538,"style":149},[148],[23,10540,10542,10545],{"className":10541},[126],[23,10543],{"className":10544,"style":131},[130],[23,10546,10473],{"className":10547},[135]," 时两种行动同样好，实践中用固定的打破平局规则即可。",[2501,10550,10552],{"id":10551},"_33-平方损失后验均值","3.3 平方损失：后验均值",[11,10554,10555,10556,10698],{},"回归问题常用平方损失 ",[23,10557,10559,10599],{"className":10558,"translate":27},[31],[23,10560,10562],{"className":10561},[35],[37,10563,10564],{"xmlns":39},[42,10565,10566,10596],{},[45,10567,10568,10570,10572,10574,10576,10578,10580,10582,10584,10586,10588,10590],{},[48,10569,555],{},[51,10571,54],{"stretchy":53},[48,10573,660],{},[51,10575,747],{"separator":122},[48,10577,495],{},[51,10579,66],{"stretchy":53},[51,10581,69],{},[51,10583,54],{"stretchy":53},[48,10585,660],{},[51,10587,4529],{},[48,10589,495],{},[4518,10591,10592,10594],{},[51,10593,66],{"stretchy":53},[2581,10595,9956],{},[114,10597,10598],{"encoding":116},"L(y,a)=(y-a)^2",[23,10600,10602,10638,10659],{"className":10601,"ariaHidden":122},[121],[23,10603,10605,10608,10611,10614,10617,10620,10623,10626,10629,10632,10635],{"className":10604},[126],[23,10606],{"className":10607,"style":131},[130],[23,10609,555],{"className":10610},[135,136],[23,10612,54],{"className":10613},[140],[23,10615,660],{"className":10616,"style":676},[135,136],[23,10618,747],{"className":10619},[824],[23,10621],{"className":10622,"style":280},[148],[23,10624,495],{"className":10625},[135,136],[23,10627,66],{"className":10628},[169],[23,10630],{"className":10631,"style":149},[148],[23,10633,69],{"className":10634},[153],[23,10636],{"className":10637,"style":149},[148],[23,10639,10641,10644,10647,10650,10653,10656],{"className":10640},[126],[23,10642],{"className":10643,"style":131},[130],[23,10645,54],{"className":10646},[140],[23,10648,660],{"className":10649,"style":676},[135,136],[23,10651],{"className":10652,"style":392},[148],[23,10654,4529],{"className":10655},[612],[23,10657],{"className":10658,"style":392},[148],[23,10660,10662,10666,10669],{"className":10661},[126],[23,10663],{"className":10664,"style":10665},[130],"height:1.0641em;vertical-align:-0.25em;",[23,10667,495],{"className":10668},[135,136],[23,10670,10672,10675],{"className":10671},[169],[23,10673,66],{"className":10674},[169],[23,10676,10678],{"className":10677},[998],[23,10679,10681],{"className":10680},[199],[23,10682,10684],{"className":10683},[204],[23,10685,10687],{"className":10686,"style":4693},[208],[23,10688,10689,10692],{"style":4696},[23,10690],{"className":10691,"style":1015},[216],[23,10693,10695],{"className":10694},[1019,1020,1021,1022],[23,10696,9956],{"className":10697},[135,1022],"。此时",[23,10700,10702],{"className":10701,"translate":27},[26],[23,10703,10705,10848],{"className":10704,"translate":27},[31],[23,10706,10708],{"className":10707},[35],[37,10709,10710],{"xmlns":39,"display":40},[42,10711,10712,10845],{},[1518,10713,10714,10772],{"rowspacing":1520,"columnalign":1521,"columnspacing":1522},[1524,10715,10716,10734],{},[1527,10717,10718],{},[1530,10719,10720],{"scriptlevel":1532,"displaystyle":122},[45,10721,10722,10724,10726,10728,10730,10732],{},[48,10723,1653],{},[51,10725,54],{"stretchy":53},[48,10727,495],{},[51,10729,747],{"separator":122},[48,10731,753],{},[51,10733,66],{"stretchy":53},[1527,10735,10736],{},[1530,10737,10738],{"scriptlevel":1532,"displaystyle":122},[45,10739,10740,10742,10744,10746,10748,10750,10752,10754,10756,10762,10764,10766,10768,10770],{},[45,10741],{},[51,10743,69],{},[48,10745,902],{"mathvariant":571},[51,10747,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,10749,54],{"stretchy":53},[48,10751,369],{},[51,10753,4529],{},[48,10755,495],{},[4518,10757,10758,10760],{},[51,10759,66],{"stretchy":53},[2581,10761,9956],{},[51,10763,60],{},[48,10765,434],{},[51,10767,69],{},[48,10769,753],{},[51,10771,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[1524,10773,10774,10780],{},[1527,10775,10776],{},[1530,10777,10778],{"scriptlevel":1532,"displaystyle":122},[45,10779],{},[1527,10781,10782],{},[1530,10783,10784],{"scriptlevel":1532,"displaystyle":122},[45,10785,10786,10788,10790,10799,10801,10803,10805,10807,10809,10811,10813,10815,10817,10819,10821,10823,10825,10827,10829,10831,10833,10835,10837,10843],{},[45,10787],{},[51,10789,69],{},[45,10791,10792,10795,10797],{},[48,10793,10794],{"mathvariant":111},"V",[48,10796,495],{"mathvariant":111},[48,10798,1653],{"mathvariant":111},[51,10800,54],{"stretchy":53},[48,10802,369],{},[51,10804,60],{},[48,10806,434],{},[51,10808,69],{},[48,10810,753],{},[51,10812,66],{"stretchy":53},[51,10814,5106],{},[51,10816,54],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,10818,495],{},[51,10820,4529],{},[48,10822,902],{"mathvariant":571},[51,10824,924],{"stretchy":53},[48,10826,369],{},[51,10828,60],{},[48,10830,434],{},[51,10832,69],{},[48,10834,753],{},[51,10836,945],{"stretchy":53},[4518,10838,10839,10841],{},[51,10840,66],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[2581,10842,9956],{},[51,10844,747],{"separator":122},[114,10846,10847],{"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}",[23,10849,10851],{"className":10850,"ariaHidden":122},[121],[23,10852,10854,10858],{"className":10853},[126],[23,10855],{"className":10856,"style":10857},[130],"height:2.9381em;vertical-align:-1.2191em;",[23,10859,10861],{"className":10860},[135],[23,10862,10864,10929],{"className":10863},[1518],[23,10865,10867],{"className":10866},[1696],[23,10868,10870,10920],{"className":10869},[199,200],[23,10871,10873,10917],{"className":10872},[204],[23,10874,10877,10908],{"className":10875,"style":10876},[208],"height:1.7191em;",[23,10878,10880,10884],{"style":10879},"top:-3.909em;",[23,10881],{"className":10882,"style":10883},[216],"height:3.054em;",[23,10885,10887,10890,10893,10896,10899,10902,10905],{"className":10886},[135],[23,10888,1653],{"className":10889,"style":144},[135,136],[23,10891,54],{"className":10892},[140],[23,10894,495],{"className":10895},[135,136],[23,10897,747],{"className":10898},[824],[23,10900],{"className":10901,"style":280},[148],[23,10903,753],{"className":10904},[135,136],[23,10906,66],{"className":10907},[169],[23,10909,10911,10914],{"style":10910},"top:-2.1949em;",[23,10912],{"className":10913,"style":10883},[216],[23,10915],{"className":10916},[135],[23,10918,297],{"className":10919},[296],[23,10921,10923],{"className":10922},[204],[23,10924,10927],{"className":10925,"style":10926},[208],"height:1.2191em;",[23,10928],{},[23,10930,10932],{"className":10931},[1753],[23,10933,10935,11211],{"className":10934},[199,200],[23,10936,10938,11208],{"className":10937},[204],[23,10939,10941,11048],{"className":10940,"style":10876},[208],[23,10942,10943,10946],{"style":10879},[23,10944],{"className":10945,"style":10883},[216],[23,10947,10949,10952,10955,10958,10961,10964,10970,10973,10976,10979,10982,10985,10988,11018,11021,11024,11027,11030,11033,11036,11039,11042],{"className":10948},[135],[23,10950],{"className":10951},[135],[23,10953],{"className":10954,"style":149},[148],[23,10956,69],{"className":10957},[153],[23,10959],{"className":10960,"style":149},[148],[23,10962,902],{"className":10963},[135,644],[23,10965,10967],{"className":10966},[135],[23,10968,924],{"className":10969},[1066,1067],[23,10971,54],{"className":10972},[140],[23,10974,369],{"className":10975,"style":392},[135,136],[23,10977],{"className":10978,"style":392},[148],[23,10980,4529],{"className":10981},[612],[23,10983],{"className":10984,"style":392},[148],[23,10986,495],{"className":10987},[135,136],[23,10989,10991,10994],{"className":10990},[169],[23,10992,66],{"className":10993},[169],[23,10995,10997],{"className":10996},[998],[23,10998,11000],{"className":10999},[199],[23,11001,11003],{"className":11002},[204],[23,11004,11007],{"className":11005,"style":11006},[208],"height:0.8641em;",[23,11008,11009,11012],{"style":4989},[23,11010],{"className":11011,"style":1015},[216],[23,11013,11015],{"className":11014},[1019,1020,1021,1022],[23,11016,9956],{"className":11017},[135,1022],[23,11019],{"className":11020,"style":149},[148],[23,11022,60],{"className":11023},[153],[23,11025],{"className":11026,"style":149},[148],[23,11028,434],{"className":11029,"style":454},[135,136],[23,11031],{"className":11032,"style":149},[148],[23,11034,69],{"className":11035},[153],[23,11037],{"className":11038,"style":149},[148],[23,11040,753],{"className":11041},[135,136],[23,11043,11045],{"className":11044},[135],[23,11046,945],{"className":11047},[1066,1067],[23,11049,11050,11053],{"style":10910},[23,11051],{"className":11052,"style":10883},[216],[23,11054,11056,11059,11062,11065,11068,11075,11078,11081,11084,11087,11090,11093,11096,11099,11102,11105,11108,11111,11114,11117,11123,11126,11129,11132,11135,11138,11141,11144,11147,11150,11153,11156,11159,11162,11165,11168,11171,11205],{"className":11055},[135],[23,11057],{"className":11058},[135],[23,11060],{"className":11061,"style":149},[148],[23,11063,69],{"className":11064},[153],[23,11066],{"className":11067,"style":149},[148],[23,11069,11071],{"className":11070},[135],[23,11072,11074],{"className":11073},[135,5391],"Var",[23,11076,54],{"className":11077},[140],[23,11079,369],{"className":11080,"style":392},[135,136],[23,11082],{"className":11083,"style":149},[148],[23,11085,60],{"className":11086},[153],[23,11088],{"className":11089,"style":149},[148],[23,11091,434],{"className":11092,"style":454},[135,136],[23,11094],{"className":11095,"style":149},[148],[23,11097,69],{"className":11098},[153],[23,11100],{"className":11101,"style":149},[148],[23,11103,753],{"className":11104},[135,136],[23,11106,66],{"className":11107},[169],[23,11109],{"className":11110,"style":392},[148],[23,11112,5106],{"className":11113},[612],[23,11115],{"className":11116,"style":392},[148],[23,11118,11120],{"className":11119},[135],[23,11121,54],{"className":11122},[1066,1067],[23,11124,495],{"className":11125},[135,136],[23,11127],{"className":11128,"style":392},[148],[23,11130,4529],{"className":11131},[612],[23,11133],{"className":11134,"style":392},[148],[23,11136,902],{"className":11137},[135,644],[23,11139,924],{"className":11140},[140],[23,11142,369],{"className":11143,"style":392},[135,136],[23,11145],{"className":11146,"style":149},[148],[23,11148,60],{"className":11149},[153],[23,11151],{"className":11152,"style":149},[148],[23,11154,434],{"className":11155,"style":454},[135,136],[23,11157],{"className":11158,"style":149},[148],[23,11160,69],{"className":11161},[153],[23,11163],{"className":11164,"style":149},[148],[23,11166,753],{"className":11167},[135,136],[23,11169,945],{"className":11170},[169],[23,11172,11174,11180],{"className":11173},[135],[23,11175,11177],{"className":11176},[135],[23,11178,66],{"className":11179},[1066,1067],[23,11181,11183],{"className":11182},[998],[23,11184,11186],{"className":11185},[199],[23,11187,11189],{"className":11188},[204],[23,11190,11193],{"className":11191,"style":11192},[208],"height:1.054em;",[23,11194,11196,11199],{"style":11195},"top:-3.3029em;margin-right:0.05em;",[23,11197],{"className":11198,"style":1015},[216],[23,11200,11202],{"className":11201},[1019,1020,1021,1022],[23,11203,9956],{"className":11204},[135,1022],[23,11206,747],{"className":11207},[824],[23,11209,297],{"className":11210},[296],[23,11212,11214],{"className":11213},[204],[23,11215,11217],{"className":11216,"style":10926},[208],[23,11218],{},[11,11220,11221,11222,11332,11333,11336],{},"第二项在 ",[23,11223,11225,11257],{"className":11224,"translate":27},[31],[23,11226,11228],{"className":11227},[35],[37,11229,11230],{"xmlns":39},[42,11231,11232,11254],{},[45,11233,11234,11236,11238,11240,11242,11244,11246,11248,11250,11252],{},[48,11235,495],{},[51,11237,69],{},[48,11239,902],{"mathvariant":571},[51,11241,924],{"stretchy":53},[48,11243,369],{},[51,11245,60],{},[48,11247,434],{},[51,11249,69],{},[48,11251,753],{},[51,11253,945],{"stretchy":53},[114,11255,11256],{"encoding":116},"a=\\mathbb E[Y\\mid X=x]",[23,11258,11260,11278,11302,11320],{"className":11259,"ariaHidden":122},[121],[23,11261,11263,11266,11269,11272,11275],{"className":11262},[126],[23,11264],{"className":11265,"style":703},[130],[23,11267,495],{"className":11268},[135,136],[23,11270],{"className":11271,"style":149},[148],[23,11273,69],{"className":11274},[153],[23,11276],{"className":11277,"style":149},[148],[23,11279,11281,11284,11287,11290,11293,11296,11299],{"className":11280},[126],[23,11282],{"className":11283,"style":131},[130],[23,11285,902],{"className":11286},[135,644],[23,11288,924],{"className":11289},[140],[23,11291,369],{"className":11292,"style":392},[135,136],[23,11294],{"className":11295,"style":149},[148],[23,11297,60],{"className":11298},[153],[23,11300],{"className":11301,"style":149},[148],[23,11303,11305,11308,11311,11314,11317],{"className":11304},[126],[23,11306],{"className":11307,"style":470},[130],[23,11309,434],{"className":11310,"style":454},[135,136],[23,11312],{"className":11313,"style":149},[148],[23,11315,69],{"className":11316},[153],[23,11318],{"className":11319,"style":149},[148],[23,11321,11323,11326,11329],{"className":11322},[126],[23,11324],{"className":11325,"style":131},[130],[23,11327,753],{"className":11328},[135,136],[23,11330,945],{"className":11331},[169]," 时取到最小值。因此平方损失下的贝叶斯行动是",[18,11334,11335],{},"后验均值","，也就是回归函数",[23,11338,11340],{"className":11339,"translate":27},[26],[23,11341,11343,11387],{"className":11342,"translate":27},[31],[23,11344,11346],{"className":11345},[35],[37,11347,11348],{"xmlns":39,"display":40},[42,11349,11350,11384],{},[45,11351,11352,11358,11360,11362,11364,11366,11368,11370,11372,11374,11376,11378,11380,11382],{},[4518,11353,11354,11356],{},[48,11355,8340],{},[51,11357,6883],{},[51,11359,54],{"stretchy":53},[48,11361,753],{},[51,11363,66],{"stretchy":53},[51,11365,69],{},[48,11367,902],{"mathvariant":571},[51,11369,924],{"stretchy":53},[48,11371,369],{},[51,11373,60],{},[48,11375,434],{},[51,11377,69],{},[48,11379,753],{},[51,11381,945],{"stretchy":53},[48,11383,112],{"mathvariant":111},[114,11385,11386],{"encoding":116},"f^*(x)=\\mathbb E[Y\\mid X=x].",[23,11388,11390,11444,11468,11486],{"className":11389,"ariaHidden":122},[121],[23,11391,11393,11396,11426,11429,11432,11435,11438,11441],{"className":11392},[126],[23,11394],{"className":11395,"style":131},[130],[23,11397,11399,11403],{"className":11398},[135],[23,11400,8340],{"className":11401,"style":11402},[135,136],"margin-right:0.1076em;",[23,11404,11406],{"className":11405},[998],[23,11407,11409],{"className":11408},[199],[23,11410,11412],{"className":11411},[204],[23,11413,11415],{"className":11414,"style":7047},[208],[23,11416,11417,11420],{"style":4989},[23,11418],{"className":11419,"style":1015},[216],[23,11421,11423],{"className":11422},[1019,1020,1021,1022],[23,11424,6883],{"className":11425},[612,1022],[23,11427,54],{"className":11428},[140],[23,11430,753],{"className":11431},[135,136],[23,11433,66],{"className":11434},[169],[23,11436],{"className":11437,"style":149},[148],[23,11439,69],{"className":11440},[153],[23,11442],{"className":11443,"style":149},[148],[23,11445,11447,11450,11453,11456,11459,11462,11465],{"className":11446},[126],[23,11448],{"className":11449,"style":131},[130],[23,11451,902],{"className":11452},[135,644],[23,11454,924],{"className":11455},[140],[23,11457,369],{"className":11458,"style":392},[135,136],[23,11460],{"className":11461,"style":149},[148],[23,11463,60],{"className":11464},[153],[23,11466],{"className":11467,"style":149},[148],[23,11469,11471,11474,11477,11480,11483],{"className":11470},[126],[23,11472],{"className":11473,"style":470},[130],[23,11475,434],{"className":11476,"style":454},[135,136],[23,11478],{"className":11479,"style":149},[148],[23,11481,69],{"className":11482},[153],[23,11484],{"className":11485,"style":149},[148],[23,11487,11489,11492,11495,11498],{"className":11488},[126],[23,11490],{"className":11491,"style":131},[130],[23,11493,753],{"className":11494},[135,136],[23,11496,945],{"className":11497},[169],[23,11499,112],{"className":11500},[135],[11,11502,11503,11504,11532,11533,11592],{},"这个结论还可以从另一个角度理解：后验均值是 ",[23,11505,11507,11520],{"className":11506,"translate":27},[31],[23,11508,11510],{"className":11509},[35],[37,11511,11512],{"xmlns":39},[42,11513,11514,11518],{},[45,11515,11516],{},[48,11517,369],{},[114,11519,369],{"encoding":116},[23,11521,11523],{"className":11522,"ariaHidden":122},[121],[23,11524,11526,11529],{"className":11525},[126],[23,11527],{"className":11528,"style":470},[130],[23,11530,369],{"className":11531,"style":392},[135,136]," 在 ",[23,11534,11536,11554],{"className":11535,"translate":27},[31],[23,11537,11539],{"className":11538},[35],[37,11540,11541],{"xmlns":39},[42,11542,11543,11551],{},[45,11544,11545],{},[4518,11546,11547,11549],{},[48,11548,555],{},[2581,11550,9956],{},[114,11552,11553],{"encoding":116},"L^2",[23,11555,11557],{"className":11556,"ariaHidden":122},[121],[23,11558,11560,11563],{"className":11559},[126],[23,11561],{"className":11562,"style":4693},[130],[23,11564,11566,11569],{"className":11565},[135],[23,11567,555],{"className":11568},[135,136],[23,11570,11572],{"className":11571},[998],[23,11573,11575],{"className":11574},[199],[23,11576,11578],{"className":11577},[204],[23,11579,11581],{"className":11580,"style":4693},[208],[23,11582,11583,11586],{"style":4696},[23,11584],{"className":11585,"style":1015},[216],[23,11587,11589],{"className":11588},[1019,1020,1021,1022],[23,11590,9956],{"className":11591},[135,1022]," 意义下最好的预测，任何其他预测都会多出一项方差。",[2501,11594,11596],{"id":11595},"_34-绝对损失后验中位数","3.4 绝对损失：后验中位数",[11,11598,11599,11600,11711,11712,11882,11883,11911,11912,11940],{},"若损失为 ",[23,11601,11603,11639],{"className":11602,"translate":27},[31],[23,11604,11606],{"className":11605},[35],[37,11607,11608],{"xmlns":39},[42,11609,11610,11636],{},[45,11611,11612,11614,11616,11618,11620,11622,11624,11626,11628,11630,11632,11634],{},[48,11613,555],{},[51,11615,54],{"stretchy":53},[48,11617,660],{},[51,11619,747],{"separator":122},[48,11621,495],{},[51,11623,66],{"stretchy":53},[51,11625,69],{},[48,11627,60],{"mathvariant":111},[48,11629,660],{},[51,11631,4529],{},[48,11633,495],{},[48,11635,60],{"mathvariant":111},[114,11637,11638],{"encoding":116},"L(y,a)=|y-a|",[23,11640,11642,11678,11699],{"className":11641,"ariaHidden":122},[121],[23,11643,11645,11648,11651,11654,11657,11660,11663,11666,11669,11672,11675],{"className":11644},[126],[23,11646],{"className":11647,"style":131},[130],[23,11649,555],{"className":11650},[135,136],[23,11652,54],{"className":11653},[140],[23,11655,660],{"className":11656,"style":676},[135,136],[23,11658,747],{"className":11659},[824],[23,11661],{"className":11662,"style":280},[148],[23,11664,495],{"className":11665},[135,136],[23,11667,66],{"className":11668},[169],[23,11670],{"className":11671,"style":149},[148],[23,11673,69],{"className":11674},[153],[23,11676],{"className":11677,"style":149},[148],[23,11679,11681,11684,11687,11690,11693,11696],{"className":11680},[126],[23,11682],{"className":11683,"style":131},[130],[23,11685,60],{"className":11686},[135],[23,11688,660],{"className":11689,"style":676},[135,136],[23,11691],{"className":11692,"style":392},[148],[23,11694,4529],{"className":11695},[612],[23,11697],{"className":11698,"style":392},[148],[23,11700,11702,11705,11708],{"className":11701},[126],[23,11703],{"className":11704,"style":131},[130],[23,11706,495],{"className":11707},[135,136],[23,11709,60],{"className":11710},[135],"，条件风险为 ",[23,11713,11715,11765],{"className":11714,"translate":27},[31],[23,11716,11718],{"className":11717},[35],[37,11719,11720],{"xmlns":39},[42,11721,11722,11762],{},[45,11723,11724,11726,11728,11730,11732,11734,11736,11738,11740,11742,11744,11746,11748,11750,11752,11754,11756,11758,11760],{},[48,11725,1653],{},[51,11727,54],{"stretchy":53},[48,11729,495],{},[51,11731,747],{"separator":122},[48,11733,753],{},[51,11735,66],{"stretchy":53},[51,11737,69],{},[48,11739,902],{"mathvariant":571},[51,11741,924],{"stretchy":53},[48,11743,60],{"mathvariant":111},[48,11745,369],{},[51,11747,4529],{},[48,11749,495],{},[48,11751,60],{"mathvariant":111},[51,11753,60],{},[48,11755,434],{},[51,11757,69],{},[48,11759,753],{},[51,11761,945],{"stretchy":53},[114,11763,11764],{"encoding":116},"r(a,x)=\\mathbb E[|Y-a|\\mid X=x]",[23,11766,11768,11804,11831,11852,11870],{"className":11767,"ariaHidden":122},[121],[23,11769,11771,11774,11777,11780,11783,11786,11789,11792,11795,11798,11801],{"className":11770},[126],[23,11772],{"className":11773,"style":131},[130],[23,11775,1653],{"className":11776,"style":144},[135,136],[23,11778,54],{"className":11779},[140],[23,11781,495],{"className":11782},[135,136],[23,11784,747],{"className":11785},[824],[23,11787],{"className":11788,"style":280},[148],[23,11790,753],{"className":11791},[135,136],[23,11793,66],{"className":11794},[169],[23,11796],{"className":11797,"style":149},[148],[23,11799,69],{"className":11800},[153],[23,11802],{"className":11803,"style":149},[148],[23,11805,11807,11810,11813,11816,11819,11822,11825,11828],{"className":11806},[126],[23,11808],{"className":11809,"style":131},[130],[23,11811,902],{"className":11812},[135,644],[23,11814,924],{"className":11815},[140],[23,11817,60],{"className":11818},[135],[23,11820,369],{"className":11821,"style":392},[135,136],[23,11823],{"className":11824,"style":392},[148],[23,11826,4529],{"className":11827},[612],[23,11829],{"className":11830,"style":392},[148],[23,11832,11834,11837,11840,11843,11846,11849],{"className":11833},[126],[23,11835],{"className":11836,"style":131},[130],[23,11838,495],{"className":11839},[135,136],[23,11841,60],{"className":11842},[135],[23,11844],{"className":11845,"style":149},[148],[23,11847,60],{"className":11848},[153],[23,11850],{"className":11851,"style":149},[148],[23,11853,11855,11858,11861,11864,11867],{"className":11854},[126],[23,11856],{"className":11857,"style":470},[130],[23,11859,434],{"className":11860,"style":454},[135,136],[23,11862],{"className":11863,"style":149},[148],[23,11865,69],{"className":11866},[153],[23,11868],{"className":11869,"style":149},[148],[23,11871,11873,11876,11879],{"className":11872},[126],[23,11874],{"className":11875,"style":131},[130],[23,11877,753],{"className":11878},[135,136],[23,11880,945],{"className":11881},[169],"。对 ",[23,11884,11886,11899],{"className":11885,"translate":27},[31],[23,11887,11889],{"className":11888},[35],[37,11890,11891],{"xmlns":39},[42,11892,11893,11897],{},[45,11894,11895],{},[48,11896,495],{},[114,11898,495],{"encoding":116},[23,11900,11902],{"className":11901,"ariaHidden":122},[121],[23,11903,11905,11908],{"className":11904},[126],[23,11906],{"className":11907,"style":703},[130],[23,11909,495],{"className":11910},[135,136]," 求次梯度，在 ",[23,11913,11915,11928],{"className":11914,"translate":27},[31],[23,11916,11918],{"className":11917},[35],[37,11919,11920],{"xmlns":39},[42,11921,11922,11926],{},[45,11923,11924],{},[48,11925,369],{},[114,11927,369],{"encoding":116},[23,11929,11931],{"className":11930,"ariaHidden":122},[121],[23,11932,11934,11937],{"className":11933},[126],[23,11935],{"className":11936,"style":470},[130],[23,11938,369],{"className":11939,"style":392},[135,136]," 的分布连续时它就是导数：",[23,11942,11944],{"className":11943,"translate":27},[26],[23,11945,11947,12016],{"className":11946,"translate":27},[31],[23,11948,11950],{"className":11949},[35],[37,11951,11952],{"xmlns":39,"display":40},[42,11953,11954,12013],{},[45,11955,11956,11963,11965,11967,11969,11971,11973,11975,11977,11979,11981,11983,11985,11987,11989,11991,11993,11995,11997,11999,12001,12003,12005,12007,12009,12011],{},[898,11957,11958,11961],{},[48,11959,11960],{"mathvariant":111},"∂",[48,11962,495],{},[48,11964,1653],{},[51,11966,54],{"stretchy":53},[48,11968,495],{},[51,11970,747],{"separator":122},[48,11972,753],{},[51,11974,66],{"stretchy":53},[51,11976,69],{},[48,11978,920],{},[51,11980,54],{"stretchy":53},[48,11982,369],{},[51,11984,9987],{},[48,11986,495],{},[51,11988,60],{},[48,11990,753],{},[51,11992,66],{"stretchy":53},[51,11994,4529],{},[48,11996,920],{},[51,11998,54],{"stretchy":53},[48,12000,369],{},[51,12002,2579],{},[48,12004,495],{},[51,12006,60],{},[48,12008,753],{},[51,12010,66],{"stretchy":53},[51,12012,747],{"separator":122},[114,12014,12015],{"encoding":116},"\\partial_a r(a,x)=P(Y\u003Ca\\mid x)-P(Y>a\\mid x),",[23,12017,12019,12097,12121,12139,12160,12184,12202],{"className":12018,"ariaHidden":122},[121],[23,12020,12022,12025,12067,12070,12073,12076,12079,12082,12085,12088,12091,12094],{"className":12021},[126],[23,12023],{"className":12024,"style":131},[130],[23,12026,12028,12032],{"className":12027},[135],[23,12029,11960],{"className":12030,"style":12031},[135],"margin-right:0.0556em;",[23,12033,12035],{"className":12034},[998],[23,12036,12038,12059],{"className":12037},[199,200],[23,12039,12041,12056],{"className":12040},[204],[23,12042,12044],{"className":12043,"style":6499},[208],[23,12045,12047,12050],{"style":12046},"top:-2.55em;margin-left:-0.0556em;margin-right:0.05em;",[23,12048],{"className":12049,"style":1015},[216],[23,12051,12053],{"className":12052},[1019,1020,1021,1022],[23,12054,495],{"className":12055},[135,136,1022],[23,12057,297],{"className":12058},[296],[23,12060,12062],{"className":12061},[204],[23,12063,12065],{"className":12064,"style":2048},[208],[23,12066],{},[23,12068,1653],{"className":12069,"style":144},[135,136],[23,12071,54],{"className":12072},[140],[23,12074,495],{"className":12075},[135,136],[23,12077,747],{"className":12078},[824],[23,12080],{"className":12081,"style":280},[148],[23,12083,753],{"className":12084},[135,136],[23,12086,66],{"className":12087},[169],[23,12089],{"className":12090,"style":149},[148],[23,12092,69],{"className":12093},[153],[23,12095],{"className":12096,"style":149},[148],[23,12098,12100,12103,12106,12109,12112,12115,12118],{"className":12099},[126],[23,12101],{"className":12102,"style":131},[130],[23,12104,920],{"className":12105,"style":1047},[135,136],[23,12107,54],{"className":12108},[140],[23,12110,369],{"className":12111,"style":392},[135,136],[23,12113],{"className":12114,"style":149},[148],[23,12116,9987],{"className":12117},[153],[23,12119],{"className":12120,"style":149},[148],[23,12122,12124,12127,12130,12133,12136],{"className":12123},[126],[23,12125],{"className":12126,"style":131},[130],[23,12128,495],{"className":12129},[135,136],[23,12131],{"className":12132,"style":149},[148],[23,12134,60],{"className":12135},[153],[23,12137],{"className":12138,"style":149},[148],[23,12140,12142,12145,12148,12151,12154,12157],{"className":12141},[126],[23,12143],{"className":12144,"style":131},[130],[23,12146,753],{"className":12147},[135,136],[23,12149,66],{"className":12150},[169],[23,12152],{"className":12153,"style":392},[148],[23,12155,4529],{"className":12156},[612],[23,12158],{"className":12159,"style":392},[148],[23,12161,12163,12166,12169,12172,12175,12178,12181],{"className":12162},[126],[23,12164],{"className":12165,"style":131},[130],[23,12167,920],{"className":12168,"style":1047},[135,136],[23,12170,54],{"className":12171},[140],[23,12173,369],{"className":12174,"style":392},[135,136],[23,12176],{"className":12177,"style":149},[148],[23,12179,2579],{"className":12180},[153],[23,12182],{"className":12183,"style":149},[148],[23,12185,12187,12190,12193,12196,12199],{"className":12186},[126],[23,12188],{"className":12189,"style":131},[130],[23,12191,495],{"className":12192},[135,136],[23,12194],{"className":12195,"style":149},[148],[23,12197,60],{"className":12198},[153],[23,12200],{"className":12201,"style":149},[148],[23,12203,12205,12208,12211,12214],{"className":12204},[126],[23,12206],{"className":12207,"style":131},[130],[23,12209,753],{"className":12210},[135,136],[23,12212,66],{"className":12213},[169],[23,12215,747],{"className":12216},[824],[11,12218,12219,12220,12248],{},"令 ",[23,12221,12223,12236],{"className":12222,"translate":27},[31],[23,12224,12226],{"className":12225},[35],[37,12227,12228],{"xmlns":39},[42,12229,12230,12234],{},[45,12231,12232],{},[2581,12233,1532],{},[114,12235,1532],{"encoding":116},[23,12237,12239],{"className":12238,"ariaHidden":122},[121],[23,12240,12242,12245],{"className":12241},[126],[23,12243],{"className":12244,"style":2770},[130],[23,12246,1532],{"className":12247},[135]," 属于次梯度，等价于",[23,12250,12252],{"className":12251,"translate":27},[26],[23,12253,12255,12322],{"className":12254,"translate":27},[31],[23,12256,12258],{"className":12257},[35],[37,12259,12260],{"xmlns":39,"display":40},[42,12261,12262,12319],{},[45,12263,12264,12266,12268,12270,12273,12275,12277,12279,12281,12283,12289,12291,12293,12295,12297,12299,12301,12303,12305,12307,12309,12311,12317],{},[48,12265,920],{},[51,12267,54],{"stretchy":53},[48,12269,369],{},[51,12271,12272],{},"≤",[48,12274,495],{},[51,12276,60],{},[48,12278,753],{},[51,12280,66],{"stretchy":53},[51,12282,7382],{},[71,12284,12285,12287],{},[2581,12286,4435],{},[2581,12288,9956],{},[51,12290,747],{"separator":122},[148,12292],{"width":750},[48,12294,920],{},[51,12296,54],{"stretchy":53},[48,12298,369],{},[51,12300,7382],{},[48,12302,495],{},[51,12304,60],{},[48,12306,753],{},[51,12308,66],{"stretchy":53},[51,12310,7382],{},[71,12312,12313,12315],{},[2581,12314,4435],{},[2581,12316,9956],{},[48,12318,112],{"mathvariant":111},[114,12320,12321],{"encoding":116},"P(Y\\le a\\mid x)\\ge \\frac12,\\qquad P(Y\\ge a\\mid x)\\ge \\frac12 .",[23,12323,12325,12349,12367,12388,12485,12503,12524],{"className":12324,"ariaHidden":122},[121],[23,12326,12328,12331,12334,12337,12340,12343,12346],{"className":12327},[126],[23,12329],{"className":12330,"style":131},[130],[23,12332,920],{"className":12333,"style":1047},[135,136],[23,12335,54],{"className":12336},[140],[23,12338,369],{"className":12339,"style":392},[135,136],[23,12341],{"className":12342,"style":149},[148],[23,12344,12272],{"className":12345},[153],[23,12347],{"className":12348,"style":149},[148],[23,12350,12352,12355,12358,12361,12364],{"className":12351},[126],[23,12353],{"className":12354,"style":131},[130],[23,12356,495],{"className":12357},[135,136],[23,12359],{"className":12360,"style":149},[148],[23,12362,60],{"className":12363},[153],[23,12365],{"className":12366,"style":149},[148],[23,12368,12370,12373,12376,12379,12382,12385],{"className":12369},[126],[23,12371],{"className":12372,"style":131},[130],[23,12374,753],{"className":12375},[135,136],[23,12377,66],{"className":12378},[169],[23,12380],{"className":12381,"style":149},[148],[23,12383,7382],{"className":12384},[153],[23,12386],{"className":12387,"style":149},[148],[23,12389,12391,12395,12458,12461,12464,12467,12470,12473,12476,12479,12482],{"className":12390},[126],[23,12392],{"className":12393,"style":12394},[130],"height:2.0074em;vertical-align:-0.686em;",[23,12396,12398,12401,12455],{"className":12397},[135],[23,12399],{"className":12400},[140,192],[23,12402,12404],{"className":12403},[71],[23,12405,12407,12447],{"className":12406},[199,200],[23,12408,12410,12444],{"className":12409},[204],[23,12411,12414,12425,12433],{"className":12412,"style":12413},[208],"height:1.3214em;",[23,12415,12416,12419],{"style":212},[23,12417],{"className":12418,"style":217},[216],[23,12420,12422],{"className":12421},[135],[23,12423,9956],{"className":12424},[135],[23,12426,12427,12430],{"style":235},[23,12428],{"className":12429,"style":217},[216],[23,12431],{"className":12432,"style":243},[242],[23,12434,12435,12438],{"style":246},[23,12436],{"className":12437,"style":217},[216],[23,12439,12441],{"className":12440},[135],[23,12442,4435],{"className":12443},[135],[23,12445,297],{"className":12446},[296],[23,12448,12450],{"className":12449},[204],[23,12451,12453],{"className":12452,"style":6287},[208],[23,12454],{},[23,12456],{"className":12457},[169,192],[23,12459,747],{"className":12460},[824],[23,12462],{"className":12463,"style":828},[148],[23,12465],{"className":12466,"style":280},[148],[23,12468,920],{"className":12469,"style":1047},[135,136],[23,12471,54],{"className":12472},[140],[23,12474,369],{"className":12475,"style":392},[135,136],[23,12477],{"className":12478,"style":149},[148],[23,12480,7382],{"className":12481},[153],[23,12483],{"className":12484,"style":149},[148],[23,12486,12488,12491,12494,12497,12500],{"className":12487},[126],[23,12489],{"className":12490,"style":131},[130],[23,12492,495],{"className":12493},[135,136],[23,12495],{"className":12496,"style":149},[148],[23,12498,60],{"className":12499},[153],[23,12501],{"className":12502,"style":149},[148],[23,12504,12506,12509,12512,12515,12518,12521],{"className":12505},[126],[23,12507],{"className":12508,"style":131},[130],[23,12510,753],{"className":12511},[135,136],[23,12513,66],{"className":12514},[169],[23,12516],{"className":12517,"style":149},[148],[23,12519,7382],{"className":12520},[153],[23,12522],{"className":12523,"style":149},[148],[23,12525,12527,12530,12592],{"className":12526},[126],[23,12528],{"className":12529,"style":12394},[130],[23,12531,12533,12536,12589],{"className":12532},[135],[23,12534],{"className":12535},[140,192],[23,12537,12539],{"className":12538},[71],[23,12540,12542,12581],{"className":12541},[199,200],[23,12543,12545,12578],{"className":12544},[204],[23,12546,12548,12559,12567],{"className":12547,"style":12413},[208],[23,12549,12550,12553],{"style":212},[23,12551],{"className":12552,"style":217},[216],[23,12554,12556],{"className":12555},[135],[23,12557,9956],{"className":12558},[135],[23,12560,12561,12564],{"style":235},[23,12562],{"className":12563,"style":217},[216],[23,12565],{"className":12566,"style":243},[242],[23,12568,12569,12572],{"style":246},[23,12570],{"className":12571,"style":217},[216],[23,12573,12575],{"className":12574},[135],[23,12576,4435],{"className":12577},[135],[23,12579,297],{"className":12580},[296],[23,12582,12584],{"className":12583},[204],[23,12585,12587],{"className":12586,"style":6287},[208],[23,12588],{},[23,12590],{"className":12591},[169,192],[23,12593,112],{"className":12594},[135],[11,12596,12597,12598,12626,12627,12655],{},"满足这两个不等式的 ",[23,12599,12601,12614],{"className":12600,"translate":27},[31],[23,12602,12604],{"className":12603},[35],[37,12605,12606],{"xmlns":39},[42,12607,12608,12612],{},[45,12609,12610],{},[48,12611,495],{},[114,12613,495],{"encoding":116},[23,12615,12617],{"className":12616,"ariaHidden":122},[121],[23,12618,12620,12623],{"className":12619},[126],[23,12621],{"className":12622,"style":703},[130],[23,12624,495],{"className":12625},[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 ",[23,12628,12630,12643],{"className":12629,"translate":27},[31],[23,12631,12633],{"className":12632},[35],[37,12634,12635],{"xmlns":39},[42,12636,12637,12641],{},[45,12638,12639],{},[48,12640,495],{},[114,12642,495],{"encoding":116},[23,12644,12646],{"className":12645,"ariaHidden":122},[121],[23,12647,12649,12652],{"className":12648},[126],[23,12650],{"className":12651,"style":703},[130],[23,12653,495],{"className":12654},[135,136]," 都不会降低期望绝对误差。",[11,12657,12658],{},"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。",[2501,12660,12662,12663],{"id":12661},"_35-非对称损失阈值不再在-121212","3.5 非对称损失：阈值不再在 ",[23,12664,12666,12683],{"className":12665,"translate":27},[31],[23,12667,12669],{"className":12668},[35],[37,12670,12671],{"xmlns":39},[42,12672,12673,12681],{},[45,12674,12675,12677,12679],{},[2581,12676,4435],{},[48,12678,2875],{"mathvariant":111},[2581,12680,9956],{},[114,12682,10473],{"encoding":116},[23,12684,12686],{"className":12685,"ariaHidden":122},[121],[23,12687,12689,12692],{"className":12688},[126],[23,12690],{"className":12691,"style":131},[130],[23,12693,10473],{"className":12694},[135],[11,12696,12697,12698,12782,12783,12865,12866,10698],{},"现实中的两类错误很少同等昂贵。设预测为正类的代价为 ",[23,12699,12701,12725],{"className":12700,"translate":27},[31],[23,12702,12704],{"className":12703},[35],[37,12705,12706],{"xmlns":39},[42,12707,12708,12722],{},[45,12709,12710],{},[898,12711,12712,12715],{},[48,12713,12714],{},"c",[45,12716,12717,12720],{},[48,12718,12719],{"mathvariant":111},"F",[48,12721,920],{"mathvariant":111},[114,12723,12724],{"encoding":116},"c_{\\mathrm{FP}}",[23,12726,12728],{"className":12727,"ariaHidden":122},[121],[23,12729,12731,12735],{"className":12730},[126],[23,12732],{"className":12733,"style":12734},[130],"height:0.5806em;vertical-align:-0.15em;",[23,12736,12738,12741],{"className":12737},[135],[23,12739,12714],{"className":12740},[135,136],[23,12742,12744],{"className":12743},[998],[23,12745,12747,12774],{"className":12746},[199,200],[23,12748,12750,12771],{"className":12749},[204],[23,12751,12753],{"className":12752,"style":2026},[208],[23,12754,12755,12758],{"style":2029},[23,12756],{"className":12757,"style":1015},[216],[23,12759,12761],{"className":12760},[1019,1020,1021,1022],[23,12762,12764],{"className":12763},[135,1022],[23,12765,12767],{"className":12766},[135,1022],[23,12768,12770],{"className":12769},[135,5391,1022],"FP",[23,12772,297],{"className":12773},[296],[23,12775,12777],{"className":12776},[204],[23,12778,12780],{"className":12779,"style":2048},[208],[23,12781],{},"（假阳性），预测为负类的代价为 ",[23,12784,12786,12809],{"className":12785,"translate":27},[31],[23,12787,12789],{"className":12788},[35],[37,12790,12791],{"xmlns":39},[42,12792,12793,12806],{},[45,12794,12795],{},[898,12796,12797,12799],{},[48,12798,12714],{},[45,12800,12801,12803],{},[48,12802,12719],{"mathvariant":111},[48,12804,12805],{"mathvariant":111},"N",[114,12807,12808],{"encoding":116},"c_{\\mathrm{FN}}",[23,12810,12812],{"className":12811,"ariaHidden":122},[121],[23,12813,12815,12818],{"className":12814},[126],[23,12816],{"className":12817,"style":12734},[130],[23,12819,12821,12824],{"className":12820},[135],[23,12822,12714],{"className":12823},[135,136],[23,12825,12827],{"className":12826},[998],[23,12828,12830,12857],{"className":12829},[199,200],[23,12831,12833,12854],{"className":12832},[204],[23,12834,12836],{"className":12835,"style":2026},[208],[23,12837,12838,12841],{"style":2029},[23,12839],{"className":12840,"style":1015},[216],[23,12842,12844],{"className":12843},[1019,1020,1021,1022],[23,12845,12847],{"className":12846},[135,1022],[23,12848,12850],{"className":12849},[135,1022],[23,12851,12853],{"className":12852},[135,5391,1022],"FN",[23,12855,297],{"className":12856},[296],[23,12858,12860],{"className":12859},[204],[23,12861,12863],{"className":12862,"style":2048},[208],[23,12864],{},"（假阴性），正确分类的代价为 ",[23,12867,12869,12882],{"className":12868,"translate":27},[31],[23,12870,12872],{"className":12871},[35],[37,12873,12874],{"xmlns":39},[42,12875,12876,12880],{},[45,12877,12878],{},[2581,12879,1532],{},[114,12881,1532],{"encoding":116},[23,12883,12885],{"className":12884,"ariaHidden":122},[121],[23,12886,12888,12891],{"className":12887},[126],[23,12889],{"className":12890,"style":2770},[130],[23,12892,1532],{"className":12893},[135],[23,12895,12897],{"className":12896,"translate":27},[26],[23,12898,12900,12994],{"className":12899,"translate":27},[31],[23,12901,12903],{"className":12902},[35],[37,12904,12905],{"xmlns":39,"display":40},[42,12906,12907,12991],{},[45,12908,12909,12911,12913,12915,12917,12919,12921,12923,12933,12935,12937,12939,12941,12943,12945,12947,12949,12951,12953,12955,12957,12959,12961,12963,12965,12975,12977,12979,12981,12983,12985,12987,12989],{},[48,12910,1653],{},[51,12912,54],{"stretchy":53},[2581,12914,4435],{},[51,12916,747],{"separator":122},[48,12918,753],{},[51,12920,66],{"stretchy":53},[51,12922,69],{},[898,12924,12925,12927],{},[48,12926,12714],{},[45,12928,12929,12931],{},[48,12930,12719],{"mathvariant":111},[48,12932,920],{"mathvariant":111},[88,12934,90],{},[48,12936,11],{},[51,12938,54],{"stretchy":53},[2581,12940,1532],{},[51,12942,60],{},[48,12944,753],{},[51,12946,66],{"stretchy":53},[51,12948,747],{"separator":122},[148,12950],{"width":750},[48,12952,1653],{},[51,12954,54],{"stretchy":53},[2581,12956,1532],{},[51,12958,747],{"separator":122},[48,12960,753],{},[51,12962,66],{"stretchy":53},[51,12964,69],{},[898,12966,12967,12969],{},[48,12968,12714],{},[45,12970,12971,12973],{},[48,12972,12719],{"mathvariant":111},[48,12974,12805],{"mathvariant":111},[88,12976,90],{},[48,12978,11],{},[51,12980,54],{"stretchy":53},[2581,12982,4435],{},[51,12984,60],{},[48,12986,753],{},[51,12988,66],{"stretchy":53},[48,12990,112],{"mathvariant":111},[114,12992,12993],{"encoding":116},"r(1,x)=c_{\\mathrm{FP}}\\,p(0\\mid x),\\qquad\nr(0,x)=c_{\\mathrm{FN}}\\,p(1\\mid x).",[23,12995,12997,13033,13106,13157,13230],{"className":12996,"ariaHidden":122},[121],[23,12998,13000,13003,13006,13009,13012,13015,13018,13021,13024,13027,13030],{"className":12999},[126],[23,13001],{"className":13002,"style":131},[130],[23,13004,1653],{"className":13005,"style":144},[135,136],[23,13007,54],{"className":13008},[140],[23,13010,4435],{"className":13011},[135],[23,13013,747],{"className":13014},[824],[23,13016],{"className":13017,"style":280},[148],[23,13019,753],{"className":13020},[135,136],[23,13022,66],{"className":13023},[169],[23,13025],{"className":13026,"style":149},[148],[23,13028,69],{"className":13029},[153],[23,13031],{"className":13032,"style":149},[148],[23,13034,13036,13039,13085,13088,13091,13094,13097,13100,13103],{"className":13035},[126],[23,13037],{"className":13038,"style":131},[130],[23,13040,13042,13045],{"className":13041},[135],[23,13043,12714],{"className":13044},[135,136],[23,13046,13048],{"className":13047},[998],[23,13049,13051,13077],{"className":13050},[199,200],[23,13052,13054,13074],{"className":13053},[204],[23,13055,13057],{"className":13056,"style":2026},[208],[23,13058,13059,13062],{"style":2029},[23,13060],{"className":13061,"style":1015},[216],[23,13063,13065],{"className":13064},[1019,1020,1021,1022],[23,13066,13068],{"className":13067},[135,1022],[23,13069,13071],{"className":13070},[135,1022],[23,13072,12770],{"className":13073},[135,5391,1022],[23,13075,297],{"className":13076},[296],[23,13078,13080],{"className":13079},[204],[23,13081,13083],{"className":13082,"style":2048},[208],[23,13084],{},[23,13086],{"className":13087,"style":280},[148],[23,13089,11],{"className":13090},[135,136],[23,13092,54],{"className":13093},[140],[23,13095,1532],{"className":13096},[135],[23,13098],{"className":13099,"style":149},[148],[23,13101,60],{"className":13102},[153],[23,13104],{"className":13105,"style":149},[148],[23,13107,13109,13112,13115,13118,13121,13124,13127,13130,13133,13136,13139,13142,13145,13148,13151,13154],{"className":13108},[126],[23,13110],{"className":13111,"style":131},[130],[23,13113,753],{"className":13114},[135,136],[23,13116,66],{"className":13117},[169],[23,13119,747],{"className":13120},[824],[23,13122],{"className":13123,"style":828},[148],[23,13125],{"className":13126,"style":280},[148],[23,13128,1653],{"className":13129,"style":144},[135,136],[23,13131,54],{"className":13132},[140],[23,13134,1532],{"className":13135},[135],[23,13137,747],{"className":13138},[824],[23,13140],{"className":13141,"style":280},[148],[23,13143,753],{"className":13144},[135,136],[23,13146,66],{"className":13147},[169],[23,13149],{"className":13150,"style":149},[148],[23,13152,69],{"className":13153},[153],[23,13155],{"className":13156,"style":149},[148],[23,13158,13160,13163,13209,13212,13215,13218,13221,13224,13227],{"className":13159},[126],[23,13161],{"className":13162,"style":131},[130],[23,13164,13166,13169],{"className":13165},[135],[23,13167,12714],{"className":13168},[135,136],[23,13170,13172],{"className":13171},[998],[23,13173,13175,13201],{"className":13174},[199,200],[23,13176,13178,13198],{"className":13177},[204],[23,13179,13181],{"className":13180,"style":2026},[208],[23,13182,13183,13186],{"style":2029},[23,13184],{"className":13185,"style":1015},[216],[23,13187,13189],{"className":13188},[1019,1020,1021,1022],[23,13190,13192],{"className":13191},[135,1022],[23,13193,13195],{"className":13194},[135,1022],[23,13196,12853],{"className":13197},[135,5391,1022],[23,13199,297],{"className":13200},[296],[23,13202,13204],{"className":13203},[204],[23,13205,13207],{"className":13206,"style":2048},[208],[23,13208],{},[23,13210],{"className":13211,"style":280},[148],[23,13213,11],{"className":13214},[135,136],[23,13216,54],{"className":13217},[140],[23,13219,4435],{"className":13220},[135],[23,13222],{"className":13223,"style":149},[148],[23,13225,60],{"className":13226},[153],[23,13228],{"className":13229,"style":149},[148],[23,13231,13233,13236,13239,13242],{"className":13232},[126],[23,13234],{"className":13235,"style":131},[130],[23,13237,753],{"className":13238},[135,136],[23,13240,66],{"className":13241},[169],[23,13243,112],{"className":13244},[135],[11,13246,13247,13248,13355],{},"选择正类当且仅当 ",[23,13249,13251,13289],{"className":13250,"translate":27},[31],[23,13252,13254],{"className":13253},[35],[37,13255,13256],{"xmlns":39},[42,13257,13258,13286],{},[45,13259,13260,13262,13264,13266,13268,13270,13272,13274,13276,13278,13280,13282,13284],{},[48,13261,1653],{},[51,13263,54],{"stretchy":53},[2581,13265,4435],{},[51,13267,747],{"separator":122},[48,13269,753],{},[51,13271,66],{"stretchy":53},[51,13273,12272],{},[48,13275,1653],{},[51,13277,54],{"stretchy":53},[2581,13279,1532],{},[51,13281,747],{"separator":122},[48,13283,753],{},[51,13285,66],{"stretchy":53},[114,13287,13288],{"encoding":116},"r(1,x)\\le r(0,x)",[23,13290,13292,13328],{"className":13291,"ariaHidden":122},[121],[23,13293,13295,13298,13301,13304,13307,13310,13313,13316,13319,13322,13325],{"className":13294},[126],[23,13296],{"className":13297,"style":131},[130],[23,13299,1653],{"className":13300,"style":144},[135,136],[23,13302,54],{"className":13303},[140],[23,13305,4435],{"className":13306},[135],[23,13308,747],{"className":13309},[824],[23,13311],{"className":13312,"style":280},[148],[23,13314,753],{"className":13315},[135,136],[23,13317,66],{"className":13318},[169],[23,13320],{"className":13321,"style":149},[148],[23,13323,12272],{"className":13324},[153],[23,13326],{"className":13327,"style":149},[148],[23,13329,13331,13334,13337,13340,13343,13346,13349,13352],{"className":13330},[126],[23,13332],{"className":13333,"style":131},[130],[23,13335,1653],{"className":13336,"style":144},[135,136],[23,13338,54],{"className":13339},[140],[23,13341,1532],{"className":13342},[135],[23,13344,747],{"className":13345},[824],[23,13347],{"className":13348,"style":280},[148],[23,13350,753],{"className":13351},[135,136],[23,13353,66],{"className":13354},[169],"，即",[23,13357,13359],{"className":13358,"translate":27},[26],[23,13360,13362,13430],{"className":13361,"translate":27},[31],[23,13363,13365],{"className":13364},[35],[37,13366,13367],{"xmlns":39,"display":40},[42,13368,13369,13427],{},[45,13370,13371,13373,13375,13377,13379,13381,13383,13385,13387,13389,13425],{},[48,13372,11],{},[51,13374,54],{"stretchy":53},[2581,13376,4435],{},[51,13378,60],{},[48,13380,753],{},[51,13382,66],{"stretchy":53},[88,13384,5360],{},[51,13386,7382],{},[88,13388,5360],{},[71,13390,13391,13401],{},[898,13392,13393,13395],{},[48,13394,12714],{},[45,13396,13397,13399],{},[48,13398,12719],{"mathvariant":111},[48,13400,920],{"mathvariant":111},[45,13402,13403,13413,13415],{},[898,13404,13405,13407],{},[48,13406,12714],{},[45,13408,13409,13411],{},[48,13410,12719],{"mathvariant":111},[48,13412,920],{"mathvariant":111},[51,13414,5106],{},[898,13416,13417,13419],{},[48,13418,12714],{},[45,13420,13421,13423],{},[48,13422,12719],{"mathvariant":111},[48,13424,12805],{"mathvariant":111},[48,13426,112],{"mathvariant":111},[114,13428,13429],{"encoding":116},"p(1\\mid x)\\ \\ge\\ \\frac{c_{\\mathrm{FP}}}{c_{\\mathrm{FP}}+c_{\\mathrm{FN}}}.",[23,13431,13433,13457,13484],{"className":13432,"ariaHidden":122},[121],[23,13434,13436,13439,13442,13445,13448,13451,13454],{"className":13435},[126],[23,13437],{"className":13438,"style":131},[130],[23,13440,11],{"className":13441},[135,136],[23,13443,54],{"className":13444},[140],[23,13446,4435],{"className":13447},[135],[23,13449],{"className":13450,"style":149},[148],[23,13452,60],{"className":13453},[153],[23,13455],{"className":13456,"style":149},[148],[23,13458,13460,13463,13466,13469,13472,13475,13478,13481],{"className":13459},[126],[23,13461],{"className":13462,"style":131},[130],[23,13464,753],{"className":13465},[135,136],[23,13467,66],{"className":13468},[169],[23,13470,5360],{"className":13471},[148],[23,13473],{"className":13474,"style":149},[148],[23,13476,7382],{"className":13477},[153],[23,13479,5360],{"className":13480},[148],[23,13482],{"className":13483,"style":149},[148],[23,13485,13487,13491,13695],{"className":13486},[126],[23,13488],{"className":13489,"style":13490},[130],"height:1.9436em;vertical-align:-0.836em;",[23,13492,13494,13497,13692],{"className":13493},[135],[23,13495],{"className":13496},[140,192],[23,13498,13500],{"className":13499},[71],[23,13501,13503,13683],{"className":13502},[199,200],[23,13504,13506,13680],{"className":13505},[204],[23,13507,13509,13618,13626],{"className":13508,"style":6056},[208],[23,13510,13511,13514],{"style":212},[23,13512],{"className":13513,"style":217},[216],[23,13515,13517,13563,13566,13569,13572],{"className":13516},[135],[23,13518,13520,13523],{"className":13519},[135],[23,13521,12714],{"className":13522},[135,136],[23,13524,13526],{"className":13525},[998],[23,13527,13529,13555],{"className":13528},[199,200],[23,13530,13532,13552],{"className":13531},[204],[23,13533,13535],{"className":13534,"style":2026},[208],[23,13536,13537,13540],{"style":2029},[23,13538],{"className":13539,"style":1015},[216],[23,13541,13543],{"className":13542},[1019,1020,1021,1022],[23,13544,13546],{"className":13545},[135,1022],[23,13547,13549],{"className":13548},[135,1022],[23,13550,12770],{"className":13551},[135,5391,1022],[23,13553,297],{"className":13554},[296],[23,13556,13558],{"className":13557},[204],[23,13559,13561],{"className":13560,"style":2048},[208],[23,13562],{},[23,13564],{"className":13565,"style":392},[148],[23,13567,5106],{"className":13568},[612],[23,13570],{"className":13571,"style":392},[148],[23,13573,13575,13578],{"className":13574},[135],[23,13576,12714],{"className":13577},[135,136],[23,13579,13581],{"className":13580},[998],[23,13582,13584,13610],{"className":13583},[199,200],[23,13585,13587,13607],{"className":13586},[204],[23,13588,13590],{"className":13589,"style":2026},[208],[23,13591,13592,13595],{"style":2029},[23,13593],{"className":13594,"style":1015},[216],[23,13596,13598],{"className":13597},[1019,1020,1021,1022],[23,13599,13601],{"className":13600},[135,1022],[23,13602,13604],{"className":13603},[135,1022],[23,13605,12853],{"className":13606},[135,5391,1022],[23,13608,297],{"className":13609},[296],[23,13611,13613],{"className":13612},[204],[23,13614,13616],{"className":13615,"style":2048},[208],[23,13617],{},[23,13619,13620,13623],{"style":235},[23,13621],{"className":13622,"style":217},[216],[23,13624],{"className":13625,"style":243},[242],[23,13627,13628,13631],{"style":246},[23,13629],{"className":13630,"style":217},[216],[23,13632,13634],{"className":13633},[135],[23,13635,13637,13640],{"className":13636},[135],[23,13638,12714],{"className":13639},[135,136],[23,13641,13643],{"className":13642},[998],[23,13644,13646,13672],{"className":13645},[199,200],[23,13647,13649,13669],{"className":13648},[204],[23,13650,13652],{"className":13651,"style":2026},[208],[23,13653,13654,13657],{"style":2029},[23,13655],{"className":13656,"style":1015},[216],[23,13658,13660],{"className":13659},[1019,1020,1021,1022],[23,13661,13663],{"className":13662},[135,1022],[23,13664,13666],{"className":13665},[135,1022],[23,13667,12770],{"className":13668},[135,5391,1022],[23,13670,297],{"className":13671},[296],[23,13673,13675],{"className":13674},[204],[23,13676,13678],{"className":13677,"style":2048},[208],[23,13679],{},[23,13681,297],{"className":13682},[296],[23,13684,13686],{"className":13685},[204],[23,13687,13690],{"className":13688,"style":13689},[208],"height:0.836em;",[23,13691],{},[23,13693],{"className":13694},[169,192],[23,13696,112],{"className":13697},[135],[11,13699,13700,13701,13733,13734,13768],{},"阈值不再固定在 ",[23,13702,13704,13721],{"className":13703,"translate":27},[31],[23,13705,13707],{"className":13706},[35],[37,13708,13709],{"xmlns":39},[42,13710,13711,13719],{},[45,13712,13713,13715,13717],{},[2581,13714,4435],{},[48,13716,2875],{"mathvariant":111},[2581,13718,9956],{},[114,13720,10473],{"encoding":116},[23,13722,13724],{"className":13723,"ariaHidden":122},[121],[23,13725,13727,13730],{"className":13726},[126],[23,13728],{"className":13729,"style":131},[130],[23,13731,10473],{"className":13732},[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 ",[23,13735,13737,13756],{"className":13736,"translate":27},[31],[23,13738,13740],{"className":13739},[35],[37,13741,13742],{"xmlns":39},[42,13743,13744,13753],{},[45,13745,13746,13748,13750],{},[2581,13747,4435],{},[48,13749,2875],{"mathvariant":111},[2581,13751,13752],{},"11",[114,13754,13755],{"encoding":116},"1\u002F11",[23,13757,13759],{"className":13758,"ariaHidden":122},[121],[23,13760,13762,13765],{"className":13761},[126],[23,13763],{"className":13764,"style":131},[130],[23,13766,13755],{"className":13767},[135]," 左右，模型会倾向于把更多样本判为正类。",[11,13770,13771,13772,13775],{},"这个式子也说明：类别不平衡本身不是问题，",[18,13773,13774],{},"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。",[11,13777,13778],{},"把常见损失与对应的贝叶斯行动放在一起：",[3764,13780,13781,13790],{},[3767,13782,13783],{},[3770,13784,13785,13787],{},[3773,13786,540],{},[3773,13788,13789],{},"贝叶斯行动",[3783,13791,13792,13800,13807,13815,14047],{},[3770,13793,13794,13797],{},[3788,13795,13796],{},"0-1 损失",[3788,13798,13799],{},"后验众数",[3770,13801,13802,13805],{},[3788,13803,13804],{},"平方损失",[3788,13806,11335],{},[3770,13808,13809,13812],{},[3788,13810,13811],{},"绝对损失",[3788,13813,13814],{},"后验中位数",[3770,13816,13817,13820],{},[3788,13818,13819],{},"非对称 0-1 损失",[3788,13821,13822,13823],{},"阈值 ",[23,13824,13826,13876],{"className":13825,"translate":27},[31],[23,13827,13829],{"className":13828},[35],[37,13830,13831],{"xmlns":39},[42,13832,13833,13873],{},[45,13834,13835,13845,13847,13849,13859,13861,13871],{},[898,13836,13837,13839],{},[48,13838,12714],{},[45,13840,13841,13843],{},[48,13842,12719],{"mathvariant":111},[48,13844,920],{"mathvariant":111},[48,13846,2875],{"mathvariant":111},[51,13848,54],{"stretchy":53},[898,13850,13851,13853],{},[48,13852,12714],{},[45,13854,13855,13857],{},[48,13856,12719],{"mathvariant":111},[48,13858,920],{"mathvariant":111},[51,13860,5106],{},[898,13862,13863,13865],{},[48,13864,12714],{},[45,13866,13867,13869],{},[48,13868,12719],{"mathvariant":111},[48,13870,12805],{"mathvariant":111},[51,13872,66],{"stretchy":53},[114,13874,13875],{"encoding":116},"c_{\\mathrm{FP}}\u002F(c_{\\mathrm{FP}}+c_{\\mathrm{FN}})",[23,13877,13879,13992],{"className":13878,"ariaHidden":122},[121],[23,13880,13882,13885,13931,13934,13937,13983,13986,13989],{"className":13881},[126],[23,13883],{"className":13884,"style":131},[130],[23,13886,13888,13891],{"className":13887},[135],[23,13889,12714],{"className":13890},[135,136],[23,13892,13894],{"className":13893},[998],[23,13895,13897,13923],{"className":13896},[199,200],[23,13898,13900,13920],{"className":13899},[204],[23,13901,13903],{"className":13902,"style":2026},[208],[23,13904,13905,13908],{"style":2029},[23,13906],{"className":13907,"style":1015},[216],[23,13909,13911],{"className":13910},[1019,1020,1021,1022],[23,13912,13914],{"className":13913},[135,1022],[23,13915,13917],{"className":13916},[135,1022],[23,13918,12770],{"className":13919},[135,5391,1022],[23,13921,297],{"className":13922},[296],[23,13924,13926],{"className":13925},[204],[23,13927,13929],{"className":13928,"style":2048},[208],[23,13930],{},[23,13932,2875],{"className":13933},[135],[23,13935,54],{"className":13936},[140],[23,13938,13940,13943],{"className":13939},[135],[23,13941,12714],{"className":13942},[135,136],[23,13944,13946],{"className":13945},[998],[23,13947,13949,13975],{"className":13948},[199,200],[23,13950,13952,13972],{"className":13951},[204],[23,13953,13955],{"className":13954,"style":2026},[208],[23,13956,13957,13960],{"style":2029},[23,13958],{"className":13959,"style":1015},[216],[23,13961,13963],{"className":13962},[1019,1020,1021,1022],[23,13964,13966],{"className":13965},[135,1022],[23,13967,13969],{"className":13968},[135,1022],[23,13970,12770],{"className":13971},[135,5391,1022],[23,13973,297],{"className":13974},[296],[23,13976,13978],{"className":13977},[204],[23,13979,13981],{"className":13980,"style":2048},[208],[23,13982],{},[23,13984],{"className":13985,"style":392},[148],[23,13987,5106],{"className":13988},[612],[23,13990],{"className":13991,"style":392},[148],[23,13993,13995,13998,14044],{"className":13994},[126],[23,13996],{"className":13997,"style":131},[130],[23,13999,14001,14004],{"className":14000},[135],[23,14002,12714],{"className":14003},[135,136],[23,14005,14007],{"className":14006},[998],[23,14008,14010,14036],{"className":14009},[199,200],[23,14011,14013,14033],{"className":14012},[204],[23,14014,14016],{"className":14015,"style":2026},[208],[23,14017,14018,14021],{"style":2029},[23,14019],{"className":14020,"style":1015},[216],[23,14022,14024],{"className":14023},[1019,1020,1021,1022],[23,14025,14027],{"className":14026},[135,1022],[23,14028,14030],{"className":14029},[135,1022],[23,14031,12853],{"className":14032},[135,5391,1022],[23,14034,297],{"className":14035},[296],[23,14037,14039],{"className":14038},[204],[23,14040,14042],{"className":14041,"style":2048},[208],[23,14043],{},[23,14045,66],{"className":14046},[169],[3770,14048,14049,14052],{},[3788,14050,14051],{},"带拒绝的 0-1 损失",[3788,14053,14054,14055,14106],{},"最大后验低于 ",[23,14056,14058,14076],{"className":14057,"translate":27},[31],[23,14059,14061],{"className":14060},[35],[37,14062,14063],{"xmlns":39},[42,14064,14065,14073],{},[45,14066,14067,14069,14071],{},[2581,14068,4435],{},[51,14070,4529],{},[48,14072,12714],{},[114,14074,14075],{"encoding":116},"1-c",[23,14077,14079,14097],{"className":14078,"ariaHidden":122},[121],[23,14080,14082,14085,14088,14091,14094],{"className":14081},[126],[23,14083],{"className":14084,"style":9404},[130],[23,14086,4435],{"className":14087},[135],[23,14089],{"className":14090,"style":392},[148],[23,14092,4529],{"className":14093},[612],[23,14095],{"className":14096,"style":392},[148],[23,14098,14100,14103],{"className":14099},[126],[23,14101],{"className":14102,"style":703},[130],[23,14104,12714],{"className":14105},[135,136]," 时拒绝",[11,14108,14109,14110,14113,14114,14188,14189,14332,14333,14361],{},"最后一行是",[18,14111,14112],{},"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 ",[23,14115,14117,14143],{"className":14116,"translate":27},[31],[23,14118,14120],{"className":14119},[35],[37,14121,14122],{"xmlns":39},[42,14123,14124,14140],{},[45,14125,14126,14128,14130,14132,14134,14136,14138],{},[48,14127,12714],{},[51,14129,372],{},[51,14131,54],{"stretchy":53},[2581,14133,1532],{},[51,14135,747],{"separator":122},[2581,14137,4435],{},[51,14139,66],{"stretchy":53},[114,14141,14142],{"encoding":116},"c\\in(0,1)",[23,14144,14146,14164],{"className":14145,"ariaHidden":122},[121],[23,14147,14149,14152,14155,14158,14161],{"className":14148},[126],[23,14150],{"className":14151,"style":513},[130],[23,14153,12714],{"className":14154},[135,136],[23,14156],{"className":14157,"style":149},[148],[23,14159,372],{"className":14160},[153],[23,14162],{"className":14163,"style":149},[148],[23,14165,14167,14170,14173,14176,14179,14182,14185],{"className":14166},[126],[23,14168],{"className":14169,"style":131},[130],[23,14171,54],{"className":14172},[140],[23,14174,1532],{"className":14175},[135],[23,14177,747],{"className":14178},[824],[23,14180],{"className":14181,"style":280},[148],[23,14183,4435],{"className":14184},[135],[23,14186,66],{"className":14187},[169],"，预测的期望代价为 ",[23,14190,14192,14230],{"className":14191,"translate":27},[31],[23,14193,14195],{"className":14194},[35],[37,14196,14197],{"xmlns":39},[42,14198,14199,14227],{},[45,14200,14201,14203,14205,14215,14217,14219,14221,14223,14225],{},[2581,14202,4435],{},[51,14204,4529],{},[898,14206,14207,14213],{},[45,14208,14209,14211],{},[48,14210,9555],{},[51,14212,6961],{},[48,14214,660],{},[48,14216,11],{},[51,14218,54],{"stretchy":53},[48,14220,660],{},[51,14222,60],{},[48,14224,753],{},[51,14226,66],{"stretchy":53},[114,14228,14229],{"encoding":116},"1-\\max_y p(y\\mid x)",[23,14231,14233,14251,14320],{"className":14232,"ariaHidden":122},[121],[23,14234,14236,14239,14242,14245,14248],{"className":14235},[126],[23,14237],{"className":14238,"style":9404},[130],[23,14240,4435],{"className":14241},[135],[23,14243],{"className":14244,"style":392},[148],[23,14246,4529],{"className":14247},[612],[23,14249],{"className":14250,"style":392},[148],[23,14252,14254,14258,14299,14302,14305,14308,14311,14314,14317],{"className":14253},[126],[23,14255],{"className":14256,"style":14257},[130],"height:1.0361em;vertical-align:-0.2861em;",[23,14259,14261,14264],{"className":14260},[1786],[23,14262,9555],{"className":14263},[1786],[23,14265,14267],{"className":14266},[998],[23,14268,14270,14290],{"className":14269},[199,200],[23,14271,14273,14287],{"className":14272},[204],[23,14274,14276],{"className":14275,"style":6499},[208],[23,14277,14278,14281],{"style":8191},[23,14279],{"className":14280,"style":1015},[216],[23,14282,14284],{"className":14283},[1019,1020,1021,1022],[23,14285,660],{"className":14286,"style":676},[135,136,1022],[23,14288,297],{"className":14289},[296],[23,14291,14293],{"className":14292},[204],[23,14294,14297],{"className":14295,"style":14296},[208],"height:0.2861em;",[23,14298],{},[23,14300],{"className":14301,"style":280},[148],[23,14303,11],{"className":14304},[135,136],[23,14306,54],{"className":14307},[140],[23,14309,660],{"className":14310,"style":676},[135,136],[23,14312],{"className":14313,"style":149},[148],[23,14315,60],{"className":14316},[153],[23,14318],{"className":14319,"style":149},[148],[23,14321,14323,14326,14329],{"className":14322},[126],[23,14324],{"className":14325,"style":131},[130],[23,14327,753],{"className":14328},[135,136],[23,14330,66],{"className":14331},[169],"，拒绝的代价为 ",[23,14334,14336,14349],{"className":14335,"translate":27},[31],[23,14337,14339],{"className":14338},[35],[37,14340,14341],{"xmlns":39},[42,14342,14343,14347],{},[45,14344,14345],{},[48,14346,12714],{},[114,14348,12714],{"encoding":116},[23,14350,14352],{"className":14351,"ariaHidden":122},[121],[23,14353,14355,14358],{"className":14354},[126],[23,14356],{"className":14357,"style":703},[130],[23,14359,12714],{"className":14360},[135,136],"，因此最优规则是",[23,14363,14365],{"className":14364,"translate":27},[26],[23,14366,14368,14422],{"className":14367,"translate":27},[31],[23,14369,14371],{"className":14370},[35],[37,14372,14373],{"xmlns":39,"display":40},[42,14374,14375,14419],{},[45,14376,14377,14380,14382,14385,14387,14397,14399,14401,14403,14405,14407,14409,14411,14413,14415,14417],{},[88,14378,14379],{},"拒绝",[148,14381],{"width":9919},[51,14383,14384],{},"⟺",[148,14386],{"width":9919},[6963,14388,14389,14395],{},[45,14390,14391,14393],{},[48,14392,9555],{},[51,14394,6961],{},[48,14396,660],{},[48,14398,11],{},[51,14400,54],{"stretchy":53},[48,14402,660],{},[51,14404,60],{},[48,14406,753],{},[51,14408,66],{"stretchy":53},[51,14410,9987],{},[2581,14412,4435],{},[51,14414,4529],{},[48,14416,12714],{},[48,14418,112],{"mathvariant":111},[114,14420,14421],{"encoding":116},"\\text{拒绝}\\quad\\Longleftrightarrow\\quad \\max_y p(y\\mid x)\u003C1-c.",[23,14423,14425,14454,14527,14548,14566],{"className":14424,"ariaHidden":122},[121],[23,14426,14428,14432,14438,14442,14445,14448,14451],{"className":14427},[126],[23,14429],{"className":14430,"style":14431},[130],"height:0.7073em;vertical-align:-0.024em;",[23,14433,14435],{"className":14434},[135,6514],[23,14436,14379],{"className":14437},[135,7240],[23,14439],{"className":14440,"style":14441},[148],"margin-right:1em;",[23,14443],{"className":14444,"style":149},[148],[23,14446,14384],{"className":14447},[153],[23,14449],{"className":14450,"style":14441},[148],[23,14452],{"className":14453,"style":149},[148],[23,14455,14457,14461,14506,14509,14512,14515,14518,14521,14524],{"className":14456},[126],[23,14458],{"className":14459,"style":14460},[130],"height:1.5861em;vertical-align:-0.8361em;",[23,14462,14464],{"className":14463},[1786,7098],[23,14465,14467,14497],{"className":14466},[199,200],[23,14468,14470,14494],{"className":14469},[204],[23,14471,14473,14484],{"className":14472,"style":703},[208],[23,14474,14475,14478],{"style":8784},[23,14476],{"className":14477,"style":217},[216],[23,14479,14481],{"className":14480},[1019,1020,1021,1022],[23,14482,660],{"className":14483,"style":676},[135,136,1022],[23,14485,14486,14489],{"style":7132},[23,14487],{"className":14488,"style":217},[216],[23,14490,14491],{},[23,14492,9555],{"className":14493},[1786],[23,14495,297],{"className":14496},[296],[23,14498,14500],{"className":14499},[204],[23,14501,14504],{"className":14502,"style":14503},[208],"height:0.8361em;",[23,14505],{},[23,14507],{"className":14508,"style":280},[148],[23,14510,11],{"className":14511},[135,136],[23,14513,54],{"className":14514},[140],[23,14516,660],{"className":14517,"style":676},[135,136],[23,14519],{"className":14520,"style":149},[148],[23,14522,60],{"className":14523},[153],[23,14525],{"className":14526,"style":149},[148],[23,14528,14530,14533,14536,14539,14542,14545],{"className":14529},[126],[23,14531],{"className":14532,"style":131},[130],[23,14534,753],{"className":14535},[135,136],[23,14537,66],{"className":14538},[169],[23,14540],{"className":14541,"style":149},[148],[23,14543,9987],{"className":14544},[153],[23,14546],{"className":14547,"style":149},[148],[23,14549,14551,14554,14557,14560,14563],{"className":14550},[126],[23,14552],{"className":14553,"style":9404},[130],[23,14555,4435],{"className":14556},[135],[23,14558],{"className":14559,"style":392},[148],[23,14561,4529],{"className":14562},[612],[23,14564],{"className":14565,"style":392},[148],[23,14567,14569,14572,14575],{"className":14568},[126],[23,14570],{"className":14571,"style":703},[130],[23,14573,12714],{"className":14574},[135,136],[23,14576,112],{"className":14577},[135],[11,14579,14580],{},"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。",[2501,14582,14584],{"id":14583},"_36-贝叶斯风险","3.6 贝叶斯风险",[11,14586,14587],{},"把定理 3.1 中的最优值单独命名：",[23,14589,14591],{"className":14590,"translate":27},[26],[23,14592,14594,14674],{"className":14593,"translate":27},[31],[23,14595,14597],{"className":14596},[35],[37,14598,14599],{"xmlns":39,"display":40},[42,14600,14601,14671],{},[45,14602,14603,14609,14611,14621,14623,14625,14627,14629,14631,14637,14639,14655,14657,14659,14661,14663,14665,14667,14669],{},[4518,14604,14605,14607],{},[48,14606,572],{},[51,14608,6883],{},[51,14610,69],{},[6963,14612,14613,14619],{},[45,14614,14615,14617],{},[48,14616,8318],{},[51,14618,6961],{},[48,14620,736],{},[48,14622,572],{},[51,14624,54],{"stretchy":53},[48,14626,736],{},[51,14628,66],{"stretchy":53},[51,14630,69],{},[898,14632,14633,14635],{},[48,14634,902],{"mathvariant":571},[48,14636,434],{},[51,14638,924],{"fence":53,"stretchy":122,"minsize":7407,"maxsize":7407},[6963,14640,14641,14647],{},[45,14642,14643,14645],{},[48,14644,6969],{},[51,14646,6961],{},[45,14648,14649,14651,14653],{},[48,14650,495],{},[51,14652,372],{},[48,14654,500],{"mathvariant":375},[48,14656,1653],{},[51,14658,54],{"stretchy":53},[48,14660,495],{},[51,14662,747],{"separator":122},[48,14664,434],{},[51,14666,66],{"stretchy":53},[51,14668,945],{"fence":53,"stretchy":122,"minsize":7407,"maxsize":7407},[48,14670,112],{"mathvariant":111},[114,14672,14673],{"encoding":116},"R^*=\\inf_{\\delta}R(\\delta)\n=\\mathbb E_X\\Big[\\min_{a\\in\\mathcal A}r(a,X)\\Big].",[23,14675,14677,14721,14803],{"className":14676,"ariaHidden":122},[121],[23,14678,14680,14683,14712,14715,14718],{"className":14679},[126],[23,14681],{"className":14682,"style":7047},[130],[23,14684,14686,14689],{"className":14685},[135],[23,14687,572],{"className":14688,"style":963},[135,136],[23,14690,14692],{"className":14691},[998],[23,14693,14695],{"className":14694},[199],[23,14696,14698],{"className":14697},[204],[23,14699,14701],{"className":14700,"style":7047},[208],[23,14702,14703,14706],{"style":4989},[23,14704],{"className":14705,"style":1015},[216],[23,14707,14709],{"className":14708},[1019,1020,1021,1022],[23,14710,6883],{"className":14711},[612,1022],[23,14713],{"className":14714,"style":149},[148],[23,14716,69],{"className":14717},[153],[23,14719],{"className":14720,"style":149},[148],[23,14722,14724,14728,14779,14782,14785,14788,14791,14794,14797,14800],{"className":14723},[126],[23,14725],{"className":14726,"style":14727},[130],"height:1.5021em;vertical-align:-0.7521em;",[23,14729,14731],{"className":14730},[1786,7098],[23,14732,14734,14770],{"className":14733},[199,200],[23,14735,14737,14767],{"className":14736},[204],[23,14738,14740,14755],{"className":14739,"style":779},[208],[23,14741,14743,14746],{"style":14742},"top:-2.3479em;margin-left:0em;",[23,14744],{"className":14745,"style":217},[216],[23,14747,14749],{"className":14748},[1019,1020,1021,1022],[23,14750,14752],{"className":14751},[135,1022],[23,14753,736],{"className":14754,"style":783},[135,136,1022],[23,14756,14757,14760],{"style":7132},[23,14758],{"className":14759,"style":217},[216],[23,14761,14762],{},[23,14763,8336,14765],{"className":14764},[1786],[23,14766,8340],{"style":8339},[23,14768,297],{"className":14769},[296],[23,14771,14773],{"className":14772},[204],[23,14774,14777],{"className":14775,"style":14776},[208],"height:0.7521em;",[23,14778],{},[23,14780],{"className":14781,"style":280},[148],[23,14783,572],{"className":14784,"style":963},[135,136],[23,14786,54],{"className":14787},[140],[23,14789,736],{"className":14790,"style":783},[135,136],[23,14792,66],{"className":14793},[169],[23,14795],{"className":14796,"style":149},[148],[23,14798,69],{"className":14799},[153],[23,14801],{"className":14802,"style":149},[148],[23,14804,14806,14809,14849,14855,14858,14911,14914,14917,14920,14923,14926,14929,14932,14935,14941],{"className":14805},[126],[23,14807],{"className":14808,"style":7538},[130],[23,14810,14812,14815],{"className":14811},[135],[23,14813,902],{"className":14814},[135,644],[23,14816,14818],{"className":14817},[998],[23,14819,14821,14841],{"className":14820},[199,200],[23,14822,14824,14838],{"className":14823},[204],[23,14825,14827],{"className":14826,"style":2026},[208],[23,14828,14829,14832],{"style":2029},[23,14830],{"className":14831,"style":1015},[216],[23,14833,14835],{"className":14834},[1019,1020,1021,1022],[23,14836,434],{"className":14837,"style":454},[135,136,1022],[23,14839,297],{"className":14840},[296],[23,14842,14844],{"className":14843},[204],[23,14845,14847],{"className":14846,"style":2048},[208],[23,14848],{},[23,14850,14852],{"className":14851},[135],[23,14853,924],{"className":14854},[1066,7585],[23,14856],{"className":14857,"style":280},[148],[23,14859,14861],{"className":14860},[1786,7098],[23,14862,14864,14903],{"className":14863},[199,200],[23,14865,14867,14900],{"className":14866},[204],[23,14868,14870,14890],{"className":14869,"style":7108},[208],[23,14871,14872,14875],{"style":7111},[23,14873],{"className":14874,"style":217},[216],[23,14876,14878],{"className":14877},[1019,1020,1021,1022],[23,14879,14881,14884,14887],{"className":14880},[135,1022],[23,14882,495],{"className":14883},[135,136,1022],[23,14885,372],{"className":14886},[153,1022],[23,14888,500],{"className":14889},[135,412,1022],[23,14891,14892,14895],{"style":7132},[23,14893],{"className":14894,"style":217},[216],[23,14896,14897],{},[23,14898,6969],{"className":14899},[1786],[23,14901,297],{"className":14902},[296],[23,14904,14906],{"className":14905},[204],[23,14907,14909],{"className":14908,"style":7150},[208],[23,14910],{},[23,14912],{"className":14913,"style":280},[148],[23,14915,1653],{"className":14916,"style":144},[135,136],[23,14918,54],{"className":14919},[140],[23,14921,495],{"className":14922},[135,136],[23,14924,747],{"className":14925},[824],[23,14927],{"className":14928,"style":280},[148],[23,14930,434],{"className":14931,"style":454},[135,136],[23,14933,66],{"className":14934},[169],[23,14936,14938],{"className":14937},[135],[23,14939,945],{"className":14940},[1066,7585],[23,14942,112],{"className":14943},[135],[11,14945,14946,15005,15006,15008,15009,15037],{},[23,14947,14949,14967],{"className":14948,"translate":27},[31],[23,14950,14952],{"className":14951},[35],[37,14953,14954],{"xmlns":39},[42,14955,14956,14964],{},[45,14957,14958],{},[4518,14959,14960,14962],{},[48,14961,572],{},[51,14963,6883],{},[114,14965,14966],{"encoding":116},"R^*",[23,14968,14970],{"className":14969,"ariaHidden":122},[121],[23,14971,14973,14976],{"className":14972},[126],[23,14974],{"className":14975,"style":6914},[130],[23,14977,14979,14982],{"className":14978},[135],[23,14980,572],{"className":14981,"style":963},[135,136],[23,14983,14985],{"className":14984},[998],[23,14986,14988],{"className":14987},[199],[23,14989,14991],{"className":14990},[204],[23,14992,14994],{"className":14993,"style":6914},[208],[23,14995,14996,14999],{"style":4696},[23,14997],{"className":14998,"style":1015},[216],[23,15000,15002],{"className":15001},[1019,1020,1021,1022],[23,15003,6883],{"className":15004},[612,1022]," 称为",[18,15007,335],{},"（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 ",[23,15010,15012,15025],{"className":15011,"translate":27},[31],[23,15013,15015],{"className":15014},[35],[37,15016,15017],{"xmlns":39},[42,15018,15019,15023],{},[45,15020,15021],{},[48,15022,753],{},[114,15024,753],{"encoding":116},[23,15026,15028],{"className":15027,"ariaHidden":122},[121],[23,15029,15031,15034],{"className":15030},[126],[23,15032],{"className":15033,"style":703},[130],[23,15035,753],{"className":15036},[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。",[11,15039,15040],{},"对二分类 0-1 损失，把 3.2 节的规则代回去，得到",[23,15042,15044],{"className":15043,"translate":27},[26],[23,15045,15047,15105],{"className":15046,"translate":27},[31],[23,15048,15050],{"className":15049},[35],[37,15051,15052],{"xmlns":39,"display":40},[42,15053,15054,15102],{},[45,15055,15056,15062,15064,15066,15068,15070,15072,15074,15076,15078,15080,15082,15084,15086,15088,15090,15092,15094,15096,15098,15100],{},[4518,15057,15058,15060],{},[48,15059,572],{},[51,15061,6883],{},[51,15063,69],{},[48,15065,902],{"mathvariant":571},[51,15067,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,15069,6969],{},[51,15071,6961],{},[51,15073,9014],{"stretchy":53},[48,15075,9769],{},[51,15077,54],{"stretchy":53},[48,15079,434],{},[51,15081,66],{"stretchy":53},[51,15083,747],{"separator":122},[2581,15085,4435],{},[51,15087,4529],{},[48,15089,9769],{},[51,15091,54],{"stretchy":53},[48,15093,434],{},[51,15095,66],{"stretchy":53},[51,15097,9024],{"stretchy":53},[51,15099,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,15101,112],{"mathvariant":111},[114,15103,15104],{"encoding":116},"R^*=\\mathbb E\\big[\\min\\{\\eta(X),1-\\eta(X)\\}\\big].",[23,15106,15108,15152,15206],{"className":15107,"ariaHidden":122},[121],[23,15109,15111,15114,15143,15146,15149],{"className":15110},[126],[23,15112],{"className":15113,"style":7047},[130],[23,15115,15117,15120],{"className":15116},[135],[23,15118,572],{"className":15119,"style":963},[135,136],[23,15121,15123],{"className":15122},[998],[23,15124,15126],{"className":15125},[199],[23,15127,15129],{"className":15128},[204],[23,15130,15132],{"className":15131,"style":7047},[208],[23,15133,15134,15137],{"style":4989},[23,15135],{"className":15136,"style":1015},[216],[23,15138,15140],{"className":15139},[1019,1020,1021,1022],[23,15141,6883],{"className":15142},[612,1022],[23,15144],{"className":15145,"style":149},[148],[23,15147,69],{"className":15148},[153],[23,15150],{"className":15151,"style":149},[148],[23,15153,15155,15158,15161,15167,15170,15173,15176,15179,15182,15185,15188,15191,15194,15197,15200,15203],{"className":15154},[126],[23,15156],{"className":15157,"style":2213},[130],[23,15159,902],{"className":15160},[135,644],[23,15162,15164],{"className":15163},[135],[23,15165,924],{"className":15166},[1066,1067],[23,15168],{"className":15169,"style":280},[148],[23,15171,6969],{"className":15172},[1786],[23,15174,9014],{"className":15175},[140],[23,15177,9769],{"className":15178,"style":676},[135,136],[23,15180,54],{"className":15181},[140],[23,15183,434],{"className":15184,"style":454},[135,136],[23,15186,66],{"className":15187},[169],[23,15189,747],{"className":15190},[824],[23,15192],{"className":15193,"style":280},[148],[23,15195,4435],{"className":15196},[135],[23,15198],{"className":15199,"style":392},[148],[23,15201,4529],{"className":15202},[612],[23,15204],{"className":15205,"style":392},[148],[23,15207,15209,15212,15215,15218,15221,15225,15231],{"className":15208},[126],[23,15210],{"className":15211,"style":2213},[130],[23,15213,9769],{"className":15214,"style":676},[135,136],[23,15216,54],{"className":15217},[140],[23,15219,434],{"className":15220,"style":454},[135,136],[23,15222,15224],{"className":15223},[169],")}",[23,15226,15228],{"className":15227},[135],[23,15229,945],{"className":15230},[1066,1067],[23,15232,112],{"className":15233},[135],[11,15235,15236],{},"对多分类，则是",[23,15238,15240],{"className":15239,"translate":27},[26],[23,15241,15243,15303],{"className":15242,"translate":27},[31],[23,15244,15246],{"className":15245},[35],[37,15247,15248],{"xmlns":39,"display":40},[42,15249,15250,15300],{},[45,15251,15252,15258,15260,15262,15264,15266,15268,15284,15286,15288,15290,15292,15294,15296,15298],{},[4518,15253,15254,15256],{},[48,15255,572],{},[51,15257,6883],{},[51,15259,69],{},[2581,15261,4435],{},[51,15263,4529],{},[48,15265,902],{"mathvariant":571},[51,15267,924],{"fence":53,"stretchy":122,"minsize":7407,"maxsize":7407},[6963,15269,15270,15276],{},[45,15271,15272,15274],{},[48,15273,9555],{},[51,15275,6961],{},[45,15277,15278,15280,15282],{},[48,15279,660],{},[51,15281,372],{},[48,15283,369],{"mathvariant":375},[48,15285,11],{},[51,15287,54],{"stretchy":53},[48,15289,660],{},[51,15291,60],{},[48,15293,434],{},[51,15295,66],{"stretchy":53},[51,15297,945],{"fence":53,"stretchy":122,"minsize":7407,"maxsize":7407},[48,15299,112],{"mathvariant":111},[114,15301,15302],{"encoding":116},"R^*=1-\\mathbb E\\Big[\\max_{y\\in\\mathcal Y}p(y\\mid X)\\Big].",[23,15304,15306,15350,15368,15461],{"className":15305,"ariaHidden":122},[121],[23,15307,15309,15312,15341,15344,15347],{"className":15308},[126],[23,15310],{"className":15311,"style":7047},[130],[23,15313,15315,15318],{"className":15314},[135],[23,15316,572],{"className":15317,"style":963},[135,136],[23,15319,15321],{"className":15320},[998],[23,15322,15324],{"className":15323},[199],[23,15325,15327],{"className":15326},[204],[23,15328,15330],{"className":15329,"style":7047},[208],[23,15331,15332,15335],{"style":4989},[23,15333],{"className":15334,"style":1015},[216],[23,15336,15338],{"className":15337},[1019,1020,1021,1022],[23,15339,6883],{"className":15340},[612,1022],[23,15342],{"className":15343,"style":149},[148],[23,15345,69],{"className":15346},[153],[23,15348],{"className":15349,"style":149},[148],[23,15351,15353,15356,15359,15362,15365],{"className":15352},[126],[23,15354],{"className":15355,"style":9404},[130],[23,15357,4435],{"className":15358},[135],[23,15360],{"className":15361,"style":392},[148],[23,15363,4529],{"className":15364},[612],[23,15366],{"className":15367,"style":392},[148],[23,15369,15371,15375,15378,15384,15387,15440,15443,15446,15449,15452,15455,15458],{"className":15370},[126],[23,15372],{"className":15373,"style":15374},[130],"height:2.0304em;vertical-align:-0.8804em;",[23,15376,902],{"className":15377},[135,644],[23,15379,15381],{"className":15380},[135],[23,15382,924],{"className":15383},[1066,7585],[23,15385],{"className":15386,"style":280},[148],[23,15388,15390],{"className":15389},[1786,7098],[23,15391,15393,15432],{"className":15392},[199,200],[23,15394,15396,15429],{"className":15395},[204],[23,15397,15399,15419],{"className":15398,"style":703},[208],[23,15400,15401,15404],{"style":7111},[23,15402],{"className":15403,"style":217},[216],[23,15405,15407],{"className":15406},[1019,1020,1021,1022],[23,15408,15410,15413,15416],{"className":15409},[135,1022],[23,15411,660],{"className":15412,"style":676},[135,136,1022],[23,15414,372],{"className":15415},[153,1022],[23,15417,369],{"className":15418,"style":413},[135,412,1022],[23,15420,15421,15424],{"style":7132},[23,15422],{"className":15423,"style":217},[216],[23,15425,15426],{},[23,15427,9555],{"className":15428},[1786],[23,15430,297],{"className":15431},[296],[23,15433,15435],{"className":15434},[204],[23,15436,15438],{"className":15437,"style":5683},[208],[23,15439],{},[23,15441],{"className":15442,"style":280},[148],[23,15444,11],{"className":15445},[135,136],[23,15447,54],{"className":15448},[140],[23,15450,660],{"className":15451,"style":676},[135,136],[23,15453],{"className":15454,"style":149},[148],[23,15456,60],{"className":15457},[153],[23,15459],{"className":15460,"style":149},[148],[23,15462,15464,15467,15470,15473,15479],{"className":15463},[126],[23,15465],{"className":15466,"style":7815},[130],[23,15468,434],{"className":15469,"style":454},[135,136],[23,15471,66],{"className":15472},[169],[23,15474,15476],{"className":15475},[135],[23,15477,945],{"className":15478},[1066,7585],[23,15480,112],{"className":15481},[135],[11,15483,15484],{},"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。",[11,15486,15487,15488,15491],{},"进一步，二分类的",[18,15489,15490],{},"超额风险","有一个很干净的形式：",[23,15493,15495],{"className":15494,"translate":27},[26],[23,15496,15498,15584],{"className":15497,"translate":27},[31],[23,15499,15501],{"className":15500},[35],[37,15502,15503],{"xmlns":39,"display":40},[42,15504,15505,15581],{},[45,15506,15507,15509,15511,15513,15515,15517,15523,15525,15527,15529,15531,15533,15535,15537,15539,15541,15543,15545,15547,15549,15551,15553,15555,15557,15559,15561,15563,15569,15571,15573,15575,15577,15579],{},[48,15508,572],{},[51,15510,54],{"stretchy":53},[48,15512,736],{},[51,15514,66],{"stretchy":53},[51,15516,4529],{},[4518,15518,15519,15521],{},[48,15520,572],{},[51,15522,6883],{},[51,15524,69],{},[48,15526,902],{"mathvariant":571},[51,15528,924],{"fence":53,"stretchy":122,"minsize":7407,"maxsize":7407},[48,15530,60],{"mathvariant":111},[2581,15532,9956],{},[48,15534,9769],{},[51,15536,54],{"stretchy":53},[48,15538,434],{},[51,15540,66],{"stretchy":53},[51,15542,4529],{},[2581,15544,4435],{},[48,15546,60],{"mathvariant":111},[88,15548,90],{},[2581,15550,4435],{"mathvariant":9011},[51,15552,9014],{"stretchy":53},[48,15554,736],{},[51,15556,54],{"stretchy":53},[48,15558,434],{},[51,15560,66],{"stretchy":53},[51,15562,9019],{"mathvariant":111},[4518,15564,15565,15567],{},[48,15566,736],{},[51,15568,6883],{},[51,15570,54],{"stretchy":53},[48,15572,434],{},[51,15574,66],{"stretchy":53},[51,15576,9024],{"stretchy":53},[51,15578,945],{"fence":53,"stretchy":122,"minsize":7407,"maxsize":7407},[48,15580,112],{"mathvariant":111},[114,15582,15583],{"encoding":116},"R(\\delta)-R^*\n=\\mathbb E\\Big[|2\\eta(X)-1|\\,\\mathbf 1\\{\\delta(X)\\ne\\delta^*(X)\\}\\Big].",[23,15585,15587,15614,15658,15698,15771],{"className":15586,"ariaHidden":122},[121],[23,15588,15590,15593,15596,15599,15602,15605,15608,15611],{"className":15589},[126],[23,15591],{"className":15592,"style":131},[130],[23,15594,572],{"className":15595,"style":963},[135,136],[23,15597,54],{"className":15598},[140],[23,15600,736],{"className":15601,"style":783},[135,136],[23,15603,66],{"className":15604},[169],[23,15606],{"className":15607,"style":392},[148],[23,15609,4529],{"className":15610},[612],[23,15612],{"className":15613,"style":392},[148],[23,15615,15617,15620,15649,15652,15655],{"className":15616},[126],[23,15618],{"className":15619,"style":7047},[130],[23,15621,15623,15626],{"className":15622},[135],[23,15624,572],{"className":15625,"style":963},[135,136],[23,15627,15629],{"className":15628},[998],[23,15630,15632],{"className":15631},[199],[23,15633,15635],{"className":15634},[204],[23,15636,15638],{"className":15637,"style":7047},[208],[23,15639,15640,15643],{"style":4989},[23,15641],{"className":15642,"style":1015},[216],[23,15644,15646],{"className":15645},[1019,1020,1021,1022],[23,15647,6883],{"className":15648},[612,1022],[23,15650],{"className":15651,"style":149},[148],[23,15653,69],{"className":15654},[153],[23,15656],{"className":15657,"style":149},[148],[23,15659,15661,15664,15667,15673,15677,15680,15683,15686,15689,15692,15695],{"className":15660},[126],[23,15662],{"className":15663,"style":7815},[130],[23,15665,902],{"className":15666},[135,644],[23,15668,15670],{"className":15669},[135],[23,15671,924],{"className":15672},[1066,7585],[23,15674,15676],{"className":15675},[135],"∣2",[23,15678,9769],{"className":15679,"style":676},[135,136],[23,15681,54],{"className":15682},[140],[23,15684,434],{"className":15685,"style":454},[135,136],[23,15687,66],{"className":15688},[169],[23,15690],{"className":15691,"style":392},[148],[23,15693,4529],{"className":15694},[612],[23,15696],{"className":15697,"style":392},[148],[23,15699,15701,15704,15708,15711,15714,15717,15720,15723,15726,15729,15732,15768],{"className":15700},[126],[23,15702],{"className":15703,"style":131},[130],[23,15705,15707],{"className":15706},[135],"1∣",[23,15709],{"className":15710,"style":280},[148],[23,15712,4435],{"className":15713},[135,9078],[23,15715,9014],{"className":15716},[140],[23,15718,736],{"className":15719,"style":783},[135,136],[23,15721,54],{"className":15722},[140],[23,15724,434],{"className":15725,"style":454},[135,136],[23,15727,66],{"className":15728},[169],[23,15730],{"className":15731,"style":149},[148],[23,15733,15735,15762,15765],{"className":15734},[153],[23,15736,15738],{"className":15737},[153],[23,15739,15741],{"className":15740},[135,9097],[23,15742,15744],{"className":15743},[9101],[23,15745,15747,15750,15759],{"className":15746},[9105],[23,15748],{"className":15749,"style":5414},[130],[23,15751,15753],{"className":15752},[9112],[23,15754,15756],{"className":15755},[135],[23,15757,9119],{"className":15758},[153],[23,15760],{"className":15761},[9123],[23,15763],{"className":15764},[148,9127],[23,15766,69],{"className":15767},[153],[23,15769],{"className":15770,"style":149},[148],[23,15772,15774,15777,15806,15809,15812,15815,15821],{"className":15773},[126],[23,15775],{"className":15776,"style":7815},[130],[23,15778,15780,15783],{"className":15779},[135],[23,15781,736],{"className":15782,"style":783},[135,136],[23,15784,15786],{"className":15785},[998],[23,15787,15789],{"className":15788},[199],[23,15790,15792],{"className":15791},[204],[23,15793,15795],{"className":15794,"style":7047},[208],[23,15796,15797,15800],{"style":4989},[23,15798],{"className":15799,"style":1015},[216],[23,15801,15803],{"className":15802},[1019,1020,1021,1022],[23,15804,6883],{"className":15805},[612,1022],[23,15807,54],{"className":15808},[140],[23,15810,434],{"className":15811,"style":454},[135,136],[23,15813,15224],{"className":15814},[169],[23,15816,15818],{"className":15817},[135],[23,15819,945],{"className":15820},[1066,7585],[23,15822,112],{"className":15823},[135],[11,15825,15826,15828,15829,15857,15858,15969,15970,15998,15999,16080,16081,16162],{},[18,15827,7679],{}," 对固定的 ",[23,15830,15832,15845],{"className":15831,"translate":27},[31],[23,15833,15835],{"className":15834},[35],[37,15836,15837],{"xmlns":39},[42,15838,15839,15843],{},[45,15840,15841],{},[48,15842,753],{},[114,15844,753],{"encoding":116},[23,15846,15848],{"className":15847,"ariaHidden":122},[121],[23,15849,15851,15854],{"className":15850},[126],[23,15852],{"className":15853,"style":703},[130],[23,15855,753],{"className":15856},[135,136],"，若 ",[23,15859,15861,15895],{"className":15860,"translate":27},[31],[23,15862,15864],{"className":15863},[35],[37,15865,15866],{"xmlns":39},[42,15867,15868,15892],{},[45,15869,15870,15872,15874,15876,15878,15880,15886,15888,15890],{},[48,15871,736],{},[51,15873,54],{"stretchy":53},[48,15875,753],{},[51,15877,66],{"stretchy":53},[51,15879,69],{},[4518,15881,15882,15884],{},[48,15883,736],{},[51,15885,6883],{},[51,15887,54],{"stretchy":53},[48,15889,753],{},[51,15891,66],{"stretchy":53},[114,15893,15894],{"encoding":116},"\\delta(x)=\\delta^*(x)",[23,15896,15898,15925],{"className":15897,"ariaHidden":122},[121],[23,15899,15901,15904,15907,15910,15913,15916,15919,15922],{"className":15900},[126],[23,15902],{"className":15903,"style":131},[130],[23,15905,736],{"className":15906,"style":783},[135,136],[23,15908,54],{"className":15909},[140],[23,15911,753],{"className":15912},[135,136],[23,15914,66],{"className":15915},[169],[23,15917],{"className":15918,"style":149},[148],[23,15920,69],{"className":15921},[153],[23,15923],{"className":15924,"style":149},[148],[23,15926,15928,15931,15960,15963,15966],{"className":15927},[126],[23,15929],{"className":15930,"style":131},[130],[23,15932,15934,15937],{"className":15933},[135],[23,15935,736],{"className":15936,"style":783},[135,136],[23,15938,15940],{"className":15939},[998],[23,15941,15943],{"className":15942},[199],[23,15944,15946],{"className":15945},[204],[23,15947,15949],{"className":15948,"style":6914},[208],[23,15950,15951,15954],{"style":4696},[23,15952],{"className":15953,"style":1015},[216],[23,15955,15957],{"className":15956},[1019,1020,1021,1022],[23,15958,6883],{"className":15959},[612,1022],[23,15961,54],{"className":15962},[140],[23,15964,753],{"className":15965},[135,136],[23,15967,66],{"className":15968},[169],"，逐点遗憾为 ",[23,15971,15973,15986],{"className":15972,"translate":27},[31],[23,15974,15976],{"className":15975},[35],[37,15977,15978],{"xmlns":39},[42,15979,15980,15984],{},[45,15981,15982],{},[2581,15983,1532],{},[114,15985,1532],{"encoding":116},[23,15987,15989],{"className":15988,"ariaHidden":122},[121],[23,15990,15992,15995],{"className":15991},[126],[23,15993],{"className":15994,"style":2770},[130],[23,15996,1532],{"className":15997},[135],"；若不等，则所选行动的后验概率是 ",[23,16000,16002,16032],{"className":16001,"translate":27},[31],[23,16003,16005],{"className":16004},[35],[37,16006,16007],{"xmlns":39},[42,16008,16009,16029],{},[45,16010,16011,16013,16015,16017,16019,16021,16023,16025,16027],{},[48,16012,9555],{},[51,16014,6961],{},[51,16016,9014],{"stretchy":53},[48,16018,9769],{},[51,16020,747],{"separator":122},[2581,16022,4435],{},[51,16024,4529],{},[48,16026,9769],{},[51,16028,9024],{"stretchy":53},[114,16030,16031],{"encoding":116},"\\max\\{\\eta,1-\\eta\\}",[23,16033,16035,16068],{"className":16034,"ariaHidden":122},[121],[23,16036,16038,16041,16044,16047,16050,16053,16056,16059,16062,16065],{"className":16037},[126],[23,16039],{"className":16040,"style":131},[130],[23,16042,9555],{"className":16043},[1786],[23,16045,9014],{"className":16046},[140],[23,16048,9769],{"className":16049,"style":676},[135,136],[23,16051,747],{"className":16052},[824],[23,16054],{"className":16055,"style":280},[148],[23,16057,4435],{"className":16058},[135],[23,16060],{"className":16061,"style":392},[148],[23,16063,4529],{"className":16064},[612],[23,16066],{"className":16067,"style":392},[148],[23,16069,16071,16074,16077],{"className":16070},[126],[23,16072],{"className":16073,"style":131},[130],[23,16075,9769],{"className":16076,"style":676},[135,136],[23,16078,9024],{"className":16079},[169],"，最优行动的后验概率是 ",[23,16082,16084,16114],{"className":16083,"translate":27},[31],[23,16085,16087],{"className":16086},[35],[37,16088,16089],{"xmlns":39},[42,16090,16091,16111],{},[45,16092,16093,16095,16097,16099,16101,16103,16105,16107,16109],{},[48,16094,6969],{},[51,16096,6961],{},[51,16098,9014],{"stretchy":53},[48,16100,9769],{},[51,16102,747],{"separator":122},[2581,16104,4435],{},[51,16106,4529],{},[48,16108,9769],{},[51,16110,9024],{"stretchy":53},[114,16112,16113],{"encoding":116},"\\min\\{\\eta,1-\\eta\\}",[23,16115,16117,16150],{"className":16116,"ariaHidden":122},[121],[23,16118,16120,16123,16126,16129,16132,16135,16138,16141,16144,16147],{"className":16119},[126],[23,16121],{"className":16122,"style":131},[130],[23,16124,6969],{"className":16125},[1786],[23,16127,9014],{"className":16128},[140],[23,16130,9769],{"className":16131,"style":676},[135,136],[23,16133,747],{"className":16134},[824],[23,16136],{"className":16137,"style":280},[148],[23,16139,4435],{"className":16140},[135],[23,16142],{"className":16143,"style":392},[148],[23,16145,4529],{"className":16146},[612],[23,16148],{"className":16149,"style":392},[148],[23,16151,16153,16156,16159],{"className":16152},[126],[23,16154],{"className":16155,"style":131},[130],[23,16157,9769],{"className":16158,"style":676},[135,136],[23,16160,9024],{"className":16161},[169],"，两者之差为",[23,16164,16166],{"className":16165,"translate":27},[26],[23,16167,16169,16235],{"className":16168,"translate":27},[31],[23,16170,16172],{"className":16171},[35],[37,16173,16174],{"xmlns":39,"display":40},[42,16175,16176,16232],{},[45,16177,16178,16180,16182,16184,16186,16188,16190,16192,16194,16196,16198,16200,16202,16204,16206,16208,16210,16212,16214,16216,16218,16220,16222,16224,16226,16228,16230],{},[48,16179,9555],{},[51,16181,6961],{},[51,16183,9014],{"stretchy":53},[48,16185,9769],{},[51,16187,747],{"separator":122},[2581,16189,4435],{},[51,16191,4529],{},[48,16193,9769],{},[51,16195,9024],{"stretchy":53},[51,16197,4529],{},[48,16199,6969],{},[51,16201,6961],{},[51,16203,9014],{"stretchy":53},[48,16205,9769],{},[51,16207,747],{"separator":122},[2581,16209,4435],{},[51,16211,4529],{},[48,16213,9769],{},[51,16215,9024],{"stretchy":53},[51,16217,69],{},[48,16219,60],{"mathvariant":111},[2581,16221,9956],{},[48,16223,9769],{},[51,16225,4529],{},[2581,16227,4435],{},[48,16229,60],{"mathvariant":111},[48,16231,112],{"mathvariant":111},[114,16233,16234],{"encoding":116},"\\max\\{\\eta,1-\\eta\\}-\\min\\{\\eta,1-\\eta\\}=|2\\eta-1|.",[23,16236,16238,16271,16292,16325,16346,16367],{"className":16237,"ariaHidden":122},[121],[23,16239,16241,16244,16247,16250,16253,16256,16259,16262,16265,16268],{"className":16240},[126],[23,16242],{"className":16243,"style":131},[130],[23,16245,9555],{"className":16246},[1786],[23,16248,9014],{"className":16249},[140],[23,16251,9769],{"className":16252,"style":676},[135,136],[23,16254,747],{"className":16255},[824],[23,16257],{"className":16258,"style":280},[148],[23,16260,4435],{"className":16261},[135],[23,16263],{"className":16264,"style":392},[148],[23,16266,4529],{"className":16267},[612],[23,16269],{"className":16270,"style":392},[148],[23,16272,16274,16277,16280,16283,16286,16289],{"className":16273},[126],[23,16275],{"className":16276,"style":131},[130],[23,16278,9769],{"className":16279,"style":676},[135,136],[23,16281,9024],{"className":16282},[169],[23,16284],{"className":16285,"style":392},[148],[23,16287,4529],{"className":16288},[612],[23,16290],{"className":16291,"style":392},[148],[23,16293,16295,16298,16301,16304,16307,16310,16313,16316,16319,16322],{"className":16294},[126],[23,16296],{"className":16297,"style":131},[130],[23,16299,6969],{"className":16300},[1786],[23,16302,9014],{"className":16303},[140],[23,16305,9769],{"className":16306,"style":676},[135,136],[23,16308,747],{"className":16309},[824],[23,16311],{"className":16312,"style":280},[148],[23,16314,4435],{"className":16315},[135],[23,16317],{"className":16318,"style":392},[148],[23,16320,4529],{"className":16321},[612],[23,16323],{"className":16324,"style":392},[148],[23,16326,16328,16331,16334,16337,16340,16343],{"className":16327},[126],[23,16329],{"className":16330,"style":131},[130],[23,16332,9769],{"className":16333,"style":676},[135,136],[23,16335,9024],{"className":16336},[169],[23,16338],{"className":16339,"style":149},[148],[23,16341,69],{"className":16342},[153],[23,16344],{"className":16345,"style":149},[148],[23,16347,16349,16352,16355,16358,16361,16364],{"className":16348},[126],[23,16350],{"className":16351,"style":131},[130],[23,16353,15676],{"className":16354},[135],[23,16356,9769],{"className":16357,"style":676},[135,136],[23,16359],{"className":16360,"style":392},[148],[23,16362,4529],{"className":16363},[612],[23,16365],{"className":16366,"style":392},[148],[23,16368,16370,16373],{"className":16369},[126],[23,16371],{"className":16372,"style":131},[130],[23,16374,16376],{"className":16375},[135],"1∣.",[11,16378,16379,16380,16408,16409],{},"对 ",[23,16381,16383,16396],{"className":16382,"translate":27},[31],[23,16384,16386],{"className":16385},[35],[37,16387,16388],{"xmlns":39},[42,16389,16390,16394],{},[45,16391,16392],{},[48,16393,434],{},[114,16395,434],{"encoding":116},[23,16397,16399],{"className":16398,"ariaHidden":122},[121],[23,16400,16402,16405],{"className":16401},[126],[23,16403],{"className":16404,"style":470},[130],[23,16406,434],{"className":16407,"style":454},[135,136]," 取期望即得。",[23,16410,16412,16425],{"className":16411,"translate":27},[31],[23,16413,16415],{"className":16414},[35],[37,16416,16417],{"xmlns":39},[42,16418,16419,16423],{},[45,16420,16421],{},[48,16422,8251],{"mathvariant":111},[114,16424,8254],{"encoding":116},[23,16426,16428],{"className":16427,"ariaHidden":122},[121],[23,16429,16431,16434],{"className":16430},[126],[23,16432],{"className":16433,"style":8264},[130],[23,16435,8251],{"className":16436},[135,8268],[11,16438,16439,16440,16515,16516,16548,16549,16577,16578,16606,16607,16635,16636,1471],{},"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 ",[23,16441,16443,16473],{"className":16442,"translate":27},[31],[23,16444,16446],{"className":16445},[35],[37,16447,16448],{"xmlns":39},[42,16449,16450,16470],{},[45,16451,16452,16454,16456,16458,16460,16462,16464,16466,16468],{},[48,16453,60],{"mathvariant":111},[2581,16455,9956],{},[48,16457,9769],{},[51,16459,54],{"stretchy":53},[48,16461,434],{},[51,16463,66],{"stretchy":53},[51,16465,4529],{},[2581,16467,4435],{},[48,16469,60],{"mathvariant":111},[114,16471,16472],{"encoding":116},"|2\\eta(X)-1|",[23,16474,16476,16506],{"className":16475,"ariaHidden":122},[121],[23,16477,16479,16482,16485,16488,16491,16494,16497,16500,16503],{"className":16478},[126],[23,16480],{"className":16481,"style":131},[130],[23,16483,15676],{"className":16484},[135],[23,16486,9769],{"className":16487,"style":676},[135,136],[23,16489,54],{"className":16490},[140],[23,16492,434],{"className":16493,"style":454},[135,136],[23,16495,66],{"className":16496},[169],[23,16498],{"className":16499,"style":392},[148],[23,16501,4529],{"className":16502},[612],[23,16504],{"className":16505,"style":392},[148],[23,16507,16509,16512],{"className":16508},[126],[23,16510],{"className":16511,"style":131},[130],[23,16513,15707],{"className":16514},[135],"。后验接近 ",[23,16517,16519,16536],{"className":16518,"translate":27},[31],[23,16520,16522],{"className":16521},[35],[37,16523,16524],{"xmlns":39},[42,16525,16526,16534],{},[45,16527,16528,16530,16532],{},[2581,16529,4435],{},[48,16531,2875],{"mathvariant":111},[2581,16533,9956],{},[114,16535,10473],{"encoding":116},[23,16537,16539],{"className":16538,"ariaHidden":122},[121],[23,16540,16542,16545],{"className":16541},[126],[23,16543],{"className":16544,"style":131},[130],[23,16546,10473],{"className":16547},[135]," 的样本位于边界附近，即使分错，代价也小；后验接近 ",[23,16550,16552,16565],{"className":16551,"translate":27},[31],[23,16553,16555],{"className":16554},[35],[37,16556,16557],{"xmlns":39},[42,16558,16559,16563],{},[45,16560,16561],{},[2581,16562,1532],{},[114,16564,1532],{"encoding":116},[23,16566,16568],{"className":16567,"ariaHidden":122},[121],[23,16569,16571,16574],{"className":16570},[126],[23,16572],{"className":16573,"style":2770},[130],[23,16575,1532],{"className":16576},[135]," 或 ",[23,16579,16581,16594],{"className":16580,"translate":27},[31],[23,16582,16584],{"className":16583},[35],[37,16585,16586],{"xmlns":39},[42,16587,16588,16592],{},[45,16589,16590],{},[2581,16591,4435],{},[114,16593,4435],{"encoding":116},[23,16595,16597],{"className":16596,"ariaHidden":122},[121],[23,16598,16600,16603],{"className":16599},[126],[23,16601],{"className":16602,"style":2770},[130],[23,16604,4435],{"className":16605},[135]," 的样本一旦分错，代价就接近 ",[23,16608,16610,16623],{"className":16609,"translate":27},[31],[23,16611,16613],{"className":16612},[35],[37,16614,16615],{"xmlns":39},[42,16616,16617,16621],{},[45,16618,16619],{},[2581,16620,4435],{},[114,16622,4435],{"encoding":116},[23,16624,16626],{"className":16625,"ariaHidden":122},[121],[23,16627,16629,16632],{"className":16628},[126],[23,16630],{"className":16631,"style":2770},[130],[23,16633,4435],{"className":16634},[135],"。换句话说，",[18,16637,16638],{},"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",[11,16640,16641,16642,16797,16798,8059],{},"回归情形也有类似的恒等式。设 ",[23,16643,16645,16687],{"className":16644,"translate":27},[31],[23,16646,16648],{"className":16647},[35],[37,16649,16650],{"xmlns":39},[42,16651,16652,16684],{},[45,16653,16654,16660,16662,16664,16666,16668,16670,16672,16674,16676,16678,16680,16682],{},[4518,16655,16656,16658],{},[48,16657,8340],{},[51,16659,6883],{},[51,16661,54],{"stretchy":53},[48,16663,753],{},[51,16665,66],{"stretchy":53},[51,16667,69],{},[48,16669,902],{"mathvariant":571},[51,16671,924],{"stretchy":53},[48,16673,369],{},[51,16675,60],{},[48,16677,434],{},[51,16679,69],{},[48,16681,753],{},[51,16683,945],{"stretchy":53},[114,16685,16686],{"encoding":116},"f^*(x)=\\mathbb E[Y\\mid X=x]",[23,16688,16690,16743,16767,16785],{"className":16689,"ariaHidden":122},[121],[23,16691,16693,16696,16725,16728,16731,16734,16737,16740],{"className":16692},[126],[23,16694],{"className":16695,"style":131},[130],[23,16697,16699,16702],{"className":16698},[135],[23,16700,8340],{"className":16701,"style":11402},[135,136],[23,16703,16705],{"className":16704},[998],[23,16706,16708],{"className":16707},[199],[23,16709,16711],{"className":16710},[204],[23,16712,16714],{"className":16713,"style":6914},[208],[23,16715,16716,16719],{"style":4696},[23,16717],{"className":16718,"style":1015},[216],[23,16720,16722],{"className":16721},[1019,1020,1021,1022],[23,16723,6883],{"className":16724},[612,1022],[23,16726,54],{"className":16727},[140],[23,16729,753],{"className":16730},[135,136],[23,16732,66],{"className":16733},[169],[23,16735],{"className":16736,"style":149},[148],[23,16738,69],{"className":16739},[153],[23,16741],{"className":16742,"style":149},[148],[23,16744,16746,16749,16752,16755,16758,16761,16764],{"className":16745},[126],[23,16747],{"className":16748,"style":131},[130],[23,16750,902],{"className":16751},[135,644],[23,16753,924],{"className":16754},[140],[23,16756,369],{"className":16757,"style":392},[135,136],[23,16759],{"className":16760,"style":149},[148],[23,16762,60],{"className":16763},[153],[23,16765],{"className":16766,"style":149},[148],[23,16768,16770,16773,16776,16779,16782],{"className":16769},[126],[23,16771],{"className":16772,"style":470},[130],[23,16774,434],{"className":16775,"style":454},[135,136],[23,16777],{"className":16778,"style":149},[148],[23,16780,69],{"className":16781},[153],[23,16783],{"className":16784,"style":149},[148],[23,16786,16788,16791,16794],{"className":16787},[126],[23,16789],{"className":16790,"style":131},[130],[23,16792,753],{"className":16793},[135,136],[23,16795,945],{"className":16796},[169],"，则对任意 ",[23,16799,16801,16821],{"className":16800,"translate":27},[31],[23,16802,16804],{"className":16803},[35],[37,16805,16806],{"xmlns":39},[42,16807,16808,16818],{},[45,16809,16810],{},[16811,16812,16813,16815],"mover",{"accent":122},[48,16814,8340],{},[51,16816,16817],{},"^",[114,16819,16820],{"encoding":116},"\\hat f",[23,16822,16824],{"className":16823,"ariaHidden":122},[121],[23,16825,16827,16831],{"className":16826},[126],[23,16828],{"className":16829,"style":16830},[130],"height:1.1523em;vertical-align:-0.1944em;",[23,16832,16835],{"className":16833},[135,16834],"accent",[23,16836,16838,16870],{"className":16837},[199,200],[23,16839,16841,16867],{"className":16840},[204],[23,16842,16845,16853],{"className":16843,"style":16844},[208],"height:0.9579em;",[23,16846,16847,16850],{"style":7132},[23,16848],{"className":16849,"style":217},[216],[23,16851,8340],{"className":16852,"style":11402},[135,136],[23,16854,16856,16859],{"style":16855},"top:-3.2634em;",[23,16857],{"className":16858,"style":217},[216],[23,16860,16864],{"className":16861,"style":16863},[16862],"accent-body","left:-0.0833em;",[23,16865,16817],{"className":16866},[135],[23,16868,297],{"className":16869},[296],[23,16871,16873],{"className":16872},[204],[23,16874,16877],{"className":16875,"style":16876},[208],"height:0.1944em;",[23,16878],{},[23,16880,16882],{"className":16881,"translate":27},[26],[23,16883,16885,16961],{"className":16884,"translate":27},[31],[23,16886,16888],{"className":16887},[35],[37,16889,16890],{"xmlns":39,"display":40},[42,16891,16892,16958],{},[45,16893,16894,16896,16898,16904,16906,16908,16914,16916,16918,16920,16922,16928,16930,16932,16934,16936,16942,16944,16946,16948,16954,16956],{},[48,16895,572],{},[51,16897,54],{"stretchy":53},[16811,16899,16900,16902],{"accent":122},[48,16901,8340],{},[51,16903,16817],{},[51,16905,66],{"stretchy":53},[51,16907,4529],{},[4518,16909,16910,16912],{},[48,16911,572],{},[51,16913,6883],{},[51,16915,69],{},[48,16917,902],{"mathvariant":571},[51,16919,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,16921,54],{"stretchy":53},[16811,16923,16924,16926],{"accent":122},[48,16925,8340],{},[51,16927,16817],{},[51,16929,54],{"stretchy":53},[48,16931,434],{},[51,16933,66],{"stretchy":53},[51,16935,4529],{},[4518,16937,16938,16940],{},[48,16939,8340],{},[51,16941,6883],{},[51,16943,54],{"stretchy":53},[48,16945,434],{},[51,16947,66],{"stretchy":53},[4518,16949,16950,16952],{},[51,16951,66],{"stretchy":53},[2581,16953,9956],{},[51,16955,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,16957,112],{"mathvariant":111},[114,16959,16960],{"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}",[23,17461,17463],{"className":17462,"ariaHidden":122},[121],[23,17464,17466,17470],{"className":17465},[126],[23,17467],{"className":17468,"style":17469},[130],"height:2.9358em;vertical-align:-1.2179em;",[23,17471,17473],{"className":17472},[135],[23,17474,17476,17520],{"className":17475},[1518],[23,17477,17479],{"className":17478},[1696],[23,17480,17482,17511],{"className":17481},[199,200],[23,17483,17485,17508],{"className":17484},[204],[23,17486,17489,17499],{"className":17487,"style":17488},[208],"height:1.7179em;",[23,17490,17492,17496],{"style":17491},"top:-3.7179em;",[23,17493],{"className":17494,"style":17495},[216],"height:2.9579em;",[23,17497],{"className":17498},[135],[23,17500,17502,17505],{"style":17501},"top:-2.1em;",[23,17503],{"className":17504,"style":17495},[216],[23,17506],{"className":17507},[135],[23,17509,297],{"className":17510},[296],[23,17512,17514],{"className":17513},[204],[23,17515,17518],{"className":17516,"style":17517},[208],"height:1.2179em;",[23,17519],{},[23,17521,17523],{"className":17522},[1753],[23,17524,17526,18116],{"className":17525},[199,200],[23,17527,17529,18113],{"className":17528},[204],[23,17530,17532,17760],{"className":17531,"style":17488},[208],[23,17533,17535,17538],{"style":17534},"top:-3.76em;",[23,17536],{"className":17537,"style":217},[216],[23,17539,17541,17544,17547,17553,17556,17559,17562,17565,17568,17610,17613,17616,17619,17648,17654,17657,17660,17663,17666,17672,17675,17678,17681,17684,17687,17716,17719,17722,17725,17754],{"className":17540},[135],[23,17542],{"className":17543},[135],[23,17545,902],{"className":17546},[135,644],[23,17548,17550],{"className":17549},[135],[23,17551,924],{"className":17552},[1066,1067],[23,17554,54],{"className":17555},[140],[23,17557,369],{"className":17558,"style":392},[135,136],[23,17560],{"className":17561,"style":392},[148],[23,17563,4529],{"className":17564},[612],[23,17566],{"className":17567,"style":392},[148],[23,17569,17571],{"className":17570},[135,16834],[23,17572,17574,17602],{"className":17573},[199,200],[23,17575,17577,17599],{"className":17576},[204],[23,17578,17580,17588],{"className":17579,"style":16844},[208],[23,17581,17582,17585],{"style":7132},[23,17583],{"className":17584,"style":217},[216],[23,17586,8340],{"className":17587,"style":11402},[135,136],[23,17589,17590,17593],{"style":16855},[23,17591],{"className":17592,"style":217},[216],[23,17594,17596],{"className":17595,"style":16863},[16862],[23,17597,16817],{"className":17598},[135],[23,17600,297],{"className":17601},[296],[23,17603,17605],{"className":17604},[204],[23,17606,17608],{"className":17607,"style":16876},[208],[23,17609],{},[23,17611,54],{"className":17612},[140],[23,17614,434],{"className":17615,"style":454},[135,136],[23,17617,66],{"className":17618},[169],[23,17620,17622,17625],{"className":17621},[169],[23,17623,66],{"className":17624},[169],[23,17626,17628],{"className":17627},[998],[23,17629,17631],{"className":17630},[199],[23,17632,17634],{"className":17633},[204],[23,17635,17637],{"className":17636,"style":11006},[208],[23,17638,17639,17642],{"style":4989},[23,17640],{"className":17641,"style":1015},[216],[23,17643,17645],{"className":17644},[1019,1020,1021,1022],[23,17646,9956],{"className":17647},[135,1022],[23,17649,17651],{"className":17650},[135],[23,17652,945],{"className":17653},[1066,1067],[23,17655],{"className":17656,"style":392},[148],[23,17658,4529],{"className":17659},[612],[23,17661],{"className":17662,"style":392},[148],[23,17664,902],{"className":17665},[135,644],[23,17667,17669],{"className":17668},[135],[23,17670,924],{"className":17671},[1066,1067],[23,17673,54],{"className":17674},[140],[23,17676,369],{"className":17677,"style":392},[135,136],[23,17679],{"className":17680,"style":392},[148],[23,17682,4529],{"className":17683},[612],[23,17685],{"className":17686,"style":392},[148],[23,17688,17690,17693],{"className":17689},[135],[23,17691,8340],{"className":17692,"style":11402},[135,136],[23,17694,17696],{"className":17695},[998],[23,17697,17699],{"className":17698},[199],[23,17700,17702],{"className":17701},[204],[23,17703,17705],{"className":17704,"style":7047},[208],[23,17706,17707,17710],{"style":4989},[23,17708],{"className":17709,"style":1015},[216],[23,17711,17713],{"className":17712},[1019,1020,1021,1022],[23,17714,6883],{"className":17715},[612,1022],[23,17717,54],{"className":17718},[140],[23,17720,434],{"className":17721,"style":454},[135,136],[23,17723,66],{"className":17724},[169],[23,17726,17728,17731],{"className":17727},[169],[23,17729,66],{"className":17730},[169],[23,17732,17734],{"className":17733},[998],[23,17735,17737],{"className":17736},[199],[23,17738,17740],{"className":17739},[204],[23,17741,17743],{"className":17742,"style":11006},[208],[23,17744,17745,17748],{"style":4989},[23,17746],{"className":17747,"style":1015},[216],[23,17749,17751],{"className":17750},[1019,1020,1021,1022],[23,17752,9956],{"className":17753},[135,1022],[23,17755,17757],{"className":17756},[135],[23,17758,945],{"className":17759},[1066,1067],[23,17761,17763,17766],{"style":17762},"top:-2.1421em;",[23,17764],{"className":17765,"style":217},[216],[23,17767,17769,17772,17775,17778,17781,17784,17790,17793,17835,17838,17841,17844,17847,17850,17853,17882,17885,17888,17891,17920,17926,17929,17932,17935,17938,17941,17944,17950,17953,17956,17959,17962,17965,17994,17997,18000,18003,18006,18035,18038,18041,18044,18047,18050,18053,18095,18098,18101,18104,18110],{"className":17768},[135],[23,17770],{"className":17771},[135],[23,17773],{"className":17774,"style":149},[148],[23,17776,69],{"className":17777},[153],[23,17779],{"className":17780,"style":149},[148],[23,17782,902],{"className":17783},[135,644],[23,17785,17787],{"className":17786},[135],[23,17788,924],{"className":17789},[1066,1067],[23,17791,54],{"className":17792},[140],[23,17794,17796],{"className":17795},[135,16834],[23,17797,17799,17827],{"className":17798},[199,200],[23,17800,17802,17824],{"className":17801},[204],[23,17803,17805,17813],{"className":17804,"style":16844},[208],[23,17806,17807,17810],{"style":7132},[23,17808],{"className":17809,"style":217},[216],[23,17811,8340],{"className":17812,"style":11402},[135,136],[23,17814,17815,17818],{"style":16855},[23,17816],{"className":17817,"style":217},[216],[23,17819,17821],{"className":17820,"style":16863},[16862],[23,17822,16817],{"className":17823},[135],[23,17825,297],{"className":17826},[296],[23,17828,17830],{"className":17829},[204],[23,17831,17833],{"className":17832,"style":16876},[208],[23,17834],{},[23,17836,54],{"className":17837},[140],[23,17839,434],{"className":17840,"style":454},[135,136],[23,17842,66],{"className":17843},[169],[23,17845],{"className":17846,"style":392},[148],[23,17848,4529],{"className":17849},[612],[23,17851],{"className":17852,"style":392},[148],[23,17854,17856,17859],{"className":17855},[135],[23,17857,8340],{"className":17858,"style":11402},[135,136],[23,17860,17862],{"className":17861},[998],[23,17863,17865],{"className":17864},[199],[23,17866,17868],{"className":17867},[204],[23,17869,17871],{"className":17870,"style":7047},[208],[23,17872,17873,17876],{"style":4989},[23,17874],{"className":17875,"style":1015},[216],[23,17877,17879],{"className":17878},[1019,1020,1021,1022],[23,17880,6883],{"className":17881},[612,1022],[23,17883,54],{"className":17884},[140],[23,17886,434],{"className":17887,"style":454},[135,136],[23,17889,66],{"className":17890},[169],[23,17892,17894,17897],{"className":17893},[169],[23,17895,66],{"className":17896},[169],[23,17898,17900],{"className":17899},[998],[23,17901,17903],{"className":17902},[199],[23,17904,17906],{"className":17905},[204],[23,17907,17909],{"className":17908,"style":11006},[208],[23,17910,17911,17914],{"style":4989},[23,17912],{"className":17913,"style":1015},[216],[23,17915,17917],{"className":17916},[1019,1020,1021,1022],[23,17918,9956],{"className":17919},[135,1022],[23,17921,17923],{"className":17922},[135],[23,17924,945],{"className":17925},[1066,1067],[23,17927],{"className":17928,"style":392},[148],[23,17930,5106],{"className":17931},[612],[23,17933],{"className":17934,"style":392},[148],[23,17936,9956],{"className":17937},[135],[23,17939],{"className":17940,"style":280},[148],[23,17942,902],{"className":17943},[135,644],[23,17945,17947],{"className":17946},[135],[23,17948,924],{"className":17949},[1066,1067],[23,17951,54],{"className":17952},[140],[23,17954,369],{"className":17955,"style":392},[135,136],[23,17957],{"className":17958,"style":392},[148],[23,17960,4529],{"className":17961},[612],[23,17963],{"className":17964,"style":392},[148],[23,17966,17968,17971],{"className":17967},[135],[23,17969,8340],{"className":17970,"style":11402},[135,136],[23,17972,17974],{"className":17973},[998],[23,17975,17977],{"className":17976},[199],[23,17978,17980],{"className":17979},[204],[23,17981,17983],{"className":17982,"style":7047},[208],[23,17984,17985,17988],{"style":4989},[23,17986],{"className":17987,"style":1015},[216],[23,17989,17991],{"className":17990},[1019,1020,1021,1022],[23,17992,6883],{"className":17993},[612,1022],[23,17995,54],{"className":17996},[140],[23,17998,434],{"className":17999,"style":454},[135,136],[23,18001,1095],{"className":18002},[169],[23,18004,54],{"className":18005},[140],[23,18007,18009,18012],{"className":18008},[135],[23,18010,8340],{"className":18011,"style":11402},[135,136],[23,18013,18015],{"className":18014},[998],[23,18016,18018],{"className":18017},[199],[23,18019,18021],{"className":18020},[204],[23,18022,18024],{"className":18023,"style":7047},[208],[23,18025,18026,18029],{"style":4989},[23,18027],{"className":18028,"style":1015},[216],[23,18030,18032],{"className":18031},[1019,1020,1021,1022],[23,18033,6883],{"className":18034},[612,1022],[23,18036,54],{"className":18037},[140],[23,18039,434],{"className":18040,"style":454},[135,136],[23,18042,66],{"className":18043},[169],[23,18045],{"className":18046,"style":392},[148],[23,18048,4529],{"className"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",[23,18128,18130,18143],{"className":18129,"translate":27},[31],[23,18131,18133],{"className":18132},[35],[37,18134,18135],{"xmlns":39},[42,18136,18137,18141],{},[45,18138,18139],{},[48,18140,434],{},[114,18142,434],{"encoding":116},[23,18144,18146],{"className":18145,"ariaHidden":122},[121],[23,18147,18149,18152],{"className":18148},[126],[23,18150],{"className":18151,"style":470},[130],[23,18153,434],{"className":18154,"style":454},[135,136]," 取条件期望后，",[23,18157,18159,18201],{"className":18158,"translate":27},[31],[23,18160,18162],{"className":18161},[35],[37,18163,18164],{"xmlns":39},[42,18165,18166,18198],{},[45,18167,18168,18170,18172,18174,18176,18182,18184,18186,18188,18190,18192,18194,18196],{},[48,18169,902],{"mathvariant":571},[51,18171,924],{"stretchy":53},[48,18173,369],{},[51,18175,4529],{},[4518,18177,18178,18180],{},[48,18179,8340],{},[51,18181,6883],{},[51,18183,54],{"stretchy":53},[48,18185,434],{},[51,18187,66],{"stretchy":53},[51,18189,60],{},[48,18191,434],{},[51,18193,945],{"stretchy":53},[51,18195,69],{},[2581,18197,1532],{},[114,18199,18200],{"encoding":116},"\\mathbb E[Y-f^*(X)\\mid X]=0",[23,18202,18204,18228,18281,18302],{"className":18203,"ariaHidden":122},[121],[23,18205,18207,18210,18213,18216,18219,18222,18225],{"className":18206},[126],[23,18208],{"className":18209,"style":131},[130],[23,18211,902],{"className":18212},[135,644],[23,18214,924],{"className":18215},[140],[23,18217,369],{"className":18218,"style":392},[135,136],[23,18220],{"className":18221,"style":392},[148],[23,18223,4529],{"className":18224},[612],[23,18226],{"className":18227,"style":392},[148],[23,18229,18231,18234,18263,18266,18269,18272,18275,18278],{"className":18230},[126],[23,18232],{"className":18233,"style":131},[130],[23,18235,18237,18240],{"className":18236},[135],[23,18238,8340],{"className":18239,"style":11402},[135,136],[23,18241,18243],{"className":18242},[998],[23,18244,18246],{"className":18245},[199],[23,18247,18249],{"className":18248},[204],[23,18250,18252],{"className":18251,"style":6914},[208],[23,18253,18254,18257],{"style":4696},[23,18255],{"className":18256,"style":1015},[216],[23,18258,18260],{"className":18259},[1019,1020,1021,1022],[23,18261,6883],{"className":18262},[612,1022],[23,18264,54],{"className":18265},[140],[23,18267,434],{"className":18268,"style":454},[135,136],[23,18270,66],{"className":18271},[169],[23,18273],{"className":18274,"style":149},[148],[23,18276,60],{"className":18277},[153],[23,18279],{"className":18280,"style":149},[148],[23,18282,18284,18287,18290,18293,18296,18299],{"className":18283},[126],[23,18285],{"className":18286,"style":131},[130],[23,18288,434],{"className":18289,"style":454},[135,136],[23,18291,945],{"className":18292},[169],[23,18294],{"className":18295,"style":149},[148],[23,18297,69],{"className":18298},[153],[23,18300],{"className":18301,"style":149},[148],[23,18303,18305,18308],{"className":18304},[126],[23,18306],{"className":18307,"style":2770},[130],[23,18309,1532],{"className":18310},[135],"，因此消失。",[23,18313,18315,18328],{"className":18314,"translate":27},[31],[23,18316,18318],{"className":18317},[35],[37,18319,18320],{"xmlns":39},[42,18321,18322,18326],{},[45,18323,18324],{},[48,18325,8251],{"mathvariant":111},[114,18327,8254],{"encoding":116},[23,18329,18331],{"className":18330,"ariaHidden":122},[121],[23,18332,18334,18337],{"className":18333},[126],[23,18335],{"className":18336,"style":8264},[130],[23,18338,8251],{"className":18339},[135,8268],[11,18341,18342],{},"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。",[2501,18344,18346],{"id":18345},"_37-map-不是贝叶斯行动","3.7 MAP 不是贝叶斯行动",[11,18348,18349],{},"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。",[11,18351,18352,18353,18421,18422,18487,18488,18516,18517,18583],{},"更麻烦的是，MAP 对重参数化不协变。若 ",[23,18354,18356,18382],{"className":18355,"translate":27},[31],[23,18357,18359],{"className":18358},[35],[37,18360,18361],{"xmlns":39},[42,18362,18363,18379],{},[45,18364,18365,18368,18370,18373,18375,18377],{},[48,18366,18367],{},"ϕ",[51,18369,69],{},[48,18371,18372],{},"h",[51,18374,54],{"stretchy":53},[48,18376,57],{},[51,18378,66],{"stretchy":53},[114,18380,18381],{"encoding":116},"\\phi=h(\\theta)",[23,18383,18385,18403],{"className":18384,"ariaHidden":122},[121],[23,18386,18388,18391,18394,18397,18400],{"className":18387},[126],[23,18389],{"className":18390,"style":5414},[130],[23,18392,18367],{"className":18393},[135,136],[23,18395],{"className":18396,"style":149},[148],[23,18398,69],{"className":18399},[153],[23,18401],{"className":18402,"style":149},[148],[23,18404,18406,18409,18412,18415,18418],{"className":18405},[126],[23,18407],{"className":18408,"style":131},[130],[23,18410,18372],{"className":18411},[135,136],[23,18413,54],{"className":18414},[140],[23,18416,57],{"className":18417,"style":144},[135,136],[23,18419,66],{"className":18420},[169]," 是一个非线性变换，",[23,18423,18425,18448],{"className":18424,"translate":27},[31],[23,18426,18428],{"className":18427},[35],[37,18429,18430],{"xmlns":39},[42,18431,18432,18446],{},[45,18433,18434,18436,18438,18440,18442,18444],{},[48,18435,11],{},[51,18437,54],{"stretchy":53},[48,18439,57],{},[51,18441,60],{},[48,18443,63],{},[51,18445,66],{"stretchy":53},[114,18447,4032],{"encoding":116},[23,18449,18451,18475],{"className":18450,"ariaHidden":122},[121],[23,18452,18454,18457,18460,18463,18466,18469,18472],{"className":18453},[126],[23,18455],{"className":18456,"style":131},[130],[23,18458,11],{"className":18459},[135,136],[23,18461,54],{"className":18462},[140],[23,18464,57],{"className":18465,"style":144},[135,136],[23,18467],{"className":18468,"style":149},[148],[23,18470,60],{"className":18471},[153],[23,18473],{"className":18474,"style":149},[148],[23,18476,18478,18481,18484],{"className":18477},[126],[23,18479],{"className":18480,"style":131},[130],[23,18482,63],{"className":18483,"style":144},[135,136],[23,18485,66],{"className":18486},[169]," 的众数经过 ",[23,18489,18491,18504],{"className":18490,"translate":27},[31],[23,18492,18494],{"className":18493},[35],[37,18495,18496],{"xmlns":39},[42,18497,18498,18502],{},[45,18499,18500],{},[48,18501,18372],{},[114,18503,18372],{"encoding":116},[23,18505,18507],{"className":18506,"ariaHidden":122},[121],[23,18508,18510,18513],{"className":18509},[126],[23,18511],{"className":18512,"style":779},[130],[23,18514,18372],{"className":18515},[135,136]," 之后，一般不是 ",[23,18518,18520,18544],{"className":18519,"translate":27},[31],[23,18521,18523],{"className":18522},[35],[37,18524,18525],{"xmlns":39},[42,18526,18527,18541],{},[45,18528,18529,18531,18533,18535,18537,18539],{},[48,18530,11],{},[51,18532,54],{"stretchy":53},[48,18534,18367],{},[51,18536,60],{},[48,18538,63],{},[51,18540,66],{"stretchy":53},[114,18542,18543],{"encoding":116},"p(\\phi\\mid D)",[23,18545,18547,18571],{"className":18546,"ariaHidden":122},[121],[23,18548,18550,18553,18556,18559,18562,18565,18568],{"className":18549},[126],[23,18551],{"className":18552,"style":131},[130],[23,18554,11],{"className":18555},[135,136],[23,18557,54],{"className":18558},[140],[23,18560,18367],{"className":18561},[135,136],[23,18563],{"className":18564,"style":149},[148],[23,18566,60],{"className":18567},[153],[23,18569],{"className":18570,"style":149},[148],[23,18572,18574,18577,18580],{"className":18573},[126],[23,18575],{"className":18576,"style":131},[130],[23,18578,63],{"className":18579,"style":144},[135,136],[23,18581,66],{"className":18582},[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。",[338,18585,18587],{"id":18586},"_4-贝叶斯错误率","4. 贝叶斯错误率",[2501,18589,18591],{"id":18590},"_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",[11,18593,18594,18595,18720,18721,18790],{},"设两类先验相等，类条件密度为 ",[23,18596,18598,18624],{"className":18597,"translate":27},[31],[23,18599,18601],{"className":18600},[35],[37,18602,18603],{"xmlns":39},[42,18604,18605,18621],{},[45,18606,18607,18613,18615],{},[898,18608,18609,18611],{},[48,18610,11],{},[2581,18612,1532],{},[51,18614,747],{"separator":122},[898,18616,18617,18619],{},[48,18618,11],{},[2581,18620,4435],{},[114,18622,18623],{"encoding":116},"p_0,p_1",[23,18625,18627],{"className":18626,"ariaHidden":122},[121],[23,18628,18630,18633,18674,18677,18680],{"className":18629},[126],[23,18631],{"className":18632,"style":672},[130],[23,18634,18636,18639],{"className":18635},[135],[23,18637,11],{"className":18638},[135,136],[23,18640,18642],{"className":18641},[998],[23,18643,18645,18666],{"className":18644},[199,200],[23,18646,18648,18663],{"className":18647},[204],[23,18649,18652],{"className":18650,"style":18651},[208],"height:0.3011em;",[23,18653,18654,18657],{"style":2029},[23,18655],{"className":18656,"style":1015},[216],[23,18658,18660],{"className":18659},[1019,1020,1021,1022],[23,18661,1532],{"className":18662},[135,1022],[23,18664,297],{"className":18665},[296],[23,18667,18669],{"className":18668},[204],[23,18670,18672],{"className":18671,"style":2048},[208],[23,18673],{},[23,18675,747],{"className":18676},[824],[23,18678],{"className":18679,"style":280},[148],[23,18681,18683,18686],{"className":18682},[135],[23,18684,11],{"className":18685},[135,136],[23,18687,18689],{"className":18688},[998],[23,18690,18692,18712],{"className":18691},[199,200],[23,18693,18695,18709],{"className":18694},[204],[23,18696,18698],{"className":18697,"style":18651},[208],[23,18699,18700,18703],{"style":2029},[23,18701],{"className":18702,"style":1015},[216],[23,18704,18706],{"className":18705},[1019,1020,1021,1022],[23,18707,4435],{"className":18708},[135,1022],[23,18710,297],{"className":18711},[296],[23,18713,18715],{"className":18714},[204],[23,18716,18718],{"className":18717,"style":2048},[208],[23,18719],{},"。由 3.2 节，贝叶斯规则在 ",[23,18722,18724,18751],{"className":18723,"translate":27},[31],[23,18725,18727],{"className":18726},[35],[37,18728,18729],{"xmlns":39},[42,18730,18731,18749],{},[45,18732,18733,18735,18737,18739,18741,18743,18745,18747],{},[48,18734,9769],{},[51,18736,54],{"stretchy":53},[48,18738,753],{},[51,18740,66],{"stretchy":53},[51,18742,69],{},[2581,18744,4435],{},[48,18746,2875],{"mathvariant":111},[2581,18748,9956],{},[114,18750,10508],{"encoding":116},[23,18752,18754,18781],{"className":18753,"ariaHidden":122},[121],[23,18755,18757,18760,18763,18766,18769,18772,18775,18778],{"className":18756},[126],[23,18758],{"className":18759,"style":131},[130],[23,18761,9769],{"className":18762,"style":676},[135,136],[23,18764,54],{"className":18765},[140],[23,18767,753],{"className":18768},[135,136],[23,18770,66],{"className":18771},[169],[23,18773],{"className":18774,"style":149},[148],[23,18776,69],{"className":18777},[153],[23,18779],{"className":18780,"style":149},[148],[23,18782,18784,18787],{"className":18783},[126],[23,18785],{"className":18786,"style":131},[130],[23,18788,10473],{"className":18789},[135]," 处切换，其中",[23,18792,18794],{"className":18793,"translate":27},[26],[23,18795,18797,18865],{"className":18796,"translate":27},[31],[23,18798,18800],{"className":18799},[35],[37,18801,18802],{"xmlns":39,"display":40},[42,18803,18804,18862],{},[45,18805,18806,18808,18810,18812,18814,18816,18860],{},[48,18807,9769],{},[51,18809,54],{"stretchy":53},[48,18811,753],{},[51,18813,66],{"stretchy":53},[51,18815,69],{},[71,18817,18818,18832],{},[45,18819,18820,18826,18828,18830],{},[898,18821,18822,18824],{},[48,18823,11],{},[2581,18825,4435],{},[51,18827,54],{"stretchy":53},[48,18829,753],{},[51,18831,66],{"stretchy":53},[45,18833,18834,18840,18842,18844,18846,18848,18854,18856,18858],{},[898,18835,18836,18838],{},[48,18837,11],{},[2581,18839,1532],{},[51,18841,54],{"stretchy":53},[48,18843,753],{},[51,18845,66],{"stretchy":53},[51,18847,5106],{},[898,18849,18850,18852],{},[48,18851,11],{},[2581,18853,4435],{},[51,18855,54],{"stretchy":53},[48,18857,753],{},[51,18859,66],{"stretchy":53},[48,18861,112],{"mathvariant":111},[114,18863,18864],{"encoding":116},"\\eta(x)=\\frac{p_1(x)}{p_0(x)+p_1(x)}.",[23,18866,18868,18895],{"className":18867,"ariaHidden":122},[121],[23,18869,18871,18874,18877,18880,18883,18886,18889,18892],{"className":18870},[126],[23,18872],{"className":18873,"style":131},[130],[23,18875,9769],{"className":18876,"style":676},[135,136],[23,18878,54],{"className":18879},[140],[23,18881,753],{"className":18882},[135,136],[23,18884,66],{"className":18885},[169],[23,18887],{"className":18888,"style":149},[148],[23,18890,69],{"className":18891},[153],[23,18893],{"className":18894,"style":149},[148],[23,18896,18898,18901,19113],{"className":18897},[126],[23,18899],{"className":18900,"style":185},[130],[23,18902,18904,18907,19110],{"className":18903},[135],[23,18905],{"className":18906},[140,192],[23,18908,18910],{"className":18909},[71],[23,18911,18913,19102],{"className":18912},[199,200],[23,18914,18916,19099],{"className":18915},[204],[23,18917,18919,19034,19042],{"className":18918,"style":209},[208],[23,18920,18921,18924],{"style":212},[23,18922],{"className":18923,"style":217},[216],[23,18925,18927,18967,18970,18973,18976,18979,18982,18985,19025,19028,19031],{"className":18926},[135],[23,18928,18930,18933],{"className":18929},[135],[23,18931,11],{"className":18932},[135,136],[23,18934,18936],{"className":18935},[998],[23,18937,18939,18959],{"className":18938},[199,200],[23,18940,18942,18956],{"className":18941},[204],[23,18943,18945],{"className":18944,"style":18651},[208],[23,18946,18947,18950],{"style":2029},[23,18948],{"className":18949,"style":1015},[216],[23,18951,18953],{"className":18952},[1019,1020,1021,1022],[23,18954,1532],{"className":18955},[135,1022],[23,18957,297],{"className":18958},[296],[23,18960,18962],{"className":18961},[204],[23,18963,18965],{"className":18964,"style":2048},[208],[23,18966],{},[23,18968,54],{"className":18969},[140],[23,18971,753],{"className":18972},[135,136],[23,18974,66],{"className":18975},[169],[23,18977],{"className":18978,"style":392},[148],[23,18980,5106],{"className":18981},[612],[23,18983],{"className":18984,"style":392},[148],[23,18986,18988,18991],{"className":18987},[135],[23,18989,11],{"className":18990},[135,136],[23,18992,18994],{"className":18993},[998],[23,18995,18997,19017],{"className":18996},[199,200],[23,18998,19000,19014],{"className":18999},[204],[23,19001,19003],{"className":19002,"style":18651},[208],[23,19004,19005,19008],{"style":2029},[23,19006],{"className":19007,"style":1015},[216],[23,19009,19011],{"className":19010},[1019,1020,1021,1022],[23,19012,4435],{"className":19013},[135,1022],[23,19015,297],{"className":19016},[296],[23,19018,19020],{"className":19019},[204],[23,19021,19023],{"className":19022,"style":2048},[208],[23,19024],{},[23,19026,54],{"className":19027},[140],[23,19029,753],{"className":19030},[135,136],[23,19032,66],{"className":19033},[169],[23,19035,19036,19039],{"style":235},[23,19037],{"className":19038,"style":217},[216],[23,19040],{"className":19041,"style":243},[242],[23,19043,19044,19047],{"style":246},[23,19045],{"className":19046,"style":217},[216],[23,19048,19050,19090,19093,19096],{"className":19049},[135],[23,19051,19053,19056],{"className":19052},[135],[23,19054,11],{"className":19055},[135,136],[23,19057,19059],{"className":19058},[998],[23,19060,19062,19082],{"className":19061},[199,200],[23,19063,19065,19079],{"className":19064},[204],[23,19066,19068],{"className":19067,"style":18651},[208],[23,19069,19070,19073],{"style":2029},[23,19071],{"className":19072,"style":1015},[216],[23,19074,19076],{"className":19075},[1019,1020,1021,1022],[23,19077,4435],{"className":19078},[135,1022],[23,19080,297],{"className":19081},[296],[23,19083,19085],{"className":19084},[204],[23,19086,19088],{"className":19087,"style":2048},[208],[23,19089],{},[23,19091,54],{"className":19092},[140],[23,19094,753],{"className":19095},[135,136],[23,19097,66],{"className":19098},[169],[23,19100,297],{"className":19101},[296],[23,19103,1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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).",[23,20718,20720,20787,21088,21174,21233],{"className":20719,"ariaHidden":122},[121],[23,20721,20723,20727,20778,20781,20784],{"className":20722},[126],[23,20724],{"className":20725,"style":20726},[130],"height:0.9857em;vertical-align:-0.247em;",[23,20728,20730,20733],{"className":20729},[135],[23,20731,11],{"className":20732},[135,136],[23,20734,20736],{"className":20735},[998],[23,20737,20739,20770],{"className":20738},[199,200],[23,20740,20742,20767],{"className":20741},[204],[23,20743,20745,20756],{"className":20744,"style":7047},[208],[23,20746,20747,20750],{"style":19349},[23,20748],{"className":20749,"style":1015},[216],[23,20751,20753],{"className":20752},[1019,1020,1021,1022],[23,20754,5342],{"className":20755},[135,136,1022],[23,20757,20758,20761],{"style":4989},[23,20759],{"className":20760,"style":1015},[216],[23,20762,20764],{"className":20763},[1019,1020,1021,1022],[23,20765,6883],{"className":20766},[612,1022],[23,20768,297],{"className":20769},[296],[23,20771,20773],{"className":20772},[204],[23,20774,20776],{"className":20775,"style":19379},[208],[23,20777],{},[23,20779],{"className":20780,"style":149},[148],[23,20782,69],{"className":20783},[153],[23,20785],{"className":20786,"style":149},[148],[23,20788,20790,20794,20856,20859,21079,21082,21085],{"className":20789},[126],[23,20791],{"className":20792,"style":20793},[130],"height:2.4em;vertical-align:-0.95em;",[23,20795,20797,20800,20853],{"className":20796},[135],[23,20798],{"className":20799},[140,192],[23,20801,20803],{"className":20802},[71],[23,20804,20806,20845],{"className":20805},[199,200],[23,20807,20809,20842],{"className":20808},[204],[23,20810,20812,20823,20831],{"className":20811,"style":12413},[208],[23,20813,20814,20817],{"style":212},[23,20815],{"className":20816,"style":217},[216],[23,20818,20820],{"className":20819},[135],[23,20821,9956],{"className":20822},[135],[23,20824,20825,20828],{"style":235},[23,20826],{"className":20827,"style":217},[216],[23,20829],{"className":20830,"style":243},[242],[23,20832,20833,20836],{"style":246},[23,20834],{"className":20835,"style":217},[216],[23,20837,20839],{"className":20838},[135],[23,20840,4435],{"className":20841},[135],[23,20843,297],{"className":20844},[296],[23,20846,20848],{"className":20847},[204],[23,20849,20851],{"className":20850,"style":6287},[208],[23,20852],{},[23,20854],{"className":20855},[169,192],[23,20857],{"className":20858,"style":280},[148],[23,20860,20862,20868,20871,20874,20877,20880,20942,20945,20948,20951,20954,20994,20997,21000,21003,21006,21009,21012,21052,21055,21058,21061,21064,21067,21070,21073],{"className":20861},[1839],[23,20863,20865],{"className":20864,"style":1844},[140,1843],[23,20866,54],{"className":20867},[1066,1021],[23,20869,4435],{"className":20870},[135],[23,20872],{"className":20873,"style":392},[148],[23,20875,4529],{"className":20876},[612],[23,20878],{"className":20879,"style":392},[148],[23,20881,20883,20886,20939],{"className":20882},[135],[23,20884],{"className":20885},[140,192],[23,20887,20889],{"className":20888},[71],[23,20890,20892,20931],{"className":20891},[199,200],[23,20893,20895,20928],{"className":20894},[204],[23,20896,20898,20909,20917],{"className":20897,"style":12413},[208],[23,20899,20900,20903],{"style":212},[23,20901],{"className":20902,"style":217},[216],[23,20904,20906],{"className":20905},[135],[23,20907,9956],{"className":20908},[135],[23,20910,20911,20914],{"style":235},[23,20912],{"className":20913,"style":217},[216],[23,20915],{"className":20916,"style":243},[242],[23,20918,20919,20922],{"style":246},[23,20920],{"className":20921,"style":217},[216],[23,20923,20925],{"className":20924},[135],[23,20926,4435],{"className":20927},[135],[23,20929,297],{"className":20930},[296],[23,20932,20934],{"className":20933},[204],[23,20935,20937],{"className":20936,"style":6287},[208],[23,20938],{},[23,20940],{"className":20941},[169,192],[23,20943],{"className":20944,"style":280},[148],[23,20946,1557],{"className":20947,"style":1792},[1786,1790,1791],[23,20949],{"className":20950,"style":280},[148],[23,20952,60],{"className":20953},[135],[23,20955,20957,20960],{"className":20956},[135],[23,20958,11],{"className":20959},[135,136],[23,20961,20963],{"className":20962},[998],[23,20964,20966,20986],{"className":20965},[199,200],[23,20967,20969,20983],{"className":20968},[204],[23,20970,20972],{"className":20971,"style":18651},[208],[23,20973,20974,20977],{"style":2029},[23,20975],{"className":20976,"style":1015},[216],[23,20978,20980],{"className":20979},[1019,1020,1021,1022],[23,20981,1532],{"className":20982},[135,1022],[23,20984,297],{"className":20985},[296],[23,20987,20989],{"className":20988},[204],[23,20990,20992],{"className":20991,"style":2048},[208],[23,20993],{},[23,20995,54],{"className":20996},[140],[23,20998,753],{"className":20999},[135,136],[23,21001,66],{"className":21002},[169],[23,21004],{"className":21005,"style":392},[148],[23,21007,4529],{"className":21008},[612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",[23,21339,21341,21367],{"className":21340,"translate":27},[31],[23,21342,21344],{"className":21343},[35],[37,21345,21346],{"xmlns":39},[42,21347,21348,21364],{},[45,21349,21350,21352,21354],{},[48,21351,20684],{"mathvariant":111},[51,21353,5758],{},[898,21355,21356,21358],{},[48,21357,20684],{"mathvariant":111},[45,21359,21360,21362],{},[48,21361,20707],{"mathvariant":111},[48,21363,10794],{"mathvariant":111},[114,21365,21366],{"encoding":116},"\\|\\cdot\\|_{\\mathrm{TV}}",[23,21368,21370,21388],{"className":21369,"ariaHidden":122},[121],[23,21371,21373,21376,21379,21382,21385],{"className":21372},[126],[23,21374],{"className":21375,"style":131},[130],[23,21377,20684],{"className":21378},[135],[23,21380],{"className":21381,"style":392},[148],[23,21383,5758],{"className":21384},[612],[23,21386],{"className":21387,"style":392},[148],[23,21389,21391,21394],{"className":21390},[126],[23,21392],{"className":21393,"style":131},[130],[23,21395,21397,21400],{"className":21396},[135],[23,21398,20684],{"className":21399},[135],[23,21401,21403],{"className":21402},[998],[23,21404,21406,21432],{"className":21405},[199,200],[23,21407,21409,21429],{"className":21408},[204],[23,21410,21412],{"className":21411,"style":2026},[208],[23,21413,21414,21417],{"style":2029},[23,21415],{"className":21416,"style":1015},[216],[23,21418,21420],{"className":21419},[1019,1020,1021,1022],[23,21421,21423],{"className":21422},[135,1022],[23,21424,21426],{"className":21425},[135,1022],[23,21427,21314],{"className":21428,"style":7090},[135,5391,1022],[23,21430,297],{"className":21431},[296],[23,21433,21435],{"className":21434},[204],[23,21436,21438],{"className":21437,"style":2048},[208],[23,21439],{}," 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 ",[23,21442,21444,21472],{"className":21443,"translate":27},[31],[23,21445,21447],{"className":21446},[35],[37,21448,21449],{"xmlns":39},[42,21450,21451,21469],{},[45,21452,21453,21461,21463,21465,21467],{},[19141,21454,21455,21457,21459],{},[48,21456,11],{},[48,21458,5342],{},[51,21460,6883],{},[51,21462,69],{},[2581,21464,4435],{},[48,21466,2875],{"mathvariant":111},[2581,21468,9956],{},[114,21470,21471],{"encoding":116},"p_e^*=1\u002F2",[23,21473,21475,21542],{"className":21474,"ariaHidden":122},[121],[23,21476,21478,21482,21533,21536,21539],{"className":21477},[126],[23,21479],{"className":21480,"style":21481},[130],"height:0.9357em;vertical-align:-0.247em;",[23,21483,21485,21488],{"className":21484},[135],[23,21486,11],{"className":21487},[135,136],[23,21489,21491],{"className":21490},[998],[23,21492,21494,21525],{"className":21493},[199,200],[23,21495,21497,21522],{"className":21496},[204],[23,21498,21500,21511],{"className":21499,"style":6914},[208],[23,21501,21502,21505],{"style":19349},[23,21503],{"className":21504,"style":1015},[216],[23,21506,21508],{"className":21507},[1019,1020,1021,1022],[23,21509,5342],{"className":21510},[135,136,1022],[23,21512,21513,21516],{"style":4696},[23,21514],{"className":21515,"style":1015},[216],[23,21517,21519],{"className":21518},[1019,1020,1021,1022],[23,21520,6883],{"className":21521},[612,1022],[23,21523,297],{"className":21524},[296],[23,21526,21528],{"className":21527},[204],[23,21529,21531],{"className":21530,"style":19379},[208],[23,21532],{},[23,21534],{"className":21535,"style":149},[148],[23,21537,69],{"className":21538},[153],[23,21540],{"className":21541,"style":149},[148],[23,21543,21545,21548],{"className":21544},[126],[23,21546],{"className":21547,"style":131},[130],[23,21549,10473],{"className":21550},[135],"，当支撑集不相交时 ",[23,21553,21555,21579],{"className":21554,"translate":27},[31],[23,21556,21558],{"className":21557},[35],[37,21559,21560],{"xmlns":39},[42,21561,21562,21576],{},[45,21563,21564,21572,21574],{},[19141,21565,21566,21568,21570],{},[48,21567,11],{},[48,21569,5342],{},[51,21571,6883],{},[51,21573,69],{},[2581,21575,1532],{},[114,21577,21578],{"encoding":116},"p_e^*=0",[23,21580,21582,21648],{"className":21581,"ariaHidden":122},[121],[23,21583,21585,21588,21639,21642,21645],{"className":21584},[126],[23,21586],{"className":21587,"style":21481},[130],[23,21589,21591,21594],{"className":21590},[135],[23,21592,11],{"className":21593},[135,136],[23,21595,21597],{"className":21596},[998],[23,21598,21600,21631],{"className":21599},[199,200],[23,21601,21603,21628],{"className":21602},[204],[23,21604,21606,21617],{"className":21605,"style":6914},[208],[23,21607,21608,21611],{"style":19349},[23,21609],{"className":21610,"style":1015},[216],[23,21612,21614],{"className":21613},[1019,1020,1021,1022],[23,21615,5342],{"className":21616},[135,136,1022],[23,21618,21619,21622],{"style":4696},[23,21620],{"className":21621,"style":1015},[216],[23,21623,21625],{"className":21624},[1019,1020,1021,1022],[23,21626,6883],{"className":21627},[612,1022],[23,21629,297],{"className":21630},[296],[23,21632,21634],{"className":21633},[204],[23,21635,21637],{"className":21636,"style":19379},[208],[23,21638],{},[23,21640],{"className":21641,"style":149},[148],[23,21643,69],{"className":21644},[153],[23,21646],{"className":21647,"style":149},[148],[23,21649,21651,21654],{"className":21650},[126],[23,21652],{"className":21653,"style":2770},[130],[23,21655,1532],{"className":21656},[135],[11,21658,21659],{},"若先验不相等，公式改为",[23,21661,21663],{"className":21662,"translate":27},[26],[23,21664,21666,21747],{"className":21665,"translate":27},[31],[23,21667,21669],{"className":21668},[35],[37,21670,21671],{"xmlns":39,"display":40},[42,21672,21673,21744],{},[45,21674,21675,21683,21685,21687,21689,21691,21693,21700,21706,21708,21710,21712,21714,21716,21722,21728,21730,21732,21734,21736,21738,21740,21742],{},[19141,21676,21677,21679,21681],{},[48,21678,11],{},[48,21680,5342],{},[51,21682,6883],{},[51,21684,69],{},[51,21686,1557],{},[48,21688,6969],{},[51,21690,6961],{},[51,21692,9014],{"stretchy":53},[898,21694,21695,21698],{},[48,21696,21697],{},"π",[2581,21699,1532],{},[898,21701,21702,21704],{},[48,21703,11],{},[2581,21705,1532],{},[51,21707,54],{"stretchy":53},[48,21709,753],{},[51,21711,66],{"stretchy":53},[51,21713,747],{"separator":122},[88,21715,5360],{},[898,21717,21718,21720],{},[48,21719,21697],{},[2581,21721,4435],{},[898,21723,21724,21726],{},[48,21725,11],{},[2581,21727,4435],{},[51,21729,54],{"stretchy":53},[48,21731,753],{},[51,21733,66],{"stretchy":53},[51,21735,9024],{"stretchy":53},[88,21737,90],{},[48,21739,1606],{},[48,21741,753],{},[51,21743,747],{"separator":122},[114,21745,21746],{"encoding":116},"p_e^*=\\int \\min\\{\\pi_0p_0(x),\\ 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高斯情形：马氏距离与 ",[23,22042,22044,22058],{"className":22043,"translate":27},[31],[23,22045,22047],{"className":22046},[35],[37,22048,22049],{"xmlns":39},[42,22050,22051,22056],{},[45,22052,22053],{},[48,22054,22055],{},"Q",[114,22057,22055],{"encoding":116},[23,22059,22061],{"className":22060,"ariaHidden":122},[121],[23,22062,22064,22067],{"className":22063},[126],[23,22065],{"className":22066,"style":817},[130],[23,22068,22055],{"className":22069},[135,136]," 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",[23,23761,23763,23781],{"className":23762,"translate":27},[31],[23,23764,23766],{"className":23765},[35],[37,23767,23768],{"xmlns":39},[42,23769,23770,23778],{},[45,23771,23772,23774,23776],{},[48,23773,369],{},[51,23775,69],{},[2581,23777,4435],{},[114,23779,23780],{"encoding":116},"Y=1",[23,23782,23784,23802],{"className":23783,"ariaHidden":122},[121],[23,23785,23787,23790,23793,23796,23799],{"className":23786},[126],[23,23788],{"className":23789,"style":470},[130],[23,23791,369],{"className":23792,"style":392},[135,136],[23,23794],{"className":23795,"style":149},[148],[23,23797,69],{"className":23798},[153],[23,23800],{"className":23801,"style":149},[148],[23,23803,23805,23808],{"className":23804},[126],[23,23806],{"className":23807,"style":2770},[130],[23,23809,4435],{"className":23810},[135]," 下，对数似然比服从 ",[23,23813,23815,23853],{"className":23814,"translate":27},[31],[23,23816,23818],{"className":23817},[35],[37,23819,23820],{"xmlns":39},[42,23821,23822,23850],{},[45,23823,23824,23826,23828,23834,23836,23838,23840,23842,23848],{},[48,23825,12805],{"mathvariant":375},[51,23827,54],{"stretchy":53},[4518,23829,23830,23832],{},[48,23831,23378],{"mathvariant":111},[2581,23833,9956],{},[48,23835,2875],{"mathvariant":111},[2581,23837,9956],{},[51,23839,747],{"separator":122},[88,23841,5360],{},[4518,23843,23844,23846],{},[48,23845,23378],{"mathvariant":111},[2581,23847,9956],{},[51,23849,66],{"stretchy":53},[114,23851,23852],{"encoding":116},"\\mathcal N(\\Delta^2\u002F2,\\ \\Delta^2)",[23,23854,23856],{"className":23855,"ariaHidden":122},[121],[23,23857,23859,23862,23865,23868,23897,23900,23903,23906,23909,23938],{"className":23858},[126],[23,23860],{"className":23861,"style":10665},[130],[23,23863,12805],{"className":23864,"style":22222},[135,412],[23,23866,54],{"className":23867},[140],[23,23869,23871,23874],{"className":23870},[135],[23,23872,23378],{"className":23873},[135],[23,23875,23877],{"className":23876},[998],[23,23878,23880],{"className":23879},[199],[23,23881,23883],{"className":23882},[204],[23,23884,23886],{"className":23885,"style":4693},[208],[23,23887,23888,23891],{"style":4696},[23,23889],{"className":23890,"style":1015},[216],[23,23892,23894],{"className":23893},[1019,1020,1021,1022],[23,23895,9956],{"className":23896},[135,1022],[23,23898,20564],{"className":23899},[135],[23,23901,747],{"className":23902},[824],[23,23904,5360],{"className":23905},[148],[23,23907],{"className":23908,"style":280},[148],[23,23910,23912,23915],{"className":23911},[135],[23,23913,23378],{"className":23914},[135],[23,23916,23918],{"className":23917},[998],[23,23919,23921],{"className":23920},[199],[23,23922,23924],{"className":23923},[204],[23,23925,23927],{"className":23926,"style":4693},[208],[23,23928,23929,23932],{"style":4696},[23,23930],{"className":23931,"style":1015},[216],[23,23933,23935],{"className":23934},[1019,1020,1021,1022],[23,23936,9956],{"className":23937},[135,1022],[23,23939,66],{"className":23940},[169],"。贝叶斯规则在似然比为零处切换，因此",[23,23943,23945],{"className":23944,"translate":27},[26],[23,23946,23948,24041],{"className":23947,"translate":27},[31],[23,23949,23951],{"className":23950},[35],[37,23952,23953],{"xmlns":39,"display":40},[42,23954,23955,24038],{},[45,23956,23957,23965,23967,23969,23971,23973,23975,23981,23983,23985,23987,23993,23995,23997,23999,24001,24003,24006,24020,24022,24024,24036],{},[19141,23958,23959,23961,23963],{},[48,23960,11],{},[48,23962,5342],{},[51,23964,6883],{},[51,23966,69],{},[48,23968,920],{},[51,23970,54],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,23972,12805],{"mathvariant":375},[51,23974,54],{"stretchy":53},[4518,23976,23977,23979],{},[48,23978,23378],{"mathvariant":111},[2581,23980,9956],{},[48,23982,2875],{"mathvariant":111},[2581,23984,9956],{},[51,23986,747],{"separator":122},[4518,23988,23989,23991],{},[48,23990,23378],{"mathvariant":111},[2581,23992,9956],{},[51,23994,66],{"stretchy":53},[51,23996,9987],{},[2581,23998,1532],{},[51,24000,66],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,24002,69],{},[48,24004,24005],{"mathvariant":111},"Φ",[45,24007,24008,24010,24012,24018],{},[51,24009,54],{"fence":122},[51,24011,4529],{},[71,24013,24014,24016],{},[48,24015,23378],{"mathvariant":111},[2581,24017,9956],{},[51,24019,66],{"fence":122},[51,24021,69],{},[48,24023,22055],{},[45,24025,24026,24028,24034],{},[51,24027,54],{"fence":122},[71,24029,24030,24032],{},[48,24031,23378],{"mathvariant":111},[2581,24033,9956],{},[51,24035,66],{"fence":122},[51,24037,747],{"separator":122},[114,24039,24040],{"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),",[23,24042,24044,24110,24210,24234,24336],{"className":24043,"ariaHidden":122},[121],[23,24045,24047,24050,24101,24104,24107],{"className":24046},[126],[23,24048],{"className":24049,"style":20726},[130],[23,24051,24053,24056],{"className":24052},[135],[23,24054,11],{"className":24055},[135,136],[23,24057,24059],{"className":24058},[998],[23,24060,24062,24093],{"className":24061},[199,200],[23,24063,24065,24090],{"className":24064},[204],[23,24066,24068,24079],{"className":24067,"style":7047},[208],[23,24069,24070,24073],{"style":19349},[23,24071],{"className":24072,"style":1015},[216],[23,24074,24076],{"className":24075},[1019,1020,1021,1022],[23,24077,5342],{"className":24078},[135,136,1022],[23,24080,24081,24084],{"style":4989},[23,24082],{"className":24083,"style":1015},[216],[23,24085,24087],{"className":24086},[1019,1020,1021,1022],[23,24088,6883],{"className":24089},[612,1022],[23,24091,297],{"className":24092},[296],[23,24094,24096],{"className":24095},[204],[23,24097,24099],{"className":24098,"style":19379},[208],[23,24100],{},[23,24102],{"className":24103,"style":149},[148],[23,24105,69],{"className":24106},[153],[23,24108],{"className":24109,"style":149},[148],[23,24111,24113,24116,24119,24125,24128,24131,24160,24163,24166,24169,24198,24201,24204,24207],{"className":24112},[126],[23,24114],{"className":24115,"style":17160},[130],[23,24117,920],{"className":24118,"style":1047},[135,136],[23,24120,24122],{"className":24121},[135],[23,24123,54],{"className":24124},[1066,1067],[23,24126,12805],{"className":24127,"style":22222},[135,412],[23,24129,54],{"className":24130},[140],[23,24132,24134,24137],{"className":24133},[135],[23,24135,23378],{"className":24136},[135],[23,24138,24140],{"className":24139},[998],[23,24141,24143],{"className":24142},[199],[23,24144,24146],{"className":24145},[204],[23,24147,24149],{"className":24148,"style":11006},[208],[23,24150,24151,24154],{"style":4989},[23,24152],{"className":24153,"style":1015},[216],[23,24155,24157],{"className":24156},[1019,1020,1021,1022],[23,24158,9956],{"className":24159},[135,1022],[23,24161,20564],{"className":24162},[135],[23,24164,747],{"className":24165},[824],[23,24167],{"className":24168,"style":280},[148],[23,24170,24172,24175],{"className":24171},[135],[23,24173,23378],{"className":24174},[135],[23,24176,24178],{"className":24177},[998],[23,24179,24181],{"className":24180},[199],[23,24182,24184],{"className":24183},[204],[23,24185,24187],{"className":24186,"style":11006},[208],[23,24188,24189,24192],{"style":4989},[23,24190],{"className":24191,"style":1015},[216],[23,24193,24195],{"className":24194},[1019,1020,1021,1022],[23,24196,9956],{"className":24197},[135,1022],[23,24199,66],{"className":24200},[169],[23,24202],{"className":24203,"style":149},[148],[23,24205,9987],{"className":24206},[153],[23,24208],{"className":24209,"style":149},[148],[23,24211,24213,24216,24219,24225,24228,24231],{"className":24212},[126],[23,24214],{"className":24215,"style":2213},[130],[23,24217,1532],{"className":24218},[135],[23,24220,24222],{"className":24221},[135],[23,24223,66],{"className":24224},[1066,1067],[23,24226],{"className":24227,"style":149},[148],[23,24229,69],{"className":24230},[153],[23,24232],{"className":24233,"style":149},[148],[23,24235,24237,24240,24243,24246,24327,24330,24333],{"className":24236},[126],[23,24238],{"className":24239,"style":20793},[130],[23,24241,24005],{"className":24242},[135],[23,24244],{"className":24245,"style":280},[148],[23,24247,24249,24255,24258,24321],{"className":24248},[1839],[23,24250,24252],{"className":24251,"style":1844},[140,1843],[23,24253,54],{"className":24254},[1066,1021],[23,24256,4529],{"className":24257},[135],[23,24259,24261,24264,24318],{"className":24260},[135],[23,24262],{"className":24263},[140,192],[23,24265,24267],{"className":24266},[71],[23,24268,24270,24310],{"className":24269},[199,200],[23,24271,24273,24307],{"className":24272},[204],[23,24274,24277,24288,24296],{"className":24275,"style":24276},[208],"height:1.3603em;",[23,24278,24279,24282],{"style":212},[23,24280],{"className":24281,"style":217},[216],[23,24283,24285],{"className":24284},[135],[23,24286,9956],{"className":24287},[135],[23,24289,24290,24293],{"style":235},[23,24291],{"className":24292,"style":217},[216],[23,24294],{"className":24295,"style":243},[242],[23,24297,24298,24301],{"style":246},[23,24299],{"className":24300,"style":217},[216],[23,24302,24304],{"className":24303},[135],[23,24305,23378],{"className":24306},[135],[23,24308,297],{"className":24309},[296],[23,24311,24313],{"className":24312},[204],[23,24314,24316],{"className":24315,"style":6287},[208],[23,24317],{},[23,24319],{"className":24320},[169,192],[23,24322,24324],{"className":24323,"style":1844},[169,1843],[23,24325,66],{"className":24326},[1066,1021],[23,24328],{"className":24329,"style":149},[148],[23,24331,69],{"className":24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",[23,24435,24437,24451],{"className":24436,"translate":27},[31],[23,24438,24440],{"className":24439},[35],[37,24441,24442],{"xmlns":39},[42,24443,24444,24448],{},[45,24445,24446],{},[48,24447,24005],{"mathvariant":111},[114,24449,24450],{"encoding":116},"\\Phi",[23,24452,24454],{"className":24453,"ariaHidden":122},[121],[23,24455,24457,24460],{"className":24456},[126],[23,24458],{"className":24459,"style":470},[130],[23,24461,24005],{"className":24462},[135]," 是标准正态分布函数，",[23,24465,24467,24489],{"className":24466,"translate":27},[31],[23,24468,24470],{"className":24469},[35],[37,24471,24472],{"xmlns":39},[42,24473,24474,24486],{},[45,24475,24476,24478,24480,24482,24484],{},[48,24477,22055],{},[51,24479,69],{},[2581,24481,4435],{},[51,24483,4529],{},[48,24485,24005],{"mathvariant":111},[114,24487,24488],{"encoding":116},"Q=1-\\Phi",[23,24490,24492,24510,24528],{"className":24491,"ariaHidden":122},[121],[23,24493,24495,24498,24501,24504,24507],{"className":24494},[126],[23,24496],{"className":24497,"style":817},[130],[23,24499,22055],{"className":24500},[135,136],[23,24502],{"className":24503,"style":149},[148],[23,24505,69],{"className":24506},[153],[23,24508],{"className":24509,"style":149},[148],[23,24511,24513,24516,24519,24522,24525],{"className":24512},[126],[23,24514],{"className":24515,"style":9404},[130],[23,24517,4435],{"className":24518},[135],[23,24520],{"className":24521,"style":392},[148],[23,24523,4529],{"className":24524},[612],[23,24526],{"className":24527,"style":392},[148],[23,24529,24531,24534],{"className":24530},[126],[23,24532],{"className":24533,"style":470},[130],[23,24535,24005],{"className":24536},[135],[11,24538,24539],{},"几个数值可以帮助建立直觉：",[3764,24541,24542,24897],{},[3767,24543,24544],{},[3770,24545,24546,24577,24704],{},[3773,24547,24548],{},[23,24549,24551,24565],{"className":24550,"translate":27},[31],[23,24552,24554],{"className":24553},[35],[37,24555,24556],{"xmlns":39},[42,24557,24558,24562],{},[45,24559,24560],{},[48,24561,23378],{"mathvariant":111},[114,24563,24564],{"encoding":116},"\\Delta",[23,24566,24568],{"className":24567,"ariaHidden":122},[121],[23,24569,24571,24574],{"className":24570},[126],[23,24572],{"className":24573,"style":470},[130],[23,24575,23378],{"className":24576},[135],[3773,24578,24579],{},[23,24580,24582,24616],{"className":24581,"translate":27},[31],[23,24583,24585],{"className":24584},[35],[37,24586,24587],{"xmlns":39},[42,24588,24589,24613],{},[45,24590,24591,24599,24601,24603,24605,24607,24609,24611],{},[19141,24592,24593,24595,24597],{},[48,24594,11],{},[48,24596,5342],{},[51,24598,6883],{},[51,24600,69],{},[48,24602,22055],{},[51,24604,54],{"stretchy":53},[48,24606,23378],{"mathvariant":111},[48,24608,2875],{"mathvariant":111},[2581,24610,9956],{},[51,24612,66],{"stretchy":53},[114,24614,24615],{"encoding":116},"p_e^*=Q(\\Delta\u002F2)",[23,24617,24619,24685],{"className":24618,"ariaHidden":122},[121],[23,24620,24622,24625,24676,24679,24682],{"className":24621},[126],[23,24623],{"className":24624,"style":21481},[130],[23,24626,24628,24631],{"className":24627},[135],[23,24629,11],{"className":24630},[135,136],[23,24632,24634],{"className":24633},[998],[23,24635,24637,24668],{"className":24636},[199,200],[23,24638,24640,24665],{"className":24639},[204],[23,24641,24643,24654],{"className":24642,"style":6914},[208],[23,24644,24645,24648],{"style":19349},[23,24646],{"className":24647,"style":1015},[216],[23,24649,24651],{"className":24650},[1019,1020,1021,1022],[23,24652,5342],{"className":24653},[135,136,1022],[23,24655,24656,24659],{"style":4696},[23,24657],{"className":24658,"style":1015},[216],[23,24660,24662],{"className":24661},[1019,1020,1021,1022],[23,24663,6883],{"className":24664},[612,1022],[23,24666,297],{"className":24667},[296],[23,24669,24671],{"className":24670},[204],[23,24672,24674],{"className":24673,"style":19379},[208],[23,24675],{},[23,24677],{"className":24678,"style":149},[148],[23,24680,69],{"className":24681},[153],[23,24683],{"className":24684,"style":149},[148],[23,24686,24688,24691,24694,24697,24701],{"className":24687},[126],[23,24689],{"className":24690,"style":131},[130],[23,24692,22055],{"className":24693},[135,136],[23,24695,54],{"className":24696},[140],[23,24698,24700],{"className":24699},[135],"Δ\u002F2",[23,24702,66],{"className":24703},[169],[3773,24705,24706,24707],{},"Bhattacharyya 上界 ",[23,24708,24710,24749],{"className":24709,"translate":27},[31],[23,24711,24713],{"className":24712},[35],[37,24714,24715],{"xmlns":39},[42,24716,24717,24746],{},[45,24718,24719,24727],{},[1530,24720,24721],{"scriptlevel":1532,"displaystyle":53},[71,24722,24723,24725],{},[2581,24724,4435],{},[2581,24726,9956],{},[4518,24728,24729,24731],{},[48,24730,5342],{},[45,24732,24733,24735,24741,24743],{},[51,24734,4529],{},[4518,24736,24737,24739],{},[48,24738,23378],{"mathvariant":111},[2581,24740,9956],{},[48,24742,2875],{"mathvariant":111},[2581,24744,24745],{},"8",[114,24747,24748],{"encoding":116},"\\tfrac12e^{-\\Delta^2\u002F8}",[23,24750,24752],{"className":24751,"ariaHidden":122},[121],[23,24753,24755,24759,24827],{"className":24754},[126],[23,24756],{"className":24757,"style":24758},[130],"height:1.3319em;vertical-align:-0.345em;",[23,24760,24762,24765,24824],{"className":24761},[135],[23,24763],{"className":24764},[140,192],[23,24766,24768],{"className":24767},[71],[23,24769,24771,24816],{"className":24770},[199,200],[23,24772,24774,24813],{"className":24773},[204],[23,24775,24777,24791,24799],{"className":24776,"style":10200},[208],[23,24778,24779,24782],{"style":10203},[23,24780],{"className":24781,"style":217},[216],[23,24783,24785],{"className":24784},[1019,1020,1021,1022],[23,24786,24788],{"className":24787},[135,1022],[23,24789,9956],{"className":24790},[135,1022],[23,24792,24793,24796],{"style":235},[23,24794],{"className":24795,"style":217},[216],[23,24797],{"className":24798,"style":243},[242],[23,24800,24801,24804],{"style":10226},[23,24802],{"className":24803,"style":217},[216],[23,24805,24807],{"className":24806},[1019,1020,1021,1022],[23,24808,24810],{"className":24809},[135,1022],[23,24811,4435],{"className":24812},[135,1022],[23,24814,297],{"className":24815},[296],[23,24817,24819],{"className":24818},[204],[23,24820,24822],{"className":24821,"style":10248},[208],[23,24823],{},[23,24825],{"className":24826},[169,192],[23,24828,24830,24833],{"className":24829},[135],[23,24831,5342],{"className":24832},[135,136],[23,24834,24836],{"className":24835},[998],[23,24837,24839],{"className":24838},[199],[23,24840,24842],{"className":24841},[204],[23,24843,24846],{"className":24844,"style":24845},[208],"height:0.9869em;",[23,24847,24848,24851],{"style":4696},[23,24849],{"className":24850,"style":1015},[216],[23,24852,24854],{"className":24853},[1019,1020,1021,1022],[23,24855,24857,24860,24893],{"className":24856},[135,1022],[23,24858,4529],{"className":24859},[135,1022],[23,24861,24863,24866],{"className":24862},[135,1022],[23,24864,23378],{"className":24865},[135,1022],[23,24867,24869],{"className":24868},[998],[23,24870,24872],{"className":24871},[199],[23,24873,24875],{"className":24874},[204],[23,24876,24879],{"className":24877,"style":24878},[208],"height:0.8913em;",[23,24880,24882,24886],{"style":24881},"top:-2.931em;margin-right:0.0714em;",[23,24883],{"className":24884,"style":24885},[216],"height:2.5em;",[23,24887,24890],{"className":24888},[1019,24889,1067,1022],"reset-size3",[23,24891,9956],{"className":24892},[135,1022],[23,24894,24896],{"className":24895},[135,1022],"\u002F8",[3783,24898,24899,24909,24919,24930],{},[3770,24900,24901,24903,24906],{},[3788,24902,4435],{},[3788,24904,24905],{},"0.3085",[3788,24907,24908],{},"0.4412",[3770,24910,24911,24913,24916],{},[3788,24912,9956],{},[3788,24914,24915],{},"0.1587",[3788,24917,24918],{},"0.3033",[3770,24920,24921,24924,24927],{},[3788,24922,24923],{},"3",[3788,24925,24926],{},"0.0668",[3788,24928,24929],{},"0.1623",[3770,24931,24932,24935,24938],{},[3788,24933,24934],{},"4",[3788,24936,24937],{},"0.0228",[3788,24939,24940],{},"0.0677",[11,24942,24943],{},"上界来自",[23,24945,24947],{"className":24946,"translate":27},[26],[23,24948,24950,25038],{"className":24949,"translate":27},[31],[23,24951,24953],{"className":24952},[35],[37,24954,24955],{"xmlns":39,"display":40},[42,24956,24957,25035],{},[45,24958,24959,24967,24969,24975,24977,24984,24986,24992,24994,25000,25002,25004,25010,25013,25015,25033],{},[19141,24960,24961,24963,24965],{},[48,24962,11],{},[48,24964,5342],{},[51,24966,6883],{},[51,24968,12272],{},[71,24970,24971,24973],{},[2581,24972,4435],{},[2581,24974,9956],{},[88,24976,90],{},[45,24978,24979,24981],{},[48,24980,2533],{"mathvariant":111},[48,24982,24983],{"mathvariant":111},"C",[51,24985,54],{"stretchy":53},[898,24987,24988,24990],{},[48,24989,920],{},[2581,24991,1532],{},[51,24993,747],{"separator":122},[898,24995,24996,24998],{},[48,24997,920],{},[2581,24999,4435],{},[51,25001,66],{"stretchy":53},[51,25003,69],{},[71,25005,25006,25008],{},[2581,25007,4435],{},[2581,25009,9956],{},[48,25011,25012],{},"exp",[51,25014,6961],{},[45,25016,25017,25019,25021,25031],{},[51,25018,54],{"fence":122},[51,25020,4529],{},[71,25022,25023,25029],{},[4518,25024,25025,25027],{},[48,25026,23378],{"mathvariant":111},[2581,25028,9956],{},[2581,25030,24745],{},[51,25032,66],{"fence":122},[51,25034,747],{"separator":122},[114,25036,25037],{"encoding"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\\frac{2}{\\Delta\\sqrt{2\\pi}}e^{-\\Delta^2\u002F8},",[23,25903,25905,26003],{"className":25904,"ariaHidden":122},[121],[23,25906,25908,25911,25914,25917,25994,25997,26000],{"className":25907},[126],[23,25909],{"className":25910,"style":20793},[130],[23,25912,22055],{"className":25913},[135,136],[23,25915],{"className":25916,"style":280},[148],[23,25918,25920,25926,25988],{"className":25919},[1839],[23,25921,25923],{"className":25922,"style":1844},[140,1843],[23,25924,54],{"className":25925},[1066,1021],[23,25927,25929,25932,25985],{"className":25928},[135],[23,25930],{"className":25931},[140,192],[23,25933,25935],{"className":25934},[71],[23,25936,25938,25977],{"className":25937},[199,200],[23,25939,25941,25974],{"className":25940},[204],[23,25942,25944,25955,25963],{"className":25943,"style":24276},[208],[23,25945,25946,25949],{"style":212},[23,25947],{"className":25948,"style":217},[216],[23,25950,25952],{"className":25951},[135],[23,25953,9956],{"className":25954},[135],[23,25956,25957,25960],{"style":235},[23,25958],{"className":25959,"style":217},[216],[23,25961],{"className":25962,"style":243},[242],[23,25964,25965,25968],{"style":246},[23,25966],{"className":25967,"style":217},[216],[23,25969,25971],{"className":25970},[135],[23,25972,23378],{"className":25973},[135],[23,25975,297],{"className":25976},[296],[23,25978,25980],{"className":25979},[204],[23,25981,25983],{"className":25982,"style":6287},[208],[23,25984],{},[23,25986],{"className":25987},[169,192],[23,25989,25991],{"className":25990,"style":1844},[169,1843],[23,25992,66],{"className":25993},[1066,1021],[23,25995],{"className":25996,"style":149},[148],[23,25998,917],{"className":25999},[153],[23,26001],{"className":26002,"style":149},[148],[23,26004,26006,26010,26127,26192],{"className":26005},[126],[23,26007],{"className":26008,"style":26009},[130],"height:2.2514em;vertical-align:-0.93em;",[23,26011,26013,26016,26124],{"className":26012},[135],[23,26014],{"className":26015},[140,192],[23,26017,26019],{"className":26018},[71],[23,26020,26022,26115],{"className":26021},[199,200],[23,26023,26025,26112],{"className":26024},[204],[23,26026,26028,26093,26101],{"className":26027,"style":12413},[208],[23,26029,26031,26034],{"style":26030},"top:-2.2028em;",[23,26032],{"className":26033,"style":217},[216],[23,26035,26037,26040],{"className":26036},[135],[23,26038,23378],{"className":26039},[135],[23,26041,26043],{"className":26042},[135,25671],[23,26044,26046,26084],{"className":26045},[199,200],[23,26047,26049,26081],{"className":26048},[204],[23,26050,26053,26068],{"className":26051,"style":26052},[208],"height:0.9072em;",[23,26054,26056,26059],{"className":26055,"style":7132},[8676],[23,26057],{"className":26058,"style":217},[216],[23,26060,26062,26065],{"className":26061,"style":25691},[135],[23,26063,9956],{"className":26064},[135],[23,26066,21697],{"className":26067,"style":676},[135,136],[23,26069,26071,26074],{"style":26070},"top:-2.8672em;",[23,26072],{"className":26073,"style":217},[216],[23,26075,26077],{"className":26076,"style":25782},[25781],[8692,26078,26079],{"xmlns":8694,"width":8695,"height":25785,"viewBox":25786,"preserveAspectRatio":8698},[8700,26080],{"d":25789},[23,26082,297],{"className":26083},[296],[23,26085,26087],{"className":26086},[204],[23,26088,26091],{"className":26089,"style":26090},[208],"height:0.1328em;",[23,26092],{},[23,26094,26095,26098],{"style":235},[23,26096],{"className":26097,"style":217},[216],[23,26099],{"className":26100,"style":243},[242],[23,26102,26103,26106],{"style":246},[23,26104],{"className":26105,"style":217},[216],[23,26107,26109],{"className":26108},[135],[23,26110,9956],{"className":26111},[135],[23,26113,297],{"className":26114},[296],[23,26116,26118],{"className":26117},[204],[23,26119,26122],{"className":26120,"style":26121},[208],"height:0.93em;",[23,26123],{},[23,26125],{"className":26126},[169,192],[23,26128,26130,26133],{"className":26129},[135],[23,26131,5342],{"className":26132},[135,136],[23,26134,26136],{"className":26135},[998],[23,26137,26139],{"className":26138},[199],[23,26140,26142],{"className":26141},[204],[23,26143,26146],{"className":26144,"style":26145},[208],"height:1.0369em;",[23,26147,26148,26151],{"style":4989},[23,26149],{"className":26150,"style":1015},[216],[23,26152,26154],{"className":26153},[1019,1020,1021,1022],[23,26155,26157,26160,26189],{"className":26156},[135,1022],[23,26158,4529],{"className":26159},[135,1022],[23,26161,26163,26166],{"className":26162},[135,1022],[23,26164,23378],{"className":26165},[135,1022],[23,26167,26169],{"className":26168},[998],[23,26170,26172],{"className":26171},[199],[23,26173,26175],{"className":26174},[204],[23,26176,26178],{"className":26177,"style":24878},[208],[23,26179,26180,26183],{"style":24881},[23,26181],{"className":26182,"style":24885},[216],[23,26184,26186],{"className":26185},[1019,24889,1067,1022],[23,26187,9956],{"className":26188},[135,1022],[23,26190,24896],{"className":26191},[135,1022],[23,26193,747],{"className":26194},[824],[11,26196,26197,26198,1471],{},"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",[18,26199,26200],{},"速率",[2501,26202,26204],{"id":26203},"_43-用散度给错误率定界","4.3 用散度给错误率定界",[11,26206,26207],{},"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：",[23,26209,26211],{"className":26210,"translate":27},[26],[23,26212,26214,26335],{"className":26213,"translate":27},[31],[23,26215,26217],{"className":26216},[35],[37,26218,26219],{"xmlns":39,"display":40},[42,26220,26221,26332],{},[45,26222,26223,26225,26231,26233,26239,26249,26251,26254,26256,26262,26264,26270,26272,26274,26276,26282,26284,26290,26292,26298,26300,26302,26304,26306,26312,26314,26320,26322,26328,26330],{},[48,26224,20684],{"mathvariant":111},[898,26226,26227,26229],{},[48,26228,920],{},[2581,26230,1532],{},[51,26232,4529],{},[898,26234,26235,26237],{},[48,26236,920],{},[2581,26238,4435],{},[898,26240,26241,26243],{},[48,26242,20684],{"mathvariant":111},[45,26244,26245,26247],{},[48,26246,20707],{"mathvariant":111},[48,26248,10794],{"mathvariant":111},[51,26250,12272],{},[48,26252,26253],{},"H",[51,26255,54],{"stretchy":53},[898,26257,26258,26260],{},[48,26259,920],{},[2581,26261,1532],{},[51,26263,747],{"separator":122},[898,26265,26266,26268],{},[48,26267,920],{},[2581,26269,4435],{},[51,26271,66],{"stretchy":53},[51,26273,747],{"separator":122},[148,26275],{"width":750},[4518,26277,26278,26280],{},[48,26279,26253],{},[2581,26281,9956],{},[51,26283,54],{"stretchy":53},[898,26285,26286,26288],{},[48,26287,920],{},[2581,26289,1532],{},[51,26291,747],{"separator":122},[898,26293,26294,26296],{},[48,26295,920],{},[2581,26297,4435],{},[51,26299,66],{"stretchy":53},[51,26301,69],{},[2581,26303,4435],{},[51,26305,4529],{},[45,26307,26308,26310],{},[48,26309,2533],{"mathvariant":111},[48,26311,24983],{"mathvariant":111},[51,26313,54],{"stretchy":53},[898,26315,26316,26318],{},[48,26317,920],{},[2581,26319,1532],{},[51,26321,747],{"separator":122},[898,26323,26324,26326],{},[48,26325,920],{},[2581,26327,4435],{},[51,26329,66],{"stretchy":53},[51,26331,747],{"separator":122},[114,26333,26334],{"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),",[23,26336,26338,26396,26497,26739,26757],{"className":26337,"ariaHidden":122},[121],[23,26339,26341,26344,26347,26387,26390,26393],{"className":26340},[126],[23,26342],{"className":26343,"style":131},[130],[23,26345,20684],{"className":26346},[135],[23,26348,26350,26353],{"className":26349},[135],[23,26351,920],{"className":26352,"style":1047},[135,136],[23,26354,26356],{"className":26355},[998],[23,26357,26359,26379],{"className":26358},[199,200],[23,26360,26362,26376],{"className":26361},[204],[23,26363,26365],{"className":26364,"style":18651},[208],[23,26366,26367,26370],{"style":21203},[23,26368],{"className":26369,"style":1015},[216],[23,26371,26373],{"className":26372},[1019,1020,1021,1022],[23,26374,1532],{"className":26375},[135,1022],[23,26377,297],{"className":26378},[296],[23,26380,26382],{"className":26381},[204],[23,26383,26385],{"className":26384,"style":2048},[208],[23,26386],{},[23,26388],{"className":26389,"style":392},[148],[23,26391,4529],{"className":26392},[612],[23,26394],{"className":26395,"style":392},[148],[23,26397,26399,26402,26442,26488,26491,26494],{"className":26398},[126],[23,26400],{"className":26401,"style":131},[130],[23,26403,26405,26408],{"className":26404},[135],[23,26406,920],{"className":26407,"style":1047},[135,136],[23,26409,26411],{"className":26410},[998],[23,26412,26414,26434],{"className":26413},[199,200],[23,26415,26417,26431],{"className":26416},[204],[23,26418,26420],{"className":26419,"style":18651},[208],[23,26421,26422,26425],{"style":21203},[23,26423],{"className":26424,"style":1015},[216],[23,26426,26428],{"className":26427},[1019,1020,1021,1022],[23,26429,4435],{"className":26430},[135,1022],[23,26432,297],{"className":26433},[296],[23,26435,26437],{"className":26436},[204],[23,26438,26440],{"className":26439,"style":2048},[208],[23,26441],{},[23,26443,26445,26448],{"className":26444},[135],[23,26446,20684],{"className":26447},[135],[23,26449,26451],{"className":26450},[998],[23,26452,26454,26480],{"className":26453},[199,200],[23,26455,26457,26477],{"className":26456},[204],[23,26458,26460],{"className":26459,"style":2026},[208],[23,26461,26462,26465],{"style":2029},[23,26463],{"className":26464,"style":1015},[216],[23,26466,26468],{"className":26467},[1019,1020,1021,1022],[23,26469,26471],{"className":26470},[135,1022],[23,26472,26474],{"className":26473},[135,1022],[23,26475,21314],{"className":26476,"style":7090},[135,5391,1022],[23,26478,297],{"className":26479},[296],[23,26481,26483],{"className":26482},[204],[23,26484,26486],{"className":26485,"style":2048},[208],[23,26487],{},[23,26489],{"className":26490,"style":149},[148],[23,26492,12272],{"className":26493},[153],[23,26495],{"className":26496,"style":149},[148],[23,26498,26500,26504,26508,26511,26551,26554,26557,26597,26600,26603,26606,26609,26638,26641,26681,26684,26687,26727,26730,26733,26736],{"className":26499},[126],[23,26501],{"className":26502,"style":26503},[130],"height:1.1141em;vertical-align:-0.25em;",[23,26505,26253],{"className":26506,"style":26507},[135,136],"margin-right:0.0813em;",[23,26509,54],{"className":26510},[140],[23,26512,26514,26517],{"className":26513},[135],[23,26515,920],{"className":26516,"style":1047},[135,136],[23,26518,26520],{"className":26519},[998],[23,26521,26523,26543],{"className":26522},[199,200],[23,26524,26526,26540],{"className":26525},[204],[23,26527,26529],{"className":26528,"style":18651},[208],[23,26530,26531,26534],{"style":21203},[23,26532],{"className":26533,"style":1015},[216],[23,26535,26537],{"className":26536},[1019,1020,1021,1022],[23,26538,1532],{"className":26539},[135,1022],[23,26541,297],{"className":26542},[296],[23,26544,26546],{"className":26545},[204],[23,26547,26549],{"className":26548,"style":2048},[208],[23,26550],{},[23,26552,747],{"className":26553},[824],[23,26555],{"className":26556,"style":280},[148],[23,26558,26560,26563],{"className":26559},[135],[23,26561,920],{"className":26562,"style":1047},[135,136],[23,26564,26566],{"className":26565},[998],[23,26567,26569,26589],{"className":26568},[199,200],[23,26570,26572,26586],{"className":26571},[204],[23,26573,26575],{"className":26574,"style":18651},[208],[23,26576,26577,26580],{"style":21203},[23,26578],{"className":26579,"style":1015},[216],[23,26581,26583],{"className":26582},[1019,1020,1021,1022],[23,26584,4435],{"className":26585},[135,1022],[23,26587,297],{"className":26588},[296],[23,26590,26592],{"className":26591},[204],[23,26593,26595],{"className":26594,"style":2048},[208],[23,26596],{},[23,26598,66],{"className":26599},[169],[23,26601,747],{"className":26602},[824],[23,26604],{"className":26605,"style":828},[148],[23,26607],{"className":26608,"style":280},[148],[23,26610,26612,26615],{"className":26611},[135],[23,26613,26253],{"className":26614,"style":26507},[135,136],[23,26616,26618],{"className":26617},[998],[23,26619,26621],{"className":26620},[199],[23,26622,26624],{"className":26623},[204],[23,26625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,4435],{},[51,26942,66],{"stretchy":53},[48,26944,112],{"mathvariant":111},[114,26946,26947],{"encoding":116},"H^2(P_0,P_1)\\le \\frac12 D_{\\mathrm{KL}}(P_0\\|P_1).",[23,26949,26951,27087],{"className":26950,"ariaHidden":122},[121],[23,26952,26954,26957,26986,26989,27029,27032,27035,27075,27078,27081,27084],{"className":26953},[126],[23,26955],{"className":26956,"style":26503},[130],[23,26958,26960,26963],{"className":26959},[135],[23,26961,26253],{"className":26962,"style":26507},[135,136],[23,26964,26966],{"className":26965},[998],[23,26967,26969],{"className":26968},[199],[23,26970,26972],{"className":26971},[204],[23,26973,26975],{"className":26974,"style":11006},[208],[23,26976,26977,26980],{"style":4989},[23,26978],{"className":26979,"style":1015},[216],[23,26981,26983],{"className":26982},[1019,1020,1021,1022],[23,26984,9956],{"className":26985},[135,1022],[23,26987,54],{"className":26988},[140],[23,26990,26992,26995],{"className":26991},[135],[23,26993,920],{"className":26994,"style":1047},[135,136],[23,26996,26998],{"className":26997},[998],[23,26999,27001,27021],{"className":27000},[199,200],[23,27002,27004,27018],{"className":27003},[204],[23,27005,27007],{"className":27006,"style":18651},[208],[23,27008,27009,27012],{"style":21203},[23,27010],{"className":27011,"style":1015},[216],[23,27013,27015],{"className":27014},[1019,1020,1021,1022],[23,27016,1532],{"className":27017},[135,1022],[23,27019,297],{"className":27020},[296],[23,27022,27024],{"className":27023},[204],[23,27025,27027],{"className":27026,"style":2048},[208],[23,27028],{},[23,27030,747],{"className":27031},[824],[23,27033],{"className":27034,"style":280},[148],[23,27036,27038,27041],{"className":27037},[135],[23,27039,920],{"className":27040,"style":1047},[135,136],[23,27042,27044],{"className":27043},[998],[23,27045,27047,27067],{"className":27046},[199,200],[23,27048,27050,27064],{"className":27049},[204],[23,27051,27053],{"className":27052,"style":18651},[208],[23,27054,27055,27058],{"style":21203},[23,27056],{"className":27057,"style":1015},[216],[23,27059,27061],{"className":27060},[1019,1020,1021,1022],[23,27062,4435],{"className":27063},[135,1022],[23,27065,297],{"className":27066},[296],[23,27068,27070],{"className":27069},[204],[23,27071,27073],{"className":27072,"style":2048},[208],[23,27074],{},[23,27076,66],{"className":27077},[169],[23,27079],{"className":27080,"style":149},[148],[23,27082,12272],{"className":27083},[153],[23,27085],{"className":27086,"style":149},[148],[23,27088,27090,27093,27155,27203,27206,27246,27249,27289,27292],{"className":27089},[126],[23,27091],{"className":27092,"style":12394},[130],[23,27094,27096,27099,27152],{"className":27095},[135],[23,27097],{"className":27098},[140,192],[23,27100,27102],{"className":27101},[71],[23,27103,27105,27144],{"className":27104},[199,200],[23,27106,27108,27141],{"className":27107},[204],[23,27109,27111,27122,27130],{"className":27110,"style":12413},[208],[23,27112,27113,27116],{"style":212},[23,27114],{"className":27115,"style":217},[216],[23,27117,27119],{"className":27118},[135],[23,27120,9956],{"className":27121},[135],[23,27123,27124,27127],{"style":235},[23,27125],{"className":27126,"style":217},[216],[23,27128],{"className":27129,"style":243},[242],[23,27131,27132,27135],{"style":246},[23,27133],{"className":27134,"style":217},[216],[23,27136,27138],{"className":27137},[135],[23,27139,4435],{"className":27140},[135],[23,27142,297],{"className":27143},[296],[23,27145,27147],{"className":27146},[204],[23,27148,27150],{"className":27149,"style":6287},[208],[23,27151],{},[23,27153],{"className":27154},[169,192],[23,27156,27158,27161],{"className":27157},[135],[23,27159,63],{"className":27160,"style":144},[135,136],[23,27162,27164],{"className":27163},[998],[23,27165,27167,27195],{"className":27166},[199,200],[23,27168,27170,27192],{"className":27169},[204],[23,27171,27173],{"className":27172,"style":2026},[208],[23,27174,27176,27179],{"style":27175},"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;",[23,27177],{"className":27178,"style":1015},[216],[23,27180,27182],{"className":27181},[1019,1020,1021,1022],[23,27183,27185],{"className":27184},[135,1022],[23,27186,27188],{"className":27187},[135,1022],[23,27189,27191],{"className":27190},[135,5391,1022],"KL",[23,27193,297],{"className":27194},[296],[23,27196,27198],{"className":27197},[204],[23,27199,27201],{"className":27200,"style":2048},[208],[23,27202],{},[23,27204,54],{"className":27205},[140],[23,27207,27209,27212],{"className":27208},[135],[23,27210,920],{"className":27211,"style":1047},[135,136],[23,27213,27215],{"className":27214},[998],[23,27216,27218,27238],{"className":27217},[199,200],[23,27219,27221,27235],{"className":27220},[204],[23,27222,27224],{"className":27223,"style":18651},[208],[23,27225,27226,27229],{"style":21203},[23,27227],{"className":27228,"style":1015},[216],[23,27230,27232],{"className":27231},[1019,1020,1021,1022],[23,27233,1532],{"className":27234},[135,1022],[23,27236,297],{"className":27237},[296],[23,27239,27241],{"className":27240},[204],[23,27242,27244],{"className":27243,"style":2048},[208],[23,272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",[23,27299,27301,27345],{"className":27300,"translate":27},[31],[23,27302,27304],{"className":27303},[35],[37,27305,27306],{"xmlns":39},[42,27307,27308,27342],{},[45,27309,27310,27318,27320,27328,27330,27332,27334,27340],{},[19141,27311,27312,27314,27316],{},[48,27313,11],{},[48,27315,5342],{},[51,27317,6883],{},[51,27319,69],{},[1530,27321,27322],{"scriptlevel":1532,"displaystyle":53},[71,27323,27324,27326],{},[2581,27325,4435],{},[2581,27327,9956],{},[51,27329,54],{"stretchy":53},[2581,27331,4435],{},[51,27333,4529],{},[45,27335,27336,27338],{},[48,27337,20707],{"mathvariant":111},[48,27339,10794],{"mathvariant":111},[51,27341,66],{"stretchy":53},[114,27343,27344],{"encoding":116},"p_e^*=\\tfrac12(1-\\mathrm{TV})",[23,27346,27348,27414,27504],{"className":27347,"ariaHidden":122},[121],[23,27349,27351,27354,27405,27408,27411],{"className":27350},[126],[23,27352],{"className":27353,"style":21481},[130],[23,27355,27357,27360],{"className":27356},[135],[23,27358,11],{"className":27359},[135,136],[23,27361,27363],{"className":27362},[998],[23,27364,27366,27397],{"className":27365},[199,200],[23,27367,27369,27394],{"className":27368},[204],[23,27370,27372,27383],{"className":27371,"style":6914},[208],[23,27373,27374,27377],{"style":19349},[23,27375],{"className":27376,"style":1015},[216],[23,27378,27380],{"className":27379},[1019,1020,1021,1022],[23,27381,5342],{"className":27382},[135,136,1022],[23,27384,27385,27388],{"style":4696},[23,27386],{"className":27387,"style":1015},[216],[23,27389,27391],{"className":27390},[1019,1020,1021,1022],[23,27392,6883],{"className":27393},[612,1022],[23,27395,297],{"className":27396},[296],[23,27398,27400],{"className":27399},[204],[23,27401,27403],{"className":27402,"style":19379},[208],[23,27404],{},[23,27406],{"className":27407,"style":149},[148],[23,27409,69],{"className":27410},[153],[23,27412],{"className":27413,"style":149},[148],[23,27415,27417,27421,27489,27492,27495,27498,27501],{"className":27416},[126],[23,27418],{"className":27419,"style":27420},[130],"height:1.1901em;vertical-align:-0.345em;",[23,27422,27424,27427,27486],{"className":27423},[135],[23,27425],{"className":27426},[140,192],[23,27428,27430],{"className":27429},[71],[23,27431,27433,27478],{"className":27432},[199,200],[23,27434,27436,27475],{"className":27435},[204],[23,27437,27439,27453,27461],{"className":27438,"style":10200},[208],[23,27440,27441,27444],{"style":10203},[23,27442],{"className":27443,"style":217},[216],[23,27445,27447],{"className":27446},[1019,1020,1021,1022],[23,27448,27450],{"className":27449},[135,1022],[23,27451,9956],{"className":27452},[135,1022],[23,27454,27455,27458],{"style":235},[23,27456],{"className":27457,"style":217},[216],[23,27459],{"className":27460,"style":243},[242],[23,27462,27463,27466],{"style":10226},[23,27464],{"className":27465,"style":217},[216],[23,27467,27469],{"className":27468},[1019,1020,1021,1022],[23,27470,27472],{"className":27471},[135,1022],[23,27473,4435],{"className":27474},[135,1022],[23,27476,297],{"className":27477},[296],[23,27479,27481],{"className":27480},[204],[23,27482,27484],{"className":27483,"style":10248},[208],[23,27485],{},[23,27487],{"className":27488},[169,192],[23,27490,54],{"className":27491},[140],[23,27493,4435],{"className":27494},[135],[23,27496],{"className":27497,"style":392},[148],[23,27499,4529],{"className":27500},[612],[23,27502],{"className":27503,"style":392},[148],[23,27505,27507,27510,27516],{"className":27506},[126],[23,27508],{"className":27509,"style":131},[130],[23,27511,27513],{"className":27512},[135],[23,27514,21314],{"className":27515,"style":7090},[135,5391],[23,27517,66],{"className":27518},[169]," 结合，得到",[23,27521,27523],{"className":27522,"translate":27},[26],[23,27524,27526,27606],{"className":27525,"translate":27},[31],[23,27527,27529],{"className":27528},[35],[37,27530,27531],{"xmlns":39,"display":40},[42,27532,27533,27603],{},[45,27534,27535,27543,27545,27547,27549,27555,27601],{},[19141,27536,27537,27539,27541],{},[48,27538,11],{},[48,27540,5342],{},[51,27542,6883],{},[88,27544,5360],{},[51,27546,7382],{},[88,27548,5360],{},[71,27550,27551,27553],{},[2581,27552,4435],{},[2581,27554,9956],{},[45,27556,27557,27559,27561,27563,27599],{},[51,27558,54],{"fence":122},[2581,27560,4435],{},[51,27562,4529],{},[25520,27564,27565],{},[71,27566,27567,27597],{},[45,27568,27569,27579,27581,27587,27589,27595],{},[898,27570,27571,27573],{},[48,27572,63],{},[45,27574,27575,27577],{},[48,27576,26922],{"mathvariant":111},[48,27578,555],{"mathvariant":111},[51,27580,54],{"stretchy":53},[898,27582,27583,27585],{},[48,27584,920],{},[2581,27586,1532],{},[48,27588,20684],{"mathvariant":111},[898,27590,27591,27593],{},[48,27592,920],{},[2581,27594,4435],{},[51,27596,66],{"stretchy":53},[2581,27598,9956],{},[51,27600,66],{"fence":122},[48,27602,112],{"mathvariant":111},[114,27604,27605],{"encoding":116},"p_e^*\\ \\ge\\ \\frac12\\left(1-\\sqrt{\\frac{D_{\\mathrm{KL}}(P_0\\|P_1)}{2}}\\right).",[23,27607,27609,27681],{"className":27608,"ariaHidden":122},[121],[23,27610,27612,27615,27666,27669,27672,27675,27678],{"className":27611},[126],[23,27613],{"className":27614,"style":20726},[130],[23,27616,27618,27621],{"className":27617},[135],[23,27619,11],{"className":27620},[135,136],[23,27622,27624],{"className":27623},[998],[23,27625,27627,27658],{"className":27626},[199,200],[23,27628,27630,27655],{"className":27629},[204],[23,27631,27633,27644],{"className":27632,"style":7047},[208],[23,27634,27635,27638],{"style":19349},[23,27636],{"className":27637,"style":1015},[216],[23,27639,27641],{"className":27640},[1019,1020,1021,1022],[23,27642,5342],{"className":27643},[135,136,1022],[23,27645,27646,27649],{"style":4989},[23,27647],{"className":27648,"style":1015},[216],[23,27650,27652],{"className":27651},[1019,1020,1021,1022],[23,27653,6883],{"className":27654},[612,1022],[23,27656,297],{"className":27657},[296],[23,27659,27661],{"className":27660},[204],[23,27662,27664],{"className":27663,"style":19379},[208],[23,27665],{},[23,27667,5360],{"className":27668},[148],[23,27670],{"className":27671,"style":149},[148],[23,27673,7382],{"className":27674},[153],[23,27676,5360],{"className":27677},[148],[23,27679],{"className":27680,"style":149},[148],[23,27682,27684,27688,27750,27753,28028,28031],{"className":27683},[126],[23,27685],{"className":27686,"style":27687},[130],"height:3em;vertical-align:-1.25em;",[23,27689,27691,27694,27747],{"className":27690},[135],[23,27692],{"className":27693},[140,192],[23,27695,27697],{"className":27696},[71],[23,27698,27700,27739],{"className":27699},[199,200],[23,27701,27703,27736],{"className":27702},[204],[23,27704,27706,27717,27725],{"className":27705,"style":12413},[208],[23,27707,27708,27711],{"style":212},[23,27709],{"className":27710,"style":217},[216],[23,27712,27714],{"className":27713},[135],[23,27715,9956],{"className":27716},[135],[23,27718,27719,27722],{"style":235},[23,27720],{"className":27721,"style":217},[216],[23,27723],{"className":27724,"style":243},[242],[23,27726,27727,27730],{"style":246},[23,27728],{"className":27729,"style":217},[216],[23,27731,27733],{"className":27732},[135],[23,27734,4435],{"className":27735},[135],[23,27737,297],{"className":27738},[296],[23,27740,27742],{"className":27741},[204],[23,27743,27745],{"className":27744,"style":6287},[208],[23,27746],{},[23,27748],{"className":27749},[169,192],[23,27751],{"className":27752,"style":280},[148],[23,27754,27756,27762,27765,27768,27771,27774,28022],{"className":27755},[1839],[23,27757,27759],{"className":27758,"style":1844},[140,1843],[23,27760,54],{"className":27761},[1066,10073],[23,27763,4435],{"className":27764},[135],[23,27766],{"className":27767,"style":392},[148],[23,27769,4529],{"className":27770},[612],[23,27772],{"className":27773,"style":392},[148],[23,27775,27777],{"className":27776},[135,25671],[23,27778,27780,28013],{"className":27779},[199,200],[23,27781,27783,28010],{"className":27782},[204],[23,27784,27787,27993],{"className":27785,"style":27786},[208],"height:1.7044em;",[23,27788,27791,27795],{"className":27789,"style":27790},[8676],"top:-4.4em;",[23,27792],{"className":27793,"style":27794},[216],"height:4.4em;",[23,27796,27799],{"className":27797,"style":27798},[135],"padding-left:1em;",[23,27800,27802,27805,27990],{"className":27801},[135],[23,27803],{"className":27804},[140,192],[23,27806,27808],{"className":27807},[71],[23,27809,27811,27982],{"className":27810},[199,200],[23,27812,27814,27979],{"className":27813},[204],[23,27815,27817,27828,27836],{"className":27816,"style":209},[208],[23,27818,27819,27822],{"style":212},[23,27820],{"className":27821,"style":217},[216],[23,27823,27825],{"className":27824},[135],[23,27826,9956],{"className":27827},[135],[23,27829,27830,27833],{"style":235},[23,27831],{"className":27832,"style":217},[216],[23,27834],{"className":27835,"style":243},[242],[23,27837,27838,27841],{"style":246},[23,27839],{"className":27840,"style":217},[216],[23,27842,27844,27890,27893,27933,27936,27976],{"className":27843},[135],[23,27845,27847,27850],{"className":27846},[135],[23,27848,63],{"className":27849,"style":144},[135,136],[23,27851,27853],{"className":27852},[998],[23,27854,27856,27882],{"className":27855},[199,200],[23,27857,27859,27879],{"className":27858},[204],[23,27860,27862],{"className":27861,"style":2026},[208],[23,27863,27864,27867],{"style":27175},[23,27865],{"className":27866,"style":1015},[216],[23,27868,27870],{"className":27869},[1019,1020,1021,1022],[23,27871,27873],{"className":27872},[135,1022],[23,27874,27876],{"className":27875},[135,1022],[23,27877,27191],{"className":27878},[135,5391,1022],[23,27880,297],{"className":27881},[296],[23,27883,27885],{"className":27884},[204],[23,27886,27888],{"className":27887,"style":2048},[208],[23,27889],{},[23,27891,54],{"className":27892},[140],[23,27894,27896,27899],{"className":27895},[135],[23,27897,920],{"className":27898,"style":1047},[135,136],[23,27900,27902],{"className":27901},[998],[23,27903,27905,27925],{"className":27904},[199,200],[23,27906,27908,27922],{"className":27907},[204],[23,27909,27911],{"className":27910,"style":18651},[208],[23,27912,27913,27916],{"style":21203},[23,27914],{"className":27915,"style":1015},[216],[23,27917,27919],{"className":27918},[1019,1020,1021,1022],[23,27920,1532],{"className":27921},[135,1022],[23,27923,297],{"className":27924},[296],[23,27926,27928],{"className":27927},[204],[23,27929,27931],{"className":27930,"style":2048},[208],[23,27932],{},[23,27934,20684],{"className":27935},[135],[23,27937,27939,27942],{"className":27938},[135],[23,27940,920],{"className":27941,"style":1047},[135,136],[23,27943,27945],{"className":27944},[998],[23,27946,27948,27968],{"className":27947},[199,200],[23,27949,27951,27965],{"className":27950},[204],[23,27952,27954],{"className":27953,"style":18651},[208],[23,27955,27956,27959],{"style":21203},[23,27957],{"className":27958,"style":1015},[216],[23,27960,27962],{"className":27961},[1019,1020,1021,1022],[23,27963,4435],{"className":27964},[135,1022],[23,27966,297],{"className":27967},[296],[23,27969,27971],{"className":27970},[204],[23,27972,27974],{"className":27973,"style":2048},[208],[23,27975],{},[23,27977,66],{"className":27978},[169],[23,27980,297],{"className":27981},[296],[23,27983,27985],{"className":27984},[204],[23,27986,27988],{"className":27987,"style":6287},[208],[23,27989],{},[23,27991],{"className":27992},[169,192],[23,27994,27996,27999],{"style":27995},"top:-3.6644em;",[23,27997],{"className":27998,"style":27794},[216],[23,28000,28003],{"className":28001,"style":28002},[25781],"min-width:1.02em;height:2.48em;",[8692,28004,28007],{"xmlns":8694,"width":8695,"height":28005,"viewBox":28006,"preserveAspectRatio":8698},"2.48em","0 0 400000 2592",[8700,28008],{"d":28009},"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",[23,28011,297],{"className":28012},[296],[23,28014,28016],{"className":28015},[204],[23,28017,28020],{"className":28018,"style":28019},[208],"height:0.7356em;",[23,28021],{},[23,28023,28025],{"className":28024,"style":1844},[169,1843],[23,28026,66],{"className":28027},[1066,10073],[23,28029],{"className":28030,"style":280},[148],[23,28032,112],{"className":28033},[135],[11,28035,28036,28037,28040],{},"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——",[18,28038,28039],{},"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。",[11,28042,28043],{},"上界则需要另一个量：",[23,28045,28047],{"className":28046,"translate":27},[26],[23,28048,28050,28106],{"className":28049,"translate":27},[31],[23,28051,28053],{"className":28052},[35],[37,28054,28055],{"xmlns":39,"display":40},[42,28056,28057,28103],{},[45,28058,28059,28067,28069,28075,28077,28083,28085,28091,28093,28099,28101],{},[19141,28060,28061,28063,28065],{},[48,28062,11],{},[48,28064,5342],{},[51,28066,6883],{},[51,28068,12272],{},[71,28070,28071,28073],{},[2581,28072,4435],{},[2581,28074,9956],{},[88,28076,90],{},[45,28078,28079,28081],{},[48,28080,2533],{"mathvariant":111},[48,28082,24983],{"mathvariant":111},[51,28084,54],{"stretchy":53},[898,28086,28087,28089],{},[48,28088,920],{},[2581,28090,1532],{},[51,28092,747],{"separator":122},[898,28094,28095,28097],{},[48,28096,920],{},[2581,28098,4435],{},[51,28100,66],{"stretchy":53},[48,28102,112],{"mathvariant":111},[114,28104,28105],{"encoding":116},"p_e^*\\le \\frac12\\,\\mathrm{BC}(P_0,P_1).",[23,28107,28109,28175],{"className":28108,"ariaHidden":122},[121],[23,28110,28112,28115,28166,28169,28172],{"className":28111},[126],[23,28113],{"className":28114,"style":20726},[130],[23,28116,28118,28121],{"className":28117},[135],[23,28119,11],{"className":28120},[135,136],[23,28122,28124],{"className":28123},[998],[23,28125,28127,28158],{"className":28126},[199,200],[23,28128,28130,28155],{"className":28129},[204],[23,28131,28133,28144],{"className":28132,"style":7047},[208],[23,28134,28135,28138],{"style":19349},[23,28136],{"className":28137,"style":1015},[216],[23,28139,28141],{"className":28140},[1019,1020,1021,1022],[23,28142,5342],{"className":28143},[135,136,1022],[23,28145,28146,28149],{"style":4989},[23,28147],{"className":28148,"style":1015},[216],[23,28150,28152],{"className":28151},[1019,1020,1021,1022],[23,28153,6883],{"className":28154},[612,1022],[23,28156,297],{"className":28157},[296],[23,28159,28161],{"className":28160},[204],[23,28162,28164],{"className":28163,"style":19379},[208],[23,28165],{},[23,28167],{"className":28168,"style":149},[148],[23,28170,12272],{"className":28171},[153],[23,28173],{"className":28174,"style":149},[148],[23,28176,28178,28181,28243,28246,28252,28255,28295,28298,28301,28341,28344],{"className":28177},[126],[23,28179],{"className":28180,"style":12394},[130],[23,28182,28184,28187,28240],{"className":28183},[135],[23,28185],{"className":28186},[140,192],[23,28188,28190],{"className":28189},[71],[23,28191,28193,28232],{"className":28192},[199,200],[23,28194,28196,28229],{"className":28195},[204],[23,28197,28199,28210,28218],{"className":28198,"style":12413},[208],[23,28200,28201,28204],{"style":212},[23,28202],{"className":28203,"style":217},[216],[23,28205,28207],{"className":28206},[135],[23,28208,9956],{"className":28209},[135],[23,28211,28212,28215],{"style":235},[23,28213],{"className":28214,"style":217},[216],[23,28216],{"className":28217,"style":243},[242],[23,28219,28220,28223],{"style":246},[23,28221],{"className":28222,"style":217},[216],[23,28224,28226],{"className":28225},[135],[23,28227,4435],{"className":28228},[135],[23,28230,297],{"className":28231},[296],[23,28233,28235],{"className":28234},[204],[23,28236,28238],{"className":28237,"style":6287},[208],[23,28239],{},[23,28241],{"className":28242},[169,192],[23,28244],{"className":28245,"style":280},[148],[23,28247,28249],{"className":28248},[135],[23,28250,25184],{"className":28251},[135,5391],[23,28253,54],{"className":28254},[140],[23,28256,28258,28261],{"className":28257},[135],[23,28259,920],{"className":28260,"style":1047},[135,136],[23,28262,28264],{"className":28263},[998],[23,28265,28267,28287],{"className":28266},[199,200],[23,28268,28270,28284],{"className":28269},[204],[23,28271,28273],{"className":28272,"style":18651},[208],[23,28274,28275,28278],{"style":21203},[23,28276],{"className":28277,"style":1015},[216],[23,28279,28281],{"className":28280},[1019,1020,1021,1022],[23,28282,1532],{"className":28283},[135,1022],[23,28285,297],{"className":28286},[296],[23,28288,28290],{"className":28289},[204],[23,28291,28293],{"className":28292,"style":2048},[208],[23,28294],{},[23,28296,747],{"className":28297},[824],[23,28299],{"className":28300,"style":280},[148],[23,28302,28304,28307],{"className":28303},[135],[23,28305,920],{"className":28306,"style":1047},[135,136],[23,28308,28310],{"className":28309},[998],[23,28311,28313,28333],{"className":28312},[199,200],[23,28314,28316,28330],{"className":28315},[204],[23,28317,28319],{"className":28318,"style":18651},[208],[23,28320,28321,28324],{"style":21203},[23,28322],{"className":28323,"style":1015},[216],[23,28325,28327],{"className":28326},[1019,1020,1021,1022],[23,28328,4435],{"className":28329},[135,1022],[23,28331,297],{"className":28332},[296],[23,28334,28336],{"className":28335},[204],[23,28337,28339],{"className":28338,"style":2048},[208],[23,28340],{},[23,28342,66],{"className":28343},[169],[23,28345,112],{"className":28346},[135],[11,28348,28349,28350,28664],{},"这是因为 ",[23,28351,28353,28409],{"className":28352,"translate":27},[31],[23,28354,28356],{"className":28355},[35],[37,28357,28358],{"xmlns":39},[42,28359,28360,28406],{},[45,28361,28362,28364,28366,28368,28370,28376,28378,28384,28386,28388,28390],{},[51,28363,1557],{},[48,28365,6969],{},[51,28367,6961],{},[51,28369,9014],{"stretchy":53},[898,28371,28372,28374],{},[48,28373,11],{},[2581,28375,1532],{},[51,28377,747],{"separator":122},[898,28379,28380,28382],{},[48,28381,11],{},[2581,28383,4435],{},[51,28385,9024],{"stretchy":53},[51,28387,12272],{},[51,28389,1557],{},[25520,28391,28392],{},[45,28393,28394,28400],{},[898,28395,28396,28398],{},[48,28397,11],{},[2581,28399,1532],{},[898,28401,28402,28404],{},[48,28403,11],{},[2581,28405,4435],{},[114,28407,28408],{"encoding":116},"\\int\\min\\{p_0,p_1\\}\\le\\int\\sqrt{p_0p_1}",[23,28410,28412,28528],{"className":28411,"ariaHidden":122},[121],[23,28413,28415,28418,28421,28424,28427,28430,28470,28473,28476,28516,28519,28522,28525],{"className":28414},[126],[23,28416],{"className":28417,"style":20454},[130],[23,28419,1557],{"className":28420,"style":20459},[1786,1790,20458],[23,28422],{"className":28423,"style":280},[148],[23,28425,6969],{"className":28426},[1786],[23,28428,9014],{"className":28429},[140],[23,28431,28433,28436],{"className":28432},[135],[23,28434,11],{"className":28435},[135,136],[23,28437,28439],{"className":28438},[998],[23,28440,28442,28462],{"className":28441},[199,200],[23,28443,28445,28459],{"className":28444},[204],[23,28446,28448],{"className":28447,"style":18651},[208],[23,28449,28450,28453],{"style":2029},[23,28451],{"className":28452,"style":1015},[216],[23,28454,28456],{"className":28455},[1019,1020,1021,1022],[23,28457,1532],{"className":28458},[135,1022],[23,28460,297],{"className":28461},[296],[23,28463,28465],{"className":28464},[204],[23,28466,28468],{"className":28467,"style":2048},[208],[23,28469],{},[23,28471,747],{"className":28472},[824],[23,28474],{"className":28475,"style":280},[148],[23,28477,28479,28482],{"className":28478},[135],[23,28480,11],{"className":28481},[135,136],[23,28483,28485],{"className":28484},[998],[23,28486,28488,28508],{"className":28487},[199,200],[23,28489,28491,28505],{"className":28490},[204],[23,28492,28494],{"className":28493,"style":18651},[208],[23,28495,28496,28499],{"style":2029},[23,28497],{"className":28498,"style":1015},[216],[23,28500,28502],{"className":28501},[1019,1020,1021,1022],[23,28503,4435],{"className":28504},[135,1022],[23,28506,297],{"className":28507},[296],[23,28509,28511],{"className":28510},[204],[23,28512,28514],{"className":28513,"style":2048},[208],[23,28515],{},[23,28517,9024],{"className":28518},[169],[23,28520],{"className":28521,"style":149},[148],[23,28523,12272],{"className":28524},[153],[23,28526],{"className":28527,"style":149},[148],[23,28529,28531,28534,28537,28540],{"className":28530},[126],[23,28532],{"className":28533,"style":25661},[130],[23,28535,1557],{"className":28536,"style":20459},[1786,1790,20458],[23,28538],{"className":28539,"style":280},[148],[23,28541,28543],{"className":28542},[135,25671],[23,28544,28546,28656],{"className":28545},[199,200],[23,28547,28549,28653],{"className":28548},[204],[23,28550,28552,28641],{"className":28551,"style":25681},[208],[23,28553,28555,28558],{"className":28554,"style":7132},[8676],[23,28556],{"className":28557,"style":217},[216],[23,28559,28561,28601],{"className":28560,"style":25691},[135],[23,28562,28564,28567],{"className":28563},[135],[23,28565,11],{"className":28566},[135,136],[23,28568,28570],{"className":28569},[998],[23,28571,28573,28593],{"className":28572},[199,200],[23,28574,28576,28590],{"className":28575},[204],[23,28577,28579],{"className":28578,"style":18651},[208],[23,28580,28581,28584],{"style":2029},[23,28582],{"className":28583,"style":1015},[216],[23,28585,28587],{"className":28586},[1019,1020,1021,1022],[23,28588,1532],{"className":28589},[135,1022],[23,28591,297],{"className":28592},[296],[23,28594,28596],{"className":28595},[204],[23,28597,28599],{"className":28598,"style":2048},[208],[23,28600],{},[23,28602,28604,28607],{"className":28603},[135],[23,28605,11],{"className":28606},[135,136],[23,28608,28610],{"className":28609},[998],[23,28611,28613,28633],{"className":28612},[199,200],[23,28614,28616,28630],{"className":28615},[204],[23,28617,28619],{"className":28618,"style":18651},[208],[23,28620,28621,28624],{"style":2029},[23,28622],{"className":28623,"style":1015},[216],[23,28625,28627],{"className":28626},[1019,1020,1021,1022],[23,28628,4435],{"className":28629},[135,1022],[23,28631,297],{"className":28632},[296],[23,28634,28636],{"className":28635},[204],[23,28637,28639],{"className":28638,"style":2048},[208],[23,28640],{},[23,28642,28643,28646],{"style":25774},[23,28644],{"className":28645,"style":217},[216],[23,28647,28649],{"className":28648,"style":25782},[25781],[8692,28650,28651],{"xmlns":8694,"width":8695,"height":25785,"viewBox":25786,"preserveAspectRatio":8698},[8700,28652],{"d":25789},[23,28654,297],{"className":28655},[296],[23,28657,28659],{"className":28658},[204],[23,28660,28662],{"className":28661,"style":25799},[208],[23,28663],{},"（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。",[11,28666,28667,28668,28671,28672,28731,28732,28736],{},"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",[18,28669,28670],{},"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 ",[23,28673,28675,28697],{"className":28674,"translate":27},[31],[23,28676,28678],{"className":28677},[35],[37,28679,28680],{"xmlns":39},[42,28681,28682,28694],{},[45,28683,28684,28690,28692],{},[45,28685,28686,28688],{},[48,28687,20707],{"mathvariant":111},[48,28689,10794],{"mathvariant":111},[51,28691,12272],{},[48,28693,26253],{},[114,28695,28696],{"encoding":116},"\\mathrm{TV}\\le H",[23,28698,28700,28722],{"className":28699,"ariaHidden":122},[121],[23,28701,28703,28707,28713,28716,28719],{"className":28702},[126],[23,28704],{"className":28705,"style":28706},[130],"height:0.8193em;vertical-align:-0.136em;",[23,28708,28710],{"className":28709},[135],[23,28711,21314],{"className":28712,"style":7090},[135,5391],[23,28714],{"className":28715,"style":149},[148],[23,28717,12272],{"className":28718},[153],[23,28720],{"className":28721,"style":149},[148],[23,28723,28725,28728],{"className":28724},[126],[23,28726],{"className":28727,"style":470},[130],[23,28729,26253],{"className":28730,"style":26507},[135,136]," 把两边连起来。",[495,28733,28735],{"href":28734},"\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。",[2501,28738,28740],{"id":28739},"_44-错误率住在哪里","4.4 错误率住在哪里",[11,28742,28743,28744,28888,28889,28963],{},"由 3.6 节的恒等式，只有满足 ",[23,28745,28747,28781],{"className":28746,"translate":27},[31],[23,28748,28750],{"className":28749},[35],[37,28751,28752],{"xmlns":39},[42,28753,28754,28778],{},[45,28755,28756,28758,28760,28762,28764,28766,28772,28774,28776],{},[48,28757,736],{},[51,28759,54],{"stretchy":53},[48,28761,434],{},[51,28763,66],{"stretchy":53},[51,28765,9019],{"mathvariant":111},[4518,28767,28768,28770],{},[48,28769,736],{},[51,28771,6883],{},[51,28773,54],{"stretchy":53},[48,28775,434],{},[51,28777,66],{"stretchy":53},[114,28779,28780],{"encoding":116},"\\delta(X)\\ne\\delta^*(X)",[23,28782,28784,28844],{"className":28783,"ariaHidden":122},[121],[23,28785,28787,28790,28793,28796,28799,28802,28805,28841],{"className":28786},[126],[23,28788],{"className":28789,"style":131},[130],[23,28791,736],{"className":28792,"style":783},[135,136],[23,28794,54],{"className":28795},[140],[23,28797,434],{"className":28798,"style":454},[135,136],[23,28800,66],{"className":28801},[169],[23,28803],{"className":28804,"style":149},[148],[23,28806,28808,28835,28838],{"className":28807},[153],[23,28809,28811],{"className":28810},[153],[23,28812,28814],{"className":28813},[135,9097],[23,28815,28817],{"className":28816},[9101],[23,28818,28820,28823,28832],{"className":28819},[9105],[23,28821],{"className":28822,"style":5414},[130],[23,28824,28826],{"className":28825},[9112],[23,28827,28829],{"className":28828},[135],[23,28830,9119],{"className":28831},[153],[23,28833],{"className":28834},[9123],[23,28836],{"className":28837},[148,9127],[23,28839,69],{"className":28840},[153],[23,28842],{"className":28843,"style":149},[148],[23,28845,28847,28850,28879,28882,28885],{"className":28846},[126],[23,28848],{"className":28849,"style":131},[130],[23,28851,28853,28856],{"className":28852},[135],[23,28854,736],{"className":28855,"style":783},[135,136],[23,28857,28859],{"className":28858},[998],[23,28860,28862],{"className":28861},[199],[23,28863,28865],{"className":28864},[204],[23,28866,28868],{"className":28867,"style":6914},[208],[23,28869,28870,28873],{"style":4696},[23,28871],{"className":28872,"style":1015},[216],[23,28874,28876],{"className":28875},[1019,1020,1021,1022],[23,28877,6883],{"className":28878},[612,1022],[23,28880,54],{"className":28881},[140],[23,28883,434],{"className":28884,"style":454},[135,136],[23,28886,66],{"className":28887},[169]," 的样本才贡献错误，且权重是 ",[23,28890,28892,28921],{"className":28891,"translate":27},[31],[23,28893,28895],{"className":28894},[35],[37,28896,28897],{"xmlns":39},[42,28898,28899,28919],{},[45,28900,28901,28903,28905,28907,28909,28911,28913,28915,28917],{},[48,28902,60],{"mathvariant":111},[2581,28904,9956],{},[48,28906,9769],{},[51,28908,54],{"stretchy":53},[48,28910,434],{},[51,28912,66],{"stretchy":53},[51,28914,4529],{},[2581,28916,4435],{},[48,28918,60],{"mathvariant":111},[114,28920,16472],{"encoding":116},[23,28922,28924,28954],{"className":28923,"ariaHidden":122},[121],[23,28925,28927,28930,28933,28936,28939,28942,28945,28948,28951],{"className":28926},[126],[23,28928],{"className":28929,"style":131},[130],[23,28931,15676],{"className":28932},[135],[23,28934,9769],{"className":28935,"style":676},[135,136],[23,28937,54],{"className":28938},[140],[23,28940,434],{"className":28941,"style":454},[135,136],[23,28943,66],{"className":28944},[169],[23,28946],{"className":28947,"style":392},[148],[23,28949,4529],{"className":28950},[612],[23,28952],{"className":28953,"style":392},[148],[23,28955,28957,28960],{"className":28956},[126],[23,28958],{"className":28959,"style":131},[130],[23,28961,15707],{"className":28962},[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。",[11,28965,28966,28967,28996,28997,16577,29025,29053,29054,29111],{},"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（",[23,28968,28970,28984],{"className":28969,"translate":27},[31],[23,28971,28973],{"className":28972},[35],[37,28974,28975],{"xmlns":39},[42,28976,28977,28981],{},[45,28978,28979],{},[48,28980,9769],{},[114,28982,28983],{"encoding":116},"\\eta",[23,28985,28987],{"className":28986,"ariaHidden":122},[121],[23,28988,28990,28993],{"className":28989},[126],[23,28991],{"className":28992,"style":672},[130],[23,28994,9769],{"className":28995,"style":676},[135,136]," 接近 ",[23,28998,29000,29013],{"className":28999,"translate":27},[31],[23,29001,29003],{"className":29002},[35],[37,29004,29005],{"xmlns":39},[42,29006,29007,29011],{},[45,29008,29009],{},[2581,29010,1532],{},[114,29012,1532],{"encoding":116},[23,29014,29016],{"className":29015,"ariaHidden":122},[121],[23,29017,29019,29022],{"className":29018},[126],[23,29020],{"className":29021,"style":2770},[130],[23,29023,1532],{"className":29024},[135],[23,29026,29028,29041],{"className":29027,"translate":27},[31],[23,29029,29031],{"className":29030},[35],[37,29032,29033],{"xmlns":39},[42,29034,29035,29039],{},[45,29036,29037],{},[2581,29038,4435],{},[114,29040,4435],{"encoding":116},[23,29042,29044],{"className":29043,"ariaHidden":122},[121],[23,29045,29047,29050],{"className":29046},[126],[23,29048],{"className":29049,"style":2770},[130],[23,29051,4435],{"className":29052},[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 ",[23,29055,29057,29080],{"className":29056,"translate":27},[31],[23,29058,29060],{"className":29059},[35],[37,29061,29062],{"xmlns":39},[42,29063,29064,29077],{},[45,29065,29066,29068,29071,29073,29075],{},[48,29067,9769],{},[51,29069,29070],{},"≈",[2581,29072,4435],{},[48,29074,2875],{"mathvariant":111},[2581,29076,9956],{},[114,29078,29079],{"encoding":116},"\\eta\\approx 1\u002F2",[23,29081,29083,29102],{"className":29082,"ariaHidden":122},[121],[23,29084,29086,29090,29093,29096,29099],{"className":29085},[126],[23,29087],{"className":29088,"style":29089},[130],"height:0.6776em;vertical-align:-0.1944em;",[23,29091,9769],{"className":29092,"style":676},[135,136],[23,29094],{"className":29095,"style":149},[148],[23,29097,29070],{"className":29098},[153],[23,29100],{"className":29101,"style":149},[148],[23,29103,29105,29108],{"className":29104},[126],[23,29106],{"className":29107,"style":131},[130],[23,29109,10473],{"className":29110},[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。",[2501,29113,29115],{"id":29114},"_45-回归的贝叶斯风险","4.5 回归的贝叶斯风险",[11,29117,29118],{},"平方损失下，贝叶斯风险为",[23,29120,29122],{"className":29121,"translate":27},[26],[23,29123,29125,29203],{"className":29124,"translate":27},[31],[23,29126,29128],{"className":29127},[35],[37,29129,29130],{"xmlns":39,"display":40},[42,29131,29132,29200],{},[45,29133,29134,29140,29142,29144,29146,29148,29150,29152,29158,29160,29162,29164,29170,29172,29174,29176,29178,29186,29188,29190,29192,29194,29196,29198],{},[4518,29135,29136,29138],{},[48,29137,572],{},[51,29139,6883],{},[51,29141,69],{},[48,29143,902],{"mathvariant":571},[51,29145,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,29147,54],{"stretchy":53},[48,29149,369],{},[51,29151,4529],{},[4518,29153,29154,29156],{},[48,29155,8340],{},[51,29157,6883],{},[51,29159,54],{"stretchy":53},[48,29161,434],{},[51,29163,66],{"stretchy":53},[4518,29165,29166,29168],{},[51,29167,66],{"stretchy":53},[2581,29169,9956],{},[51,29171,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,29173,69],{},[48,29175,902],{"mathvariant":571},[51,29177,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[45,29179,29180,29182,29184],{},[48,29181,10794],{"mathvariant":111},[48,29183,495],{"mathvariant":111},[48,29185,1653],{"mathvariant":111},[51,29187,54],{"stretchy":53},[48,29189,369],{},[51,29191,60],{},[48,29193,434],{},[51,29195,66],{"stretchy":53},[51,29197,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,29199,112],{"mathvariant":111},[114,29201,29202],{"encoding":116},"R^*= \\mathbb E\\big[(Y-f^*(X))^2\\big]=\\mathbb E\\big[\\mathrm{Var}(Y\\mid X)\\big].",[23,29204,29206,29250,29280,29368,29404],{"className":29205,"ariaHidden":122},[121],[23,29207,29209,29212,29241,29244,29247],{"className":29208},[126],[23,29210],{"className":29211,"style":7047},[130],[23,29213,29215,29218],{"className":29214},[135],[23,29216,572],{"className":29217,"style":963},[135,136],[23,29219,29221],{"className":29220},[998],[23,29222,29224],{"className":29223},[199],[23,29225,29227],{"className":29226},[204],[23,29228,29230],{"className":29229,"style":7047},[208],[23,29231,29232,29235],{"style":4989},[23,29233],{"className":29234,"style":1015},[216],[23,29236,29238],{"className":29237},[1019,1020,1021,1022],[23,29239,6883],{"className":29240},[612,1022],[23,29242],{"className":29243,"style":149},[148],[23,29245,69],{"className":29246},[153],[23,29248],{"className":29249,"style":149},[148],[23,29251,29253,29256,29259,29265,29268,29271,29274,29277],{"className":29252},[126],[23,29254],{"className":29255,"style":2213},[130],[23,29257,902],{"className":29258},[135,644],[23,29260,29262],{"className":29261},[135],[23,29263,924],{"className":29264},[1066,1067],[23,29266,54],{"className":29267},[140],[23,29269,369],{"className":29270,"style":392},[135,136],[23,29272],{"className":29273,"style":392},[148],[23,29275,4529],{"className":29276},[612],[23,29278],{"className":29279,"style":392},[148],[23,29281,29283,29286,29315,29318,29321,29324,29353,29359,29362,29365],{"className":29282},[126],[23,29284],{"className":29285,"style":17160},[130],[23,29287,29289,29292],{"className":29288},[135],[23,29290,8340],{"className":29291,"style":11402},[135,136],[23,29293,29295],{"className":29294},[998],[23,29296,29298],{"className":29297},[199],[23,29299,29301],{"className":29300},[204],[23,29302,29304],{"className":29303,"style":7047},[208],[23,29305,29306,29309],{"style":4989},[23,29307],{"className":29308,"style":1015},[216],[23,29310,29312],{"className":29311},[1019,1020,1021,1022],[23,29313,6883],{"className":29314},[612,1022],[23,29316,54],{"className":29317},[140],[23,29319,434],{"className":29320,"style":454},[135,136],[23,29322,66],{"className":29323},[169],[23,29325,29327,29330],{"className":29326},[169],[23,29328,66],{"className":29329},[169],[23,29331,29333],{"className":29332},[998],[23,29334,29336],{"className":29335},[199],[23,29337,29339],{"className":29338},[204],[23,29340,29342],{"className":29341,"style":11006},[208],[23,29343,29344,29347],{"style":4989},[23,29345],{"className":29346,"style":1015},[216],[23,29348,29350],{"className":29349},[1019,1020,1021,1022],[23,29351,9956],{"className":29352},[135,1022],[23,29354,29356],{"className":29355},[135],[23,29357,945],{"className":29358},[1066,1067],[23,29360],{"className":29361,"style":149},[148],[23,29363,69],{"className":29364},[153],[23,29366],{"className":29367,"style":149},[148],[23,29369,29371,29374,29377,29383,29389,29392,29395,29398,29401],{"className":29370},[126],[23,29372],{"className":29373,"style":2213},[130],[23,29375,902],{"className":29376},[135,644],[23,29378,29380],{"className":29379},[135],[23,29381,924],{"className":29382},[1066,1067],[23,29384,29386],{"className":29385},[135],[23,29387,11074],{"className":29388},[135,5391],[23,29390,54],{"className":29391},[140],[23,29393,369],{"className":29394,"style":392},[135,136],[23,29396],{"className":29397,"style":149},[148],[23,29399,60],{"className":29400},[153],[23,29402],{"className":29403,"style":149},[148],[23,29405,29407,29410,29413,29416,29422],{"className":29406},[126],[23,29408],{"className":29409,"style":2213},[130],[23,29411,434],{"className":29412,"style":454},[135,136],[23,29414,66],{"className":29415},[169],[23,29417,29419],{"className":29418},[135],[23,29420,945],{"className":29421},[1066,1067],[23,29423,112],{"className":29424},[135],[11,29426,29427,29428,29456,29457,29571],{},"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 ",[23,29429,29431,29444],{"className":29430,"translate":27},[31],[23,29432,29434],{"className":29433},[35],[37,29435,29436],{"xmlns":39},[42,29437,29438,29442],{},[45,29439,29440],{},[48,29441,63],{},[114,29443,63],{"encoding":116},[23,29445,29447],{"className":29446,"ariaHidden":122},[121],[23,29448,29450,29453],{"className":29449},[126],[23,29451],{"className":29452,"style":470},[130],[23,29454,63],{"className":29455,"style":144},[135,136]," 上学到的回归器 ",[23,29458,29460,29482],{"className":29459,"translate":27},[31],[23,29461,29463],{"className":29462},[35],[37,29464,29465],{"xmlns":39},[42,29466,29467,29479],{},[45,29468,29469],{},[898,29470,29471,29477],{},[16811,29472,29473,29475],{"accent":122},[48,29474,8340],{},[51,29476,16817],{},[48,29478,63],{},[114,29480,29481],{"encoding":116},"\\hat f_D",[23,29483,29485],{"className":29484,"ariaHidden":122},[121],[23,29486,29488,29491],{"className":29487},[126],[23,29489],{"className":29490,"style":16830},[130],[23,29492,29494,29536],{"className":29493},[135],[23,29495,29497],{"className":29496},[135,16834],[23,29498,29500,29528],{"className":29499},[199,200],[23,29501,29503,29525],{"className":29502},[204],[23,29504,29506,29514],{"className":29505,"style":16844},[208],[23,29507,29508,29511],{"style":7132},[23,29509],{"className":29510,"style":217},[216],[23,29512,8340],{"className":29513,"style":11402},[135,136],[23,29515,29516,29519],{"style":16855},[23,29517],{"className":29518,"style":217},[216],[23,29520,29522],{"className":29521,"style":16863},[16862],[23,29523,16817],{"className":29524},[135],[23,29526,297],{"className":29527},[296],[23,29529,29531],{"className":29530},[204],[23,29532,29534],{"className":29533,"style":16876},[208],[23,29535],{},[23,29537,29539],{"className":29538},[998],[23,29540,29542,29563],{"className":29541},[199,200],[23,29543,29545,29560],{"className":29544},[204],[23,29546,29548],{"className":29547,"style":2026},[208],[23,29549,29551,29554],{"style":29550},"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;",[23,29552],{"className":29553,"style":1015},[216],[23,29555,29557],{"className":29556},[1019,1020,1021,1022],[23,29558,63],{"className":29559,"style":144},[135,136,1022],[23,29561,297],{"className":29562},[296],[23,29564,29566],{"className":29565},[204],[23,29567,29569],{"className":29568,"style":2048},[208],[23,29570],{},"，有标准的偏差-方差分解：",[23,29573,29575],{"className":29574,"translate":27},[26],[23,29576,29578,29759],{"className":29577,"translate":27},[31],[23,29579,29581],{"className":29580},[35],[37,29582,29583],{"xmlns":39,"display":40},[42,29584,29585,29756],{},[45,29586,29587,29593,29595,29597,29599,29601,29611,29613,29615,29617,29623,29625,29627,29630,29648,29650,29709,29711,29754],{},[898,29588,29589,29591],{},[48,29590,902],{"mathvariant":571},[48,29592,63],{},[51,29594,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,29596,54],{"stretchy":53},[48,29598,369],{},[51,29600,4529],{},[898,29602,29603,29609],{},[16811,29604,29605,29607],{"accent":122},[48,29606,8340],{},[51,29608,16817],{},[48,29610,63],{},[51,29612,54],{"stretchy":53},[48,29614,434],{},[51,29616,66],{"stretchy":53},[4518,29618,29619,29621],{},[51,29620,66],{"stretchy":53},[2581,29622,9956],{},[51,29624,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,29626,69],{},[148,29628],{"linebreak":29629},"newline",[6963,29631,29632,29646],{},[6963,29633,29634,29644],{},[19141,29635,29636,29639,29642],{},[48,29637,29638],{},"σ",[48,29640,29641],{},"ε",[2581,29643,9956],{},[51,29645,8481],{"stretchy":122},[88,29647,335],{},[51,29649,5106],{},[6963,29651,29652,29702],{},[6963,29653,29654,29700],{},[45,29655,29656,29658,29664,29666,29668,29670,29672,29678,29688,29690,29692,29694],{},[51,29657,54],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[4518,29659,29660,29662],{},[48,29661,8340],{},[51,29663,6883],{},[51,29665,54],{"stretchy":53},[48,29667,434],{},[51,29669,66],{"stretchy":53},[51,29671,4529],{},[898,29673,29674,29676],{},[48,29675,902],{"mathvariant":571},[48,29677,63],{},[898,29679,29680,29686],{},[16811,29681,29682,29684],{"accent":122},[48,29683,8340],{},[51,29685,16817],{},[48,29687,63],{},[51,29689,54],{"stretchy":53},[48,29691,434],{},[51,29693,66],{"stretchy":53},[4518,29695,29696,29698],{},[51,29697,66],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[2581,29699,9956],{},[51,29701,8481],{"stretchy":122},[4518,29703,29704,29707],{},[88,29705,29706],{},"偏差",[2581,29708,9956],{},[51,29710,5106],{},[6963,29712,29713,29751],{},[6963,29714,29715,29749],{},[45,29716,29717,29729,29731,29741,29743,29745,29747],{},[898,29718,29719,29727],{},[45,29720,29721,29723,29725],{},[48,29722,10794],{"mathvariant":111},[48,29724,495],{"mathvariant":111},[48,29726,1653],{"mathvariant":111},[48,29728,63],{},[51,29730,54],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[898,29732,29733,29739],{},[16811,29734,29735,29737],{"accent":122},[48,29736,8340],{},[51,29738,16817],{},[48,29740,63],{},[51,29742,54],{"stretchy":53},[48,29744,434],{},[51,29746,66],{"stretchy":53},[51,29748,66],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,29750,8481],{"stretchy":122},[88,29752,29753],{},"方差",[48,29755,112],{"mathvariant":111},[114,29757,29758],{"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",[2501,30786,30788],{"id":30787},"_51-生成式与判别式","5.1 生成式与判别式",[11,30790,30791,30792,30916,30917,30945,30946,31012,31013,31057,31058,31061,31062,31065],{},"贝叶斯规则只依赖后验概率 ",[23,30793,30795,30832],{"className":30794,"translate":27},[31],[23,30796,30798],{"className":30797},[35],[37,30799,30800],{"xmlns":39},[42,30801,30802,30830],{},[45,30803,30804,30806,30808,30810,30812,30814,30816,30818,30820,30822,30824,30826,30828],{},[48,30805,9769],{},[51,30807,54],{"stretchy":53},[48,30809,753],{},[51,30811,66],{"stretchy":53},[51,30813,69],{},[48,30815,11],{},[51,30817,54],{"stretchy":53},[48,30819,369],{},[51,30821,69],{},[2581,30823,4435],{},[51,30825,60],{},[48,30827,753],{},[51,30829,66],{"stretchy":53},[114,30831,9796],{"encoding":116},[23,30833,30835,30862,30886,30904],{"className":30834,"ariaHidden":122},[121],[23,30836,30838,30841,30844,30847,30850,30853,30856,30859],{"className":30837},[126],[23,30839],{"className":30840,"style":131},[130],[23,30842,9769],{"className":30843,"style":676},[135,136],[23,30845,54],{"className":30846},[140],[23,30848,753],{"className":30849},[135,136],[23,30851,66],{"className":30852},[169],[23,30854],{"className":30855,"style":149},[148],[23,30857,69],{"className":30858},[153],[23,30860],{"className":30861,"style":149},[148],[23,30863,30865,30868,30871,30874,30877,30880,30883],{"className":30864},[126],[23,30866],{"className":30867,"style":131},[130],[23,30869,11],{"className":30870},[135,136],[23,30872,54],{"className":30873},[140],[23,30875,369],{"className":30876,"style":392},[135,136],[23,30878],{"className":30879,"style":149},[148],[23,30881,69],{"className":30882},[153],[23,30884],{"className":30885,"style":149},[148],[23,30887,30889,30892,30895,30898,30901],{"className":30888},[126],[23,30890],{"className":30891,"style":131},[130],[23,30893,4435],{"className":30894},[135],[23,30896],{"className":30897,"style":149},[148],[23,30899,60],{"className":30900},[153],[23,30902],{"className":30903,"style":149},[148],[23,30905,30907,30910,30913],{"className":30906},[126],[23,30908],{"className":30909,"style":131},[130],[23,30911,753],{"className":30912},[135,136],[23,30914,66],{"className":30915},[169],"。因此有两条路可以走：直接建模 ",[23,30918,30920,30933],{"className":30919,"translate":27},[31],[23,30921,30923],{"className":30922},[35],[37,30924,30925],{"xmlns":39},[42,30926,30927,30931],{},[45,30928,30929],{},[48,30930,9769],{},[114,30932,28983],{"encoding":116},[23,30934,30936],{"className":30935,"ariaHidden":122},[121],[23,30937,30939,30942],{"className":30938},[126],[23,30940],{"className":30941,"style":672},[130],[23,30943,9769],{"className":30944,"style":676},[135,136],"，或者分别建模类条件密度 ",[23,30947,30949,30973],{"className":30948,"translate":27},[31],[23,30950,30952],{"className":30951},[35],[37,30953,30954],{"xmlns":39},[42,30955,30956,30970],{},[45,30957,30958,30960,30962,30964,30966,30968],{},[48,30959,11],{},[51,30961,54],{"stretchy":53},[48,30963,753],{},[51,30965,60],{},[48,30967,660],{},[51,30969,66],{"stretchy":53},[114,30971,30972],{"encoding":116},"p(x\\mid y)",[23,30974,30976,31000],{"className":30975,"ariaHidden":122},[121],[23,30977,30979,30982,30985,30988,30991,30994,30997],{"className":30978},[126],[23,30980],{"className":30981,"style":131},[130],[23,30983,11],{"className":30984},[135,136],[23,30986,54],{"className":30987},[140],[23,30989,753],{"className":30990},[135,136],[23,30992],{"className":30993,"style":149},[148],[23,30995,60],{"className":30996},[153],[23,30998],{"className":30999,"style":149},[148],[23,31001,31003,31006,31009],{"className":31002},[126],[23,31004],{"className":31005,"style":131},[130],[23,31007,660],{"className":31008,"style":676},[135,136],[23,31010,66],{"className":31011},[169]," 与先验 ",[23,31014,31016,31036],{"className":31015,"translate":27},[31],[23,31017,31019],{"className":31018},[35],[37,31020,31021],{"xmlns":39},[42,31022,31023,31033],{},[45,31024,31025,31027,31029,31031],{},[48,31026,11],{},[51,31028,54],{"stretchy":53},[48,31030,660],{},[51,31032,66],{"stretchy":53},[114,31034,31035],{"encoding":116},"p(y)",[23,31037,31039],{"className":31038,"ariaHidden":122},[121],[23,31040,31042,31045,31048,31051,31054],{"className":31041},[126],[23,31043],{"className":31044,"style":131},[130],[23,31046,11],{"className":31047},[135,136],[23,31049,54],{"className":31050},[140],[23,31052,660],{"className":31053,"style":676},[135,136],[23,31055,66],{"className":31056},[169],"，再用贝叶斯公式合成。前者叫",[18,31059,31060],{},"判别式","（discriminative），后者叫",[18,31063,31064],{},"生成式","（generative）。",[11,31067,31068],{},"生成式的合成公式是",[23,31070,31072],{"className":31071,"translate":27},[26],[23,31073,31075,31161],{"className":31074,"translate":27},[31],[23,31076,31078],{"className":31077},[35],[37,31079,31080],{"xmlns":39,"display":40},[42,31081,31082,31158],{},[45,31083,31084,31086,31088,31090,31092,31094,31156],{},[48,31085,9769],{},[51,31087,54],{"stretchy":53},[48,31089,753],{},[51,31091,66],{"stretchy":53},[51,31093,69],{},[71,31095,31096,31116],{},[45,31097,31098,31104,31110,31112,31114],{},[898,31099,31100,31102],{},[48,31101,21697],{},[2581,31103,4435],{},[898,31105,31106,31108],{},[48,31107,11],{},[2581,31109,4435],{},[51,31111,54],{"stretchy":53},[48,31113,753],{},[51,31115,66],{"stretchy":53},[45,31117,31118,31124,31130,31132,31134,31136,31138,31144,31150,31152,31154],{},[898,31119,31120,31122],{},[48,31121,21697],{},[2581,31123,1532],{},[898,31125,31126,31128],{},[48,31127,11],{},[2581,31129,1532],{},[51,31131,54],{"stretchy":53},[48,31133,753],{},[51,31135,66],{"stretchy":53},[51,31137,5106],{},[898,31139,31140,31142],{},[48,31141,21697],{},[2581,31143,4435],{},[898,31145,31146,31148],{},[48,31147,11],{},[2581,31149,4435],{},[51,31151,54],{"stretchy":53},[48,31153,753],{},[51,31155,66],{"stretchy":53},[48,31157,112],{"mathvariant":111},[114,31159,31160],{"encoding":116},"\\eta(x)=\\frac{\\pi_1p_1(x)}{\\pi_0p_0(x)+\\pi_1p_1(x)}.",[23,31162,31164,31191],{"className":31163,"ariaHidden":122},[121],[23,31165,31167,31170,31173,31176,31179,31182,31185,31188],{"className":31166},[126],[23,31168],{"className":31169,"style":131},[130],[23,31171,9769],{"className":31172,"style":676},[135,136],[23,31174,54],{"className":31175},[140],[23,31177,753],{"className":31178},[135,136],[23,31180,66],{"className":31181},[169],[23,31183],{"className":31184,"style":149},[148],[23,31186,69],{"className":31187},[153],[23,31189],{"className":31190,"style":149},[148],[23,31192,31194,31197,31529],{"className":31193},[126],[23,31195],{"className":31196,"style":185},[130],[23,31198,31200,31203,31526],{"className":31199},[135],[23,31201],{"className":31202},[140,192],[23,31204,31206],{"className":31205},[71],[23,31207,31209,31518],{"className":31208},[199,200],[23,31210,31212,31515],{"className":31211},[204],[23,31213,31215,31410,31418],{"className":31214,"style":209},[208],[23,31216,31217,31220],{"style":212},[23,31218],{"className":31219,"style":217},[216],[23,31221,31223,31263,31303,31306,31309,31312,31315,31318,31321,31361,31401,31404,31407],{"className":31222},[135],[23,31224,31226,31229],{"className":31225},[135],[23,31227,21697],{"className":31228,"style":676},[135,136],[23,31230,31232],{"className":31231},[998],[23,31233,31235,31255],{"className":31234},[199,200],[23,31236,31238,31252],{"className":31237},[204],[23,31239,31241],{"className":31240,"style":18651},[208],[23,31242,31243,31246],{"style":21854},[23,31244],{"className":31245,"style":1015},[216],[23,31247,31249],{"className":31248},[1019,1020,1021,1022],[23,31250,1532],{"className":31251},[135,1022],[23,31253,297],{"className":31254},[296],[23,31256,31258],{"className":31257},[204],[23,31259,31261],{"className":31260,"style":2048},[208],[23,31262],{},[23,31264,31266,31269],{"className":31265},[135],[23,31267,11],{"className":31268},[135,136],[23,31270,31272],{"className":31271},[998],[23,31273,31275,31295],{"className":31274},[199,200],[23,31276,31278,31292],{"className":31277},[204],[23,31279,31281],{"className":31280,"style":18651},[208],[23,31282,31283,31286],{"style":2029},[23,31284],{"className":31285,"style":1015},[216],[23,31287,31289],{"className":31288},[1019,1020,1021,1022],[23,31290,1532],{"className":31291},[135,1022],[23,31293,297],{"className":31294},[296],[23,31296,31298],{"className":31297},[204],[23,31299,31301],{"className":31300,"style":2048},[208],[23,31302],{},[23,31304,54],{"className":31305},[140],[23,31307,753],{"className":31308},[135,136],[23,31310,66],{"className":31311},[169],[23,31313],{"className":31314,"style":392},[148],[23,31316,5106],{"className":31317},[612],[23,31319],{"className":31320,"style":392},[148],[23,31322,31324,31327],{"className":31323},[135],[23,31325,21697],{"className":31326,"style":676},[135,136],[23,31328,31330],{"className":31329},[998],[23,31331,31333,31353],{"className":31332},[199,200],[23,31334,31336,31350],{"className":31335},[204],[23,31337,31339],{"className":31338,"style":18651},[208],[23,31340,31341,31344],{"style":21854},[23,31342],{"className":31343,"style":1015},[216],[23,31345,31347],{"className":31346},[1019,1020,1021,1022],[23,31348,4435],{"className":31349},[135,1022],[23,31351,297],{"className":31352},[296],[23,31354,31356],{"className":31355},[204],[23,31357,31359],{"className":31358,"style":2048},[208],[23,31360],{},[23,31362,31364,31367],{"className":31363},[135],[23,31365,11],{"className":31366},[135,136],[23,31368,31370],{"className":31369},[998],[23,31371,31373,31393],{"className":31372},[199,200],[23,31374,31376,31390],{"className":31375},[204],[23,31377,31379],{"className":31378,"style":18651},[208],[23,31380,31381,31384],{"style":2029},[23,31382],{"className":31383,"style":1015},[216],[23,31385,31387],{"className":31386},[1019,1020,1021,1022],[23,31388,4435],{"className":31389},[135,1022],[23,31391,297],{"className":31392},[296],[23,31394,31396],{"className":31395},[204],[23,31397,31399],{"className":31398,"style":2048},[208],[23,31400],{},[23,31402,54],{"className":31403},[140],[23,31405,753],{"className":31406},[135,136],[23,31408,66],{"className":31409},[169],[23,31411,31412,31415],{"style":235},[23,31413],{"className":31414,"style":217},[216],[23,31416],{"className":31417,"style":243},[242],[23,31419,31420,31423],{"style":246},[23,31421],{"className":31422,"style":217},[216],[23,31424,31426,31466,31506,31509,31512],{"className":31425},[135],[23,31427,31429,31432],{"className":31428},[135],[23,31430,21697],{"className":31431,"style":676},[135,136],[23,31433,31435],{"className":31434},[998],[23,31436,31438,31458],{"className":31437},[199,200],[23,31439,31441,31455],{"className":31440},[204],[23,31442,31444],{"className":31443,"style":18651},[208],[23,31445,31446,31449],{"style":21854},[23,31447],{"className":31448,"style":1015},[216],[23,31450,31452],{"className":31451},[1019,1020,1021,1022],[23,31453,4435],{"className":31454},[135,1022],[23,31456,297],{"className":31457},[296],[23,31459,31461],{"className":31460},[204],[23,31462,31464],{"className":31463,"style":2048},[208],[23,31465],{},[23,31467,31469,31472],{"className":31468},[135],[23,31470,11],{"className":31471},[135,136],[23,31473,31475],{"className":31474},[998],[23,31476,31478,31498],{"className":31477},[199,200],[23,31479,31481,31495],{"className":31480},[204],[23,31482,31484],{"className":31483,"style":18651},[208],[23,31485,31486,31489],{"style":2029},[23,31487],{"className":31488,"style":1015},[216],[23,31490,31492],{"className":31491},[1019,1020,1021,1022],[23,31493,4435],{"className":31494},[135,1022],[23,31496,297],{"className":31497},[296],[23,31499,31501],{"className":31500},[204],[23,31502,31504],{"className":31503,"style":2048},[208],[23,31505],{},[23,31507,54],{"className":31508},[140],[23,31510,753],{"className":31511},[135,136],[23,31513,66],{"className":31514},[169],[23,31516,297],{"className":31517},[296],[23,31519,31521],{"className":31520},[204],[23,31522,31524],{"className":31523,"style":304},[208],[23,31525],{},[23,31527],{"className":31528},[169,192],[23,31530,112],{"className":31531},[135],[11,31533,31534,31535,31564,31565,31593],{},"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 ",[23,31536,31538,31552],{"className":31537,"translate":27},[31],[23,31539,31541],{"className":31540},[35],[37,31542,31543],{"xmlns":39},[42,31544,31545,31549],{},[45,31546,31547],{},[48,31548,22116],{"mathvariant":111},[114,31550,31551],{"encoding":116},"\\Sigma",[23,31553,31555],{"className":31554,"ariaHidden":122},[121],[23,31556,31558,31561],{"className":31557},[126],[23,31559],{"className":31560,"style":470},[130],[23,31562,22116],{"className":31563},[135],"，此时对数似然比是 ",[23,31566,31568,31581],{"className":31567,"translate":27},[31],[23,31569,31571],{"className":31570},[35],[37,31572,31573],{"xmlns":39},[42,31574,31575,31579],{},[45,31576,31577],{},[48,31578,753],{},[114,31580,753],{"encoding":116},[23,31582,31584],{"className":31583,"ariaHidden":122},[121],[23,31585,31587,31590],{"className":31586},[126],[23,31588],{"className":31589,"style":703},[130],[23,31591,753],{"className":31592},[135,136]," 的线性函数：",[23,31595,31597],{"className":31596,"translate":27},[26],[23,31598,31600,31689],{"className":31599,"translate":27},[31],[23,31601,31603],{"className":31602},[35],[37,31604,31605],{"xmlns":39,"display":40},[42,31606,31607,31686],{},[45,31608,31609,31611,31613,31643,31645,31647,31653,31655,31661,31667,31677,31679,31681,31684],{},[48,31610,22422],{},[51,31612,6961],{},[71,31614,31615,31629],{},[45,31616,31617,31623,31625,31627],{},[898,31618,31619,31621],{},[48,31620,11],{},[2581,31622,4435],{},[51,31624,54],{"stretchy":53},[48,31626,753],{},[51,31628,66],{"stretchy":53},[45,31630,31631,31637,31639,31641],{},[898,31632,31633,31635],{},[48,31634,11],{},[2581,31636,1532],{},[51,31638,54],{"stretchy":53},[48,31640,753],{},[51,31642,66],{"stretchy":53},[51,31644,69],{},[51,31646,54],{"stretchy":53},[898,31648,31649,31651],{},[48,31650,22109],{},[2581,31652,4435],{},[51,31654,4529],{},[898,31656,31657,31659],{},[48,31658,22109],{},[2581,31660,1532],{},[4518,31662,31663,31665],{},[51,31664,66],{"stretchy":53},[48,31666,22475],{"mathvariant":111},[4518,31668,31669,31671],{},[48,31670,22116],{"mathvariant":111},[45,31672,31673,31675],{},[51,31674,4529],{},[2581,31676,4435],{},[48,31678,753],{},[51,31680,5106],{},[88,31682,31683],{},"常数",[51,31685,747],{"separator":122},[114,31687,31688],{"encoding":116},"\\log\\frac{p_1(x)}{p_0(x)}\n=(\\mu_1-\\mu_0)^\\top\\Sigma^{-1}x+\\text{常数},",[23,31690,31692,31869,31927,32049],{"className":31691,"ariaHidden":122},[121],[23,31693,31695,31698,31703,31706,31860,31863,31866],{"className":31694},[126],[23,31696],{"className":31697,"style":185},[130],[23,31699,22579,31701],{"className":31700},[1786],[23,31702,7091],{"style":7090},[23,31704],{"className":31705,"style":280},[148],[23,31707,31709,31712,31857],{"className":31708},[135],[23,31710],{"className":31711},[140,192],[23,31713,31715],{"className":31714},[71],[23,31716,31718,31849],{"className":31717},[199,200],[23,31719,31721,31846],{"className":31720},[204],[23,31722,31724,31781,31789],{"className":31723,"style":209},[208],[23,31725,31726,31729],{"style":212},[23,31727],{"className":31728,"style":217},[216],[23,31730,31732,31772,31775,31778],{"className":31731},[135],[23,31733,31735,31738],{"className":31734},[135],[23,31736,11],{"className":31737},[135,136],[23,31739,31741],{"className":31740},[998],[23,31742,31744,31764],{"className":31743},[199,200],[23,31745,31747,31761],{"className":31746},[204],[23,31748,31750],{"className":31749,"style":18651},[208],[23,31751,31752,31755],{"style":2029},[23,31753],{"className":31754,"style":1015},[216],[23,31756,31758],{"className":31757},[1019,1020,1021,1022],[23,31759,1532],{"className":31760},[135,1022],[23,31762,297],{"className":31763},[296],[23,31765,31767],{"className":31766},[204],[23,31768,31770],{"className":31769,"style":2048},[208],[23,31771],{},[23,31773,54],{"className":31774},[140],[23,31776,753],{"className":31777},[135,136],[23,31779,66],{"className":31780},[169],[23,31782,31783,31786],{"style":235},[23,31784],{"className":31785,"style":217},[216],[23,31787],{"className":31788,"style":243},[242],[23,31790,31791,31794],{"style":246},[23,31792],{"className":31793,"style":217},[216],[23,31795,31797,31837,31840,31843],{"className":31796},[135],[23,31798,31800,31803],{"className":31799},[135],[23,31801,11],{"className":31802},[135,136],[23,31804,31806],{"className":31805},[998],[23,31807,31809,31829],{"className":31808},[199,200],[23,31810,31812,31826],{"className":31811},[204],[23,31813,31815],{"className":31814,"style":18651},[208],[23,31816,31817,31820],{"style":2029},[23,31818],{"className":31819,"style":1015},[216],[23,31821,31823],{"className":31822},[1019,1020,1021,1022],[23,31824,4435],{"className":31825},[135,1022],[23,31827,297],{"className":31828},[296],[23,31830,31832],{"className":31831},[204],[23,31833,31835],{"className":31834,"style":2048},[208],[23,31836],{},[23,31838,54],{"className":31839},[140],[23,31841,753],{"className":31842},[135,136],[23,31844,66],{"className":31845},[169],[23,31847,297],{"className":31848},[296],[23,31850,31852],{"className":31851},[204],[23,31853,31855],{"className":31854,"style":304},[208],[23,31856],{},[23,31858],{"className":31859},[169,192],[23,31861],{"className":31862,"style":149},[148],[23,31864,69],{"className":31865},[153],[23,31867],{"className":31868,"style":149},[148],[23,31870,31872,31875,31878,31918,31921,31924],{"className":31871},[126],[23,31873],{"className":31874,"style":131},[130],[23,31876,54],{"className":31877},[140],[23,31879,31881,31884],{"className":31880},[135],[23,31882,22109],{"className":31883},[135,136],[23,31885,31887],{"className":31886},[998],[23,31888,31890,31910],{"className":31889},[199,200],[23,31891,31893,31907],{"className":31892},[204],[23,31894,31896],{"className":31895,"style":18651},[208],[23,31897,31898,31901],{"style":2029},[23,31899],{"className":31900,"style":1015},[216],[23,31902,31904],{"className":31903},[1019,1020,1021,1022],[23,31905,4435],{"className":31906},[135,1022],[23,3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",[23,32068,32070,32096],{"className":32069,"translate":27},[31],[23,32071,32073],{"className":32072},[35],[37,32074,32075],{"xmlns":39},[42,32076,32077,32093],{},[45,32078,32079,32085,32087],{},[898,32080,32081,32083],{},[48,32082,22116],{"mathvariant":111},[2581,32084,1532],{},[51,32086,747],{"separator":122},[898,32088,32089,32091],{},[48,32090,22116],{"mathvariant":111},[2581,32092,4435],{},[114,32094,32095],{"encoding":116},"\\Sigma_0,\\Sigma_1",[23,32097,32099],{"className":32098,"ariaHidden":122},[121],[23,32100,32102,32105,32145,32148,32151],{"className":32101},[126],[23,32103],{"className":32104,"style":817},[130],[23,32106,32108,32111],{"className":32107},[135],[23,32109,22116],{"className":32110},[135],[23,32112,32114],{"className":32113},[998],[23,32115,32117,32137],{"className":32116},[199,200],[23,32118,32120,32134],{"className":32119},[204],[23,32121,32123],{"className":32122,"style":18651},[208],[23,32124,32125,32128],{"style":2029},[23,32126],{"className":32127,"style":1015},[216],[23,32129,32131],{"className":32130},[1019,1020,1021,1022],[23,32132,1532],{"className":32133},[135,1022],[23,32135,297],{"className":32136},[296],[23,32138,32140],{"className":32139},[204],[23,32141,32143],{"className":32142,"style":2048},[208],[23,32144],{},[23,32146,747],{"className":32147},[824],[23,32149],{"className":32150,"style":280},[148],[23,32152,32154,32157],{"className":32153},[135],[23,32155,22116],{"className":32156},[135],[23,32158,32160],{"className":32159},[998],[23,32161,32163,32183],{"className":32162},[199,200],[23,32164,32166,32180],{"className":32165},[204],[23,32167,32169],{"className":32168,"style":18651},[208],[23,32170,32171,32174],{"style":2029},[23,32172],{"className":32173,"style":1015},[216],[23,32175,32177],{"className":32176},[1019,1020,1021,1022],[23,32178,4435],{"className":32179},[135,1022],[23,32181,297],{"className":32182},[296],[23,32184,32186],{"className":32185},[204],[23,32187,32189],{"className":32188,"style":2048},[208],[23,32190],{},"，对数似然比出现 ",[23,32193,32195,32245],{"className":32194,"translate":27},[31],[23,32196,32198],{"className":32197},[35],[37,32199,32200],{"xmlns":39},[42,32201,32202,32242],{},[45,32203,32204,32210,32212,32224,32226,32238,32240],{},[4518,32205,32206,32208],{},[48,32207,753],{},[48,32209,22475],{"mathvariant":111},[51,32211,54],{"stretchy":53},[19141,32213,32214,32216,32218],{},[48,32215,22116],{"mathvariant":111},[2581,32217,4435],{},[45,32219,32220,32222],{},[51,32221,4529],{},[2581,32223,4435],{},[51,32225,4529],{},[19141,32227,32228,32230,32232],{},[48,32229,22116],{"mathvariant":111},[2581,32231,1532],{},[45,32233,32234,32236],{},[51,32235,4529],{},[2581,32237,4435],{},[51,32239,66],{"stretchy":53},[48,32241,753],{},[114,32243,32244],{"encoding":116},"x^\\top(\\Sigma_1^{-1}-\\Sigma_0^{-1})x",[23,32246,32248,32357],{"className":32247,"ariaHidden":122},[121],[23,32249,32251,32255,32284,32287,32348,32351,32354],{"className":32250},[126],[23,32252],{"className":32253,"style":32254},[130],"height:1.1205em;vertical-align:-0.2663em;",[23,32256,32258,32261],{"className":32257},[135],[23,32259,753],{"className":32260},[135,136],[23,32262,32264],{"className":32263},[998],[23,32265,32267],{"className":32266},[199],[23,32268,32270],{"className":32269},[204],[23,32271,32273],{"className":32272,"style":4751},[208],[23,32274,32275,32278],{"style":4696},[23,32276],{"className":32277,"style":1015},[216],[23,32279,32281],{"className":32280},[1019,1020,1021,1022],[23,32282,22475],{"className":32283},[135,1022],[23,32285,54],{"className":32286},[140],[23,32288,32290,32293],{"className":32289},[135],[23,32291,22116],{"className":32292},[135],[23,32294,32296],{"className":32295},[998],[23,32297,32299,32339],{"className":32298},[199,200],[23,32300,32302,32336],{"className":32301},[204],[23,32303,32306,32318],{"className":32304,"style":32305},[208],"height:0.8542em;",[23,32307,32309,32312],{"style":32308},"top:-2.4337em;margin-left:0em;margin-right:0.05em;",[23,32310],{"className":32311,"style":1015},[216],[23,32313,32315],{"className":32314},[1019,1020,1021,1022],[23,32316,4435],{"className":32317},[135,1022],[23,32319,32321,32324],{"style":32320},"top:-3.1031em;margin-right:0.05em;",[23,32322],{"className":32323,"style":1015},[216],[23,32325,32327],{"className":32326},[1019,1020,1021,1022],[23,32328,32330,32333],{"className":32329},[135,1022],[23,32331,4529],{"className":32332},[135,1022],[23,32334,4435],{"className":32335},[135,1022],[23,32337,297],{"className":32338},[296],[23,32340,32342],{"className":32341},[204],[23,32343,32346],{"className":32344,"style":32345},[208],"height:0.2663em;",[23,32347],{},[23,32349],{"className":32350,"style":392},[148],[23,32352,4529],{"className":32353},[612],[23,32355],{"className":32356,"style":392},[148],[23,32358,32360,32363,32420,32423],{"className":32359},[126],[23,32361],{"className":32362,"style":32254},[130],[23,32364,32366,32369],{"className":32365},[135],[23,32367,22116],{"className":32368},[135],[23,32370,32372],{"className":32371},[998],[23,32373,32375,32412],{"className":32374},[199,200],[23,32376,32378,32409],{"className":32377},[204],[23,32379,32381,32392],{"className":32380,"style":32305},[208],[23,32382,32383,32386],{"style":32308},[23,32384],{"className":32385,"style":1015},[216],[23,32387,32389],{"className":32388},[1019,1020,1021,1022],[23,32390,1532],{"className":32391},[135,1022],[23,32393,32394,32397],{"style":32320},[23,32395],{"className":32396,"style":1015},[216],[23,32398,32400],{"className":32399},[1019,1020,1021,1022],[23,32401,32403,32406],{"className":32402},[135,1022],[23,32404,4529],{"className":32405},[135,1022],[23,32407,4435],{"className":32408},[135,1022],[23,32410,297],{"className":32411},[296],[23,32413,32415],{"className":32414},[204],[23,32416,32418],{"className":32417,"style":32345},[208],[23,32419],{},[23,32421,66],{"className":32422},[169],[23,32424,753],{"className":32425},[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 ",[23,32428,32430,32453],{"className":32429,"translate":27},[31],[23,32431,32433],{"className":32432},[35],[37,32434,32435],{"xmlns":39},[42,32436,32437,32451],{},[45,32438,32439,32441,32443,32445,32447,32449],{},[48,32440,11],{},[51,32442,54],{"stretchy":53},[48,32444,753],{},[51,32446,60],{},[48,32448,660],{},[51,32450,66],{"stretchy":53},[114,32452,30972],{"encoding":116},[23,32454,32456,32480],{"className":32455,"ariaHidden":122},[121],[23,32457,32459,32462,32465,32468,32471,32474,32477],{"className":32458},[126],[23,32460],{"className":32461,"style":131},[130],[23,32463,11],{"className":32464},[135,136],[23,32466,54],{"className":32467},[140],[23,32469,753],{"className":32470},[135,136],[23,32472],{"className":32473,"style":149},[148],[23,32475,60],{"className":32476},[153],[23,32478],{"className":32479,"style":149},[148],[23,32481,32483,32486,32489],{"className":32482},[126],[23,32484],{"className":32485,"style":131},[130],[23,32487,660],{"className":32488,"style":676},[135,136],[23,32490,66],{"className":32491},[169]," 拆成各个特征的乘积，用很大的偏差换取很小的方差。",[11,32494,32495,32496,32524,32525,32590],{},"两条路的目标是同一个 ",[23,32497,32499,32512],{"className":32498,"translate":27},[31],[23,32500,32502],{"className":32501},[35],[37,32503,32504],{"xmlns":39},[42,32505,32506,32510],{},[45,32507,32508],{},[48,32509,9769],{},[114,32511,28983],{"encoding":116},[23,32513,32515],{"className":32514,"ariaHidden":122},[121],[23,32516,32518,32521],{"className":32517},[126],[23,32519],{"className":32520,"style":672},[130],[23,32522,9769],{"className":32523,"style":676},[135,136],"，但统计性质不同。生成式利用了 ",[23,32526,32528,32551],{"className":32527,"translate":27},[31],[23,32529,32531],{"className":32530},[35],[37,32532,32533],{"xmlns":39},[42,32534,32535,32549],{},[45,32536,32537,32539,32541,32543,32545,32547],{},[48,32538,11],{},[51,32540,54],{"stretchy":53},[48,32542,753],{},[51,32544,60],{},[48,32546,660],{},[51,32548,66],{"stretchy":53},[114,32550,30972],{"encoding":116},[23,32552,32554,32578],{"className":32553,"ariaHidden":122},[121],[23,32555,32557,32560,32563,32566,32569,32572,32575],{"className":32556},[126],[23,32558],{"className":32559,"style":131},[130],[23,32561,11],{"className":32562},[135,136],[23,32564,54],{"className":32565},[140],[23,32567,753],{"className":32568},[135,136],[23,32570],{"className":32571,"style":149},[148],[23,32573,60],{"className":32574},[153],[23,32576],{"className":32577,"style":149},[148],[23,32579,32581,32584,32587],{"className":32580},[126],[23,32582],{"className":32583,"style":131},[130],[23,32585,660],{"className":32586,"style":676},[135,136],[23,32588,66],{"className":32589},[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。",[11,32592,32593,32594,32597,32598,32626,32627,32700,32701,32759],{},"这里说的是 ",[18,32595,32596],{},"oracle 贝叶斯规则","：它假设 ",[23,32599,32601,32614],{"className":32600,"translate":27},[31],[23,32602,32604],{"className":32603},[35],[37,32605,32606],{"xmlns":39},[42,32607,32608,32612],{},[45,32609,32610],{},[48,32611,9769],{},[114,32613,28983],{"encoding":116},[23,32615,32617],{"className":32616,"ariaHidden":122},[121],[23,32618,32620,32623],{"className":32619},[126],[23,32621],{"className":32622,"style":672},[130],[23,32624,9769],{"className":32625,"style":676},[135,136]," 已知。实践中我们把估计的 ",[23,32628,32630,32648],{"className":32629,"translate":27},[31],[23,32631,32633],{"className":32632},[35],[37,32634,32635],{"xmlns":39},[42,32636,32637,32645],{},[45,32638,32639],{},[16811,32640,32641,32643],{"accent":122},[48,32642,9769],{},[51,32644,16817],{},[114,32646,32647],{"encoding":116},"\\hat\\eta",[23,32649,32651],{"className":32650,"ariaHidden":122},[121],[23,32652,32654,32657],{"className":32653},[126],[23,32655],{"className":32656,"style":5414},[130],[23,32658,32660],{"className":32659},[135,16834],[23,32661,32663,32692],{"className":32662},[199,200],[23,32664,32666,32689],{"className":32665},[204],[23,32667,32669,32677],{"className":32668,"style":779},[208],[23,32670,32671,32674],{"style":7132},[23,32672],{"className":32673,"style":217},[216],[23,32675,9769],{"className":32676,"style":676},[135,136],[23,32678,32679,32682],{"style":7132},[23,32680],{"className":32681,"style":217},[216],[23,32683,32686],{"className":32684,"style":32685},[16862],"left:-0.1944em;",[23,32687,16817],{"className":32688},[135],[23,32690,297],{"className":32691},[296],[23,32693,32695],{"className":32694},[204],[23,32696,32698],{"className":32697,"style":16876},[208],[23,32699],{}," 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 ",[23,32702,32704,32721],{"className":32703,"translate":27},[31],[23,32705,32707],{"className":32706},[35],[37,32708,32709],{"xmlns":39},[42,32710,32711,32719],{},[45,32712,32713],{},[4518,32714,32715,32717],{},[48,32716,572],{},[51,32718,6883],{},[114,32720,14966],{"encoding":116},[23,32722,32724],{"className":32723,"ariaHidden":122},[121],[23,32725,32727,32730],{"className":32726},[126],[23,32728],{"className":32729,"style":6914},[130],[23,32731,32733,32736],{"className":32732},[135],[23,32734,572],{"className":32735,"style":963},[135,136],[23,32737,32739],{"className":32738},[998],[23,32740,32742],{"className":32741},[199],[23,32743,32745],{"className":32744},[204],[23,32746,32748],{"className":32747,"style":6914},[208],[23,32749,32750,32753],{"style":4696},[23,32751],{"className":32752,"style":1015},[216],[23,32754,32756],{"className":32755},[1019,1020,1021,1022],[23,32757,6883],{"className":32758},[612,1022]," 之间的差距，正是第 7 节要分解的估计误差。",[11,32761,32762,32763,32807,32808,32840,32841,32869,32870,1471],{},"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 ",[23,32764,32766,32786],{"className":32765,"translate":27},[31],[23,32767,32769],{"className":32768},[35],[37,32770,32771],{"xmlns":39},[42,32772,32773,32783],{},[45,32774,32775,32777,32779,32781],{},[48,32776,9769],{},[51,32778,54],{"stretchy":53},[48,32780,753],{},[51,32782,66],{"stretchy":53},[114,32784,32785],{"encoding":116},"\\eta(x)",[23,32787,32789],{"className":32788,"ariaHidden":122},[121],[23,32790,32792,32795,32798,32801,32804],{"className":32791},[126],[23,32793],{"className":32794,"style":131},[130],[23,32796,9769],{"className":32797,"style":676},[135,136],[23,32799,54],{"className":32800},[140],[23,32802,753],{"className":32803},[135,136],[23,32805,66],{"className":32806},[169]," 是否越过 ",[23,32809,32811,32828],{"className":32810,"translate":27},[31],[23,32812,32814],{"className":32813},[35],[37,32815,32816],{"xmlns":39},[42,32817,32818,32826],{},[45,32819,32820,32822,32824],{},[2581,32821,4435],{},[48,32823,2875],{"mathvariant":111},[2581,32825,9956],{},[114,32827,10473],{"encoding":116},[23,32829,32831],{"className":32830,"ariaHidden":122},[121],[23,32832,32834,32837],{"className":32833},[126],[23,32835],{"className":32836,"style":131},[130],[23,32838,10473],{"className":32839},[135],"，而不要求 ",[23,32842,32844,32857],{"className":32843,"translate":27},[31],[23,32845,32847],{"className":32846},[35],[37,32848,32849],{"xmlns":39},[42,32850,32851,32855],{},[45,32852,32853],{},[48,32854,9769],{},[114,32856,28983],{"encoding":116},[23,32858,32860],{"className":32859,"ariaHidden":122},[121],[23,32861,32863,32866],{"className":32862},[126],[23,32864],{"className":32865,"style":672},[130],[23,32867,9769],{"className":32868,"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：",[18,32871,32872],{},"概率估计的质量与决策的质量不是同一件事",[11,32874,32875,32876,32919,32920,32923,32924,32989],{},"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 ",[23,32877,32879,32898],{"className":32878,"translate":27},[31],[23,32880,32882],{"className":32881},[35],[37,32883,32884],{"xmlns":39},[42,32885,32886,32896],{},[45,32887,32888,32890,32892,32894],{},[48,32889,9769],{},[51,32891,54],{"stretchy":53},[48,32893,753],{},[51,32895,66],{"stretchy":53},[114,32897,32785],{"encoding":116},[23,32899,32901],{"className":32900,"ariaHidden":122},[121],[23,32902,32904,32907,32910,32913,32916],{"className":32903},[126],[23,32905],{"className":32906,"style":131},[130],[23,32908,9769],{"className":32909,"style":676},[135,136],[23,32911,54],{"className":32912},[140],[23,32914,753],{"className":32915},[135,136],[23,32917,66],{"className":32918},[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",[18,32921,32922],{},"严格 proper scoring rule","，它的总体最优解恰好是真实的条件分布 ",[23,32925,32927,32950],{"className":32926,"translate":27},[31],[23,32928,32930],{"className":32929},[35],[37,32931,32932],{"xmlns":39},[42,32933,32934,32948],{},[45,32935,32936,32938,32940,32942,32944,32946],{},[48,32937,11],{},[51,32939,54],{"stretchy":53},[48,32941,660],{},[51,32943,60],{},[48,32945,753],{},[51,32947,66],{"stretchy":53},[114,32949,6815],{"encoding":116},[23,32951,32953,32977],{"className":32952,"ariaHidden":122},[121],[23,32954,32956,32959,32962,32965,32968,32971,32974],{"className":32955},[126],[23,32957],{"className":32958,"style":131},[130],[23,32960,11],{"className":32961},[135,136],[23,32963,54],{"className":32964},[140],[23,32966,660],{"className":32967,"style":676},[135,136],[23,32969],{"className":32970,"style":149},[148],[23,32972,60],{"className":32973},[153],[23,32975],{"className":32976,"style":149},[148],[23,32978,32980,32983,32986],{"className":32979},[126],[23,32981],{"className":32982,"style":131},[130],[23,32984,753],{"className":32985},[135,136],[23,32987,66],{"className":32988},[169],"：",[23,32991,32993],{"className":32992,"translate":27},[26],[23,32994,32996,33105],{"className":32995,"translate":27},[31],[23,32997,32999],{"className":32998},[35],[37,33000,33001],{"xmlns":39,"display":40},[42,33002,33003,33102],{},[45,33004,33005,33007,33009,33011,33013,33015,33018,33020,33022,33024,33026,33028,33030,33032,33034,33036,33038,33040,33042,33044,33046,33048,33050,33052,33054,33056,33058,33060,33062,33068,33070,33072,33074,33076,33078,33080,33082,33084,33086,33088,33090,33092,33094,33096,33098,33100],{},[48,33006,902],{"mathvariant":571},[51,33008,924],{"stretchy":53},[51,33010,4529],{},[48,33012,22422],{},[51,33014,6961],{},[48,33016,33017],{},"q",[51,33019,54],{"stretchy":53},[48,33021,369],{},[51,33023,60],{},[48,33025,434],{},[51,33027,66],{"stretchy":53},[51,33029,945],{"stretchy":53},[51,33031,69],{},[48,33033,902],{"mathvariant":571},[51,33035,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,33037,26253],{"mathvariant":571},[51,33039,54],{"stretchy":53},[48,33041,11],{},[51,33043,54],{"stretchy":53},[51,33045,5758],{},[51,33047,60],{},[48,33049,434],{},[51,33051,66],{"stretchy":53},[51,33053,66],{"stretchy":53},[51,33055,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,33057,5106],{},[48,33059,902],{"mathvariant":571},[51,33061,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[45,33063,33064,33066],{},[48,33065,26922],{"mathvariant":111},[48,33067,555],{"mathvariant":111},[51,33069,54],{"stretchy":53},[48,33071,11],{},[51,33073,54],{"stretchy":53},[51,33075,5758],{},[51,33077,60],{},[48,33079,434],{},[51,33081,66],{"stretchy":53},[48,33083,20684],{"mathvariant":111},[48,33085,33017],{},[51,33087,54],{"stretchy":53},[51,33089,5758],{},[51,33091,60],{},[48,33093,434],{},[51,33095,66],{"stretchy":53},[51,33097,66],{"stretchy":53},[51,33099,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,33101,747],{"separator":122},[114,33103,33104],{"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],",[23,33106,33108,33152,33174,33213,33240,33282,33315],{"className":33107,"ariaHidden":122},[121],[23,33109,33111,33114,33117,33120,33123,33126,33131,33134,33137,33140,33143,33146,33149],{"className":33110},[126],[23,33112],{"className":33113,"style":131},[130],[23,33115,902],{"className":33116},[135,644],[23,33118,924],{"className":33119},[140],[23,33121,4529],{"className":33122},[135],[23,33124],{"className":33125,"style":280},[148],[23,33127,22579,33129],{"className":33128},[1786],[23,33130,7091],{"style":7090},[23,33132],{"className":33133,"style":280},[148],[23,33135,33017],{"className":33136,"style":676},[135,136],[23,33138,54],{"className":33139},[140],[23,33141,369],{"className":33142,"style":392},[135,136],[23,33144],{"className":33145,"style":149},[148],[23,33147,60],{"className":33148},[153],[23,33150],{"className":33151,"style":149},[148],[23,33153,33155,33158,33161,33165,33168,33171],{"className":33154},[126],[23,33156],{"className":33157,"style":131},[130],[23,33159,434],{"className":33160,"style":454},[135,136],[23,33162,33164],{"className":33163},[169],")]",[23,33166],{"className":33167,"style":149},[148],[23,33169,69],{"className":33170},[153],[23,33172],{"className":33173,"style":149},[148],[23,33175,33177,33180,33183,33189,33192,33195,33198,33201,33204,33207,33210],{"className":33176},[126],[23,33178],{"className":33179,"style":2213},[130],[23,33181,902],{"className":33182},[135,644],[23,33184,33186],{"className":33185},[135],[23,33187,924],{"className":33188},[1066,1067],[23,33190,26253],{"className":33191},[135,644],[23,33193,54],{"className":33194},[140],[23,33196,11],{"className":33197},[135,136],[23,33199,54],{"className":33200},[140],[23,33202,5758],{"className":33203},[135],[23,33205],{"className":33206,"style":149},[148],[23,33208,60],{"className":33209},[153],[23,33211],{"className":33212,"style":149},[148],[23,33214,33216,33219,33222,33225,33231,33234,33237],{"className":33215},[126],[23,33217],{"className":33218,"style":2213},[130],[23,33220,434],{"className":33221,"style":454},[135,136],[23,33223,1095],{"className":33224},[169],[23,33226,33228],{"className":33227},[135],[23,33229,945],{"className":33230},[1066,1067],[23,33232],{"className":33233,"style":392},[148],[23,33235,5106],{"className":33236},[612],[23,33238],{"className":33239,"style":392},[148],[23,33241,33243,33246,33249,33255,33261,33264,33267,33270,33273,33276,33279],{"className":33242},[126],[23,33244],{"className":33245,"style":2213},[130],[23,33247,902],{"className":33248},[135,644],[23,33250,33252],{"className":33251},[135],[23,33253,924],{"className":33254},[1066,1067],[23,33256,33258],{"className":33257},[135],[23,33259,27191],{"className":33260},[135,5391],[23,33262,54],{"className":33263},[140],[23,33265,11],{"className":33266},[135,136],[23,33268,54],{"className":33269},[140],[23,33271,5758],{"className":33272},[135],[23,33274],{"className":33275,"style":149},[148],[23,33277,60],{"className":33278},[153],[23,33280],{"className":33281,"style":149},[148],[23,33283,33285,33288,33291,33294,33297,33300,33303,33306,33309,33312],{"className":33284},[126],[23,33286],{"className":33287,"style":131},[130],[23,33289,434],{"className":33290,"style":454},[135,136],[23,33292,66],{"className":33293},[169],[23,33295,20684],{"className":33296},[135],[23,33298,33017],{"className":33299,"style":676},[135,136],[23,33301,54],{"className":33302},[140],[23,33304,5758],{"className":33305},[135],[23,33307],{"className":33308,"style":149},[148],[23,33310,60],{"className":33311},[153],[23,33313],{"className":33314,"style":149},[148],[23,33316,33318,33321,33324,33327,33333],{"className":33317},[126],[23,33319],{"className":33320,"style":2213},[130],[23,33322,434],{"className":33323,"style":454},[135,136],[23,33325,1095],{"className":33326},[169],[23,33328,33330],{"className":33329},[135],[23,33331,945],{"className":33332},[1066,1067],[23,33334,747],{"className":33335},[824],[11,33337,33338,33339,33367,33368,33456],{},"右边第一项与 ",[23,33340,33342,33355],{"className":33341,"translate":27},[31],[23,33343,33345],{"className":33344},[35],[37,33346,33347],{"xmlns":39},[42,33348,33349,33353],{},[45,33350,33351],{},[48,33352,33017],{},[114,33354,33017],{"encoding":116},[23,33356,33358],{"className":33357,"ariaHidden":122},[121],[23,33359,33361,33364],{"className":33360},[126],[23,33362],{"className":33363,"style":672},[130],[23,33365,33017],{"className":33366,"style":676},[135,136]," 无关，第二项在 ",[23,33369,33371,33399],{"className":33370,"translate":27},[31],[23,33372,33374],{"className":33373},[35],[37,33375,33376],{"xmlns":39},[42,33377,33378,33396],{},[45,33379,33380,33382,33384,33386,33388,33390,33392,33394],{},[48,33381,33017],{},[51,33383,69],{},[48,33385,11],{},[51,33387,54],{"stretchy":53},[51,33389,5758],{},[51,33391,60],{},[48,33393,434],{},[51,33395,66],{"stretchy":53},[114,33397,33398],{"encoding":116},"q=p(\\cdot\\mid X)",[23,33400,33402,33420,33444],{"className":33401,"ariaHidden":122},[121],[23,33403,33405,33408,33411,33414,33417],{"className":33404},[126],[23,33406],{"className":33407,"style":672},[130],[23,33409,33017],{"className":33410,"style":676},[135,136],[23,33412],{"className":33413,"style":149},[148],[23,33415,69],{"className":33416},[153],[23,33418],{"className":33419,"style":149},[148],[23,33421,33423,33426,33429,33432,33435,33438,33441],{"className":33422},[126],[23,33424],{"className":33425,"style":131},[130],[23,33427,11],{"className":33428},[135,136],[23,33430,54],{"className":33431},[140],[23,33433,5758],{"className":33434},[135],[23,33436],{"className":33437,"style":149},[148],[23,33439,60],{"className":33440},[153],[23,33442],{"className":33443,"style":149},[148],[23,33445,33447,33450,33453],{"className":33446},[126],[23,33448],{"className":33449,"style":131},[130],[23,33451,434],{"className":33452,"style":454},[135,136],[23,33454,66],{"className":33455},[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。",[2501,33458,33460],{"id":33459},"_52-正则化就是先验","5.2 正则化就是先验",[11,33462,33463,33464,33741,33742,33801],{},"设参数化模型 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",[23,33743,33745,33763],{"className":33744,"translate":27},[31],[23,33746,33748],{"className":33747},[35],[37,33749,33750],{"xmlns":39},[42,33751,33752,33760],{},[45,33753,33754],{},[4518,33755,33756,33758],{},[48,33757,29638],{},[2581,33759,9956],{},[114,33761,33762],{"encoding":116},"\\sigma^2",[23,33764,33766],{"className":33765,"ariaHidden":122},[121],[23,33767,33769,33772],{"className":33768},[126],[23,33770],{"className":33771,"style":4693},[130],[23,33773,33775,33778],{"className":33774},[135],[23,33776,29638],{"className":33777,"style":676},[135,136],[23,33779,33781],{"className":33780},[998],[23,33782,33784],{"className":33783},[199],[23,33785,33787],{"className":33786},[204],[23,33788,33790],{"className":33789,"style":4693},[208],[23,33791,33792,33795],{"style":4696},[23,33793],{"className":33794,"style":1015},[216],[23,33796,33798],{"className":33797},[1019,1020,1021,1022],[23,33799,9956],{"className":33800},[135,1022],"。则负对数似然正比于平方误差：",[23,33803,33805],{"className":33804,"translate":27},[26],[23,33806,33808,33899],{"className":33807,"translate":27},[31],[23,33809,33811],{"className":33810},[35],[37,33812,33813],{"xmlns":39,"display":40},[42,33814,33815,33896],{},[45,33816,33817,33819,33821,33823,33825,33827,33829,33831,33833,33835,33837,33851,33858,33860,33866,33868,33874,33876,33882,33884,33890,33892,33894],{},[51,33818,4529],{},[48,33820,22422],{},[51,33822,6961],{},[48,33824,11],{},[51,33826,54],{"stretchy":53},[48,33828,63],{},[51,33830,60],{},[48,33832,57],{},[51,33834,66],{"stretchy":53},[51,33836,69],{},[71,33838,33839,33841],{},[2581,33840,4435],{},[45,33842,33843,33845],{},[2581,33844,9956],{},[4518,33846,33847,33849],{},[48,33848,29638],{},[2581,33850,9956],{},[6963,33852,33853,33856],{},[51,33854,33855],{},"∑",[48,33857,33497],{},[51,33859,54],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[898,33861,33862,33864],{},[48,33863,660],{},[48,33865,33497],{},[51,33867,4529],{},[898,33869,33870,33872],{},[48,33871,8340],{},[48,33873,57],{},[51,33875,54],{"stretchy":53},[898,33877,33878,33880],{},[48,33879,753],{},[48,33881,33497],{},[51,33883,66],{"stretchy":53},[4518,33885,33886,33888],{},[51,33887,66],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[2581,33889,9956],{},[51,33891,5106],{},[88,33893,31683],{},[48,33895,112],{"mathvariant":111},[114,33897,33898],{"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}.",[23,33900,33902,33940,33961,34171,34306],{"className":33901,"ariaHidden":122},[121],[23,33903,33905,33908,33911,33914,33919,33922,33925,33928,33931,33934,33937],{"className":33904},[126],[23,33906],{"className":33907,"style":131},[130],[23,33909,4529],{"className":33910},[135],[23,33912],{"className":33913,"style":280},[148],[23,33915,22579,33917],{"className":33916},[1786],[23,33918,7091],{"style":7090},[23,33920],{"className":33921,"style":280},[148],[23,33923,11],{"className":33924},[135,136],[23,33926,54],{"className":33927},[140],[23,33929,63],{"className":33930,"style":144},[135,136],[23,33932],{"className":33933,"style":149},[148],[23,33935,60],{"className":33936},[153],[23,33938],{"className":33939,"style":149},[148],[23,33941,33943,33946,33949,33952,33955,33958],{"className":33942},[126],[23,33944],{"className":33945,"style":131},[130],[23,33947,57],{"className":33948,"style":144},[135,136],[23,33950,66],{"className":33951},[169],[23,33953],{"className":33954,"style":149},[148],[23,33956,69],{"className":33957},[153],[23,33959],{"className":33960,"style":149},[148],[23,33962,33964,33968,34061,34064,34113,34116,34122,34162,34165,34168],{"className":33963},[126],[23,33965],{"className":33966,"style":33967},[130],"height:2.5991em;vertical-align:-1.2777em;",[23,33969,33971,33974,34058],{"className":33970},[135],[23,33972],{"className":33973},[140,192],[23,33975,33977],{"className":33976},[71],[23,33978,33980,34050],{"className":33979},[199,200],[23,33981,33983,34047],{"className":33982},[204],[23,33984,33986,34028,34036],{"className":33985,"style":12413},[208],[23,33987,33988,33991],{"style":212},[23,33989],{"className":33990,"style":217},[216],[23,33992,33994,33997],{"className":33993},[135],[23,33995,9956],{"className":33996},[135],[23,33998,34000,34003],{"className":33999},[135],[23,34001,29638],{"className":34002,"style":676},[135,136],[23,34004,34006],{"className":34005},[998],[23,34007,34009],{"className":34008},[199],[23,34010,34012],{"className":34011},[204],[23,34013,34016],{"className":34014,"style":34015},[208],"height:0.7401em;",[23,34017,34019,34022],{"style":34018},"top:-2.989em;margin-right:0.05em;",[23,34020],{"className":34021,"style":1015},[216],[23,34023,34025],{"className":34024},[1019,1020,1021,1022],[23,34026,9956],{"className":34027},[135,1022],[23,34029,34030,34033],{"style":235},[23,34031],{"className":34032,"style":217},[216],[23,34034],{"className":34035,"style":243},[242],[23,34037,34038,34041],{"style":246},[23,34039],{"className":34040,"style":217},[216],[23,34042,34044],{"className":34043},[135],[23,34045,4435],{"className":34046},[135],[23,34048,297],{"className":34049},[296],[23,34051,34053],{"className":34052},[204],[23,34054,34056],{"className":34055,"style":6287},[208],[23,34057],{},[23,34059],{"className":34060},[169,192],[23,34062],{"className":34063,"style":280},[148],[23,34065,34067],{"className":34066},[1786,7098],[23,34068,34070,34104],{"className":34069},[199,200],[23,34071,34073,34101],{"className":34072},[204],[23,34074,34077,34090],{"className":34075,"style":34076},[208],"height:1.05em;",[23,34078,34080,34084],{"style":34079},"top:-1.8723em;margin-left:0em;",[23,34081],{"className":34082,"style":34083},[216],"height:3.05em;",[23,34085,34087],{"className":34086},[1019,1020,1021,1022],[23,34088,33497],{"className":34089},[135,136,1022],[23,34091,34093,34096],{"style":34092},"top:-3.05em;",[23,34094],{"className":34095,"style":34083},[216],[23,34097,34098],{},[23,34099,33855],{"className":34100},[1786,1790,1791],[23,34102,297],{"className":34103},[296],[23,34105,34107],{"className":34106},[204],[23,34108,34111],{"className":34109,"style":34110},[208],"height:1.2777em;",[23,34112],{},[23,34114],{"className":34115,"style":280},[148],[23,34117,34119],{"className":34118},[135],[23,34120,54],{"className":34121},[1066,1067],[23,34123,34125,34128],{"className":34124},[135],[23,34126,660],{"className":34127,"style":676},[135,136],[23,34129,34131],{"className":34130},[998],[23,34132,34134,34154],{"className":34133},[199,200],[23,34135,34137,34151],{"className":34136},[204],[23,34138,34140],{"className":34139,"style":33651},[208],[23,34141,34142,34145],{"style":21854},[23,34143],{"className":34144,"style":1015},[216],[23,34146,34148],{"className":34147},[1019,1020,1021,1022],[23,34149,33497],{"className":34150},[135,136,1022],[23,34152,297],{"className":34153},[296],[23,34155,34157],{"className":34156},[204],[23,34158,34160],{"className":34159,"style":2048},[208],[23,34161],{},[23,34163],{"className":34164,"style":392},[148],[23,34166,4529],{"className":34167},[612],[23,34169],{"className":34170,"style":392},[148],[23,34172,34174,34178,34219,34222,34262,34265,34297,34300,34303],{"className":34173},[126],[23,34175],{"className":34176,"style":34177},[130],"height:1.404em;vertical-align:-0.35em;",[23,34179,34181,34184],{"className":34180},[135],[23,34182,8340],{"className":34183,"s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",[23,35386,35388,35420],{"className":35387,"translate":27},[31],[23,35389,35391],{"className":35390},[35],[37,35392,35393],{"xmlns":39},[42,35394,35395,35417],{},[45,35396,35397,35403,35405,35407,35409,35415],{},[4518,35398,35399,35401],{},[48,35400,29638],{},[2581,35402,9956],{},[48,35404,2875],{"mathvariant":111},[51,35406,54],{"stretchy":53},[48,35408,4806],{},[4518,35410,35411,35413],{},[48,35412,34352],{},[2581,35414,9956],{},[51,35416,66],{"stretchy":53},[114,35418,35419],{"encoding":116},"\\sigma^2\u002F(n\\tau^2)",[23,35421,35423],{"className":35422,"ariaHidden":122},[121],[23,35424,35426,35429,35458,35461,35464,35467,35496],{"className":35425},[126],[23,35427],{"className":35428,"style":10665},[130],[23,35430,35432,35435],{"className":35431},[135],[23,35433,29638],{"className":35434,"style":676},[135,136],[23,35436,35438],{"className":35437},[998],[23,35439,35441],{"className":35440},[199],[23,35442,35444],{"className":35443},[204],[23,35445,35447],{"className":35446,"style":4693},[208],[23,35448,35449,35452],{"style":4696},[23,35450],{"className":35451,"style":1015},[216],[23,35453,35455],{"className":35454},[1019,1020,1021,1022],[23,35456,9956],{"className":35457},[135,1022],[23,35459,2875],{"className":35460},[135],[23,35462,54],{"className":35463},[140],[23,35465,4806],{"className":35466},[135,136],[23,35468,35470,35473],{"className":35469},[135],[23,35471,34352],{"className":35472,"style":34411},[135,136],[23,35474,35476],{"className":35475},[998],[23,35477,35479],{"className":35478},[199],[23,35480,35482],{"className":35481},[204],[23,35483,35485],{"className":35484,"style":4693},[208],[23,35486,35487,35490],{"style":4696},[23,35488],{"className":35489,"style":1015},[216],[23,35491,35493],{"className":35492},[1019,1020,1021,1022],[23,35494,9956],{"className":35495},[135,1022],[23,35497,66],{"className":35498},[169],"。若先验换成 Laplace 分布 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",[23,35797,35799,35821],{"className":35798,"translate":27},[31],[23,35800,35802],{"className":35801},[35],[37,35803,35804],{"xmlns":39},[42,35805,35806,35818],{},[45,35807,35808,35810,35812],{},[48,35809,20684],{"mathvariant":111},[48,35811,57],{},[898,35813,35814,35816],{},[48,35815,20684],{"mathvariant":111},[2581,35817,4435],{},[114,35819,35820],{"encoding":116},"\\|\\theta\\|_1",[23,35822,35824],{"className":35823,"ariaHidden":122},[121],[23,35825,35827,35830,35833,35836],{"className":35826},[126],[23,35828],{"className":35829,"style":131},[130],[23,35831,20684],{"className":35832},[135],[23,35834,57],{"className":35835,"style":144},[135,136],[23,35837,35839,35842],{"className":35838},[135],[23,35840,20684],{"className":35841},[135],[23,35843,35845],{"className":35844},[998],[23,35846,35848,35868],{"className":35847},[199,200],[23,35849,35851,35865],{"className":35850},[204],[23,35852,35854],{"className":35853,"style":18651},[208],[23,35855,35856,35859],{"style":2029},[23,35857],{"className":35858,"style":1015},[216],[23,35860,35862],{"className":35861},[1019,1020,1021,1022],[23,35863,4435],{"className":35864},[135,1022],[23,35866,297],{"className":35867},[296],[23,35869,35871],{"className":35870},[204],[23,35872,35874],{"className":35873,"style":2048},[208],[23,35875],{},"，MAP 就变成 Lasso。",[3764,35878,35879,35891],{},[3767,35880,35881],{},[3770,35882,35883,35885,35888],{},[3773,35884,3790],{},[3773,35886,35887],{},"负对数先验",[3773,35889,35890],{},"等价惩罚",[3783,35892,35893,35903,36124,36276],{},[3770,35894,35895,35898,35900],{},[3788,35896,35897],{},"均匀先验",[3788,35899,31683],{},[3788,35901,35902],{},"无（MLE）",[3770,35904,35905,35995,36121],{},[3788,35906,35907,35908],{},"高斯 ",[23,35909,35911,35939],{"className":35910,"translate":27},[31],[23,35912,35914],{"className":35913},[35],[37,35915,35916],{"xmlns":39},[42,35917,35918,35936],{},[45,35919,35920,35922,35924,35926,35928,35934],{},[48,35921,12805],{"mathvariant":375},[51,35923,54],{"stretchy":53},[2581,35925,1532],{},[51,35927,747],{"separator":122},[4518,35929,35930,35932],{},[48,35931,34352],{},[2581,35933,9956],{},[51,35935,66],{"stretchy":53},[114,35937,35938],{"encoding":116},"\\mathcal N(0,\\tau^2)",[23,35940,35942],{"className":35941,"ariaHidden":122},[121],[23,35943,35945,35948,35951,35954,35957,35960,35963,35992],{"className":35944},[126],[23,35946],{"className":35947,"style":10665},[130],[23,35949,12805],{"className":35950,"style":22222},[135,412],[23,35952,54],{"className":35953},[140],[23,35955,1532],{"className":35956},[135],[23,35958,747],{"className":35959},[824],[23,35961],{"className":35962,"style":280},[148],[23,35964,35966,35969],{"className":35965},[135],[23,35967,34352],{"className":35968,"style":34411},[135,136],[23,35970,35972],{"className":35971},[998],[23,35973,35975],{"className":35974},[199],[23,35976,35978],{"className":35977},[204],[23,35979,35981],{"className":35980,"style":4693},[208],[23,35982,35983,35986],{"style":4696},[23,35984],{"className":35985,"style":1015},[216],[23,35987,35989],{"className":35988},[1019,1020,1021,1022],[23,35990,9956],{"className":35991},[135,1022],[23,35993,66],{"className":35994},[169],[3788,35996,35997],{},[23,35998,36000,36036],{"className":35999,"translate":27},[31],[23,36001,36003],{"className":36002},[35],[37,36004,36005],{"xmlns":39},[42,36006,36007,36033],{},[45,36008,36009,36011,36013,36019,36021,36023,36025,36031],{},[48,36010,20684],{"mathvariant":111},[48,36012,57],{},[4518,36014,36015,36017],{},[48,36016,20684],{"mathvariant":111},[2581,36018,9956],{},[48,36020,2875],{"mathvariant":111},[51,36022,54],{"stretchy":53},[2581,36024,9956],{},[4518,36026,36027,36029],{},[48,36028,34352],{},[2581,36030,9956],{},[51,36032,66],{"stretchy":53},[114,36034,36035],{"encoding":116},"\\|\\theta\\|^2\u002F(2\\tau^2)",[23,36037,36039],{"className":36038,"ariaHidden":122},[121],[23,36040,36042,36045,36048,36051,36080,36083,36086,36089,36118],{"className":36041},[126],[23,36043],{"className":36044,"style":10665},[130],[23,36046,20684],{"className":36047},[135],[23,36049,57],{"className":36050,"style":144},[135,136],[23,36052,36054,36057],{"className":36053},[135],[23,36055,20684],{"className":36056},[135],[23,36058,36060],{"className":36059},[998],[23,36061,36063],{"className":36062},[199],[23,36064,36066],{"className":36065},[204],[23,36067,36069],{"className":36068,"style":4693},[208],[23,36070,36071,36074],{"style":4696},[23,36072],{"className":36073,"style":1015},[216],[23,36075,36077],{"className":36076},[1019,1020,1021,1022],[23,36078,9956],{"className":36079},[135,1022],[23,36081,2875],{"className":36082},[135],[23,36084,54],{"className":36085},[140],[23,36087,9956],{"className":36088},[135],[23,36090,36092,36095],{"className":36091},[135],[23,36093,34352],{"className":36094,"style":34411},[135,136],[23,36096,36098],{"className":36097},[998],[23,36099,36101],{"className":36100},[199],[23,36102,36104],{"className":36103},[204],[23,36105,36107],{"className":36106,"style":4693},[208],[23,36108,36109,36112],{"style":4696},[23,36110],{"className":36111,"style":1015},[216],[23,36113,36115],{"className":36114},[1019,1020,1021,1022],[23,36116,9956],{"className":36117},[135,1022],[23,36119,66],{"className":36120},[169],[3788,36122,36123],{},"岭回归、权重衰减",[3770,36125,36126,36181,36273],{},[3788,36127,36128,36129],{},"Laplace ",[23,36130,36132,36154],{"className":36131,"translate":27},[31],[23,36133,36135],{"className":36134},[35],[37,36136,36137],{"xmlns":39},[42,36138,36139,36151],{},[45,36140,36141,36143,36145,36147,36149],{},[51,36142,54],{"stretchy":53},[2581,36144,1532],{},[51,36146,747],{"separator":122},[48,36148,35537],{},[51,36150,66],{"stretchy":53},[114,36152,36153],{"encoding":116},"(0,b)",[23,36155,36157],{"className":36156,"ariaHidden":122},[121],[23,36158,36160,36163,36166,36169,36172,36175,36178],{"className":36159},[126],[23,36161],{"className":36162,"style":131},[130],[23,36164,54],{"className":36165},[140],[23,36167,1532],{"className":36168},[135],[23,36170,747],{"className":36171},[824],[23,36173],{"className":36174,"style":280},[148],[23,36176,35537],{"className":36177},[135,136],[23,36179,66],{"className":36180},[169],[3788,36182,36183],{},[23,36184,36186,36212],{"className":36185,"translate":27},[31],[23,36187,36189],{"className":36188},[35],[37,36190,36191],{"xmlns":39},[42,36192,36193,36209],{},[45,36194,36195,36197,36199,36205,36207],{},[48,36196,20684],{"mathvariant":111},[48,36198,57],{},[898,36200,36201,36203],{},[48,36202,20684],{"mathvariant":111},[2581,36204,4435],{},[48,36206,2875],{"mathvariant":111},[48,36208,35537],{},[114,36210,36211],{"encoding":116},"\\|\\theta\\|_1\u002Fb",[23,36213,36215],{"className":36214,"ariaHidden":122},[121],[23,36216,36218,36221,36224,36227,36267,36270],{"className":36217},[126],[23,36219],{"className":36220,"style":131},[130],[23,36222,20684],{"className":36223},[135],[23,36225,57],{"className":36226,"style":144},[135,136],[23,36228,36230,36233],{"className":36229},[135],[23,36231,20684],{"className":36232},[135],[23,36234,36236],{"className":36235},[998],[23,36237,36239,36259],{"className":36238},[199,200],[23,36240,36242,36256],{"className":36241},[204],[23,36243,36245],{"className":36244,"style":18651},[208],[23,36246,36247,36250],{"style":2029},[23,36248],{"className":36249,"style":1015},[216],[23,36251,36253],{"className":36252},[1019,1020,1021,1022],[23,36254,4435],{"className":36255},[135,1022],[23,36257,297],{"className":36258},[296],[23,36260,36262],{"className":36261},[204],[23,36263,36265],{"className":36264,"style":2048},[208],[23,36266],{},[23,36268,2875],{"className":36269},[135],[23,36271,35537],{"className":36272},[135,136],[3788,36274,36275],{},"Lasso",[3770,36277,36278,36281,36284],{},[3788,36279,36280],{},"稀疏诱导先验",[3788,36282,36283],{},"非凸惩罚",[3788,36285,36286],{},"特征选择",[11,36288,36289],{},"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。",[11,36291,36292],{},"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。",[2501,36294,36296],{"id":36295},"_53-似然比roc-与代价","5.3 似然比、ROC 与代价",[11,36298,36299],{},"二分类的贝叶斯规则可以写成似然比检验：",[23,36301,36303],{"className":36302,"translate":27},[26],[23,36304,36306,36412],{"className":36305,"translate":27},[31],[23,36307,36309],{"className":36308},[35],[37,36310,36311],{"xmlns":39,"display":40},[42,36312,36313,36409],{},[45,36314,36315,36317,36319,36321,36323,36325,36327,36329,36331,36333,36363,36365,36407],{},[48,36316,736],{},[51,36318,54],{"stretchy":53},[48,36320,753],{},[51,36322,66],{"stretchy":53},[51,36324,69],{},[2581,36326,4435],{},[148,36328],{"width":9919},[51,36330,14384],{},[148,36332],{"width":9919},[71,36334,36335,36349],{},[45,36336,36337,36343,36345,36347],{},[898,36338,36339,36341],{},[48,36340,11],{},[2581,36342,4435],{},[51,36344,54],{"stretchy":53},[48,36346,753],{},[51,36348,66],{"stretchy":53},[45,36350,36351,36357,36359,36361],{},[898,36352,36353,36355],{},[48,36354,11],{},[2581,36356,1532],{},[51,36358,54],{"stretchy":53},[48,36360,753],{},[51,36362,66],{"stretchy":53},[51,36364,7382],{},[71,36366,36367,36387],{},[45,36368,36369,36375,36377],{},[898,36370,36371,36373],{},[48,36372,21697],{},[2581,36374,1532],{},[88,36376,90],{},[898,36378,36379,36381],{},[48,36380,12714],{},[45,36382,36383,36385],{},[48,36384,12719],{"mathvariant":111},[48,36386,920],{"mathvariant":111},[45,36388,36389,36395,36397],{},[898,36390,36391,36393],{},[48,36392,21697],{},[2581,36394,4435],{},[88,36396,90],{},[898,36398,36399,36401],{},[48,36400,12714],{},[45,36402,36403,36405],{},[48,36404,12719],{"mathvariant":111},[48,36406,12805],{"mathvariant":111},[48,36408,112],{"mathvariant":111},[114,36410,36411],{"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}}}.",[23,36413,36415,36442,36467,36636],{"className":36414,"ariaHidden":122},[121],[23,36416,36418,36421,36424,36427,36430,36433,36436,36439],{"className":36417},[126],[23,36419],{"className":36420,"style":131},[130],[23,36422,736],{"className":36423,"style":783},[135,136],[23,36425,54],{"className":36426},[140],[23,36428,753],{"className":36429},[135,136],[23,36431,66],{"className":36432},[169],[23,36434],{"className":36435,"style":149},[148],[23,36437,69],{"className":36438},[153],[23,36440],{"className":36441,"style":149},[148],[23,36443,36445,36449,36452,36455,36458,36461,36464],{"className":36444},[126],[23,36446],{"className":36447,"style":36448},[130],"height:0.6684em;vertical-align:-0.024em;",[23,36450,4435],{"className":36451},[135],[23,36453],{"className":36454,"style":14441},[148],[23,36456],{"className":36457,"style":149},[148],[23,36459,14384],{"className":36460},[153],[23,36462],{"className":36463,"style":14441},[148],[23,36465],{"className":36466,"style":149},[148],[23,36468,36470,36473,36627,36630,36633],{"className":36469},[126],[23,36471],{"className":36472,"style":185},[130],[23,36474,36476,36479,36624],{"className":36475},[135],[23,36477],{"className":36478},[140,192],[23,36480,36482],{"className":36481},[71],[23,36483,36485,36616],{"className":36484},[199,200],[23,36486,36488,36613],{"className":36487},[204],[23,36489,36491,36548,36556],{"className":36490,"style":209},[208],[23,36492,36493,36496],{"style":212},[23,36494],{"className":36495,"style":217},[216],[23,36497,36499,36539,36542,36545],{"className":36498},[135],[23,36500,36502,36505],{"className":36501},[135],[23,36503,11],{"className":36504},[135,136],[23,36506,36508],{"className":36507},[998],[23,36509,36511,36531],{"className":36510},[199,200],[23,36512,36514,36528],{"className":36513},[204],[23,36515,36517],{"className":36516,"style":18651},[208],[23,36518,36519,36522],{"style":2029},[23,36520],{"className":36521,"style":1015},[216],[23,36523,36525],{"className":36524},[1019,1020,1021,1022],[23,36526,1532],{"className":36527},[135,1022],[23,36529,297],{"className":36530},[296],[23,36532,36534],{"className":36533},[204],[23,36535,36537],{"className":36536,"style":2048},[208],[23,36538],{},[23,36540,54],{"className":36541},[140],[23,36543,753],{"className":36544},[135,136],[23,36546,66],{"className":36547},[169],[23,36549,36550,36553],{"style":235},[23,36551],{"className":36552,"style":217},[216],[23,36554],{"className":36555,"style":243},[242],[23,36557,36558,36561],{"style":246},[23,36559],{"className":36560,"style":217},[216],[23,36562,36564,36604,36607,36610],{"className":36563},[135],[23,36565,36567,36570],{"className":36566},[135],[23,36568,11],{"className":36569},[135,136],[23,36571,36573],{"className":36572},[998],[23,36574,36576,36596],{"className":36575},[199,200],[23,36577,36579,36593],{"className":36578},[204],[23,36580,36582],{"className":36581,"style":18651},[208],[23,36583,36584,36587],{"style":2029},[23,36585],{"className":36586,"style":1015},[216],[23,36588,36590],{"className":36589},[1019,1020,1021,1022],[23,36591,4435],{"className":36592},[135,1022],[23,36594,297],{"className":36595},[296],[23,36597,36599],{"className":36598},[204],[23,36600,36602],{"className":36601,"style":2048},[208],[23,36603],{},[23,36605,54],{"className":36606},[140],[23,36608,753],{"className":36609},[135,136],[23,36611,66],{"className":36612},[169],[23,36614,297],{"className":36615},[296],[23,36617,36619],{"className":36618},[204],[23,36620,36622],{"className":36621,"style":304},[208],[23,36623],{},[23,36625],{"className":36626},[169,192],[23,36628],{"className":36629,"style":149},[148],[23,36631,7382],{"className":36632},[153],[23,36634],{"className":36635,"style":149},[148],[23,36637,36639,36642,36876],{"className":36638},[126],[23,36640],{"className":36641,"style":13490},[130],[23,36643,36645,36648,36873],{"className":36644},[135],[23,36646],{"className":36647},[140,192],[23,36649,36651],{"className":36650},[71],[23,36652,36654,36865],{"className":36653},[199,200],[23,36655,36657,36862],{"className":36656},[204],[23,36658,36660,36757,36765],{"className":36659,"style":6056},[208],[23,36661,36662,36665],{"style":212},[23,36663],{"className":36664,"style":217},[216],[23,36666,36668,36708,36711],{"className":36667},[135],[23,36669,36671,36674],{"className":36670},[135],[23,36672,21697],{"className":36673,"style":676},[135,136],[23,36675,36677],{"className":36676},[998],[23,36678,36680,36700],{"className":36679},[199,200],[23,36681,36683,36697],{"className":36682},[204],[23,36684,36686],{"className":36685,"style":18651},[208],[23,36687,36688,36691],{"style":21854},[23,36689],{"className":36690,"style":1015},[216],[23,36692,36694],{"className":36693},[1019,1020,1021,1022],[23,36695,4435],{"className":36696},[135,1022],[23,36698,297],{"className":36699},[296],[23,36701,36703],{"className":36702},[204],[23,36704,36706],{"className":36705,"style":2048},[208],[23,36707],{},[23,36709],{"className":36710,"style":280},[148],[23,36712,36714,36717],{"className":36713},[135],[23,36715,12714],{"className":36716},[135,136],[23,36718,36720],{"className":36719},[998],[23,36721,36723,36749],{"className":36722},[199,200],[23,36724,36726,36746],{"className":36725},[204],[23,36727,36729],{"className":36728,"style":2026},[208],[23,36730,36731,36734],{"style":2029},[23,36732],{"className":36733,"style":1015},[216],[23,36735,36737],{"className":36736},[1019,1020,1021,1022],[23,36738,36740],{"className":36739},[135,1022],[23,36741,36743],{"className":36742},[135,1022],[23,36744,12853],{"className":36745},[135,5391,1022],[23,36747,297],{"className":36748},[296],[23,36750,36752],{"className":36751},[204],[23,36753,36755],{"className":36754,"style":2048},[208],[23,36756],{},[23,36758,36759,36762],{"style":235},[23,36760],{"className":36761,"style":217},[216],[23,36763],{"className":36764,"style":243},[242],[23,36766,36767,36770],{"style":246},[23,36768],{"className":36769,"style":217},[216],[23,36771,36773,36813,36816],{"className":36772},[135],[23,36774,36776,36779],{"className":36775},[135],[23,36777,21697],{"className":36778,"style":676},[135,136],[23,36780,36782],{"className":36781},[998],[23,36783,36785,36805],{"className":36784},[199,200],[23,36786,36788,36802],{"className":36787},[204],[23,36789,36791],{"className":36790,"style":18651},[208],[23,36792,36793,36796],{"style":21854},[23,36794],{"className":36795,"style":1015},[216],[23,36797,36799],{"className":36798},[1019,1020,1021,1022],[23,36800,1532],{"className":36801},[135,1022],[23,36803,297],{"className":36804},[296],[23,36806,36808],{"className":36807},[204],[23,36809,36811],{"className":36810,"style":2048},[208],[23,36812],{},[23,36814],{"className":36815,"style":280},[148],[23,36817,36819,36822],{"className":36818},[135],[23,36820,12714],{"className":36821},[135,136],[23,36823,36825],{"className":36824},[998],[23,36826,36828,36854],{"className":36827},[199,200],[23,36829,36831,36851],{"className":36830},[204],[23,36832,36834],{"className":36833,"style":2026},[208],[23,36835,36836,36839],{"style":2029},[23,36837],{"className":36838,"style":1015},[216],[23,36840,36842],{"className":36841},[1019,1020,1021,1022],[23,36843,36845],{"className":36844},[135,1022],[23,36846,36848],{"className":36847},[135,1022],[23,36849,12770],{"className":36850},[135,5391,1022],[23,36852,297],{"className":36853},[296],[23,36855,36857],{"className":36856},[204],[23,36858,36860],{"className":36859,"style":2048},[208],[23,36861],{},[23,36863,297],{"className":36864},[296],[23,36866,36868],{"className":36867},[204],[23,36869,36871],{"className":36870,"style":13689},[208],[23,36872],{},[23,36874],{"className":36875},[169,192],[23,36877,112],{"className":36878},[135],[11,36880,36881],{},"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。",[11,36883,36884,36885,36914,36915,36969,36970,37024],{},"把阈值从低到高扫一遍，就得到 ROC 曲线。设阈值为 ",[23,36886,36888,36901],{"className":36887,"translate":27},[31],[23,36889,36891],{"className":36890},[35],[37,36892,36893],{"xmlns":39},[42,36894,36895,36899],{},[45,36896,36897],{},[48,36898,5345],{},[114,36900,5345],{"encoding":116},[23,36902,36904],{"className":36903,"ariaHidden":122},[121],[23,36905,36907,36911],{"className":36906},[126],[23,36908],{"className":36909,"style":36910},[130],"height:0.6151em;",[23,36912,5345],{"className":36913},[135,136],"，记真正率为 ",[23,36916,36918,36944],{"className":36917,"translate":27},[31],[23,36919,36921],{"className":36920},[35],[37,36922,36923],{"xmlns":39},[42,36924,36925,36941],{},[45,36926,36927,36935,36937,36939],{},[45,36928,36929,36931,36933],{},[48,36930,20707],{"mathvariant":111},[48,36932,920],{"mathvariant":111},[48,36934,572],{"mathvariant":111},[51,36936,54],{"stretchy":53},[48,36938,5345],{},[51,36940,66],{"stretchy":53},[114,36942,36943],{"encoding":116},"\\mathrm{TPR}(t)",[23,36945,36947],{"className":36946,"ariaHidden":122},[121],[23,36948,36950,36953,36960,36963,36966],{"className":36949},[126],[23,36951],{"className":36952,"style":131},[130],[23,36954,36956],{"className":36955},[135],[23,36957,36959],{"className":36958},[135,5391],"TPR",[23,36961,54],{"className":36962},[140],[23,36964,5345],{"className":36965},[135,136],[23,36967,66],{"className":36968},[169],"、假正率为 ",[23,36971,36973,36999],{"className":36972,"translate":27},[31],[23,36974,36976],{"className":36975},[35],[37,36977,36978],{"xmlns":39},[42,36979,36980,36996],{},[45,36981,36982,36990,36992,36994],{},[45,36983,36984,36986,36988],{},[48,36985,12719],{"mathvariant":111},[48,36987,920],{"mathvariant":111},[48,36989,572],{"mathvariant":111},[51,36991,54],{"stretchy":53},[48,36993,5345],{},[51,36995,66],{"stretchy":53},[114,36997,36998],{"encoding":116},"\\mathrm{FPR}(t)",[23,37000,37002],{"className":37001,"ariaHidden":122},[121],[23,37003,37005,37008,37015,37018,37021],{"className":37004},[126],[23,37006],{"className":37007,"style":131},[130],[23,37009,37011],{"className":37010},[135],[23,37012,37014],{"className":37013},[135,5391],"FPR",[23,37016,54],{"className":37017},[140],[23,37019,5345],{"className":37020},[135,136],[23,37022,66],{"className":37023},[169],"，则期望代价为",[23,37026,37028],{"className":37027,"translate":27},[26],[23,37029,37031,37133],{"className":37030,"translate":27},[31],[23,37032,37034],{"className":37033},[35],[37,37035,37036],{"xmlns":39,"display":40},[42,37037,37038,37130],{},[45,37039,37040,37042,37044,37046,37048,37050,37056,37058,37068,37070,37078,37080,37082,37084,37086,37092,37094,37104,37106,37108,37110,37112,37120,37122,37124,37126,37128],{},[48,37041,24983],{},[51,37043,54],{"stretchy":53},[48,37045,5345],{},[51,37047,66],{"stretchy":53},[51,37049,69],{},[898,37051,37052,37054],{},[48,37053,21697],{},[2581,37055,1532],{},[88,37057,90],{},[898,37059,37060,37062],{},[48,37061,12714],{},[45,37063,37064,37066],{},[48,37065,12719],{"mathvariant":111},[48,37067,920],{"mathvariant":111},[88,37069,90],{},[45,37071,37072,37074,37076],{},[48,37073,12719],{"mathvariant":111},[48,37075,920],{"mathvariant":111},[48,37077,572],{"mathvariant":111},[51,37079,54],{"stretchy":53},[48,37081,5345],{},[51,37083,66],{"stretchy":53},[51,37085,5106],{},[898,37087,37088,37090],{},[48,37089,21697],{},[2581,37091,4435],{},[88,37093,90],{},[898,37095,37096,37098],{},[48,37097,12714],{},[45,37099,37100,37102],{},[48,37101,12719],{"mathvariant":111},[48,37103,12805],{"mathvariant":111},[88,37105,90],{},[51,37107,54],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[2581,37109,4435],{},[51,37111,4529],{},[45,37113,37114,37116,37118],{},[48,37115,20707],{"mathvariant":111},[48,37117,920],{"mathvariant":111},[48,37119,572],{"mathvariant":111},[51,37121,54],{"stretchy":53},[48,37123,5345],{},[51,37125,66],{"stretchy":53},[51,37127,66],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,37129,112],{"mathvariant":111},[114,37131,37132],{"encoding":116},"C(t)=\\pi_0\\,c_{\\mathrm{FP}}\\,\\mathrm{FPR}(t)\n+\\pi_1\\,c_{\\mathrm{FN}}\\,\\big(1-\\mathrm{TPR}(t)\\big).",[23,37134,37136,37164,37286,37402],{"className":37135,"ariaHidden":122},[121],[23,37137,37139,37142,37146,37149,37152,37155,37158,37161],{"className":37138},[126],[23,37140],{"className":37141,"style":131},[130],[23,37143,24983],{"className":37144,"style":37145},[135,136],"margin-right:0.0715em;",[23,37147,54],{"className":37148},[140],[23,37150,5345],{"className":37151},[135,136],[23,37153,66],{"className":37154},[169],[23,37156],{"className":37157,"style":149},[148],[23,37159,69],{"className":37160},[153],[23,37162],{"className":37163,"style":149},[148],[23,37165,37167,37170,37210,37213,37259,37262,37268,37271,37274,37277,37280,37283],{"className":37166},[126],[23,37168],{"className":37169,"style":131},[130],[23,37171,37173,37176],{"className":37172},[135],[23,37174,21697],{"className":37175,"style":676},[135,136],[23,37177,37179],{"className":37178},[998],[23,37180,37182,37202],{"className":37181},[199,200],[23,37183,37185,37199],{"className":37184},[204],[23,37186,37188],{"className":37187,"style":18651},[208],[23,37189,37190,37193],{"style":21854},[23,37191],{"className":37192,"style":1015},[216],[23,37194,37196],{"className":37195},[1019,1020,1021,1022],[23,37197,1532],{"className":37198},[135,1022],[23,37200,297],{"className":37201},[296],[23,37203,37205],{"className":37204},[204],[23,37206,37208],{"className":37207,"style":2048},[208],[23,37209],{},[23,37211],{"className":37212,"style":280},[148],[23,37214,37216,37219],{"className":37215},[135],[23,37217,12714],{"className":37218},[135,136],[23,37220,37222],{"className":37221},[998],[23,37223,37225,37251],{"className":37224},[199,200],[23,37226,37228,37248],{"className":37227},[204],[23,37229,37231],{"className":37230,"style":2026},[208],[23,37232,37233,37236],{"style":2029},[23,37234],{"className":37235,"style":1015},[216],[23,37237,37239],{"className":37238},[1019,1020,1021,1022],[23,37240,37242],{"className":37241},[135,1022],[23,37243,37245],{"className":37244},[135,1022],[23,37246,12770],{"className":37247},[135,5391,1022],[23,37249,297],{"className":37250},[296],[23,37252,37254],{"className":37253},[204],[23,37255,37257],{"className":37256,"style":2048},[208],[23,37258],{},[23,37260],{"className":37261,"style":280},[148],[23,37263,37265],{"className":37264},[135],[23,37266,370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ROC 上选一个使这个线性函数最小的点。等代价线的斜率是",[23,37436,37438],{"className":37437,"translate":27},[26],[23,37439,37441,37529],{"className":37440,"translate":27},[31],[23,37442,37444],{"className":37443},[35],[37,37445,37446],{"xmlns":39,"display":40},[42,37447,37448,37526],{},[45,37449,37450,37480,37482,37524],{},[71,37451,37452,37466],{},[45,37453,37454,37456,37458],{},[48,37455,1606],{},[88,37457,90],{},[45,37459,37460,37462,37464],{},[48,37461,20707],{"mathvariant":111},[48,37463,920],{"mathvariant":111},[48,37465,572],{"mathvariant":111},[45,37467,37468,37470,37472],{},[48,37469,1606],{},[88,37471,90],{},[45,37473,37474,37476,37478],{},[48,37475,12719],{"mathvariant":111},[48,37477,920],{"mathvariant":111},[48,37479,572],{"mathvariant":111},[51,37481,69],{},[71,37483,37484,37504],{},[45,37485,37486,37492,37494],{},[898,37487,37488,37490],{},[48,37489,21697],{},[2581,37491,1532],{},[88,37493,90],{},[898,37495,37496,37498],{},[48,37497,12714],{},[45,37499,37500,37502],{},[48,37501,12719],{"mathvariant":111},[48,37503,920],{"mathvariant":111},[45,37505,37506,37512,37514],{},[898,37507,37508,37510],{},[48,37509,21697],{},[2581,37511,4435],{},[88,37513,90],{},[898,37515,37516,37518],{},[48,37517,12714],{},[45,37519,37520,37522],{},[48,37521,12719],{"mathvariant":111},[48,37523,12805],{"mathvariant":111},[48,37525,112],{"mathvariant":111},[114,37527,37528],{"encoding":116},"\\frac{d\\,\\mathrm{TPR}}{d\\,\\mathrm{FPR}}\n=\\frac{\\pi_0\\,c_{\\mathrm{FP}}}{\\pi_1\\,c_{\\mathrm{FN}}}.",[23,37530,37532,37627],{"className":37531,"ariaHidden":122},[121],[23,37533,37535,37538,37618,37621,37624],{"className":37534},[126],[23,37536],{"className":37537,"style":6229},[130],[23,37539,37541,37544,37615],{"className":37540},[135],[23,37542],{"className":37543},[140,192],[23,37545,37547],{"className":37546},[71],[23,37548,37550,37607],{"className":37549},[199,200],[23,37551,37553,37604],{"className":37552},[204],[23,37554,37556,37576,37584],{"className":37555,"style":5607},[208],[23,37557,37558,37561],{"style":212},[23,37559],{"className":37560,"style":217},[216],[23,37562,37564,37567,37570],{"className":37563},[135],[23,37565,1606],{"className":37566},[135,136],[23,37568],{"className":37569,"style":280},[148],[23,37571,37573],{"className":37572},[135],[23,37574,37014],{"className":37575},[135,5391],[23,37577,37578,37581],{"style":235},[23,37579],{"className":37580,"style":217},[216],[23,37582],{"className":37583,"style":243},[242],[23,37585,37586,37589],{"style":246},[23,37587],{"className":37588,"style":217},[216],[23,37590,37592,37595,37598],{"className":37591},[135],[23,37593,1606],{"className":37594},[135,136],[23,37596],{"className":37597,"style":280},[148],[23,37599,37601],{"className":37600},[135],[23,37602,36959],{"className":37603},[135,5391],[23,37605,297],{"className":37606},[296],[23,37608,37610],{"className":37609},[204],[23,37611,37613],{"className":37612,"style":6287},[208],[23,37614],{},[23,37616],{"className":37617},[169,192],[23,37619],{"className":37620,"style":149},[148],[23,37622,69],{"className":37623},[153],[23,37625],{"className":37626,"style":149},[148],[23,37628,37630,37633,37867],{"className":37629},[126],[23,37631],{"className":37632,"style":13490},[130],[23,37634,37636,37639,37864],{"className":37635},[135],[23,37637],{"className":37638},[140,192],[23,37640,37642],{"className":37641},[71],[23,37643,37645,37856],{"className":37644},[199,200],[23,37646,37648,37853],{"className":37647},[204],[23,37649,37651,37748,37756],{"className":37650,"style":6056},[208],[23,37652,37653,37656],{"style":212},[23,37654],{"className":37655,"style":217},[216],[23,37657,37659,37699,37702],{"className":37658},[135],[23,37660,37662,37665],{"className":37661},[135],[23,37663,21697],{"className":37664,"style":676},[135,136],[23,37666,37668],{"className":37667},[998],[23,37669,37671,37691],{"className":37670},[199,200],[23,37672,37674,37688],{"className":37673},[204],[23,37675,37677],{"className":37676,"style":18651},[208],[23,37678,37679,37682],{"style":21854},[23,37680],{"className":37681,"style":1015},[216],[23,37683,37685],{"className":37684},[1019,1020,1021,1022],[23,37686,4435],{"className":37687},[135,1022],[23,37689,297],{"className":37690},[296],[23,37692,37694],{"className":37693},[204],[23,37695,37697],{"className":37696,"style":2048},[208],[23,37698],{},[23,37700],{"className":37701,"style":280},[148],[23,37703,37705,37708],{"className":37704},[135],[23,37706,12714],{"className":37707},[135,136],[23,37709,37711],{"className":37710},[998],[23,37712,37714,37740],{"className":37713},[199,200],[23,37715,37717,37737],{"className":37716},[204],[23,37718,37720],{"className":37719,"style":2026},[208],[23,37721,37722,37725],{"style":2029},[23,37723],{"className":37724,"style":1015},[216],[23,37726,37728],{"className":37727},[1019,1020,1021,1022],[23,37729,37731],{"className":37730},[135,1022],[23,37732,37734],{"className":37733},[135,1022],[23,37735,12853],{"className":37736},[135,5391,1022],[23,37738,297],{"className":37739},[296],[23,37741,37743],{"className":37742},[204],[23,37744,37746],{"className":37745,"style":2048},[208],[23,37747],{},[23,37749,37750,37753],{"style":235},[23,37751],{"className":37752,"style":217},[216],[23,37754],{"className":37755,"style":243},[242],[23,37757,37758,37761],{"style":246},[23,37759],{"className":37760,"style":217},[216],[23,37762,37764,37804,37807],{"className":37763},[135],[23,37765,37767,37770],{"className":37766},[135],[23,37768,21697],{"className":37769,"style":676},[135,136],[23,37771,37773],{"className":37772},[998],[23,37774,37776,37796],{"className":37775},[199,200],[23,37777,37779,37793],{"className":37778},[204],[23,37780,37782],{"className":37781,"style":18651},[208],[23,37783,37784,37787],{"style":21854},[23,37785],{"className":37786,"style":1015},[216],[23,37788,37790],{"className":37789},[1019,1020,1021,1022],[23,37791,1532],{"className":37792},[135,1022],[23,37794,297],{"className":37795},[296],[23,37797,37799],{"className":37798},[204],[23,37800,37802],{"className":37801,"style":2048},[208],[23,37803],{},[23,37805],{"className":37806,"style":280},[148],[23,37808,37810,37813],{"className":37809},[135],[23,37811,12714],{"className":37812},[135,136],[23,37814,37816],{"className":37815},[998],[23,37817,37819,37845],{"className":37818},[199,200],[23,37820,37822,37842],{"className":37821},[204],[23,37823,37825],{"className":37824,"style":2026},[208],[23,37826,37827,37830],{"style":2029},[23,37828],{"className":37829,"style":1015},[216],[23,37831,37833],{"className":37832},[1019,1020,1021,1022],[23,37834,37836],{"className":37835},[135,1022],[23,37837,37839],{"className":37838},[135,1022],[23,37840,12770],{"className":37841},[135,5391,1022],[23,37843,297],{"className":37844},[296],[23,37846,37848],{"className":37847},[204],[23,37849,37851],{"className":37850,"style":2048},[208],[23,37852],{},[23,37854,297],{"className":37855},[296],[23,37857,37859],{"className":37858},[204],[23,37860,37862],{"className":37861,"style":13689},[208],[23,37863],{},[23,37865],{"className":37866},[169,192],[23,37868,112],{"className":37869},[135],[11,37871,37872,37873,1471],{},"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。",[18,37874,37875],{},"脱离代价谈「最好的分类器」是没有意义的",[2501,37877,37879],{"id":37878},"_54-拒绝选项与选择性分类","5.4 拒绝选项与选择性分类",[11,37881,37882,37883,37933,37934,37937],{},"3.5 节已经导出了 Chow 规则：当最大后验低于 ",[23,37884,37886,37903],{"className":37885,"translate":27},[31],[23,37887,37889],{"className":37888},[35],[37,37890,37891],{"xmlns":39},[42,37892,37893,37901],{},[45,37894,37895,37897,37899],{},[2581,37896,4435],{},[51,37898,4529],{},[48,37900,12714],{},[114,37902,14075],{"encoding":116},[23,37904,37906,37924],{"className":37905,"ariaHidden":122},[121],[23,37907,37909,37912,37915,37918,37921],{"className":37908},[126],[23,37910],{"className":37911,"style":9404},[130],[23,37913,4435],{"className":37914},[135],[23,37916],{"className":37917,"style":392},[148],[23,37919,4529],{"className":37920},[612],[23,37922],{"className":37923,"style":392},[148],[23,37925,37927,37930],{"className":37926},[126],[23,37928],{"className":37929,"style":703},[130],[23,37931,12714],{"className":37932},[135,136]," 时拒绝。把这句话放到整个数据集上，就得到一条",[18,37935,37936],{},"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。",[11,37939,37940],{},"选择性分类的理论目标不是最大化准确率，而是最小化",[23,37942,37944],{"className":37943,"translate":27},[26],[23,37945,37947,37974],{"className":37946,"translate":27},[31],[23,37948,37950],{"className":37949},[35],[37,37951,37952],{"xmlns":39,"display":40},[42,37953,37954,37971],{},[45,37955,37956,37959,37961,37969],{},[88,37957,37958],{},"选择性风险",[51,37960,69],{},[71,37962,37963,37966],{},[88,37964,37965],{},"被接受样本上的错误数",[88,37967,37968],{},"被接受样本数",[51,37970,747],{"separator":122},[114,37972,37973],{"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}},",[23,37975,37977,37998],{"className":37976,"ariaHidden":122},[121],[23,37978,37980,37983,37989,37992,37995],{"className":37979},[126],[23,37981],{"className":37982,"style":470},[130],[23,37984,37986],{"className":37985},[135,6514],[23,37987,37958],{"className":37988},[135,7240],[23,37990],{"className":37991,"style":149},[148],[23,37993,69],{"className":37994},[153],[23,37996],{"className":37997,"style":149},[148],[23,37999,38001,38005,38073],{"className":38000},[126],[23,38002],{"className":38003,"style":38004},[130],"height:2.0463em;vertical-align:-0.686em;",[23,38006,38008,38011,38070],{"className":38007},[135],[23,38009],{"className":38010},[140,192],[23,38012,38014],{"className":38013},[71],[23,38015,38017,38062],{"className":38016},[199,200],[23,38018,38020,38059],{"className":38019},[204],[23,38021,38023,38037,38045],{"className":38022,"style":24276},[208],[23,38024,38025,38028],{"style":212},[23,38026],{"className":38027,"style":217},[216],[23,38029,38031],{"className":38030},[135],[23,38032,38034],{"className":38033},[135,6514],[23,38035,37968],{"className":38036},[135,7240],[23,38038,38039,38042],{"style":235},[23,38040],{"className":38041,"style":217},[216],[23,38043],{"className":38044,"style":243},[242],[23,38046,38047,38050],{"style":246},[23,38048],{"className":38049,"style":217},[216],[23,38051,38053],{"className":38052},[135],[23,38054,38056],{"className":38055},[135,6514],[23,38057,37965],{"className":38058},[135,7240],[23,38060,297],{"className":38061},[296],[23,38063,38065],{"className":38064},[204],[23,38066,38068],{"className":38067,"style":6287},[208],[23,38069],{},[23,38071],{"className":38072},[169,192],[23,38074,747],{"className":38075},[824],[11,38077,38078,38079,38107,38108,38136],{},"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 ",[23,38080,38082,38095],{"className":38081,"translate":27},[31],[23,38083,38085],{"className":38084},[35],[37,38086,38087],{"xmlns":39},[42,38088,38089,38093],{},[45,38090,38091],{},[48,38092,12714],{},[114,38094,12714],{"encoding":116},[23,38096,38098],{"className":38097,"ariaHidden":122},[121],[23,38099,38101,38104],{"className":38100},[126],[23,38102],{"className":38103,"style":703},[130],[23,38105,12714],{"className":38106},[135,136]," 越小，模型越倾向于拒绝；",[23,38109,38111,38124],{"className":38110,"translate":27},[31],[23,38112,38114],{"className":38113},[35],[37,38115,38116],{"xmlns":39},[42,38117,38118,38122],{},[45,38119,38120],{},[48,38121,12714],{},[114,38123,12714],{"encoding":116},[23,38125,38127],{"className":38126,"ariaHidden":122},[121],[23,38128,38130,38133],{"className":38129},[126],[23,38131],{"className":38132,"style":703},[130],[23,38134,12714],{"className":38135},[135,136]," 越大，模型越倾向于冒险。",[11,38138,38139],{},"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。",[2501,38141,38143],{"id":38142},"_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",[11,38145,38146],{},"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。",[11,38148,38149,38150,38274,38275,38436,38437,38588],{},"设原分布的类先验为 ",[23,38151,38153,38179],{"className":38152,"translate":27},[31],[23,38154,38156],{"className":38155},[35],[37,38157,38158],{"xmlns":39},[42,38159,38160,38176],{},[45,38161,38162,38168,38170],{},[898,38163,38164,38166],{},[48,38165,21697],{},[2581,38167,1532],{},[51,38169,747],{"separator":122},[898,38171,38172,38174],{},[48,38173,21697],{},[2581,38175,4435],{},[114,38177,38178],{"encoding":116},"\\pi_0,\\pi_1",[23,38180,38182],{"className":38181,"ariaHidden":122},[121],[23,38183,38185,38188,38228,38231,38234],{"className":38184},[126],[23,38186],{"className":38187,"style":672},[130],[23,38189,38191,38194],{"className":38190},[135],[23,38192,21697],{"className":38193,"style":676},[135,136],[23,38195,38197],{"className":38196},[998],[23,38198,38200,38220],{"className":38199},[199,200],[23,38201,38203,38217],{"className":38202},[204],[23,38204,38206],{"className":38205,"style":18651},[208],[23,38207,38208,38211],{"style":21854},[23,38209],{"className":38210,"style":1015},[216],[23,38212,38214],{"className":38213},[1019,1020,1021,1022],[23,38215,1532],{"className":38216},[135,1022],[23,38218,297],{"className":38219},[296],[23,38221,38223],{"className":38222},[204],[23,38224,38226],{"className":38225,"style":2048},[208],[23,38227],{},[23,38229,747],{"className":38230},[824],[23,38232],{"className":38233,"style":280},[148],[23,38235,38237,38240],{"className":38236},[135],[23,38238,21697],{"className":38239,"style":676},[135,136],[23,38241,38243],{"className":38242},[998],[23,38244,38246,38266],{"className":38245},[199,200],[23,38247,38249,38263],{"className":38248},[204],[23,38250,38252],{"className":38251,"style":18651},[208],[23,38253,38254,38257],{"style":21854},[23,38255],{"className":38256,"style":1015},[216],[23,38258,38260],{"className":38259},[1019,1020,1021,1022],[23,38261,4435],{"className":38262},[135,1022],[23,38264,297],{"className":38265},[296],[23,38267,38269],{"className":38268},[204],[23,38270,38272],{"className":38271,"style":2048},[208],[23,38273],{},"，重采样后的先验为 ",[23,38276,38278,38309],{"className":38277,"translate":27},[31],[23,38279,38281],{"className":38280},[35],[37,38282,38283],{"xmlns":39},[42,38284,38285,38306],{},[45,38286,38287,38296,38298],{},[19141,38288,38289,38291,38293],{},[48,38290,21697],{},[2581,38292,1532],{},[51,38294,38295],{"mathvariant":111,"lspace":1522,"rspace":1522},"′",[51,38297,747],{"separator":122},[19141,38299,38300,38302,38304],{},[48,38301,21697],{},[2581,38303,4435],{},[51,38305,38295],{"mathvariant":111,"lspace":1522,"rspace":1522},[114,38307,38308],{"encoding":116},"\\pi'_0,\\pi'_1",[23,38310,38312],{"className":38311,"ariaHidden":122},[121],[23,38313,38315,38319,38376,38379,38382],{"className":38314},[126],[23,38316],{"className":38317,"style":38318},[130],"height:1em;vertical-align:-0.2481em;",[23,38320,38322,38325],{"className":38321},[135],[23,38323,21697],{"className":38324,"style":676},[135,136],[23,38326,38328],{"className":38327},[998],[23,38329,38331,38367],{"className":38330},[199,200],[23,38332,38334,38364],{"className":38333},[204],[23,38335,38338,38350],{"className":38336,"style":38337},[208],"height:0.7519em;",[23,38339,38341,38344],{"style":38340},"top:-2.4519em;margin-left:-0.0359em;margin-right:0.05em;",[23,38342],{"className":38343,"style":1015},[216],[23,38345,38347],{"className":38346},[1019,1020,1021,1022],[23,38348,1532],{"className":38349},[135,1022],[23,38351,38352,38355],{"style":4696},[23,38353],{"className":38354,"style":1015},[216],[23,38356,38358],{"className":38357},[1019,1020,1021,1022],[23,38359,38361],{"className":38360},[135,1022],[23,38362,38295],{"className":38363},[135,1022],[23,38365,297],{"className":38366},[296],[23,38368,38370],{"className":38369},[204],[23,38371,38374],{"className":38372,"style":38373},[208],"height:0.2481em;",[23,38375],{},[23,38377,747],{"className":38378},[824],[23,38380],{"className":38381,"style":280},[148],[23,38383,38385,38388],{"className":38384},[135],[23,38386,21697],{"className":38387,"style":676},[135,136],[23,38389,38391],{"className":38390},[998],[23,38392,38394,38428],{"className":38393},[199,200],[23,38395,38397,38425],{"className":38396},[204],[23,38398,38400,38411],{"className":38399,"style":38337},[208],[23,38401,38402,38405],{"style":38340},[23,38403],{"className":38404,"style":1015},[216],[23,38406,38408],{"className":38407},[1019,1020,1021,1022],[23,38409,4435],{"className":38410},[135,1022],[23,38412,38413,38416],{"style":4696},[23,38414],{"className":38415,"style":1015},[216],[23,38417,38419],{"className":38418},[1019,1020,1021,1022],[23,38420,38422],{"className":38421},[135,1022],[23,38423,38295],{"className":38424},[135,1022],[23,38426,297],{"className":38427},[296],[23,38429,38431],{"className":38430},[204],[23,38432,38434],{"className":38433,"style":38373},[208],[23,38435],{},"。同一个似然比 ",[23,38438,38440,38478],{"className":38439,"translate":27},[31],[23,38441,38443],{"className":38442},[35],[37,38444,38445],{"xmlns":39},[42,38446,38447,38475],{},[45,38448,38449,38455,38457,38459,38461,38463,38469,38471,38473],{},[898,38450,38451,38453],{},[48,38452,11],{},[2581,38454,4435],{},[51,38456,54],{"stretchy":53},[48,38458,753],{},[51,38460,66],{"stretchy":53},[48,38462,2875],{"mathvariant":111},[898,38464,38465,38467],{},[48,38466,11],{},[2581,38468,1532],{},[51,38470,54],{"stretchy":53},[48,38472,753],{},[51,38474,66],{"stretchy":53},[114,38476,38477],{"encoding":116},"p_1(x)\u002Fp_0(x)",[23,38479,38481],{"className":38480,"ariaHidden":122},[121],[23,38482,38484,38487,38527,38530,38533,38536,38539,38579,38582,38585],{"className":38483},[126],[23,38485],{"className":38486,"style":131},[130],[23,38488,38490,38493],{"className":38489},[135],[23,38491,11],{"className":38492},[135,136],[23,38494,38496],{"className":38495},[998],[23,38497,38499,38519],{"className":38498},[199,200],[23,38500,38502,38516],{"className":38501},[204],[23,38503,38505],{"className":38504,"style":18651},[208],[23,38506,38507,38510],{"style":2029},[23,38508],{"className":38509,"style":1015},[216],[23,38511,38513],{"className":38512},[1019,1020,1021,1022],[23,38514,4435],{"className":38515},[135,1022],[23,38517,297],{"className":38518},[296],[23,38520,38522],{"className":38521},[204],[23,38523,38525],{"className":38524,"style":2048},[208],[23,38526],{},[23,38528,54],{"className":38529},[140],[23,38531,753],{"className":38532},[135,136],[23,38534,66],{"className":38535},[169],[23,38537,2875],{"className":38538},[135],[23,38540,38542,38545],{"className":38541},[135],[23,38543,11],{"className":38544},[135,136],[23,38546,38548],{"className":38547},[998],[23,38549,38551,38571],{"className":38550},[199,200],[23,38552,38554,38568],{"className":38553},[204],[23,38555,38557],{"className":38556,"style":18651},[208],[23,38558,38559,38562],{"style":2029},[23,38560],{"className":38561,"style":1015},[216],[23,38563,38565],{"className":38564},[1019,1020,1021,1022],[23,38566,1532],{"className":38567},[135,1022],[23,38569,297],{"className":38570},[296],[23,38572,38574],{"className":38573},[204],[23,38575,38577],{"className":38576,"style":2048},[208],[23,38578],{},[23,38580,54],{"className":38581},[140],[23,38583,753],{"className":38584},[135,136],[23,38586,66],{"className":38587},[169]," 在两种分布下给出的后验几率不同：",[23,38590,38592],{"className":38591,"translate":27},[26],[23,38593,38595,38729],{"className":38594,"translate":27},[31],[23,38596,38598],{"className":38597},[35],[37,38599,38600],{"xmlns":39,"display":40},[42,38601,38602,38726],{},[45,38603,38604,38643,38645,38684,38686,38724],{},[71,38605,38606,38625],{},[45,38607,38608,38615,38617,38619,38621,38623],{},[898,38609,38610,38612],{},[48,38611,11],{},[88,38613,38614],{},"orig",[51,38616,54],{"stretchy":53},[2581,38618,4435],{},[51,38620,60],{},[48,38622,753],{},[51,38624,66],{"stretchy":53},[45,38626,38627,38633,38635,38637,38639,38641],{},[898,38628,38629,38631],{},[48,38630,11],{},[88,38632,38614],{},[51,38634,54],{"stretchy":53},[2581,38636,1532],{},[51,38638,60],{},[48,38640,753],{},[51,38642,66],{"stretchy":53},[51,38644,69],{},[71,38646,38647,38666],{},[45,38648,38649,38656,38658,38660,38662,38664],{},[898,38650,38651,38653],{},[48,38652,11],{},[88,38654,38655],{},"res",[51,38657,54],{"stretchy":53},[2581,38659,4435],{},[51,38661,60],{},[48,38663,753],{},[51,38665,66],{"stretchy":53},[45,38667,38668,38674,38676,38678,38680,38682],{},[898,38669,38670,38672],{},[48,38671,11],{},[88,38673,38655],{},[51,38675,54],{"stretchy":53},[2581,38677,1532],{},[51,38679,60],{},[48,38681,753],{},[51,38683,66],{"stretchy":53},[51,38685,5758],{},[71,38687,38688,38706],{},[45,38689,38690,38696,38698],{},[898,38691,38692,38694],{},[48,38693,21697],{},[2581,38695,4435],{},[88,38697,90],{},[19141,38699,38700,38702,38704],{},[48,38701,21697],{},[2581,38703,1532],{},[51,38705,38295],{"mathvariant":111,"lspace":1522,"rspace":1522},[45,38707,38708,38716,38718],{},[19141,38709,38710,38712,38714],{},[48,38711,21697],{},[2581,38713,4435],{},[51,38715,38295],{"mathvariant":111,"lspace":1522,"rspace":1522},[88,38717,90],{},[898,38719,38720,38722],{},[48,38721,21697],{},[2581,38723,1532],{},[48,38725,112],{"mathvariant":111},[114,38727,38728],{"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}.",[23,38730,38732,38940,39145],{"className":38731,"ariaHidden":122},[121],[23,38733,38735,38739,38931,38934,38937],{"className":38734},[126],[23,38736],{"className":38737,"style":38738},[130],"height:2.3991em;vertical-align:-0.9721em;",[23,38740,38742,38745,38928],{"className":38741},[135],[23,38743],{"className":38744},[140,192],[23,38746,38748],{"className":38747},[71],[23,38749,38751,38919],{"className":38750},[199,200],[23,38752,38754,38916],{"className":38753},[204],[23,38755,38757,38833,38841],{"className":38756,"style":209},[208],[23,38758,38759,38762],{"style":212},[23,38760],{"className":38761,"style":217},[216],[23,38763,38765,38812,38815,38818,38821,38824,38827,38830],{"className":38764},[135],[23,38766,38768,38771],{"className":38767},[135],[23,38769,11],{"className":38770},[135,136],[23,38772,38774],{"className":38773},[998],[23,38775,38777,38804],{"className":38776},[199,200],[23,38778,38780,38801],{"className":38779},[204],[23,38781,38784],{"className":38782,"style":38783},[208],"height:0.3175em;",[23,38785,38786,38789],{"style":2029},[23,38787],{"className":38788,"style":1015},[216],[23,38790,38792],{"className":38791},[1019,1020,1021,1022],[23,38793,38795],{"className":38794},[135,1022],[23,38796,38798],{"className":38797},[135,6514,1022],[23,38799,38614],{"className":38800},[135,1022],[23,38802,297],{"className":38803},[296],[23,38805,38807],{"className":38806},[204],[23,38808,38810],{"className":38809,"style":14296},[208],[23,38811],{},[23,38813,54],{"className":38814},[140],[23,38816,1532],{"className":38817},[135],[23,38819],{"className":38820,"style":149},[148],[23,38822,60],{"className":38823},[153],[23,38825],{"className":38826,"style":149},[148],[23,38828,753],{"className":38829},[135,136],[23,38831,66],{"className":38832},[169],[23,38834,38835,38838],{"style":235},[23,38836],{"className":38837,"style":217},[216],[23,38839],{"className":38840,"style":243},[242],[23,38842,38843,38846],{"style":246},[23,38844],{"className":38845,"style":217},[216],[23,38847,38849,38895,38898,38901,38904,38907,38910,38913],{"className":38848},[135],[23,38850,38852,38855],{"className":38851},[135],[23,38853,11],{"className":38854},[135,136],[23,38856,38858],{"className":38857},[998],[23,38859,38861,38887],{"className":38860},[199,200],[23,38862,38864,38884],{"className":38863},[204],[23,38865,38867],{"className":38866,"style":38783},[208],[23,38868,38869,38872],{"style":2029},[23,38870],{"className":38871,"style":1015},[216],[23,38873,38875],{"className":38874},[1019,1020,1021,1022],[23,38876,38878],{"className":38877},[135,1022],[23,38879,38881],{"className":38880},[135,6514,1022],[23,38882,38614],{"className":38883},[135,1022],[23,38885,297],{"className":38886},[296],[23,38888,38890],{"className":38889},[204],[23,38891,38893],{"className":38892,"style":14296},[208],[23,38894],{},[23,38896,54],{"className":38897},[140],[23,38899,4435],{"className":38900},[135],[23,38902],{"className":38903,"style":149},[148],[23,38905,60],{"className":38906},[153],[23,38908],{"className":38909,"style":149},[148],[23,38911,753],{"className":38912},[135,136],[23,38914,66],{"className":38915},[169],[23,38917,297],{"className":38918},[296],[23,38920,38922],{"className":38921},[204],[23,38923,38926],{"className":38924,"style":38925},[208],"height:0.9721em;",[23,38927],{},[23,38929],{"className":38930},[169,192],[23,38932],{"className":38933,"style":149},[148],[23,38935,69],{"className":38936},[153],[23,38938],{"className":38939,"style":149},[148],[23,38941,38943,38946,39136,39139,39142],{"className":38942},[126],[23,38944],{"className":38945,"style":185},[130],[23,38947,38949,38952,39133],{"className":38948},[135],[23,38950],{"className":38951},[140,192],[23,38953,38955],{"className":38954},[71],[23,38956,38958,39125],{"className":38957},[199,200],[23,38959,38961,39122],{"className":38960},[204],[23,38962,38964,39039,39047],{"className":38963,"style":209},[208],[23,38965,38966,38969],{"style":212},[23,38967],{"className":38968,"style":217},[216],[23,38970,38972,39018,39021,39024,39027,39030,39033,39036],{"className":38971},[135],[23,38973,38975,38978],{"className":38974},[135],[23,38976,11],{"className":38977},[135,136],[23,38979,38981],{"className":38980},[998],[23,38982,38984,39010],{"className":38983},[199,200],[23,38985,38987,39007],{"className":38986},[204],[23,38988,38990],{"className":38989,"style":6499},[208],[23,38991,38992,38995],{"style":2029},[23,38993],{"className":38994,"style":1015},[216],[23,38996,38998],{"className":38997},[1019,1020,1021,1022],[23,38999,39001],{"className":39000},[135,1022],[23,39002,39004],{"className":39003},[135,6514,1022],[23,39005,38655],{"className":39006},[135,1022],[23,39008,297],{"className":39009},[296],[23,39011,39013],{"className":39012},[204],[23,39014,39016],{"className":39015,"style":2048},[208],[23,39017],{},[23,39019,54],{"className":39020},[140],[23,39022,1532],{"className":39023},[135],[23,39025],{"className":39026,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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。",[11,39414,39415],{},"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。",[338,39417,39419],{"id":39418},"_6-从参数到分布","6. 从参数到分布",[2501,39421,39423],{"id":39422},"_61-三个预测器","6.1 三个预测器",[11,39425,39426,39427,39455,39456,39484],{},"给定训练集 ",[23,39428,39430,39443],{"className":39429,"translate":27},[31],[23,39431,39433],{"className":39432},[35],[37,39434,39435],{"xmlns":39},[42,39436,39437,39441],{},[45,39438,39439],{},[48,39440,63],{},[114,39442,63],{"encoding":116},[23,39444,39446],{"className":39445,"ariaHidden":122},[121],[23,39447,39449,39452],{"className":39448},[126],[23,39450],{"className":39451,"style":470},[130],[23,39453,63],{"className":39454,"style":144},[135,136],"，预测 ",[23,39457,39459,39472],{"className":39458,"translate":27},[31],[23,39460,39462],{"className":39461},[35],[37,39463,39464],{"xmlns":39},[42,39465,39466,39470],{},[45,39467,39468],{},[48,39469,369],{},[114,39471,369],{"encoding":116},[23,39473,39475],{"className":39474,"ariaHidden":122},[121],[23,39476,39478,39481],{"className":39477},[126],[23,39479],{"className":39480,"style":470},[130],[23,39482,369],{"className":39483,"style":392},[135,136]," 至少有三层做法：",[39486,39487,39488,39668,39936],"ol",{},[349,39489,39490,39493,39494,39555,39556,39667],{},[18,39491,39492],{},"代入式","：先估计一个参数 ",[23,39495,39497,39515],{"className":39496,"translate":27},[31],[23,39498,39500],{"className":39499},[35],[37,39501,39502],{"xmlns":39},[42,39503,39504,39512],{},[45,39505,39506],{},[16811,39507,39508,39510],{"accent":122},[48,39509,57],{},[51,39511,16817],{},[114,39513,39514],{"encoding":116},"\\hat\\theta",[23,39516,39518],{"className":39517,"ariaHidden":122},[121],[23,39519,39521,39524],{"className":39520},[126],[23,39522],{"className":39523,"style":16844},[130],[23,39525,39527],{"className":39526},[135,16834],[23,39528,39530],{"className":39529},[199],[23,39531,39533],{"className":39532},[204],[23,39534,39536,39544],{"className":39535,"style":16844},[208],[23,39537,39538,39541],{"style":7132},[23,39539],{"className":39540,"style":217},[216],[23,39542,57],{"className":39543,"style":144},[135,136],[23,39545,39546,39549],{"style":16855},[23,39547],{"className":39548,"style":217},[216],[23,39550,39552],{"className":39551,"style":34869},[16862],[23,39553,16817],{"className":39554},[135],"，再用 ",[23,39557,39559,39591],{"className":39558,"translate":27},[31],[23,39560,39562],{"className":39561},[35],[37,39563,39564],{"xmlns":39},[42,39565,39566,39588],{},[45,39567,39568,39570,39572,39574,39576,39578,39580,39586],{},[48,39569,11],{},[51,39571,54],{"stretchy":53},[48,39573,660],{},[51,39575,60],{},[48,39577,753],{},[51,39579,747],{"separator":122},[16811,39581,39582,39584],{"accent":122},[48,39583,57],{},[51,39585,16817],{},[51,39587,66],{"stretchy":53},[114,39589,39590],{"encoding":116},"p(y\\mid x,\\hat\\theta)",[23,39592,39594,39618],{"className":39593,"ariaHidden":122},[121],[23,39595,39597,39600,39603,39606,39609,39612,39615],{"className":39596},[126],[23,39598],{"className":39599,"style":131},[130],[23,39601,11],{"className":39602},[135,136],[23,39604,54],{"className":39605},[140],[23,39607,660],{"className":39608,"style":676},[135,136],[23,39610],{"className":39611,"style":149},[148],[23,39613,60],{"className":39614},[153],[23,39616],{"className":39617,"style":149},[148],[23,39619,39621,39624,39627,39630,39633,39664],{"className":39620},[126],[23,39622],{"className":39623,"style":16970},[130],[23,39625,753],{"className":39626},[135,136],[23,39628,747],{"className":39629},[824],[23,39631],{"className":39632,"style":280},[148],[23,39634,39636],{"className":39635},[135,16834],[23,39637,39639],{"className":39638},[199],[23,39640,39642],{"className":39641},[204],[23,39643,39645,39653],{"className":39644,"style":16844},[208],[23,39646,39647,39650],{"style":7132},[23,39648],{"className":39649,"style":217},[216],[23,39651,57],{"className":39652,"style":144},[135,136],[23,39654,39655,39658],{"style":16855},[23,39656],{"className":39657,"style":217},[216],[23,39659,39661],{"className":39660,"style":34869},[16862],[23,39662,16817],{"className":39663},[135],[23,39665,66],{"className":39666},[169]," 预测；",[349,39669,39670,39673,39674],{},[18,39671,39672],{},"后验预测","：对参数积分，\n",[23,39675,39677],{"className":39676,"translate":27},[26],[23,39678,39680,39752],{"className":39679,"translate":27},[31],[23,39681,39683],{"className":39682},[35],[37,39684,39685],{"xmlns":39,"display":40},[42,39686,39687,39749],{},[45,39688,39689,39696,39698,39700,39702,39704,39706,39708,39710,39712,39714,39716,39718,39720,39722,39724,39726,39728,39730,39732,39734,39736,39738,39740,39742,39744,39746],{},[16811,39690,39691,39693],{"accent":122},[48,39692,11],{},[51,39694,39695],{},"ˉ",[51,39697,54],{"stretchy":53},[48,39699,660],{},[51,39701,60],{},[48,39703,753],{},[51,39705,66],{"stretchy":53},[51,39707,69],{},[51,39709,1557],{},[48,39711,11],{},[51,39713,54],{"stretchy":53},[48,39715,660],{},[51,39717,60],{},[48,39719,753],{},[51,39721,747],{"separator":122},[48,39723,57],{},[51,39725,66],{"stretchy":53},[88,39727,90],{},[48,39729,11],{},[51,39731,54],{"stretchy":53},[48,39733,57],{},[51,39735,60],{},[48,39737,63],{},[51,39739,66],{"stretchy":53},[88,39741,90],{},[48,39743,1606],{},[48,39745,57],{},[51,39747,39748],{"separator":122},";",[114,39750,39751],{"encoding":116},"\\bar p(y\\mid x)=\\int p(y\\mid x,\\theta)\\,p(\\theta\\mid D)\\,d\\theta;",[23,39753,39755,39819,39840,39870,39912],{"className":39754,"ariaHidden":122},[121],[23,39756,39758,39761,39804,39807,39810,39813,39816],{"className":39757},[126],[23,39759],{"className":39760,"style":131},[130],[23,39762,39764],{"className":39763},[135,16834],[23,39765,39767,39796],{"className":39766},[199,200],[23,39768,39770,39793],{"className":39769},[204],[23,39771,39774,39782],{"className":39772,"style":39773},[208],"height:0.5678em;",[23,39775,39776,39779],{"style":7132},[23,39777],{"className":39778,"style":217},[216],[23,39780,11],{"className":39781},[135,136],[23,39783,39784,39787],{"style":7132},[23,39785],{"className":39786,"style":217},[216],[23,39788,39790],{"className":39789,"style":34869},[16862],[23,39791,39695],{"className":39792},[135],[23,39794,297],{"className":39795},[296],[23,39797,39799],{"className":39798},[204],[23,39800,39802],{"className":39801,"style":16876},[208],[23,39803],{},[23,39805,54],{"className":39806},[140],[23,39808,660],{"className":39809,"style":676},[135,136],[23,39811],{"className":39812,"style":149},[148],[23,39814,60],{"className":39815},[153],[23,39817],{"className":39818,"style":149},[148],[23,39820,39822,39825,39828,39831,39834,39837],{"className":39821},[126],[23,39823],{"className":39824,"style":131},[130],[23,39826,753],{"className":39827},[135,136],[23,39829,66],{"className":39830},[169],[23,39832],{"className":39833,"style":149},[148],[23,39835,69],{"className":39836},[153],[23,39838],{"className":39839,"style":149},[148],[23,39841,39843,39846,39849,39852,39855,39858,39861,39864,39867],{"className":39842},[126],[23,39844],{"className":39845,"style":3696},[130],[23,39847,1557],{"className":39848,"style":1792},[1786,1790,1791],[23,39850],{"className":39851,"style":280},[148],[23,39853,11],{"className":39854},[135,136],[23,39856,54],{"className":39857},[140],[23,39859,660],{"className":39860,"style":676},[135,136],[23,39862],{"className":39863,"style":149},[148],[23,39865,60],{"className":39866},[153],[23,39868],{"className":39869,"style":149},[148],[23,39871,39873,39876,39879,39882,39885,39888,39891,39894,39897,39900,39903,39906,39909],{"className":39872},[126],[23,39874],{"className":39875,"style":131},[130],[23,39877,753],{"className":39878},[135,136],[23,39880,747],{"className":39881},[824],[23,39883],{"className":39884,"style":280},[148],[23,39886,57],{"className":39887,"style":144},[135,136],[23,39889,66],{"className":39890},[169],[23,39892],{"className":39893,"style":280},[148],[23,39895,11],{"className":39896},[135,136],[23,39898,54],{"className":39899},[140],[23,39901,57],{"className":39902,"style":144},[135,136],[23,39904],{"className":39905,"style":149},[148],[23,39907,60],{"className":39908},[153],[23,39910],{"className":39911,"style":149},[148],[23,39913,39915,39918,39921,39924,39927,39930,39933],{"className":39914},[126],[23,39916],{"className":39917,"style":131},[130],[23,39919,63],{"className":39920,"style":144},[135,136],[23,39922,66],{"className":39923},[169],[23,39925],{"className":39926,"style":280},[148],[23,39928,1606],{"className":39929},[135,136],[23,39931,57],{"className":39932,"style":144},[135,136],[23,39934,39748],{"className":39935},[824],[349,39937,39938,39940,39941],{},[18,39939,13789],{},"：在预测分布上再做一次决策，\n",[23,39942,39944],{"className":39943,"translate":27},[26],[23,39945,39947,40027],{"className":39946,"translate":27},[31],[23,39948,39950],{"className":39949},[35],[37,39951,39952],{"xmlns":39,"display":40},[42,39953,39954,40024],{},[45,39955,39956,39962,39964,39966,39968,39970,39972,39974,39984,39986,39988,39990,39992,39994,39996,39998,40000,40006,40008,40010,40012,40014,40016,40018,40020,40022],{},[4518,39957,39958,39960],{},[48,39959,495],{},[51,39961,6883],{},[51,39963,54],{"stretchy":53},[48,39965,753],{},[51,39967,66],{"stretchy":53},[51,39969,69],{},[48,39971,6958],{},[51,39973,6961],{},[6963,39975,39976,39982],{},[45,39977,39978,39980],{},[48,39979,6969],{},[51,39981,6961],{},[48,39983,495],{},[51,39985,1557],{},[48,39987,555],{},[51,39989,54],{"stretchy":53},[48,39991,660],{},[51,39993,747],{"separator":122},[48,39995,495],{},[51,39997,66],{"stretchy":53},[88,39999,90],{},[16811,40001,40002,40004],{"accent":122},[48,40003,11],{},[51,40005,39695],{},[51,40007,54],{"stretchy":53},[48,40009,660],{},[51,40011,60],{},[48,40013,753],{},[51,40015,66],{"stretchy":53},[88,40017,90],{},[48,40019,1606],{},[48,40021,660],{},[48,40023,112],{"mathvariant":111},[114,40025,40026],{"encoding":116},"a^*(x)=\\arg\\min_a\\int L(y,a)\\,\\bar p(y\\mid x)\\,dy.",[23,40028,40030,40083,40231],{"className":40029,"ariaHidden":122},[121],[23,40031,40033,40036,40065,40068,40071,40074,40077,40080],{"className":40032},[126],[23,40034],{"className":40035,"style":131},[130],[23,40037,40039,40042],{"className":40038},[135],[23,40040,495],{"className":40041},[135,136],[23,40043,40045],{"className":40044},[998],[23,40046,40048],{"className":40047},[199],[23,40049,40051],{"className":40050},[204],[23,40052,40054],{"className":40053,"style":7047},[208],[23,40055,40056,40059],{"style":4989},[23,40057],{"className":40058,"style":1015},[216],[23,40060,40062],{"className":40061},[1019,1020,1021,1022],[23,40063,6883],{"className":40064},[612,1022],[23,40066,54],{"className":40067},[140],[23,40069,753],{"className":40070},[135,136],[23,40072,66],{"className":40073},[169],[23,40075],{"className":40076,"style":149},[148],[23,40078,69],{"className":40079},[153],[23,40081],{"className":40082,"style":149},[148],[23,40084,40086,40089,40094,40097,40141,40144,40147,40150,40153,40156,40159,40162,40165,40168,40171,40174,40216,40219,40222,40225,40228],{"className":40085},[126],[23,40087],{"className":40088,"style":3696},[130],[23,40090,7087,40092],{"className":40091},[1786],[23,40093,7091],{"style":7090},[23,40095],{"className":40096,"style":280},[148],[23,40098,40100],{"className":40099},[1786,7098],[23,40101,40103,40133],{"className":40102},[199,200],[23,40104,40106,40130],{"className":40105},[204],[23,40107,40109,40120],{"className":40108,"style":7108},[208],[23,40110,40111,40114],{"style":8784},[23,40112],{"className":40113,"style":217},[216],[23,40115,40117],{"className":40116},[1019,1020,1021,1022],[23,40118,495],{"className":40119},[135,136,1022],[23,40121,40122,40125],{"style":7132},[23,40123],{"className":40124,"style":217},[216],[23,40126,40127],{},[23,40128,6969],{"className":40129},[1786],[23,40131,297],{"className":40132},[296],[23,40134,40136],{"className":40135},[204],[23,40137,40139],{"className":40138,"style":8816},[208],[23,40140],{},[23,40142],{"className":40143,"style":280},[148],[23,40145,1557],{"className":40146,"style":1792},[1786,1790,1791],[23,40148],{"className":40149,"style":280},[148],[23,40151,555],{"className":40152},[135,136],[23,40154,54],{"className":40155},[140],[23,40157,660],{"className":40158,"style":676},[135,136],[23,40160,747],{"className":40161},[824],[23,40163],{"className":40164,"style":280},[148],[23,40166,495],{"className":40167},[135,136],[23,40169,66],{"className":40170},[169],[23,40172],{"className":40173,"style":280},[148],[23,40175,40177],{"className":40176},[135,16834],[23,40178,40180,40208],{"className":40179},[199,200],[23,40181,40183,40205],{"className":40182},[204],[23,40184,40186,40194],{"className":40185,"style":39773},[208],[23,40187,40188,40191],{"style":7132},[23,40189],{"className":40190,"style":217},[216],[23,40192,11],{"className":40193},[135,136],[23,40195,40196,40199],{"style":7132},[23,40197],{"className":40198,"style":217},[216],[23,40200,40202],{"className":40201,"style":34869},[16862],[23,40203,39695],{"className":40204},[135],[23,40206,297],{"className":40207},[296],[23,40209,40211],{"className":40210},[204],[23,40212,40214],{"className":40213,"style":16876},[208],[23,40215],{},[23,40217,54],{"className":40218},[140],[23,40220,660],{"className":40221,"style":676},[135,136],[23,40223],{"className":40224,"style":149},[148],[23,40226,60],{"className":40227},[153],[23,40229],{"className":40230,"style":149},[148],[23,40232,40234,40237,40240,40243,40246,40249,40252],{"className":40233},[126],[23,40235],{"className":40236,"style":131},[130],[23,40238,753],{"className":40239},[135,136],[23,40241,66],{"className":40242},[169],[23,40244],{"className":40245,"style":280},[148],[23,40247,1606],{"className":40248},[135,136],[23,40250,660],{"className":40251,"style":676},[135,136],[23,40253,112],{"className":40254},[135],[11,40256,40257],{},"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。",[11,40259,40260,40261,40340,40341,40369],{},"后验预测是一个混合分布，不是「把后验均值代进去」。由于 ",[23,40262,40264,40292],{"className":40263,"translate":27},[31],[23,40265,40267],{"className":40266},[35],[37,40268,40269],{"xmlns":39},[42,40270,40271,40289],{},[45,40272,40273,40275,40277,40279,40281,40283,40285,40287],{},[48,40274,11],{},[51,40276,54],{"stretchy":53},[48,40278,660],{},[51,40280,60],{},[48,40282,753],{},[51,40284,747],{"separator":122},[48,40286,57],{},[51,40288,66],{"stretchy":53},[114,40290,40291],{"encoding":116},"p(y\\mid x,\\theta)",[23,40293,40295,40319],{"className":40294,"ariaHidden":122},[121],[23,40296,40298,40301,40304,40307,40310,40313,40316],{"className":40297},[126],[23,40299],{"className":40300,"style":131},[130],[23,40302,11],{"className":40303},[135,136],[23,40305,54],{"className":40306},[140],[23,40308,660],{"className":40309,"style":676},[135,136],[23,40311],{"className":40312,"style":149},[148],[23,40314,60],{"className":40315},[153],[23,40317],{"className":40318,"style":149},[148],[23,40320,40322,40325,40328,40331,40334,40337],{"className":40321},[126],[23,40323],{"className":40324,"style":131},[130],[23,40326,753],{"className":40327},[135,136],[23,40329,747],{"className":40330},[824],[23,40332],{"className":40333,"style":280},[148],[23,40335,57],{"className":40336,"style":144},[135,136],[23,40338,66],{"className":40339},[169]," 通常对 ",[23,40342,40344,40357],{"className":40343,"translate":27},[31],[23,40345,40347],{"className":40346},[35],[37,40348,40349],{"xmlns":39},[42,40350,40351,40355],{},[45,40352,40353],{},[48,40354,57],{},[114,40356,3371],{"encoding":116},[23,40358,40360],{"className":40359,"ariaHidden":122},[121],[23,40361,40363,40366],{"className":40362},[126],[23,40364],{"className":40365,"style":779},[130],[23,40367,57],{"className":40368,"style":144},[135,136]," 非线性，",[23,40371,40373],{"className":40372,"translate":27},[26],[23,40374,40376,40456],{"className":40375,"translate":27},[31],[23,40377,40379],{"className":40378},[35],[37,40380,40381],{"xmlns":39,"display":40},[42,40382,40383,40453],{},[45,40384,40385,40387,40389,40391,40393,40395,40397,40399,40401,40403,40405,40407,40409,40411,40413,40415,40417,40419,40421,40423,40425,40427,40429,40431,40433,40435,40437,40439,40441,40443,40445,40447,40449,40451],{},[51,40386,1557],{},[48,40388,11],{},[51,40390,54],{"stretchy":53},[48,40392,660],{},[51,40394,60],{},[48,40396,753],{},[51,40398,747],{"separator":122},[48,40400,57],{},[51,40402,66],{"stretchy":53},[88,40404,90],{},[48,40406,11],{},[51,40408,54],{"stretchy":53},[48,40410,57],{},[51,40412,60],{},[48,40414,63],{},[51,40416,66],{"stretchy":53},[88,40418,90],{},[48,40420,1606],{},[48,40422,57],{},[51,40424,9019],{"mathvariant":111},[48,40426,11],{},[51,40428,54],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,40430,660],{},[51,40432,60],{},[48,40434,753],{},[51,40436,747],{"separator":122},[48,40438,902],{"mathvariant":571},[51,40440,924],{"stretchy":53},[48,40442,57],{},[51,40444,60],{},[48,40446,63],{},[51,40448,945],{"stretchy":53},[51,40450,66],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,40452,112],{"mathvariant":111},[114,40454,40455],{"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).",[23,40457,40459,40489,40531,40594,40621,40654],{"className":40458,"ariaHidden":122},[121],[23,40460,40462,40465,40468,40471,40474,40477,40480,40483,40486],{"className":40461},[126],[23,40463],{"className":40464,"style":3696},[130],[23,40466,1557],{"className":40467,"style":1792},[1786,1790,1791],[23,40469],{"className":40470,"style":280},[148],[23,40472,11],{"className":40473},[135,136],[23,40475,54],{"className":40476},[140],[23,40478,660],{"className":40479,"style":676},[135,136],[23,40481],{"className":40482,"style":149},[148],[23,40484,60],{"className":40485},[153],[23,40487],{"className":40488,"style":149},[148],[23,40490,40492,40495,40498,40501,40504,40507,40510,40513,40516,40519,40522,40525,40528],{"className":40491},[126],[23,40493],{"className":40494,"style":131},[130],[23,40496,753],{"className":40497},[135,136],[23,40499,747],{"className":40500},[824],[23,40502],{"className":40503,"style":280},[148],[23,40505,57],{"className":40506,"style":144},[135,136],[23,40508,66],{"className":40509},[169],[23,40511],{"className":40512,"style":280},[148],[23,40514,11],{"className":40515},[135,136],[23,40517,54],{"className":40518},[140],[23,40520,57],{"className":40521,"style":144},[135,136],[23,40523],{"className":40524,"style":149},[148],[23,40526,60],{"className":40527},[153],[23,40529],{"className":40530,"style":149},[148],[23,40532,40534,40537,40540,40543,40546,40549,40552,40555,40591],{"className":40533},[126],[23,40535],{"className":40536,"style":131},[130],[23,40538,63],{"className":40539,"style":144},[135,136],[23,40541,66],{"className":40542},[169],[23,40544],{"className":40545,"style":280},[148],[23,40547,1606],{"className":40548},[135,136],[23,40550,57],{"className":40551,"style":144},[135,136],[23,40553],{"className":40554,"style":149},[148],[23,40556,40558,40585,40588],{"className":40557},[153],[23,40559,40561],{"className":40560},[153],[23,40562,40564],{"className":40563},[135,9097],[23,40565,40567],{"className":40566},[9101],[23,40568,40570,40573,40582],{"className":40569},[9105],[23,40571],{"className":40572,"style":5414},[130],[23,40574,40576],{"className":40575},[9112],[23,40577,40579],{"className":40578},[135],[23,40580,9119],{"className":40581},[153],[23,40583],{"className":40584},[9123],[23,40586],{"className":40587},[148,9127],[23,40589,69],{"className":40590},[153],[23,40592],{"className":40593,"style":149},[148],[23,40595,40597,40600,40603,40609,40612,40615,40618],{"className":40596},[126],[23,40598],{"className":40599,"style":2213},[130],[23,40601,11],{"className":40602},[135,136],[23,40604,40606],{"className":40605},[135],[23,40607,54],{"className":40608},[1066,1067],[23,40610,660],{"className":40611,"style":676},[135,136],[23,40613],{"className":40614,"style":149},[148],[23,40616,60],{"className":40617},[153],[23,40619],{"className":40620,"style":149},[148],[23,40622,40624,40627,40630,40633,40636,40639,40642,40645,40648,40651],{"className":40623},[126],[23,40625],{"className":40626,"style":131},[130],[23,40628,753],{"className":40629},[135,136],[23,40631,747],{"className":40632},[824],[23,40634],{"className":40635,"style":280},[148],[23,40637,902],{"className":40638},[135,644],[23,40640,924],{"className":40641},[140],[23,40643,57],{"className":40644,"style":144},[135,136],[23,40646],{"className":40647,"style":149},[148],[23,40649,60],{"className":40650},[153],[23,40652],{"className":40653,"style":149},[148],[23,40655,40657,40660,40663,40666,40672],{"className":40656},[126],[23,40658],{"className":40659,"style":2213},[130],[23,40661,63],{"className":40662,"style":144},[135,136],[23,40664,945],{"className":40665},[169],[23,40667,40669],{"className":40668},[135],[23,40670,66],{"className":40671},[1066,1067],[23,40673,112],{"className":40674},[135],[11,40676,40677],{},"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」。",[2501,40679,40681],{"id":40680},"_62-两种不确定性","6.2 两种不确定性",[11,40683,40684,40685,40688,40689,40717,40718,40746,40747,40750],{},"预测的不确定性可以分成两类。",[18,40686,40687],{},"偶然不确定性","（aleatoric uncertainty）来自数据生成过程本身的噪声：同样的 ",[23,40690,40692,40705],{"className":40691,"translate":27},[31],[23,40693,40695],{"className":40694},[35],[37,40696,40697],{"xmlns":39},[42,40698,40699,40703],{},[45,40700,40701],{},[48,40702,753],{},[114,40704,753],{"encoding":116},[23,40706,40708],{"className":40707,"ariaHidden":122},[121],[23,40709,40711,40714],{"className":40710},[126],[23,40712],{"className":40713,"style":703},[130],[23,40715,753],{"className":40716},[135,136]," 可能对应不同的 ",[23,40719,40721,40734],{"className":40720,"translate":27},[31],[23,40722,40724],{"className":40723},[35],[37,40725,40726],{"xmlns":39},[42,40727,40728,40732],{},[45,40729,40730],{},[48,40731,660],{},[114,40733,660],{"encoding":116},[23,40735,40737],{"className":40736,"ariaHidden":122},[121],[23,40738,40740,40743],{"className":40739},[126],[23,40741],{"className":40742,"style":672},[130],[23,40744,660],{"className":40745,"style":676},[135,136],"，即使知道真实参数也无法消除。",[18,40748,40749],{},"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。",[11,40752,40753],{},"对平方损失，这两种不确定性由全方差公式精确分解：",[23,40755,40757],{"className":40756,"translate":27},[26],[23,40758,40760,40898],{"className":40759,"translate":27},[31],[23,40761,40763],{"className":40762},[35],[37,40764,40765],{"xmlns":39,"display":40},[42,40766,40767,40895],{},[45,40768,40769,40777,40779,40781,40783,40785,40787,40789,40791,40793,40842,40844,40893],{},[45,40770,40771,40773,40775],{},[48,40772,10794],{"mathvariant":111},[48,40774,495],{"mathvariant":111},[48,40776,1653],{"mathvariant":111},[51,40778,54],{"stretchy":53},[48,40780,369],{},[51,40782,60],{},[48,40784,753],{},[51,40786,747],{"separator":122},[48,40788,63],{},[51,40790,66],{"stretchy":53},[51,40792,69],{},[6963,40794,40795,40839],{},[6963,40796,40797,40837],{},[45,40798,40799,40811,40813,40821,40823,40825,40827,40829,40831,40833,40835],{},[898,40800,40801,40803],{},[48,40802,902],{"mathvariant":571},[45,40804,40805,40807,40809],{},[48,40806,57],{},[51,40808,60],{},[48,40810,63],{},[51,40812,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[45,40814,40815,40817,40819],{},[48,40816,10794],{"mathvariant":111},[48,40818,495],{"mathvariant":111},[48,40820,1653],{"mathvariant":111},[51,40822,54],{"stretchy":53},[48,40824,369],{},[51,40826,60],{},[48,40828,753],{},[51,40830,747],{"separator":122},[48,40832,57],{},[51,40834,66],{"stretchy":53},[51,40836,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,40838,8481],{"stretchy":122},[88,40840,40841],{},"偶然",[51,40843,5106],{},[6963,40845,40846,40890],{},[6963,40847,40848,40888],{},[45,40849,40850,40868,40870,40872,40874,40876,40878,40880,40882,40884,40886],{},[898,40851,40852,40860],{},[45,40853,40854,40856,40858],{},[48,40855,10794],{"mathvariant":111},[48,40857,495],{"mathvariant":111},[48,40859,1653],{"mathvariant":111},[45,40861,40862,40864,40866],{},[48,40863,57],{},[51,40865,60],{},[48,40867,63],{},[51,40869,54],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,40871,902],{"mathvariant":571},[51,40873,924],{"stretchy":53},[48,40875,369],{},[51,40877,60],{},[48,40879,753],{},[51,40881,747],{"separator":122},[48,40883,57],{},[51,40885,945],{"stretchy":53},[51,40887,66],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,40889,8481],{"stretchy":122},[88,40891,40892],{},"认知",[48,40894,112],{"mathvariant":111},[114,40896,40897],{"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}}.",[23,40899,40901,40928,40958,41182],{"className":40900,"ariaHidden":122},[121],[23,40902,40904,40907,40913,40916,40919,40922,40925],{"className":40903},[126],[23,40905],{"className":40906,"style":131},[130],[23,40908,40910],{"className":40909},[135],[23,40911,11074],{"className":40912},[135,5391],[23,40914,54],{"className":40915},[140],[23,40917,369],{"className":40918,"style":392},[135,136],[23,40920],{"className":40921,"style":149},[148],[23,40923,60],{"className":40924},[153],[23,40926],{"className":40927,"style":149},[148],[23,40929,40931,40934,40937,40940,40943,40946,40949,40952,40955],{"className":40930},[126],[23,40932],{"className":40933,"style":131},[130],[23,40935,753],{"className":40936},[135,136],[23,40938,747],{"className":40939},[824],[23,40941],{"className":40942,"style":280},[148],[23,40944,63],{"className":40945,"style":144},[135,136],[23,40947,66],{"className":40948},[169],[23,40950],{"className":40951,"style":149},[148],[23,40953,69],{"className":40954},[153],[23,40956],{"className":40957,"style":149},[148],[23,40959,40961,40965,41173,41176,41179],{"className":40960},[126],[23,40962],{"className":40963,"style":40964},[130],"height:2.5315em;vertical-align:-1.6815em;",[23,40966,40968],{"className":40967},[1839,6963],[23,40969,40971,41164],{"className":40970},[199,200],[23,40972,40974,41161],{"className":40973},[204],[23,40975,40978,40996],{"className":40976,"style":40977},[208],"height:0.85em;",[23,40979,40981,40984],{"style":40980},"top:-1.3185em;",[23,40982],{"className":40983,"style":217},[216],[23,40985,40987],{"className":40986},[1019,1020,1021,1022],[23,40988,40990],{"className":40989},[135,1022],[23,40991,40993],{"className":40992},[135,6514,1022],[23,40994,40841],{"className":40995},[135,7240,1022],[23,40997,40998,41001],{"style":7132},[23,40999],{"className":41000,"style":217},[216],[23,41002,41004],{"className":41003},[1839,6963],[23,41005,41007,41152],{"className":41006},[199,200],[23,41008,41010,41149],{"className":41009},[204],[23,41011,41013,41044],{"className":41012,"style":40977},[208],[23,41014,41017,41020],{"className":41015,"style":41016},[8676],"top:-1.9968em;",[23,41018],{"className":41019,"style":217},[216],[23,41021,41023,41030,41037],{"className":41022,"style":8685},[8684],[23,41024,41026],{"className":41025,"style":8690},[8689],[8692,41027,41028],{"xmlns":8694,"width":8695,"height":8696,"viewBox":8697,"preserveAspectRatio":8698},[8700,41029],{"d":8702},[23,41031,41033],{"className":41032,"style":8690},[8706],[8692,41034,41035],{"xmlns":8694,"width":8695,"height":8696,"viewBox":8697,"preserveAspectRatio":8709},[8700,41036],{"d":8712},[23,41038,41040],{"className":41039,"style":8690},[8716],[8692,41041,41042],{"xmlns":8694,"width":8695,"height":8696,"viewBox":8697,"preserveAspectRatio":8719},[8700,41043],{"d":8722},[23,41045,41046,41049],{"style":7132},[23,41047],{"className":41048,"style":217},[216],[23,41050,41052,41101,41107,41113,41116,41119,41122,41125,41128,41131,41134,41137,41140,41143],{"className":41051},[135],[23,41053,41055,41058],{"className":41054},[135],[23,41056,902],{"className":41057},[135,644],[23,41059,41061],{"className":41060},[998],[23,41062,41064,41093],{"className":41063},[199,200],[23,41065,41067,41090],{"className":41066},[204],[23,41068,41070],{"className":41069,"style":1008},[208],[23,41071,41072,41075],{"style":1011},[23,41073],{"className":41074,"style":1015},[216],[23,41076,41078],{"className":41077},[1019,1020,1021,1022],[23,41079,41081,41084,41087],{"className":41080},[135,1022],[23,41082,57],{"className":41083,"style":144},[135,136,1022],[23,41085,60],{"className":41086},[153,1022],[23,41088,63],{"className":41089,"style":144},[135,136,1022],[23,41091,297],{"className":41092},[296],[23,41094,41096],{"className":41095},[204],[23,41097,41099],{"className":41098,"style":1057},[208],[23,41100],{},[23,41102,41104],{"className":41103},[135],[23,41105,924],{"className":41106},[1066,1067],[23,41108,41110],{"className":41109},[135],[23,41111,11074],{"className":41112},[135,5391],[23,41114,54],{"className":41115},[140],[23,41117,369],{"className":41118,"style":392},[135,136],[23,41120],{"className":41121,"style":149},[148],[23,41123,60],{"className":41124},[153],[23,41126],{"className":41127,"style":149},[148],[23,41129,753],{"className":41130},[135,136],[23,41132,747],{"className":41133},[824],[23,41135],{"className":41136,"style":280},[148],[23,41138,57],{"className":41139,"style":144},[135,136],[23,41141,66],{"className":41142},[169],[23,41144,41146],{"className":41145},[135],[23,41147,945],{"className":41148},[1066,1067],[23,41150,297],{"className":41151},[296],[23,41153,41155],{"className":41154},[204],[23,41156,41159],{"className":41157,"style":41158},[208],"height:1.0032em;",[23,41160],{},[23,41162,297],{"className":41163},[296],[23,41165,41167],{"className":41166},[204],[23,41168,41171],{"className":41169,"style":41170},[208],"height:1.6815em;",[23,41172],{},[23,41174],{"className":41175,"style":392},[148],[23,41177,5106],{"className":41178},[612],[23,41180],{"className":41181,"style":392},[148],[23,41183,41185,41188,41392,41395],{"className":41184},[126],[23,41186],{"className":41187,"style":40964},[130],[23,41189,41191],{"className":41190},[1839,6963],[23,41192,41194,41384],{"className":41193},[199,200],[23,41195,41197,41381],{"className":41196},[204],[23,41198,41200,41217],{"className":41199,"style":40977},[208],[23,41201,41202,41205],{"style":40980},[23,41203],{"className":41204,"style":217},[216],[23,41206,41208],{"className":41207},[1019,1020,1021,1022],[23,41209,41211],{"className":41210},[135,1022],[23,41212,41214],{"className":41213},[135,6514,1022],[23,41215,40892],{"className":41216},[135,7240,1022],[23,41218,41219,41222],{"style":7132},[23,41220],{"className":41221,"style":217},[216],[23,41223,41225],{"className":41224},[1839,6963],[23,41226,41228,41373],{"className":41227},[199,200],[23,41229,41231,41370],{"className":41230},[204],[23,41232,41234,41264],{"className":41233,"style":40977},[208],[23,41235,41237,41240],{"className":41236,"style":41016},[8676],[23,41238],{"className":41239,"style":217},[216],[23,41241,41243,41250,41257],{"className":41242,"style":8685},[8684],[23,41244,41246],{"className":41245,"style":8690},[8689],[8692,41247,41248],{"xmlns":8694,"width":8695,"height":8696,"viewBox":8697,"preserveAspectRatio":8698},[8700,41249],{"d":8702},[23,41251,41253],{"className":41252,"style":8690},[8706],[8692,41254,41255],{"xmlns":8694,"width":8695,"height":8696,"viewBox":8697,"preserveAspectRatio":8709},[8700,41256],{"d":8712},[23,41258,41260],{"className":41259,"style":8690},[8716],[8692,41261,41262],{"xmlns":8694,"width":8695,"height":8696,"viewBox":8697,"preserveAspectRatio":8719},[8700,41263],{"d":8722},[23,41265,41266,41269],{"style":7132},[23,41267],{"className":41268,"style":217},[216],[23,41270,41272,41325,41331,41334,41337,41340,41343,41346,41349,41352,41355,41358,41361,41364],{"className":41271},[135],[23,41273,41275,41281],{"className":41274},[135],[23,41276,41278],{"className":41277},[135],[23,41279,11074],{"className":41280},[135,5391],[23,41282,41284],{"className":41283},[998],[23,41285,41287,41317],{"className":41286},[199,200],[23,41288,41290,41314],{"className":41289},[204],[23,41291,41293],{"className":41292,"style":1008},[208],[23,41294,41296,41299],{"style":41295},"top:-2.5198em;margin-right:0.05em;",[23,41297],{"className":41298,"style":1015},[216],[23,41300,41302],{"className":41301},[1019,1020,1021,1022],[23,41303,41305,41308,41311],{"className":41304},[135,1022],[23,41306,57],{"className":41307,"style":144},[135,136,1022],[23,41309,60],{"className":41310},[153,1022],[23,41312,63],{"className":41313,"style":144},[135,136,1022],[23,41315,297],{"className":41316},[296],[23,41318,41320],{"className":41319},[204],[23,41321,41323],{"className":41322,"style":1057},[208],[23,41324],{},[23,41326,41328],{"className":41327},[135],[23,41329,54],{"className":41330},[1066,1067],[23,41332,902],{"className":41333},[135,644],[23,41335,924],{"className":41336},[140],[23,41338,369],{"className":41339,"style":392},[135,136],[23,41341],{"className":41342,"style":149},[148],[23,41344,60],{"className":41345},[153],[23,41347],{"className":41348,"style":149},[148],[23,41350,753],{"className":41351},[135,136],[23,41353,747],{"className":41354},[824],[23,41356],{"className":41357,"style":280},[148],[23,41359,57],{"className":41360,"style":144},[135,136],[23,41362,945],{"className":41363},[169],[23,41365,41367],{"className":41366},[135],[23,41368,66],{"className":41369},[1066,1067],[23,41371,297],{"className":41372},[296],[23,41374,41376],{"className":41375},[204],[23,41377,41379],{"className":41378,"style":41158},[208],[23,41380],{},[23,41382,297],{"className":41383},[296],[23,41385,41387],{"className":41386},[204],[23,41388,41390],{"className":41389,"style":41170},[208],[23,41391],{},[23,41393],{"className":41394,"style":280},[148],[23,41396,112],{"className":41397},[135],[11,41399,41400],{},"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。",[11,41402,41403],{},"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。",[2501,41405,41407],{"id":41406},"_63-推断本身也是决策","6.3 推断本身也是决策",[11,41409,41410],{},"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：",[3764,41412,41413,41423],{},[3767,41414,41415],{},[3770,41416,41417,41420],{},[3773,41418,41419],{},"参数空间上的损失",[3773,41421,41422],{},"最优估计",[3783,41424,41425,41431,41437],{},[3770,41426,41427,41429],{},[3788,41428,13804],{},[3788,41430,11335],{},[3770,41432,41433,41435],{},[3788,41434,13811],{},[3788,41436,13814],{},[3770,41438,41439,41441],{},[3788,41440,13796],{},[3788,41442,41443],{},"后验众数（MAP）",[11,41445,41446],{},"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。",[11,41448,41449],{},"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。",[2501,41451,41453],{"id":41452},"_64-后验的集中","6.4 后验的集中",[11,41455,41456,41457,41460],{},"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。",[18,41458,41459],{},"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，",[23,41462,41464],{"className":41463,"translate":27},[26],[23,41465,41467,41548],{"className":41466,"translate":27},[31],[23,41468,41470],{"className":41469},[35],[37,41471,41472],{"xmlns":39,"display":40},[42,41473,41474,41545],{},[45,41475,41476,41480,41482,41484,41486,41488,41498,41500,41502,41504,41506,41509,41511,41513,41515,41517,41519,41521,41523,41525,41531,41541,41543],{},[25520,41477,41478],{},[48,41479,4806],{},[88,41481,90],{},[51,41483,54],{"stretchy":53},[48,41485,57],{},[51,41487,4529],{},[898,41489,41490,41496],{},[16811,41491,41492,41494],{"accent":122},[48,41493,57],{},[51,41495,16817],{},[48,41497,4806],{},[51,41499,66],{"stretchy":53},[51,41501,60],{},[48,41503,63],{},[88,41505,5360],{},[51,41507,41508],{},"⇒",[88,41510,5360],{},[48,41512,12805],{"mathvariant":375},[51,41514,54],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[2581,41516,1532],{},[51,41518,747],{"separator":122},[88,41520,5360],{},[48,41522,34357],{},[51,41524,54],{"stretchy":53},[898,41526,41527,41529],{},[48,41528,57],{},[2581,41530,1532],{},[4518,41532,41533,41535],{},[51,41534,66],{"stretchy":53},[45,41536,41537,41539],{},[51,41538,4529],{},[2581,41540,4435],{},[51,41542,66],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,41544,747],{"separator":122},[114,41546,41547],{"encoding":116},"\\sqrt n\\,(\\theta-\\hat\\theta_n)\\mid D\n\\ \\Rightarrow\\\n\\mathcal N\\big(0,\\ I(\\theta_0)^{-1}\\big),",[23,41549,41551,41623,41709,41733],{"className":41550,"ariaHidden":122},[121],[23,41552,41554,41558,41605,41608,41611,41614,41617,41620],{"className":41553},[126],[23,41555],{"className":41556,"style":41557},[130],"height:1.0992em;vertical-align:-0.25em;",[23,41559,41561],{"className":41560},[135,25671],[23,41562,41564,41596],{"className":41563},[199,200],[23,41565,41567,41593],{"className":41566},[204],[23,41568,41571,41580],{"className":41569,"style":41570},[208],"height:0.8492em;",[23,41572,41574,41577],{"className":41573,"style":7132},[8676],[23,41575],{"className":41576,"style":217},[216],[23,41578,4806],{"className":41579,"style":25691},[135,136],[23,41581,41583,41586],{"style":41582},"top:-2.8092em;",[23,41584],{"className":41585,"style":217},[216],[23,41587,41589],{"className":41588,"style":25782},[25781],[8692,41590,41591],{"xmlns":8694,"width":8695,"height":25785,"viewBox":25786,"preserveAspectRatio":8698},[8700,41592],{"d":25789},[23,41594,297],{"className":41595},[296],[23,41597,41599],{"className":41598},[204],[23,41600,41603],{"className":41601,"style":41602},[208],"height:0.1908em;",[23,41604],{},[23,41606],{"className":41607,"style":280},[148],[23,41609,54],{"className":41610},[140],[23,41612,57],{"className":41613,"style":144},[135,136],[23,41615],{"className":41616,"style":392},[148],[23,41618,4529],{"className":41619},[612],[23,41621],{"className":41622,"style":392},[148],[23,41624,41626,41629,41697,41700,41703,41706],{"className":41625},[126],[23,41627],{"className":41628,"style":16970},[130],[23,41630,41632,41663],{"className":41631},[135],[23,41633,41635],{"className":41634},[135,16834],[23,41636,41638],{"className":41637},[199],[23,41639,41641],{"className":41640},[204],[23,41642,41644,41652],{"className":41643,"style":16844},[208],[23,41645,41646,41649],{"style":7132},[23,41647],{"className":41648,"style":217},[216],[23,41650,57],{"className":41651,"style":144},[135,136],[23,41653,41654,41657],{"style":16855},[23,41655],{"className":41656,"style":217},[216],[23,41658,41660],{"className":41659,"style":34869},[16862],[23,41661,16817],{"className":41662},[135],[23,41664,41666],{"className":41665},[998],[23,41667,41669,41689],{"className":41668},[199,200],[23,41670,41672,41686],{"className":41671},[204],[23,41673,41675],{"className":41674,"style":6499},[208],[23,41676,41677,41680],{"style":27175},[23,41678],{"className":41679,"style":1015},[216],[23,41681,41683],{"className":41682},[1019,1020,1021,1022],[23,41684,4806],{"className":41685},[135,136,1022],[23,41687,297],{"className":41688},[296],[23,41690,41692],{"className":41691},[204],[23,41693,41695],{"className":41694,"style":2048},[208],[23,41696],{},[23,41698,66],{"className":41699},[169],[23,41701],{"className":41702,"style":149},[148],[23,41704,60],{"className":41705},[153],[23,41707],{"className":41708,"style":149},[148],[23,41710,41712,41715,41718,41721,41724,41727,41730],{"className":41711},[126],[23,41713],{"className":41714,"style":470},[130],[23,41716,63],{"className":41717,"style":144},[135,136],[23,41719,5360],{"className":41720},[148],[23,41722],{"className":41723,"style":149},[148],[23,41725,41508],{"className":41726},[153],[23,41728,5360],{"className":41729},[148],[23,41731],{"className":41732,"style":149},[148],[23,41734,41736,41739,41742,41748,41751,41754,41757,41760,41763,41766,41806,41841,41847],{"className":41735},[126],[23,41737],{"className":41738,"style":17160},[130],[23,41740,12805],{"className":41741,"style":22222},[135,412],[23,41743,41745],{"className":41744},[135],[23,41746,54],{"className":41747},[1066,1067],[23,41749,1532],{"className":41750},[135],[23,41752,747],{"className":41753},[824],[23,41755,5360],{"className":41756},[148],[23,41758],{"className":41759,"style":280},[148],[23,41761,34357],{"className":41762,"style":454},[135,136],[23,41764,54],{"className":41765},[140],[23,41767,41769,41772],{"className":41768},[135],[23,41770,57],{"className":41771,"style":144},[135,136],[23,41773,41775],{"className":41774},[998],[23,41776,41778,41798],{"className":41777},[199,200],[23,41779,41781,41795],{"className":41780},[204],[23,41782,41784],{"className":41783,"style":18651},[208],[23,41785,41786,41789],{"style":27175},[23,41787],{"className":41788,"style":1015},[216],[23,41790,41792],{"className":41791},[1019,1020,1021,1022],[23,41793,1532],{"className":41794},[135,1022],[23,41796,297],{"className":41797},[296],[23,41799,41801],{"className":41800},[204],[23,41802,41804],{"className":41803,"style":2048},[208],[23,41805],{},[23,41807,41809,41812],{"className":41808},[169],[23,41810,66],{"className":41811},[169],[23,41813,41815],{"className":41814},[998],[23,41816,41818],{"className":41817},[199],[23,41819,41821],{"className":41820},[204],[23,41822,41824],{"className":41823,"style":11006},[208],[23,41825,41826,41829],{"style":4989},[23,41827],{"className":41828,"style":1015},[216],[23,41830,41832],{"className":41831},[1019,1020,1021,1022],[23,41833,41835,41838],{"className":41834},[135,1022],[23,41836,4529],{"className":41837},[135,1022],[23,41839,4435],{"className":41840},[135,1022],[23,41842,41844],{"className":41843},[135],[23,41845,66],{"className":41846},[1066,1067],[23,41848,747],{"className":41849},[824],[11,41851,24433,41852,41937],{},[23,41853,41855,41879],{"className":41854,"translate":27},[31],[23,41856,41858],{"className":41857},[35],[37,41859,41860],{"xmlns":39},[42,41861,41862,41876],{},[45,41863,41864,41866,41868,41874],{},[48,41865,34357],{},[51,41867,54],{"stretchy":53},[898,41869,41870,41872],{},[48,41871,57],{},[2581,41873,1532],{},[51,41875,66],{"stretchy":53},[114,41877,41878],{"encoding":116},"I(\\theta_0)",[23,41880,41882],{"className":41881,"ariaHidden":122},[121],[23,41883,41885,41888,41891,41894,41934],{"className":41884},[126],[23,41886],{"className":41887,"style":131},[130],[23,41889,34357],{"className":41890,"style":454},[135,136],[23,41892,54],{"className":41893},[140],[23,41895,41897,41900],{"className":41896},[135],[23,41898,57],{"className":41899,"style":144},[135,136],[23,41901,41903],{"className":41902},[998],[23,41904,41906,41926],{"className":41905},[199,200],[23,41907,41909,41923],{"className":41908},[204],[23,41910,41912],{"className":41911,"style":18651},[208],[23,41913,41914,41917],{"style":27175},[23,41915],{"className":41916,"style":1015},[216],[23,41918,41920],{"className":41919},[1019,1020,1021,1022],[23,41921,1532],{"className":41922},[135,1022],[23,41924,297],{"className":41925},[296],[23,41927,41929],{"className":41928},[204],[23,41930,41932],{"className":41931,"style":2048},[208],[23,41933],{},[23,41935,66],{"className":41936},[169]," 是 Fisher 信息。等价地，后验近似为",[23,41939,41941],{"className":41940,"translate":27},[26],[23,41942,41944,42020],{"className":41943,"translate":27},[31],[23,41945,41947],{"className":41946},[35],[37,41948,41949],{"xmlns":39,"display":40},[42,41950,41951,42017],{},[45,41952,41953,41955,41957,41959,41961,41963,41965,41967,41969,42015],{},[48,41954,11],{},[51,41956,54],{"stretchy":53},[48,41958,57],{},[51,41960,60],{},[48,41962,63],{},[51,41964,66],{"stretchy":53},[51,41966,29070],{},[48,41968,12805],{"mathvariant":375},[45,41970,41971,41973,41983,41985,41987,41993,41995,41997,42003,42013],{},[51,41972,54],{"fence":122},[898,41974,41975,41981],{},[16811,41976,41977,41979],{"accent":122},[48,41978,57],{},[51,41980,16817],{},[48,41982,4806],{},[51,41984,747],{"separator":122},[88,41986,5360],{},[71,41988,41989,41991],{},[2581,41990,4435],{},[48,41992,4806],{},[48,41994,34357],{},[51,41996,54],{"stretchy":53},[898,41998,41999,42001],{},[48,42000,57],{},[2581,42002,1532],{},[4518,42004,42005,42007],{},[51,42006,66],{"stretchy":53},[45,42008,42009,42011],{},[51,42010,4529],{},[2581,42012,4435],{},[51,42014,66],{"fence":122},[48,42016,112],{"mathvariant":111},[114,42018,42019],{"encoding":116},"p(\\theta\\mid D)\\approx\n\\mathcal N\\left(\\hat\\theta_n,\\ \\frac{1}{n}I(\\theta_0)^{-1}\\right).",[23,42021,42023,42047,42068],{"className":42022,"ariaHidden":122},[121],[23,42024,42026,42029,42032,42035,42038,42041,42044],{"className":42025},[126],[23,42027],{"className":42028,"style":131},[130],[23,42030,11],{"className":42031},[135,136],[23,42033,54],{"className":42034},[140],[23,42036,57],{"className":42037,"style":144},[135,136],[23,42039],{"className":42040,"style":149},[148],[23,42042,60],{"className":42043},[153],[23,42045],{"className":42046,"style":149},[148],[23,42048,42050,42053,42056,42059,42062,42065],{"className":42049},[126],[23,42051],{"className":42052,"style":131},[130],[23,42054,63],{"className":42055,"style":144},[135,136],[23,42057,66],{"className":42058},[169],[23,42060],{"className":42061,"style":149},[148],[23,42063,29070],{"className":42064},[153],[23,42066],{"className":42067,"style":149},[148],[23,42069,42071,42074,42077,42080,42315,42318],{"className":42070},[126],[23,42072],{"className":42073,"style":20793},[130],[23,42075,12805],{"className":42076,"style":22222},[135,412],[23,42078],{"className":42079,"style":280},[148],[23,42081,42083,42089,42157,42160,42163,42166,42228,42231,42234,42274,42309],{"className":42082},[1839],[23,42084,42086],{"className":42085,"style":1844},[140,1843],[23,42087,54],{"className":42088},[1066,1021],[23,42090,42092,42123],{"className":42091},[135],[23,42093,42095],{"className":42094},[135,16834],[23,42096,42098],{"className":42097},[199],[23,42099,42101],{"className":42100},[204],[23,42102,42104,42112],{"className":42103,"style":16844},[208],[23,42105,42106,42109],{"style":7132},[23,42107],{"className":42108,"style":217},[216],[23,42110,57],{"className":42111,"style":144},[135,136],[23,42113,42114,42117],{"style":16855},[23,42115],{"className":42116,"style":217},[216],[23,42118,42120],{"className":42119,"style":34869},[16862],[23,42121,16817],{"className":42122},[135],[23,42124,42126],{"className":42125},[998],[23,42127,42129,42149],{"className":42128},[199,200],[23,42130,42132,42146],{"className":42131},[204],[23,42133,42135],{"className":42134,"style":6499},[208],[23,42136,42137,42140],{"style":27175},[23,42138],{"className":42139,"style":1015},[216],[23,42141,42143],{"className":42142},[1019,1020,1021,1022],[23,42144,4806],{"className":42145},[135,136,1022],[23,42147,297],{"className":42148},[296],[23,42150,42152],{"className":42151},[204],[23,42153,42155],{"className":42154,"style":2048},[208],[23,42156],{},[23,42158,747],{"className":42159},[824],[23,42161,5360],{"className":42162},[148],[23,42164],{"className":42165,"style":280},[148],[23,42167,42169,42172,42225],{"className":42168},[135],[23,42170],{"className":42171},[140,192],[23,42173,42175],{"className":42174},[71],[23,42176,42178,42217],{"className":42177},[199,200],[23,42179,42181,42214],{"className":42180},[204],[23,42182,42184,42195,42203],{"className":42183,"style":12413},[208],[23,42185,42186,42189],{"style":212},[23,42187],{"className":42188,"style":217},[216],[23,42190,42192],{"className":42191},[135],[23,42193,4806],{"className":42194},[135,136],[23,42196,42197,42200],{"style":235},[23,42198],{"className":42199,"style":217},[216],[23,42201],{"className":42202,"style":243},[242],[23,42204,42205,42208],{"style":246},[23,42206],{"className":42207,"style":217},[216],[23,42209,42211],{"className":42210},[135],[23,42212,4435],{"className":42213},[135],[23,42215,297],{"className":42216},[296],[23,42218,42220],{"className":42219},[204],[23,42221,42223],{"className":42222,"style":6287},[208],[23,42224],{},[23,42226],{"className":42227},[169,192],[23,42229,34357],{"className":42230,"style":454},[135,136],[23,42232,54],{"className":42233},[140],[23,42235,42237,42240],{"className":42236},[135],[23,42238,57],{"className":42239,"style":144},[135,136],[23,42241,42243],{"className":42242},[998],[23,42244,42246,42266],{"className":42245},[199,200],[23,42247,42249,42263],{"className":42248},[204],[23,42250,42252],{"className":42251,"style":18651},[208],[23,42253,42254,42257],{"style":27175},[23,42255],{"className":42256,"style":1015},[216],[23,42258,42260],{"className":42259},[1019,1020,1021,1022],[23,42261,1532],{"className":42262},[135,1022],[23,42264,297],{"className":42265},[296],[23,42267,42269],{"className":42268},[204],[23,42270,42272],{"className":42271,"style":2048},[208],[23,42273],{},[23,42275,42277,42280],{"className":42276},[169],[23,42278,66],{"className":42279},[169],[23,42281,42283],{"className":42282},[998],[23,42284,42286],{"className":42285},[199],[23,42287,42289],{"className":42288},[204],[23,42290,42292],{"className":42291,"style":11006},[208],[23,42293,42294,42297],{"style":4989},[23,42295],{"className":42296,"style":1015},[216],[23,42298,42300],{"className":42299},[1019,1020,1021,1022],[23,42301,42303,42306],{"className":42302},[135,1022],[23,42304,4529],{"className":42305},[135,1022],[23,42307,4435],{"className":42308},[135,1022],[23,42310,42312],{"className":42311,"style":1844},[169,1843],[23,42313,66],{"className":42314},[1066,1021],[23,42316],{"className":42317,"style":280},[148],[23,42319,112],{"className":42320},[135],[11,42322,42323,42326],{},[18,42324,42325],{},"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。",[11,42328,42329,42330,42430],{},"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 ",[23,42331,42333,42360],{"className":42332,"translate":27},[31],[23,42334,42336],{"className":42335},[35],[37,42337,42338],{"xmlns":39},[42,42339,42340,42357],{},[45,42341,42342,42345,42347,42349,42351,42355],{},[48,42343,42344],{},"O",[51,42346,54],{"stretchy":53},[2581,42348,4435],{},[48,42350,2875],{"mathvariant":111},[25520,42352,42353],{},[48,42354,4806],{},[51,42356,66],{"stretchy":53},[114,42358,42359],{"encoding":116},"O(1\u002F\\sqrt n)",[23,42361,42363],{"className":42362,"ariaHidden":122},[121],[23,42364,42366,42370,42373,42376,42380,42427],{"className":42365},[126],[23,42367],{"className":42368,"style":42369},[130],"height:1.0503em;vertical-align:-0.25em;",[23,42371,42344],{"className":42372,"style":144},[135,136],[23,42374,54],{"className":42375},[140],[23,42377,42379],{"className":42378},[135],"1\u002F",[23,42381,42383],{"className":42382},[135,25671],[23,42384,42386,42418],{"className":42385},[199,200],[23,42387,42389,42415],{"className":42388},[204],[23,42390,42393,42402],{"className":42391,"style":42392},[208],"height:0.8003em;",[23,42394,42396,42399],{"className":42395,"style":7132},[8676],[23,42397],{"className":42398,"style":217},[216],[23,42400,4806],{"className":42401,"style":25691},[135,136],[23,42403,42405,42408],{"style":42404},"top:-2.7603em;",[23,42406],{"className":42407,"style":217},[216],[23,42409,42411],{"className":42410,"style":25782},[25781],[8692,42412,42413],{"xmlns":8694,"width":8695,"height":25785,"viewBox":25786,"preserveAspectRatio":8698},[8700,42414],{"d":25789},[23,42416,297],{"className":42417},[296],[23,42419,42421],{"className":42420},[204],[23,42422,42425],{"className":42423,"style":42424},[208],"height:0.2397em;",[23,42426],{},[23,42428,66],{"className":42429},[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。",[2501,42432,42434],{"id":42433},"_65-边缘似然与奥卡姆剃刀","6.5 边缘似然与奥卡姆剃刀",[11,42436,42437,42438,42441],{},"贝叶斯模型比较的核心量是",[18,42439,42440],{},"边缘似然","（marginal likelihood），也就是贝叶斯公式的分母：",[23,42443,42445],{"className":42444,"translate":27},[26],[23,42446,42448,42514],{"className":42447,"translate":27},[31],[23,42449,42451],{"className":42450},[35],[37,42452,42453],{"xmlns":39,"display":40},[42,42454,42455,42511],{},[45,42456,42457,42459,42461,42463,42465,42467,42469,42471,42473,42475,42477,42479,42481,42483,42485,42487,42489,42491,42493,42495,42497,42499,42501,42503,42505,42507,42509],{},[48,42458,11],{},[51,42460,54],{"stretchy":53},[48,42462,63],{},[51,42464,60],{},[48,42466,34736],{},[51,42468,66],{"stretchy":53},[51,42470,69],{},[51,42472,1557],{},[48,42474,11],{},[51,42476,54],{"stretchy":53},[48,42478,63],{},[51,42480,60],{},[48,42482,57],{},[51,42484,747],{"separator":122},[48,42486,34736],{},[51,42488,66],{"stretchy":53},[88,42490,90],{},[48,42492,11],{},[51,42494,54],{"stretchy":53},[48,42496,57],{},[51,42498,60],{},[48,42500,34736],{},[51,42502,66],{"stretchy":53},[88,42504,90],{},[48,42506,1606],{},[48,42508,57],{},[48,42510,112],{"mathvariant":111},[114,42512,42513],{"encoding":116},"p(D\\mid M)=\\int p(D\\mid\\theta,M)\\,p(\\theta\\mid M)\\,d\\theta .",[23,42515,42517,42541,42563,42593,42635],{"className":42516,"ariaHidden":122},[121],[23,42518,42520,42523,42526,42529,42532,42535,42538],{"className":42519},[126],[23,42521],{"className":42522,"style":131},[130],[23,42524,11],{"className":42525},[135,136],[23,42527,54],{"className":42528},[140],[23,42530,63],{"className":42531,"style":144},[135,136],[23,42533],{"className":42534,"style":149},[148],[23,42536,60],{"className":42537},[153],[23,42539],{"className":42540,"style":149},[148],[23,42542,42544,42547,42551,42554,42557,42560],{"className":42543},[126],[23,42545],{"className":42546,"style":131},[130],[23,42548,34736],{"className":42549,"style":42550},[135,136],"margin-right:0.109em;",[23,42552,66],{"className":42553},[169],[23,42555],{"className":42556,"style":149},[148],[23,42558,69],{"className":42559},[153],[23,42561],{"className":42562,"style":149},[148],[23,42564,42566,42569,42572,42575,42578,42581,42584,42587,42590],{"className":42565},[126],[23,42567],{"className":42568,"style":3696},[130],[23,42570,1557],{"className":42571,"style":1792},[1786,1790,1791],[23,42573],{"className":42574,"style":280},[148],[23,42576,11],{"className":42577},[135,136],[23,42579,54],{"className":42580},[140],[23,42582,63],{"className":42583,"style":144},[135,136],[23,42585],{"className":42586,"style":149},[148],[23,42588,60],{"className":42589},[153],[23,42591],{"className":42592,"style":149},[148],[23,42594,42596,42599,42602,42605,42608,42611,42614,42617,42620,42623,42626,42629,42632],{"className":42595},[126],[23,42597],{"className":42598,"style":131},[130],[23,42600,57],{"className":42601,"style":144},[135,136],[23,42603,747],{"className":42604},[824],[23,42606],{"className":42607,"style":280},[148],[23,42609,34736],{"className":42610,"style":42550},[135,136],[23,42612,66],{"className":42613},[169],[23,42615],{"className":42616,"style":280},[148],[23,42618,11],{"className":42619},[135,136],[23,42621,54],{"className":42622},[140],[23,42624,57],{"className":42625,"style":144},[135,136],[23,42627],{"className":42628,"style":149},[148],[23,42630,60],{"className":42631},[153],[23,42633],{"className":42634,"style":149},[148],[23,42636,42638,42641,42644,42647,42650,42653,42656],{"className":42637},[126],[23,42639],{"className":42640,"style":131},[130],[23,42642,34736],{"className":42643,"style":42550},[135,136],[23,42645,66],{"className":42646},[169],[23,42648],{"className":42649,"style":280},[148],[23,42651,1606],{"className":42652},[135,136],[23,42654,57],{"className":42655,"style":144},[135,136],[23,42657,112],{"className":42658},[135],[11,42660,42661,42662,32989],{},"两个模型的比较用",[18,42663,42664],{},"贝叶斯因子",[23,42666,42668],{"className":42667,"translate":27},[26],[23,42669,42671,42822],{"className":42670,"translate":27},[31],[23,42672,42674],{"className":42673},[35],[37,42675,42676],{"xmlns":39,"display":40},[42,42677,42678,42819],{},[45,42679,42680,42691,42693,42731,42733,42735,42773,42775,42785,42787,42817],{},[898,42681,42682,42688],{},[45,42683,42684,42686],{},[48,42685,2533],{"mathvariant":111},[48,42687,12719],{"mathvariant":111},[2581,42689,42690],{},"12",[51,42692,69],{},[71,42694,42695,42713],{},[45,42696,42697,42699,42701,42703,42705,42711],{},[48,42698,11],{},[51,42700,54],{"stretchy":53},[48,42702,63],{},[51,42704,60],{},[898,42706,42707,42709],{},[48,42708,34736],{},[2581,42710,4435],{},[51,42712,66],{"stretchy":53},[45,42714,42715,42717,42719,42721,42723,42729],{},[48,42716,11],{},[51,42718,54],{"stretchy":53},[48,42720,63],{},[51,42722,60],{},[898,42724,42725,42727],{},[48,42726,34736],{},[2581,42728,9956],{},[51,42730,66],{"stretchy":53},[51,42732,747],{"separator":122},[148,42734],{"width":750},[71,42736,42737,42755],{},[45,42738,42739,42741,42743,42749,42751,42753],{},[48,42740,11],{},[51,42742,54],{"stretchy":53},[898,42744,42745,42747],{},[48,42746,34736],{},[2581,42748,4435],{},[51,42750,60],{},[48,42752,63],{},[51,42754,66],{"stretchy":53},[45,42756,42757,42759,42761,42767,42769,42771],{},[48,42758,11],{},[51,42760,54],{"stretchy":53},[898,42762,42763,42765],{},[48,42764,34736],{},[2581,42766,9956],{},[51,42768,60],{},[48,42770,63],{},[51,42772,66],{"stretchy":53},[51,42774,69],{},[898,42776,42777,42783],{},[45,42778,42779,42781],{},[48,42780,2533],{"mathvariant":111},[48,42782,12719],{"mathvariant":111},[2581,42784,42690],{},[51,42786,5758],{},[71,42788,42789,42803],{},[45,42790,42791,42793,42795,42801],{},[48,42792,11],{},[51,42794,54],{"stretchy":53},[898,42796,42797,42799],{},[48,42798,34736],{},[2581,42800,4435],{},[51,42802,66],{"stretchy":53},[45,42804,42805,42807,42809,42815],{},[48,42806,11],{},[51,42808,54],{"stretchy":53},[898,42810,42811,42813],{},[48,42812,34736],{},[2581,42814,9956],{},[51,42816,66],{"stretchy":53},[48,42818,112],{"mathvariant":111},[114,42820,42821],{"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)}.",[23,42823,42825,42888,43269,43330],{"className":42824,"ariaHidden":122},[121],[23,42826,42828,42832,42879,42882,42885],{"className":42827},[126],[23,42829],{"className":42830,"style":42831},[130],"height:0.8333em;vertical-align:-0.15em;",[23,42833,42835,42842],{"className":42834},[135],[23,42836,42838],{"className":42837},[135],[23,42839,42841],{"className":42840},[135,5391],"BF",[23,42843,42845],{"className":42844},[998],[23,42846,42848,42871],{"className":42847},[199,200],[23,42849,42851,42868],{"className":42850},[204],[23,42852,42854],{"className":42853,"style":18651},[208],[23,42855,42856,42859],{"style":8191},[23,42857],{"className":42858,"style":1015},[216],[23,42860,42862],{"className":42861},[1019,1020,1021,1022],[23,42863,42865],{"className":42864},[135,1022],[23,42866,42690],{"className":42867},[135,1022],[23,42869,297],{"className":42870},[296],[23,42872,42874],{"className":42873},[204],[23,42875,42877],{"className":42876,"style":2048},[208],[23,42878],{},[23,42880],{"className":42881,"style":149},[148],[23,42883,69],{"className":42884},[153],[23,42886],{"className":42887,"style":149},[148],[23,42889,42891,42894,43073,43076,43079,43082,43260,43263,43266],{"className":42890},[126],[23,42892],{"className":42893,"style":185},[130],[23,42895,42897,42900,43070],{"className":42896},[135],[23,42898],{"className":42899},[140,192],[23,42901,42903],{"className":42902},[71],[23,42904,42906,43062],{"className":42905},[199,200],[23,42907,42909,43059],{"className":42908},[204],[23,42910,42912,42982,42990],{"className":42911,"style":209},[208],[23,42913,42914,42917],{"style":212},[23,42915],{"className":42916,"style":217},[216],[23,42918,42920,42923,42926,42929,42932,42935,42938,42979],{"className":42919},[135],[23,42921,11],{"className":42922},[135,136],[23,42924,54],{"className":42925},[140],[23,42927,63],{"className":42928,"style":144},[135,136],[23,42930],{"className":42931,"style":149},[148],[23,42933,60],{"className":42934},[153],[23,42936],{"className":42937,"style":149},[148],[23,42939,42941,42944],{"className":42940},[135],[23,42942,34736],{"className":42943,"style":42550},[135,136],[23,42945,42947],{"className":42946},[998],[23,42948,42950,42971],{"className":42949},[199,200],[23,42951,42953,42968],{"className":42952},[204],[23,42954,42956],{"className":42955,"style":18651},[208],[23,42957,42959,42962],{"style":42958},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[23,42960],{"className":42961,"style":1015},[216],[23,42963,42965],{"className":42964},[1019,1020,1021,1022],[23,42966,9956],{"className":42967},[135,1022],[23,42969,297],{"className":42970},[296],[23,42972,42974],{"className":42973},[204],[23,42975,42977],{"className":42976,"style":2048},[208],[23,42978],{},[23,42980,66],{"className":42981},[169],[23,42983,42984,42987],{"style":235},[23,42985],{"className":42986,"style":217},[216],[23,42988],{"className":42989,"style":243},[242],[23,42991,42992,42995],{"style":246},[23,42993],{"className":42994,"style":217},[216],[23,42996,42998,43001,43004,43007,43010,43013,43016,43056],{"className":42997},[135],[23,42999,11],{"className":43000},[135,136],[23,43002,54],{"className":43003},[140],[23,43005,63],{"className":43006,"style":144},[135,136],[23,43008],{"className":43009,"style":149},[148],[23,43011,60],{"className":43012},[153],[23,43014],{"className":43015,"style":149},[148],[23,43017,43019,43022],{"className":43018},[135],[23,43020,34736],{"className":43021,"style":42550},[135,136],[23,43023,43025],{"className":43024},[998],[23,43026,43028,43048],{"className":43027},[199,200],[23,43029,43031,43045],{"className":43030},[204],[23,43032,43034],{"className":43033,"style":18651},[208],[23,43035,43036,43039],{"style":42958},[23,43037],{"className":43038,"style":1015},[216],[23,43040,43042],{"className":43041},[1019,1020,1021,1022],[23,43043,4435],{"className":43044},[135,1022],[23,43046,297],{"className":43047},[296],[23,43049,43051],{"className":43050},[204],[23,43052,43054],{"className":43053,"style":2048},[208],[23,43055],{},[23,43057,66],{"className":43058},[169],[23,43060,297],{"className":43061},[296],[23,43063,43065],{"className":43064},[204],[23,43066,43068],{"className":43067,"style":304},[208],[23,43069],{},[23,43071],{"className":43072},[169,192],[23,43074,747],{"className":43075},[824],[23,43077],{"className":43078,"style":828},[148],[23,43080],{"className":43081,"style":280},[148],[23,43083,43085,43088,43257],{"className":43084},[135],[23,43086],{"className":43087},[140,192],[23,43089,43091],{"className":43090},[71],[23,43092,43094,43249],{"className":43093},[199,200],[23,43095,43097,43246],{"className":43096},[204],[23,43098,43100,43169,43177],{"className":43099,"style":209},[208],[23,43101,43102,43105],{"style":212},[23,43103],{"className":43104,"style":217},[216],[23,43106,43108,43111,43114,43154,43157,43160,43163,43166],{"className":43107},[135],[23,43109,11],{"className":43110},[135,136],[23,43112,54],{"className":43113}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420,43421,43424],{"style":246},[23,43422],{"className":43423,"style":217},[216],[23,43425,43427,43430,43433,43473],{"className":43426},[135],[23,43428,11],{"className":43429},[135,136],[23,43431,54],{"className":43432},[140],[23,43434,43436,43439],{"className":43435},[135],[23,43437,34736],{"className":43438,"style":42550},[135,136],[23,43440,43442],{"className":43441},[998],[23,43443,43445,43465],{"className":43444},[199,200],[23,43446,43448,43462],{"className":43447},[204],[23,43449,43451],{"className":43450,"style":18651},[208],[23,43452,43453,43456],{"style":42958},[23,43454],{"className":43455,"style":1015},[216],[23,43457,43459],{"className":43458},[1019,1020,1021,1022],[23,43460,4435],{"className":43461},[135,1022],[23,43463,297],{"className":43464},[296],[23,43466,43468],{"className":43467},[204],[23,43469,43471],{"className":43470,"style":2048},[208],[23,43472],{},[23,43474,66],{"className":43475},[169],[23,43477,297],{"className":43478},[296],[23,43480,43482],{"className":43481},[204],[23,43483,43485],{"className":43484,"style":304},[208],[23,43486],{},[23,43488],{"className":43489},[169,192],[23,43491,112],{"className":43492},[135],[11,43494,43495,43496,43556,43557,34441],{},"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}.",[23,43983,43985,44009,44030,44054,44143],{"className":43984,"ariaHidden":122},[121],[23,43986,43988,43991,43994,43997,44000,44003,44006],{"className":43987},[126],[23,43989],{"className":43990,"style":131},[130],[23,43992,11],{"className":43993},[135,136],[23,43995,54],{"className":43996},[140],[23,43998,63],{"className":43999,"style":144},[135,136],[23,44001],{"className":44002,"style":149},[148],[23,44004,60],{"className":44005},[153],[23,44007],{"className":44008,"style":149},[148],[23,44010,44012,44015,44018,44021,44024,44027],{"className":44011},[126],[23,44013],{"className":44014,"style":131},[130],[23,44016,34736],{"className":44017,"style":42550},[135,136],[23,44019,66],{"className":44020},[169],[23,44022],{"className":44023,"style":149},[148],[23,44025,29070],{"className":44026},[153],[23,44028],{"className":44029,"style":149},[148],[23,44031,44033,44036,44039,44042,44045,44048,44051],{"className":44032},[126],[23,44034],{"className":44035,"style":131},[130],[23,44037,11],{"className":44038},[135,136],[23,44040,54],{"className":44041},[140],[23,44043,63],{"className":44044,"style":144},[135,136],[23,44046],{"className":44047,"style":149},[148],[23,44049,60],{"className":44050},[153],[23,44052],{"className":44053,"style":149},[148],[23,44055,44057,44060,44091,44094,44097,44100,44103,44134,44137,44140],{"className":44056},[126],[23,44058],{"className":44059,"style":16970},[130],[23,44061,44063],{"className":44062},[135,16834],[23,44064,44066],{"className":44065},[199],[23,44067,44069],{"className":44068},[204],[23,44070,44072,44080],{"className":44071,"style":16844},[208],[23,44073,44074,44077],{"style":7132},[23,44075],{"className":44076,"style":217},[216],[23,44078,57],{"className":44079,"style":144},[135,136],[23,44081,44082,44085],{"style":16855},[23,44083],{"className":44084,"style":217},[216],[23,44086,44088],{"className":44087,"style":34869},[16862],[23,44089,16817],{"className":44090},[135],[23,44092,66],{"className":44093},[169],[23,44095],{"className":44096,"style":280},[148],[23,44098,11],{"className":44099},[135,136],[23,44101,54],{"className":44102},[140],[23,44104,44106],{"className":44105},[135,16834],[23,44107,44109],{"className":44108},[199],[23,44110,44112],{"className":44111},[204],[23,44113,44115,44123],{"className":44114,"style":16844},[208],[23,44116,44117,44120],{"style":7132},[23,44118],{"className":44119,"style":217},[216],[23,44121,57],{"className":44122,"style":144},[135,136],[23,44124,44125,44128],{"style":16855},[23,44126],{"className":44127,"style":217},[216],[23,44129,44131],{"className":44130,"style":34869},[16862],[23,44132,16817],{"className":44133},[135],[23,44135],{"className":44136,"style":149},[148],[23,44138,60],{"className":44139},[153],[23,44141],{"className":44142,"style":149},[148],[23,44144,44146,44150,44153,44156,44159,44162,44165,44168,44204,44207,44210,44245],{"className":44145},[126],[23,44147],{"className":44148,"style":44149},[130],"height:1.188em;vertical-align:-0.25em;",[23,44151,34736],{"className":44152,"style":42550},[135,136],[23,44154,66],{"className":44155},[169],[23,44157],{"className":44158,"style":280},[148],[23,44160,54],{"className":44161},[140],[23,44163,9956],{"className":44164},[135],[23,44166,21697],{"className":44167,"style":676},[135,136],[23,44169,44171,44174],{"className":44170},[169],[23,44172,66],{"className":44173},[169],[23,44175,44177],{"className":44176},[998],[23,44178,44180],{"className":44179},[199],[23,44181,44183],{"className":44182},[204],[23,44184,44187],{"className":44185,"style":44186},[208],"height:0.938em;",[23,44188,44189,44192],{"style":4989},[23,44190],{"className":44191,"style":1015},[216],[23,44193,44195],{"className":44194},[1019,1020,1021,1022],[23,44196,44198,44201],{"className":44197},[135,1022],[23,44199,1606],{"className":44200},[135,136,1022],[23,44202,20564],{"className":44203},[135,1022],[23,44205,60],{"className":44206},[135],[23,44208,26253],{"className":44209,"style":26507},[135,136],[23,44211,44213,44216],{"className":44212},[135],[23,44214,60],{"className":44215},[135],[23,44217,44219],{"className":44218},[998],[23,44220,44222],{"className":44221},[199],[23,44223,44225],{"className":44224},[204],[23,44226,44228],{"className":44227,"style":44186},[208],[23,44229,44230,44233],{"style":4989},[23,44231],{"className":44232,"style":1015},[216],[23,44234,44236],{"className":44235},[1019,1020,1021,1022],[23,44237,44239,44242],{"className":44238},[135,1022],[23,44240,4529],{"className":44241},[135,1022],[23,44243,10473],{"className":44244},[135,1022],[23,44246,112],{"className":44247},[135],[11,44249,44250,44251,44354],{},"在正则模型里 ",[23,44252,44254,44284],{"className":44253,"translate":27},[31],[23,44255,44257],{"className":44256},[35],[37,44258,44259],{"xmlns":39},[42,44260,44261,44281],{},[45,44262,44263,44265,44267,44269,44271,44273,44279],{},[48,44264,26253],{},[51,44266,29070],{},[48,44268,4806],{},[48,44270,34357],{},[51,44272,54],{"stretchy":53},[16811,44274,44275,44277],{"accent":122},[48,44276,57],{},[51,44278,16817],{},[51,44280,66],{"stretchy":53},[114,44282,44283],{"encoding":116},"H\\approx 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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。",[11,44984,44985,44986,1471],{},"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。",[18,44987,44988],{},"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",[338,44990,44992],{"id":44991},"_7-机器学习理论中的位置","7. 机器学习理论中的位置",[2501,44994,44996],{"id":44995},"_71-风险分解","7.1 风险分解",[11,44998,4410,44999,45029,45030,8059,45212,45274],{},[23,45000,45002,45016],{"className":45001,"translate":27},[31],[23,45003,45005],{"className":45004},[35],[37,45006,45007],{"xmlns":39},[42,45008,45009,45013],{},[45,45010,45011],{},[48,45012,26253],{"mathvariant":375},[114,45014,45015],{"encoding":116},"\\mathcal H",[23,45017,45019],{"className":45018,"ariaHidden":122},[121],[23,45020,45022,45025],{"className":45021},[126],[23,45023],{"className":45024,"style":470},[130],[23,45026,26253],{"className":45027,"style":45028},[135,412],"margin-right:0.0097em;"," 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是从数据中学到的模型。则",[23,45276,45278],{"className":45277,"translate":27},[26],[23,45279,45281,45383],{"className":45280,"translate":27},[31],[23,45282,45284],{"className":45283},[35],[37,45285,45286],{"xmlns":39,"display":40},[42,45287,45288,45380],{},[45,45289,45290,45292,45294,45300,45302,45304,45310,45312,45341,45343,45378],{},[48,45291,572],{},[51,45293,54],{"stretchy":53},[16811,45295,45296,45298],{"accent":122},[48,45297,18372],{},[51,45299,16817],{},[51,45301,66],{"stretchy":53},[51,45303,4529],{},[4518,45305,45306,45308],{},[48,45307,572],{},[51,45309,6883],{},[51,45311,69],{},[6963,45313,45314,45338],{},[6963,45315,45316,45336],{},[45,45317,45318,45320,45326,45328,45334],{},[51,45319,54],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[898,45321,45322,45324],{},[48,45323,572],{},[48,45325,26253],{"mathvariant":375},[51,45327,4529],{},[4518,45329,45330,45332],{},[48,45331,572],{},[51,45333,6883],{},[51,45335,66],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,45337,8481],{"stretchy":122},[88,45339,45340],{},"近似误差",[51,45342,5106],{},[6963,45344,45345,45375],{},[6963,45346,45347,45373],{},[45,45348,45349,45351,45353,45355,45361,45363,45365,45371],{},[51,45350,54],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,45352,572],{},[51,45354,54],{"stretchy":53},[16811,45356,45357,45359],{"accent":122},[48,45358,18372],{},[51,45360,16817],{},[51,45362,66],{"stretchy":53},[51,45364,4529],{},[898,45366,45367,45369],{},[48,45368,572],{},[48,45370,26253],{"mathvariant":375},[51,45372,66],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,45374,8481],{"stretchy":122},[88,45376,45377],{},"估计误差",[48,45379,112],{"mathvariant":111},[114,45381,45382],{"encoding":116},"R(\\hat 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的情况下，分别研究这两项能压到多低。",[11,46047,46048],{},"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。",[2501,46050,46052],{"id":46051},"_72-一致性","7.2 一致性",[11,46054,46055,46056,15005,46128,46131,46132,8059],{},"一个决策规则序列 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\\xrightarrow{\\ P\\ }\\ R^*",[23,46219,46221,46361],{"className":46220,"ariaHidden":122},[121],[23,46222,46224,46228,46231,46234,46274,46277,46280,46283,46355,46358],{"className":46223},[126],[23,46225],{"className":46226,"style":46227},[130],"height:1.3503em;vertical-align:-0.25em;",[23,46229,572],{"className":46230,"style":963},[135,136],[23,46232,54],{"className":46233},[140],[23,46235,46237,46240],{"className":46236},[135],[23,46238,736],{"className":46239,"style":783},[135,136],[23,46241,46243],{"className":46242},[998],[23,46244,46246,46266],{"className":46245},[199,200],[23,46247,46249,46263],{"className":46248},[204],[23,46250,46252],{"className":46251,"style":6499},[208],[23,46253,46254,46257],{"style":46107},[23,46255],{"className":46256,"style":1015},[216],[23,46258,46260],{"className":46259},[1019,1020,1021,1022],[23,46261,4806],{"className":46262},[135,136,1022],[23,46264,297],{"className":46265},[296],[23,46267,46269],{"className":46268},[204],[23,46270,46272],{"className":46271,"style":2048},[208],[23,46273],{},[23,46275,66],{"className":46276},[169],[23,46278,5360],{"className":46279},[148],[23,46281],{"className":46282,"style":149},[148],[23,46284,46287],{"className":46285},[153,46286],"x-arrow",[23,46288,46290,46346],{"className":46289},[199,200],[23,46291,46293,46343],{"className":46292},[204],[23,46294,46297,46325],{"className":46295,"style":46296},[208],"height:1.1003em;",[23,46298,46300,46303],{"style":46299},"top:-3.322em;",[23,46301],{"className":46302,"style":1015},[216],[23,46304,46307],{"className":46305},[1019,1020,1021,1022,46306],"x-arrow-pad",[23,46308,46310,46316,46319],{"className":46309},[135,1022],[23,46311,46313],{"className":46312},[148,1022],[23,46314,5360],{"className":46315},[1022],[23,46317,920],{"className":46318,"style":1047},[135,136,1022],[23,46320,46322],{"className":46321},[148,1022],[23,46323,5360],{"className":46324},[1022],[23,46326,46329,46332],{"className":46327,"style":46328},[8676],"top:-2.689em;",[23,46330],{"className":46331,"style":1015},[216],[23,46333,46336],{"className":46334,"style":46335},[25781],"height:0.522em;min-width:1.469em;",[8692,46337,46340],{"xmlns":8694,"width":8695,"height":46338,"viewBox":46339,"preserveAspectRatio":8719},"0.522em","0 0 400000 522",[8700,46341],{"d":46342},"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",[23,46344,297],{"className":46345},[296],[23,46347,46349],{"className":46348},[204],[23,46350,46353],{"className":46351,"style":46352},[208],"height:0.011em;",[23,46354],{},[23,46356,5360],{"className":46357},[148],[23,46359],{"className":46360,"style":149},[148],[23,46362,46364,46367],{"className":46363},[126],[23,46365],{"className":46366,"style":7047},[130],[23,46368,46370,46373],{"className":46369},[135],[23,46371,572],{"className":46372,"style":963},[135,136],[23,46374,46376],{"className":46375},[998],[23,46377,46379],{"className":46378},[199],[23,46380,46382],{"className":46381},[204],[23,46383,46385],{"className":46384,"style":7047},[208],[23,46386,46387,46390],{"style":4989},[23,46388],{"className":46389,"style":1015},[216],[23,46391,46393],{"className":46392},[1019,1020,1021,1022],[23,46394,6883],{"className":46395},[612,1022],[11,46397,46398,46399,46401],{},"依概率成立。注意这里收敛的是",[18,46400,870],{},"，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。",[11,46403,46404,46432,46433,46485,46486,46547,46548,46576,46577,46605,46606,46670],{},[23,46405,46407,46420],{"className":46406,"translate":27},[31],[23,46408,46410],{"className":46409},[35],[37,46411,46412],{"xmlns":39},[42,46413,46414,46418],{},[45,46415,46416],{},[48,46417,4836],{},[114,46419,4836],{"encoding":116},[23,46421,46423],{"className":46422,"ariaHidden":122},[121],[23,46424,46426,46429],{"className":46425},[126],[23,46427],{"className":46428,"style":779},[130],[23,46430,4836],{"className":46431,"style":4851},[135,136]," 近邻是最经典的例子。Stone 定理说：若 ",[23,46434,46436,46455],{"className":46435,"translate":27},[31],[23,46437,46439],{"className":46438},[35],[37,46440,46441],{"xmlns":39},[42,46442,46443,46452],{},[45,46444,46445,46447,46449],{},[48,46446,4836],{},[51,46448,568],{},[48,46450,46451],{"mathvariant":111},"∞",[114,46453,46454],{"encoding":116},"k\\to\\infty",[23,46456,46458,46476],{"className":46457,"ariaHidden":122},[121],[23,46459,46461,46464,46467,46470,46473],{"className":46460},[126],[23,46462],{"className":46463,"style":779},[130],[23,46465,4836],{"className":46466,"style":4851},[135,136],[23,46468],{"className":46469,"style":149},[148],[23,46471,568],{"className":46472},[153],[23,46474],{"className":46475,"style":149},[148],[23,46477,46479,46482],{"className":46478},[126],[23,46480],{"className":46481,"style":703},[130],[23,46483,46451],{"className":46484},[135]," 且 ",[23,46487,46489,46511],{"className":46488,"translate":27},[31],[23,46490,46492],{"className":46491},[35],[37,46493,46494],{"xmlns":39},[42,46495,46496,46508],{},[45,46497,46498,46500,46502,46504,46506],{},[48,46499,4836],{},[48,46501,2875],{"mathvariant":111},[48,46503,4806],{},[51,46505,568],{},[2581,46507,1532],{},[114,46509,46510],{"encoding":116},"k\u002Fn\\to0",[23,46512,46514,46538],{"className":46513,"ariaHidden":122},[121],[23,46515,46517,46520,46523,46526,46529,46532,46535],{"className":46516},[126],[23,46518],{"className":46519,"style":131},[130],[23,46521,4836],{"className":46522,"style":4851},[135,136],[23,46524,2875],{"className":46525},[135],[23,46527,4806],{"className":46528},[135,136],[23,46530],{"className":46531,"style":149},[148],[23,46533,568],{"className":46534},[153],[23,46536],{"className":46537,"style":149},[148],[23,46539,46541,46544],{"className":46540},[126],[23,46542],{"className":46543,"style":2770},[130],[23,46545,1532],{"className":46546},[135],"，则 ",[23,46549,46551,46564],{"className":46550,"translate":27},[31],[23,46552,46554],{"className":46553},[35],[37,46555,46556],{"xmlns":39},[42,46557,46558,46562],{},[45,46559,46560],{},[48,46561,4836],{},[114,46563,4836],{"encoding":116},[23,46565,46567],{"className":46566,"ariaHidden":122},[121],[23,46568,46570,46573],{"className":46569},[126],[23,46571],{"className":46572,"style":779},[130],[23,46574,4836],{"className":46575,"style":4851},[135,136],"-NN 是普遍一致的。",[23,46578,46580,46593],{"className":46579,"translate":27},[31],[23,46581,46583],{"className":46582},[35],[37,46584,46585],{"xmlns":39},[42,46586,46587,46591],{},[45,46588,46589],{},[2581,46590,4435],{},[114,46592,4435],{"encoding":116},[23,46594,46596],{"className":46595,"ariaHidden":122},[121],[23,46597,46599,46602],{"className":46598},[126],[23,46600],{"className":46601,"style":2770},[130],[23,46603,4435],{"className":46604},[135],"-NN 更极端：它的渐近错误率不超过 ",[23,46607,46609,46629],{"className":46608,"translate":27},[31],[23,46610,46612],{"className":46611},[35],[37,46613,46614],{"xmlns":39},[42,46615,46616,46626],{},[45,46617,46618,46620],{},[2581,46619,9956],{},[4518,46621,46622,46624],{},[48,46623,572],{},[51,46625,6883],{},[114,46627,46628],{"encoding":116},"2R^*",[23,46630,46632],{"className":46631,"ariaHidden":122},[121],[23,46633,46635,46638,46641],{"className":46634},[126],[23,46636],{"className":46637,"style":6914},[130],[23,46639,9956],{"className":46640},[135],[23,46642,46644,46647],{"className":46643},[135],[23,46645,572],{"className":46646,"style":963},[135,136],[23,46648,46650],{"className":46649},[998],[23,46651,46653],{"className":46652},[199],[23,46654,46656],{"className":46655},[204],[23,46657,46659],{"className":46658,"style":6914},[208],[23,46660,46661,46664],{"style":4696},[23,46662],{"className":46663,"style":1015},[216],[23,46665,46667],{"className":46666},[1019,1020,1021,1022],[23,46668,6883],{"className":46669},[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。",[11,46672,46673,46674,46760],{},"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 ",[23,46675,46677,46701],{"className":46676,"translate":27},[31],[23,46678,46680],{"className":46679},[35],[37,46681,46682],{"xmlns":39},[42,46683,46684,46698],{},[45,46685,46686,46688,46690],{},[48,46687,4806],{},[51,46689,69],{},[4518,46691,46692,46695],{},[2581,46693,46694],{},"10",[2581,46696,46697],{},"6",[114,46699,46700],{"encoding":116},"n=10^6",[23,46702,46704,46722],{"className":46703,"ariaHidden":122},[121],[23,46705,46707,46710,46713,46716,46719],{"className":46706},[126],[23,46708],{"className":46709,"style":703},[130],[23,46711,4806],{"className":46712},[135,136],[23,46714],{"className":46715,"style":149},[148],[23,46717,69],{"className":46718},[153],[23,46720],{"className":46721,"style":149},[148],[23,46723,46725,46728,46731],{"className":46724},[126],[23,46726],{"className":46727,"style":4693},[130],[23,46729,4435],{"className":46730},[135],[23,46732,46734,46737],{"className":46733},[135],[23,46735,1532],{"className":46736},[135],[23,46738,46740],{"className":46739},[998],[23,46741,46743],{"className":46742},[199],[23,46744,46746],{"className":46745},[204],[23,46747,46749],{"className":46748,"style":4693},[208],[23,46750,46751,46754],{"style":4696},[23,46752],{"className":46753,"style":1015},[216],[23,46755,46757],{"className":46756},[1019,1020,1021,1022],[23,46758,46697],{"className":46759},[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。",[2501,46762,46764],{"id":46763},"_73-pac-bayes把信念放进泛化界","7.3 PAC-Bayes：把「信念」放进泛化界",[11,46766,46767,46768,46810],{},"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 ",[23,46769,46771,46789],{"className":46770,"translate":27},[31],[23,46772,46774],{"className":46773},[35],[37,46775,46776],{"xmlns":39},[42,46777,46778,46786],{},[45,46779,46780,46782,46784],{},[48,46781,920],{},[51,46783,747],{"separator":122},[48,46785,22055],{},[114,46787,46788],{"encoding":116},"P,Q",[23,46790,46792],{"className":46791,"ariaHidden":122},[121],[23,46793,46795,46798,46801,46804,46807],{"className":46794},[126],[23,46796],{"className":46797,"style":817},[130],[23,46799,920],{"className":46800,"style":1047},[135,136],[23,46802,747],{"className":46803},[824],[23,46805],{"className":46806,"style":280},[148],[23,46808,22055],{"className":46809},[135,136]," 是假设空间上的分布，不要与数据分布混淆。",[6857,46812,46813,47057,47646],{},[11,46814,46815,46818,46819,46847,46848,46900,46901,46975,46976,47027,47028,47056],{},[18,46816,46817],{},"定理 7.1（McAllester 界）"," 设先验 ",[23,46820,46822,46835],{"className":46821,"translate":27},[31],[23,46823,46825],{"className":46824},[35],[37,46826,46827],{"xmlns":39},[42,46828,46829,46833],{},[45,46830,46831],{},[48,46832,920],{},[114,46834,920],{"encoding":116},[23,46836,46838],{"className":46837,"ariaHidden":122},[121],[23,46839,46841,46844],{"className":46840},[126],[23,46842],{"className":46843,"style":470},[130],[23,46845,920],{"className":46846,"style":1047},[135,136]," 在看到数据前固定，损失取值于 ",[23,46849,46851,46873],{"className":46850,"translate":27},[31],[23,46852,46854],{"className":46853},[35],[37,46855,46856],{"xmlns":39},[42,46857,46858,46870],{},[45,46859,46860,46862,46864,46866,46868],{},[51,46861,924],{"stretchy":53},[2581,46863,1532],{},[51,46865,747],{"separator":122},[2581,46867,4435],{},[51,46869,945],{"stretchy":53},[114,46871,46872],{"encoding":116},"[0,1]",[23,46874,46876],{"className":46875,"ariaHidden":122},[121],[23,46877,46879,46882,46885,46888,46891,46894,46897],{"className":46878},[126],[23,46880],{"className":46881,"style":131},[130],[23,46883,924],{"className":46884},[140],[23,46886,1532],{"className":46887},[135],[23,46889,747],{"className":46890},[824],[23,46892],{"className":46893,"style":280},[148],[23,46895,4435],{"className":46896},[135],[23,46898,945],{"className":46899},[169],"。对任意 ",[23,46902,46904,46930],{"className":46903,"translate":27},[31],[23,46905,46907],{"className":46906},[35],[37,46908,46909],{"xmlns":39},[42,46910,46911,46927],{},[45,46912,46913,46915,46917,46919,46921,46923,46925],{},[48,46914,736],{},[51,46916,372],{},[51,46918,54],{"stretchy":53},[2581,46920,1532],{},[51,46922,747],{"separator":122},[2581,46924,4435],{},[51,46926,66],{"stretchy":53},[114,46928,46929],{"encoding":116},"\\delta\\in(0,1)",[23,46931,46933,46951],{"className":46932,"ariaHidden":122},[121],[23,46934,46936,46939,46942,46945,46948],{"className":46935},[126],[23,46937],{"className":46938,"style":4450},[130],[23,46940,736],{"className":46941,"style":783},[135,136],[23,46943],{"className":46944,"style":149},[148],[23,46946,372],{"className":46947},[153],[23,46949],{"className":46950,"style":149},[148],[23,46952,46954,46957,46960,46963,46966,46969,46972],{"className":46953},[126],[23,46955],{"className":46956,"style":131},[130],[23,46958,54],{"className":46959},[140],[23,46961,1532],{"className":46962},[135],[23,46964,747],{"className":46965},[824],[23,46967],{"className":46968,"style":280},[148],[23,46970,4435],{"className":46971},[135],[23,46973,66],{"className":46974},[169],"，以至少 ",[23,46977,46979,46997],{"className":46978,"translate":27},[31],[23,46980,46982],{"className":46981},[35],[37,46983,46984],{"xmlns":39},[42,46985,46986,46994],{},[45,46987,46988,46990,46992],{},[2581,46989,4435],{},[51,46991,4529],{},[48,46993,736],{},[114,46995,46996],{"encoding":116},"1-\\delta",[23,46998,47000,47018],{"className":46999,"ariaHidden":122},[121],[23,47001,47003,47006,47009,47012,47015],{"className":47002},[126],[23,47004],{"className":47005,"style":9404},[130],[23,47007,4435],{"className":47008},[135],[23,47010],{"className":47011,"style":392},[148],[23,47013,4529],{"className":47014},[612],[23,47016],{"className":47017,"style":392},[148],[23,47019,47021,47024],{"className":47020},[126],[23,47022],{"className":47023,"style":779},[130],[23,47025,736],{"className":47026,"style":783},[135,136]," 的概率，对所有后验分布 ",[23,47029,47031,47044],{"className":47030,"translate":27},[31],[23,47032,47034],{"className":47033},[35],[37,47035,47036],{"xmlns":39},[42,47037,47038,47042],{},[45,47039,47040],{},[48,47041,22055],{},[114,47043,22055],{"encoding":116},[23,47045,47047],{"className":47046,"ariaHidden":122},[121],[23,47048,47050,47053],{"className":47049},[126],[23,47051],{"className":47052,"style":817},[130],[23,47054,22055],{"className":47055},[135,136]," 同时成立：",[23,47058,47060],{"className":47059,"translate":27},[26],[23,47061,47063,47185],{"className":47062,"translate":27},[31],[23,47064,47066],{"className":47065},[35],[37,47067,47068],{"xmlns":39,"display":40},[42,47069,47070,47182],{},[45,47071,47072,47084,47086,47088,47090,47092,47094,47096,47098,47110,47112,47122,47124,47126,47128,47130,47132,47180],{},[898,47073,47074,47076],{},[48,47075,902],{"mathvariant":571},[45,47077,47078,47080,47082],{},[48,47079,18372],{},[51,47081,917],{},[48,47083,22055],{},[51,47085,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,47087,572],{},[51,47089,54],{"stretchy":53},[48,47091,18372],{},[51,47093,66],{"stretchy":53},[51,47095,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,47097,12272],{},[898,47099,47100,47102],{},[48,47101,902],{"mathvariant":571},[45,47103,47104,47106,47108],{},[48,47105,18372],{},[51,47107,917],{},[48,47109,22055],{},[51,47111,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[898,47113,47114,47120],{},[16811,47115,47116,47118],{"accent":122},[48,47117,572],{},[51,47119,16817],{},[48,47121,4806],{},[51,47123,54],{"stretchy":53},[48,47125,18372],{},[51,47127,66],{"stretchy":53},[51,47129,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,47131,5106],{},[25520,47133,47134],{},[71,47135,47136,47174],{},[45,47137,47138,47144,47146,47148,47150,47152,47154,47156,47158,47160,47162,47164,47168,47170,47172],{},[45,47139,47140,47142],{},[48,47141,26922],{"mathvariant":111},[48,47143,555],{"mathvariant":111},[51,47145,54],{"stretchy":53},[48,47147,22055],{},[48,47149,20684],{"mathvariant":111},[48,47151,920],{},[51,47153,66],{"stretchy":53},[51,47155,5106],{},[48,47157,22422],{},[51,47159,6961],{},[51,47161,54],{"stretchy":53},[2581,47163,9956],{},[25520,47165,47166],{},[48,47167,4806],{},[48,47169,2875],{"mathvariant":111},[48,47171,736],{},[51,47173,66],{"stretchy":53},[45,47175,47176,47178],{},[2581,47177,9956],{},[48,47179,4806],{},[51,47181,747],{"separator":122},[114,47183,47184],{"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}},",[23,47186,47188,47276,47432],{"className":47187,"ariaHidden":122},[121],[23,47189,47191,47194,47243,47249,47252,47255,47258,47261,47267,47270,47273],{"className":47190},[126],[23,47192],{"className":47193,"style":2213},[130],[23,47195,47197,47200],{"className":47196},[135],[23,47198,902],{"className":47199},[135,644],[23,47201,47203],{"className":47202},[998],[23,47204,47206,47235],{"className":47205},[199,200],[23,47207,47209,47232],{"className":47208},[204],[23,47210,47212],{"className":47211,"style":34196},[208],[23,47213,47214,47217],{"style":2029},[23,47215],{"className":47216,"style":1015},[216],[23,47218,47220],{"className":47219},[1019,1020,1021,1022],[23,47221,47223,47226,47229],{"className":47222},[135,1022],[23,47224,18372],{"className":47225},[135,136,1022],[23,47227,917],{"className":47228},[153,1022],[23,47230,22055],{"className":47231},[135,136,1022],[23,47233,297],{"className":47234},[296],[23,47236,47238],{"className":47237},[204],[23,47239,47241],{"className":47240,"style":14296},[208],[23,47242],{},[23,47244,47246],{"className":47245},[135],[23,47247,924],{"className":47248},[1066,1067],[23,47250,572],{"className":47251,"style":963},[135,136],[23,47253,54],{"className":47254},[140],[23,47256,18372],{"className":47257},[135,136],[23,47259,66],{"className":47260},[169],[23,47262,47264],{"className":47263},[135],[23,47265,945],{"className":47266},[1066,1067],[23,47268],{"className":47269,"style":149},[148],[23,47271,12272],{"className":47272},[153],[23,47274],{"className":47275,"style":149},[148],[23,47277,47279,47283,47332,47338,47408,47411,47414,47417,47423,47426,47429],{"className":47278},[126],[23,47280],{"className":47281,"style":47282},[130],"height:1.2968em;vertical-align:-0.35em;",[23,47284,47286,47289],{"className":47285},[135],[23,47287,902],{"className":47288},[135,644],[23,47290,47292],{"className":47291},[998],[23,47293,47295,47324],{"className":47294},[199,200],[23,47296,47298,47321],{"className":47297},[204],[23,47299,47301],{"className":47300,"style":34196},[208],[23,47302,47303,47306],{"style":2029},[23,47304],{"className":47305,"style":1015},[216],[23,47307,47309],{"className":47308},[1019,1020,1021,1022],[23,47310,47312,47315,47318],{"className":47311},[135,1022],[23,47313,18372],{"className":47314},[135,136,1022],[23,47316,917],{"className":47317},[153,1022],[23,47319,22055],{"className":47320},[135,136,1022],[23,47322,297],{"className":47323},[296],[23,47325,47327],{"className":47326},[204],[23,47328,47330],{"className":47329,"style":14296},[208],[23,47331],{},[23,47333,47335],{"className":47334},[135],[23,47336,924],{"className":47337},[1066,1067],[23,47339,47341,47374],{"className":47340},[135],[23,47342,47344],{"className":47343},[135,16834],[23,47345,47347],{"className":47346},[199],[23,47348,47350],{"className":47349},[204],[23,47351,47354,47362],{"className":47352,"style":47353},[208],"height:0.9468em;",[23,47355,47356,47359],{"style":7132},[23,47357],{"className":47358,"style":217},[216],[23,47360,572],{"className":47361,"style":963},[135,136],[23,47363,47365,47368],{"style":47364},"top:-3.2523em;",[23,47366],{"className":47367,"style":217},[216],[23,47369,47371],{"className":47370,"style":34869},[16862],[23,47372,16817],{"className":47373},[135],[23,47375,47377],{"className":47376},[998],[23,47378,47380,47400],{"className":47379},[199,200],[23,47381,47383,47397],{"className":47382},[204],[23,47384,47386],{"className":47385,"style":6499},[208],[23,47387,47388,47391],{"style":45106},[23,47389],{"className":47390,"style":1015},[216],[23,47392,47394],{"className":47393},[1019,1020,1021,1022],[23,47395,4806],{"className":47396},[135,136,1022],[23,47398,297],{"className":47399},[296],[23,47401,47403],{"className":47402},[204],[23,47404,47406],{"className":47405,"style":2048},[208],[23,47407],{},[23,47409,54],{"className":47410},[140],[23,47412,18372],{"className":47413},[135,136],[23,47415,66],{"className":47416},[169],[23,47418,47420],{"className":47419},[135],[23,47421,945],{"className":47422},[1066,1067],[23,47424],{"className":47425,"style":392},[148],[23,47427,5106],{"className":47428},[612],[23,47430],{"className":47431,"style":392},[148],[23,47433,47435,47439,47643],{"className":47434},[126],[23,47436],{"className":47437,"style":47438},[130],"height:2.44em;vertical-align:-0.7105em;",[23,47440,47442],{"className":47441},[135,25671],[23,47443,47445,47634],{"className":47444},[199,200],[23,47446,47448,47631],{"className":47447},[204],[23,47449,47452,47618],{"className":47450,"style":47451},[208],"height:1.7295em;",[23,47453,47455,47458],{"className":47454,"style":27790},[8676],[23,47456],{"className":47457,"style":27794},[216],[23,47459,47461],{"className":47460,"style":27798},[135],[23,47462,47464,47467,47615],{"className":47463},[135],[23,47465],{"className":47466},[140,192],[23,47468,47470],{"className":47469},[71],[23,47471,47473,47607],{"className":47472},[199,200],[23,47474,47476,47604],{"className":47475},[204],[23,47477,47480,47494,4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R_n",[23,47674,47676],{"className":47675,"ariaHidden":122},[121],[23,47677,47679,47683],{"className":47678},[126],[23,47680],{"className":47681,"style":47682},[130],"height:1.0968em;vertical-align:-0.15em;",[23,47684,47686,47717],{"className":47685},[135],[23,47687,47689],{"className":47688},[135,16834],[23,47690,47692],{"className":47691},[199],[23,47693,47695],{"className":47694},[204],[23,47696,47698,47706],{"className":47697,"style":47353},[208],[23,47699,47700,47703],{"style":7132},[23,47701],{"className":47702,"style":217},[216],[23,47704,572],{"className":47705,"style":963},[135,136],[23,47707,47708,47711],{"style":47364},[23,47709],{"className":47710,"style":217},[216],[23,47712,47714],{"className":47713,"style":34869},[16862],[23,47715,16817],{"className":47716},[135],[23,47718,47720],{"className":47719},[998],[23,47721,47723,47743],{"className":47722},[199,200],[23,47724,47726,47740],{"className":47725},[204],[23,47727,47729],{"className":47728,"style":6499},[208],[23,47730,47731,47734],{"style":45106},[23,47732],{"className":47733,"style":1015},[216],[23,47735,47737],{"className":47736},[1019,1020,1021,1022],[23,47738,4806],{"className":47739},[135,136,1022],[23,47741,297],{"className":47742},[296],[23,47744,47746],{"className":47745},[204],[23,47747,47749],{"className":47748,"style":2048},[208],[23,47750],{}," 是经验风险。",[11,47753,47754,47755,47783,47784,47812],{},"这个界的美妙之处在于：",[23,47756,47758,47771],{"className":47757,"translate":27},[31],[23,47759,47761],{"className":47760},[35],[37,47762,47763],{"xmlns":39},[42,47764,47765,47769],{},[45,47766,47767],{},[48,47768,920],{},[114,47770,920],{"encoding":116},[23,47772,47774],{"className":47773,"ariaHidden":122},[121],[23,47775,47777,47780],{"className":47776},[126],[23,47778],{"className":47779,"style":470},[130],[23,47781,920],{"className":47782,"style":1047},[135,136]," 必须在看到数据前固定，",[23,47785,47787,47800],{"className":47786,"translate":27},[31],[23,47788,47790],{"className":47789},[35],[37,47791,47792],{"xmlns":39},[42,47793,47794,47798],{},[45,47795,47796],{},[48,47797,22055],{},[114,47799,22055],{"encoding":116},[23,47801,47803],{"className":47802,"ariaHidden":122},[121],[23,47804,47806,47809],{"className":47805},[126],[23,47807],{"className":47808,"style":817},[130],[23,47810,22055],{"className":47811},[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。",[11,47814,47815,47816,47819,47820,47848,47849,47877],{},"这正是奥卡姆剃刀的概率版本，只不过它作用在",[18,47817,47818],{},"分布","而不是单个假设上。一个在大量假设上分散的 ",[23,47821,47823,47836],{"className":47822,"translate":27},[31],[23,47824,47826],{"className":47825},[35],[37,47827,47828],{"xmlns":39},[42,47829,47830,47834],{},[45,47831,47832],{},[48,47833,22055],{},[114,47835,22055],{"encoding":116},[23,47837,47839],{"className":47838,"ariaHidden":122},[121],[23,47840,47842,47845],{"className":47841},[126],[23,47843],{"className":47844,"style":817},[130],[23,47846,22055],{"className":47847},[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 ",[23,47850,47852,47865],{"className":47851,"translate":27},[31],[23,47853,47855],{"className":47854},[35],[37,47856,47857],{"xmlns":39},[42,47858,47859,47863],{},[45,47860,47861],{},[48,47862,22055],{},[114,47864,22055],{"encoding":116},[23,47866,47868],{"className":47867,"ariaHidden":122},[121],[23,47869,47871,47874],{"className":47870},[126],[23,47872],{"className":47873,"style":817},[130],[23,47875,22055],{"className":47876},[135,136]," 则可以用较小的 KL 换取较好的界。",[11,47879,47880],{},"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。",[2501,47882,47884],{"id":47883},"_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",[11,47886,47887,47888,47931,47932,47961],{},"精确后验通常不可行，因为归一化常数 ",[23,47889,47891,47910],{"className":47890,"translate":27},[31],[23,47892,47894],{"className":47893},[35],[37,47895,47896],{"xmlns":39},[42,47897,47898,47908],{},[45,47899,47900,47902,47904,47906],{},[48,47901,11],{},[51,47903,54],{"stretchy":53},[48,47905,63],{},[51,47907,66],{"stretchy":53},[114,47909,3945],{"encoding":116},[23,47911,47913],{"className":47912,"ariaHidden":122},[121],[23,47914,47916,47919,47922,47925,47928],{"className":47915},[126],[23,47917],{"className":47918,"style":131},[130],[23,47920,11],{"className":47921},[135,136],[23,47923,54],{"className":47924},[140],[23,47926,63],{"className":47927,"style":144},[135,136],[23,47929,66],{"className":47930},[169]," 是一个高维积分。变分推断用一个受限制的分布族 ",[23,47933,47935,47949],{"className":47934,"translate":27},[31],[23,47936,47938],{"className":47937},[35],[37,47939,47940],{"xmlns":39},[42,47941,47942,47946],{},[45,47943,47944],{},[48,47945,22055],{"mathvariant":375},[114,47947,47948],{"encoding":116},"\\mathcal Q",[23,47950,47952],{"className":47951,"ariaHidden":122},[121],[23,47953,47955,47958],{"className":47954},[126],[23,47956],{"className":47957,"style":408},[130],[23,47959,22055],{"className":47960},[135,412]," 去逼近后验，目标是",[23,47963,47965],{"className":47964,"translate":27},[26],[23,47966,47968,48048],{"className":47967,"translate":27},[31],[23,47969,47971],{"className":47970},[35],[37,47972,47973],{"xmlns":39,"display":40},[42,47974,47975,48045],{},[45,47976,47977,47983,47985,47987,47989,48005,48007,48013,48015,48017,48019,48021,48023,48025,48027,48029,48031,48033,48035,48037,48039,48041,48043],{},[4518,47978,47979,47981],{},[48,47980,33017],{},[51,47982,6883],{},[51,47984,69],{},[48,47986,6958],{},[51,47988,6961],{},[6963,47990,47991,47997],{},[45,47992,47993,47995],{},[48,47994,6969],{},[51,47996,6961],{},[45,47998,47999,48001,48003],{},[48,48000,33017],{},[51,48002,372],{},[48,48004,22055],{"mathvariant":375},[88,48006,5360],{},[45,48008,48009,48011],{},[48,48010,26922],{"mathvariant":111},[48,48012,555],{"mathvariant":111},[51,48014,54],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,48016,33017],{},[51,48018,54],{"stretchy":53},[48,48020,57],{},[51,48022,66],{"stretchy":53},[88,48024,5360],{},[48,48026,20684],{"mathvariant":111},[88,48028,5360],{},[48,48030,11],{},[51,48032,54],{"stretchy":53},[48,48034,57],{},[51,48036,60],{},[48,48038,63],{},[51,48040,66],{"stretchy":53},[51,48042,66],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,48044,112],{"mathvariant":111},[114,48046,48047],{"encoding":116},"q^*=\\arg\\min_{q\\in\\mathcal Q}\\ \\mathrm{KL}\\big(q(\\theta)\\ \\|\\ p(\\theta\\mid D)\\big).",[23,48049,48051,48096,48221],{"className":48050,"ariaHidden":122},[121],[23,48052,48054,48058,48087,48090,48093],{"className":48053},[126],[23,48055],{"className":48056,"style":48057},[130],"height:0.9331em;vertical-align:-0.1944em;",[23,48059,48061,48064],{"className":48060},[135],[23,48062,33017],{"className":48063,"style":676},[135,136],[23,48065,48067],{"className":48066},[998],[23,48068,48070],{"className":48069},[199],[23,48071,48073],{"className":48072},[204],[23,48074,48076],{"className":48075,"style":7047},[208],[23,48077,48078,48081],{"style":4989},[23,48079],{"className":48080,"style":1015},[216],[23,48082,48084],{"className":48083},[1019,1020,1021,1022],[23,48085,6883],{"className":48086},[612,1022],[23,48088],{"className":48089,"style":149},[148],[23,48091,69],{"className":48092},[153],[23,48094],{"className":48095,"style":149},[148],[23,48097,48099,48103,48108,48111,48164,48167,48170,48176,48182,48185,48188,48191,48194,48197,48200,48203,48206,48209,48212,48215,48218],{"className":48098},[126],[23,48100],{"className":48101,"style":48102},[130],"height:1.7304em;vertical-align:-0.8804em;",[23,48104,7087,48106],{"className":48105},[1786],[23,48107,7091],{"style":7090},[23,48109],{"className":48110,"style":280},[148],[23,48112,48114],{"className":48113},[1786,7098],[23,48115,48117,48156],{"className":48116},[199,200],[23,48118,48120,48153],{"className":48119},[204],[23,48121,48123,48143],{"className":48122,"style":7108},[208],[23,48124,48125,48128],{"style":7111},[23,48126],{"className":48127,"style":217},[216],[23,48129,48131],{"className":48130},[1019,1020,1021,1022],[23,48132,48134,48137,48140],{"className":48133},[135,1022],[23,48135,33017],{"className":48136,"style":676},[135,136,1022],[23,48138,372],{"className":48139},[153,1022],[23,48141,22055],{"className":48142},[135,412,1022],[23,48144,48145,48148],{"style":7132},[23,48146],{"className":48147,"style":217},[216],[23,48149,48150],{},[23,48151,6969],{"className":48152},[1786],[23,48154,297],{"className":48155},[296],[23,48157,48159],{"className":48158},[204],[23,48160,48162],{"className":48161,"style":5683},[208],[23,48163],{},[23,48165,5360],{"className":48166},[148],[23,48168],{"className":48169,"style":280},[148],[23,48171,48173],{"className":48172},[135],[23,48174,27191],{"className":48175},[135,5391],[23,48177,48179],{"className":48178},[135],[23,48180,54],{"className":48181},[1066,1067],[23,48183,33017],{"className":48184,"style":676},[135,136],[23,48186,54],{"className":48187},[140],[23,48189,57],{"className":48190,"style":144},[135,136],[23,48192,66],{"className":48193},[169],[23,48195,5360],{"className":48196},[148],[23,48198,20684],{"className":48199},[135],[23,48201,5360],{"className":48202},[148],[23,48204,11],{"className":48205},[135,136],[23,48207,54],{"className":48208},[140],[23,48210,57],{"className":48211,"style":144},[135,136],[23,48213],{"className":48214,"style":149},[148],[23,48216,60],{"className":48217},[153],[23,48219],{"className":48220,"style":149},[148],[23,48222,48224,48227,48230,48233,48239],{"className":48223},[126],[23,48225],{"className":48226,"style":2213},[130],[23,48228,63],{"className":48229,"style":144},[135,136],[23,48231,66],{"className":48232},[169],[23,48234,48236],{"className":48235},[135],[23,48237,66],{"className":48238},[1066,1067],[23,48240,112],{"className":48241},[135],[11,48243,48244,48245,48288],{},"直接最小化这个 KL 仍然需要 ",[23,48246,48248,48267],{"className":48247,"translate":27},[31],[23,48249,48251],{"className":48250},[35],[37,48252,48253],{"xmlns":39},[42,48254,48255,48265],{},[45,48256,48257,48259,48261,48263],{},[48,48258,11],{},[51,48260,54],{"stretchy":53},[48,48262,63],{},[51,48264,66],{"stretchy":53},[114,48266,3945],{"encoding":116},[23,48268,48270],{"className":48269,"ariaHidden":122},[121],[23,48271,48273,48276,48279,48282,48285],{"className":48272},[126],[23,48274],{"className":48275,"style":131},[130],[23,48277,11],{"className":48278},[135,136],[23,48280,54],{"className":48281},[140],[23,48283,63],{"className":48284,"style":144},[135,136],[23,48286,66],{"className":48287},[169],"，于是利用恒等式",[23,48290,48292],{"className":48291,"translate":27},[26],[23,48293,48295,48429],{"className":48294,"translate":27},[31],[23,48296,48298],{"className":48297},[35],[37,48299,48300],{"xmlns":39,"display":40},[42,48301,48302,48426],{},[45,48303,48304,48306,48308,48310,48312,48314,48316,48318,48386,48388,48394,48396,48398,48400,48402,48404,48406,48408,48410,48412,48414,48416,48418,48420,48422,48424],{},[48,48305,22422],{},[51,48307,6961],{},[48,48309,11],{},[51,48311,54],{"stretchy":53},[48,48313,63],{},[51,48315,66],{"stretchy":53},[51,48317,69],{},[6963,48319,48320,48368],{},[6963,48321,48322,48366],{},[45,48323,48324,48330],{},[898,48325,48326,48328],{},[48,48327,902],{"mathvariant":571},[48,48329,33017],{},[45,48331,48332,48334,48336,48338,48364],{},[51,48333,924],{"fence":122},[48,48335,22422],{},[51,48337,6961],{},[71,48339,48340,48354],{},[45,48341,48342,48344,48346,48348,48350,48352],{},[48,48343,11],{},[51,48345,54],{"stretchy":53},[48,48347,63],{},[51,48349,747],{"separator":122},[48,48351,57],{},[51,48353,66],{"stretchy":53},[45,48355,48356,48358,48360,48362],{},[48,48357,33017],{},[51,48359,54],{"stretchy":53},[48,48361,57],{},[51,48363,66],{"stretchy":53},[51,48365,945],{"fence":122},[51,48367,8481],{"stretchy":122},[45,48369,48370,48380,48382,48384],{},[45,48371,48372,48374,48376,48378],{},[48,48373,902],{"mathvariant":111},[48,48375,555],{"mathvariant":111},[48,48377,2533],{"mathvariant":111},[48,48379,42344],{"mathvariant":111},[51,48381,54],{"stretchy":53},[48,48383,33017],{},[51,48385,66],{"stretchy":53},[51,48387,5106],{},[45,48389,48390,48392],{},[48,48391,26922],{"mathvariant":111},[48,48393,555],{"mathvariant":111},[51,48395,54],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,48397,33017],{},[51,48399,54],{"stretchy":53},[48,48401,57],{},[51,48403,66],{"stretchy":53},[88,48405,5360],{},[48,48407,20684],{"mathvariant":111},[88,48409,5360],{},[48,48411,11],{},[51,48413,54],{"stretchy":53},[48,48415,57],{},[51,48417,60],{},[48,48419,63],{},[51,48421,66],{"stretchy":53},[51,48423,66],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,48425,747],{"separator":122},[114,48427,48428],{"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),",[23,48430,48432,48467,48763,48820],{"className":48431,"ariaHidden":122},[121],[23,48433,48435,48438,48443,48446,48449,48452,48455,48458,48461,48464],{"className":48434},[126],[23,48436],{"className":48437,"style":131},[130],[23,48439,22579,48441],{"className":48440},[1786],[23,48442,7091],{"style":7090},[23,48444],{"className":48445,"style":280},[148],[23,48447,11],{"className":48448},[135,136],[23,48450,54],{"className":48451},[140],[23,48453,63],{"className":48454,"style":144},[135,136],[23,48456,66],{"className":48457},[169],[23,48459],{"className":48460,"style":149},[148],[23,48462,69],{"className":48463},[153],[23,48465],{"className":48466,"style":149},[148],[23,48468,48470,48474,48754,48757,48760],{"className":48469},[126],[23,48471],{"className":48472,"style":48473},[130],"height:3.948em;vertical-align:-2.498em;",[23,48475,48477],{"className":48476},[1839,6963],[23,48478,48480,48745],{"className":48479},[199,200],[23,48481,48483,48742],{"className":48482},[204],[23,48484,48487,48515],{"className":48485,"style":48486},[208],"height:1.45em;",[23,48488,48490,48493],{"style":48489},"top:-1.127em;",[23,48491],{"className":48492,"style":1713},[216],[23,48494,48496],{"className":48495},[1019,1020,1021,1022],[23,48497,48499,48506,48509,48512],{"className":48498},[135,1022],[23,48500,48502],{"className":48501},[135,1022],[23,48503,48505],{"className":48504},[135,5391,1022],"ELBO",[23,48507,54],{"className":48508},[140,1022],[23,48510,33017],{"className":48511,"style":676},[135,136,1022],[23,48513,66],{"className":48514},[169,1022],[23,48516,48518,48521],{"style":48517},"top:-3.45em;",[23,48519],{"className":48520,"style":1713},[216],[23,48522,48524],{"className":48523},[1839,6963],[23,48525,48527,48733],{"className":48526},[199,200],[23,48528,48530,48730],{"className":48529},[204],[23,48531,48533,48564],{"className":48532,"style":48486},[208],[23,48534,48537,48540],{"className":48535,"style":48536},[8676],"top:-1.852em;",[23,48538],{"className":48539,"style":1713},[216],[23,48541,48543,48550,48557],{"className":48542,"style":8685},[8684],[23,48544,48546],{"className":48545,"style":8690},[8689],[8692,48547,48548],{"xmlns":8694,"width":8695,"height":8696,"viewBox":8697,"preserveAspectRatio":8698},[8700,48549],{"d":8702},[23,48551,48553],{"className":48552,"style":8690},[8706],[8692,48554,48555],{"xmlns":8694,"width":8695,"height":8696,"viewBox":8697,"preserveAspectRatio":8709},[8700,48556],{"d":8712},[23,48558,48560],{"className":48559,"style":8690},[8716],[8692,48561,48562],{"xmlns":8694,"width":8695,"height":8696,"viewBox":8697,"preserveAspectRatio":8719},[8700,48563],{"d":8722},[23,48565,48566,48569],{"style":48517},[23,48567],{"className":48568,"style":1713},[216],[23,48570,48572,48615,48618],{"className":48571},[135],[23,48573,48575,48578],{"className":48574},[135],[23,48576,902],{"className":48577},[135,644],[23,48579,48581],{"className":48580},[998],[23,48582,48584,48607],{"className":48583},[199,200],[23,48585,48587,48604],{"className":48586},[204],[23,48588,48590],{"className":48589,"style":6499},[208],[23,48591,48592,48595],{"style":2029},[23,48593],{"className":48594,"style":1015},[216],[23,48596,48598],{"className":48597},[1019,1020,1021,1022],[23,48599,48601],{"className":48600},[135,1022],[23,48602,33017],{"className":48603,"style":676},[135,136,1022],[23,48605,297],{"className":48606},[296],[23,48608,48610],{"className":48609},[204],[23,48611,48613],{"className":48612,"style":14296},[208],[23,48614],{},[23,48616],{"className":48617,"style":280},[148],[23,48619,48621,48627,48632,48635,48724],{"className":48620},[1839],[23,48622,48624],{"className":48623,"style":1844},[140,1843],[23,48625,924],{"className":48626},[1066,1021],[23,48628,22579,48630],{"className":48629},[1786],[23,48631,7091],{"style":7090},[23,48633],{"className":48634,"style":280},[148],[23,48636,48638,48641,48721],{"className":48637},[135],[23,48639],{"className":48640},[140,192],[23,48642,48644],{"className":48643},[71],[23,48645,48647,48713],{"className":48646},[199,200],[23,48648,48650,48710],{"className":48649},[204],[23,48651,48653,48673,48681],{"className":48652,"style":209},[208],[23,48654,48655,48658],{"style":212},[23,48656],{"className":48657,"style":217},[216],[23,48659,48661,48664,48667,48670],{"className":48660},[135],[23,48662,33017],{"className":48663,"style":676},[135,136],[23,48665,54],{"className":48666},[140],[23,48668,57],{"className":48669,"style":144},[135,136],[23,48671,66],{"className":48672},[169],[23,48674,48675,48678],{"style":235},[23,48676],{"className":48677,"style":217},[216],[23,48679],{"className":48680,"style":243},[242],[23,48682,48683,48686],{"style":246},[23,48684],{"className":48685,"style":217},[216],[23,48687,48689,48692,48695,48698,48701,48704,48707],{"className":48688},[135],[23,48690,11],{"className":48691},[135,136],[23,48693,54],{"className":48694},[140],[23,48696,63],{"className":48697,"style":144},[135,136],[23,48699,747],{"className":48700},[824],[23,48702],{"className":48703,"style":280},[148],[23,48705,57],{"className":48706,"style":144},[135,136],[23,48708,66],{"className":48709},[169],[23,48711,297],{"className":48712},[296],[23,48714,48716],{"className":48715},[204],[23,48717,48719],{"className":48718,"style":304},[208],[23,48720],{},[23,48722],{"className":48723},[169,192],[23,48725,48727],{"className":48726,"style":1844},[169,1843],[23,48728,945],{"className":48729},[1066,1021],[23,48731,297],{"className":48732},[296],[23,48734,48736],{"className":48735},[204],[23,48737,48740],{"className":48738,"style":48739},[208],"height:1.598em;",[23,48741],{},[23,48743,297],{"className":48744},[296],[23,48746,48748],{"className":48747},[204],[23,48749,48752],{"className":48750,"style":48751},[208],"height:2.498em;",[23,48753],{},[23,48755],{"className":48756,"style":392},[148],[23,48758,5106],{"className":48759},[612],[23,48761],{"className":48762,"style":392},[148],[23,48764,48766,48769,48775,48781,48784,48787,48790,48793,48796,48799,48802,48805,48808,48811,48814,48817],{"className":48765},[126],[23,48767],{"className":48768,"style":2213},[130],[23,48770,48772],{"className":48771},[135],[23,48773,27191],{"className":48774},[135,5391],[23,48776,48778],{"className":48777},[135],[23,48779,54],{"className":48780},[1066,1067],[23,48782,33017],{"className":48783,"style":676},[135,136],[23,48785,54],{"className":48786},[140],[23,48788,57],{"className":48789,"style":144},[135,136],[23,48791,66],{"className":48792},[169],[23,48794,5360],{"className":48795},[148],[23,48797,20684],{"className":48798},[135],[23,48800,5360],{"className":48801},[148],[23,48803,11],{"className":48804},[135,136],[23,48806,54],{"className":48807},[140],[23,48809,57],{"className":48810,"style":144},[135,136],[23,48812],{"className":48813,"style":149},[148],[23,48815,60],{"className":48816},[153],[23,48818],{"className":48819,"style":149},[148],[23,48821,48823,48826,48829,48832,48838],{"className":48822},[126],[23,48824],{"className":48825,"style":2213},[130],[23,48827,63],{"className":48828,"style":144},[135,136],[23,48830,66],{"className":48831},[169],[23,48833,48835],{"className":48834},[135],[23,48836,66],{"className":48837},[1066,1067],[23,48839,747],{"className":48840},[824],[11,48842,48843,48844,48847],{},"把问题转化为最大化",[18,48845,48846],{},"证据下界","（ELBO）：",[23,48849,48851],{"className":48850,"translate":27},[26],[23,48852,48854,48946],{"className":48853,"translate":27},[31],[23,48855,48857],{"className":48856},[35],[37,48858,48859],{"xmlns":39,"display":40},[42,48860,48861,48943],{},[45,48862,48863,48873,48875,48877,48879,48881,48887,48889,48891,48893,48895,48897,48899,48901,48903,48905,48907,48909,48915,48917,48919,48921,48923,48925,48927,48929,48931,48933,48935,48937,48939,48941],{},[45,48864,48865,48867,48869,48871],{},[48,48866,902],{"mathvariant":111},[48,48868,555],{"mathvariant":111},[48,48870,2533],{"mathvariant":111},[48,48872,42344],{"mathvariant":111},[51,48874,54],{"stretchy":53},[48,48876,33017],{},[51,48878,66],{"stretchy":53},[51,48880,69],{},[898,48882,48883,48885],{},[48,48884,902],{"mathvariant":571},[48,48886,33017],{},[51,48888,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,48890,22422],{},[51,48892,6961],{},[48,48894,11],{},[51,48896,54],{"stretchy":53},[48,48898,63],{},[51,48900,60],{},[48,48902,57],{},[51,48904,66],{"stretchy":53},[51,48906,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,48908,4529],{},[45,48910,48911,48913],{},[48,48912,26922],{"mathvariant":111},[48,48914,555],{"mathvariant":111},[51,48916,54],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,48918,33017],{},[51,48920,54],{"stretchy":53},[48,48922,57],{},[51,48924,66],{"stretchy":53},[88,48926,5360],{},[48,48928,20684],{"mathvariant":111},[88,48930,5360],{},[48,48932,11],{},[51,48934,54],{"stretchy":53},[48,48936,57],{},[51,48938,66],{"stretchy":53},[51,48940,66],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,48942,112],{"mathvariant":111},[114,48944,48945],{"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).",[23,48947,48949,48979,49060,49087],{"className":48948,"ariaHidden":122},[121],[23,48950,48952,48955,48961,48964,48967,48970,48973,48976],{"className":48951},[126],[23,48953],{"className":48954,"style":131},[130],[23,48956,48958],{"className":48957},[135],[23,48959,48505],{"className":48960},[135,5391],[23,48962,54],{"className":48963},[140],[23,48965,33017],{"className":48966,"style":676},[135,136],[23,48968,66],{"className":48969},[169],[23,48971],{"className":48972,"style":149},[148],[23,48974,69],{"className":48975},[153],[23,48977],{"className":48978,"style":149},[148],[23,48980,48982,48985,49025,49031,49034,49039,49042,49045,49048,49051,49054,49057],{"className":48981},[126],[23,48983],{"className":48984,"style":2213},[130],[23,48986,48988,48991],{"className":48987},[135],[23,48989,902],{"className":48990},[135,644],[23,48992,48994],{"className":48993},[998],[23,48995,48997,49017],{"className":48996},[199,200],[23,48998,49000,49014],{"className":48999},[204],[23,49001,49003],{"className":49002,"style":6499},[208],[23,49004,49005,49008],{"style":2029},[23,49006],{"className":49007,"style":1015},[216],[23,49009,49011],{"className":49010},[1019,1020,1021,1022],[23,49012,33017],{"className":49013,"style":676},[135,136,1022],[23,49015,297],{"className":49016},[296],[23,49018,49020],{"className":49019},[204],[23,49021,49023],{"className":49022,"style":14296},[208],[23,49024],{},[23,49026,49028],{"className":49027},[135],[23,49029,924],{"className":49030},[1066,1067],[23,49032],{"className":49033,"style":280},[148],[23,49035,22579,49037],{"className":49036},[1786],[23,49038,7091],{"style":7090},[23,49040],{"className":49041,"style":280},[148],[23,49043,11],{"className":49044},[135,136],[23,49046,54],{"className":49047},[140],[23,49049,63],{"className":49050,"style":144},[135,136],[23,49052],{"className":49053,"style":149},[148],[23,49055,60],{"className":49056},[153],[23,49058],{"className":49059,"style":149},[148],[23,49061,49063,49066,49069,49072,49078,49081,49084],{"className":49062},[126],[23,49064],{"className":49065,"style":2213},[130],[23,49067,57],{"className":49068,"style":144},[135,136],[23,49070,66],{"className":49071},[169],[23,49073,49075],{"className":49074},[135],[23,49076,945],{"className":49077},[1066,1067],[23,49079],{"className":49080,"style":392},[148],[23,49082,4529],{"className":49083},[612],[23,49085],{"className":49086,"style":392},[148],[23,49088,49090,49093,49099,49105,49108,49111,49114,49117,49120,49123,49126,49129,49132,49135,49138,49144],{"className":49089},[126],[23,49091],{"className":49092,"style":2213},[130],[23,49094,49096],{"className":49095},[135],[23,49097,27191],{"className":49098},[135,5391],[23,49100,49102],{"className":49101},[135],[23,49103,54],{"className":49104},[1066,1067],[23,49106,33017],{"className":49107,"style":676},[135,136],[23,49109,54],{"className":49110},[140],[23,49112,57],{"className":49113,"style":144},[135,136],[23,49115,66],{"className":49116},[169],[23,49118,5360],{"className":49119},[148],[23,49121,20684],{"className":49122},[135],[23,49124,5360],{"className":49125},[148],[23,49127,11],{"className":49128},[135,136],[23,49130,54],{"className":49131},[140],[23,49133,57],{"className":49134,"style":144},[135,136],[23,49136,66],{"className":49137},[169],[23,49139,49141],{"className":49140},[135],[23,49142,66],{"className":49143},[1066,1067],[23,49145,112],{"className":49146},[135],[11,49148,49149,49150,49178,49179,49207,49208,49303,49304,4136,49360,49388],{},"第一项鼓励 ",[23,49151,49153,49166],{"className":49152,"translate":27},[31],[23,49154,49156],{"className":49155},[35],[37,49157,49158],{"xmlns":39},[42,49159,49160,49164],{},[45,49161,49162],{},[48,49163,33017],{},[114,49165,33017],{"encoding":116},[23,49167,49169],{"className":49168,"ariaHidden":122},[121],[23,49170,49172,49175],{"className":49171},[126],[23,49173],{"className":49174,"style":672},[130],[23,49176,33017],{"className":49177,"style":676},[135,136]," 解释数据，第二项惩罚 ",[23,49180,49182,49195],{"className":49181,"translate":27},[31],[23,49183,49185],{"className":49184},[35],[37,49186,49187],{"xmlns":39},[42,49188,49189,49193],{},[45,49190,49191],{},[48,49192,33017],{},[114,49194,33017],{"encoding":116},[23,49196,49198],{"className":49197,"ariaHidden":122},[121],[23,49199,49201,49204],{"className":49200},[126],[23,49202],{"className":49203,"style":672},[130],[23,49205,33017],{"className":49206,"style":676},[135,136]," 偏离先验。最大化 ELBO 等价于最小化 ",[23,49209,49211,49249],{"className":49210,"translate":27},[31],[23,49212,49214],{"className":49213},[35],[37,49215,49216],{"xmlns":39},[42,49217,49218,49246],{},[45,49219,49220,49226,49228,49230,49232,49234,49236,49238,49240,49242,49244],{},[45,49221,49222,49224],{},[48,49223,26922],{"mathvariant":111},[48,49225,555],{"mathvariant":111},[51,49227,54],{"stretchy":53},[48,49229,33017],{},[48,49231,20684],{"mathvariant":111},[48,49233,11],{},[51,49235,54],{"stretchy":53},[48,49237,57],{},[51,49239,60],{},[48,49241,63],{},[51,49243,66],{"stretchy":53},[51,49245,66],{"stretchy":53},[114,49247,49248],{"encoding":116},"\\mathrm{KL}(q\\|p(\\theta\\mid D))",[23,49250,49252,49291],{"className":49251,"ariaHidden":122},[121],[23,49253,49255,49258,49264,49267,49270,49273,49276,49279,49282,49285,49288],{"className":49254},[126],[23,49256],{"className":49257,"style":131},[130],[23,49259,49261],{"className":49260},[135],[23,49262,27191],{"className":49263},[135,5391],[23,49265,54],{"className":49266},[140],[23,49268,33017],{"className":49269,"style":676},[135,136],[23,49271,20684],{"className":49272},[135],[23,49274,11],{"className":49275},[135,136],[23,49277,54],{"className":49278},[140],[23,49280,57],{"className":49281,"style":144},[135,136],[23,49283],{"className":49284,"style":149},[148],[23,49286,60],{"className":49287},[153],[23,49289],{"className":49290,"style":149},[148],[23,49292,49294,49297,49300],{"className":49293},[126],[23,49295],{"className":49296,"style":131},[130],[23,49298,63],{"className":49299,"style":144},[135,136],[23,49301,1095],{"className":49302},[169],"，因为 ",[23,49305,49307,49331],{"className":49306,"translate":27},[31],[23,49308,49310],{"className":49309},[35],[37,49311,49312],{"xmlns":39},[42,49313,49314,49328],{},[45,49315,49316,49318,49320,49322,49324,49326],{},[48,49317,22422],{},[51,49319,6961],{},[48,49321,11],{},[51,49323,54],{"stretchy":53},[48,49325,63],{},[51,49327,66],{"stretchy":53},[114,49329,49330],{"encoding":116},"\\log p(D)",[23,49332,49334],{"className":49333,"ariaHidden":122},[121],[23,49335,49337,49340,49345,49348,49351,49354,49357],{"className":49336},[126],[23,49338],{"className":49339,"style":131},[130],[23,49341,22579,49343],{"className":49342},[1786],[23,49344,7091],{"style":7090},[23,49346],{"className":49347,"style":280},[148],[23,49349,11],{"className":49350},[135,136],[23,49352,54],{"className":49353},[140],[23,49355,63],{"className":49356,"style":144},[135,136],[23,49358,66],{"className":49359},[169],[23,49361,49363,49376],{"className":49362,"translate":27},[31],[23,49364,49366],{"className":49365},[35],[37,49367,49368],{"xmlns":39},[42,49369,49370,49374],{},[45,49371,49372],{},[48,49373,33017],{},[114,49375,33017],{"encoding":116},[23,49377,49379],{"className":49378,"ariaHidden":122},[121],[23,49380,49382,49385],{"className":49381},[126],[23,49383],{"className":49384,"style":672},[130],[23,49386,33017],{"className":49387,"style":676},[135,136]," 无关。",[11,49390,49391,49392,49453,49454,49457,49458,49486,49487,49453,49548,49551,49552,49617],{},"KL 的方向很重要。",[23,49393,49395,49423],{"className":49394,"translate":27},[31],[23,49396,49398],{"className":49397},[35],[37,49399,49400],{"xmlns":39},[42,49401,49402,49420],{},[45,49403,49404,49410,49412,49414,49416,49418],{},[45,49405,49406,49408],{},[48,49407,26922],{"mathvariant":111},[48,49409,555],{"mathvariant":111},[51,49411,54],{"stretchy":53},[48,49413,33017],{},[48,49415,20684],{"mathvariant":111},[48,49417,11],{},[51,49419,66],{"stretchy":53},[114,49421,49422],{"encoding":116},"\\mathrm{KL}(q\\|p)",[23,49424,49426],{"className":49425,"ariaHidden":122},[121],[23,49427,49429,49432,49438,49441,49444,49447,49450],{"className":49428},[126],[23,49430],{"className":49431,"style":131},[130],[23,49433,49435],{"className":49434},[135],[23,49436,27191],{"className":49437},[135,5391],[23,49439,54],{"className":49440},[140],[23,49442,33017],{"className":49443,"style":676},[135,136],[23,49445,20684],{"className":49446},[135],[23,49448,11],{"className":49449},[135,136],[23,49451,66],{"className":49452},[169]," 倾向",[18,49455,49456],{},"寻模","（mode-seeking）：只要 ",[23,49459,49461,49474],{"className":49460,"translate":27},[31],[23,49462,49464],{"className":49463},[35],[37,49465,49466],{"xmlns":39},[42,49467,49468,49472],{},[45,49469,49470],{},[48,49471,33017],{},[114,49473,33017],{"encoding":116},[23,49475,49477],{"className":49476,"ariaHidden":122},[121],[23,49478,49480,49483],{"className":49479},[126],[23,49481],{"className":49482,"style":672},[130],[23,49484,33017],{"className":49485,"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 ",[23,49488,49490,49518],{"className":49489,"translate":27},[31],[23,49491,49493],{"className":49492},[35],[37,49494,49495],{"xmlns":39},[42,49496,49497,49515],{},[45,49498,49499,49505,49507,49509,49511,49513],{},[45,49500,49501,49503],{},[48,49502,26922],{"mathvariant":111},[48,49504,555],{"mathvariant":111},[51,49506,54],{"stretchy":53},[48,49508,11],{},[48,49510,20684],{"mathvariant":111},[48,49512,33017],{},[51,49514,66],{"stretchy":53},[114,49516,49517],{"encoding":116},"\\mathrm{KL}(p\\|q)",[23,49519,49521],{"className":49520,"ariaHidden":122},[121],[23,49522,49524,49527,49533,49536,49539,49542,49545],{"className":49523},[126],[23,49525],{"className":49526,"style":131},[130],[23,49528,49530],{"className":49529},[135],[23,49531,27191],{"className":49532},[135,5391],[23,49534,54],{"className":49535},[140],[23,49537,11],{"className":49538},[135,136],[23,49540,20684],{"className":49541},[135],[23,49543,33017],{"className":49544,"style":676},[135,136],[23,49546,66],{"className":49547},[169],[18,49549,49550],{},"覆盖质量","（mass-covering），但它的计算需要 ",[23,49553,49555,49578],{"className":49554,"translate":27},[31],[23,49556,49558],{"className":49557},[35],[37,49559,49560],{"xmlns":39},[42,49561,49562,49576],{},[45,49563,49564,49566,49568,49570,49572,49574],{},[48,49565,11],{},[51,49567,54],{"stretchy":53},[48,49569,57],{},[51,49571,60],{},[48,49573,63],{},[51,49575,66],{"stretchy":53},[114,49577,4032],{"encoding":116},[23,49579,49581,49605],{"className":49580,"ariaHidden":122},[121],[23,49582,49584,49587,49590,49593,49596,49599,49602],{"className":49583},[126],[23,49585],{"className":49586,"style":131},[130],[23,49588,11],{"className":49589},[135,136],[23,49591,54],{"className":49592},[140],[23,49594,57],{"className":49595,"style":144},[135,136],[23,49597],{"className":49598,"style":149},[148],[23,49600,60],{"className":49601},[153],[23,49603],{"className":49604,"style":149},[148],[23,49606,49608,49611,49614],{"className":49607},[126],[23,49609],{"className":49610,"style":131},[130],[23,49612,63],{"className":49613,"style":144},[135,136],[23,49615,66],{"className":49616},[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。",[11,49619,49620,49621,49650,49651,1471],{},"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 ",[23,49622,49624,49638],{"className":49623,"translate":27},[31],[23,49625,49627],{"className":49626},[35],[37,49628,49629],{"xmlns":39},[42,49630,49631,49635],{},[45,49632,49633],{},[48,49634,500],{"mathvariant":375},[114,49636,49637],{"encoding":116},"\\mathcal A",[23,49639,49641],{"className":49640,"ariaHidden":122},[121],[23,49642,49644,49647],{"className":49643},[126],[23,49645],{"className":49646,"style":470},[130],[23,49648,500],{"className":49649},[135,412]," 变成了 ",[23,49652,49654,49667],{"className":49653,"translate":27},[31],[23,49655,49657],{"className":49656},[35],[37,49658,49659],{"xmlns":39},[42,49660,49661,49665],{},[45,49662,49663],{},[48,49664,22055],{"mathvariant":375},[114,49666,47948],{"encoding":116},[23,49668,49670],{"className":49669,"ariaHidden":122},[121],[23,49671,49673,49676],{"className":49672},[126],[23,49674],{"className":49675,"style":408},[130],[23,49677,22055],{"className":49678},[135,412],[2501,49680,49682],{"id":49681},"_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",[11,49684,49685,49686,49716,49717,49783],{},"把参数换成网络权重 ",[23,49687,49689,49703],{"className":49688,"translate":27},[31],[23,49690,49692],{"className":49691},[35],[37,49693,49694],{"xmlns":39},[42,49695,49696,49701],{},[45,49697,49698],{},[48,49699,49700],{},"w",[114,49702,49700],{"encoding":116},[23,49704,49706],{"className":49705,"ariaHidden":122},[121],[23,49707,49709,49712],{"className":49708},[126],[23,49710],{"className":49711,"style":703},[130],[23,49713,49700],{"className":49714,"style":49715},[135,136],"margin-right:0.0269em;","，后验 ",[23,49718,49720,49744],{"className":49719,"translate":27},[31],[23,49721,49723],{"className":49722},[35],[37,49724,49725],{"xmlns":39},[42,49726,49727,49741],{},[45,49728,49729,49731,49733,49735,49737,49739],{},[48,49730,11],{},[51,49732,54],{"stretchy":53},[48,49734,49700],{},[51,49736,60],{},[48,49738,63],{},[51,49740,66],{"stretchy":53},[114,49742,49743],{"encoding":116},"p(w\\mid D)",[23,49745,49747,49771],{"className":49746,"ariaHidden":122},[121],[23,49748,49750,49753,49756,49759,49762,49765,49768],{"className":49749},[126],[23,49751],{"className":49752,"style":131},[130],[23,49754,11],{"className":49755},[135,136],[23,49757,54],{"className":49758},[140],[23,49760,49700],{"className":49761,"style":49715},[135,136],[23,49763],{"className":49764,"style":149},[148],[23,49766,60],{"className":49767},[153],[23,49769],{"className":49770,"style":149},[148],[23,49772,49774,49777,49780],{"className":49773},[126],[23,49775],{"className":49776,"style":131},[130],[23,49778,63],{"className":49779,"style":144},[135,136],[23,49781,66],{"className":49782},[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。",[11,49785,49786],{},"对分类问题，预测熵可以分解为",[23,49788,49790],{"className":49789,"translate":27},[26],[23,49791,49793,49879],{"className":49792,"translate":27},[31],[23,49794,49796],{"className":49795},[35],[37,49797,49798],{"xmlns":39,"display":40},[42,49799,49800,49876],{},[45,49801,49802,49804,49806,49808,49810,49812,49814,49816,49818,49820,49832,49834,49836,49838,49840,49842,49844,49846,49848,49850,49852,49854,49856,49858,49860,49862,49864,49866,49868,49870,49872,49874],{},[48,49803,26253],{"mathvariant":571},[51,49805,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,49807,369],{},[51,49809,60],{},[48,49811,753],{},[51,49813,747],{"separator":122},[48,49815,63],{},[51,49817,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,49819,69],{},[898,49821,49822,49824],{},[48,49823,902],{"mathvariant":571},[45,49825,49826,49828,49830],{},[48,49827,49700],{},[51,49829,60],{},[48,49831,63],{},[51,49833,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,49835,26253],{"mathvariant":571},[51,49837,924],{"stretchy":53},[48,49839,369],{},[51,49841,60],{},[48,49843,753],{},[51,49845,747],{"separator":122},[48,49847,49700],{},[51,49849,945],{"stretchy":53},[51,49851,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,49853,5106],{},[48,49855,34357],{"mathvariant":571},[51,49857,54],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,49859,369],{},[51,49861,39748],{"separator":122},[48,49863,49700],{},[51,49865,60],{},[48,49867,753],{},[51,49869,747],{"separator":122},[48,49871,63],{},[51,49873,66],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,49875,112],{"mathvariant":111},[114,49877,49878],{"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).",[23,49880,49882,49909,49942,50021,50057,50093],{"className":49881,"ariaHidden":122},[121],[23,49883,49885,49888,49891,49897,49900,49903,49906],{"className":49884},[126],[23,49886],{"className":49887,"style":2213},[130],[23,49889,26253],{"className":49890},[135,644],[23,49892,49894],{"className":49893},[135],[23,49895,924],{"className":49896},[1066,1067],[23,49898,369],{"className":49899,"style":392},[135,136],[23,49901],{"className":49902,"style":149},[148],[23,49904,60],{"className":49905},[153],[23,49907],{"className":49908,"style":149},[148],[23,49910,49912,49915,49918,49921,49924,49927,49933,49936,49939],{"className":49911},[126],[23,49913],{"className":49914,"style":2213},[130],[23,49916,753],{"className":49917},[135,136],[23,49919,747],{"className":49920},[824],[23,49922],{"className":49923,"style":280},[148],[23,49925,63],{"className":49926,"style":144},[135,136],[23,49928,49930],{"className":49929},[135],[23,49931,945],{"className":49932},[1066,1067],[23,49934],{"className":49935,"style":149},[148],[23,49937,69],{"className":49938},[153],[23,49940],{"className":49941,"style":149},[148],[23,49943,49945,49948,49997,50003,50006,50009,50012,50015,50018],{"className":49944},[126],[23,49946],{"className":49947,"style":988},[130],[23,49949,49951,49954],{"className":49950},[135],[23,49952,902],{"className":49953},[135,644],[23,49955,49957],{"className":49956},[998],[23,49958,49960,49989],{"className":49959},[199,200],[23,49961,49963,49986],{"className":49962},[204],[23,49964,49966],{"className":49965,"style":1008},[208],[23,49967,49968,49971],{"style":1011},[23,49969],{"className":49970,"style":1015},[216],[23,49972,49974],{"className":49973},[1019,1020,1021,1022],[23,49975,49977,49980,49983],{"className":49976},[135,1022],[23,49978,49700],{"className":49979,"style":49715},[135,136,1022],[23,49981,60],{"className":49982},[153,1022],[23,49984,63],{"className":49985,"style":144},[135,136,1022],[23,49987,297],{"className":49988},[296],[23,49990,49992],{"className":49991},[204],[23,49993,49995],{"className":49994,"style":1057},[208],[23,49996],{},[23,49998,50000],{"className":49999},[135],[23,50001,924],{"className":50002},[1066,1067],[23,50004,26253],{"className":50005},[135,644],[23,50007,924],{"className":50008},[140],[23,50010,369],{"className":50011,"style":392},[135,136],[23,50013],{"className":50014,"style":149},[148],[23,50016,60],{"className":50017},[153],[23,50019],{"className":50020,"style":149},[148],[23,50022,50024,50027,50030,50033,50036,50039,50042,50048,50051,50054],{"className":50023},[126],[23,50025],{"className":50026,"style":2213},[130],[23,50028,753],{"className":50029},[135,136],[23,50031,747],{"className":50032},[824],[23,50034],{"className":50035,"style":280},[148],[23,50037,49700],{"className":50038,"style":49715},[135,136],[23,50040,945],{"className":50041},[169],[23,50043,50045],{"className":50044},[135],[23,50046,945],{"className":50047},[1066,1067],[23,50049],{"className":50050,"style":392},[148],[23,50052,5106],{"className":50053},[612],[23,50055],{"className":50056,"style":392},[148],[23,50058,50060,50063,50066,50072,50075,50078,50081,50084,50087,50090],{"className":50059},[126],[23,50061],{"className":50062,"style":2213},[130],[23,50064,34357],{"className":50065},[135,644],[23,50067,50069],{"className":50068},[135],[23,50070,54],{"className":50071},[1066,1067],[23,50073,369],{"className":50074,"style":392},[135,136],[23,50076,39748],{"className":50077},[824],[23,50079],{"className":50080,"style":280},[148],[23,50082,49700],{"className":50083,"style":49715},[135,136],[23,50085],{"className":50086,"style":149},[148],[23,50088,60],{"className":50089},[153],[23,50091],{"className":50092,"style":149},[148],[23,50094,50096,50099,50102,50105,50108,50111,50117],{"className":50095},[126],[23,50097],{"className":50098,"style":2213},[130],[23,50100,753],{"className":50101},[135,136],[23,50103,747],{"className":50104},[824],[23,50106],{"className":50107,"style":280},[148],[23,50109,63],{"className":50110,"style":144},[135,136],[23,50112,50114],{"className":50113},[135],[23,50115,66],{"className":50116},[1066,1067],[23,50118,112],{"className":50119},[135],[11,50121,50122],{},"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。",[11,50124,50125],{},"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。",[2501,50127,50129],{"id":50128},"_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",[11,50131,50132,50133,50136,50137,50302],{},"在序贯决策里，后验不只是用来「估计」，还用来「探索」。",[18,50134,50135],{},"Thompson 采样","的规则很简单：每一步从后验中采样一个环境参数 ",[23,50138,50140,50177],{"className":50139,"translate":27},[31],[23,50141,50143],{"className":50142},[35],[37,50144,50145],{"xmlns":39},[42,50146,50147,50174],{},[45,50148,50149,50156,50158,50160,50162,50164,50166,50172],{},[16811,50150,50151,50153],{"accent":122},[48,50152,57],{},[51,50154,50155],{},"~",[51,50157,917],{},[48,50159,11],{},[51,50161,54],{"stretchy":53},[48,50163,57],{},[51,50165,60],{},[898,50167,50168,50170],{},[48,50169,63],{},[48,50171,5345],{},[51,50173,66],{"stretchy":53},[114,50175,50176],{"encoding":116},"\\tilde\\theta\\sim p(\\theta\\mid D_t)",[23,50178,50180,50228,50252],{"className":50179,"ariaHidden":122},[121],[23,50181,50183,50187,50219,50222,50225],{"className":50182},[126],[23,50184],{"className":50185,"style":50186},[130],"height:0.9313em;",[23,50188,50190],{"className":50189},[135,16834],[23,50191,50193],{"className":50192},[199],[23,50194,50196],{"className":50195},[204],[23,50197,50199,50207],{"className":50198,"style":50186},[208],[23,50200,50201,50204],{"style":7132},[23,50202],{"className":50203,"style":217},[216],[23,50205,57],{"className":50206,"style":144},[135,136],[23,50208,50210,50213],{"style":50209},"top:-3.6134em;",[23,50211],{"className":50212,"style":217},[216],[23,50214,50216],{"className":50215,"style":34869},[16862],[23,50217,50155],{"className":50218},[135],[23,50220],{"className":50221,"style":149},[148],[23,50223,917],{"className":50224},[153],[23,50226],{"className":50227,"style":149},[148],[23,50229,50231,50234,50237,50240,50243,50246,50249],{"className":50230},[126],[23,50232],{"className":50233,"style":131},[130],[23,50235,11],{"className":50236},[135,136],[23,50238,54],{"className":50239},[140],[23,50241,57],{"className":50242,"style":144},[135,136],[23,50244],{"className":50245,"style":149},[148],[23,50247,60],{"className":50248},[153],[23,50250],{"className":50251,"style":149},[148],[23,50253,50255,50258,50299],{"className":50254},[126],[23,50256],{"className":50257,"style":131},[130],[23,50259,50261,50264],{"className":50260},[135],[23,50262,63],{"className":50263,"style":144},[135,136],[23,50265,50267],{"className":50266},[998],[23,50268,50270,50291],{"className":50269},[199,200],[23,50271,50273,50288],{"className":50272},[204],[23,50274,50277],{"className":50275,"style":50276},[208],"height:0.2806em;",[23,50278,50279,50282],{"style":27175},[23,50280],{"className":50281,"style":1015},[216],[23,50283,50285],{"className":50284},[1019,1020,1021,1022],[23,50286,5345],{"className":50287},[135,136,1022],[23,50289,297],{"className":50290},[296],[23,50292,50294],{"className":50293},[204],[23,50295,50297],{"className":50296,"style":2048},[208],[23,50298],{},[23,50300,66],{"className":50301},[169],"，然后按这个采样出的环境做最优动作。",[11,50304,50305],{},"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。",[11,50307,50308],{},"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。",[2501,50310,50312],{"id":50311},"_77-主动学习与贝叶斯优化","7.7 主动学习与贝叶斯优化",[11,50314,50315],{},"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」：",[23,50317,50319],{"className":50318,"translate":27},[26],[23,50320,50322,50422],{"className":50321,"translate":27},[31],[23,50323,50325],{"className":50324},[35],[37,50326,50327],{"xmlns":39,"display":40},[42,50328,50329,50419],{},[45,50330,50331,50338,50340,50342,50344,50354,50356,50382,50417],{},[898,50332,50333,50335],{},[48,50334,753],{},[88,50336,50337],{},"next",[51,50339,69],{},[48,50341,6958],{},[51,50343,6961],{},[6963,50345,50346,50352],{},[45,50347,50348,50350],{},[48,50349,6969],{},[51,50351,6961],{},[48,50353,753],{},[88,50355,5360],{},[898,50357,50358,50360],{},[48,50359,902],{"mathvariant":571},[45,50361,50362,50364,50366,50368,50370,50372,50374,50376,50378,50380],{},[48,50363,660],{},[51,50365,917],{},[48,50367,11],{},[51,50369,54],{"stretchy":53},[48,50371,660],{},[51,50373,60],{},[48,50375,753],{},[51,50377,747],{"separator":122},[48,50379,63],{},[51,50381,66],{"stretchy":53},[45,50383,50384,50386,50392,50394,50396,50399,50401,50403,50405,50407,50409,50411,50413,50415],{},[51,50385,924],{"fence":122},[4518,50387,50388,50390],{},[48,50389,572],{},[51,50391,6883],{},[51,50393,54],{"stretchy":53},[48,50395,63],{},[51,50397,50398],{},"∪",[51,50400,9014],{"stretchy":53},[51,50402,54],{"stretchy":53},[48,50404,753],{},[51,50406,747],{"separator":122},[48,50408,660],{},[51,50410,66],{"stretchy":53},[51,50412,9024],{"stretchy":53},[51,50414,66],{"stretchy":53},[51,50416,945],{"fence":122},[51,50418,747],{"separator":122},[114,50420,50421],{"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],",[23,50423,50425,50486],{"className":50424,"ariaHidden":122},[121],[23,50426,50428,50431,50477,50480,50483],{"className":50427},[126],[23,50429],{"className":50430,"style":12734},[130],[23,50432,50434,50437],{"className":50433},[135],[23,50435,753],{"className":50436},[135,136],[23,50438,50440],{"className":50439},[998],[23,50441,50443,50469],{"className":50442},[199,200],[23,50444,50446,50466],{"className":50445},[204],[23,50447,50449],{"className":50448,"style":50276},[208],[23,50450,50451,50454],{"style":2029},[23,50452],{"className":50453,"style":1015},[216],[23,50455,50457],{"className":50456},[1019,1020,1021,1022],[23,50458,50460],{"className":50459},[135,1022],[23,50461,50463],{"className":50462},[135,6514,1022],[23,50464,50337],{"className":50465},[135,1022],[23,50467,297],{"className":50468},[296],[23,50470,50472],{"className":50471},[204],[23,50473,50475],{"className":50474,"style":2048},[208],[23,50476],{},[23,50478],{"className":50479,"style":149},[148],[23,50481,69],{"className":50482},[153],[23,50484],{"className":50485,"style":149},[148],[23,50487,50489,50493,50498,50501,50548,50551,50554,50624,50627,50700,50703],{"className":50488},[126],[23,50490],{"className":50491,"style":50492},[130],"height:1.45em;vertical-align:-0.7em;",[23,50494,7087,50496],{"className":50495},[1786],[23,50497,7091],{"style":7090},[23,50499],{"className":50500,"style":280},[148],[23,50502,50504],{"className":50503},[1786,7098],[23,50505,50507,50540],{"className":50506},[199,200],[23,50508,50510,50537],{"className":50509},[204],[23,50511,50513,50527],{"className":50512,"style":7108},[208],[23,50514,50515,50518],{"style":8784},[23,50516],{"className":50517,"style":217},[216],[23,50519,50521],{"className":50520},[1019,1020,1021,1022],[23,50522,50524],{"className":50523},[135,1022],[23,50525,753],{"className":50526},[135,136,1022],[23,50528,50529,50532],{"style":7132},[23,50530],{"className":50531,"style":217},[216],[23,50533,50534],{},[23,50535,6969],{"className":50536},[1786],[23,50538,297],{"className":50539},[296],[23,50541,50543],{"className":50542},[204],[23,50544,50546],{"className":50545,"style":8816},[208],[23,50547],{},[23,50549,5360],{"className":50550},[148],[23,50552],{"className":50553,"style":280},[148],[23,50555,50557,50560],{"className":50556},[135],[23,50558,902],{"className":50559},[135,644],[23,50561,50563],{"className":50562},[998],[23,50564,50566,50616],{"className":50565},[199,200],[23,50567,50569,50613],{"className":50568},[204],[23,50570,50572],{"className":50571,"style":1008},[208],[23,50573,50574,50577],{"style":1011},[23,50575],{"className":50576,"style":1015},[216],[23,50578,50580],{"className":50579},[1019,1020,1021,1022],[23,50581,50583,50586,50589,50592,50595,50598,50601,50604,50607,50610],{"className":50582},[135,1022],[23,50584,660],{"className":50585,"style":676},[135,136,1022],[23,50587,917],{"className":50588},[153,1022],[23,50590,11],{"className":50591},[135,136,1022],[23,50593,54],{"className":50594},[140,1022],[23,50596,660],{"className":50597,"style":676},[135,136,1022],[23,50599,60],{"className":50600},[153,1022],[23,50602,753],{"className":50603},[135,136,1022],[23,50605,747],{"className":50606},[824,1022],[23,50608,63],{"className":50609,"style":144},[135,136,1022],[23,50611,66],{"className":50612},[169,1022],[23,50614,297],{"className":50615},[296],[23,50617,50619],{"className":50618},[204],[23,50620,50622],{"className":50621,"style":1057},[208],[23,50623],{},[23,50625],{"className":50626,"style":280},[148],[23,50628,50630,50633,50662,50665,50668,50671,50674,50677,50681,50684,50687,50690,50693,50697],{"className":50629},[1839],[23,50631,924],{"className":50632,"style":1844},[140,1843],[23,50634,50636,50639],{"className":50635},[135],[23,50637,572],{"className":50638,"style":963},[135,136],[23,50640,50642],{"className":50641},[998],[23,50643,50645],{"className":50644},[199],[23,50646,50648],{"className":50647},[204],[23,50649,50651],{"className":50650,"style":7047},[208],[23,50652,50653,50656],{"style":4989},[23,50654],{"className":50655,"style":1015},[216],[23,50657,50659],{"className":50658},[1019,1020,1021,1022],[23,50660,6883],{"className":50661},[612,1022],[23,50663,54],{"className":50664},[140],[23,50666,63],{"className":50667,"style":144},[135,136],[23,50669],{"className":50670,"style":392},[148],[23,50672,50398],{"className":50673},[612],[23,50675],{"className":50676,"style":392},[148],[23,50678,50680],{"className":50679},[140],"{(",[23,50682,753],{"className":50683},[135,136],[23,50685,747],{"className":50686},[824],[23,50688],{"className":50689,"style":280},[148],[23,50691,660],{"className":50692,"style":676},[135,136],[23,50694,50696],{"className":50695},[169],")})",[23,50698,945],{"className":50699,"style":1844},[169,1843],[23,50701],{"className":50702,"style":280},[148],[23,50704,747],{"className":50705},[824],[11,50707,24433,50708,50782],{},[23,50709,50711,50735],{"className":50710,"translate":27},[31],[23,50712,50714],{"className":50713},[35],[37,50715,50716],{"xmlns":39},[42,50717,50718,50732],{},[45,50719,50720,50726,50728,50730],{},[4518,50721,50722,50724],{},[48,50723,572],{},[51,50725,6883],{},[51,50727,54],{"stretchy":53},[48,50729,63],{},[51,50731,66],{"stretchy":53},[114,50733,50734],{"encoding":116},"R^*(D)",[23,50736,50738],{"className":50737,"ariaHidden":122},[121],[23,50739,50741,50744,50773,50776,50779],{"className":50740},[126],[23,50742],{"className":50743,"style":131},[130],[23,50745,50747,50750],{"className":50746},[135],[23,50748,572],{"className":50749,"style":963},[135,136],[23,50751,50753],{"className":50752},[998],[23,50754,50756],{"className":50755},[199],[23,50757,50759],{"className":50758},[204],[23,50760,50762],{"className":50761,"style":6914},[208],[23,50763,50764,50767],{"style":4696},[23,50765],{"className":50766,"style":1015},[216],[23,50768,50770],{"className":50769},[1019,1020,1021,1022],[23,50771,6883],{"className":50772},[612,1022],[23,50774,54],{"className":50775},[140],[23,50777,63],{"className":50778,"style":144},[135,136],[23,50780,66],{"className":50781},[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。",[11,50784,50785],{},"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",[23,50787,50789],{"className":50788,"translate":27},[26],[23,50790,50792,50862],{"className":50791,"translate":27},[31],[23,50793,50795],{"className":50794},[35],[37,50796,50797],{"xmlns":39,"display":40},[42,50798,50799,50859],{},[45,50800,50801,50807,50809,50811,50813,50815,50817,50819,50821,50823,50825,50827,50829,50831,50839,50841,50843,50845,50847,50849,50851,50853,50855,50857],{},[45,50802,50803,50805],{},[48,50804,902],{"mathvariant":111},[48,50806,34357],{"mathvariant":111},[51,50808,54],{"stretchy":53},[48,50810,753],{},[51,50812,66],{"stretchy":53},[51,50814,69],{},[48,50816,902],{"mathvariant":571},[51,50818,924],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[48,50820,9555],{},[51,50822,6961],{},[51,50824,9014],{"stretchy":53},[2581,50826,1532],{},[51,50828,747],{"separator":122},[88,50830,5360],{},[898,50832,50833,50835,50837],{},[48,50834,8340],{},[48,50836,6969],{},[51,50838,6961],{},[51,50840,4529],{},[48,50842,8340],{},[51,50844,54],{"stretchy":53},[48,50846,753],{},[51,50848,66],{"stretchy":53},[51,50850,9024],{"stretchy":53},[51,50852,60],{},[48,50854,63],{},[51,50856,945],{"fence":53,"stretchy":122,"minsize":923,"maxsize":923},[51,50858,747],{"separator":122},[114,50860,50861],{"encoding":116},"\\mathrm{EI}(x)=\\mathbb E\\big[\\max\\{0,\\ f_{\\min}-f(x)\\}\\mid D\\big],",[23,50863,50865,50896,50994,51021],{"className":50864,"ariaHidden":122},[121],[23,50866,50868,50871,50878,50881,50884,50887,50890,50893],{"className":50867},[126],[23,50869],{"className":50870,"style":131},[130],[23,50872,50874],{"className":50873},[135],[23,50875,50877],{"className":50876},[135,5391],"EI",[23,50879,54],{"className":50880},[140],[23,50882,753],{"className":50883},[135,136],[23,50885,66],{"className":50886},[169],[23,50888],{"className":50889,"style":149},[148],[23,50891,69],{"className":50892},[153],[23,50894],{"className":50895,"style":149},[148],[23,50897,50899,50902,50905,50911,50914,50917,50920,50923,50926,50929,50932,50985,50988,50991],{"className":50898},[126],[23,50900],{"className":50901,"style":2213},[130],[23,50903,902],{"className":50904},[135,644],[23,50906,50908],{"className":50907},[135],[23,50909,924],{"className":50910},[1066,1067],[23,50912],{"className":50913,"style":280},[148],[23,50915,9555],{"className":50916},[1786],[23,50918,9014],{"className":50919},[140],[23,50921,1532],{"className":50922},[135],[23,50924,747],{"className":50925},[824],[23,50927,5360],{"className":50928},[148],[23,50930],{"className":50931,"style":280},[148],[23,50933,50935,50938],{"className":50934},[135],[23,50936,8340],{"className":50937,"style":11402},[135,136],[23,50939,50941],{"className":50940},[998],[23,50942,50944,50977],{"className":50943},[199,200],[23,50945,50947,50974],{"className":50946},[204],[23,50948,50950],{"className":50949,"style":38783},[208],[23,50951,50952,50955],{"style":29550},[23,50953],{"className":50954,"style":1015},[216],[23,50956,50958],{"className":50957},[1019,1020,1021,1022],[23,50959,50961],{"className":50960},[135,1022],[23,50962,50964,50968,50971],{"className":50963},[1786,1022],[23,50965,50967],{"className":50966},[1022],"m",[23,50969,33497],{"className":50970},[1022],[23,50972,4806],{"className":50973},[1022],[23,50975,297],{"className":50976},[296],[23,50978,50980],{"className":50979},[204],[23,50981,50983],{"className":50982,"style":2048},[208],[23,50984],{},[23,50986],{"className":50987,"style":392},[148],[23,50989,4529],{"className":50990},[612],[23,50992],{"className":50993,"style":392},[148],[23,50995,50997,51000,51003,51006,51009,51012,51015,51018],{"className":50996},[126],[23,50998],{"className":50999,"style":131},[130],[23,51001,8340],{"className":51002,"style":11402},[135,136],[23,51004,54],{"className":51005},[140],[23,51007,753],{"className":51008},[135,136],[23,51010,15224],{"className":51011},[169],[23,51013],{"className":51014,"style":149},[148],[23,51016,60],{"className":51017},[153],[23,51019],{"className":51020,"style":149},[148],[23,51022,51024,51027,51030,51036],{"className":51023},[126],[23,51025],{"className":51026,"style":2213},[130],[23,51028,63],{"className":51029,"style":144},[135,136],[23,51031,51033],{"className":51032},[135],[23,51034,945],{"className":51035},[1066,1067],[23,51037,747],{"className":51038},[824],[11,51040,51041],{},"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。",[2501,51043,51045],{"id":51044},"_78-可容许性与完整类定理","7.8 可容许性与完整类定理",[11,51047,51048,51049,15005,51077,51080,51081,51143],{},"最后回到 2.4 节留下的那个说法。一个决策规则 ",[23,51050,51052,51065],{"className":51051,"translate":27},[31],[23,51053,51055],{"className":51054},[35],[37,51056,51057],{"xmlns":39},[42,51058,51059,51063],{},[45,51060,51061],{},[48,51062,736],{},[114,51064,7341],{"encoding":116},[23,51066,51068],{"className":51067,"ariaHidden":122},[121],[23,51069,51071,51074],{"className":51070},[126],[23,51072],{"className":51073,"style":779},[130],[23,51075,736],{"className":51076,"style":783},[135,136],[18,51078,51079],{},"可容许的","（admissible），若不存在另一个规则 ",[23,51082,51084,51102],{"className":51083,"translate":27},[31],[23,51085,51087],{"className":51086},[35],[37,51088,51089],{"xmlns":39},[42,51090,51091,51099],{},[45,51092,51093],{},[4518,51094,51095,51097],{},[48,51096,736],{},[51,51098,38295],{"mathvariant":111,"lspace":1522,"rspace":1522},[114,51100,51101],{"encoding":116},"\\delta'",[23,51103,51105],{"className":51104,"ariaHidden":122},[121],[23,51106,51108,51111],{"className":51107},[126],[23,51109],{"className":51110,"style":38337},[130],[23,51112,51114,51117],{"className":51113},[135],[23,51115,736],{"className":51116,"style":783},[135,136],[23,51118,51120],{"className":51119},[998],[23,51121,51123],{"className":51122},[199],[23,51124,51126],{"className":51125},[204],[23,51127,51129],{"className":51128,"style":38337},[208],[23,51130,51131,51134],{"style":4696},[23,51132],{"className":51133,"style":1015},[216],[23,51135,51137],{"className":51136},[1019,1020,1021,1022],[23,51138,51140],{"className":51139},[135,1022],[23,51141,38295],{"className":51142},[135,1022],"，使得",[23,51145,51147],{"className":51146,"translate":27},[26],[23,51148,51150,51193],{"className":51149,"translate":27},[31],[23,51151,51153],{"className":51152},[35],[37,51154,51155],{"xmlns":39,"display":40},[42,51156,51157,51190],{},[45,51158,51159,51161,51163,51169,51171,51173,51175,51177,51179,51181,51183,51186,51188],{},[48,51160,572],{},[51,51162,54],{"stretchy":53},[4518,51164,51165,51167],{},[48,51166,736],{},[51,51168,38295],{"mathvariant":111,"lspace":1522,"rspace":1522},[51,51170,66],{"stretchy":53},[51,51172,12272],{},[48,51174,572],{},[51,51176,54],{"stretchy":53},[48,51178,736],{},[51,51180,66],{"stretchy":53},[148,51182],{"width":9919},[88,51184,51185],{},"对所有分布 ",[48,51187,920],{},[51,51189,747],{"separator":122},[114,51191,51192],{"encoding":116},"R(\\delta')\\le R(\\delta)\\quad\\text{对所有分布 }P,",[23,51194,51196,51254],{"className":51195,"ariaHidden":122},[121],[23,51197,51199,51203,51206,51209,51242,51245,51248,51251],{"className":51198},[126],[23,51200],{"className":51201,"style":51202},[130],"height:1.0519em;vertical-align:-0.25em;",[23,51204,572],{"className":51205,"style":963},[135,136],[23,51207,54],{"className":51208},[140],[23,51210,51212,51215],{"className":51211},[135],[23,51213,736],{"className":51214,"style":783},[135,136],[23,51216,51218],{"className":51217},[998],[23,51219,51221],{"className":51220},[199],[23,51222,51224],{"className":51223},[204],[23,51225,51228],{"className":51226,"style":51227},[208],"height:0.8019em;",[23,51229,51230,51233],{"style":4989},[23,51231],{"className":51232,"style":1015},[216],[23,51234,51236],{"className":51235},[1019,1020,1021,1022],[23,51237,51239],{"className":51238},[135,1022],[23,51240,38295],{"className":51241},[135,1022],[23,51243,66],{"className":51244},[169],[23,51246],{"className":51247,"style":149},[148],[23,51249,12272],{"className":51250},[153],[23,51252],{"className":51253,"style":149},[148],[23,51255,51257,51260,51263,51266,51269,51272,51275,51285,51288],{"className":51256},[126],[23,51258],{"className":51259,"style":131},[130],[23,51261,572],{"className":51262,"style":963},[135,136],[23,51264,54],{"className":51265},[140],[23,51267,736],{"className":51268,"style":783},[135,136],[23,51270,66],{"className":51271},[169],[23,51273],{"className":51274,"style":14441},[148],[23,51276,51278,51282],{"className":51277},[135,6514],[23,51279,51281],{"className":51280},[135,7240],"对所有分布",[23,51283,5360],{"className":51284},[135],[23,51286,920],{"className":51287,"style":1047},[135,136],[23,51289,747],{"className":51290},[824],[11,51292,51293,51294,51322],{},"且至少对一个 ",[23,51295,51297,51310],{"className":51296,"translate":27},[31],[23,51298,51300],{"className":51299},[35],[37,51301,51302],{"xmlns":39},[42,51303,51304,51308],{},[45,51305,51306],{},[48,51307,920],{},[114,51309,920],{"encoding":116},[23,51311,51313],{"className":51312,"ariaHidden":122},[121],[23,51314,51316,51319],{"className":51315},[126],[23,51317],{"className":51318,"style":470},[130],[23,51320,920],{"className":51321,"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。",[6857,51324,51325],{},[11,51326,51327,51330,51331,51360],{},[18,51328,51329],{},"定理 7.2（Wald 完整类定理）"," 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 ",[23,51332,51334,51348],{"className":51333,"translate":27},[31],[23,51335,51337],{"className":51336},[35],[37,51338,51339],{"xmlns":39},[42,51340,51341,51345],{},[45,51342,51343],{},[48,51344,29638],{},[114,51346,51347],{"encoding":116},"\\sigma",[23,51349,51351],{"className":51350,"ariaHidden":122},[121],[23,51352,51354,51357],{"className":51353},[126],[23,51355],{"className":51356,"style":703},[130],[23,51358,29638],{"className":51359,"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。",[11,51362,51363],{},"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。",[11,51365,51366,51367,1471],{},"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：",[18,51368,51369],{},"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",[11,51371,51372],{},"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。",[338,51374,51376],{"id":51375},"_8-几个可以手算的例子","8. 几个可以手算的例子",[2501,51378,51380],{"id":51379},"_81-betabernoulli-的完整更新","8.1 Beta–Bernoulli 的完整更新",[11,51382,51383,51384,51452,51453,4821,51481,51510],{},"取先验 ",[23,51385,51387,51419],{"className":51386,"translate":27},[31],[23,51388,51390],{"className":51389},[35],[37,51391,51392],{"xmlns":39},[42,51393,51394,51416],{},[45,51395,51396,51406,51408,51410,51412,51414],{},[45,51397,51398,51400,51402,51404],{},[48,51399,2533],{"mathvariant":111},[48,51401,5342],{"mathvariant":111},[48,51403,5345],{"mathvariant":111},[48,51405,495],{"mathvariant":111},[51,51407,54],{"stretchy":53},[2581,51409,24923],{},[51,51411,747],{"separator":122},[2581,51413,24923],{},[51,51415,66],{"stretchy":53},[114,51417,51418],{"encoding":116},"\\mathrm{Beta}(3,3)",[23,51420,51422],{"className":51421,"ariaHidden":122},[121],[23,51423,51425,51428,51434,51437,51440,51443,51446,51449],{"className":51424},[126],[23,51426],{"className":51427,"style":131},[130],[23,51429,51431],{"className":51430},[135],[23,51432,5392],{"className":51433},[135,5391],[23,51435,54],{"className":51436},[140],[23,51438,24923],{"className":51439},[135],[23,51441,747],{"className":51442},[824],[23,51444],{"className":51445,"style":280},[148],[23,51447,24923],{"className":51448},[135],[23,51450,66],{"className":51451},[169],"，观测到 ",[23,51454,51456,51469],{"className":51455,"translate":27},[31],[23,51457,51459],{"className":51458},[35],[37,51460,51461],{"xmlns":39},[42,51462,51463,51467],{},[45,51464,51465],{},[2581,51466,46694],{},[114,51468,46694],{"encoding":116},[23,51470,51472],{"className":51471,"ariaHidden":122},[121],[23,51473,51475,51478],{"className":51474},[126],[23,51476],{"className":51477,"style":2770},[130],[23,51479,46694],{"className":51480},[135],[23,51482,51484,51498],{"className":51483,"translate":27},[31],[23,51485,51487],{"className":51486},[35],[37,51488,51489],{"xmlns":39},[42,51490,51491,51496],{},[45,51492,51493],{},[2581,51494,51495],{},"7",[114,51497,51495],{"encoding":116},[23,51499,51501],{"className":51500,"ariaHidden":122},[121],[23,51502,51504,51507],{"className":51503},[126],[23,51505],{"className":51506,"style":2770},[130],[23,51508,51495],{"className":51509},[135]," 次正面。后验为",[23,51512,51514],{"className":51513,"translate":27},[26],[23,51515,51517,51583],{"className":51516,"translate":27},[31],[23,51518,51520],{"className":51519},[35],[37,51521,51522],{"xmlns":39,"display":40},[42,51523,51524,51580],{},[45,51525,51526,51536,51538,51540,51542,51544,51546,51548,51550,51552,51554,51556,51558,51568,51570,51572,51574,51576,51578],{},[45,51527,51528,51530,51532,51534],{},[48,51529,2533],{"mathvariant":111},[48,51531,5342],{"mathvariant":111},[48,51533,5345],{"mathvariant":111},[48,51535,495],{"mathvariant":111},[51,51537,54],{"stretchy":53},[2581,51539,24923],{},[51,51541,5106],{},[2581,51543,51495],{},[51,51545,747],{"separator":122},[88,51547,5360],{},[2581,51549,24923],{},[51,51551,5106],{},[2581,51553,24923],{},[51,51555,66],{"stretchy":53},[51,51557,69],{},[45,51559,51560,51562,51564,51566],{},[48,51561,2533],{"mathvariant":111},[48,51563,5342],{"mathvariant":111},[48,51565,5345],{"mathvariant":111},[48,51567,495],{"mathvariant":111},[51,51569,54],{"stretchy":53},[2581,51571,46694],{},[51,51573,747],{"separator":122},[2581,51575,46697],{},[51,51577,66],{"stretchy":53},[48,51579,112],{"mathvariant":111},[114,51581,51582],{"encoding":116},"\\mathrm{Beta}(3+7,\\ 3+3)=\\mathrm{Beta}(10,6).",[23,51584,51586,51613,51644,51665],{"className":51585,"ariaHidden":122},[121],[23,51587,51589,51592,51598,51601,51604,51607,51610],{"className":51588},[126],[23,51590],{"className":51591,"style":131},[130],[23,51593,51595],{"className":51594},[135],[23,51596,5392],{"className":51597},[135,5391],[23,51599,54],{"className":51600},[140],[23,51602,24923],{"className":51603},[135],[23,51605],{"className":51606,"style":392},[148],[23,51608,5106],{"className":51609},[612],[23,51611],{"className":51612,"style":392},[148],[23,51614,51616,51620,51623,51626,51629,51632,51635,51638,51641],{"className":51615},[126],[23,51617],{"className":51618,"style":51619},[130],"height:0.8389em;vertical-align:-0.1944em;",[23,51621,51495],{"className":51622},[135],[23,51624,747],{"className":51625},[824],[23,51627,5360],{"className":51628},[148],[23,51630],{"className":51631,"style":280},[148],[23,51633,24923],{"className":51634},[135],[23,51636],{"className":51637,"style":392},[148],[23,51639,5106],{"className":51640},[612],[23,51642],{"className":51643,"style":392},[148],[23,51645,51647,51650,51653,51656,51659,51662],{"className":51646},[126],[23,51648],{"className":51649,"style":131},[130],[23,51651,24923],{"className":51652},[135],[23,51654,66],{"className":51655},[169],[23,51657],{"className":51658,"style":149},[148],[23,51660,69],{"className":51661},[153],[23,51663],{"className":51664,"style":149},[148],[23,51666,51668,51671,51677,51680,51683,51686,51689,51692,51695],{"className":51667},[126],[23,51669],{"className":51670,"style":131},[130],[23,51672,51674],{"className":51673},[135],[23,51675,5392],{"className":51676},[135,5391],[23,51678,54],{"className":51679},[140],[23,51681,46694],{"className":51682},[135],[23,51684,747],{"className":51685},[824],[23,51687],{"className":51688,"style":280},[148],[23,51690,46697],{"className":51691},[135],[23,51693,66],{"className":51694},[169],[23,51696,112],{"className":51697},[135],[11,51699,51700],{},"把几个估计量放在一起：",[3764,51702,51703,51713],{},[3767,51704,51705],{},[3770,51706,51707,51710],{},[3773,51708,51709],{},"量",[3773,51711,51712],{},"值",[3783,51714,51715,51779,51843,51907,51972],{},[3770,51716,51717,51720],{},[3788,51718,51719],{},"先验均值",[3788,51721,51722],{},[23,51723,51725,51748],{"className":51724,"translate":27},[31],[23,51726,51728],{"className":51727},[35],[37,51729,51730],{"xmlns":39},[42,51731,51732,51745],{},[45,51733,51734,51736,51738,51740,51742],{},[2581,51735,24923],{},[48,51737,2875],{"mathvariant":111},[2581,51739,46697],{},[51,51741,69],{},[2581,51743,51744],{},"0.5",[114,51746,51747],{"encoding":116},"3\u002F6=0.5",[23,51749,51751,51770],{"className":51750,"ariaHidden":122},[121],[23,51752,51754,51757,51761,51764,51767],{"className":51753},[126],[23,51755],{"className":51756,"style":131},[130],[23,51758,51760],{"className":51759},[135],"3\u002F6",[23,51762],{"className":51763,"style":149},[148],[23,51765,69],{"className":51766},[153],[23,51768],{"className":51769,"style":149},[148],[23,51771,51773,51776],{"className":51772},[126],[23,51774],{"className":51775,"style":2770},[130],[23,51777,51744],{"className":51778},[135],[3770,51780,51781,51784],{},[3788,51782,51783],{},"最大似然估计",[3788,51785,51786],{},[23,51787,51789,51812],{"className":51788,"translate":27},[31],[23,51790,51792],{"className":51791},[35],[37,51793,51794],{"xmlns":39},[42,51795,51796,51809],{},[45,51797,51798,51800,51802,51804,51806],{},[2581,51799,51495],{},[48,51801,2875],{"mathvariant":111},[2581,51803,46694],{},[51,51805,69],{},[2581,51807,51808],{},"0.7",[114,51810,51811],{"encoding":116},"7\u002F10=0.7",[23,51813,51815,51834],{"className":51814,"ariaHidden":122},[121],[23,51816,51818,51821,51825,51828,51831],{"className":51817},[126],[23,51819],{"className":51820,"style":131},[130],[23,51822,51824],{"className":51823},[135],"7\u002F10",[23,51826],{"className":51827,"style":149},[148],[23,51829,69],{"className":51830},[153],[23,51832],{"className":51833,"style":149},[148],[23,51835,51837,51840],{"className":51836},[126],[23,51838],{"className":51839,"style":2770},[130],[23,51841,51808],{"className":51842},[135],[3770,51844,51845,51847],{},[3788,51846,11335],{},[3788,51848,51849],{},[23,51850,51852,51876],{"className":51851,"translate":27},[31],[23,51853,51855],{"className":51854},[35],[37,51856,51857],{"xmlns":39},[42,51858,51859,51873],{},[45,51860,51861,51863,51865,51868,51870],{},[2581,51862,46694],{},[48,51864,2875],{"mathvariant":111},[2581,51866,51867],{},"16",[51,51869,69],{},[2581,51871,51872],{},"0.625",[114,51874,51875],{"encoding":116},"10\u002F16=0.625",[23,51877,51879,51898],{"className":51878,"ariaHidden":122},[121],[23,51880,51882,51885,51889,51892,51895],{"className":51881},[126],[23,51883],{"className":51884,"style":131},[130],[23,51886,51888],{"className":51887},[135],"10\u002F16",[23,51890],{"className":51891,"style":149},[148],[23,51893,69],{"className":51894},[153],[23,51896],{"className":51897,"style":149},[148],[23,51899,51901,51904],{"className":51900},[126],[23,51902],{"className":51903,"style":2770},[130],[23,51905,51872],{"className":51906},[135],[3770,51908,51909,51911],{},[3788,51910,13799],{},[3788,51912,51913],{},[23,51914,51916,51941],{"className":51915,"translate":27},[31],[23,51917,51919],{"className":51918},[35],[37,51920,51921],{"xmlns":39},[42,51922,51923,51938],{},[45,51924,51925,51928,51930,51933,51935],{},[2581,51926,51927],{},"9",[48,51929,2875],{"mathvariant":111},[2581,51931,51932],{},"14",[51,51934,29070],{},[2581,51936,51937],{},"0.643",[114,51939,51940],{"encoding":116},"9\u002F14\\approx0.643",[23,51942,51944,51963],{"className":51943,"ariaHidden":122},[121],[23,51945,51947,51950,51954,51957,51960],{"className":51946},[126],[23,51948],{"className":51949,"style":131},[130],[23,51951,51953],{"className":51952},[135],"9\u002F14",[23,51955],{"className":51956,"style":149},[148],[23,51958,29070],{"className":51959},[153],[23,51961],{"className":51962,"style":149},[148],[23,51964,51966,51969],{"className":51965},[126],[23,51967],{"className":51968,"style":2770},[130],[23,51970,51937],{"className":51971},[135],[3770,51973,51974,52112],{},[3788,51975,51976,51977],{},"后验预测 ",[23,51978,51980,52012],{"className":51979,"translate":27},[31],[23,51981,51983],{"className":51982},[35],[37,51984,51985],{"xmlns":39},[42,51986,51987,52009],{},[45,51988,51989,51991,51993,51999,52001,52003,52005,52007],{},[48,51990,11],{},[51,51992,54],{"stretchy":53},[898,51994,51995,51997],{},[48,51996,753],{},[88,51998,6412],{},[51,52000,69],{},[2581,52002,4435],{},[51,52004,60],{},[48,52006,63],{},[51,52008,66],{"stretchy":53},[114,52010,52011],{"encoding":116},"p(x_{\\text{new}}=1\\mid D)",[23,52013,52015,52082,52100],{"className":52014,"ariaHidden":122},[121],[23,52016,52018,52021,52024,52027,52073,52076,52079],{"className":52017},[126],[23,52019],{"className":52020,"style":131},[130],[23,52022,11],{"className":52023},[135,136],[23,52025,54],{"className":52026},[140],[23,52028,52030,52033],{"className":52029},[135],[23,52031,753],{"className":52032},[135,136],[23,52034,52036],{"className":52035},[998],[23,52037,52039,52065],{"className":52038},[199,200],[23,52040,52042,52062],{"className":52041},[204],[23,52043,52045],{"className":52044,"style":6499},[208],[23,52046,52047,52050],{"style":2029},[23,52048],{"className":52049,"style":1015},[216],[23,52051,52053],{"className":52052},[1019,1020,1021,1022],[23,52054,52056],{"className":52055},[135,1022],[23,52057,52059],{"className":52058},[135,6514,1022],[23,52060,6412],{"className":52061},[135,1022],[23,52063,297],{"className":52064},[296],[23,52066,52068],{"className":52067},[204],[23,52069,52071],{"className":52070,"style":2048},[208],[23,52072],{},[23,52074],{"className":52075,"style":149},[148],[23,52077,69],{"className":52078},[153],[23,52080],{"className":52081,"style":149},[148],[23,52083,52085,52088,52091,52094,52097],{"className":52084},[126],[23,52086],{"className":52087,"style":131},[130],[23,52089,4435],{"className":52090},[135],[23,52092],{"className":52093,"style":149},[148],[23,52095,60],{"className":52096},[153],[23,52098],{"className":52099,"style":149},[148],[23,52101,52103,52106,52109],{"className":52102},[126],[23,52104],{"className":52105,"style":131},[130],[23,52107,63],{"className":52108,"style":144},[135,136],[23,52110,66],{"className":52111},[169],[3788,52113,52114],{},[23,52115,52117,52130],{"className":52116,"translate":27},[31],[23,52118,52120],{"className":52119},[35],[37,52121,52122],{"xmlns":39},[42,52123,52124,52128],{},[45,52125,52126],{},[2581,52127,51872],{},[114,52129,51872],{"encoding":116},[23,52131,52133],{"className":52132,"ariaHidden":122},[121],[23,52134,52136,52139],{"className":52135},[126],[23,52137],{"className":52138,"style":2770},[130],[23,52140,51872],{"className":52141},[135],[11,52143,52144,52145,52173,52174,52202,52203,52232,52233,52262,52263,52322,52323,1471],{},"后验均值被先验从 ",[23,52146,52148,52161],{"className":52147,"translate":27},[31],[23,52149,52151],{"className":52150},[35],[37,52152,52153],{"xmlns":39},[42,52154,52155,52159],{},[45,52156,52157],{},[2581,52158,51808],{},[114,52160,51808],{"encoding":116},[23,52162,52164],{"className":52163,"ariaHidden":122},[121],[23,52165,52167,52170],{"className":52166},[126],[23,52168],{"className":52169,"style":2770},[130],[23,52171,51808],{"className":52172},[135]," 拉回到 ",[23,52175,52177,52190],{"className":52176,"translate":27},[31],[23,52178,52180],{"className":52179},[35],[37,52181,52182],{"xmlns":39},[42,52183,52184,52188],{},[45,52185,52186],{},[2581,52187,51872],{},[114,52189,51872],{"encoding":116},[23,52191,52193],{"className":52192,"ariaHidden":122},[121],[23,52194,52196,52199],{"className":52195},[126],[23,52197],{"className":52198,"style":2770},[130],[23,52200,51872],{"className":52201},[135],"，这就是收缩。数据越多，拉力越弱：若观测到 ",[23,52204,52206,52220],{"className":52205,"translate":27},[31],[23,52207,52209],{"className":52208},[35],[37,52210,52211],{"xmlns":39},[42,52212,52213,52218],{},[45,52214,52215],{},[2581,52216,52217],{},"100",[114,52219,52217],{"encoding":116},[23,52221,52223],{"className":52222,"ariaHidden":122},[121],[23,52224,52226,52229],{"className":52225},[126],[23,52227],{"className":52228,"style":2770},[130],[23,52230,52217],{"className":52231},[135]," 次中 ",[23,52234,52236,52250],{"className":52235,"translate":27},[31],[23,52237,52239],{"className":52238},[35],[37,52240,52241],{"xmlns":39},[42,52242,52243,52248],{},[45,52244,52245],{},[2581,52246,52247],{},"70",[114,52249,52247],{"encoding":116},[23,52251,52253],{"className":52252,"ariaHidden":122},[121],[23,52254,52256,52259],{"className":52255},[126],[23,52257],{"className":52258,"style":2770},[130],[23,52260,52247],{"className":52261},[135]," 次正面，后验均值变为 ",[23,52264,52266,52291],{"className":52265,"translate":27},[31],[23,52267,52269],{"className":52268},[35],[37,52270,52271],{"xmlns":39},[42,52272,52273,52288],{},[45,52274,52275,52278,52280,52283,52285],{},[2581,52276,52277],{},"73",[48,52279,2875],{"mathvariant":111},[2581,52281,52282],{},"106",[51,52284,29070],{},[2581,52286,52287],{},"0.689",[114,52289,52290],{"encoding":116},"73\u002F106\\approx0.689",[23,52292,52294,52313],{"className":52293,"ariaHidden":122},[121],[23,52295,52297,52300,52304,52307,52310],{"className":52296},[126],[23,52298],{"className":52299,"style":131},[130],[23,52301,52303],{"className":52302},[135],"73\u002F106",[23,52305],{"className":52306,"style":149},[148],[23,52308,29070],{"className":52309},[153],[23,52311],{"className":52312,"style":149},[148],[23,52314,52316,52319],{"className":52315},[126],[23,52317],{"className":52318,"style":2770},[130],[23,52320,52287],{"className":52321},[135],"，已经非常接近 ",[23,52324,52326,52339],{"className":52325,"translate":27},[31],[23,52327,52329],{"className":52328},[35],[37,52330,52331],{"xmlns":39},[42,52332,52333,52337],{},[45,52334,52335],{},[2581,52336,51808],{},[114,52338,51808],{"encoding":116},[23,52340,52342],{"className":52341,"ariaHidden":122},[121],[23,52343,52345,52348],{"className":52344},[126],[23,52346],{"className":52347,"style":2770},[130],[23,52349,51808],{"className":52350},[135],[11,52352,52353,52354,52386,52387,52415,52416,52444,52445,52473],{},"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 ",[23,52355,52357,52374],{"className":52356,"translate":27},[31],[23,52358,52360],{"className":52359},[35],[37,52361,52362],{"xmlns":39},[42,52363,52364,52372],{},[45,52365,52366,52368,52370],{},[2581,52367,4435],{},[48,52369,2875],{"mathvariant":111},[2581,52371,9956],{},[114,52373,10473],{"encoding":116},[23,52375,52377],{"className":52376,"ariaHidden":122},[121],[23,52378,52380,52383],{"className":52379},[126],[23,52381],{"className":52382,"style":131},[130],[23,52384,10473],{"className":52385},[135]," 就猜正面，",[23,52388,52390,52403],{"className":52389,"translate":27},[31],[23,52391,52393],{"className":52392},[35],[37,52394,52395],{"xmlns":39},[42,52396,52397,52401],{},[45,52398,52399],{},[2581,52400,51872],{},[114,52402,51872],{"encoding":116},[23,52404,52406],{"className":52405,"ariaHidden":122},[121],[23,52407,52409,52412],{"className":52408},[126],[23,52410],{"className":52411,"style":2770},[130],[23,52413,51872],{"className":52414},[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 ",[23,52417,52419,52432],{"className":52418,"translate":27},[31],[23,52420,52422],{"className":52421},[35],[37,52423,52424],{"xmlns":39},[42,52425,52426,52430],{},[45,52427,52428],{},[2581,52429,9956],{},[114,52431,9956],{"encoding":116},[23,52433,52435],{"className":52434,"ariaHidden":122},[121],[23,52436,52438,52441],{"className":52437},[126],[23,52439],{"className":52440,"style":2770},[130],[23,52442,9956],{"className":52443},[135],"，把正面误判为反面的代价为 ",[23,52446,52448,52461],{"className":52447,"translate":27},[31],[23,52449,52451],{"className":52450},[35],[37,52452,52453],{"xmlns":39},[42,52454,52455,52459],{},[45,52456,52457],{},[2581,52458,4435],{},[114,52460,4435],{"encoding":116},[23,52462,52464],{"className":52463,"ariaHidden":122},[121],[23,52465,52467,52470],{"className":52466},[126],[23,52468],{"className":52469,"style":2770},[130],[23,52471,4435],{"className":52472},[135],"，则阈值变为",[23,52475,52477],{"className":52476,"translate":27},[26],[23,52478,52480,52552],{"className":52479,"translate":27},[31],[23,52481,52483],{"className":52482},[35],[37,52484,52485],{"xmlns":39,"display":40},[42,52486,52487,52549],{},[45,52488,52489,52525,52527,52539,52541,52547],{},[71,52490,52491,52501],{},[898,52492,52493,52495],{},[48,52494,12714],{},[45,52496,52497,52499],{},[48,52498,12719],{"mathvariant":111},[48,52500,920],{"mathvariant":111},[45,52502,52503,52513,52515],{},[898,52504,52505,52507],{},[48,52506,12714],{},[45,52508,52509,52511],{},[48,52510,12719],{"mathvariant":111},[48,52512,920],{"mathvariant":111},[51,52514,5106],{},[898,52516,52517,52519],{},[48,52518,12714],{},[45,52520,52521,52523],{},[48,52522,12719],{"mathvariant":111},[48,52524,12805],{"mathvariant":111},[51,52526,69],{},[71,52528,52529,52531],{},[2581,52530,9956],{},[45,52532,52533,52535,52537],{},[2581,52534,9956],{},[51,52536,5106],{},[2581,52538,4435],{},[51,52540,69],{},[71,52542,52543,52545],{},[2581,52544,9956],{},[2581,52546,24923],{},[51,52548,747],{"separator":122},[114,52550,52551],{"encoding":116},"\\frac{c_{\\mathrm{FP}}}{c_{\\mathrm{FP}}+c_{\\mathrm{FN}}}\n=\\frac{2}{2+1}=\\frac23,",[23,52553,52555,52773,52864],{"className":52554,"ariaHidden":122},[121],[23,52556,52558,52561,52764,52767,52770],{"className":52557},[126],[23,52559],{"className":52560,"style":13490},[130],[23,52562,52564,52567,52761],{"className":52563},[135],[23,52565],{"className":52566},[140,192],[23,52568,52570],{"className":52569},[71],[23,52571,52573,52753],{"className":52572},[199,200],[23,52574,52576,52750],{"className":52575},[204],[23,52577,52579,52688,52696],{"className":52578,"style":6056},[208],[23,52580,52581,52584],{"style":212},[23,52582],{"className":52583,"style":217},[216],[23,52585,52587,52633,52636,52639,52642],{"className":52586},[135],[23,52588,52590,52593],{"className":52589},[135],[23,52591,12714],{"className":52592},[135,136],[23,52594,52596],{"className":52595},[998],[23,52597,52599,52625],{"className":52598},[199,200],[23,52600,52602,52622],{"className":52601},[204],[23,52603,52605],{"className":52604,"style":2026},[208],[23,52606,52607,52610],{"style":2029},[23,52608],{"className":52609,"style":1015},[216],[23,52611,52613],{"className":52612},[1019,1020,1021,1022],[23,52614,52616],{"className":52615},[135,1022],[23,52617,52619],{"className":52618},[135,1022],[23,52620,12770],{"className":52621},[135,5391,1022],[23,52623,297],{"className":52624},[296],[23,52626,52628],{"className":52627},[204],[23,52629,52631],{"className":52630,"style":2048},[208],[23,52632],{},[23,52634],{"className":52635,"style":392},[148],[23,52637,5106],{"className":52638},[612],[23,52640],{"className":52641,"style":392},[148],[23,52643,52645,52648],{"className":52644},[135],[23,52646,12714],{"className":52647},[135,136],[23,52649,52651],{"className":52650},[998],[23,52652,52654,52680],{"className":52653},[199,200],[23,52655,52657,52677],{"className":52656},[204],[23,52658,52660],{"className":52659,"style":2026},[208],[23,52661,52662,52665],{"style":2029},[23,52663],{"className":52664,"style":1015},[216],[23,52666,52668],{"className":52667},[1019,1020,1021,1022],[23,52669,52671],{"className":52670},[135,1022],[23,52672,52674],{"className":52673},[135,1022],[23,52675,12853],{"className":52676},[135,5391,1022],[23,52678,297],{"className":52679},[296],[23,52681,52683],{"className":52682},[204],[23,52684,52686],{"className":52685,"style":2048},[208],[23,52687],{},[23,52689,52690,52693],{"style":235},[23,52691],{"className":52692,"style":217},[216],[23,52694],{"className":52695,"style":243},[242],[23,52697,52698,52701],{"style":246},[23,52699],{"className":52700,"style":217},[216],[23,52702,52704],{"className":52703},[135],[23,52705,52707,52710],{"className":52706},[135],[23,52708,12714],{"className":52709},[135,136],[23,52711,52713],{"className":52712},[998],[23,52714,52716,52742],{"className":52715},[199,200],[23,52717,52719,52739],{"className":52718},[204],[23,52720,52722],{"className":52721,"style":2026},[208],[23,52723,52724,52727],{"style":2029},[23,52725],{"className":52726,"style":1015},[216],[23,52728,52730],{"className":52729},[1019,1020,1021,1022],[23,52731,52733],{"className":52732},[135,1022],[23,52734,52736],{"className":52735},[135,1022],[23,52737,12770],{"className":52738},[135,5391,1022],[23,52740,297],{"className":52741},[296],[23,52743,52745],{"className":52744},[204],[23,52746,52748],{"className":52747,"style":2048},[208],[23,52749],{},[23,52751,297],{"className":52752},[296],[23,52754,52756],{"className":52755},[204],[23,52757,52759],{"className":52758,"style":13689},[208],[23,52760],{},[23,52762],{"className":52763},[169,192],[23,52765],{"className":52766,"style":149},[148],[23,52768,69],{"className":52769},[153],[23,52771],{"className":52772,"style":149},[148],[23,52774,52776,52780,52855,52858,52861],{"className":52775},[126],[23,52777],{"className":52778,"style":52779},[130],"height:2.0908em;vertical-align:-0.7693em;",[23,52781,52783,52786,52852],{"className":52782},[135],[23,52784],{"className":52785},[140,192],[23,52787,52789],{"className":52788},[71],[23,52790,52792,52843],{"className":52791},[199,200],[23,52793,52795,52840],{"className":52794},[204],[23,52796,52798,52821,52829],{"className":52797,"style":12413},[208],[23,52799,52800,52803],{"style":212},[23,52801],{"className":52802,"style":217},[216],[23,52804,52806,52809,52812,52815,52818],{"className":52805},[135],[23,52807,9956],{"className":52808},[135],[23,52810],{"className":52811,"style":392},[148],[23,52813,5106],{"className":52814},[612],[23,52816],{"className":52817,"style":392},[148],[23,52819,4435],{"className":52820},[135],[23,52822,52823,52826],{"style":235},[23,52824],{"className":52825,"style":217},[216],[23,52827],{"className":52828,"style":243},[242],[23,52830,52831,52834],{"style":246},[23,52832],{"className":52833,"style":217},[216],[23,52835,52837],{"className":52836},[135],[23,52838,9956],{"className":52839},[135],[23,52841,297],{"className":52842},[296],[23,52844,52846],{"className":52845},[204],[23,52847,52850],{"className":52848,"style":52849},[208],"height:0.7693em;",[23,52851],{},[23,52853],{"className":52854},[169,192],[23,52856],{"className":52857,"style":149},[148],[23,52859,69],{"className":52860},[153],[23,52862],{"className":52863,"style":149},[148],[23,52865,52867,52870,52932],{"className":52866},[126],[23,52868],{"className":52869,"style":12394},[130],[23,52871,52873,52876,52929],{"className":52872},[135],[23,52874],{"className":52875},[140,192],[23,52877,52879],{"className":52878},[71],[23,52880,52882,52921],{"className":52881},[199,200],[23,52883,52885,52918],{"className":52884},[204],[23,52886,52888,52899,52907],{"className":52887,"style":12413},[208],[23,52889,52890,52893],{"style":212},[23,52891],{"className":52892,"style":217},[216],[23,52894,52896],{"className":52895},[135],[23,52897,24923],{"className":52898},[135],[23,52900,52901,52904],{"style":235},[23,52902],{"className":52903,"style":217},[216],[23,52905],{"className":52906,"style":243},[242],[23,52908,52909,52912],{"style":246},[23,52910],{"className":52911,"style":217},[216],[23,52913,52915],{"className":52914},[135],[23,52916,9956],{"className":52917},[135],[23,52919,297],{"className":52920},[296],[23,52922,52924],{"className":52923},[204],[23,52925,52927],{"className":52926,"style":6287},[208],[23,52928],{},[23,52930],{"className":52931},[169,192],[23,52933,747],{"className":52934},[824],[11,52936,52937,52938,52995],{},"而 ",[23,52939,52941,52963],{"className":52940,"translate":27},[31],[23,52942,52944],{"className":52943},[35],[37,52945,52946],{"xmlns":39},[42,52947,52948,52960],{},[45,52949,52950,52952,52954,52956,52958],{},[2581,52951,51872],{},[51,52953,9987],{},[2581,52955,9956],{},[48,52957,2875],{"mathvariant":111},[2581,52959,24923],{},[114,52961,52962],{"encoding":116},"0.625\u003C2\u002F3",[23,52964,52966,52985],{"className":52965,"ariaHidden":122},[121],[23,52967,52969,52973,52976,52979,52982],{"className":52968},[126],[23,52970],{"className":52971,"style":52972},[130],"height:0.6835em;vertical-align:-0.0391em;",[23,52974,51872],{"className":52975},[135],[23,52977],{"className":52978,"style":149},[148],[23,52980,9987],{"className":52981},[153],[23,52983],{"className":52984,"style":149},[148],[23,52986,52988,52991],{"className":52987},[126],[23,52989],{"className":52990,"style":131},[130],[23,52992,52994],{"className":52993},[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。",[2501,52997,52999],{"id":52998},"_82-高斯先验与收缩","8.2 高斯先验与收缩",[11,53001,53002,53003,53153,53154,53304,53305,53367,53368,53396],{},"设观测模型 ",[23,53004,53006,53042],{"className":53005,"translate":27},[31],[23,53007,53009],{"className":53008},[35],[37,53010,53011],{"xmlns":39},[42,53012,53013,53039],{},[45,53014,53015,53021,53023,53025,53027,53029,53031,53037],{},[898,53016,53017,53019],{},[48,53018,660],{},[48,53020,33497],{},[51,53022,917],{},[48,53024,12805],{"mathvariant":375},[51,53026,54],{"stretchy":53},[48,53028,57],{},[51,53030,747],{"separator":122},[4518,53032,53033,53035],{},[48,53034,29638],{},[2581,53036,9956],{},[51,53038,66],{"stretchy":53},[114,53040,53041],{"encoding":116},"y_i\\sim\\mathcal N(\\theta,\\sigma^2)",[23,53043,53045,53100],{"className":53044,"ariaHidden":122},[121],[23,53046,53048,53051,53091,53094,53097],{"className":53047},[126],[23,53049],{"className":53050,"style":672},[130],[23,53052,53054,53057],{"className":53053},[135],[23,53055,660],{"className":53056,"style":676},[135,136],[23,53058,53060],{"className":53059},[998],[23,53061,53063,53083],{"className":53062},[199,200],[23,53064,53066,53080],{"className":53065},[204],[23,53067,53069],{"className":53068,"style":33651},[208],[23,53070,53071,53074],{"style":21854},[23,53072],{"className":53073,"style":1015},[216],[23,53075,53077],{"className":53076},[1019,1020,1021,1022],[23,53078,33497],{"className":53079},[135,136,1022],[23,53081,297],{"className":53082},[296],[23,53084,53086],{"className":53085},[204],[23,53087,53089],{"className":53088,"style":2048},[208],[23,53090],{},[23,53092],{"className":53093,"style":149},[148],[23,53095,917],{"className":53096},[153],[23,53098],{"className":53099,"style":149},[148],[23,53101,53103,53106,53109,53112,53115,53118,53121,53150],{"className":53102},[126],[23,53104],{"className":53105,"style":10665},[130],[23,53107,12805],{"className":53108,"style":22222},[135,412],[23,53110,54],{"className":53111},[140],[23,53113,57],{"className":53114,"style":144},[135,136],[23,53116,747],{"className":53117},[824],[23,53119],{"className":53120,"style":280},[148],[23,53122,53124,53127],{"className":53123},[135],[23,53125,29638],{"className":53126,"style":676},[135,136],[23,53128,53130],{"className":53129},[998],[23,53131,53133],{"className":53132},[199],[23,53134,53136],{"className":53135},[204],[23,53137,53139],{"className":53138,"style":4693},[208],[23,53140,53141,53144],{"style":4696},[23,53142],{"className":53143,"style":1015},[216],[23,53145,53147],{"className":53146},[1019,1020,1021,1022],[23,53148,9956],{"className":53149},[135,1022],[23,53151,66],{"className":53152},[169],"，先验 ",[23,53155,53157,53193],{"className":53156,"translate":27},[31],[23,53158,53160],{"className":53159},[35],[37,53161,53162],{"xmlns":39},[42,53163,53164,53190],{},[45,53165,53166,53168,53170,53172,53174,53180,53182,53188],{},[48,53167,57],{},[51,53169,917],{},[48,53171,12805],{"mathvariant":375},[51,53173,54],{"stretchy":53},[898,53175,53176,53178],{},[48,53177,22109],{},[2581,53179,1532],{},[51,53181,747],{"separator":122},[4518,53183,53184,53186],{},[48,53185,34352],{},[2581,53187,9956],{},[51,53189,66],{"stretchy":53},[114,53191,53192],{"encoding":116},"\\theta\\sim\\mathcal N(\\mu_0,\\tau^2)",[23,53194,53196,53214],{"className":53195,"ariaHidden":122},[121],[23,53197,53199,53202,53205,53208,53211],{"className":53198},[126],[23,53200],{"className":53201,"style":779},[130],[23,53203,57],{"className":53204,"style":144},[135,136],[23,53206],{"className":53207,"style":149},[148],[23,53209,917],{"className":53210},[153],[23,53212],{"className":53213,"style":149},[148],[23,53215,53217,53220,53223,53226,53266,53269,53272,53301],{"className":53216},[126],[23,53218],{"className":53219,"style":10665},[130],[23,53221,12805],{"className":53222,"style":22222},[135,412],[23,53224,54],{"className":53225},[140],[23,53227,53229,53232],{"className":53228},[135],[23,53230,22109],{"className":53231},[135,136],[23,53233,53235],{"className":53234},[998],[23,53236,53238,53258],{"className":53237},[199,200],[23,53239,53241,53255],{"className":53240},[204],[23,53242,53244],{"className":53243,"style":18651},[208],[23,53245,53246,53249],{"style":2029},[23,53247],{"className":53248,"style":1015},[216],[23,53250,53252],{"className":53251},[1019,1020,1021,1022],[23,53253,1532],{"className":53254},[135,1022],[23,53256,297],{"className":53257},[296],[23,53259,53261],{"className":53260},[204],[23,53262,53264],{"className":53263,"style":2048},[208],[23,53265],{},[23,53267,747],{"className":53268},[824],[23,53270],{"className":53271,"style":280},[148],[23,53273,53275,53278],{"className":53274},[135],[23,53276,34352],{"className":53277,"style":34411},[135,136],[23,53279,53281],{"className":53280},[998],[23,53282,53284],{"className":53283},[199],[23,53285,53287],{"className":53286},[204],[23,53288,53290],{"className":53289,"style":4693},[208],[23,53291,53292,53295],{"style":4696},[23,53293],{"className":53294,"style":1015},[216],[23,53296,53298],{"className":53297},[1019,1020,1021,1022],[23,53299,9956],{"className":53300},[135,1022],[23,53302,66],{"className":53303},[169],"，样本均值为 ",[23,53306,53308,53326],{"className":53307,"translate":27},[31],[23,53309,53311],{"className":53310},[35],[37,53312,53313],{"xmlns":39},[42,53314,53315,53323],{},[45,53316,53317],{},[16811,53318,53319,53321],{"accent":122},[48,53320,753],{},[51,53322,39695],{},[114,53324,53325],{"encoding":116},"\\bar x",[23,53327,53329],{"className":53328,"ariaHidden":122},[121],[23,53330,53332,53335],{"className":53331},[126],[23,53333],{"className":53334,"style":39773},[130],[23,53336,53338],{"className":53337},[135,16834],[23,53339,53341],{"className":53340},[199],[23,53342,53344],{"className":53343},[204],[23,53345,53347,53355],{"className":53346,"style":39773},[208],[23,53348,53349,53352],{"style":7132},[23,53350],{"className":53351,"style":217},[216],[23,53353,753],{"className":53354},[135,136],[23,53356,53357,53360],{"style":7132},[23,53358],{"className":53359,"style":217},[216],[23,53361,53364],{"className":53362,"style":53363},[16862],"left:-0.2222em;",[23,53365,39695],{"className":53366},[135],"，样本量为 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5776,"style":149},[148],[23,55778,55780,55783],{"className":55779},[126],[23,55781],{"className":55782,"style":2770},[130],[23,55784,55751],{"className":55785},[135],[3770,55787,55788,55791,55851],{},[3788,55789,55790],{},"25",[3788,55792,55793],{},[23,55794,55796,55820],{"className":55795,"translate":27},[31],[23,55797,55799],{"className":55798},[35],[37,55800,55801],{"xmlns":39},[42,55802,55803,55817],{},[45,55804,55805,55807,55809,55812,55814],{},[2581,55806,55790],{},[48,55808,2875],{"mathvariant":111},[2581,55810,55811],{},"26",[51,55813,29070],{},[2581,55815,55816],{},"0.962",[114,55818,55819],{"encoding":116},"25\u002F26\\approx0.962",[23,55821,55823,55842],{"className":55822,"ariaHidden":122},[121],[23,55824,55826,55829,55833,55836,55839],{"className":55825},[126],[23,55827],{"className":55828,"style":131},[130],[23,55830,55832],{"className":55831},[135],"25\u002F26",[23,55834],{"className":55835,"style":149},[148],[23,55837,29070],{"className":55838},[153],[23,55840],{"className":55841,"style":149},[148],[23,55843,55845,55848],{"className":55844},[126],[23,55846],{"className":55847,"style":2770},[130],[23,55849,55816],{"className":55850},[135],[3788,55852,55853],{},[23,55854,55856,55879],{"className":55855,"translate":27},[31],[23,55857,55859],{"className":55858},[35],[37,55860,55861],{"xmlns":39},[42,55862,55863,55876],{},[45,55864,55865,55867,55869,55871,55873],{},[2581,55866,4435],{},[48,55868,2875],{"mathvariant":111},[2581,55870,55811],{},[51,55872,29070],{},[2581,55874,55875],{},"0.038",[114,55877,55878],{"encoding":116},"1\u002F26\\approx0.038",[23,55880,55882,55901],{"className":55881,"ariaHidden":122},[121],[23,55883,55885,55888,55892,55895,55898],{"className":55884},[126],[23,55886],{"className":55887,"style":131},[130],[23,55889,55891],{"className":55890},[135],"1\u002F26",[23,55893],{"className":55894,"style":149},[148],[23,55896,29070],{"className":55897},[153],[23,55899],{"className":55900,"style":149},[148],[23,55902,55904,55907],{"className":55903},[126],[23,55905],{"className":55906,"style":2770},[130],[23,55908,55875],{"className":55909},[135],[3770,55911,55912,55914,55974],{},[3788,55913,52217],{},[3788,55915,55916],{},[23,55917,55919,55943],{"className":55918,"translate":27},[31],[23,55920,55922],{"className":55921},[35],[37,55923,55924],{"xmlns":39},[42,55925,55926,55940],{},[45,55927,55928,55930,55932,55935,55937],{},[2581,55929,52217],{},[48,55931,2875],{"mathvariant":111},[2581,55933,55934],{},"101",[51,55936,29070],{},[2581,55938,55939],{},"0.990",[114,55941,55942],{"encoding":116},"100\u002F101\\approx0.990",[23,55944,55946,55965],{"className":55945,"ariaHidden":122},[121],[23,55947,55949,55952,55956,55959,55962],{"className":55948},[126],[23,55950],{"className":55951,"style":131},[130],[23,55953,55955],{"className":55954},[135],"100\u002F101",[23,55957],{"className":55958,"style":149},[148],[23,55960,29070],{"className":55961},[153],[23,55963],{"className":55964,"style":149},[148],[23,55966,55968,55971],{"className":55967},[126],[23,55969],{"className":55970,"style":2770},[130],[23,55972,55939],{"className":55973},[135],[3788,55975,55976],{},[23,55977,55979,56002],{"className":55978,"translate":27},[31],[23,55980,55982],{"className":55981},[35],[37,55983,55984],{"xmlns":39},[42,55985,55986,55999],{},[45,55987,55988,55990,55992,55994,55996],{},[2581,55989,4435],{},[48,55991,2875],{"mathvariant":111},[2581,55993,55934],{},[51,55995,29070],{},[2581,55997,55998],{},"0.0099",[114,56000,56001],{"encoding":116},"1\u002F101\\approx0.0099",[23,56003,56005,56024],{"className":56004,"ariaHidden":122},[121],[23,56006,56008,56011,56015,56018,56021],{"className":56007},[126],[23,56009],{"className":56010,"style":131},[130],[23,56012,56014],{"className":56013},[135],"1\u002F101",[23,56016],{"className":56017,"style":149},[148],[23,56019,29070],{"className":56020},[153],[23,56022],{"className":56023,"style":149},[148],[23,56025,56027,56030],{"className":56026},[126],[23,56028],{"className":56029,"style":2770},[130],[23,56031,55998],{"className":56032},[135],[11,56034,56035],{},"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。",[2501,56037,56039],{"id":56038},"_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",[11,56041,56042,56043,56077,56078,56111,56112,56145],{},"设某疾病的患病率为 ",[23,56044,56046,56063],{"className":56045,"translate":27},[31],[23,56047,56049],{"className":56048},[35],[37,56050,56051],{"xmlns":39},[42,56052,56053,56060],{},[45,56054,56055,56057],{},[2581,56056,4435],{},[48,56058,56059],{"mathvariant":111},"%",[114,56061,56062],{"encoding":116},"1\\%",[23,56064,56066],{"className":56065,"ariaHidden":122},[121],[23,56067,56069,56073],{"className":56068},[126],[23,56070],{"className":56071,"style":56072},[130],"height:0.8056em;vertical-align:-0.0556em;",[23,56074,56076],{"className":56075},[135],"1%","，检测的灵敏度为 ",[23,56079,56081,56098],{"className":56080,"translate":27},[31],[23,56082,56084],{"className":56083},[35],[37,56085,56086],{"xmlns":39},[42,56087,56088,56095],{},[45,56089,56090,56093],{},[2581,56091,56092],{},"99",[48,56094,56059],{"mathvariant":111},[114,56096,56097],{"encoding":116},"99\\%",[23,56099,56101],{"className":56100,"ariaHidden":122},[121],[23,56102,56104,56107],{"className":56103},[126],[23,56105],{"className":56106,"style":56072},[130],[23,56108,56110],{"className":56109},[135],"99%","，特异度为 ",[23,56113,56115,56132],{"className":56114,"translate":27},[31],[23,56116,56118],{"className":56117},[35],[37,56119,56120],{"xmlns":39},[42,56121,56122,56129],{},[45,56123,56124,56127],{},[2581,56125,56126],{},"95",[48,56128,56059],{"mathvariant":111},[114,56130,56131],{"encoding":116},"95\\%",[23,56133,56135],{"className":56134,"ariaHidden":122},[121],[23,56136,56138,56141],{"className":56137},[126],[23,56139],{"className":56140,"style":56072},[130],[23,56142,56144],{"className":56143},[135],"95%","。一个人检测阳性，患病概率是多少？",[11,56147,56148],{},"用贝叶斯公式：",[23,56150,56152],{"className":56151,"translate":27},[26],[23,56153,56155,56353],{"className":56154,"translate":27},[31],[23,56156,56158],{"className":56157},[35],[37,56159,56160],{"xmlns":39,"display":40},[42,56161,56162,56350],{},[1518,56163,56164,56267,56314],{"rowspacing":1520,"columnalign":1521,"columnspacing":1522},[1524,56165,56166,56184],{},[1527,56167,56168],{},[1530,56169,56170],{"scriptlevel":1532,"displaystyle":122},[45,56171,56172,56174,56176,56178,56180,56182],{},[48,56173,920],{},[51,56175,54],{"stretchy":53},[48,56177,63],{},[51,56179,60],{},[51,56181,5106],{},[51,56183,66],{"stretchy":53},[1527,56185,56186],{},[1530,56187,56188],{"scriptlevel":1532,"displaystyle":122},[45,56189,56190,56192,56194],{},[45,56191],{},[51,56193,69],{},[71,56195,56196,56218],{},[45,56197,56198,56200,56202,56204,56206,56208,56210,56212,56214,56216],{},[48,56199,920],{},[51,56201,54],{"stretchy":53},[51,56203,5106],{},[51,56205,60],{},[48,56207,63],{},[51,56209,66],{"stretchy":53},[48,56211,920],{},[51,56213,54],{"stretchy":53},[48,56215,63],{},[51,56217,66],{"stretchy":53},[45,56219,56220,56222,56224,56226,56228,56230,56232,56234,56236,56238,56240,56242,56244,56246,56248,56250,56253,56255,56257,56259,56261,56263,56265],{},[48,56221,920],{},[51,56223,54],{"stretchy":53},[51,56225,5106],{},[51,56227,60],{},[48,56229,63],{},[51,56231,66],{"stretchy":53},[48,56233,920],{},[51,56235,54],{"stretchy":53},[48,56237,63],{},[51,56239,66],{"stretchy":53},[51,56241,5106],{},[48,56243,920],{},[51,56245,54],{"stretchy":53},[51,56247,5106],{},[51,56249,60],{},[48,56251,56252],{"mathvariant":111},"¬",[48,56254,63],{},[51,56256,66],{"stretchy":53},[48,56258,920],{},[51,56260,54],{"stretchy":53},[48,56262,56252],{"mathvariant":111},[48,56264,63],{},[51,56266,66],{"stretchy":53},[1524,56268,56269,56275],{},[1527,56270,56271],{},[1530,56272,56273],{"scriptlevel":1532,"displaystyle":122},[45,56274],{},[1527,56276,56277],{},[1530,56278,56279],{"scriptlevel":1532,"displaystyle":122},[45,56280,56281,56283,56285],{},[45,56282],{},[51,56284,69],{},[71,56286,56287,56297],{},[45,56288,56289,56292,56294],{},[2581,56290,56291],{},"0.99",[51,56293,563],{},[2581,56295,56296],{},"0.01",[45,56298,56299,56301,56303,56305,56307,56310,56312],{},[2581,56300,56291],{},[51,56302,563],{},[2581,56304,56296],{},[51,56306,5106],{},[2581,56308,56309],{},"0.05",[51,56311,563],{},[2581,56313,56291],{},[1524,56315,56316,56322],{},[1527,56317,56318],{},[1530,56319,56320],{"scriptlevel":1532,"displaystyle":122},[45,56321],{},[1527,56323,56324],{},[1530,56325,56326],{"scriptlevel":1532,"displaystyle":122},[45,56327,56328,56330,56332,56345,56347],{},[45,56329],{},[51,56331,69],{},[71,56333,56334,56336],{},[2581,56335,55998],{},[45,56337,56338,56340,56342],{},[2581,56339,55998],{},[51,56341,5106],{},[2581,56343,56344],{},"0.0495",[51,56346,29070],{},[2581,56348,56349],{},"0.167.",[114,56351,56352],{"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}",[23,56354,56356],{"className":56355,"ariaHidden":122},[121],[23,56357,56359,56363],{"className":56358},[126],[23,56360],{"className":56361,"style":56362},[130],"height:7.1445em;vertical-align:-3.3223em;",[23,56364,56366],{"className":56365},[135],[23,56367,56369,56446],{"className":56368},[1518],[23,56370,56372],{"className":56371},[1696],[23,56373,56375,56437],{"className":56374},[199,200],[23,56376,56378,56434],{"className":56377},[204],[23,56379,56382,56416,56425],{"className":56380,"style":56381},[208],"height:3.8223em;",[23,56383,56385,56389],{"style":56384},"top:-5.8223em;",[23,56386],{"className":56387,"style":56388},[216],"height:3.427em;",[23,56390,56392,56395,56398,56401,56404,56407,56410,56413],{"className":56391},[135],[23,56393,920],{"className":56394,"style":1047},[135,136],[23,56396,54],{"className":56397},[140],[23,56399,63],{"className":56400,"style":144},[135,136],[23,56402],{"className":56403,"style":149},[148],[23,56405,60],{"className":56406},[153],[23,56408],{"className":56409,"style":149},[148],[23,56411,5106],{"className":56412},[135],[23,56414,66],{"className":56415},[169],[23,56417,56419,56422],{"style":56418},"top:-3.2648em;",[23,56420],{"className":56421,"style":56388},[216],[23,56423],{"className":56424},[135],[23,56426,56428,56431],{"style":56427},"top:-0.8741em;",[23,56429],{"className":56430,"style":56388},[216],[23,56432],{"className":56433},[135],[23,56435,297],{"className":56436},[296],[23,56438,56440],{"className":56439},[204],[23,56441,56444],{"className":56442,"style":56443},[208],"height:3.3223em;",[23,56445],{},[23,56447,56449],{"className":56448},[1753],[23,56450,56452,56896],{"className":56451},[199,200],[23,56453,56455,56893],{"className":56454},[204],[23,56456,56458,56657,56787],{"className":56457,"style":56381},[208],[23,56459,56460,56463],{"style":56384},[23,56461],{"className":56462,"style":56388},[216],[23,56464,56466,56469,56472,56475,56478],{"className":56465},[135],[23,56467],{"className":56468},[135],[23,56470],{"className":56471,"style":149},[148],[23,56473,69],{"className":56474},[153],[23,56476],{"className":56477,"style":149},[148],[23,56479,56481,56484,56654],{"className":56480},[135],[23,56482],{"className":56483},[140,192],[23,56485,56487],{"className":56486},[71],[23,56488,56490,56646],{"className":56489},[199,200],[23,56491,56493,56643],{"className":56492},[204],[23,56494,56496,56591,56599],{"className":56495,"style":209},[208],[23,56497,56498,56501],{"style":212},[23,56499],{"className":56500,"style":217},[216],[23,56502,56504,56507,56510,56513,56516,56519,56522,56525,56528,56531,56534,56537,56540,56543,56546,56549,56552,56555,56558,56561,56564,56567,56570,56573,56576,56579,56582,56585,56588],{"className":56503},[135],[23,56505,920],{"className":56506,"style":1047},[135,136],[23,56508,54],{"className":56509},[140],[23,56511,5106],{"className":56512},[135],[23,56514],{"className":56515,"style":149},[148],[23,56517,60],{"className":56518},[153],[23,56520],{"className":56521,"style":149},[148],[23,56523,63],{"className":56524,"style":144},[135,136],[23,56526,66],{"className":56527},[169],[23,56529,920],{"className":56530,"style":1047},[135,136],[23,56532,54],{"className":56533},[140],[23,56535,63],{"className":56536,"style":144},[135,136],[23,56538,66],{"className":56539},[169],[23,56541],{"className":56542,"style":392},[148],[23,56544,5106],{"className":56545},[612],[23,56547],{"className":56548,"style":392},[148],[23,56550,920],{"className":56551,"style":1047},[135,136],[23,56553,54],{"className":56554},[140],[23,56556,5106],{"className":56557},[135],[23,56559],{"className":56560,"style":149},[148],[23,56562,60],{"className":56563},[153],[23,56565],{"className":56566,"style":149},[148],[23,56568,56252],{"className":56569},[135],[23,56571,63],{"className":56572,"style":144},[135,136],[23,56574,66],{"className":56575},[169],[23,56577,920],{"className":56578,"style":1047},[135,136],[23,56580,54],{"className":56581},[140],[23,56583,56252],{"className":56584},[135],[23,56586,63],{"className":56587,"style":144},[135,136],[23,56589,66],{"className":56590},[169],[23,56592,56593,56596],{"style":235},[23,56594],{"className":56595,"style":217},[216],[23,56597],{"className":56598,"style":243},[242],[23,56600,56601,56604],{"style":246},[23,56602],{"className":56603,"style":217},[216],[23,56605,56607,56610,56613,56616,56619,56622,56625,56628,56631,56634,56637,56640],{"className":56606},[135],[23,56608,920],{"className":56609,"style":1047},[135,136],[23,56611,54],{"className":56612},[140],[23,56614,5106],{"className":56615},[135],[23,56617],{"className":56618,"style":149},[148],[23,56620,60],{"className":56621},[153],[23,56623],{"className":56624,"style":149},[148],[23,56626,63],{"className":56627,"style":144},[135,136],[23,56629,66],{"className":56630},[169],[23,56632,920],{"className":56633,"style":1047},[135,136],[23,56635,54],{"className":56636},[140],[23,56638,63],{"className":56639,"style":144},[135,136],[23,56641,66],{"className":56642},[169],[23,56644,297],{"className":56645},[296],[23,56647,56649],{"className":56648},[204],[23,56650,56652],{"className":56651,"style":304},[208],[23,56653],{},[23,56655],{"className":56656},[169,192],[23,56658,56659,56662],{"style":56418},[23,56660],{"className":56661,"style":56388},[216],[23,56663,56665,56668,56671,56674,56677],{"className":56664},[135],[23,56666],{"className":56667},[135],[23,56669],{"className":56670,"style":149},[148],[23,56672,69],{"className":56673},[153],[23,56675],{"className":56676,"style":149},[148],[23,56678,56680,56683,56784],{"className":56679},[135],[23,56681],{"className":56682},[140,192],[23,56684,56686],{"className":56685},[71],[23,56687,56689,56776],{"className":56688},[199,200],[23,56690,56692,56773],{"className":56691},[204],[23,56693,56695,56742,56750],{"className":56694,"style":12413},[208],[23,56696,56697,56700],{"style":212},[23,56698],{"className":56699,"style":217},[216],[23,56701,56703,56706,56709,56712,56715,56718,56721,56724,56727,56730,56733,56736,56739],{"className":56702},[135],[23,56704,56291],{"className":56705},[135],[23,56707],{"className":56708,"style":392},[148],[23,56710,563],{"className":56711},[612],[23,56713],{"className":56714,"style":392},[148],[23,56716,56296],{"className":56717},[135],[23,56719],{"className":56720,"style":392},[148],[23,56722,5106],{"className":56723},[612],[23,56725],{"className":56726,"style":392},[148],[23,56728,56309],{"className":56729},[135],[23,56731],{"className":56732,"style":392},[148],[23,56734,563],{"className":56735},[612],[23,56737],{"className":56738,"style":392},[148],[23,56740,56291],{"className":56741},[135],[23,56743,56744,56747],{"style":235},[23,56745],{"className":56746,"style":217},[216],[23,56748],{"className":56749,"style":243},[242],[23,56751,56752,56755],{"style":246},[23,56753],{"className":56754,"style":217},[216],[23,56756,56758,56761,56764,56767,56770],{"className":56757},[135],[23,56759,56291],{"className":56760},[135],[23,56762],{"className":56763,"style":392},[148],[23,56765,563],{"className":56766},[612],[23,56768],{"className":56769,"style":392},[148],[23,56771,56296],{"className":56772},[135],[23,56774,297],{"className":56775},[296],[23,56777,56779],{"className":56778},[204],[23,56780,56782],{"className":56781,"style":52849},[208],[23,56783],{},[23,56785],{"className":56786},[169,192],[23,56788,56789,56792],{"style":56427},[23,56790],{"className":56791,"style":56388},[216],[23,56793,56795,56798,56801,56804,56807,56881,56884,56887,56890],{"className":56794},[135],[23,56796],{"className":56797},[135],[23,56799],{"className":56800,"style":149},[148],[23,56802,69],{"className":56803},[153],[23,56805],{"className":56806,"style":149},[148],[23,56808,56810,56813,56878],{"className":56809},[135],[23,56811],{"className":56812},[140,192],[23,56814,56816],{"className":56815},[71],[23,56817,56819,56870],{"className":56818},[199,200],[23,56820,56822,56867],{"className":56821},[204],[23,56823,56825,56848,56856],{"className":56824,"style":12413},[208],[23,56826,56827,56830],{"style":212},[23,56828],{"className":56829,"style":217},[216],[23,56831,56833,56836,56839,56842,56845],{"className":56832},[135],[23,56834,55998],{"className":56835},[135],[23,56837],{"className":56838,"style":392},[148],[23,56840,5106],{"className":56841},[612],[23,56843],{"className":56844,"style":392},[148],[23,56846,56344],{"className":56847},[135],[23,56849,56850,56853],{"style":235},[23,56851],{"className":56852,"style":217},[216],[23,56854],{"className":56855,"style":243},[242],[23,56857,56858,56861],{"style":246},[23,56859],{"className":56860,"style":217},[216],[23,56862,56864],{"className":56863},[135],[23,56865,55998],{"className":56866},[135],[23,56868,297],{"className":56869},[296],[23,56871,56873],{"className":56872},[204],[23,56874,56876],{"className":56875,"style":52849},[208],[23,56877],{},[23,56879],{"className":56880},[169,192],[23,56882],{"className":56883,"style":149},[148],[23,56885,29070],{"className":56886},[153],[23,56888],{"className":56889,"style":149},[148],[23,56891,56349],{"className":56892},[135],[23,56894,297],{"className":56895},[296],[23,56897,56899],{"className":56898},[204],[23,56900,56902],{"className":56901,"style":56443},[208],[23,56903],{},[11,56905,56906,56907,56940,56941,56973,56974,57004,57005,4136,57071,57137],{},"尽管检测看起来很准，阳性者真正患病的概率只有 ",[23,56908,56910,56927],{"className":56909,"translate":27},[31],[23,56911,56913],{"className":56912},[35],[37,56914,56915],{"xmlns":39},[42,56916,56917,56924],{},[45,56918,56919,56922],{},[2581,56920,56921],{},"16.7",[48,56923,56059],{"mathvariant":111},[114,56925,56926],{"encoding":116},"16.7\\%",[23,56928,56930],{"className":56929,"ariaHidden":122},[121],[23,56931,56933,56936],{"className":56932},[126],[23,56934],{"className":56935,"style":56072},[130],[23,56937,56939],{"className":56938},[135],"16.7%","，因为健康人占了绝大多数，",[23,56942,56944,56960],{"className":56943,"translate":27},[31],[23,56945,56947],{"className":56946},[35],[37,56948,56949],{"xmlns":39},[42,56950,56951,56957],{},[45,56952,56953,56955],{},[2581,56954,55687],{},[48,56956,56059],{"mathvariant":111},[114,56958,56959],{"encoding":116},"5\\%",[23,56961,56963],{"className":56962,"ariaHidden":122},[121],[23,56964,56966,56969],{"className":56965},[126],[23,56967],{"className":56968,"style":56072},[130],[23,56970,56972],{"className":56971},[135],"5%"," 的假阳性率乘以 ",[23,56975,56977,56992],{"className":56976,"translate":27},[31],[23,56978,56980],{"className":56979},[35],[37,56981,56982],{"xmlns":39},[42,56983,56984,56990],{},[45,56985,56986,56988],{},[2581,56987,56092],{},[48,56989,56059],{"mathvariant":111},[114,56991,56097],{"encoding":116},[23,56993,56995],{"className":56994,"ariaHidden":122},[121],[23,56996,56998,57001],{"className":56997},[126],[23,56999],{"className":57000,"style":56072},[130],[23,57002,56110],{"className":57003},[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 ",[23,57006,57008,57032],{"className":57007,"translate":27},[31],[23,57009,57011],{"className":57010},[35],[37,57012,57013],{"xmlns":39},[42,57014,57015,57029],{},[45,57016,57017,57019,57021,57023,57025,57027],{},[48,57018,920],{},[51,57020,54],{"stretchy":53},[51,57022,5106],{},[51,57024,60],{},[48,57026,63],{},[51,57028,66],{"stretchy":53},[114,57030,57031],{"encoding":116},"P(+\\mid D)",[23,57033,57035,57059],{"className":57034,"ariaHidden":122},[121],[23,57036,57038,57041,57044,57047,57050,57053,57056],{"className":57037},[126],[23,57039],{"className":57040,"style":131},[130],[23,57042,920],{"className":57043,"style":1047},[135,136],[23,57045,54],{"className":57046},[140],[23,57048,5106],{"className":57049},[135],[23,57051],{"className":57052,"style":149},[148],[23,57054,60],{"className":57055},[153],[23,57057],{"className":57058,"style":149},[148],[23,57060,57062,57065,57068],{"className":57061},[126],[23,57063],{"className":57064,"style":131},[130],[23,57066,63],{"className":57067,"style":144},[135,136],[23,57069,66],{"className":57070},[169],[23,57072,57074,57098],{"className":57073,"translate":27},[31],[23,57075,57077],{"className":57076},[35],[37,57078,57079],{"xmlns":39},[42,57080,57081,57095],{},[45,57082,57083,57085,57087,57089,57091,57093],{},[48,57084,920],{},[51,57086,54],{"stretchy":53},[48,57088,63],{},[51,57090,60],{},[51,57092,5106],{},[51,57094,66],{"stretchy":53},[114,57096,57097],{"encoding":116},"P(D\\mid +)",[23,57099,57101,57125],{"className":57100,"ariaHidden":122},[121],[23,57102,57104,57107,57110,57113,57116,57119,57122],{"className":57103},[126],[23,57105],{"className":57106,"style":131},[130],[23,57108,920],{"className":57109,"style":1047},[135,136],[23,57111,54],{"className":57112},[140],[23,57114,63],{"className":57115,"style":144},[135,136],[23,57117],{"className":57118,"style":149},[148],[23,57120,60],{"className":57121},[153],[23,57123],{"className":57124,"style":149},[148],[23,57126,57128,57131,57134],{"className":57127},[126],[23,57129],{"className":57130,"style":131},[130],[23,57132,5106],{"className":57133},[135],[23,57135,66],{"className":57136},[169]," 混为一谈。",[11,57139,57140],{},"患病率变化时，同一个检测给出的后验完全不同：",[3764,57142,57143,57217],{},[3767,57144,57145],{},[3770,57146,57147,57150],{},[3773,57148,57149],{},"患病率",[3773,57151,57152],{},[23,57153,57155,57178],{"className":57154,"translate":27},[31],[23,57156,57158],{"className":57157},[35],[37,57159,57160],{"xmlns":39},[42,57161,57162,57176],{},[45,57163,57164,57166,57168,57170,57172,57174],{},[48,57165,920],{},[51,57167,54],{"stretchy":53},[48,57169,63],{},[51,57171,60],{},[51,57173,5106],{},[51,57175,66],{"stretchy":53},[114,57177,57097],{"encoding":116},[23,57179,57181,57205],{"className":57180,"ariaHidden":122},[121],[23,57182,57184,57187,57190,57193,57196,57199,57202],{"className":57183},[126],[23,57185],{"className":57186,"style":131},[130],[23,57188,920],{"className":57189,"style":1047},[135,136],[23,57191,54],{"className":57192},[140],[23,57194,63],{"className":57195,"style":144},[135,136],[23,57197],{"className":57198,"style":149},[148],[23,57200,60],{"className":57201},[153],[23,57203],{"className":57204,"style":149},[148],[23,57206,57208,57211,57214],{"className":57207},[126],[23,57209],{"className":57210,"style":131},[130],[23,57212,5106],{"className":57213},[135],[23,57215,66],{"className":57216},[169],[3783,57218,57219,57286,57351,57416],{},[3770,57220,57221,57255],{},[3788,57222,57223],{},[23,57224,57226,57242],{"className":57225,"translate":27},[31],[23,57227,57229],{"className":57228},[35],[37,57230,57231],{"xmlns":39},[42,57232,57233,57239],{},[45,57234,57235,57237],{},[2581,57236,51744],{},[48,57238,56059],{"mathvariant":111},[114,57240,57241],{"encoding":116},"0.5\\%",[23,57243,57245],{"className":57244,"ariaHidden":122},[121],[23,57246,57248,57251],{"className":57247},[126],[23,57249],{"className":57250,"style":56072},[130],[23,57252,57254],{"className":57253},[135],"0.5%",[3788,57256,57257],{},[23,57258,57260,57274],{"className":57259,"translate":27},[31],[23,57261,57263],{"className":57262},[35],[37,57264,57265],{"xmlns":39},[42,57266,57267,57272],{},[45,57268,57269],{},[2581,57270,57271],{},"0.090",[114,57273,57271],{"encoding":116},[23,57275,57277],{"className":57276,"ariaHidden":122},[121],[23,57278,57280,57283],{"className":57279},[126],[23,57281],{"className":57282,"style":2770},[130],[23,57284,57271],{"className":57285},[135],[3770,57287,57288,57320],{},[3788,57289,57290],{},[23,57291,57293,57308],{"className":57292,"translate":27},[31],[23,57294,57296],{"className":57295},[35],[37,57297,57298],{"xmlns":39},[42,57299,57300,57306],{},[45,57301,57302,57304],{},[2581,57303,4435],{},[48,57305,56059],{"mathvariant":111},[114,57307,56062],{"encoding":116},[23,57309,57311],{"className":57310,"ariaHidden":122},[121],[23,57312,57314,57317],{"className":57313},[126],[23,57315],{"className":57316,"style":56072},[130],[23,57318,56076],{"className":57319},[135],[3788,57321,57322],{},[23,57323,57325,57339],{"className":57324,"translate":27},[31],[23,57326,57328],{"className":57327},[35],[37,57329,57330],{"xmlns":39},[42,57331,57332,57337],{},[45,57333,57334],{},[2581,57335,57336],{},"0.167",[114,57338,57336],{"encoding":116},[23,57340,57342],{"className":57341,"ariaHidden":122},[121],[23,57343,57345,57348],{"className":57344},[126],[23,57346],{"className":57347,"style":2770},[130],[23,57349,57336],{"className":57350},[135],[3770,57352,57353,57385],{},[3788,57354,57355],{},[23,57356,57358,57373],{"className":57357,"translate":27},[31],[23,57359,57361],{"className":57360},[35],[37,57362,57363],{"xmlns":39},[42,57364,57365,57371],{},[45,57366,57367,57369],{},[2581,57368,55687],{},[48,57370,56059],{"mathvariant":111},[114,57372,56959],{"encoding":116},[23,57374,57376],{"className":57375,"ariaHidden":122},[121],[23,57377,57379,57382],{"className":57378},[126],[23,57380],{"className":57381,"style":56072},[130],[23,57383,56972],{"className":57384},[135],[3788,57386,57387],{},[23,57388,57390,57404],{"className":57389,"translate":27},[31],[23,57391,57393],{"className":57392},[35],[37,57394,57395],{"xmlns":39},[42,57396,57397,57402],{},[45,57398,57399],{},[2581,57400,57401],{},"0.510",[114,57403,57401],{"encoding":116},[23,57405,57407],{"className":57406,"ariaHidden":122},[121],[23,57408,57410,57413],{"className":57409},[126],[23,57411],{"className":57412,"style":2770},[130],[23,57414,57401],{"className":57415},[135],[3770,57417,57418,57452],{},[3788,57419,57420],{},[23,57421,57423,57439],{"className":57422,"translate":27},[31],[23,57424,57426],{"className":57425},[35],[37,57427,57428],{"xmlns":39},[42,57429,57430,57436],{},[45,57431,57432,57434],{},[2581,57433,46694],{},[48,57435,56059],{"mathvariant":111},[114,57437,57438],{"encoding":116},"10\\%",[23,57440,57442],{"className":57441,"ariaHidden":122},[121],[23,57443,57445,57448],{"className":57444},[126],[23,57446],{"className":57447,"style":56072},[130],[23,57449,57451],{"className":57450},[135],"10%",[3788,57453,57454],{},[23,57455,57457,57471],{"className":57456,"translate":27},[31],[23,57458,57460],{"className":57459},[35],[37,57461,57462],{"xmlns":39},[42,57463,57464,57469],{},[45,57465,57466],{},[2581,57467,57468],{},"0.688",[114,57470,57468],{"encoding":116},[23,57472,57474],{"className":57473,"ariaHidden":122},[121],[23,57475,57477,57480],{"className":57476},[126],[23,57478],{"className":57479,"style":2770},[130],[23,57481,57468],{"className":57482},[135],[11,57484,57485,57486,57514,57515,55163,57617,57719],{},"现在加入代价。设漏诊的代价是误诊的 ",[23,57487,57489,57502],{"className":57488,"translate":27},[31],[23,57490,57492],{"className":57491},[35],[37,57493,57494],{"xmlns":39},[42,57495,57496,57500],{},[45,57497,57498],{},[2581,57499,46694],{},[114,57501,46694],{"encoding":116},[23,57503,57505],{"className":57504,"ariaHidden":122},[121],[23,57506,57508,57511],{"className":57507},[126],[23,57509],{"className":57510,"style":2770},[130],[23,57512,46694],{"className":57513},[135]," 倍，即 ",[23,57516,57518,57544],{"className":57517,"translate":27},[31],[23,57519,57521],{"className":57520},[35],[37,57522,57523],{"xmlns":39},[42,57524,57525,57541],{},[45,57526,57527,57537,57539],{},[898,57528,57529,57531],{},[48,57530,12714],{},[45,57532,57533,57535],{},[48,57534,12719],{"mathvariant":111},[48,57536,12805],{"mathvariant":111},[51,57538,69],{},[2581,57540,46694],{},[114,57542,57543],{"encoding":116},"c_{\\mathrm{FN}}=10",[23,57545,57547,57608],{"className":57546,"ariaHidden":122},[121],[23,57548,57550,57553,57599,57602,57605],{"className":57549},[126],[23,57551],{"className":57552,"style":12734},[130],[23,57554,57556,57559],{"className":57555},[135],[23,57557,12714],{"className":57558},[135,136],[23,57560,57562],{"className":57561},[998],[23,57563,57565,57591],{"className":57564},[199,200],[23,57566,57568,57588],{"className":57567},[204],[23,57569,57571],{"className":57570,"style":2026},[208],[23,57572,57573,57576],{"style":2029},[23,57574],{"className":57575,"style":1015},[216],[23,57577,57579],{"className":57578},[1019,1020,1021,1022],[23,57580,57582],{"className":57581},[135,1022],[23,57583,57585],{"className":57584},[135,1022],[23,57586,12853],{"className":57587},[135,5391,1022],[23,57589,297],{"className":57590},[296],[23,57592,57594],{"className":57593},[204],[23,57595,57597],{"className":57596,"style":2048},[208],[23,57598],{},[23,57600],{"className":57601,"style":149},[148],[23,57603,69],{"className":57604},[153],[23,57606],{"className":57607,"style":149},[148],[23,57609,57611,57614],{"className":57610},[126],[23,57612],{"className":57613,"style":2770},[130],[23,57615,46694],{"className":57616},[135],[23,57618,57620,57646],{"className":57619,"translate":27},[31],[23,57621,57623],{"className":57622},[35],[37,57624,57625],{"xmlns":39},[42,57626,57627,57643],{},[45,57628,57629,57639,57641],{},[898,57630,57631,57633],{},[48,57632,12714],{},[45,57634,57635,57637],{},[48,57636,12719],{"mathvariant":111},[48,57638,920],{"mathvariant":111},[51,57640,69],{},[2581,57642,4435],{},[114,57644,57645],{"encoding":116},"c_{\\mathrm{FP}}=1",[23,57647,57649,57710],{"className":57648,"ariaHidden":122},[121],[23,57650,57652,57655,57701,57704,57707],{"className":57651},[126],[23,57653],{"className":57654,"style":12734},[130],[23,57656,57658,57661],{"className":57657},[135],[23,57659,12714],{"className":57660},[135,136],[23,57662,57664],{"className":57663},[998],[23,57665,57667,57693],{"className":57666},[199,200],[23,57668,57670,57690],{"className":57669},[204],[23,57671,57673],{"className":57672,"style":2026},[208],[23,57674,57675,57678],{"style":2029},[23,57676],{"className":57677,"style":1015},[216],[23,57679,57681],{"className":57680},[1019,1020,1021,1022],[23,57682,57684],{"className":57683},[135,1022],[23,57685,57687],{"className":57686},[135,1022],[23,57688,12770],{"className":57689},[135,5391,1022],[23,57691,297],{"className":57692},[296],[23,57694,57696],{"className":57695},[204],[23,57697,57699],{"className":57698,"style":2048},[208],[23,57700],{},[23,57702],{"className":57703,"style":149},[148],[23,57705,69],{"className":57706},[153],[23,57708],{"className":57709,"style":149},[148],[23,57711,57713,57716],{"className":57712},[126],[23,57714],{"className":57715,"style":2770},[130],[23,57717,4435],{"className":57718},[135],"，则治疗阈值是",[23,57721,57723],{"className":57722,"translate":27},[26],[23,57724,57726,57787],{"className":57725,"translate":27},[31],[23,57727,57729],{"className":57728},[35],[37,57730,57731],{"xmlns":39,"display":40},[42,57732,57733,57784],{},[45,57734,57735,57771,57773,57779,57781],{},[71,57736,57737,57747],{},[898,57738,57739,57741],{},[48,57740,12714],{},[45,57742,57743,57745],{},[48,57744,12719],{"mathvariant":111},[48,57746,920],{"mathvariant":111},[45,57748,57749,57759,57761],{},[898,57750,57751,57753],{},[48,57752,12714],{},[45,57754,57755,57757],{},[48,57756,12719],{"mathvariant":111},[48,57758,920],{"mathvariant":111},[51,57760,5106],{},[898,57762,57763,57765],{},[48,57764,12714],{},[45,57766,57767,57769],{},[48,57768,12719],{"mathvariant":111},[48,57770,12805],{"mathvariant":111},[51,57772,69],{},[71,57774,57775,57777],{},[2581,57776,4435],{},[2581,57778,13752],{},[51,57780,29070],{},[2581,57782,57783],{},"0.091.",[114,57785,57786],{"encoding":116},"\\frac{c_{\\mathrm{FP}}}{c_{\\mathrm{FP}}+c_{\\mathrm{FN}}}\n=\\frac{1}{11}\\approx0.091.",[23,57788,57790,58008,58085],{"className":57789,"ariaHidden":122},[121],[23,57791,57793,57796,57999,58002,58005],{"className":57792},[126],[23,57794],{"className":57795,"style":13490},[130],[23,57797,57799,57802,57996],{"className":57798},[135],[23,57800],{"className":57801},[140,192],[23,57803,57805],{"className":57804},[71],[23,57806,57808,57988],{"className":57807},[199,200],[23,57809,57811,57985],{"className":57810},[204],[23,57812,57814,57923,57931],{"className":57813,"style":6056},[208],[23,57815,57816,57819],{"style":212},[23,57817],{"className":57818,"style":217},[216],[23,57820,57822,57868,57871,57874,57877],{"className":57821},[135],[23,57823,57825,57828],{"className":57824},[135],[23,57826,12714],{"className":57827},[135,136],[23,57829,57831],{"className":57830},[998],[23,57832,57834,57860],{"className":57833},[199,200],[23,57835,57837,57857],{"className":57836},[204],[23,57838,57840],{"className":57839,"style":2026},[208],[23,57841,57842,57845],{"style":2029},[23,57843],{"className":57844,"style":1015},[216],[23,57846,57848],{"className":57847},[1019,1020,1021,1022],[23,57849,57851],{"className":57850},[135,1022],[23,57852,57854],{"className":57853},[135,1022],[23,57855,12770],{"className":57856},[135,5391,1022],[23,57858,297],{"className":57859},[296],[23,57861,57863],{"className":57862},[204],[23,57864,57866],{"className":57865,"style":2048},[208],[23,57867],{},[23,57869],{"className":57870,"style":392},[148],[23,57872,5106],{"className":57873},[612],[23,57875],{"className":57876,"style":392},[148],[23,57878,57880,57883],{"className":57879},[135],[23,57881,12714],{"className":57882},[135,136],[23,57884,57886],{"className":57885},[998],[23,57887,57889,57915],{"className":57888},[199,200],[23,57890,57892,57912],{"className":57891},[204],[23,57893,57895],{"className":57894,"style":2026},[208],[23,57896,57897,57900],{"style":2029},[23,57898],{"className":57899,"style":1015},[216],[23,57901,57903],{"className":57902},[1019,1020,1021,1022],[23,57904,57906],{"className":57905},[135,1022],[23,57907,57909],{"className":57908},[135,1022],[23,57910,12853],{"className":57911},[135,5391,1022],[23,57913,297],{"className":57914},[296],[23,57916,57918],{"className":57917},[204],[23,57919,57921],{"className":57920,"style":2048},[208],[23,57922],{},[23,57924,57925,57928],{"style":235},[23,57926],{"className":57927,"style":217},[216],[23,57929],{"className":57930,"style":243},[242],[23,57932,57933,57936],{"style":246},[23,57934],{"className":57935,"style":217},[216],[23,57937,57939],{"className":57938},[135],[23,57940,57942,57945],{"className":57941},[135],[23,57943,12714],{"className":57944},[135,136],[23,57946,57948],{"className":57947},[998],[23,57949,57951,57977],{"className":57950},[199,200],[23,57952,57954,57974],{"className":57953},[204],[23,57955,57957],{"className":57956,"style":2026},[208],[23,57958,57959,57962],{"style":2029},[23,57960],{"className":57961,"style":1015},[216],[23,57963,57965],{"className":57964},[1019,1020,1021,1022],[23,57966,57968],{"className":57967},[135,1022],[23,57969,57971],{"className":57970},[135,1022],[23,57972,12770],{"className":57973},[135,5391,1022],[23,57975,297],{"className":57976},[296],[23,57978,57980],{"className":57979},[204],[23,57981,57983],{"className":57982,"style":2048},[208],[23,57984],{},[23,57986,297],{"className":57987},[296],[23,57989,57991],{"className":57990},[204],[23,57992,57994],{"className":57993,"style":13689},[208],[23,57995],{},[23,57997],{"className":57998},[169,192],[23,58000],{"className":58001,"style":149},[148],[23,58003,69],{"className":58004},[153],[23,58006],{"className":58007,"style":149},[148],[23,58009,58011,58014,58076,58079,58082],{"className":58010},[126],[23,58012],{"className":58013,"style":12394},[130],[23,58015,58017,58020,58073],{"className":58016},[135],[23,58018],{"className":58019},[140,192],[23,58021,58023],{"className":58022},[71],[23,58024,58026,58065],{"className":58025},[199,200],[23,58027,58029,58062],{"className":58028},[204],[23,58030,58032,58043,58051],{"className":58031,"style":12413},[208],[23,58033,58034,58037],{"style":212},[23,58035],{"className":58036,"style":217},[216],[23,58038,58040],{"className":58039},[135],[23,58041,13752],{"className":58042},[135],[23,58044,58045,58048],{"style":235},[23,58046],{"className":58047,"style":217},[216],[23,58049],{"className":58050,"style":243},[242],[23,58052,58053,58056],{"style":246},[23,58054],{"className":58055,"style":217},[216],[23,58057,58059],{"className":58058},[135],[23,58060,4435],{"className":58061},[135],[23,58063,297],{"className":58064},[296],[23,58066,58068],{"className":58067},[204],[23,58069,58071],{"className":58070,"style":6287},[208],[23,58072],{},[23,58074],{"className":58075},[169,192],[23,58077],{"className":58078,"style":149},[148],[23,58080,29070],{"className":58081},[153],[23,58083],{"className":58084,"style":149},[148],[23,58086,58088,58091],{"className":58087},[126],[23,58089],{"className":58090,"style":2770},[130],[23,58092,57783],{"className":58093},[135],[11,58095,58096,58097,58127,58128,58239,58240,58127,58270,58358],{},"在患病率 ",[23,58098,58100,58115],{"className":58099,"translate":27},[31],[23,58101,58103],{"className":58102},[35],[37,58104,58105],{"xmlns":39},[42,58106,58107,58113],{},[45,58108,58109,58111],{},[2581,58110,4435],{},[48,58112,56059],{"mathvariant":111},[114,58114,56062],{"encoding":116},[23,58116,58118],{"className":58117,"ariaHidden":122},[121],[23,58119,58121,58124],{"className":58120},[126],[23,58122],{"className":58123,"style":56072},[130],[23,58125,56076],{"className":58126},[135]," 时，",[23,58129,58131,58164],{"className":58130,"translate":27},[31],[23,58132,58134],{"className":58133},[35],[37,58135,58136],{"xmlns":39},[42,58137,58138,58161],{},[45,58139,58140,58142,58144,58146,58148,58150,58152,58154,58156,58158],{},[48,58141,920],{},[51,58143,54],{"stretchy":53},[48,58145,63],{},[51,58147,60],{},[51,58149,5106],{},[51,58151,66],{"stretchy":53},[51,58153,29070],{},[2581,58155,57336],{},[51,58157,2579],{},[2581,58159,58160],{},"0.091",[114,58162,58163],{"encoding":116},"P(D\\mid +)\\approx0.167>0.091",[23,58165,58167,58191,58212,58230],{"className":58166,"ariaHidden":122},[121],[23,58168,58170,58173,58176,58179,58182,58185,58188],{"className":58169},[126],[23,58171],{"className":58172,"style":131},[130],[23,58174,920],{"className":58175,"style":1047},[135,136],[23,58177,54],{"className":58178},[140],[23,58180,63],{"className":58181,"style":144},[135,136],[23,58183],{"className":58184,"style":149},[148],[23,58186,60],{"className":58187},[153],[23,58189],{"className":58190,"style":149},[148],[23,58192,58194,58197,58200,58203,58206,58209],{"className":58193},[126],[23,58195],{"className":58196,"style":131},[130],[23,58198,5106],{"className":58199},[135],[23,58201,66],{"className":58202},[169],[23,58204],{"className":58205,"style":149},[148],[23,58207,29070],{"className":58208},[153],[23,58210],{"className":58211,"style":149},[148],[23,58213,58215,58218,58221,58224,58227],{"className":58214},[126],[23,58216],{"className":58217,"style":52972},[130],[23,58219,57336],{"className":58220},[135],[23,58222],{"className":58223,"style":149},[148],[23,58225,2579],{"className":58226},[153],[23,58228],{"className":58229,"style":149},[148],[23,58231,58233,58236],{"className":58232},[126],[23,58234],{"className":58235,"style":2770},[130],[23,58237,58160],{"className":58238},[135],"，应当治疗；在患病率 ",[23,58241,58243,58258],{"className":58242,"translate":27},[31],[23,58244,58246],{"className":58245},[35],[37,58247,58248],{"xmlns":39},[42,58249,58250,58256],{},[45,58251,58252,58254],{},[2581,58253,51744],{},[48,58255,56059],{"mathvariant":111},[114,58257,57241],{"encoding":116},[23,58259,58261],{"className":58260,"ariaHidden":122},[121],[23,58262,58264,58267],{"className":58263},[126],[23,58265],{"className":58266,"style":56072},[130],[23,58268,57254],{"className":58269},[135],[23,58271,58273,58301],{"className":58272,"translate":27},[31],[23,58274,58276],{"className":58275},[35],[37,58277,58278],{"xmlns":39},[42,58279,58280,58298],{},[45,58281,58282,58284,58286,58288,58290,58292,58294,58296],{},[48,58283,920],{},[51,58285,54],{"stretchy":53},[48,58287,63],{},[51,58289,60],{},[51,58291,5106],{},[51,58293,66],{"stretchy":53},[51,58295,29070],{},[2581,58297,57271],{},[114,58299,58300],{"encoding":116},"P(D\\mid +)\\approx0.090",[23,58302,58304,58328,58349],{"className":58303,"ariaHidden":122},[121],[23,58305,58307,58310,58313,58316,58319,58322,58325],{"className":58306},[126],[23,58308],{"className":58309,"style":131},[130],[23,58311,920],{"className":58312,"style":1047},[135,136],[23,58314,54],{"className":58315},[140],[23,58317,63],{"className":58318,"style":144},[135,136],[23,58320],{"className":58321,"style":149},[148],[23,58323,60],{"className":58324},[153],[23,58326],{"className":58327,"style":149},[148],[23,58329,58331,58334,58337,58340,58343,58346],{"className":58330},[126],[23,58332],{"className":58333,"style":131},[130],[23,58335,5106],{"className":58336},[135],[23,58338,66],{"className":58339},[169],[23,58341],{"className":58342,"style":149},[148],[23,58344,29070],{"className":58345},[153],[23,58347],{"className":58348,"style":149},[148],[23,58350,58352,58355],{"className":58351},[126],[23,58353],{"className":58354,"style":2770},[130],[23,58356,57271],{"className":58357},[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。",[11,58360,58361],{},"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",[23,58363,58365],{"className":58364,"translate":27},[26],[23,58366,58368,58402],{"className":58367,"translate":27},[31],[23,58369,58371],{"className":58370},[35],[37,58372,58373],{"xmlns":39,"display":40},[42,58374,58375,58399],{},[45,58376,58377,58379,58381,58383,58385,58388,58390,58392,58394,58397],{},[2581,58378,56291],{},[51,58380,563],{},[2581,58382,56296],{},[51,58384,5106],{},[2581,58386,58387],{},"0.95",[51,58389,563],{},[2581,58391,56291],{},[51,58393,29070],{},[2581,58395,58396],{},"0.9504",[51,58398,747],{"separator":122},[114,58400,58401],{"encoding":116},"0.99\\times0.01+0.95\\times0.99\\approx0.9504,",[23,58403,58405,58423,58441,58459,58477],{"className":58404,"ariaHidden":122},[121],[23,58406,58408,58411,58414,58417,58420],{"className":58407},[126],[23,58409],{"className":58410,"style":9404},[130],[23,58412,56291],{"className":58413},[135],[23,58415],{"className":58416,"style":392},[148],[23,58418,563],{"className":58419},[612],[23,58421],{"className":58422,"style":392},[148],[23,58424,58426,58429,58432,58435,58438],{"className":58425},[126],[23,58427],{"className":58428,"style":9404},[130],[23,58430,56296],{"className":58431},[135],[23,58433],{"className":58434,"style":392},[148],[23,58436,5106],{"className":58437},[612],[23,58439],{"className":58440,"style":392},[148],[23,58442,58444,58447,58450,58453,58456],{"className":58443},[126],[23,58445],{"className":58446,"style":9404},[130],[23,58448,58387],{"className":58449},[135],[23,58451],{"className":58452,"style":392},[148],[23,58454,563],{"className":58455},[612],[23,58457],{"className":58458,"style":392},[148],[23,58460,58462,58465,58468,58471,58474],{"className":58461},[126],[23,58463],{"className":58464,"style":2770},[130],[23,58466,56291],{"className":58467},[135],[23,58469],{"className":58470,"style":149},[148],[23,58472,29070],{"className":58473},[153],[23,58475],{"className":58476,"style":149},[148],[23,58478,58480,58483,58486],{"className":58479},[126],[23,58481],{"className":58482,"style":51619},[130],[23,58484,58396],{"className":58485},[135],[23,58487,747],{"className":58488},[824],[11,58490,58491],{},"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。",[2501,58493,58495],{"id":58494},"_84-高斯两类问题","8.4 高斯两类问题",[11,58497,58498,58499,58580,58581,4136,58609,58637,58638,58689],{},"设两类共享方差 ",[23,58500,58502,58524],{"className":58501,"translate":27},[31],[23,58503,58505],{"className":58504},[35],[37,58506,58507],{"xmlns":39},[42,58508,58509,58521],{},[45,58510,58511,58517,58519],{},[4518,58512,58513,58515],{},[48,58514,29638],{},[2581,58516,9956],{},[51,58518,69],{},[2581,58520,4435],{},[114,58522,58523],{"encoding":116},"\\sigma^2=1",[23,58525,58527,58571],{"className":58526,"ariaHidden":122},[121],[23,58528,58530,58533,58562,58565,58568],{"className":58529},[126],[23,58531],{"className":58532,"style":4693},[130],[23,58534,58536,58539],{"className":58535},[135],[23,58537,29638],{"className":58538,"style":676},[135,136],[23,58540,58542],{"className":58541},[998],[23,58543,58545],{"className":58544},[199],[23,58546,58548],{"className":58547},[204],[23,58549,58551],{"className":58550,"style":4693},[208],[23,58552,58553,58556],{"style":4696},[23,58554],{"className":58555,"style":1015},[216],[23,58557,58559],{"className":58558},[1019,1020,1021,1022],[23,58560,9956],{"className":58561},[135,1022],[23,58563],{"className":58564,"style":149},[148],[23,58566,69],{"className":58567},[153],[23,58569],{"className":58570,"style":149},[148],[23,58572,58574,58577],{"className":58573},[126],[23,58575],{"className":58576,"style":2770},[130],[23,58578,4435],{"className":58579},[135],"，均值分别为 ",[23,58582,58584,58597],{"className":58583,"translate":27},[31],[23,58585,58587],{"className":58586},[35],[37,58588,58589],{"xmlns":39},[42,58590,58591,58595],{},[45,58592,58593],{},[2581,58594,1532],{},[114,58596,1532],{"encoding":116},[23,58598,58600],{"className":58599,"ariaHidden":122},[121],[23,58601,58603,58606],{"className":58602},[126],[23,58604],{"className":58605,"style":2770},[130],[23,58607,1532],{"className":58608},[135],[23,58610,58612,58625],{"className":58611,"translate":27},[31],[23,58613,58615],{"className":58614},[35],[37,58616,58617],{"xmlns":39},[42,58618,58619,58623],{},[45,58620,58621],{},[2581,58622,9956],{},[114,58624,9956],{"encoding":116},[23,58626,58628],{"className":58627,"ariaHidden":122},[121],[23,58629,58631,58634],{"className":58630},[126],[23,58632],{"className":58633,"style":2770},[130],[23,58635,9956],{"className":58636},[135],"，先验相等。则 ",[23,58639,58641,58659],{"className":58640,"translate":27},[31],[23,58642,58644],{"className":58643},[35],[37,58645,58646],{"xmlns":39},[42,58647,58648,58656],{},[45,58649,58650,58652,58654],{},[48,58651,23378],{"mathvariant":111},[51,58653,69],{},[2581,58655,9956],{},[114,58657,58658],{"encoding":116},"\\Delta=2",[23,58660,58662,58680],{"className":58661,"ariaHidden":122},[121],[23,58663,58665,58668,58671,58674,58677],{"className":58664},[126],[23,58666],{"className":58667,"style":470},[130],[23,58669,23378],{"className":58670},[135],[23,58672],{"className":58673,"style":149},[148],[23,58675,69],{"className":58676},[153],[23,58678],{"className":58679,"style":149},[148],[23,58681,58683,58686],{"className":58682},[126],[23,58684],{"className":58685,"style":2770},[130],[23,58687,9956],{"className":58688},[135],"，贝叶斯错误率为",[23,58691,58693],{"className":58692,"translate":27},[26],[23,58694,58696,58731],{"className":58695,"translate":27},[31],[23,58697,58699],{"className":58698},[35],[37,58700,58701],{"xmlns":39,"display":40},[42,58702,58703,58728],{},[45,58704,58705,58713,58715,58717,58719,58721,58723,58725],{},[19141,58706,58707,58709,58711],{},[48,58708,11],{},[48,58710,5342],{},[51,58712,6883],{},[51,58714,69],{},[48,58716,22055],{},[51,58718,54],{"stretchy":53},[2581,58720,4435],{},[51,58722,66],{"stretchy":53},[51,58724,29070],{},[2581,58726,58727],{},"0.1587.",[114,58729,58730],{"encoding":116},"p_e^*=Q(1)\\approx0.1587.",[23,58732,58734,58800,58827],{"className":58733,"ariaHidden":122},[121],[23,58735,58737,58740,58791,58794,58797],{"className":58736},[126],[23,58738],{"className":58739,"style":20726},[130],[23,58741,58743,58746],{"className":58742},[135],[23,58744,11],{"className":58745},[135,136],[23,58747,58749],{"className":58748},[998],[23,58750,58752,58783],{"className":58751},[199,200],[23,58753,58755,58780],{"className":58754},[204],[23,58756,58758,58769],{"className":58757,"style":7047},[208],[23,58759,58760,58763],{"style":19349},[23,58761],{"className":58762,"style":1015},[216],[23,58764,58766],{"className":58765},[1019,1020,1021,1022],[23,58767,5342],{"className":58768},[135,136,1022],[23,58770,58771,58774],{"style":4989},[23,58772],{"className":58773,"style":1015},[216],[23,58775,58777],{"className":58776},[1019,1020,1021,1022],[23,58778,6883],{"className":58779},[612,1022],[23,58781,297],{"className":58782},[296],[23,58784,58786],{"className":58785},[204],[23,58787,58789],{"className":58788,"style":19379},[208],[23,58790],{},[23,58792],{"className":58793,"style":149},[148],[23,58795,69],{"className":58796},[153],[23,58798],{"className":58799,"style":149},[148],[23,58801,58803,58806,58809,58812,58815,58818,58821,58824],{"className":58802},[126],[23,58804],{"className":58805,"style":131},[130],[23,58807,22055],{"className":58808},[135,136],[23,58810,54],{"className":58811},[140],[23,58813,4435],{"className":58814},[135],[23,58816,66],{"className":58817},[169],[23,58819],{"className":58820,"style":149},[148],[23,58822,29070],{"className":58823},[153],[23,58825],{"className":58826,"style":149},[148],[23,58828,58830,58833],{"className":58829},[126],[23,58831],{"className":58832,"style":2770},[130],[23,58834,58727],{"className":58835},[135],[11,58837,58838,58839,8059,58867,58918,58919,58985,58986,59092],{},"若把均值差扩大到 ",[23,58840,58842,58855],{"className":58841,"translate":27},[31],[23,58843,58845],{"className":58844},[35],[37,58846,58847],{"xmlns":39},[42,58848,58849,58853],{},[45,58850,58851],{},[2581,58852,24934],{},[114,58854,24934],{"encoding":116},[23,58856,58858],{"className":58857,"ariaHidden":122},[121],[23,58859,58861,58864],{"className":58860},[126],[23,58862],{"className":58863,"style":2770},[130],[23,58865,24934],{"className":58866},[135],[23,58868,58870,58888],{"className":58869,"translate":27},[31],[23,58871,58873],{"className":58872},[35],[37,58874,58875],{"xmlns":39},[42,58876,58877,58885],{},[45,58878,58879,58881,58883],{},[48,58880,23378],{"mathvariant":111},[51,58882,69],{},[2581,58884,24934],{},[114,58886,58887],{"encoding":116},"\\Delta=4",[23,58889,58891,58909],{"className":58890,"ariaHidden":122},[121],[23,58892,58894,58897,58900,58903,58906],{"className":58893},[126],[23,58895],{"className":58896,"style":470},[130],[23,58898,23378],{"className":58899},[135],[23,58901],{"className":58902,"style":149},[148],[23,58904,69],{"className":58905},[153],[23,58907],{"className":58908,"style":149},[148],[23,58910,58912,58915],{"className":58911},[126],[23,58913],{"className":58914,"style":2770},[130],[23,58916,24934],{"className":58917},[135],"，错误率降为 ",[23,58920,58922,58946],{"className":58921,"translate":27},[31],[23,58923,58925],{"className":58924},[35],[37,58926,58927],{"xmlns":39},[42,58928,58929,58943],{},[45,58930,58931,58933,58935,58937,58939,58941],{},[48,58932,22055],{},[51,58934,54],{"stretchy":53},[2581,58936,9956],{},[51,58938,66],{"stretchy":53},[51,58940,29070],{},[2581,58942,24937],{},[114,58944,58945],{"encoding":116},"Q(2)\\approx0.0228",[23,58947,58949,58976],{"className":58948,"ariaHidden":122},[121],[23,58950,58952,58955,58958,58961,58964,58967,58970,58973],{"className":58951},[126],[23,58953],{"className":58954,"style":131},[130],[23,58956,22055],{"className":58957},[135,136],[23,58959,54],{"className":58960},[140],[23,58962,9956],{"className":58963},[135],[23,58965,66],{"className":58966},[169],[23,58968],{"className":58969,"style":149},[148],[23,58971,29070],{"className":58972},[153],[23,58974],{"className":58975,"style":149},[148],[23,58977,58979,58982],{"className":58978},[126],[23,58980],{"className":58981,"style":2770},[130],[23,58983,24937],{"className":58984},[135],"。错误率随马氏距离以 ",[23,58987,58989,59019],{"className":58988,"translate":27},[31],[23,58990,58992],{"className":58991},[35],[37,58993,58994],{"xmlns":39},[42,58995,58996,59016],{},[45,58997,58998],{},[4518,58999,59000,59002],{},[48,59001,5342],{},[45,59003,59004,59006,59012,59014],{},[51,59005,4529],{},[4518,59007,59008,59010],{},[48,59009,23378],{"mathvariant":111},[2581,59011,9956],{},[48,59013,2875],{"mathvariant":111},[2581,59015,24745],{},[114,59017,59018],{"encoding":116},"e^{-\\Delta^2\u002F8}",[23,59020,59022],{"className":59021,"ariaHidden":122},[121],[23,59023,59025,59028],{"className":59024},[126],[23,59026],{"className":59027,"style":24845},[130],[23,59029,59031,59034],{"className":59030},[135],[23,59032,5342],{"className":59033},[135,136],[23,59035,59037],{"className":59036},[998],[23,59038,59040],{"className":59039},[199],[23,59041,59043],{"className":59042},[204],[23,59044,59046],{"className":59045,"style":24845},[208],[23,59047,59048,59051],{"style":4696},[23,59049],{"className":59050,"style":1015},[216],[23,59052,59054],{"className":59053},[1019,1020,1021,1022],[23,59055,59057,59060,59089],{"className":59056},[135,1022],[23,59058,4529],{"className":59059},[135,1022],[23,59061,59063,59066],{"className":59062},[135,1022],[23,59064,23378],{"className":59065},[135,1022],[23,59067,59069],{"className":59068},[998],[23,59070,59072],{"className":59071},[199],[23,59073,59075],{"className":59074},[204],[23,59076,59078],{"className":59077,"style":24878},[208],[23,59079,59080,59083],{"style":24881},[23,59081],{"className":59082,"style":24885},[216],[23,59084,59086],{"className":59085},[1019,24889,1067,1022],[23,59087,9956],{"className":59088},[135,1022],[23,59090,24896],{"className":59091},[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。",[11,59094,59095],{},"Bhattacharyya 上界给出",[23,59097,59099],{"className":59098,"translate":27},[26],[23,59100,59102,59150],{"className":59101,"translate":27},[31],[23,59103,59105],{"className":59104},[35],[37,59106,59107],{"xmlns":39,"display":40},[42,59108,59109,59147],{},[45,59110,59111,59119,59121,59127,59145],{},[19141,59112,59113,59115,59117],{},[48,59114,11],{},[48,59116,5342],{},[51,59118,6883],{},[51,59120,12272],{},[71,59122,59123,59125],{},[2581,59124,4435],{},[2581,59126,9956],{},[4518,59128,59129,59131],{},[48,59130,5342],{},[45,59132,59133,59135,59141,59143],{},[51,59134,4529],{},[4518,59136,59137,59139],{},[48,59138,23378],{"mathvariant":111},[2581,59140,9956],{},[48,59142,2875],{"mathvariant":111},[2581,59144,24745],{},[51,59146,747],{"separator":122},[114,59148,59149],{"encoding":116},"p_e^*\\le\\frac12e^{-\\Delta^2\u002F8},",[23,59151,59153,59219],{"className":59152,"ariaHidden":122},[121],[23,59154,59156,59159,59210,59213,59216],{"className":59155},[126],[23,59157],{"className":59158,"style":20726},[130],[23,59160,59162,59165],{"className":59161},[135],[23,59163,11],{"className":59164},[135,136],[23,59166,59168],{"className":59167},[998],[23,59169,59171,59202],{"className":59170},[199,200],[23,59172,59174,59199],{"className":59173},[204],[23,59175,59177,59188],{"className":59176,"style":7047},[208],[23,59178,59179,59182],{"style":19349},[23,59180],{"className":59181,"style":1015},[216],[23,59183,59185],{"className":59184},[1019,1020,1021,1022],[23,59186,5342],{"className":59187},[135,136,1022],[23,59189,59190,59193],{"style":4989},[23,59191],{"className":59192,"style":1015},[216],[23,59194,59196],{"className":59195},[1019,1020,1021,1022],[23,59197,6883],{"className":59198},[612,1022],[23,59200,297],{"className":59201},[296],[23,59203,59205],{"className":59204},[204],[23,59206,59208],{"className":59207,"style":19379},[208],[23,59209],{},[23,59211],{"className":59212,"style":149},[148],[23,59214,12272],{"className":59215},[153],[23,59217],{"className":59218,"style":149},[148],[23,59220,59222,59225,59287,59351],{"className":59221},[126],[23,59223],{"className":59224,"style":12394},[130],[23,59226,59228,59231,59284],{"className":59227},[135],[23,59229],{"className":59230},[140,192],[23,59232,59234],{"className":59233},[71],[23,59235,59237,59276],{"className":59236},[199,200],[23,59238,59240,59273],{"className":59239},[204],[23,59241,59243,59254,59262],{"className":59242,"style":12413},[208],[23,59244,59245,59248],{"style":212},[23,59246],{"className":59247,"style":217},[216],[23,59249,59251],{"className":59250},[135],[23,59252,9956],{"className":59253},[135],[23,59255,59256,59259],{"style":235},[23,59257],{"className":59258,"style":217},[216],[23,59260],{"className":59261,"style":243},[242],[23,59263,59264,59267],{"style":246},[23,59265],{"className":59266,"style":217},[216],[23,59268,59270],{"className":59269},[135],[23,59271,4435],{"className":59272},[135],[23,59274,297],{"className":59275},[296],[23,59277,59279],{"className":59278},[204],[23,59280,59282],{"className":59281,"style":6287},[208],[23,59283],{},[23,59285],{"className":59286},[169,192],[23,59288,59290,59293],{"className":59289},[135],[23,59291,5342],{"className":59292},[135,136],[23,59294,59296],{"className":59295},[998],[23,59297,59299],{"className":59298},[199],[23,59300,59302],{"className":59301},[204],[23,59303,59305],{"className":59304,"style":26145},[208],[23,59306,59307,59310],{"style":4989},[23,59308],{"className":59309,"style":1015},[216],[23,59311,59313],{"className":59312},[1019,1020,1021,1022],[23,59314,59316,59319,59348],{"className":59315},[135,1022],[23,59317,4529],{"className":59318},[135,1022],[23,59320,59322,59325],{"className":59321},[135,1022],[23,59323,23378],{"className":59324},[135,1022],[23,59326,59328],{"className":59327},[998],[23,59329,59331],{"className":59330},[199],[23,59332,59334],{"className":59333},[204],[23,59335,59337],{"className":59336,"style":24878},[208],[23,59338,59339,59342],{"style":24881},[23,59340],{"className":59341,"style":24885},[216],[23,59343,59345],{"className":59344},[1019,24889,1067,1022],[23,59346,9956],{"className":59347},[135,1022],[23,59349,24896],{"className":59350},[135,1022],[23,59352,747],{"className":59353},[824],[11,59355,59356,59357,59407,59408,59437,59438,59407,59488,59516],{},"在 ",[23,59358,59360,59377],{"className":59359,"translate":27},[31],[23,59361,59363],{"className":59362},[35],[37,59364,59365],{"xmlns":39},[42,59366,59367,59375],{},[45,59368,59369,59371,59373],{},[48,59370,23378],{"mathvariant":111},[51,59372,69],{},[2581,59374,9956],{},[114,59376,58658],{"encoding":116},[23,59378,59380,59398],{"className":59379,"ariaHidden":122},[121],[23,59381,59383,59386,59389,59392,59395],{"className":59382},[126],[23,59384],{"className":59385,"style":470},[130],[23,59387,23378],{"className":59388},[135],[23,59390],{"className":59391,"style":149},[148],[23,59393,69],{"className":59394},[153],[23,59396],{"className":59397,"style":149},[148],[23,59399,59401,59404],{"className":59400},[126],[23,59402],{"className":59403,"style":2770},[130],[23,59405,9956],{"className":59406},[135]," 时为 ",[23,59409,59411,59425],{"className":59410,"translate":27},[31],[23,59412,59414],{"className":59413},[35],[37,59415,59416],{"xmlns":39},[42,59417,59418,59423],{},[45,59419,59420],{},[2581,59421,59422],{},"0.303",[114,59424,59422],{"encoding":116},[23,59426,59428],{"className":59427,"ariaHidden":122},[121],[23,59429,59431,59434],{"className":59430},[126],[23,59432],{"className":59433,"style":2770},[130],[23,59435,59422],{"className":59436},[135],"，在 ",[23,59439,59441,59458],{"className":59440,"translate":27},[31],[23,59442,59444],{"className":59443},[35],[37,59445,59446],{"xmlns":39},[42,59447,59448,59456],{},[45,59449,59450,59452,59454],{},[48,59451,23378],{"mathvariant":111},[51,59453,69],{},[2581,59455,24934],{},[114,59457,58887],{"encoding":116},[23,59459,59461,59479],{"className":59460,"ariaHidden":122},[121],[23,59462,59464,59467,59470,59473,59476],{"className":59463},[126],[23,59465],{"className":59466,"style":470},[130],[23,59468,23378],{"className":59469},[135],[23,59471],{"className":59472,"style":149},[148],[23,59474,69],{"className":59475},[153],[23,59477],{"className":59478,"style":149},[148],[23,59480,59482,59485],{"className":59481},[126],[23,59483],{"className":59484,"style":2770},[130],[23,59486,24934],{"className":59487},[135],[23,59489,59491,59504],{"className":59490,"translate":27},[31],[23,59492,59494],{"className":59493},[35],[37,59495,59496],{"xmlns":39},[42,59497,59498,59502],{},[45,59499,59500],{},[2581,59501,24940],{},[114,59503,24940],{"encoding":116},[23,59505,59507],{"className":59506,"ariaHidden":122},[121],[23,59508,59510,59513],{"className":59509},[126],[23,59511],{"className":59512,"style":2770},[130],[23,59514,24940],{"className":59515},[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」。",[338,59518,59519],{"id":59519},"结语",[11,59521,59522],{},"把全文的线索压缩成一行：",[23,59524,59526],{"className":59525,"translate":27},[26],[23,59527,59529,59695],{"className":59528,"translate":27},[31],[23,59530,59532],{"className":59531},[35],[37,59533,59534],{"xmlns":39,"display":40},[42,59535,59536,59692],{},[45,59537,59538,59556,59558,59580,59582,59584,59587,59589,59611,59613,59615,59617,59619,59667,59669,59671,59673,59675,59690],{},[6963,59539,59540,59554],{},[6963,59541,59542,59552],{},[45,59543,59544,59546,59548,59550],{},[48,59545,11],{},[51,59547,54],{"stretchy":53},[48,59549,57],{},[51,59551,66],{"stretchy":53},[51,59553,8481],{"stretchy":122},[88,59555,3790],{},[51,59557,563],{},[6963,59559,59560,59578],{},[6963,59561,59562,59576],{},[45,59563,59564,59566,59568,59570,59572,59574],{},[48,59565,11],{},[51,59567,54],{"stretchy":53},[48,59569,63],{},[51,59571,60],{},[48,59573,57],{},[51,59575,66],{"stretchy":53},[51,59577,8481],{"stretchy":122},[88,59579,3844],{},[88,59581,5360],{},[148,59583],{"linebreak":29629},[51,59585,59586],{},"⟶",[88,59588,5360],{},[6963,59590,59591,59609],{},[6963,59592,59593,59607],{},[45,59594,59595,59597,59599,59601,59603,59605],{},[48,59596,11],{},[51,59598,54],{"stretchy":53},[48,59600,57],{},[51,59602,60],{},[48,59604,63],{},[51,59606,66],{"stretchy":53},[51,59608,8481],{"stretchy":122},[88,59610,4003],{},[88,59612,5360],{},[148,59614],{"linebreak":29629},[51,59616,59586],{},[88,59618,5360],{},[6963,59620,59621,59665],{},[6963,59622,59623,59663],{},[45,59624,59625,59627,59629,59639,59641,59643,59645,59647,59649,59651,59653,59655,59657,59659,59661],{},[48,59626,6958],{},[51,59628,6961],{},[6963,59630,59631,59637],{},[45,59632,59633,59635],{},[48,59634,6969],{},[51,59636,6961],{},[48,59638,495],{},[88,59640,5360],{},[48,59642,902],{"mathvariant":571},[51,59644,924],{"stretchy":53},[48,59646,555],{},[51,59648,54],{"stretchy":53},[48,59650,369],{},[51,59652,747],{"separator":122},[48,59654,495],{},[51,59656,66],{"stretchy":53},[51,59658,60],{},[48,59660,63],{},[51,59662,945],{"stretchy":53},[51,59664,8481],{"stretchy":122},[88,59666,480],{},[88,59668,5360],{},[148,59670],{"linebreak":29629},[51,59672,59586],{},[88,59674,5360],{},[6963,59676,59677,59687],{},[6963,59678,59679,59685],{},[4518,59680,59681,59683],{},[48,59682,572],{},[51,59684,6883],{},[51,59686,8481],{"stretchy":122},[88,59688,59689],{},"下限",[48,59691,112],{"mathvariant":111},[114,59693,59694],{"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}}.",[23,59696,59698,59836,59975,59978,59994,60133,60136,60151,60369,60372,60387],{"className":59697,"ariaHidden":122},[121],[23,59699,59701,59705,59827,59830,59833],{"className":59700},[126],[23,59702],{"className":59703,"style":59704},[130],"height:2.3263em;vertical-align:-1.5763em;",[23,59706,59708],{"className":59707},[1839,6963],[23,59709,59711,59818],{"className":59710},[199,200],[23,59712,59714,59815],{"className":59713},[204],[23,59715,59717,59735],{"className":59716,"style":8637},[208],[23,59718,59720,59723],{"style":59719},"top:-1.4237em;",[23,59721],{"className":59722,"style":217},[216],[23,59724,59726],{"className":59725},[1019,1020,1021,1022],[23,59727,59729],{"className":59728},[135,1022],[23,59730,59732],{"className":59731},[135,6514,1022],[23,59733,3790],{"className":59734},[135,7240,1022],[23,59736,59737,59740],{"style":7132},[23,59738],{"className":59739,"style":217},[216],[23,59741,59743],{"className":59742},[1839,6963],[23,59744,59746,59806],{"className":59745},[199,200],[23,59747,59749,59803],{"className":59748},[204],[23,59750,59752,59783],{"className":59751,"style":8637},[208],[23,59753,59756,59759],{"className":59754,"style":59755},[8676],"top:-2.102em;",[23,59757],{"className":59758,"style":217},[216],[23,59760,59762,59769,59776],{"className":59761,"style":8685},[8684],[23,59763,59765],{"className":59764,"style":8690},[8689],[8692,59766,59767],{"xmlns":8694,"width":8695,"height":8696,"viewBox":8697,"preserveAspectRatio":8698},[8700,59768],{"d":8702},[23,59770,59772],{"className":59771,"style":8690},[8706],[8692,59773,59774],{"xmlns":8694,"width":8695,"height":8696,"viewBox":8697,"preserveAspectRatio":8709},[8700,59775],{"d":8712},[23,59777,59779],{"className":59778,"style":8690},[8716],[8692,59780,59781],{"xmlns":8694,"width":8695,"height":8696,"viewBox":8697,"preserveAspectRatio":8719},[8700,59782],{"d":8722},[23,59784,59785,59788],{"style":7132},[23,59786],{"className":59787,"style":217},[216],[23,59789,59791,59794,59797,59800],{"className":59790},[135],[23,59792,11],{"className":59793},[135,136],[23,59795,54],{"className":59796},[140],[23,59798,57],{"className":59799,"style":144},[135,136],[23,59801,66],{"className":59802},[169],[23,59804,297],{"className":59805},[296],[23,59807,59809],{"className":59808},[204],[23,59810,59813],{"className":59811,"style":59812},[208],"height:0.898em;",[23,59814],{},[23,59816,297],{"className":59817},[296],[23,59819,59821],{"className":59820},[204],[23,59822,59825],{"className":59823,"style":59824},[208],"height:1.5763em;",[23,59826],{},[23,59828],{"className":59829,"style":392},[148],[23,59831,563],{"className":59832},[612],[23,59834],{"className":59835,"style":392},[148],[23,59837,59839,59842,59972],{"className":59838},[126],[23,59840],{"className":59841,"style":59704},[130],[23,59843,59845],{"className":59844},[1839,6963],[23,59846,59848,59964],{"className":59847},[199,200],[23,59849,59851,59961],{"className":59850},[204],[23,59852,59854,59871],{"className":59853,"style":8637},[208],[23,59855,59856,59859],{"style":59719},[23,59857],{"className":59858,"style":217},[216],[23,59860,59862],{"className":59861},[1019,1020,1021,1022],[23,59863,59865],{"className":59864},[135,1022],[23,59866,59868],{"className":59867},[135,6514,1022],[23,59869,3844],{"className":59870},[135,7240,1022],[23,59872,59873,59876],{"style":7132},[23,59874],{"className":59875,"style":217},[216],[23,59877,59879],{"className":59878},[1839,6963],[23,59880,59882,59953],{"className":59881},[199,200],[23,59883,59885,59950],{"className":59884},[204],[23,59886,59888,59918],{"className":59887,"style":8637},[208],[23,59889,59891,59894],{"className":59890,"style":59755},[8676],[23,59892],{"className":59893,"style":217},[216],[23,59895,59897,59904,59911],{"className":59896,"style":8685},[8684],[23,59898,59900],{"className":59899,"style":8690},[8689],[8692,59901,59902],{"xmlns":8694,"width":8695,"height":8696,"viewBox":8697,"preserveAspectRatio":8698},[8700,59903],{"d":8702},[23,59905,59907],{"className":59906,"style":8690},[8706],[8692,59908,59909],{"xmlns":8694,"width":8695,"height":8696,"viewBox":8697,"preserveAspectRatio":8709},[8700,59910],{"d":8712},[23,59912,59914],{"className":59913,"style":8690},[8716],[8692,59915,59916],{"xmlns":8694,"width":8695,"height":8696,"viewBox":8697,"preserveAspectRatio":8719},[8700,59917],{"d":8722},[23,59919,59920,59923],{"style":7132},[23,59921],{"className":59922,"style":217},[216],[23,59924,59926,59929,59932,59935,59938,59941,59944,59947],{"className":59925},[135],[23,59927,11],{"className":59928},[135,136],[23,59930,54],{"className":59931},[140],[23,59933,63],{"className":59934,"style":144},[135,136],[23,59936],{"className":59937,"style":149},[148],[23,59939,60],{"className":59940},[153],[23,59942],{"className":59943,"style":149},[148],[23,59945,57],{"className":59946,"style":144},[135,136],[23,59948,66],{"className":59949},[169],[23,59951,297],{"className":59952},[296],[23,59954,59956],{"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。",[338,60695,60697],{"id":60696},"一无穷维优化","一、无穷维优化",[11,60699,60700,60701,60802,60803,60831,60832,60860],{},"线性模型 ",[23,60702,60704,60734],{"className":60703,"translate":27},[31],[23,60705,60707],{"className":60706},[35],[37,60708,60709],{"xmlns":39},[42,60710,60711,60731],{},[45,60712,60713,60715,60717,60719,60721,60723,60729],{},[48,60714,8340],{},[51,60716,54],{"stretchy":53},[48,60718,753],{},[51,60720,66],{"stretchy":53},[51,60722,69],{},[4518,60724,60725,60727],{},[48,60726,49700],{},[48,60728,22475],{"mathvariant":111},[48,60730,753],{},[114,60732,60733],{"encoding":116},"f(x)=w^\\top x",[23,60735,60737,60764],{"className":60736,"ariaHidden":122},[121],[23,60738,60740,60743,60746,60749,60752,60755,60758,60761],{"className":60739},[126],[23,60741],{"className":60742,"style":131},[130],[23,60744,8340],{"className":60745,"style":11402},[135,136],[23,60747,54],{"className":60748},[140],[23,60750,753],{"className":60751},[135,136],[23,60753,66],{"className":60754},[169],[23,60756],{"className":60757,"style":149},[148],[23,60759,69],{"className":60760},[153],[23,60762],{"className":60763,"style":149},[148],[23,60765,60767,60770,60799],{"className":60766},[126],[23,60768],{"className":60769,"style":4751},[130],[23,60771,60773,60776],{"className":60772},[135],[23,60774,49700],{"className":60775,"style":49715},[135,136],[23,60777,60779],{"className":60778},[998],[23,60780,60782],{"className":60781},[199],[23,60783,60785],{"className":60784},[204],[23,60786,60788],{"className":60787,"style":4751},[208],[23,60789,60790,60793],{"style":4696},[23,60791],{"className":60792,"style":1015},[216],[23,60794,60796],{"className":60795},[1019,1020,1021,1022],[23,60797,22475],{"className":60798},[135,1022],[23,60800,753],{"className":60801},[135,136]," 的表达能力受限于特征 ",[23,60804,60806,60819],{"className":60805,"translate":27},[31],[23,60807,60809],{"className":60808},[35],[37,60810,60811],{"xmlns":39},[42,60812,60813,60817],{},[45,60814,60815],{},[48,60816,753],{},[114,60818,753],{"encoding":116},[23,60820,60822],{"className":60821,"ariaHidden":122},[121],[23,60823,60825,60828],{"className":60824},[126],[23,60826],{"className":60827,"style":703},[130],[23,60829,753],{"className":60830},[135,136]," 的坐标。想拟合非线性关系，一个标准做法是先把 ",[23,60833,60835,60848],{"className":60834,"translate":27},[31],[23,60836,60838],{"className":60837},[35],[37,60839,60840],{"xmlns":39},[42,60841,60842,60846],{},[45,60843,60844],{},[48,60845,753],{},[114,60847,753],{"encoding":116},[23,60849,60851],{"className":60850,"ariaHidden":122},[121],[23,60852,60854,60857],{"className":60853},[126],[23,60855],{"className":60856,"style":703},[130],[23,60858,753],{"className":60859},[135,136]," 映射到一个更大的特征空间：",[23,60862,60864],{"className":60863,"translate":27},[26],[23,60865,60867,60906],{"className":60866,"translate":27},[31],[23,60868,60870],{"className":60869},[35],[37,60871,60872],{"xmlns":39,"display":40},[42,60873,60874,60903],{},[45,60875,60876,60878,60880,60882,60884,60886,60892,60895,60897,60899,60901],{},[48,60877,8340],{},[51,60879,54],{"stretchy":53},[48,60881,753],{},[51,60883,66],{"stretchy":53},[51,60885,69],{},[4518,60887,60888,60890],{},[48,60889,49700],{},[48,60891,22475],{"mathvariant":111},[48,60893,60894],{},"φ",[51,60896,54],{"stretchy":53},[48,60898,753],{},[51,60900,66],{"stretchy":53},[51,60902,747],{"separator":122},[114,60904,60905],{"encoding":116},"f(x)=w^\\top \\varphi(x),",[23,60907,60909,60936],{"className":60908,"ariaHidden":122},[121],[23,60910,60912,60915,60918,60921,60924,60927,60930,60933],{"className":60911},[126],[23,60913],{"className":60914,"style":131},[130],[23,60916,8340],{"className":60917,"style":11402},[135,136],[23,60919,54],{"className":60920},[140],[23,60922,753],{"className":60923},[135,136],[23,60925,66],{"className":60926},[169],[23,60928],{"className":60929,"style":149},[148],[23,60931,69],{"className":60932},[153],[23,60934],{"className":60935,"style":149},[148],[23,60937,60939,60942,60971,60974,60977,60980,60983],{"className":60938},[126],[23,60940],{"className":60941,"style":4967},[130],[23,60943,60945,60948],{"className":60944},[135],[23,60946,49700],{"className":60947,"style":49715},[135,136],[23,60949,60951],{"className":60950},[998],[23,60952,60954],{"className":60953},[199],[23,60955,60957],{"className":60956},[204],[23,60958,60960],{"className":60959,"style":4986},[208],[23,60961,60962,60965],{"style":4989},[23,60963],{"className":60964,"style":1015},[216],[23,60966,60968],{"className":60967},[1019,1020,1021,1022],[23,60969,22475],{"className":60970},[135,1022],[23,60972,60894],{"className":60973},[135,136],[23,60975,54],{"className":60976},[140],[23,60978,753],{"className":60979},[135,136],[23,60981,66],{"className":60982},[169],[23,60984,747],{"className":60985},[824],[11,60987,24433,60988,61091,61092,61121,61122,61150],{},[23,60989,60991,61017],{"className":60990,"translate":27},[31],[23,60992,60994],{"className":60993},[35],[37,60995,60996],{"xmlns":39},[42,60997,60998,61014],{},[45,60999,61000,61002,61004,61006,61008],{},[48,61001,60894],{},[51,61003,558],{},[48,61005,434],{"mathvariant":375},[51,61007,568],{},[4518,61009,61010,61012],{},[48,61011,572],{"mathvariant":571},[48,61013,1606],{},[114,61015,61016],{"encoding":116},"\\varphi:\\mathcal X\\to\\mathbb R^d",[23,61018,61020,61038,61056],{"className":61019,"ariaHidden":122},[121],[23,61021,61023,61026,61029,61032,61035],{"className":61022},[126],[23,61024],{"className":61025,"style":672},[130],[23,61027,60894],{"className":61028},[135,136],[23,61030],{"className":61031,"style":149},[148],[23,61033,558],{"className":61034},[153],[23,61036],{"className":61037,"style":149},[148],[23,61039,61041,61044,61047,61050,61053],{"className":61040},[126],[23,61042],{"className":61043,"style":470},[130],[23,61045,434],{"className":61046,"style":474},[135,412],[23,61048],{"className":61049,"style":149},[148],[23,61051,568],{"className":61052},[153],[23,61054],{"className":61055,"style":149},[148],[23,61057,61059,61062],{"className":61058},[126],[23,61060],{"className":61061,"style":4751},[130],[23,61063,61065,61068],{"className":61064},[135],[23,61066,572],{"className":61067},[135,644],[23,61069,61071],{"className":61070},[998],[23,61072,61074],{"className":61073},[199],[23,61075,61077],{"className":61076},[204],[23,61078,61080],{"className":61079,"style":4751},[208],[23,61081,61082,61085],{"style":4696},[23,61083],{"className":61084,"style":1015},[216],[23,61086,61088],{"className":61087},[1019,1020,1021,1022],[23,61089,1606],{"className":61090},[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，",[23,61093,61095,61109],{"className":61094,"translate":27},[31],[23,61096,61098],{"className":61097},[35],[37,61099,61100],{"xmlns":39},[42,61101,61102,61106],{},[45,61103,61104],{},[48,61105,60894],{},[114,61107,61108],{"encoding":116},"\\varphi",[23,61110,61112],{"className":61111,"ariaHidden":122},[121],[23,61113,61115,61118],{"className":61114},[126],[23,61116],{"className":61117,"style":672},[130],[23,61119,60894],{"className":61120},[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 ",[23,61123,61125,61138],{"className":61124,"translate":27},[31],[23,61126,61128],{"className":61127},[35],[37,61129,61130],{"xmlns":39},[42,61131,61132,61136],{},[45,61133,61134],{},[48,61135,60894],{},[114,61137,61108],{"encoding":116},[23,61139,61141],{"className":61140,"ariaHidden":122},[121],[23,61142,61144,61147],{"className":61143},[126],[23,61145],{"className":61146,"style":672},[130],[23,61148,60894],{"className":61149},[135,136],"，优化根本无从下手。",[11,61152,61153],{},"核方法绕开了显式特征：它只通过内积",[23,61155,61157],{"className":61156,"translate":27},[26],[23,61158,61160,61218],{"className":61159,"translate":27},[31],[23,61161,61163],{"className":61162},[35],[37,61164,61165],{"xmlns":39,"display":40},[42,61166,61167,61215],{},[45,61168,61169,61171,61173,61175,61177,61183,61185,61187,61190,61192,61194,61196,61198,61200,61202,61204,61210,61212],{},[48,61170,4836],{},[51,61172,54],{"stretchy":53},[48,61174,753],{},[51,61176,747],{"separator":122},[4518,61178,61179,61181],{},[48,61180,753],{},[51,61182,38295],{"mathvariant":111,"lspace":1522,"rspace":1522},[51,61184,66],{"stretchy":53},[51,61186,69],{},[51,61188,61189],{"stretchy":53},"⟨",[48,61191,60894],{},[51,61193,54],{"stretchy":53},[48,61195,753],{},[51,61197,66],{"stretchy":53},[51,61199,747],{"separator":122},[48,61201,60894],{},[51,61203,54],{"stretchy":53},[4518,61205,61206,61208],{},[48,61207,753],{},[51,61209,38295],{"mathvariant":111,"lspace":1522,"rspace":1522},[51,61211,66],{"stretchy":53},[51,61213,61214],{"stretchy":53},"⟩",[114,61216,61217],{"encoding":116},"k(x,x')=\\langle \\varphi(x),\\varphi(x')\\rangle",[23,61219,61221,61286],{"className":61220,"ariaHidden":122},[121],[23,61222,61224,61227,61230,61233,61236,61239,61242,61274,61277,61280,61283],{"className":61223},[126],[23,61225],{"className":61226,"style":51202},[130],[23,61228,4836],{"className":61229,"style":4851},[135,136],[23,61231,54],{"className":61232},[140],[23,61234,753],{"className":61235},[135,136],[23,61237,747],{"className":61238},[824],[23,61240],{"className":61241,"style":280},[148],[23,61243,61245,61248],{"className":61244},[135],[23,61246,753],{"className":61247},[135,136],[23,61249,61251],{"className":61250},[998],[23,61252,61254],{"className":61253},[199],[23,61255,61257],{"className":61256},[204],[23,61258,61260],{"className":61259,"style":51227},[208],[23,61261,61262,61265],{"style":4989},[23,61263],{"className":61264,"style":1015},[216],[23,61266,61268],{"className":61267},[1019,1020,1021,1022],[23,61269,61271],{"className":61270},[135,1022],[23,61272,38295],{"className":61273},[135,1022],[23,61275,66],{"className":61276},[169],[23,61278],{"className":61279,"style":149},[148],[23,61281,69],{"className":61282},[153],[23,61284],{"className":61285,"style":149},[148],[23,61287,61289,61292,61295,61298,61301,61304,61307,61310,61313,61316,61319,61351],{"className":61288},[126],[23,61290],{"className":61291,"style":51202},[130],[23,61293,61189],{"className":61294},[140],[23,61296,60894],{"className":61297},[135,136],[23,61299,54],{"className":61300},[140],[23,61302,753],{"className":61303},[135,136],[23,61305,66],{"className":61306},[169],[23,61308,747],{"className":61309},[824],[23,61311],{"className":61312,"style":280},[148],[23,61314,60894],{"className":61315},[135,136],[23,61317,54],{"className":61318},[140],[23,61320,61322,61325],{"className":61321},[135],[23,61323,753],{"className":61324},[135,136],[23,61326,61328],{"className":61327},[998],[23,61329,61331],{"className":61330},[199],[23,61332,61334],{"className":61333},[204],[23,61335,61337],{"className":61336,"style":51227},[208],[23,61338,61339,61342],{"style":4989},[23,61340],{"className":61341,"style":1015},[216],[23,61343,61345],{"className":61344},[1019,1020,1021,1022],[23,61346,61348],{"className":61347},[135,1022],[23,61349,38295],{"className":61350},[135,1022],[23,61352,61354],{"className":61353},[169],")⟩",[11,61356,61357,61358,61386],{},"来访问特征空间。这个 ",[23,61359,61361,61374],{"className":61360,"translate":27},[31],[23,61362,61364],{"className":61363},[35],[37,61365,61366],{"xmlns":39},[42,61367,61368,61372],{},[45,61369,61370],{},[48,61371,4836],{},[114,61373,4836],{"encoding":116},[23,61375,61377],{"className":61376,"ariaHidden":122},[121],[23,61378,61380,61383],{"className":61379},[126],[23,61381],{"className":61382,"style":779},[130],[23,61384,4836],{"className":61385,"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",[23,61388,61390],{"className":61389,"translate":27},[26],[23,61391,61393,61433],{"className":61392,"translate":27},[31],[23,61394,61396],{"className":61395},[35],[37,61397,61398],{"xmlns":39,"display":40},[42,61399,61400,61430],{},[45,61401,61402,61404,61406,61408,61410,61412,61414,61416,61418,61420,61422,61424,61426,61428],{},[48,61403,8340],{},[51,61405,54],{"stretchy":53},[48,61407,753],{},[51,61409,66],{"stretchy":53},[51,61411,69],{},[51,61413,61189],{"stretchy":53},[48,61415,49700],{},[51,61417,747],{"separator":122},[48,61419,60894],{},[51,61421,54],{"stretchy":53},[48,61423,753],{},[51,61425,66],{"stretchy":53},[51,61427,61214],{"stretchy":53},[51,61429,747],{"separator":122},[114,61431,61432],{"encoding":116},"f(x)=\\langle w,\\varphi(x)\\rangle,",[23,61434,61436,61463],{"className":61435,"ariaHidden":122},[121],[23,61437,61439,61442,61445,61448,61451,61454,61457,61460],{"className":61438},[126],[23,61440],{"className":61441,"style":131},[130],[23,61443,8340],{"className":61444,"style":11402},[135,136],[23,61446,54],{"className":61447},[140],[23,61449,753],{"className":61450},[135,136],[23,61452,66],{"className":61453},[169],[23,61455],{"className":61456,"style":149},[148],[23,61458,69],{"className":61459},[153],[23,61461],{"className":61462,"style":149},[148],[23,61464,61466,61469,61472,61475,61478,61481,61484,61487,61490,61493],{"className":61465},[126],[23,61467],{"className":61468,"style":131},[130],[23,61470,61189],{"className":61471},[140],[23,61473,49700],{"className":61474,"style":49715},[135,136],[23,61476,747],{"className":61477},[824],[23,61479],{"className":61480,"style":280},[148],[23,61482,60894],{"className":61483},[135,136],[23,61485,54],{"className":61486},[140],[23,61488,753],{"className":61489},[135,136],[23,61491,61354],{"className":61492},[169],[23,61494,747],{"className":61495},[824],[11,61497,52937,61498,61526,61527,61555,61556,61584,61585,1471],{},[23,61499,61501,61514],{"className":61500,"translate":27},[31],[23,61502,61504],{"className":61503},[35],[37,61505,61506],{"xmlns":39},[42,61507,61508,61512],{},[45,61509,61510],{},[48,61511,49700],{},[114,61513,49700],{"encoding":116},[23,61515,61517],{"className":61516,"ariaHidden":122},[121],[23,61518,61520,61523],{"className":61519},[126],[23,61521],{"className":61522,"style":703},[130],[23,61524,49700],{"className":61525,"style":49715},[135,136]," 所在的希尔伯特空间（更准确地说，是 ",[23,61528,61530,61543],{"className":61529,"translate":27},[31],[23,61531,61533],{"className":61532},[35],[37,61534,61535],{"xmlns":39},[42,61536,61537,61541],{},[45,61538,61539],{},[48,61540,60894],{},[114,61542,61108],{"encoding":116},[23,61544,61546],{"className":61545,"ariaHidden":122},[121],[23,61547,61549,61552],{"className":61548},[126],[23,61550],{"className":61551,"style":672},[130],[23,61553,60894],{"className":61554},[135,136]," 张成的闭包）就是核 ",[23,61557,61559,61572],{"className":61558,"translate":27},[31],[23,61560,61562],{"className":61561},[35],[37,61563,61564],{"xmlns":39},[42,61565,61566,61570],{},[45,61567,61568],{},[48,61569,4836],{},[114,61571,4836],{"encoding":116},[23,61573,61575],{"className":61574,"ariaHidden":122},[121],[23,61576,61578,61581],{"className":61577},[126],[23,61579],{"className":61580,"style":779},[130],[23,61582,4836],{"className":61583,"style":4851},[135,136]," 对应的再生核希尔伯特空间 ",[23,61586,61588,61606],{"className":61587,"translate":27},[31],[23,61589,61591],{"className":61590},[35],[37,61592,61593],{"xmlns":39},[42,61594,61595,61603],{},[45,61596,61597],{},[898,61598,61599,61601],{},[48,61600,26253],{"mathvariant":375},[48,61602,4836],{},[114,61604,61605],{"encoding":116},"\\mathcal H_k",[23,61607,61609],{"className":61608,"ariaHidden":122},[121],[23,61610,61612,61615],{"className":61611},[126],[23,61613],{"className":61614,"style":42831},[130],[23,61616,61618,61621],{"className":61617},[135],[23,61619,26253],{"className":61620,"style":45028},[135,412],[23,61622,61624],{"className":61623},[998],[23,61625,61627,61648],{"className":61626},[199,200],[23,61628,61630,61645],{"className":61629},[204],[23,61631,61633],{"className":61632,"style":34196},[208],[23,61634,61636,61639],{"style":61635},"top:-2.55em;margin-left:-0.0097em;margin-right:0.05em;",[23,61637],{"className":61638,"style":1015},[216],[23,61640,61642],{"className":61641},[1019,1020,1021,1022],[23,61643,4836],{"className":61644,"style":4851},[135,136,1022],[23,61646,297],{"className":61647},[296],[23,61649,61651],{"className":61650},[204],[23,61652,61654],{"className":61653,"style":2048},[208],[23,61655],{},[11,61657,61658],{},"正则化经验风险最小化（regularized empirical risk minimization）的目标是",[23,61660,61662],{"className":61661,"translate":27},[26],[23,61663,61665,61788],{"className":61664,"translate":27},[31],[23,61666,61668],{"className":61667},[35],[37,61669,61670],{"xmlns":39,"display":40},[42,61671,61672,61785],{},[1518,61673,61675],{"width":61674},"100%",[1524,61676,61677,61680,61778,61780],{},[1527,61678],{"width":61679},"50%",[1527,61681,61682],{},[45,61683,61684,61690,61692,61694,61696,61712,61776],{},[4518,61685,61686,61688],{},[48,61687,8340],{},[51,61689,6883],{},[51,61691,69],{},[48,61693,6958],{},[51,61695,6961],{},[6963,61697,61698,61704],{},[45,61699,61700,61702],{},[48,61701,6969],{},[51,61703,6961],{},[45,61705,61706,61708,61710],{},[48,61707,8340],{},[51,61709,372],{},[48,61711,26253],{"mathvariant":375},[45,61713,61714,61716,61731,61733,61735,61741,61743,61745,61747,61753,61755,61757,61759,61762,61764,61766,61774],{},[51,61715,924],{"fence":122},[61717,61718,61719,61721,61729],"munderover",{},[51,61720,33855],{},[45,61722,61723,61725,61727],{},[48,61724,33497],{},[51,61726,69],{},[2581,61728,4435],{},[48,61730,4806],{},[48,61732,555],{},[51,61734,54],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[898,61736,61737,61739],{},[48,61738,660],{},[48,61740,33497],{},[51,61742,747],{"separator":122},[48,61744,8340],{},[51,61746,54],{"stretchy":53},[898,61748,61749,61751],{},[48,61750,753],{},[48,61752,33497],{},[51,61754,66],{"stretchy":53},[51,61756,66],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[51,61758,5106],{},[48,61760,61761],{},"λ",[48,61763,20684],{"mathvariant":111},[48,61765,8340],{},[19141,61767,61768,61770,61772],{},[48,61769,20684],{"mathvariant":111},[48,61771,26253],{"mathvariant":375},[2581,61773,9956],{},[51,61775,945],{"fence":122},[51,61777,747],{"separator":122},[1527,61779],{"width":61679},[1527,61781,61782],{},[88,61783,61784],{},"(1)",[114,61786,61787],{"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}",[23,61789,61791,61835,62179],{"className":61790,"ariaHidden":122},[121],[23,61792,61794,61797,61826,61829,61832],{"className":61793},[126],[23,61795],{"className":61796,"style":48057},[130],[23,61798,61800,61803],{"className":61799},[135],[23,61801,8340],{"className":61802,"style":11402},[135,136],[23,61804,61806],{"className":61805},[998],[23,61807,61809],{"className":61808},[199],[23,61810,61812],{"className":61811},[204],[23,61813,61815],{"className":61814,"style":7047},[208],[23,61816,61817,61820],{"style":4989},[23,61818],{"className":61819,"style":1015},[216],[23,61821,61823],{"className":61822},[1019,1020,1021,1022],[23,61824,6883],{"className":61825},[612,1022],[23,61827],{"className":61828,"style":149},[148],[23,61830,69],{"className":61831},[153],[23,61833],{"className":61834,"style":149},[148],[23,61836,61838,61842,61847,61850,61904,61907,62173,62176],{"className":61837},[126],[23,61839],{"className":61840,"style":61841},[130],"height:3.0277em;vertical-align:-1.2777em;",[23,61843,7087,61845],{"className":61844},[1786],[23,61846,7091],{"style":7090},[23,61848],{"className":61849,"style":280},[148],[23,61851,61853],{"className":61852},[1786,7098],[23,61854,61856,61895],{"className":61855},[199,200],[23,61857,61859,61892],{"className":61858},[204],[23,61860,61862,61882],{"className":61861,"style":7108},[208],[23,61863,61864,61867],{"style":14742},[23,61865],{"className":61866,"style":217},[216],[23,61868,61870],{"className":61869},[1019,1020,1021,1022],[23,61871,61873,61876,61879],{"className":61872},[135,1022],[23,61874,8340],{"className":61875,"style":11402},[135,136,1022],[23,61877,372],{"className":61878},[153,1022],[23,61880,26253],{"className":61881,"style":45028},[135,412,1022],[23,61883,61884,61887],{"style":7132},[23,61885],{"className":61886,"style":217},[216],[23,61888,61889],{},[23,61890,6969],{"className":61891},[1786],[23,61893,297],{"className":61894},[296],[23,61896,61898],{"className":61897},[204],[23,61899,61902],{"className":61900,"style":61901},[208],"height:0.8882em;",[23,61903],{},[23,61905],{"className":61906,"style":280},[148],[23,61908,61910,61916,61982,61985,61988,61994,62034,62037,62040,62043,62046,62086,62089,62095,62098,62101,62104,62107,62110,62113,62167],{"className":61909},[1839],[23,61911,61913],{"className":61912,"style":1844},[140,1843],[23,61914,924],{"className":61915},[1066,10073],[23,61917,61919],{"className":61918},[1786,7098],[23,61920,61922,61974],{"className":61921},[199,200],[23,61923,61925,61971],{"className":61924},[204],[23,61926,61929,61949,61959],{"className":61927,"style":61928},[208],"height:1.6514em;",[23,61930,61931,61934],{"style":34079},[23,61932],{"className":61933,"style":34083},[216],[23,61935,61937],{"className":61936},[1019,1020,1021,1022],[23,61938,61940,61943,61946],{"className":61939},[135,1022],[23,61941,33497],{"className":61942},[135,136,1022],[23,61944,69],{"className":61945},[153,1022],[23,61947,4435],{"className":61948},[135,1022],[23,61950,61951,61954],{"style":34092},[23,61952],{"className":61953,"style":34083},[216],[23,61955,61956],{},[23,61957,33855],{"className":61958},[1786,1790,1791],[23,61960,61962,61965],{"style":61961},"top:-4.3em;margin-left:0em;",[23,61963],{"className":61964,"style":34083},[216],[23,61966,61968],{"className":61967},[1019,1020,1021,1022],[23,61969,4806],{"className":61970},[135,136,1022],[23,61972,297],{"className":61973},[296],[23,61975,61977],{"className":61976},[204],[23,61978,61980],{"className":61979,"style":34110},[208],[23,61981],{},[23,61983],{"className":61984,"style":280},[148],[23,61986,555],{"className":61987},[135,136],[23,61989,61991],{"className":61990},[140],[23,61992,54],{"className":61993},[1066,1067],[23,61995,61997,62000],{"className":61996},[135],[23,61998,660],{"className":61999,"style":676},[135,136],[23,62001,62003],{"className":62002},[998],[23,62004,62006,62026],{"className":62005},[199,200],[23,62007,62009,62023],{"className":62008},[204],[23,62010,62012],{"className":62011,"style":33651},[208],[23,62013,62014,62017],{"style":21854},[23,62015],{"className":62016,"style":1015},[216],[23,62018,62020],{"className":62019},[1019,1020,1021,1022],[23,62021,33497],{"className":62022},[135,136,1022],[23,62024,297],{"className":62025},[296],[23,62027,62029],{"className":62028},[204],[23,62030,62032],{"className":62031,"style":2048},[208],[23,62033],{},[23,62035,747],{"className":62036},[824],[23,62038],{"className":62039,"style":280},[148],[23,62041,8340],{"className":62042,"style":11402},[135,136],[23,62044,54],{"className":62045},[140],[23,62047,62049,62052],{"className":62048},[135],[23,62050,753],{"className":62051},[135,136],[23,62053,62055],{"className":62054},[998],[23,62056,62058,62078],{"className":62057},[199,200],[23,62059,62061,62075],{"className":62060},[204],[23,62062,62064],{"className":62063,"style":33651},[208],[23,62065,62066,62069],{"style":2029},[23,62067],{"className":62068,"style":1015},[216],[23,62070,62072],{"className":62071},[1019,1020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是核 ",[23,62233,62235,62248],{"className":62234,"translate":27},[31],[23,62236,62238],{"className":62237},[35],[37,62239,62240],{"xmlns":39},[42,62241,62242,62246],{},[45,62243,62244],{},[48,62245,4836],{},[114,62247,4836],{"encoding":116},[23,62249,62251],{"className":62250,"ariaHidden":122},[121],[23,62252,62254,62257],{"className":62253},[126],[23,62255],{"className":62256,"style":779},[130],[23,62258,4836],{"className":62259,"style":4851},[135,136]," 对应的 RKHS，",[23,62262,62264,62282],{"className":62263,"translate":27},[31],[23,62265,62267],{"className":62266},[35],[37,62268,62269],{"xmlns":39},[42,62270,62271,62279],{},[45,62272,62273,62275,62277],{},[48,62274,61761],{},[51,62276,7382],{},[2581,62278,1532],{},[114,62280,62281],{"encoding":116},"\\lambda\\ge 0",[23,62283,62285,62304],{"className":62284,"ariaHidden":122},[121],[23,62286,62288,62292,62295,62298,62301],{"className":62287},[126],[23,62289],{"className":62290,"style":62291},[130],"height:0.8304em;vertical-align:-0.136em;",[23,62293,61761],{"className":62294},[135,136],[23,62296],{"className":62297,"style":149},[148],[23,62299,7382],{"className":62300},[153],[23,62302],{"className":62303,"style":149},[148],[23,62305,62307,62310],{"className":62306},[126],[23,62308],{"className":62309,"style":2770},[130],[23,62311,1532],{"className":62312},[135]," 是正则化参数。若 ",[23,62315,62317,62330],{"className":62316,"translate":27},[31],[23,62318,62320],{"className":62319},[35],[37,62321,62322],{"xmlns":39},[42,62323,62324,62328],{},[45,62325,62326],{},[48,62327,26253],{"mathvariant":375},[114,62329,45015],{"encoding":116},[23,62331,62333],{"className":62332,"ariaHidden":122},[121],[23,62334,62336,62339],{"className":62335},[126],[23,62337],{"className":62338,"style":470},[130],[23,62340,26253],{"className":62341,"style":45028},[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 ",[23,62344,62346,62359],{"className":62345,"translate":27},[31],[23,62347,62349],{"className":62348},[35],[37,62350,62351],{"xmlns":39},[42,62352,62353,62357],{},[45,62354,62355],{},[48,62356,4806],{},[114,62358,4806],{"encoding":116},[23,62360,62362],{"className":62361,"ariaHidden":122},[121],[23,62363,62365,62368],{"className":62364},[126],[23,62366],{"className":62367,"style":703},[130],[23,62369,4806],{"className":62370},[135,136]," 维问题。",[338,62373,62375],{"id":62374},"二rkhs-速览","二、RKHS 速览",[2501,62377,62379],{"id":62378},"_21-正定核与再生性","2.1 正定核与再生性",[11,62381,4410,62382,62411,62412,15005,62508,62511,62512,62540,62541,62705,62706,7354],{},[23,62383,62385,62399],{"className":62384,"translate":27},[31],[23,62386,62388],{"className":62387},[35],[37,62389,62390],{"xmlns":39},[42,62391,62392,62396],{},[45,62393,62394],{},[48,62395,434],{"mathvariant":375},[114,62397,62398],{"encoding":116},"\\mathcal X",[23,62400,62402],{"className":62401,"ariaHidden":122},[121],[23,62403,62405,62408],{"className":62404},[126],[23,62406],{"className":62407,"style":470},[130],[23,62409,434],{"className":62410,"style":474},[135,412]," 是一个非空集合。一个对称函数 ",[23,62413,62415,62441],{"className":62414,"translate":27},[31],[23,62416,62418],{"className":62417},[35],[37,62419,62420],{"xmlns":39},[42,62421,62422,62438],{},[45,62423,62424,62426,62428,62430,62432,62434,62436],{},[48,62425,4836],{},[51,62427,558],{},[48,62429,434],{"mathvariant":375},[51,62431,563],{},[48,62433,434],{"mathvariant":375},[51,62435,568],{},[48,62437,572],{"mathvariant":571},[114,62439,62440],{"encoding":116},"k:\\mathcal X\\times\\mathcal X\\to\\mathbb R",[23,62442,62444,62462,62481,62499],{"className":62443,"ariaHidden":122},[121],[23,62445,62447,62450,62453,62456,62459],{"className":62446},[126],[23,62448],{"className":62449,"style":779},[130],[23,62451,4836],{"className":62452,"style":4851},[135,136],[23,62454],{"className":62455,"style":149},[148],[23,62457,558],{"className":62458},[153],[23,62460],{"className":62461,"style":149},[148],[23,62463,62465,62469,62472,62475,62478],{"className":62464},[126],[23,62466],{"className":62467,"style":62468},[130],"height:0.7667em;vertical-align:-0.0833em;",[23,62470,434],{"className":62471,"style":474},[135,412],[23,62473],{"className":62474,"style":392},[148],[23,62476,563],{"className":62477},[612],[23,62479],{"className":62480,"style":392},[148],[23,62482,62484,62487,62490,62493,62496],{"className":62483},[126],[23,62485],{"className":62486,"style":470},[130],[23,62488,434],{"className":62489,"style":474},[135,412],[23,62491],{"className":62492,"style":149},[148],[23,62494,568],{"className":62495},[153],[23,62497],{"className":62498,"style":149},[148],[23,62500,62502,62505],{"className":62501},[126],[23,62503],{"className":62504,"style":640},[130],[23,62506,572],{"className":62507},[135,644],[18,62509,62510],{},"正定核","（positive definite kernel），如果对任意 ",[23,62513,62515,62528],{"className":62514,"translate":27},[31],[23,62516,62518],{"className":62517},[35],[37,62519,62520],{"xmlns":39},[42,62521,62522,62526],{},[45,62523,62524],{},[48,62525,4806],{},[114,62527,4806],{"encoding":116},[23,62529,62531],{"className":62530,"ariaHidden":122},[121],[23,62532,62534,62537],{"className":62533},[126],[23,62535],{"className":62536,"style":703},[130],[23,62538,4806],{"className":62539},[135,136],"、任意 ",[23,62542,62544,62579],{"className":62543,"translate":27},[31],[23,62545,62547],{"className":62546},[35],[37,62548,62549],{"xmlns":39},[42,62550,62551,62576],{},[45,62552,62553,62559,62561,62564,62566,62572,62574],{},[898,62554,62555,62557],{},[48,62556,753],{},[2581,62558,4435],{},[51,62560,747],{"separator":122},[51,62562,62563],{},"…",[51,62565,747],{"separator":122},[898,62567,62568,62570],{},[48,62569,753],{},[48,62571,4806],{},[51,62573,372],{},[48,62575,434],{"mathvariant":375},[114,62577,62578],{"encoding":116},"x_1,\\dots,x_n\\in\\mathcal X",[23,62580,62582,62696],{"className":62581,"ariaHidden":122},[121],[23,62583,62585,62589,62629,62632,62635,62638,62641,62644,62647,62687,62690,62693],{"className":62584},[126],[23,62586],{"className":62587,"style":62588},[130],"height:0.7335em;vertical-align:-0.1944em;",[23,62590,62592,62595],{"className":62591},[135],[23,62593,753],{"className":62594},[135,136],[23,62596,62598],{"className":62597},[998],[23,62599,62601,62621],{"className":62600},[199,200],[23,62602,62604,62618],{"className":62603},[204],[23,62605,62607],{"className":62606,"style":18651},[208],[23,62608,62609,62612],{"style":2029},[23,62610],{"className":62611,"style":1015},[216],[23,62613,62615],{"className":62614},[1019,1020,1021,1022],[23,62616,4435],{"className":62617},[135,1022],[23,62619,297],{"className":62620},[296],[23,62622,62624],{"className":62623},[204],[23,62625,62627],{"className":62626,"style":2048},[208],[23,62628],{},[23,62630,747],{"className":62631},[824],[23,62633],{"className":62634,"style":280},[148],[23,62636,62563],{"className":62637},[1839],[23,62639],{"className":62640,"style":280},[148],[23,62642,747],{"className":62643},[824],[23,62645],{"className":62646,"style":280},[148],[23,62648,62650,62653],{"className":62649},[135],[23,62651,753],{"className":62652},[135,136],[23,62654,62656],{"className":62655},[998],[23,62657,62659,62679],{"className":62658},[199,200],[23,62660,62662,62676],{"className":62661},[204],[23,62663,62665],{"className":62664,"style":6499},[208],[23,62666,62667,62670],{"style":2029},[23,62668],{"className":62669,"style":1015},[216],[23,62671,62673],{"className":62672},[1019,1020,1021,1022],[23,62674,4806],{"className":62675},[135,136,1022],[23,62677,297],{"className":62678},[296],[23,62680,62682],{"className":62681},[204],[23,62683,62685],{"className":62684,"style":2048},[208],[23,62686],{},[23,62688],{"className":62689,"style":149},[148],[23,62691,372],{"className":62692},[153],[23,62694],{"className":62695,"style":149},[148],[23,62697,62699,62702],{"className":62698},[126],[23,62700],{"className":62701,"style":470},[130],[23,62703,434],{"className":62704,"style":474},[135,412]," 和任意 ",[23,62707,62709,62743],{"className":62708,"translate":27},[31],[23,62710,62712],{"className":62711},[35],[37,62713,62714],{"xmlns":39},[42,62715,62716,62740],{},[45,62717,62718,62724,62726,62728,62730,62736,62738],{},[898,62719,62720,62722],{},[48,62721,12714],{},[2581,62723,4435],{},[51,62725,747],{"separator":122},[51,62727,62563],{},[51,62729,747],{"separator":122},[898,62731,62732,62734],{},[48,62733,12714],{},[48,62735,4806],{},[51,62737,372],{},[48,62739,572],{"mathvariant":571},[114,62741,62742],{"encoding":116},"c_1,\\dots,c_n\\in\\mathbb R",[23,62744,62746,62859],{"className":62745,"ariaHidden":122},[121],[23,62747,62749,62752,62792,62795,62798,62801,62804,62807,62810,62850,62853,62856],{"className":62748},[126],[23,62750],{"className":62751,"style":62588},[130],[23,62753,62755,62758],{"className":62754},[135],[23,62756,12714],{"className":62757},[135,136],[23,62759,62761],{"className":62760},[998],[23,62762,62764,62784],{"className":62763},[199,200],[23,62765,62767,62781],{"className":62766},[204],[23,62768,62770],{"className":62769,"style":18651},[208],[23,62771,62772,62775],{"style":2029},[23,62773],{"className":62774,"style":1015},[216],[23,62776,62778],{"className":62777},[1019,1020,1021,1022],[23,62779,4435],{"className":62780},[135,1022],[23,62782,297],{"className":62783},[296],[23,62785,62787],{"className":62786},[204],[23,62788,62790],{"className":62789,"style":2048},[208],[23,62791],{},[23,62793,747],{"className":62794},[824],[23,62796],{"className":62797,"style":280},[148],[23,62799,62563],{"className":62800},[1839],[23,62802],{"className":62803,"style":280},[148],[23,62805,747],{"className":62806},[824],[23,62808],{"className":62809,"style":280},[148],[23,62811,62813,62816],{"className":62812},[135],[23,62814,12714],{"className":62815},[135,136],[23,62817,62819],{"className":62818},[998],[23,62820,62822,62842],{"className":62821},[199,200],[23,62823,62825,62839],{"className":62824},[204],[23,62826,62828],{"className":62827,"style":6499},[208],[23,62829,62830,62833],{"style":2029},[23,62831],{"className":62832,"style":1015},[216],[23,62834,62836],{"className":62835},[1019,1020,1021,1022],[23,62837,4806],{"className":62838},[135,136,1022],[23,62840,297],{"className":62841},[296],[23,62843,62845],{"className":62844},[204],[23,62846,62848],{"className":62847,"style":2048},[208],[23,62849],{},[23,62851],{"className":62852,"style":149},[148],[23,62854,372],{"className":62855},[153],[23,62857],{"className":62858,"style":149},[148],[23,62860,62862,62865],{"className":62861},[126],[23,62863],{"className":62864,"style":640},[130],[23,62866,572],{"className":62867},[135,644],[23,62869,62871],{"className":62870,"translate":27},[26],[23,62872,62874,62952],{"className":62873,"translate":27},[31],[23,62875,62877],{"className":62876},[35],[37,62878,62879],{"xmlns":39,"display":40},[42,62880,62881,62949],{},[45,62882,62883,62897,62911,62917,62923,62925,62927,62929,62935,62937,62943,62945,62947],{},[61717,62884,62885,62887,62895],{},[51,62886,33855],{},[45,62888,62889,62891,62893],{},[48,62890,33497],{},[51,62892,69],{},[2581,62894,4435],{},[48,62896,4806],{},[61717,62898,62899,62901,62909],{},[51,62900,33855],{},[45,62902,62903,62905,62907],{},[48,62904,35522],{},[51,62906,69],{},[2581,62908,4435],{},[48,62910,4806],{},[898,62912,62913,62915],{},[48,62914,12714],{},[48,62916,33497],{},[898,62918,62919,62921],{},[48,62920,12714],{},[48,62922,35522],{},[88,62924,90],{},[48,62926,4836],{},[51,62928,54],{"stretchy":53},[898,62930,62931,62933],{},[48,62932,753],{},[48,62934,33497],{},[51,62936,747],{"separator":122},[898,62938,62939,62941],{},[48,62940,753],{},[48,62942,35522],{},[51,62944,66],{"stretchy":53},[51,62946,7382],{},[2581,62948,2583],{},[114,62950,62951],{"encoding":116},"\\sum_{i=1}^n\\sum_{j=1}^n c_i c_j\\,k(x_i,x_j)\\ge 0.",[23,62953,62955,63284],{"className":62954,"ariaHidden":122},[121],[23,62956,62958,62962,63026,63029,63094,63097,63137,63177,63180,63183,63186,63226,63229,63232,63272,63275,63278,63281],{"className":62957},[126],[23,62959],{"className":62960,"style":62961},[130],"height:3.0652em;vertical-align:-1.4138em;",[23,62963,62965],{"className":62964},[1786,7098],[23,62966,62968,63018],{"className":62967},[199,200],[23,62969,62971,63015],{"className":62970},[204],[23,62972,62974,62994,63004],{"className":62973,"style":61928},[208],[23,62975,62976,62979],{"style":34079},[23,62977],{"className":62978,"style":34083},[216],[23,62980,62982],{"className":62981},[1019,1020,1021,1022],[23,62983,62985,62988,62991],{"className":62984},[135,1022],[23,62986,33497],{"className":62987},[135,136,1022],[23,62989,69],{"className":62990},[153,1022],[23,62992,4435],{"className":62993},[135,1022],[23,62995,62996,62999],{"style":34092},[23,62997],{"className":62998,"style":34083},[216],[23,63000,63001],{},[23,63002,33855],{"className":63003},[1786,1790,1791],[23,63005,63006,63009],{"style":61961},[23,63007],{"className":63008,"style":34083},[216],[23,63010,63012],{"className":63011},[1019,1020,1021,1022],[23,63013,4806],{"className":63014},[135,136,1022],[23,63016,297],{"className":63017},[296],[23,63019,63021],{"className":63020},[204],[23,63022,63024],{"className":63023,"style":34110},[208],[23,63025],{},[23,63027],{"className":63028,"style":280},[148],[23,63030,63032],{"className":63031},[1786,7098],[23,63033,63035,63085],{"className":63034},[199,200],[23,63036,63038,63082],{"className":63037},[204],[23,63039,63041,63061,63071],{"className":63040,"style":61928},[208],[23,63042,63043,63046],{"style":34079},[23,63044],{"className":63045,"style":34083},[216],[23,63047,63049],{"className":63048},[1019,1020,1021,1022],[23,63050,63052,63055,63058],{"className":63051},[135,1022],[23,63053,35522],{"className":63054,"style":35607},[135,136,1022],[23,63056,69],{"className":63057},[153,1022],[23,63059,4435],{"className":63060},[135,1022],[23,63062,63063,63066],{"style":34092},[23,63064],{"className":63065,"style":34083},[216],[23,63067,63068],{},[23,63069,33855],{"className":63070},[1786,1790,1791],[23,63072,63073,63076],{"style":61961},[23,63074],{"className":63075,"style":34083},[216],[23,63077,63079],{"className":63078},[1019,1020,1021,1022],[23,63080,4806],{"className":63081},[135,136,1022],[23,63083,297],{"className":63084},[296],[23,63086,63088],{"className":63087},[204],[23,63089,63092],{"className":63090,"style":63091},[208],"height:1.4138em;",[23,63093],{},[23,63095],{"className":63096,"style":280},[148],[23,63098,63100,63103],{"className":63099},[135],[23,63101,12714],{"className":63102},[135,136],[23,63104,63106],{"className":63105},[998],[23,63107,63109,63129],{"className":63108},[199,200],[23,63110,63112,63126],{"className":63111},[204],[23,63113,63115],{"className":63114,"style":33651},[208],[23,63116,63117,63120],{"style":2029},[23,63118],{"className":63119,"style":1015},[216],[23,63121,63123],{"className":63122},[1019,1020,1021,1022],[23,63124,33497],{"className":63125},[135,136,1022],[23,63127,297],{"className":63128},[296],[23,63130,63132],{"className":63131},[204],[23,63133,63135],{"className":63134,"style":2048},[208],[23,63136],{},[23,63138,63140,63143],{"className":63139},[135],[23,63141,12714],{"className":63142},[135,136],[23,63144,63146],{"className":63145},[998],[23,63147,63149,63169],{"className":63148},[199,200],[23,63150,63152,63166],{"className":63151},[204],[23,63153,63155],{"className":63154,"style":33651},[208],[23,63156,63157,63160],{"style":2029},[23,63158],{"className":63159,"style":1015},[216],[23,63161,63163],{"className":63162},[1019,1020,1021,1022],[23,63164,35522],{"className":63165,"style":35607},[135,136,1022],[23,63167,297],{"className":63168},[296],[23,63170,63172],{"className":63171},[204],[23,63173,63175],{"className":63174,"style":14296},[208],[23,63176],{},[23,63178],{"className":63179,"style":280},[148],[23,63181,4836],{"className":63182,"style":4851},[135,136],[23,63184,54],{"className":63185},[140],[23,63187,63189,63192],{"className":63188},[135],[23,63190,753],{"className":63191},[135,136],[23,63193,63195],{"className":63194},[998],[23,63196,63198,63218],{"className":63197},[199,200],[23,63199,63201,63215],{"className":63200},[204],[23,63202,63204],{"className":63203,"style":33651},[208],[23,63205,63206,63209],{"style":2029},[23,63207],{"className":63208,"style":1015},[216],[23,63210,63212],{"className":63211},[1019,1020,1021,1022],[23,63213,33497],{"className":63214},[135,136,1022],[23,63216,297],{"className":63217},[296],[23,63219,63221],{"className":63220},[204],[23,63222,63224],{"className":63223,"style":2048},[208],[23,63225],{},[23,63227,747],{"className":63228},[824],[23,63230],{"className":63231,"style":280},[148],[23,63233,63235,63238],{"className":63234},[135],[23,63236,753],{"className":63237},[135,136],[23,63239,63241],{"className":63240},[998],[23,63242,63244,63264],{"className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",[23,63297,63299,63343],{"className":63298,"translate":27},[31],[23,63300,63302],{"className":63301},[35],[37,63303,63304],{"xmlns":39},[42,63305,63306,63340],{},[45,63307,63308,63318,63320,63322,63324,63330,63332,63338],{},[898,63309,63310,63312],{},[48,63311,26922],{},[45,63313,63314,63316],{},[48,63315,33497],{},[48,63317,35522],{},[51,63319,69],{},[48,63321,4836],{},[51,63323,54],{"stretchy":53},[898,63325,63326,63328],{},[48,63327,753],{},[48,63329,33497],{},[51,63331,747],{"separator":122},[898,63333,63334,63336],{},[48,63335,753],{},[48,63337,35522],{},[51,63339,66],{"stretchy":53},[114,63341,63342],{"encoding":116},"K_{ij}=k(x_i,x_j)",[23,63344,63346,63407],{"className":63345,"ariaHidden":122},[121],[23,63347,63349,63353,63398,63401,63404],{"className":63348},[126],[23,63350],{"className":63351,"style":63352},[130],"height:0.9694em;vertical-align:-0.2861em;",[23,63354,63356,63359],{"className":63355},[135],[23,63357,26922],{"className":63358,"style":37145},[135,136],[23,63360,63362],{"className":63361},[998],[23,63363,63365,63390],{"className":63364},[199,200],[23,63366,63368,63387],{"className":63367},[204],[23,63369,63371],{"className":63370,"style":33651},[208],[23,63372,63374,63377],{"style":63373},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[23,63375],{"className":63376,"style":1015},[216],[23,63378,63380],{"className":63379},[1019,1020,1021,1022],[23,63381,63383],{"className":63382},[135,1022],[23,63384,63386],{"className":63385,"style":35607},[135,136,1022],"ij",[23,63388,297],{"className":63389},[296],[23,63391,63393],{"className":63392},[204],[23,63394,63396],{"className":63395,"style":14296},[208],[23,63397],{},[23,63399],{"className":63400,"style":149},[148],[23,63402,69],{"className":63403},[153],[23,63405],{"className":63406,"style":149},[148],[23,63408,63410,63413,63416,63419,63459,63462,63465,63505],{"className":63409},[126],[23,63411],{"className":63412,"style":14257},[130],[23,63414,4836],{"className":63415,"style":4851},[135,136],[23,63417,54],{"className":63418},[140],[23,63420,63422,63425],{"className":63421},[135],[23,63423,753],{"className":63424},[135,136],[23,63426,63428],{"className":63427},[998],[23,63429,63431,63451],{"className":63430},[199,200],[23,63432,63434,63448],{"className":63433},[204],[23,63435,63437],{"className":63436,"style":33651},[208],[23,63438,63439,63442],{"style":2029},[23,63440],{"className":63441,"style":1015},[216],[23,63443,63445],{"className":63444},[1019,1020,1021,1022],[23,63446,33497],{"className":63447},[135,136,1022],[23,63449,297],{"className":63450},[296],[23,63452,63454],{"className":63453},[204],[23,63455,63457],{"className":63456,"style":2048},[208],[23,63458],{},[23,63460,747],{"className":63461},[824],[23,63463],{"className":63464,"style":280},[148],[23,63466,63468,63471],{"className":63467},[135],[23,63469,753],{"className":63470},[135,136],[23,63472,63474],{"className":63473},[998],[23,63475,63477,63497],{"className":63476},[199,200],[23,63478,63480,63494],{"className":63479},[204],[23,63481,63483],{"className":63482,"style":33651},[208],[23,63484,63485,63488],{"style":2029},[23,63486],{"className":63487,"style":1015},[216],[23,63489,63491],{"className":63490},[1019,1020,1021,1022],[23,63492,35522],{"className":63493,"style":35607},[135,136,1022],[23,63495,297],{"className":63496},[296],[23,63498,63500],{"className":63499},[204],[23,63501,63503],{"className":63502,"style":14296},[208],[23,63504],{},[23,63506,66],{"className":63507},[169]," 总是半正定的。",[6857,63510,63511,63645,63966,64180],{},[11,63512,63513,63516,63517,63545,63546,63615,63616,63644],{},[18,63514,63515],{},"定理 2.1（Moore–Aronszajn）"," 对任意正定核 ",[23,63518,63520,63533],{"className":63519,"translate":27},[31],[23,63521,63523],{"className":63522},[35],[37,63524,63525],{"xmlns":39},[42,63526,63527,63531],{},[45,63528,63529],{},[48,63530,4836],{},[114,63532,4836],{"encoding":116},[23,63534,63536],{"className":63535,"ariaHidden":122},[121],[23,63537,63539,63542],{"className":63538},[126],[23,63540],{"className":63541,"style":779},[130],[23,63543,4836],{"className":63544,"style":4851},[135,136],"，存在唯一的希尔伯特空间 ",[23,63547,63549,63566],{"className":63548,"translate":27},[31],[23,63550,63552],{"className":63551},[35],[37,63553,63554],{"xmlns":39},[42,63555,63556,63564],{},[45,63557,63558],{},[898,63559,63560,63562],{},[48,63561,26253],{"mathvariant":375},[48,63563,4836],{},[114,63565,61605],{"encoding":116},[23,63567,63569],{"className":63568,"ariaHidden":122},[121],[23,63570,63572,63575],{"className":63571},[126],[23,63573],{"className":63574,"style":42831},[130],[23,63576,63578,63581],{"className":63577},[135],[23,63579,26253],{"className":63580,"style":45028},[135,412],[23,63582,63584],{"className":63583},[998],[23,63585,63587,63607],{"className":63586},[199,200],[23,63588,63590,63604],{"className":63589},[204],[23,63591,63593],{"className":63592,"style":34196},[208],[23,63594,63595,63598],{"style":61635},[23,63596],{"className":63597,"style":1015},[216],[23,63599,63601],{"className":63600},[1019,1020,1021,1022],[23,63602,4836],{"className":63603,"style":4851},[135,136,1022],[23,63605,297],{"className":63606},[296],[23,63608,63610],{"className":63609},[204],[23,63611,63613],{"className":63612,"style":2048},[208],[23,63614],{},"，由 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X",[23,63671,63673,63691],{"className":63672,"ariaHidden":122},[121],[23,63674,63676,63679,63682,63685,63688],{"className":63675},[126],[23,63677],{"className":63678,"style":513},[130],[23,63680,753],{"className":63681},[135,136],[23,63683],{"className":63684,"style":149},[148],[23,63686,372],{"className":63687},[153],[23,63689],{"className":63690,"style":149},[148],[23,63692,63694,63697],{"className":63693},[126],[23,63695],{"className":63696,"style":470},[130],[23,63698,434],{"className":63699,"style":474},[135,412],[23,63701,63703,63735],{"className":63702,"translate":27},[31],[23,63704,63706],{"className":63705},[35],[37,63707,63708],{"xmlns":39},[42,63709,63710,63732],{},[45,63711,63712,63714,63716,63718,63720,63722,63724,63726],{},[48,63713,4836],{},[51,63715,54],{"stretchy":53},[51,63717,5758],{},[51,63719,747],{"separator":122},[48,63721,753],{},[51,63723,66],{"stretchy":53},[51,63725,372],{},[898,63727,63728,63730],{},[48,63729,26253],{"mathvariant":375},[48,63731,4836],{},[114,63733,63734],{"encoding":116},"k(\\cdot,x)\\in\\mathcal 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和 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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。",[2501,65732,65734],{"id":65733},"_22-特征映射与核技巧","2.2 特征映射与核技巧",[11,65736,65737],{},"令",[23,65739,65741],{"className":65740,"translate":27},[26],[23,65742,65744,65788],{"className":65743,"translate":27},[31],[23,65745,65747],{"className":65746},[35],[37,65748,65749],{"xmlns":39,"display":40},[42,65750,65751,65785],{},[45,65752,65753,65755,65757,65759,65761,65763,65765,65767,65769,65771,65773,65775,65777,65783],{},[48,65754,60894],{},[51,65756,54],{"stretchy":53},[48,65758,753],{},[51,65760,66],{"stretchy":53},[51,65762,69],{},[48,65764,4836],{},[51,65766,54],{"stretchy":53},[51,65768,5758],{},[51,65770,747],{"separator":122},[48,65772,753],{},[51,65774,66],{"stretchy":53},[51,65776,372],{},[898,65778,65779,65781],{},[48,65780,26253],{"mathvariant":375},[48,65782,4836],{},[48,65784,112],{"mathvariant":111},[114,65786,65787],{"encoding":116},"\\varphi(x)=k(\\cdot,x)\\in\\mathcal H_k.",[23,65789,65791,65818,65854],{"className":65790,"ariaHidden":122},[121],[23,65792,65794,65797,65800,65803,65806,65809,65812,65815],{"className":65793},[126],[23,65795],{"className":65796,"style":131},[130],[23,65798,60894],{"className":65799},[135,136],[23,65801,54],{"className":65802},[140],[23,65804,753],{"className":65805},[135,136],[23,65807,66],{"className":65808},[169],[23,65810],{"className":65811,"style":149},[148],[23,65813,69],{"className":65814},[153],[23,65816],{"className":65817,"style":149},[148],[23,65819,65821,65824,65827,65830,65833,65836,65839,65842,65845,65848,65851],{"className":65820},[126],[23,65822],{"className":65823,"style":131},[130],[23,65825,4836],{"className":65826,"style":4851},[135,136],[23,65828,54],{"className":65829},[140],[23,65831,5758],{"className":65832},[135],[23,65834,747],{"className":65835},[824],[23,65837],{"className":65838,"style":280},[148],[23,65840,753],{"className":65841},[135,136],[23,65843,66],{"className":65844},[169],[23,65846],{"className":65847,"style":149},[148],[23,65849,372],{"className":65850},[153],[23,65852],{"className":65853,"style":149},[148],[23,65855,65857,65860,65900],{"className":65856},[126],[23,65858],{"className":65859,"style":42831},[130],[23,65861,65863,65866],{"className":65862},[135],[23,65864,26253],{"className":65865,"style":45028},[135,412],[23,65867,65869],{"className":65868},[998],[23,65870,65872,65892],{"className":65871},[199,200],[23,65873,65875,65889],{"className":65874},[204],[23,65876,65878],{"className":65877,"style":34196},[208],[23,65879,65880,65883],{"style":61635},[23,65881],{"className":65882,"style":1015},[216],[23,65884,65886],{"className":65885},[1019,1020,1021,1022],[23,65887,4836],{"className":65888,"style":4851},[135,136,1022],[23,65890,297],{"className":65891},[296],[23,65893,65895],{"className":65894},[204],[23,65896,65898],{"className":65897,"style":2048},[208],[23,65899],{},[23,65901,112],{"className":65902},[135],[11,65904,65905,65906,65934,65935,65963,65964,66033],{},"则 ",[23,65907,65909,65922],{"className":65908,"translate":27},[31],[23,65910,65912],{"className":65911},[35],[37,65913,65914],{"xmlns":39},[42,65915,65916,65920],{},[45,65917,65918],{},[48,65919,60894],{},[114,65921,61108],{"encoding":116},[23,65923,65925],{"className":65924,"ariaHidden":122},[121],[23,65926,65928,65931],{"className":65927},[126],[23,65929],{"className":65930,"style":672},[130],[23,65932,60894],{"className":65933},[135,136]," 就是一个从 ",[23,65936,65938,65951],{"className":65937,"translate":27},[31],[23,65939,65941],{"className":65940},[35],[37,65942,65943],{"xmlns":39},[42,65944,65945,65949],{},[45,65946,65947],{},[48,65948,434],{"mathvariant":375},[114,65950,62398],{"encoding":116},[23,65952,65954],{"className":65953,"ariaHidden":122},[121],[23,65955,65957,65960],{"className":65956},[126],[23,65958],{"className":65959,"style":470},[130],[23,65961,434],{"className":65962,"style":474},[135,412]," 到 ",[23,65965,65967,65984],{"className":65966,"translate":27},[31],[23,65968,65970],{"className":65969},[35],[37,65971,65972],{"xmlns":39},[42,65973,65974,65982],{},[45,65975,65976],{},[898,65977,65978,65980],{},[48,65979,26253],{"mathvariant":375},[48,65981,4836],{},[114,65983,61605],{"encoding":116},[23,65985,65987],{"className":65986,"ariaHidden":122},[121],[23,65988,65990,65993],{"className":65989},[126],[23,65991],{"className":65992,"style":42831},[130],[23,65994,65996,65999],{"className":65995},[135],[23,65997,26253],{"className":65998,"style":45028},[135,412],[23,66000,66002],{"className":66001},[998],[23,66003,66005,66025],{"className":66004},[199,200],[23,66006,66008,66022],{"className":66007},[204],[23,66009,66011],{"className":66010,"style":34196},[208],[23,66012,66013,66016],{"style":61635},[23,66014],{"className":66015,"style":1015},[216],[23,66017,66019],{"className":66018},[1019,1020,1021,1022],[23,66020,4836],{"className":66021,"style":4851},[135,136,1022],[23,66023,297],{"className":66024},[296],[23,66026,66028],{"className":66027},[204],[23,66029,66031],{"className":66030,"style":2048},[208],[23,66032],{}," 的特征映射，并且",[23,66035,66037],{"className":66036,"translate":27},[26],[23,66038,66040,66106],{"className":66039,"translate":27},[31],[23,66041,66043],{"className":66042},[35],[37,66044,66045],{"xmlns":39,"display":40},[42,66046,66047,66103],{},[45,66048,66049,66051,66053,66055,66057,66063,66065,66067,66069,66071,66073,66075,66077,66079,66081,66083,66089,66091,66101],{},[48,66050,4836],{},[51,66052,54],{"stretchy":53},[48,66054,753],{},[51,66056,747],{"separator":122},[4518,66058,66059,66061],{},[48,66060,753],{},[51,66062,38295],{"mathvariant":111,"lspace":1522,"rspace":1522},[51,66064,66],{"stretchy":53},[51,66066,69],{},[51,66068,61189],{"stretchy":53},[48,66070,60894],{},[51,66072,54],{"stretchy":53},[48,66074,753],{},[51,66076,66],{"stretchy":53},[51,66078,747],{"separator":122},[48,66080,60894],{},[51,66082,54],{"stretchy":53},[4518,66084,66085,66087],{},[48,66086,753],{},[51,66088,38295],{"mathvariant":111,"lspace":1522,"rspace":1522},[51,66090,66],{"stretchy":53},[898,66092,66093,66095],{},[51,66094,61214],{"stretchy":53},[898,66096,66097,66099],{},[48,66098,26253],{"mathvariant":375},[48,66100,4836],{},[48,66102,112],{"mathvariant":111},[114,66104,66105],{"encoding":116},"k(x,x')=\\langle \\varphi(x),\\varphi(x')\\rangle_{\\mathcal H_k}.",[23,66107,66109,66174],{"className":66108,"ariaHidden":122},[121],[23,66110,66112,66115,66118,66121,66124,66127,66130,66162,66165,66168,66171],{"className":66111},[126],[23,66113],{"className":66114,"style":51202},[130],[23,66116,4836],{"className":66117,"style":4851},[135,136],[23,66119,54],{"className":66120},[140],[23,66122,753],{"className":66123},[135,136],[23,66125,747],{"className":66126},[824],[23,66128],{"className":66129,"style":280},[148],[23,66131,66133,66136],{"className":66132},[135],[23,66134,753],{"className":66135},[135,136],[23,66137,66139],{"className":66138},[998],[23,66140,66142],{"className":66141},[199],[23,66143,66145],{"className":66144},[204],[23,66146,66148],{"className":66147,"style":51227},[208],[23,66149,66150,66153],{"style":4989},[23,66151],{"className":66152,"style":1015},[216],[23,66154,66156],{"className":66155},[1019,1020,1021,1022],[23,66157,66159],{"className":66158},[135,1022],[23,66160,38295],{"className":66161},[135,1022],[23,66163,66],{"className":66164},[169],[23,66166],{"className":66167,"style":149},[148],[23,66169,69],{"className":66170},[153],[23,66172],{"className":66173,"style":149},[148],[23,66175,66177,66181,66184,66187,66190,66193,66196,66199,66202,66205,66208,66240,66243,66323],{"className":66176},[126],[23,66178],{"className":66179,"style":66180},[130],"height:1.0578em;vertical-align:-0.2559em;",[23,66182,61189],{"className":66183},[140],[23,66185,60894],{"className":66186},[135,136],[23,66188,54],{"className":66189},[140],[23,66191,753],{"className":66192},[135,136],[23,66194,66],{"className":66195},[169],[23,66197,747],{"className":66198},[824],[23,66200],{"className":66201,"style":280},[148],[23,66203,60894],{"className":66204},[135,136],[23,66206,54],{"className":66207},[140],[23,66209,66211,66214],{"className":66210},[135],[23,66212,753],{"className":66213},[135,136],[23,66215,66217],{"className":66216},[998],[23,66218,66220],{"className":66219},[199],[23,66221,66223],{"className":66222},[204],[23,66224,66226],{"className":66225,"style":51227},[208],[23,66227,66228,66231],{"style":4989},[23,66229],{"className":66230,"style":1015},[216],[23,66232,66234],{"className":66233},[1019,1020,1021,1022],[23,66235,66237],{"className":66236},[135,1022],[23,66238,38295],{"className":66239},[135,1022],[23,66241,66],{"className":66242},[169],[23,66244,66246,66249],{"className":66245},[169],[23,66247,61214],{"className":66248},[169],[23,66250,66252],{"className":66251},[998],[23,66253,66255,66315],{"className":66254},[199,200],[23,66256,66258,66312],{"className":66257},[204],[23,66259,66261],{"className":66260,"style":2026},[208],[23,66262,66263,66266],{"style":2029},[23,66264],{"className":66265,"style":1015},[216],[23,66267,66269],{"className":66268},[1019,1020,1021,1022],[23,66270,66272],{"className":66271},[135,1022],[23,66273,66275,66278],{"className":66274},[135,1022],[23,66276,26253],{"className":66277,"style":45028},[135,412,1022],[23,66279,66281],{"className":66280},[998],[23,66282,66284,66304],{"className":66283},[199,200],[23,66285,66287,66301],{"className":66286},[204],[23,66288,66290],{"className":66289,"style":1008},[208],[23,66291,66292,66295],{"style":64116},[23,66293],{"className":66294,"style":24885},[216],[23,66296,66298],{"className":66297},[1019,24889,1067,1022],[23,66299,4836],{"className":66300,"style":4851},[135,136,1022],[23,66302,297],{"className":66303},[296],[23,66305,66307],{"className":66306},[204],[23,66308,66310],{"className":66309,"style":64135},[208],[23,66311],{},[23,66313,297],{"className":66314},[296],[23,66316,66318],{"className":66317},[204],[23,66319,66321],{"className":66320,"style":64147},[208],[23,66322],{},[23,66324,112],{"className":66325},[135],[11,66327,9750,66328,66331,66332,66445,66446,66536,66537,66565],{},[18,66329,66330],{},"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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",[23,66538,66540,66553],{"className":66539,"translate":27},[31],[23,66541,66543],{"className":66542},[35],[37,66544,66545],{"xmlns":39},[42,66546,66547,66551],{},[45,66548,66549],{},[48,66550,60894],{},[114,66552,61108],{"encoding":116},[23,66554,66556],{"className":66555,"ariaHidden":122},[121],[23,66557,66559,66562],{"className":66558},[126],[23,66560],{"className":66561,"style":672},[130],[23,66563,60894],{"className":66564},[135,136]," 的情况下运行。SVM、核岭回归、核 PCA 都是这样。",[11,66567,66568],{},"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。",[2501,66570,66572],{"id":66571},"_23-常见核与它们的-rkhs","2.3 常见核与它们的 RKHS",[3764,66574,66575,66588],{},[3767,66576,66577],{},[3770,66578,66579,66582,66585],{},[3773,66580,66581],{},"核",[3773,66583,66584],{},"形式",[3773,66586,66587],{},"特征维数 \u002F RKHS",[3783,66589,66590,66713,66986,67235,67423],{},[3770,66591,66592,66595,66682],{},[3788,66593,66594],{},"线性核",[3788,66596,66597],{},[23,66598,66600,66626],{"className":66599,"translate":27},[31],[23,66601,66603],{"className":66602},[35],[37,66604,66605],{"xmlns":39},[42,66606,66607,66623],{},[45,66608,66609,66611,66613,66615,66621],{},[51,66610,61189],{"stretchy":53},[48,66612,753],{},[51,66614,747],{"separator":122},[4518,66616,66617,66619],{},[48,66618,753],{},[51,66620,38295],{"mathvariant":111,"lspace":1522,"rspace":1522},[51,66622,61214],{"stretchy":53},[114,66624,66625],{"encoding":116},"\\langle 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维",[3770,66714,66715,66718,66866],{},[3788,66716,66717],{},"多项式核",[3788,66719,66720],{},[23,66721,66723,66761],{"className":66722,"translate":27},[31],[23,66724,66726],{"className":66725},[35],[37,66727,66728],{"xmlns":39},[42,66729,66730,66758],{},[45,66731,66732,66734,66736,66738,66740,66746,66748,66750,66752],{},[51,66733,54],{"stretchy":53},[51,66735,61189],{"stretchy":53},[48,66737,753],{},[51,66739,747],{"separator":122},[4518,66741,66742,66744],{},[48,66743,753],{},[51,66745,38295],{"mathvariant":111,"lspace":1522,"rspace":1522},[51,66747,61214],{"stretchy":53},[51,66749,5106],{},[48,66751,12714],{},[4518,66753,66754,66756],{},[51,66755,66],{"stretchy":53},[48,66757,11],{},[114,66759,66760],{"encoding":116},"(\\langle x,x'\\rangle+c)^p",[23,66762,66764,66827],{"className":66763,"ariaHidden":122},[121],[23,66765,66767,66770,66774,66777,66780,66783,66815,66818,66821,66824],{"className":66766},[126],[23,66768],{"className":66769,"style":66382},[130],[23,66771,66773],{"className":66772},[140],"(⟨",[23,66775,753],{"className":66776},[135,136],[23,66778,747],{"className":66779},[824],[23,66781],{"className":66782,"style":280},[148],[23,66784,66786,66789],{"className":66785},[135],[23,66787,753],{"className":66788},[135,136],[23,66790,66792],{"className":66791},[998],[23,66793,66795],{"className":66794},[199],[23,66796,66798],{"className":66797},[204],[23,66799,66801],{"className":66800,"style":38337},[208],[23,66802,66803,66806],{"style":4696},[23,66804],{"className":66805,"style":1015},[216],[23,66807,66809],{"className":66808},[1019,1020,1021,1022],[23,66810,66812],{"className":66811},[135,1022],[23,66813,38295],{"className":66814},[135,1022],[23,66816,61214],{"className":66817},[169],[23,66819],{"className":66820,"style":392},[148],[23,66822,5106],{"className":66823},[612],[23,66825],{"className":66826,"style":392},[148],[23,66828,66830,66833,66836],{"className":66829},[126],[23,66831],{"className":66832,"style":131},[130],[23,66834,12714],{"className":66835},[135,136],[23,66837,66839,66842],{"className":66838},[169],[23,66840,66],{"className":66841},[169],[23,66843,66845],{"className":66844},[998],[23,66846,66848],{"className":66847},[199],[23,66849,66851],{"className":66850},[204],[23,66852,66855],{"className":66853,"style":66854},[208],"height:0.6644em;",[23,66856,66857,66860],{"style":4696},[23,66858],{"className":66859,"style":1015},[216],[23,66861,66863],{"className":66862},[1019,1020,1021,1022],[23,66864,11],{"className":66865},[135,136,1022],[3788,66867,66868,66712],{},[23,66869,66871,66900],{"className":66870,"translate":27},[31],[23,66872,66874],{"className":66873},[35],[37,66875,66876],{"xmlns":39},[42,66877,66878,66897],{},[45,66879,66880,66882,66895],{},[51,66881,54],{"fence":122},[71,66883,66885,66893],{"linethickness":66884},"0px",[45,66886,66887,66889,66891],{},[48,66888,1606],{},[51,66890,5106],{},[48,66892,11],{},[48,66894,11],{},[51,66896,66],{"fence":122},[114,66898,66899],{"encoding":116},"\\binom{d+p}{p}",[23,66901,66903],{"className":66902,"ariaHidden":122},[121],[23,66904,66906,66910],{"className":66905},[126],[23,66907],{"className":66908,"style":66909},[130],"height:1.4112em;vertical-align:-0.4811em;",[23,66911,66913,66919,66980],{"className":66912},[135],[23,66914,66916],{"className":66915,"style":1844},[140,1843],[23,66917,54],{"className":66918},[1066,1067],[23,66920,66922],{"className":66921},[71],[23,66923,66925,66971],{"className":66924},[199,200],[23,66926,66928,66968],{"className":66927},[204],[23,66929,66932,66947],{"className":66930,"style":66931},[208],"height:0.9301em;",[23,66933,66935,66938],{"style":66934},"top:-2.355em;",[23,66936],{"className":66937,"style":1015},[216],[23,66939,66941],{"className":66940},[1019,1020,1021,1022],[23,66942,66944],{"className":66943},[135,1022],[23,66945,11],{"className":66946},[135,136,1022],[23,66948,66950,66953],{"style":66949},"top:-3.144em;",[23,66951],{"className":66952,"style":1015},[216],[23,66954,66956],{"className":66955},[1019,1020,1021,1022],[23,66957,66959,66962,66965],{"className":66958},[135,1022],[23,66960,1606],{"className":66961},[135,136,1022],[23,66963,5106],{"className":66964},[612,1022],[23,66966,11],{"className":66967},[135,136,1022],[23,66969,297],{"className":66970},[296],[23,66972,66974],{"className":66973},[204],[23,66975,66978],{"className":66976,"style":66977},[208],"height:0.4811em;",[23,66979],{},[23,66981,66983],{"className":66982,"style":1844},[169,1843],[23,66984,66],{"className":66985},[1066,1067],[3770,66987,66988,66991,67232],{},[3788,66989,66990],{},"高斯核",[3788,66992,66993],{},[23,66994,66996,67055],{"className":66995,"translate":27},[31],[23,66997,66999],{"className":66998},[35],[37,67000,67001],{"xmlns":39},[42,67002,67003,67052],{},[45,67004,67005,67007,67009,67012,67014,67016,67018,67020,67022,67028,67036,67038,67040,67042,67048,67050],{},[48,67006,25012],{},[51,67008,6961],{},[88,67010,67011],{}," ⁣",[51,67013,54],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[51,67015,4529],{},[48,67017,20684],{"mathvariant":111},[48,67019,753],{},[51,67021,4529],{},[4518,67023,67024,67026],{},[48,67025,753],{},[51,67027,38295],{"mathvariant":111,"lspace":1522,"rspace":1522},[19141,67029,67030,67032,67034],{},[48,67031,20684],{"mathvariant":111},[2581,67033,9956],{},[2581,67035,9956],{},[48,67037,2875],{"mathvariant":111},[51,67039,54],{"stretchy":53},[2581,67041,9956],{},[4518,67043,67044,67046],{},[48,67045,29638],{},[2581,67047,9956],{},[51,67049,66],{"stretchy":53},[51,67051,66],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[114,67053,67054],{"encoding":116},"\\exp\\!\\bigl(-\\|x-x'\\|_2^2\u002F(2\\sigma^2)\\bigr)",[23,67056,67058,67095],{"className":67057,"ariaHidden":122},[121],[23,67059,67061,67064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核",[3788,67241,67242],{},[23,67243,67245,67291],{"className":67244,"translate":27},[31],[23,67246,67248],{"className":67247},[35],[37,67249,67250],{"xmlns":39},[42,67251,67252,67288],{},[45,67253,67254,67256,67258,67260,67262,67264,67266,67268,67270,67276,67282,67284,67286],{},[48,67255,25012],{},[51,67257,6961],{},[88,67259,67011],{},[51,67261,54],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[51,67263,4529],{},[48,67265,20684],{"mathvariant":111},[48,67267,753],{},[51,67269,4529],{},[4518,67271,67272,67274],{},[48,67273,753],{},[51,67275,38295],{"mathvariant":111,"lspace":1522,"rspace":1522},[898,67277,67278,67280],{},[48,67279,20684],{"mathvariant":111},[2581,67281,4435],{},[48,67283,2875],{"mathvariant":111},[48,67285,29638],{},[51,67287,66],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[114,67289,67290],{"encoding":116},"\\exp\\!\\bigl(-\\|x-x'\\|_1\u002F\\sigma\\bigr)",[23,67292,67294,67330],{"className":67293,"ariaHidden":122},[121],[23,67295,67297,67300,67303,67306,67312,67315,67318,67321,67324,67327],{"className":67296},[126],[23,67298],{"className":67299,"style":2213},[130],[23,67301,25012],{"className":67302},[1786],[23,67304],{"className":67305,"style":67070},[148],[23,67307,67309],{"className":67308},[140],[23,67310,54],{"className":67311},[1066,1067],[23,67313,4529],{"className":67314},[135],[23,67316,20684],{"className":67317},[135],[23,67319,753],{"className":67320},[135,136],[23,67322],{"className":67323,"style":392},[148],[23,67325,4529],{"className":67326},[612],[23,67328],{"className":67329,"style":392},[148],[23,67331,67333,67336,67368,67408,67411,67414],{"className":67332},[126],[23,67334],{"className":67335,"style":2213},[130],[23,67337,67339,67342],{"className":67338},[135],[23,67340,753],{"className":67341},[135,136],[23,67343,67345],{"className":67344},[998],[23,67346,67348],{"className":67347},[199],[23,67349,67351],{"className":67350},[204],[23,67352,67354],{"className":67353,"style":38337},[208],[23,67355,67356,67359],{"style":4696},[23,67357],{"className":67358,"style":1015},[216],[23,67360,67362],{"className":67361},[1019,1020,1021,1022],[23,67363,67365],{"className":67364},[135,1022],[23,67366,38295],{"className":67367},[135,1022],[23,67369,67371,67374],{"className":67370},[135],[23,67372,20684],{"className":67373},[135],[23,67375,67377],{"className":67376},[998],[23,67378,67380,67400],{"className":67379},[199,200],[23,67381,67383,67397],{"className":67382},[204],[23,67384,67386],{"className":67385,"style":18651},[208],[23,67387,67388,67391],{"style":2029},[23,67389],{"className":67390,"style":1015},[216],[23,67392,67394],{"className":67393},[1019,1020,1021,1022],[23,67395,4435],{"className":67396},[135,1022],[23,67398,297],{"className":67399},[296],[23,67401,67403],{"className":67402},[204],[23,67404,67406],{"className":67405,"style":2048},[208],[23,67407],{},[23,67409,2875],{"className":67410},[135],[23,67412,29638],{"className":67413,"style":676},[135,136],[23,67415,67417],{"className":67416},[169],[23,67418,66],{"className":67419},[1066,1067],[3788,67421,67422],{},"无穷维，Sobolev 型",[3770,67424,67425,67428,67431],{},[3788,67426,67427],{},"Matérn 核",[3788,67429,67430],{},"由修正 Bessel 函数给出",[3788,67432,67433,67434],{},"Sobolev 空间 ",[23,67435,67437,67466],{"className":67436,"translate":27},[31],[23,67438,67440],{"className":67439},[35],[37,67441,67442],{"xmlns":39},[42,67443,67444,67463],{},[45,67445,67446],{},[4518,67447,67448,67450],{},[48,67449,26253],{},[45,67451,67452,67455,67457,67459,67461],{},[48,67453,67454],{},"ν",[51,67456,5106],{},[48,67458,1606],{},[48,67460,2875],{"mathvariant":111},[2581,67462,9956],{},[114,67464,67465],{"encoding":116},"H^{\\nu+d\u002F2}",[23,67467,67469],{"className":67468,"ariaHidden":122},[121],[23,67470,67472,67475],{"className":67471},[126],[23,67473],{"className":67474,"style":35727},[130],[23,67476,67478,67481],{"className":67477},[135],[23,67479,26253],{"className":67480,"style":26507},[135,136],[23,67482,67484],{"className":67483},[998],[23,67485,67487],{"className":67486},[199],[23,67488,67490],{"className":67489},[204],[23,67491,67493],{"className":67492,"style":35727},[208],[23,67494,67495,67498],{"style":4696},[23,67496],{"className":67497,"style":1015},[216],[23,67499,67501],{"className":67500},[1019,1020,1021,1022],[23,67502,67504,67508,67511,67514],{"className":67503},[135,1022],[23,67505,67454],{"className":67506,"style":67507},[135,136,1022],"margin-right:0.0637em;",[23,67509,5106],{"className":67510},[612,1022],[23,67512,1606],{"className":67513},[135,136,1022],[23,67515,20564],{"className":67516},[135,1022],[11,67518,67519,67520,67589,67590,1471],{},"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 ",[23,67521,67523,67540],{"className":67522,"translate":27},[31],[23,67524,67526],{"className":67525},[35],[37,67527,67528],{"xmlns":39},[42,67529,67530,67538],{},[45,67531,67532],{},[898,67533,67534,67536],{},[48,67535,26253],{"mathvariant":375},[48,67537,4836],{},[114,67539,61605],{"encoding":116},[23,67541,67543],{"className":67542,"ariaHidden":122},[121],[23,67544,67546,67549],{"className":67545},[126],[23,67547],{"className":67548,"style":42831},[130],[23,67550,67552,67555],{"className":67551},[135],[23,67553,26253],{"className":67554,"style":45028},[135,412],[23,67556,67558],{"className":67557},[998],[23,67559,67561,67581],{"className":67560},[199,200],[23,67562,67564,67578],{"className":67563},[204],[23,67565,67567],{"className":67566,"style":34196},[208],[23,67568,67569,67572],{"style":61635},[23,67570],{"className":67571,"style":1015},[216],[23,67573,67575],{"className":67574},[1019,1020,1021,1022],[23,67576,4836],{"className":67577,"style":4851},[135,136,1022],[23,67579,297],{"className":67580},[296],[23,67582,67584],{"className":67583},[204],[23,67585,67587],{"className":67586,"style":2048},[208],[23,67588],{}," 降到 ",[23,67591,67593,67611],{"className":67592,"translate":27},[31],[23,67594,67596],{"className":67595},[35],[37,67597,67598],{"xmlns":39},[42,67599,67600,67608],{},[45,67601,67602],{},[4518,67603,67604,67606],{},[48,67605,572],{"mathvariant":571},[48,67607,4806],{},[114,67609,67610],{"encoding":116},"\\mathbb R^n",[23,67612,67614],{"className":67613,"ariaHidden":122},[121],[23,67615,67617,67620],{"className":67616},[126],[23,67618],{"className":67619,"style":640},[130],[23,67621,67623,67626],{"className":67622},[135],[23,67624,572],{"className":67625},[135,644],[23,67627,67629],{"className":67628},[998],[23,67630,67632],{"className":67631},[199],[23,67633,67635],{"className":67634},[204],[23,67636,67638],{"className":67637,"style":66854},[208],[23,67639,67640,67643],{"style":4696},[23,67641],{"className":67642,"style":1015},[216],[23,67644,67646],{"className":67645},[1019,1020,1021,1022],[23,67647,4806],{"className":67648},[135,136,1022],[338,67650,67652],{"id":67651},"三表示定理","三、表示定理",[2501,67654,67656],{"id":67655},"_31-定理","3.1 定理",[6857,67658,67659,68230,68730,68733,68739,69082],{},[11,67660,67661,67664,67665,62231,67693,67721,67722,67990,67991,68131,68132,68229],{},[18,67662,67663],{},"定理 3.1（表示定理）"," 设 ",[23,67666,67668,67681],{"className":67667,"translate":27},[31],[23,67669,67671],{"className":67670},[35],[37,67672,67673],{"xmlns":39},[42,67674,67675,67679],{},[45,67676,67677],{},[48,67678,26253],{"mathvariant":375},[114,67680,45015],{"encoding":116},[23,67682,67684],{"className":67683,"ariaHidden":122},[121],[23,67685,67687,67690],{"className":67686},[126],[23,67688],{"className":67689,"style":470},[130],[23,67691,26253],{"className":67692,"style":45028},[135,412],[23,67694,67696,67709],{"className":67695,"translate":27},[31],[23,67697,67699],{"className":67698},[35],[37,67700,67701],{"xmlns":39},[42,67702,67703,67707],{},[45,67704,67705],{},[48,67706,4836],{},[114,67708,4836],{"encoding":116},[23,67710,67712],{"className":67711,"ariaHidden":122},[121],[23,67713,67715,67718],{"className":67714},[126],[23,67716],{"className":67717,"style":779},[130],[23,67719,4836],{"className":67720,"style":4851},[135,136]," 对应的 RKHS，给定训练数据 ",[23,67723,67725,67779],{"className":67724,"translate":27},[31],[23,67726,67728],{"className":67727},[35],[37,67729,67730],{"xmlns":39},[42,67731,67732,67776],{},[45,67733,67734,67736,67742,67744,67750,67752,67754,67756,67758,67760,67766,67768,67774],{},[51,67735,54],{"stretchy":53},[898,67737,67738,67740],{},[48,67739,753],{},[2581,67741,4435],{},[51,67743,747],{"separator":122},[898,67745,67746,67748],{},[48,67747,660],{},[2581,67749,4435],{},[51,67751,66],{"stretchy":53},[51,67753,747],{"separator":122},[51,67755,62563],{},[51,67757,747],{"separator":122},[51,67759,54],{"stretchy":53},[898,67761,67762,67764],{},[48,67763,753],{},[48,67765,4806],{},[51,67767,747],{"separator":122},[898,67769,67770,67772],{},[48,67771,660],{},[48,67773,4806],{},[51,67775,66],{"stretchy":53},[114,67777,67778],{"encoding":116},"(x_1,y_1),\\dots,(x_n,y_n)",[23,67780,67782],{"className":67781,"ariaHidden":122},[121],[23,67783,67785,67788,67791,67831,67834,67837,67877,67880,67883,67886,67889,67892,67895,67898,67901,67941,67944,67947,67987],{"className":67784},[126],[23,67786],{"className":67787,"style":131},[130],[23,67789,54],{"className":67790},[140],[23,67792,67794,67797],{"className":67793},[135],[23,67795,753],{"className":67796},[135,136],[23,67798,67800],{"className":67799},[998],[23,67801,67803,67823],{"className":67802},[199,200],[23,67804,67806,67820],{"className":67805},[204],[23,67807,67809],{"className":67808,"style":18651},[208],[23,67810,67811,67814],{"style":2029},[23,67812],{"className":67813,"style":1015},[216],[23,67815,67817],{"className":67816},[1019,1020,1021,1022],[23,67818,4435],{"className":67819},[135,1022],[23,67821,297],{"className":67822},[296],[23,67824,67826],{"className":67825},[204],[23,67827,67829],{"className":67828,"style":2048},[208],[23,67830],{},[23,67832,747],{"className":67833},[824],[23,67835],{"className":67836,"style":280},[148],[23,67838,67840,67843],{"className":67839},[135],[23,67841,660],{"className":67842,"style":676},[135,136],[23,67844,67846],{"className":67845},[998],[23,67847,67849,67869],{"className":67848},[199,200],[23,67850,67852,67866],{"className":67851},[204],[23,67853,67855],{"className":67854,"style":18651},[208],[23,67856,67857,67860],{"style":21854},[23,67858],{"className":67859,"style":1015},[216],[23,67861,67863],{"className":67862},[1019,1020,1021,1022],[23,67864,4435],{"className":67865},[135,1022],[23,67867,297],{"className":67868},[296],[23,67870,67872],{"className":67871},[204],[23,67873,67875],{"className":67874,"style":2048},[208],[23,67876],{},[23,67878,66],{"className":67879},[169],[23,67881,747],{"className":67882},[824],[23,67884],{"className":67885,"style":280},[148],[23,67887,62563],{"className":67888},[1839],[23,67890],{"className":67891,"style":280},[148],[23,67893,747],{"className":67894},[824],[23,67896],{"className":67897,"style":280},[148],[23,67899,54],{"className":67900},[140],[23,67902,67904,67907],{"className":67903},[135],[23,67905,753],{"className":67906},[135,136],[23,67908,67910],{"className":67909},[998],[23,67911,67913,67933],{"className":67912},[199,200],[23,67914,67916,67930],{"className":67915},[204],[23,67917,67919],{"className":67918,"style":6499},[208],[23,67920,67921,67924],{"style":2029},[23,67922],{"className":67923,"style":1015},[216],[23,67925,67927],{"className":67926},[1019,1020,1021,1022],[23,67928,4806],{"className":67929},[135,136,1022],[23,67931,297],{"className":67932},[296],[23,67934,67936],{"className":67935},[204],[23,67937,67939],{"className":67938,"style":2048},[208],[23,67940],{},[23,67942,747],{"className":67943},[824],[23,67945],{"className":67946,"style":280},[148],[23,67948,67950,67953],{"className":67949},[135],[23,67951,660],{"className":67952,"style":676},[135,136],[23,67954,67956],{"className":67955},[998],[23,67957,67959,67979],{"className":67958},[199,200],[23,67960,67962,67976],{"className":67961},[204],[23,67963,67965],{"className":67964,"style":6499},[208],[23,67966,67967,67970],{"style":21854},[23,67968],{"className":67969,"style":1015},[216],[23,67971,67973],{"className":67972},[1019,1020,1021,1022],[23,67974,4806],{"className":67975},[135,136,1022],[23,67977,297],{"className":67978},[296],[23,67980,67982],{"className":67981},[204],[23,67983,67985],{"className":67984,"style":2048},[208],[23,67986],{},[23,67988,66],{"className":67989},[169],"。设 ",[23,67992,67994,68030],{"className":67993,"translate":27},[31],[23,67995,67997],{"className":67996},[35],[37,67998,67999],{"xmlns":39},[42,68000,68001,68027],{},[45,68002,68003,68005,68007,68013,68015,68017,68019,68021,68023,68025],{},[48,68004,555],{},[51,68006,558],{},[4518,68008,68009,68011],{},[48,68010,572],{"mathvariant":571},[48,68012,4806],{},[51,68014,568],{},[48,68016,572],{"mathvariant":571},[51,68018,50398],{},[51,68020,9014],{"stretchy":53},[51,68022,5106],{},[48,68024,46451],{"mathvariant":111},[51,68026,9024],{"stretchy":53},[114,68028,68029],{"encoding":116},"L:\\mathbb R^n\\to\\mathbb R\\cup\\{+\\infty\\}",[23,68031,68033,68051,68095,68113],{"className":68032,"ariaHidden":122},[121],[23,68034,68036,68039,68042,68045,68048],{"className":68035},[126],[23,68037],{"className":68038,"style":470},[130],[23,68040,555],{"className":68041},[135,136],[23,68043],{"className":68044,"style":149},[148],[23,68046,558],{"className":68047},[153],[23,68049],{"className":68050,"style":149},[148],[23,68052,68054,68057,68086,68089,68092],{"className":68053},[126],[23,68055],{"className":68056,"style":640},[130],[23,68058,68060,68063],{"className":68059},[135],[23,68061,572],{"className":68062},[135,644],[23,68064,68066],{"className":68065},[998],[23,68067,68069],{"className":68068},[199],[23,68070,68072],{"className":68071},[204],[23,68073,68075],{"className":68074,"style":66854},[208],[23,68076,68077,68080],{"style":4696},[23,68078],{"className":68079,"style":1015},[216],[23,68081,68083],{"className":68082},[1019,1020,1021,1022],[23,68084,4806],{"className":68085},[135,136,1022],[23,68087],{"className":68088,"style":149},[148],[23,68090,568],{"className":68091},[153],[23,68093],{"className":68094,"style":149},[148],[23,68096,68098,68101,68104,68107,68110],{"className":68097},[126],[23,68099],{"className":68100,"style":640},[130],[23,68102,572],{"className":68103},[135,644],[23,68105],{"className":68106,"style":392},[148],[23,68108,50398],{"className":68109},[612],[23,68111],{"className":68112,"style":392},[148],[23,68114,68116,68119,68122,68125,68128],{"className":68115},[126],[23,68117],{"className":68118,"style":131},[130],[23,68120,9014],{"className":68121},[140],[23,68123,5106],{"className":68124},[135],[23,68126,46451],{"className":68127},[135],[23,68129,9024],{"className":68130},[169]," 是任意损失函数，",[23,68133,68135,68166],{"className":68134,"translate":27},[31],[23,68136,68138],{"className":68137},[35],[37,68139,68140],{"xmlns":39},[42,68141,68142,68163],{},[45,68143,68144,68147,68149,68151,68153,68155,68157,68159,68161],{},[48,68145,68146],{"mathvariant":111},"Ω",[51,68148,558],{},[51,68150,924],{"stretchy":53},[2581,68152,1532],{},[51,68154,747],{"separator":122},[48,68156,46451],{"mathvariant":111},[51,68158,66],{"stretchy":53},[51,68160,568],{},[48,68162,572],{"mathvariant":571},[114,68164,68165],{"encoding":116},"\\Omega:[0,\\infty)\\to\\mathbb R",[23,68167,68169,68187,68220],{"className":68168,"ariaHidden":122},[121],[23,68170,68172,68175,68178,68181,68184],{"className":68171},[126],[23,68173],{"className":68174,"style":470},[130],[23,68176,68146],{"className":68177},[135],[23,68179],{"className":68180,"style":149},[148],[23,68182,558],{"className":68183},[153],[23,68185],{"className":68186,"style":149},[148],[23,68188,68190,68193,68196,68199,68202,68205,68208,68211,68214,68217],{"className":68189},[126],[23,68191],{"className":68192,"style":131},[130],[23,68194,924],{"className":68195},[140],[23,68197,1532],{"className":68198},[135],[23,68200,747],{"className":68201},[824],[23,68203],{"className":68204,"style":280},[148],[23,68206,46451],{"className":68207},[135],[23,68209,66],{"className":68210},[169],[23,68212],{"className":68213,"style":149},[148],[23,68215,568],{"className":68216},[153],[23,68218],{"className":68219,"style":149},[148],[23,68221,68223,68226],{"className":68222},[126],[23,68224],{"className":68225,"style":640},[130],[23,68227,572],{"className":68228},[135,644]," 是非减函数。考虑",[23,68231,68233],{"className":68232,"translate":27},[26],[23,68234,68236,68355],{"className":68235,"translate":27},[31],[23,68237,68239],{"className":68238},[35],[37,68240,68241],{"xmlns":39,"display":40},[42,68242,68243,68352],{},[1518,68244,68245],{"width":61674},[1524,68246,68247,68249,68345,68347],{},[1527,68248],{"width":61679},[1527,68250,68251],{},[45,68252,68253,68259,68261,68263,68265,68281,68343],{},[4518,68254,68255,68257],{},[48,68256,8340],{},[51,68258,6883],{},[51,68260,372],{},[48,68262,6958],{},[51,68264,6961],{},[6963,68266,68267,68273],{},[45,68268,68269,68271],{},[48,68270,6969],{},[51,68272,6961],{},[45,68274,68275,68277,68279],{},[48,68276,8340],{},[51,68278,372],{},[48,68280,26253],{"mathvariant":375},[45,68282,68283,68285,68287,68289,68291,68293,68299,68301,68303,68305,68307,68309,68311,68317,68319,68321,68323,68325,68327,68329,68331,68339,68341],{},[51,68284,9014],{"fence":122},[48,68286,555],{},[51,68288,54],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[48,68290,8340],{},[51,68292,54],{"stretchy":53},[898,68294,68295,68297],{},[48,68296,753],{},[2581,68298,4435],{},[51,68300,66],{"stretchy":53},[51,68302,747],{"separator":122},[51,68304,62563],{},[51,68306,747],{"separator":122},[48,68308,8340],{},[51,68310,54],{"stretchy":53},[898,68312,68313,68315],{},[48,68314,753],{},[48,68316,4806],{},[51,68318,66],{"stretchy":53},[51,68320,66],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[51,68322,5106],{},[48,68324,68146],{"mathvariant":111},[51,68326,54],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[48,68328,20684],{"mathvariant":111},[48,68330,8340],{},[19141,68332,68333,68335,68337],{},[48,68334,20684],{"mathvariant":111},[48,68336,26253],{"mathvariant":375},[2581,68338,9956],{},[51,68340,66],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[51,68342,9024],{"fence":122},[48,68344,112],{"mathvariant":111},[1527,68346],{"width":61679},[1527,68348,68349],{},[88,68350,68351],{},"(2)",[114,68353,68354],{"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}",[23,68356,68358,68402,68709],{"className":68357,"ariaHidden":122},[121],[23,68359,68361,68364,68393,68396,68399],{"className":68360},[126],[23,68362],{"className":68363,"style":48057},[130],[23,68365,68367,68370],{"className":68366},[135],[23,68368,8340],{"className":68369,"style":11402},[135,136],[23,68371,68373],{"className":68372},[998],[23,68374,68376],{"className":68375},[199],[23,68377,68379],{"className":68378},[204],[23,68380,68382],{"className":68381,"style":7047},[208],[23,68383,68384,68387],{"style":4989},[23,68385],{"className":68386,"style":1015},[216],[23,68388,68390],{"className":68389},[1019,1020,1021,1022],[23,68391,6883],{"className":68392},[612,1022],[23,68394],{"className":68395,"style":149},[148],[23,68397,372],{"className":68398},[153],[23,68400],{"className":68401,"style":149},[148],[23,68403,68405,68409,68414,68417,68470,68473,68703,68706],{"className":68404},[126],[23,68406],{"className":68407,"style":68408},[130],"height:1.7523em;vertical-align:-0.8882em;",[23,68410,7087,68412],{"className":68411},[1786],[23,68413,7091],{"style":7090},[23,68415],{"className":68416,"style":280},[148],[23,68418,68420],{"className":68419},[1786,7098],[23,68421,68423,68462],{"className":68422},[199,200],[23,68424,68426,68459],{"className":68425},[204],[23,68427,68429,68449],{"className":68428,"style":7108},[208],[23,68430,68431,68434],{"style":14742},[23,68432],{"className":68433,"style":217},[216],[23,68435,68437],{"className":68436},[1019,1020,1021,1022],[23,68438,68440,68443,68446],{"className":68439},[135,1022],[23,68441,8340],{"className":68442,"style":11402},[135,136,1022],[23,68444,372],{"className":68445},[153,1022],[23,68447,26253],{"className":68448,"style":45028},[135,412,1022],[23,68450,68451,68454],{"style":7132},[23,68452],{"className":68453,"style":217},[216],[23,68455,68456],{},[23,68457,6969],{"className":68458},[1786],[23,68460,297],{"className":68461},[296],[23,68463,68465],{"className":68464},[204],[23,68466,68468],{"className":68467,"style":61901},[208],[23,68469],{},[23,68471],{"className":68472,"style":280},[148],[23,68474,68476,68482,68485,68491,68494,68497,68537,68540,68543,68546,68549,68552,68555,68558,68561,68564,68604,68607,68613,68616,68619,68622,68625,68631,68634,68637,68691,68697],{"className":68475},[1839],[23,68477,68479],{"className":68478,"style":1844},[140,1843],[23,68480,9014],{"className":68481},[1066,1067],[23,68483,555],{"className":68484},[135,136],[23,68486,68488],{"className":68487},[140],[23,68489,54],{"className":68490},[1066,1067],[23,68492,8340],{"className":68493,"style":11402},[135,136],[23,68495,54],{"className":68496},[140],[23,68498,68500,68503],{"className":68499},[135],[23,68501,753],{"className":68502},[135,136],[23,68504,68506],{"className":68505},[998],[23,68507,68509,68529],{"className":68508},[199,200],[23,68510,68512,68526],{"className":68511},[204],[23,68513,68515],{"className":68514,"style":18651},[208],[23,68516,68517,68520],{"style":2029},[23,68518],{"className":68519,"style":1015},[216],[23,68521,68523],{"className":68522},[1019,1020,1021,1022],[23,68524,4435],{"className":68525},[135,1022],[23,68527,297],{"className":68528},[296],[23,68530,68532],{"className":68531},[204],[23,68533,68535],{"className":68534,"style":2048},[208],[23,68536],{},[23,68538,66],{"className":68539},[169],[23,68541,747],{"className":68542},[824],[23,68544],{"className":68545,"style":280},[148],[23,68547,62563],{"className":68548},[1839],[23,68550],{"className":68551,"style":280},[148],[23,68553,747],{"className":68554},[824],[23,68556],{"className":68557,"style":280},[148],[23,68559,8340],{"className":68560,"style":11402},[135,136],[23,68562,54],{"className":68563},[140],[23,68565,68567,68570],{"className":68566},[135],[23,68568,753],{"className":68569},[135,136],[23,68571,68573],{"className":68572},[998],[23,68574,68576,68596],{"className":68575},[199,200],[23,68577,68579,68593],{"className":68578},[204],[23,68580,68582],{"className":68581,"style":6499},[208],[23,68583,68584,68587],{"style":2029},[23,68585],{"className":68586,"style":1015},[216],[23,68588,68590],{"className":68589},[1019,1020,1021,1022],[23,68591,4806],{"className":68592},[135,136,1022],[23,68594,297],{"className":68595},[296],[23,68597,68599],{"className":68598},[204],[23,68600,68602],{"className":68601,"style":2048},[208],[23,68603],{},[23,68605,66],{"className":68606},[169],[23,68608,68610],{"className":68609},[169],[23,68611,66],{"className":68612},[1066,1067],[23,68614],{"className":68615,"style":392},[148],[23,68617,5106],{"className":68618},[612],[23,68620],{"className":68621,"style":392},[148],[23,68623,68146],{"className":68624},[135],[23,68626,68628],{"className":68627},[140],[23,68629,54],{"className":68630},[1066,1067],[23,68632,20684],{"className":68633},[135],[23,68635,8340],{"className":68636,"style":11402},[135,136],[23,68638,68640,68643],{"className":68639},[135],[23,68641,20684],{"className":68642},[135],[23,68644,68646],{"className":68645},[998],[23,68647,68649,68683],{"className":68648},[199,200],[23,68650,68652,68680],{"className":68651},[204],[23,68653,68655,68669],{"className":68654,"style":11006},[208],[23,68656,68657,68660],{"style":19349},[23,68658],{"className":68659,"style":1015},[216],[23,68661,68663],{"className":68662},[1019,1020,1021,1022],[23,68664,68666],{"className":68665},[135,1022],[23,68667,26253],{"className":68668,"style":45028},[135,412,1022],[23,68670,68671,68674],{"style":4989},[23,68672],{"className":68673,"style":1015},[216],[23,68675,68677],{"className":68676},[1019,1020,1021,1022],[23,68678,9956],{"className":68679},[135,1022],[23,68681,297],{"className":68682},[296],[23,68684,68686],{"className":68685},[204],[23,68687,68689],{"className":68688,"style":19379},[208],[23,68690],{},[23,68692,68694],{"className":68693},[169],[23,68695,66],{"className":68696},[1066,1067],[23,68698,68700],{"className":68699,"style":1844},[169,1843],[23,68701,9024],{"className":68702},[1066,1067],[23,68704],{"className":68705,"style":280},[148],[23,68707,112],{"className":68708},[135],[23,68710,68712,68715],{"className":68711},[62182],[23,68713],{"className":68714,"style":68408},[130],[23,68716,68718,68721,68727],{"className":68717},[135,6514],[23,68719,54],{"className":68720},[135],[23,68722,68724],{"className":68723},[135],[23,68725,9956],{"className":68726},[135],[23,68728,66],{"className":68729},[135],[11,68731,68732],{},"则：",[11,68734,68735,68738],{},[18,68736,68737],{},"(i)"," 存在一个最优解具有形式",[23,68740,68742],{"className":68741,"translate":27},[26],[23,68743,68745,68826],{"className":68744,"translate":27},[31],[23,68746,68748],{"className":68747},[35],[37,68749,68750],{"xmlns":39,"display":40},[42,68751,68752,68823],{},[1518,68753,68754],{"width":61674},[1524,68755,68756,68758,68816,68818],{},[1527,68757],{"width":61679},[1527,68759,68760],{},[45,68761,68762,68768,68770,68772,68774,68776,68790,68796,68798,68800,68802,68804,68806,68812,68814],{},[4518,68763,68764,68766],{},[48,68765,8340],{},[51,68767,6883],{},[51,68769,54],{"stretchy":53},[48,68771,753],{},[51,68773,66],{"stretchy":53},[51,68775,69],{},[61717,68777,68778,68780,68788],{},[51,68779,33855],{},[45,68781,68782,68784,68786],{},[48,68783,33497],{},[51,68785,69],{},[2581,68787,4435],{},[48,68789,4806],{},[898,68791,68792,68794],{},[48,68793,4526],{},[48,68795,33497],{},[88,68797,90],{},[48,68799,4836],{},[51,68801,54],{"stretchy":53},[48,68803,753],{},[51,68805,747],{"separator":122},[898,68807,68808,68810],{},[48,68809,753],{},[48,68811,33497],{},[51,68813,66],{"stretchy":53},[51,68815,39748],{"separator":122},[1527,68817],{"width":61679},[1527,68819,68820],{},[88,68821,68822],{},"(3)",[114,68824,68825],{"encoding":116},"f^*(x)=\\sum_{i=1}^n\\alpha_i\\,k(x,x_i);\n\\tag{3}",[23,68827,68829,68882,69061],{"className":68828,"ariaHidden":122},[121],[23,68830,68832,68835,68864,68867,68870,68873,68876,68879],{"className":68831},[126],[23,68833],{"className":68834,"style":131},[130],[23,68836,68838,68841],{"className":68837},[135],[23,68839,8340],{"className":68840,"style":11402},[135,136],[23,68842,68844],{"className":68843},[998],[23,68845,68847],{"className":68846},[199],[23,68848,68850],{"className":68849},[204],[23,68851,68853],{"className":68852,"style":7047},[208],[23,68854,68855,68858],{"style":4989},[23,68856],{"className":68857,"style":1015},[216],[23,68859,68861],{"className":68860},[1019,1020,1021,1022],[23,68862,6883],{"className":68863},[612,1022],[23,68865,54],{"className":68866},[140],[23,68868,753],{"className":68869},[135,136],[23,68871,66],{"className":68872},[169],[23,68874],{"className":68875,"style":149},[148],[23,68877,69],{"className":68878},[153],[23,68880],{"className":68881,"style":149},[148],[23,68883,68885,68889,68953,68956,68997,69000,69003,69006,69009,69012,69015,69055,69058],{"className":68884},[126],[23,68886],{"className":68887,"style":68888},[130],"height:2.9291em;vertical-align:-1.2777em;",[23,68890,68892],{"className":68891},[1786,7098],[23,68893,68895,68945],{"className":68894},[199,200],[23,68896,68898,68942],{"className":68897},[204],[23,68899,68901,68921,68931],{"className":68900,"style":61928},[208],[23,68902,68903,68906],{"style":34079},[23,68904],{"className":68905,"style":34083},[216],[23,68907,68909],{"className":68908},[1019,1020,1021,1022],[23,68910,68912,68915,68918],{"className":68911},[135,1022],[23,68913,33497],{"className":68914},[135,136,1022],[23,68916,69],{"className":68917},[153,1022],[23,68919,4435],{"className":68920},[135,1022],[23,68922,68923,68926],{"style":34092},[23,68924],{"className":68925,"style":34083},[216],[23,68927,68928],{},[23,68929,33855],{"className":68930},[1786,1790,1791],[23,68932,68933,68936],{"style":61961},[23,68934],{"className":68935,"style":34083},[216],[23,68937,68939],{"className":68938},[1019,1020,1021,1022],[23,68940,4806],{"className":68941},[135,136,1022],[23,68943,297],{"className":68944},[296],[23,68946,68948],{"className":68947},[204],[23,68949,68951],{"className":68950,"style":34110},[208],[23,68952],{},[23,68954],{"className":68955,"style":280},[148],[23,68957,68959,68962],{"className":68958},[135],[23,68960,4526],{"className":68961,"style":4645},[135,136],[23,68963,68965],{"className":68964},[998],[23,68966,68968,68989],{"className":68967},[199,200],[23,68969,68971,68986],{"className":68970},[204],[23,68972,68974],{"className":68973,"style":33651},[208],[23,68975,68977,68980],{"style":68976},"top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;",[23,68978],{"className":68979,"style":1015},[216],[23,68981,68983],{"className":68982},[1019,1020,1021,1022],[23,68984,33497],{"className":68985},[135,136,1022],[23,68987,297],{"className":68988},[296],[23,68990,68992],{"className":68991},[204],[23,68993,68995],{"className":68994,"style":2048},[208],[23,68996],{},[23,68998],{"className":68999,"style":280},[148],[23,69001,4836],{"className":69002,"style":4851},[135,136],[23,69004,54],{"className":69005},[140],[23,69007,753],{"className":69008},[135,136],[23,69010,747],{"className":69011},[824],[23,69013],{"className":69014,"style":280},[148],[23,69016,69018,69021],{"className":69017},[135],[23,69019,753],{"className":69020},[135,136],[23,69022,69024],{"className":69023},[998],[23,69025,69027,69047],{"className":69026},[199,200],[23,69028,69030,69044],{"className":69029},[204],[23,69031,69033],{"className":69032,"style":33651},[208],[23,69034,69035,69038],{"style":2029},[23,69036],{"className":69037,"style":1015},[216],[23,69039,69041],{"className":69040},[1019,1020,1021,1022],[23,69042,33497],{"className":69043},[135,136,1022],[23,69045,297],{"className":69046},[296],[23,69048,69050],{"className":69049},[204],[23,69051,69053],{"className":69052,"style":2048},[208],[23,69054],{},[23,69056,66],{"className":69057},[169],[23,69059,39748],{"className":69060},[824],[23,69062,69064,69067],{"className":69063},[62182],[23,69065],{"className":69066,"style":68888},[130],[23,69068,69070,69073,69079],{"className":69069},[135,6514],[23,69071,54],{"className":69072},[135],[23,69074,69076],{"className":69075},[135],[23,69077,24923],{"className":69078},[135],[23,69080,66],{"className":69081},[135],[11,69083,69084,69087,69088,69117,69118,69121],{},[18,69085,69086],{},"(ii)"," 若 ",[23,69089,69091,69105],{"className":69090,"translate":27},[31],[23,69092,69094],{"className":69093},[35],[37,69095,69096],{"xmlns":39},[42,69097,69098,69102],{},[45,69099,69100],{},[48,69101,68146],{"mathvariant":111},[114,69103,69104],{"encoding":116},"\\Omega",[23,69106,69108],{"className":69107,"ariaHidden":122},[121],[23,69109,69111,69114],{"className":69110},[126],[23,69112],{"className":69113,"style":470},[130],[23,69115,68146],{"className":69116},[135]," 严格递增，则",[18,69119,69120],{},"每个","最优解都具有这个形式。",[11,69123,69124],{},"这里有三个细节值得强调。",[11,69126,69127,69128,69156,69157,69327,69328,69356,69357,69385,69386,69456,69457,69485],{},"第一，",[23,69129,69131,69144],{"className":69130,"translate":27},[31],[23,69132,69134],{"className":69133},[35],[37,69135,69136],{"xmlns":39},[42,69137,69138,69142],{},[45,69139,69140],{},[48,69141,555],{},[114,69143,555],{"encoding":116},[23,69145,69147],{"className":69146,"ariaHidden":122},[121],[23,69148,69150,69153],{"className":69149},[126],[23,69151],{"className":69152,"style":470},[130],[23,69154,555],{"className":69155},[135,136]," 只通过 ",[23,69158,69160,69202],{"className":69159,"translate":27},[31],[23,69161,69163],{"className":69162},[35],[37,69164,69165],{"xmlns":39},[42,69166,69167,69199],{},[45,69168,69169,69171,69173,69179,69181,69183,69185,69187,69189,69191,69197],{},[48,69170,8340],{},[51,69172,54],{"stretchy":53},[898,69174,69175,69177],{},[48,69176,753],{},[2581,69178,4435],{},[51,69180,66],{"stretchy":53},[51,69182,747],{"separator":122},[51,69184,62563],{},[51,69186,747],{"separator":122},[48,69188,8340],{},[51,69190,54],{"stretchy":53},[898,69192,69193,69195],{},[48,69194,753],{},[48,69196,4806],{},[51,69198,66],{"stretchy":53},[114,69200,69201],{"encoding":116},"f(x_1),\\dots,f(x_n)",[23,69203,69205],{"className":69204,"ariaHidden":122},[121],[23,69206,69208,69211,69214,69217,69257,69260,69263,69266,69269,69272,69275,69278,69281,69284,69324],{"className":69207},[126],[23,69209],{"className":69210,"style":131},[130],[23,69212,8340],{"className":69213,"style":11402},[135,136],[23,69215,54],{"className":69216},[140],[23,69218,69220,69223],{"className":69219},[135],[23,69221,753],{"className":69222},[135,136],[23,69224,69226],{"className":69225},[998],[23,69227,69229,69249],{"className":69228},[199,200],[23,69230,69232,69246],{"className":69231},[204],[23,69233,69235],{"className":69234,"style":18651},[208],[23,69236,69237,69240],{"style":2029},[23,69238],{"className":69239,"style":1015},[216],[23,69241,69243],{"className":69242},[1019,1020,1021,1022],[23,69244,4435],{"className":69245},[135,1022],[23,69247,297],{"className":69248},[296],[23,69250,69252],{"className":69251},[204],[23,69253,69255],{"className":69254,"style":2048},[208],[23,69256],{},[23,69258,66],{"className":69259},[169],[23,69261,747],{"className":69262},[824],[23,69264],{"className":69265,"style":280},[148],[23,69267,62563],{"className":69268},[1839],[23,69270],{"className":69271,"style":280},[148],[23,69273,747],{"className":69274},[824],[23,69276],{"className":69277,"style":280},[148],[23,69279,8340],{"className":69280,"style":11402},[135,136],[23,69282,54],{"className":69283},[140],[23,69285,69287,69290],{"className":69286},[135],[23,69288,753],{"className":69289},[135,136],[23,69291,69293],{"className":69292},[998],[23,69294,69296,69316],{"className":69295},[199,200],[23,69297,69299,69313],{"className":69298},[204],[23,69300,69302],{"className":69301,"style":6499},[208],[23,69303,69304,69307],{"style":2029},[23,69305],{"className":69306,"style":1015},[216],[23,69308,69310],{"className":69309},[1019,1020,1021,1022],[23,69311,4806],{"className":69312},[135,136,1022],[23,69314,297],{"className":69315},[296],[23,69317,69319],{"className":69318},[204],[23,69320,69322],{"className":69321,"style":2048},[208],[23,69323],{},[23,69325,66],{"className":69326},[169]," 这 ",[23,69329,69331,69344],{"className":69330,"translate":27},[31],[23,69332,69334],{"className":69333},[35],[37,69335,69336],{"xmlns":39},[42,69337,69338,69342],{},[45,69339,69340],{},[48,69341,4806],{},[114,69343,4806],{"encoding":116},[23,69345,69347],{"className":69346,"ariaHidden":122},[121],[23,69348,69350,69353],{"className":69349},[126],[23,69351],{"className":69352,"style":703},[130],[23,69354,4806],{"className":69355},[135,136]," 个值依赖 ",[23,69358,69360,69373],{"className":69359,"translate":27},[31],[23,69361,69363],{"className":69362},[35],[37,69364,69365],{"xmlns":39},[42,69366,69367,69371],{},[45,69368,69369],{},[48,69370,8340],{},[114,69372,8340],{"encoding":116},[23,69374,69376],{"className":69375,"ariaHidden":122},[121],[23,69377,69379,69382],{"className":69378},[126],[23,69380],{"className":69381,"style":5414},[130],[23,69383,8340],{"className":69384,"style":11402},[135,136],"（标签 ",[23,69387,69389,69407],{"className":69388,"translate":27},[31],[23,69390,69392],{"className":69391},[35],[37,69393,69394],{"xmlns":39},[42,69395,69396,69404],{},[45,69397,69398],{},[898,69399,69400,69402],{},[48,69401,660],{},[48,69403,33497],{},[114,69405,69406],{"encoding":116},"y_i",[23,69408,69410],{"className":69409,"ariaHidden":122},[121],[23,69411,69413,69416],{"className":69412},[126],[23,69414],{"className":69415,"style":672},[130],[23,69417,69419,69422],{"className":69418},[135],[23,69420,660],{"className":69421,"style":676},[135,136],[23,69423,69425],{"className":69424},[998],[23,69426,69428,69448],{"className":69427},[199,200],[23,69429,69431,69445],{"className":69430},[204],[23,69432,69434],{"className":69433,"style":33651},[208],[23,69435,69436,69439],{"style":21854},[23,69437],{"className":69438,"style":1015},[216],[23,69440,69442],{"className":69441},[1019,1020,1021,1022],[23,69443,33497],{"className":69444},[135,136,1022],[23,69446,297],{"className":69447},[296],[23,69449,69451],{"className":69450},[204],[23,69452,69454],{"className":69453,"style":2048},[208],[23,69455],{}," 已吸收进 ",[23,69458,69460,69473],{"className":69459,"translate":27},[31],[23,69461,69463],{"className":69462},[35],[37,69464,69465],{"xmlns":39},[42,69466,69467,69471],{},[45,69468,69469],{},[48,69470,555],{},[114,69472,555],{"encoding":116},[23,69474,69476],{"className":69475,"ariaHidden":122},[121],[23,69477,69479,69482],{"className":69478},[126],[23,69480],{"className":69481,"style":470},[130],[23,69483,555],{"className":69484},[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。",[11,69487,69488,69489,69517,69518,69592,69593,69664,69665,69716],{},"第二，",[23,69490,69492,69505],{"className":69491,"translate":27},[31],[23,69493,69495],{"className":69494},[35],[37,69496,69497],{"xmlns":39},[42,69498,69499,69503],{},[45,69500,69501],{},[48,69502,68146],{"mathvariant":111},[114,69504,69104],{"encoding":116},[23,69506,69508],{"className":69507,"ariaHidden":122},[121],[23,69509,69511,69514],{"className":69510},[126],[23,69512],{"className":69513,"style":470},[130],[23,69515,68146],{"className":69516},[135]," 只要求非减，不要求凸、不要求可微。式 (1) 中的 ",[23,69519,69521,69545],{"className":69520,"translate":27},[31],[23,69522,69524],{"className":69523},[35],[37,69525,69526],{"xmlns":39},[42,69527,69528,69542],{},[45,69529,69530,69532,69534,69536],{},[48,69531,61761],{},[48,69533,20684],{"mathvariant":111},[48,69535,8340],{},[4518,69537,69538,69540],{},[48,69539,20684],{"mathvariant":111},[2581,69541,9956],{},[114,69543,69544],{"encoding":116},"\\lambda\\|f\\|^2",[23,69546,69548],{"className":69547,"ariaHidden":122},[121],[23,69549,69551,69554,69557,69560,69563],{"className":69550},[126],[23,69552],{"className":69553,"style":10665},[130],[23,69555,61761],{"className":69556},[135,136],[23,69558,20684],{"className":69559},[135],[23,69561,8340],{"className":69562,"style":11402},[135,136],[23,69564,69566,69569],{"className":69565},[135],[23,69567,20684],{"className":69568},[135],[23,69570,69572],{"className":69571},[998],[23,69573,69575],{"className":69574},[199],[23,69576,69578],{"className":69577},[204],[23,69579,69581],{"className":69580,"style":4693},[208],[23,69582,69583,69586],{"style":4696},[23,69584],{"className":69585,"style":1015},[216],[23,69587,69589],{"className":69588},[1019,1020,1021,1022],[23,69590,9956],{"className":69591},[135,1022]," 对应 ",[23,69594,69596,69622],{"className":69595,"translate":27},[31],[23,69597,69599],{"className":69598},[35],[37,69600,69601],{"xmlns":39},[42,69602,69603,69619],{},[45,69604,69605,69607,69609,69611,69613,69615,69617],{},[48,69606,68146],{"mathvariant":111},[51,69608,54],{"stretchy":53},[48,69610,5345],{},[51,69612,66],{"stretchy":53},[51,69614,69],{},[48,69616,61761],{},[48,69618,5345],{},[114,69620,69621],{"encoding":116},"\\Omega(t)=\\lambda t",[23,69623,69625,69652],{"className":69624,"ariaHidden":122},[121],[23,69626,69628,69631,69634,69637,69640,69643,69646,69649],{"className":69627},[126],[23,69629],{"className":69630,"style":131},[130],[23,69632,68146],{"className":69633},[135],[23,69635,54],{"className":69636},[140],[23,69638,5345],{"className":69639},[135,136],[23,69641,66],{"className":69642},[169],[23,69644],{"className":69645,"style":149},[148],[23,69647,69],{"className":69648},[153],[23,69650],{"className":69651,"style":149},[148],[23,69653,69655,69658,69661],{"className":69654},[126],[23,69656],{"className":69657,"style":779},[130],[23,69659,61761],{"className":69660},[135,136],[23,69662,5345],{"className":69663},[135,136],"（",[23,69666,69668,69686],{"className":69667,"translate":27},[31],[23,69669,69671],{"className":69670},[35],[37,69672,69673],{"xmlns":39},[42,69674,69675,69683],{},[45,69676,69677,69679,69681],{},[48,69678,61761],{},[51,69680,2579],{},[2581,69682,1532],{},[114,69684,69685],{"encoding":116},"\\lambda>0",[23,69687,69689,69707],{"className":69688,"ariaHidden":122},[121],[23,69690,69692,69695,69698,69701,69704],{"className":69691},[126],[23,69693],{"className":69694,"style":4450},[130],[23,69696,61761],{"className":69697},[135,136],[23,69699],{"className":69700,"style":149},[148],[23,69702,2579],{"className":69703},[153],[23,69705],{"className":69706,"style":149},[148],[23,69708,69710,69713],{"className":69709},[126],[23,69711],{"className":69712,"style":2770},[130],[23,69714,1532],{"className":69715},[135]," 时严格递增），是最常见的特例。",[11,69718,69719,69720,69723,69724,69752],{},"第三，第 1 条是",[18,69721,69722],{},"存在性","，第 2 条才是「任意最优解」。如果 ",[23,69725,69727,69740],{"className":69726,"translate":27},[31],[23,69728,69730],{"className":69729},[35],[37,69731,69732],{"xmlns":39},[42,69733,69734,69738],{},[45,69735,69736],{},[48,69737,68146],{"mathvariant":111},[114,69739,69104],{"encoding":116},[23,69741,69743],{"className":69742,"ariaHidden":122},[121],[23,69744,69746,69749],{"className":69745},[126],[23,69747],{"className":69748,"style":470},[130],[23,69750,68146],{"className":69751},[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。",[2501,69754,69756],{"id":69755},"_32-证明正交分解","3.2 证明：正交分解",[11,69758,69759],{},"证明的核心是一个正交分解。令",[23,69761,69763],{"className":69762,"translate":27},[26],[23,69764,69766,69839],{"className":69765,"translate":27},[31],[23,69767,69769],{"className":69768},[35],[37,69770,69771],{"xmlns":39,"display":40},[42,69772,69773,69836],{},[45,69774,69775,69781,69783,69785,69787,69789,69791,69793,69795,69797,69803,69805,69807,69809,69811,69813,69815,69817,69819,69825,69827,69829,69832,69834],{},[898,69776,69777,69779],{},[48,69778,26253],{"mathvariant":375},[2581,69780,1532],{},[51,69782,69],{},[48,69784,23],{"mathvariant":111},[51,69786,6961],{},[51,69788,9014],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[48,69790,4836],{},[51,69792,54],{"stretchy":53},[51,69794,5758],{},[51,69796,747],{"separator":122},[898,69798,69799,69801],{},[48,69800,753],{},[2581,69802,4435],{},[51,69804,66],{"stretchy":53},[51,69806,747],{"separator":122},[51,69808,62563],{},[51,69810,747],{"separator":122},[48,69812,4836],{},[51,69814,54],{"stretchy":53},[51,69816,5758],{},[51,69818,747],{"separator":122},[898,69820,69821,69823],{},[48,69822,753],{},[48,69824,4806],{},[51,69826,66],{"stretchy":53},[51,69828,9024],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[51,69830,69831],{},"⊆",[48,69833,26253],{"mathvariant":375},[48,69835,112],{"mathvariant":111},[114,69837,69838],{"encoding":116},"\\mathcal H_0=\\operatorname{span}\\bigl\\{k(\\cdot,x_1),\\dots,k(\\cdot,x_n)\\bigr\\}\\subseteq\\mathcal H.",[23,69840,69842,69897,70064],{"className":69841,"ariaHidden":122},[121],[23,69843,69845,69848,69888,69891,69894],{"className":69844},[126],[23,69846],{"className":69847,"style":42831},[130],[23,69849,69851,69854],{"className":69850},[135],[23,69852,26253],{"className":69853,"style":45028},[135,412],[23,69855,69857],{"className":69856},[998],[23,69858,69860,69880],{"className":69859},[199,200],[23,69861,69863,69877],{"className":69862},[204],[23,69864,69866],{"className":69865,"style":18651},[208],[23,69867,69868,69871],{"style":61635},[23,69869],{"className":69870,"style":1015},[216],[23,69872,69874],{"className":69873},[1019,1020,1021,1022],[23,69875,1532],{"className":69876},[135,1022],[23,69878,297],{"className":69879},[296],[23,69881,69883],{"className":69882},[204],[23,69884,69886],{"className":69885,"style":2048},[208],[23,69887],{},[23,69889],{"className":69890,"style":149},[148],[23,69892,69],{"className":69893},[153],[23,69895],{"className":69896,"style":149},[148],[23,69898,69900,69903,69909,69915,69918,69921,69924,69927,69930,69970,69973,69976,69979,69982,69985,69988,69991,69994,69997,70000,70003,70006,70046,70049,70055,70058,70061],{"className":69899},[126],[23,69901],{"className":69902,"style":2213},[130],[23,69904,69906],{"className":69905},[1786],[23,69907,23],{"className":69908},[135,5391],[23,69910,69912],{"className":69911},[140],[23,69913,9014],{"className":69914},[1066,1067],[23,69916,4836],{"className":69917,"style":4851},[135,136],[23,69919,54],{"className":69920},[140],[23,69922,5758],{"className":69923},[135],[23,69925,747],{"className":69926},[824],[23,69928],{"className":69929,"style":280},[148],[23,69931,69933,69936],{"className":69932},[135],[23,69934,753],{"className":69935},[135,136],[23,69937,69939],{"className":69938},[998],[23,69940,69942,69962],{"className":69941},[199,200],[23,69943,69945,69959],{"className":69944},[204],[23,69946,69948],{"className":69947,"style":18651},[208],[23,69949,69950,69953],{"style":2029},[23,69951],{"className":69952,"style":1015},[216],[23,69954,69956],{"className":69955},[1019,1020,1021,1022],[23,69957,4435],{"className":69958},[135,1022],[23,69960,297],{"className":69961},[296],[23,69963,69965],{"className":69964},[204],[23,69966,69968],{"className":69967,"style":2048},[208],[23,69969],{},[23,69971,66],{"className":69972},[169],[23,69974,747],{"className":69975},[824],[23,69977],{"className":69978,"style":280},[148],[23,69980,62563],{"className":69981},[1839],[23,69983],{"className":69984,"style":280},[148],[23,69986,747],{"className":69987},[824],[23,69989],{"className":69990,"style":280},[148],[23,69992,4836],{"className":69993,"style":4851},[135,136],[23,69995,54],{"className":69996},[140],[23,69998,5758],{"className":69999},[135],[23,70001,747],{"className":70002},[824],[23,70004],{"className":70005,"style":280},[148],[23,70007,70009,70012],{"className":70008},[135],[23,70010,753],{"className":70011},[135,136],[23,70013,70015],{"className":70014},[998],[23,70016,70018,70038],{"className":70017},[199,200],[23,70019,70021,70035],{"className":70020},[204],[23,70022,70024],{"className":70023,"style":6499},[208],[23,70025,70026,70029],{"style":2029},[23,70027],{"className":70028,"style":1015},[216],[23,70030,70032],{"className":70031},[1019,1020,1021,1022],[23,70033,4806],{"className":70034},[135,136,1022],[23,70036,297],{"className":70037},[296],[23,70039,70041],{"className":70040},[204],[23,70042,70044],{"className":70043,"style":2048},[208],[23,70045],{},[23,70047,66],{"className":70048},[169],[23,70050,70052],{"className":70051},[169],[23,70053,9024],{"className":70054},[1066,1067],[23,70056],{"className":70057,"style":149},[148],[23,70059,69831],{"className":70060},[153],[23,70062],{"className":70063,"style":149},[148],[23,70065,70067,70070,70073],{"className":70066},[126],[23,70068],{"className":70069,"style":470},[130],[23,70071,26253],{"className":70072,"style":45028},[135,412],[23,70074,112],{"className":70075},[135],[11,70077,70078,70079,70130],{},"对任意 ",[23,70080,70082,70100],{"className":70081,"translate":27},[31],[23,70083,70085],{"className":70084},[35],[37,70086,70087],{"xmlns":39},[42,70088,70089,70097],{},[45,70090,70091,70093,70095],{},[48,70092,8340],{},[51,70094,372],{},[48,70096,26253],{"mathvariant":375},[114,70098,70099],{"encoding":116},"f\\in\\mathcal H",[23,70101,70103,70121],{"className":70102,"ariaHidden":122},[121],[23,70104,70106,70109,70112,70115,70118],{"className":70105},[126],[23,70107],{"className":70108,"style":5414},[130],[23,70110,8340],{"className":70111,"style":11402},[135,136],[23,70113],{"className":70114,"style":149},[148],[23,70116,372],{"className":70117},[153],[23,70119],{"className":70120,"style":149},[148],[23,70122,70124,70127],{"className":70123},[126],[23,70125],{"className":70126,"style":470},[130],[23,70128,26253],{"className":70129,"style":45028},[135,412],"，把它分解为",[23,70132,70134],{"className":70133,"translate":27},[26],[23,70135,70137,70208],{"className":70136,"translate":27},[31],[23,70138,70140],{"className":70139},[35],[37,70141,70142],{"xmlns":39,"display":40},[42,70143,70144,70205],{},[45,70145,70146,70148,70150,70156,70158,70165,70167,70169,70175,70177,70183,70185,70187,70193,70195,70203],{},[48,70147,8340],{},[51,70149,69],{},[898,70151,70152,70154],{},[48,70153,8340],{},[51,70155,20684],{},[51,70157,5106],{},[898,70159,70160,70162],{},[48,70161,8340],{},[51,70163,70164],{},"⊥",[51,70166,747],{"separator":122},[148,70168],{"width":750},[898,70170,70171,70173],{},[48,70172,8340],{},[51,70174,20684],{},[51,70176,372],{},[898,70178,70179,70181],{},[48,70180,26253],{"mathvariant":375},[2581,70182,1532],{},[51,70184,747],{"separator":122},[148,70186],{"width":9919},[898,70188,70189,70191],{},[48,70190,8340],{},[51,70192,70164],{},[51,70194,372],{},[19141,70196,70197,70199,70201],{},[48,70198,26253],{"mathvariant":375},[2581,70200,1532],{},[51,70202,70164],{},[48,70204,112],{"mathvariant":111},[114,70206,70207],{"encoding":116},"f=f_\\parallel+f_\\perp,\\qquad f_\\parallel\\in\\mathcal H_0,\\quad f_\\perp\\in\\mathcal H_0^\\perp.",[23,70209,70211,70229,70286,70390,70494],{"className":70210,"ariaHidden":122},[121],[23,70212,70214,70217,70220,70223,70226],{"className":70213},[126],[23,70215],{"className":70216,"style":5414},[130],[23,70218,8340],{"className":70219,"style":11402},[135,136],[23,70221],{"className":70222,"style":149},[148],[23,70224,69],{"className":70225},[153],[23,70227],{"className":70228,"style":149},[148],[23,70230,70232,70236,70277,70280,70283],{"className":70231},[126],[23,70233],{"className":70234,"style":70235},[130],"height:1.0496em;vertical-align:-0.3552em;",[23,70237,70239,70242],{"className":70238},[135],[23,70240,8340],{"className":70241,"style":11402},[135,136],[23,70243,70245],{"className":70244},[998],[23,70246,70248,70269],{"className":70247},[199,200],[23,70249,70251,70266],{"className":70250},[204],[23,70252,70254],{"className":70253,"style":1008},[208],[23,70255,70257,70260],{"style":70256},"top:-2.5198em;margin-left:-0.1076em;margin-right:0.05em;",[23,70258],{"className":70259,"style":1015},[216],[23,70261,70263],{"className":70262},[1019,1020,1021,1022],[23,70264,20684],{"className":70265},[153,1022],[23,70267,297],{"className":70268},[296],[23,70270,70272],{"className":70271},[204],[23,70273,70275],{"className":70274,"style":1057},[208],[23,70276],{},[23,70278],{"className":70279,"style":392},[148],[23,70281,5106],{"className":70282},[612],[23,70284],{"className":70285,"style":392},[148],[23,70287,70289,70292,70332,70335,70338,70341,70381,70384,70387],{"className":70288},[126],[23,70290],{"className":70291,"style":70235},[130],[23,70293,70295,70298],{"className":70294},[135],[23,70296,8340],{"className":70297,"style":11402},[135,136],[23,70299,70301],{"className":70300},[998],[23,70302,70304,70324],{"className":70303},[199,200],[23,70305,70307,70321],{"className":70306},[204],[23,70308,70310],{"className":70309,"style":34196},[208],[23,70311,70312,70315],{"style":29550},[23,70313],{"className":70314,"style":1015},[216],[23,70316,70318],{"className":70317},[1019,1020,1021,1022],[23,70319,70164],{"className":70320},[153,1022],[23,70322,297],{"className":70323},[296],[23,70325,70327],{"className":70326},[204],[23,70328,70330],{"className":70329,"style":2048},[208],[23,70331],{},[23,70333,747],{"className":70334},[824],[23,70336],{"className":70337,"style":828},[148],[23,70339],{"className":70340,"style":280},[148],[23,70342,70344,70347],{"className":70343},[135],[23,70345,8340],{"className":70346,"style":11402},[135,136],[23,70348,70350],{"className":70349},[998],[23,70351,70353,70373],{"className":70352},[199,200],[23,70354,70356,70370],{"className":70355},[204],[23,70357,70359],{"className":70358,"style":1008},[208],[23,70360,70361,70364],{"style":70256},[23,70362],{"className":70363,"style":1015},[216],[23,70365,70367],{"className":70366},[1019,1020,1021,1022],[23,70368,20684],{"className":70369},[153,1022],[23,70371,297],{"className":70372},[296],[23,70374,70376],{"className":70375},[204],[23,70377,70379],{"className":70378,"style":1057},[208],[23,70380],{},[23,70382],{"className":70383,"style":149},[148],[23,70385,372],{"className":70386},[153],[23,70388],{"className":70389,"style":149},[148],[23,70391,70393,70396,70436,70439,70442,70445,70485,70488,70491],{"className":70392},[126],[23,70394],{"className":70395,"style":5414},[130],[23,70397,70399,70402],{"className":70398},[135],[23,70400,26253],{"className":70401,"style":45028},[135,412],[23,70403,70405],{"className":70404},[998],[23,70406,70408,70428],{"className":70407},[199,200],[23,70409,70411,70425],{"className":70410},[204],[23,70412,70414],{"className":70413,"style":18651},[208],[23,70415,70416,70419],{"style":61635},[23,70417],{"className":70418,"style":1015},[216],[23,70420,70422],{"className":70421},[1019,1020,1021,1022],[23,70423,1532],{"className":70424},[135,1022],[23,70426,297],{"className":70427},[296],[23,70429,70431],{"className":70430},[204],[23,70432,70434],{"className":70433,"style":2048},[208],[23,70435],{},[23,70437,747],{"className":70438},[824],[23,70440],{"className":70441,"style":14441},[148],[23,70443],{"className":70444,"style":280},[148],[23,70446,70448,70451],{"className":70447},[135],[23,70449,8340],{"className":70450,"style":11402},[135,136],[23,70452,70454],{"className":70453},[998],[23,70455,70457,70477],{"className":70456},[199,200],[23,70458,70460,70474],{"className":70459},[204],[23,70461,70463],{"className":70462,"style":34196},[208],[23,70464,70465,70468],{"style":29550},[23,70466],{"className":70467,"style":1015},[216],[23,70469,70471],{"className":70470},[1019,1020,1021,1022],[23,70472,70164],{"className":70473},[153,1022],[23,70475,297],{"className":70476},[296],[23,70478,70480],{"className":70479},[204],[23,70481,70483],{"className":70482,"style":2048},[208],[23,70484],{},[23,70486],{"className":70487,"style":149},[148],[23,70489,372],{"className":70490},[153],[23,70492],{"className":70493,"style":149},[148],[23,70495,70497,70501,70553],{"className":70496},[126],[23,70498],{"className":70499,"style":70500},[130],"height:1.1461em;vertical-align:-0.247em;",[23,70502,70504,70507],{"className":70503},[135],[23,70505,26253],{"className":70506,"style":45028},[135,412],[23,70508,70510],{"className":70509},[998],[23,70511,70513,70545],{"className":70512},[199,200],[23,70514,70516,70542],{"className":70515},[204],[23,70517,70519,70531],{"className":70518,"style":4986},[208],[23,70520,70522,70525],{"style":70521},"top:-2.453em;margin-left:-0.0097em;margin-right:0.05em;",[23,70523],{"className":70524,"style":1015},[216],[23,70526,70528],{"className":70527},[1019,1020,1021,1022],[23,70529,1532],{"className":70530},[135,1022],[23,70532,70533,70536],{"style":4989},[23,70534],{"className":70535,"style":1015},[216],[23,70537,70539],{"className":70538},[1019,1020,1021,1022],[23,70540,70164],{"className":70541},[153,1022],[23,70543,297],{"className":70544},[296],[23,70546,70548],{"className":70547},[204],[23,70549,70551],{"className":70550,"style":19379},[208],[23,70552],{},[23,70554,112],{"className":70555},[135],[11,70557,70558,70559,49303,70629,70790],{},"对每个训练点 ",[23,70560,70562,70580],{"className":70561,"translate":27},[31],[23,70563,70565],{"className":70564},[35],[37,70566,70567],{"xmlns":39},[42,70568,70569,70577],{},[45,70570,70571],{},[898,70572,70573,70575],{},[48,70574,753],{},[48,70576,33497],{},[114,70578,70579],{"encoding":116},"x_i",[23,70581,70583],{"className":70582,"ariaHidden":122},[121],[23,70584,70586,70589],{"className":70585},[126],[23,70587],{"className":70588,"style":12734},[130],[23,70590,70592,70595],{"className":70591},[135],[23,70593,753],{"className":70594},[135,136],[23,70596,70598],{"className":70597},[998],[23,70599,70601,70621],{"className":70600},[199,200],[23,70602,70604,70618],{"className":70603},[204],[23,70605,70607],{"className":70606,"style":33651},[208],[23,70608,70609,70612],{"style":2029},[23,70610],{"className":70611,"style":1015},[216],[23,70613,70615],{"className":70614},[1019,1020,1021,1022],[23,70616,33497],{"className":70617},[135,136,1022],[23,70619,297],{"className":70620},[296],[23,70622,70624],{"className":70623},[204],[23,70625,70627],{"className":70626,"style":2048},[208],[23,70628],{},[23,70630,70632,70668],{"className":70631,"translate":27},[31],[23,70633,70635],{"className":70634},[35],[37,70636,70637],{"xmlns":39},[42,70638,70639,70665],{},[45,70640,70641,70643,70645,70647,70649,70655,70657,70659],{},[48,70642,4836],{},[51,70644,54],{"stretchy":53},[51,70646,5758],{},[51,70648,747],{"separator":122},[898,70650,70651,70653],{},[48,70652,753],{},[48,70654,33497],{},[51,70656,66],{"stretchy":53},[51,70658,372],{},[898,70660,70661,70663],{},[48,70662,26253],{"mathvariant":375},[2581,70664,1532],{},[114,70666,70667],{"encoding":116},"k(\\cdot,x_i)\\in\\mathcal H_0",[23,70669,70671,70744],{"className":70670,"ariaHidden":122},[121],[23,70672,70674,70677,70680,70683,70686,70689,70692,70732,70735,70738,70741],{"className":70673},[126],[23,70675],{"className":70676,"style":131},[130],[23,70678,4836],{"className":70679,"style":4851},[135,136],[23,70681,54],{"className":70682},[140],[23,70684,5758],{"className":70685},[135],[23,70687,747],{"className":70688},[824],[23,70690],{"className":70691,"style":280},[148],[23,70693,70695,70698],{"className":70694},[135],[23,70696,753],{"className":70697},[135,136],[23,70699,70701],{"className":70700},[998],[23,70702,70704,70724],{"className":70703},[199,200],[23,70705,70707,70721],{"className":70706},[204],[23,70708,70710],{"className":70709,"style":33651},[208],[23,70711,70712,70715],{"style":2029},[23,70713],{"className":70714,"style":1015},[216],[23,70716,70718],{"className":70717},[1019,1020,1021,1022],[23,70719,33497],{"className":70720},[135,136,1022],[23,70722,297],{"className":70723},[296],[23,70725,70727],{"className":70726},[204],[23,70728,70730],{"className":70729,"style":2048},[208],[23,70731],{},[23,70733,66],{"className":70734},[169],[23,70736],{"className":70737,"style":149},[148],[23,70739,372],{"className":70740},[153],[23,70742],{"className":70743,"style":149},[148],[23,70745,70747,70750],{"className":70746},[126],[23,70748],{"className":70749,"style":42831},[130],[23,70751,70753,70756],{"className":70752},[135],[23,70754,26253],{"className":70755,"style":45028},[135,412],[23,70757,70759],{"className":70758},[998],[23,70760,70762,70782],{"className":70761},[199,200],[23,70763,70765,70779],{"className":70764},[204],[23,70766,70768],{"className":70767,"style":18651},[208],[23,70769,70770,70773],{"style":61635},[23,70771],{"className":70772,"style":1015},[216],[23,70774,70776],{"className":70775},[1019,1020,1021,1022],[23,70777,1532],{"className":70778},[135,1022],[23,70780,297],{"className":70781},[296],[23,70783,70785],{"className":70784},[204],[23,70786,70788],{"className":70787,"style":2048},[208],[23,70789],{},"，所以由再生性",[23,70792,70794],{"className":70793,"translate":27},[26],[23,70795,70797,70929],{"className":70796,"translate":27},[31],[23,70798,70800],{"className":70799},[35],[37,70801,70802],{"xmlns":39,"display":40},[42,70803,70804,70926],{},[45,70805,70806,70808,70810,70816,70818,70820,70822,70824,70826,70828,70830,70832,70834,70840,70842,70844,70846,70848,70850,70856,70858,70860,70862,70864,70866,70872,70874,70876,70878,70880,70886,70888,70890,70892,70894,70896,70902,70904,70906,70908,70914,70916,70922,70924],{},[48,70807,8340],{},[51,70809,54],{"stretchy":53},[898,70811,70812,70814],{},[48,70813,753],{},[48,70815,33497],{},[51,70817,66],{"stretchy":53},[51,70819,69],{},[51,70821,61189],{"stretchy":53},[48,70823,8340],{},[51,70825,747],{"separator":122},[48,70827,4836],{},[51,70829,54],{"stretchy":53},[51,70831,5758],{},[51,70833,747],{"separator":122},[898,70835,70836,70838],{},[48,70837,753],{},[48,70839,33497],{},[51,70841,66],{"stretchy":53},[51,70843,61214],{"stretchy":53},[148,70845],{"linebreak":29629},[51,70847,69],{},[51,70849,61189],{"stretchy":53},[898,70851,70852,70854],{},[48,70853,8340],{},[51,70855,20684],{},[51,70857,747],{"separator":122},[48,70859,4836],{},[51,70861,54],{"stretchy":53},[51,70863,5758],{},[51,70865,747],{"separator":122},[898,70867,70868,70870],{},[48,70869,753],{},[48,70871,33497],{},[51,70873,66],{"stretchy":53},[51,70875,61214],{"stretchy":53},[51,70877,5106],{},[51,70879,61189],{"stretchy":53},[898,70881,70882,70884],{},[48,70883,8340],{},[51,70885,70164],{},[51,70887,747],{"separator":122},[48,70889,4836],{},[51,70891,54],{"stretchy":53},[51,70893,5758],{},[51,70895,747],{"separator":122},[898,70897,70898,70900],{},[48,70899,753],{},[48,70901,33497],{},[51,70903,66],{"stretchy":53},[51,70905,61214],{"stretchy":53},[51,70907,69],{},[898,70909,70910,70912],{},[48,70911,8340],{},[51,70913,20684],{},[51,70915,54],{"stretchy":53},[898,70917,70918,70920],{},[48,70919,753],{},[48,70921,33497],{},[51,70923,66],{"stretchy":53},[48,70925,112],{"mathvariant":111},[114,70927,70928],{"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).",[23,70930,70932,70996,71072,71075,71088,71211,71333],{"className":70931,"ariaHidden":122},[121],[23,70933,70935,70938,70941,70944,70984,70987,70990,70993],{"className":70934},[126],[23,70936],{"className":70937,"style":131},[130],[23,70939,8340],{"className":70940,"style":11402},[135,136],[23,70942,54],{"className":70943},[140],[23,70945,70947,70950],{"className":70946},[135],[23,70948,753],{"className":70949},[135,136],[23,70951,70953],{"className":70952},[998],[23,70954,70956,70976],{"className":70955},[199,200],[23,70957,70959,70973],{"className":70958},[204],[23,70960,70962],{"className":70961,"style":33651},[208],[23,70963,70964,70967],{"style":2029},[23,70965],{"className":70966,"style":1015},[216],[23,70968,70970],{"className":70969},[1019,1020,1021,1022],[23,70971,33497],{"className":70972},[135,136,1022],[23,70974,297],{"className":70975},[296],[23,70977,70979],{"className":70978},[204],[23,70980,70982],{"className":70981,"style":2048},[208],[23,70983],{},[23,70985,66],{"className":70986},[169],[23,70988],{"className":70989,"style":149},[148],[23,70991,69],{"className":70992},[153],[23,70994],{"className":70995,"style":149},[148],[23,70997,70999,71002,71005,71008,71011,71014,71017,71020,71023,71026,71029,71069],{"className":70998},[126],[23,71000],{"className":71001,"style":131},[130],[23,71003,61189],{"className":71004},[140],[23,71006,8340],{"className":71007,"style":11402},[135,136],[23,71009,747],{"className":71010},[824],[23,71012],{"className":71013,"style":280},[148],[23,71015,4836],{"className":71016,"style":4851},[135,136],[23,71018,54],{"className":71019},[140],[23,71021,5758],{"className":71022},[135],[23,71024,747],{"className":71025},[824],[23,71027],{"className":71028,"style":280},[148],[23,71030,71032,71035],{"className":71031},[135],[23,71033,753],{"className":71034},[135,136],[23,71036,71038],{"className":71037},[998],[23,71039,71041,71061],{"className":71040},[199,200],[23,71042,71044,71058],{"className":71043},[204],[23,71045,71047],{"className":71046,"style":33651},[208],[23,71048,71049,71052],{"style":2029},[23,71050],{"className":71051,"style":1015},[216],[23,71053,71055],{"className":71054},[1019,1020,1021,1022],[23,71056,33497],{"className":71057},[135,136,1022],[23,71059,297],{"className":71060},[296],[23,71062,71064],{"className":71063},[204],[23,71065,71067],{"className":71066,"style":2048},[208],[23,71068],{},[23,71070,61354],{"className":71071},[169],[23,71073],{"className":71074},[148,29629],[23,71076,71078,71082,71085],{"className":71077},[126],[23,71079],{"className":71080,"style":71081},[130],"height:0.3669em;",[23,71083,69],{"className":71084},[153],[23,71086],{"className":71087,"style":149},[148],[23,71089,71091,71095,71098,71138,71141,71144,71147,71150,71153,71156,71159,71199,71202,71205,71208],{"className":71090},[126],[23,71092],{"className":71093,"style":71094},[130],"height:1.1052em;vertical-align:-0.3552em;",[23,71096,61189],{"className":71097},[140],[23,71099,71101,71104],{"className":71100},[135],[23,71102,8340],{"className":71103,"style":11402},[135,136],[23,71105,71107],{"className":71106},[998],[23,71108,71110,71130],{"className":71109},[199,200],[23,71111,71113,71127],{"className":71112},[204],[23,71114,71116],{"className":71115,"style":1008},[208],[23,71117,71118,71121],{"style":70256},[23,71119],{"className":71120,"style":1015},[216],[23,71122,71124],{"className":71123},[1019,1020,1021,1022],[23,71125,20684],{"className":71126},[153,1022],[23,71128,297],{"className":71129},[296],[23,71131,71133],{"className":71132},[204],[23,71134,71136],{"className":71135,"style":1057},[208],[23,71137],{},[23,71139,747],{"className":71140},[824],[23,71142],{"className":71143,"style":280},[148],[23,71145,4836],{"className":71146,"style":4851},[135,136],[23,71148,54],{"className":71149},[140],[23,71151,5758],{"className":71152},[135],[23,71154,747],{"className":71155},[824],[23,71157],{"className":71158,"style":280},[148],[23,71160,71162,71165],{"className":71161},[135],[23,71163,753],{"className":71164},[135,136],[23,71166,71168],{"className":71167},[998],[23,71169,71171,71191],{"className":71170},[199,200],[23,71172,71174,71188],{"className":71173},[204],[23,71175,71177],{"className":71176,"style":33651},[208],[23,71178,71179,71182],{"style":2029},[23,71180],{"className":71181,"style":1015},[216],[23,71183,71185],{"className":71184},[1019,1020,1021,1022],[23,71186,33497],{"className":71187},[135,136,1022],[23,71189,297],{"className":71190},[296],[23,71192,71194],{"className":71193},[204],[23,71195,71197],{"className":71196,"style":2048},[208],[23,71198],{},[23,71200,61354],{"className":71201},[169],[23,71203],{"className":71204,"style":392},[148],[23,71206,5106],{"className":71207},[612],[23,71209],{"className":71210,"style":392},[148],[23,71212,71214,71217,71220,71260,71263,71266,71269,71272,71275,71278,71281,71321,71324,71327,71330],{"className":71213},[126],[23,71215],{"className":71216,"style":131},[130],[23,71218,61189],{"className":71219},[140],[23,71221,71223,71226],{"className":71222},[135],[23,71224,8340],{"className":71225,"style":11402},[135,136],[23,71227,71229],{"className":71228},[998],[23,71230,71232,71252],{"className":71231},[199,200],[23,71233,71235,71249],{"className":71234},[204],[23,71236,71238],{"className":71237,"style":34196},[208],[23,71239,71240,71243],{"style":29550},[23,71241],{"className":71242,"style":1015},[216],[23,71244,71246],{"className":71245},[1019,1020,1021,1022],[23,71247,70164],{"className":71248},[153,1022],[23,71250,297],{"className":71251},[296],[23,71253,71255],{"className":71254},[204],[23,71256,71258],{"className":71257,"style":2048},[208],[23,71259],{},[23,71261,747],{"className":71262},[824],[23,71264],{"className":71265,"style":280},[148],[23,71267,4836],{"className":71268,"style":4851},[135,136],[23,71270,54],{"className":71271},[140],[23,71273,5758],{"className":71274},[135],[23,71276,747],{"className":71277},[824],[23,71279],{"className":71280,"style":280},[148],[23,71282,71284,71287],{"className":71283},[135],[23,71285,753],{"className":71286},[135,136],[23,71288,71290],{"className":71289},[998],[23,71291,71293,71313],{"className":71292},[199,200],[23,71294,71296,71310],{"className":71295},[204],[23,71297,71299],{"className":71298,"style":33651},[208],[23,71300,71301,71304],{"style":2029},[23,71302],{"className":71303,"style":1015},[216],[23,71305,71307],{"className":71306},[1019,1020,1021,1022],[23,71308,33497],{"className":71309},[135,136,1022],[23,71311,297],{"className":71312},[296],[23,71314,71316],{"className":71315},[204],[23,71317,71319],{"className":71318,"style":2048},[208],[23,71320],{},[23,71322,61354],{"className":71323},[169],[23,71325],{"className":71326,"style":149},[148],[23,71328,69],{"className":71329},[153],[23,71331],{"className":71332,"style":149},[148],[23,71334,71336,71339,71379,71382,71422,71425],{"className":71335},[126],[23,71337],{"className":71338,"style":71094},[130],[23,71340,71342,71345],{"className":71341},[135],[23,71343,8340],{"className":71344,"style":11402},[135,136],[23,71346,71348],{"className":71347},[998],[23,71349,71351,71371],{"className":71350},[199,200],[23,71352,71354,71368],{"className":71353},[204],[23,71355,71357],{"className":71356,"style":1008},[208],[23,71358,71359,71362],{"style":70256},[23,71360],{"className":71361,"style":1015},[216],[23,71363,71365],{"className":71364},[1019,1020,1021,1022],[23,71366,20684],{"className":71367},[153,1022],[23,71369,297],{"className":71370},[296],[23,71372,71374],{"className":71373},[204],[23,71375,71377],{"className":71376,"style":1057},[208],[23,71378],{},[23,71380,54],{"className":71381},[140],[23,71383,71385,71388],{"className":71384},[135],[23,71386,753],{"className":71387},[135,136],[23,71389,71391],{"className":71390},[998],[23,71392,71394,71414],{"className":71393},[199,200],[23,71395,71397,71411],{"className":71396},[204],[23,71398,71400],{"className":71399,"style":33651},[208],[23,71401,71402,71405],{"style":2029},[23,71403],{"className":71404,"style":1015},[216],[23,71406,71408],{"className":71407},[1019,1020,1021,1022],[23,71409,33497],{"className":71410},[135,136,1022],[23,71412,297],{"className":71413},[296],[23,71415,71417],{"className":71416},[204],[23,71418,71420],{"className":71419,"style":2048},[208],[23,71421],{},[23,71423,66],{"className":71424},[169],[23,71426,112],{"className":71427},[135],[11,71429,71430,71431,71504],{},"也就是说，",[18,71432,71433,71503],{},[23,71434,71436,71454],{"className":71435,"translate":27},[31],[23,71437,71439],{"className":71438},[35],[37,71440,71441],{"xmlns":39},[42,71442,71443,71451],{},[45,71444,71445],{},[898,71446,71447,71449],{},[48,71448,8340],{},[51,71450,70164],{},[114,71452,71453],{"encoding":116},"f_\\perp",[23,71455,71457],{"className":71456,"ariaHidden":122},[121],[23,71458,71460,71463],{"className":71459},[126],[23,71461],{"className":71462,"style":5414},[130],[23,71464,71466,71469],{"className":71465},[135],[23,71467,8340],{"className":71468,"style":11402},[135,136],[23,71470,71472],{"className":71471},[998],[23,71473,71475,71495],{"className":71474},[199,200],[23,71476,71478,71492],{"className":71477},[204],[23,71479,71481],{"className":71480,"style":34196},[208],[23,71482,71483,71486],{"style":29550},[23,71484],{"className":71485,"style":1015},[216],[23,71487,71489],{"className":71488},[1019,1020,1021,1022],[23,71490,70164],{"className":71491},[153,1022],[23,71493,297],{"className":71494},[296],[23,71496,71498],{"className":71497},[204],[23,71499,71501],{"className":71500,"style":2048},[208],[23,71502],{}," 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。",[11,71506,71507],{},"再看正则项。由勾股定理，",[23,71509,71511],{"className":71510,"translate":27},[26],[23,71512,71514,71594],{"className":71513,"translate":27},[31],[23,71515,71517],{"className":71516},[35],[37,71518,71519],{"xmlns":39,"display":40},[42,71520,71521,71591],{},[45,71522,71523,71525,71527,71535,71537,71539,71545,71553,71555,71557,71563,71571,71573,71575,71581,71589],{},[48,71524,20684],{"mathvariant":111},[48,71526,8340],{},[19141,71528,71529,71531,71533],{},[48,71530,20684],{"mathvariant":111},[48,71532,26253],{"mathvariant":375},[2581,71534,9956],{},[51,71536,69],{},[48,71538,20684],{"mathvariant":111},[898,71540,71541,71543],{},[48,71542,8340],{},[51,71544,20684],{},[19141,71546,71547,71549,71551],{},[48,71548,20684],{"mathvariant":111},[48,71550,26253],{"mathvariant":375},[2581,71552,9956],{},[51,71554,5106],{},[48,71556,20684],{"mathvariant":111},[898,71558,71559,71561],{},[48,71560,8340],{},[51,71562,70164],{},[19141,71564,71565,71567,71569],{},[48,71566,20684],{"mathvariant":111},[48,71568,26253],{"mathvariant":375},[2581,71570,9956],{},[51,71572,7382],{},[48,71574,20684],{"mathvariant":111},[898,71576,71577,71579],{},[48,71578,8340],{},[51,71580,20684],{},[19141,71582,71583,71585,71587],{},[48,71584,20684],{"mathvariant":111},[48,71586,26253],{"mathvariant":375},[2581,71588,9956],{},[48,71590,112],{"mathvariant":111},[114,71592,71593],{"encoding":116},"\\|f\\|_{\\mathcal H}^2=\\|f_\\parallel\\|_{\\mathcal H}^2+\\|f_\\perp\\|_{\\mathcal H}^2\n\\ge \\|f_\\parallel\\|_{\\mathcal H}^2.",[23,71595,71597,71672,71785,71897],{"className":71596,"ariaHidden":122},[121],[23,71598,71600,71603,71606,71609,71663,71666,71669],{"className":71599},[126],[23,71601],{"className":71602,"style":26503},[130],[23,71604,20684],{"className":71605},[135],[23,71607,8340],{"className":71608,"style":11402},[135,136],[23,71610,71612,71615],{"className":71611},[135],[23,71613,20684],{"className":71614},[135],[23,71616,71618],{"className":71617},[998],[23,71619,71621,71655],{"className":71620},[199,200],[23,71622,71624,71652],{"className":71623},[204],[23,71625,71627,71641],{"className":71626,"style":11006},[208],[23,71628,71629,71632],{"style":19349},[23,71630],{"className":71631,"style":1015},[216],[23,71633,71635],{"className":71634},[1019,1020,1021,1022],[23,71636,71638],{"className":71637},[135,1022],[23,71639,26253],{"className":71640,"style":45028},[135,412,1022],[23,71642,71643,71646],{"style":4989},[23,71644],{"className":71645,"style":1015},[216],[23,71647,71649],{"className":71648},[1019,1020,1021,1022],[23,71650,9956],{"className":71651},[135,1022],[23,71653,297],{"className":71654},[296],[23,71656,71658],{"className":71657},[204],[23,71659,71661],{"className":71660,"style":19379},[208],[23,71662],{},[23,71664],{"className":71665,"style":149},[148],[23,71667,69],{"className":71668},[153],[23,71670],{"className":71671,"style":149},[148],[23,71673,71675,71679,71682,71722,71776,71779,71782],{"className":71674},[126],[23,71676],{"className":71677,"style":71678},[130],"height:1.2193em;vertical-align:-0.3552em;",[23,71680,20684],{"className":71681},[135],[23,71683,71685,71688],{"className":71684},[135],[23,71686,8340],{"className":71687,"style":11402},[135,136],[23,71689,71691],{"className":71690},[998],[23,71692,71694,71714],{"className":71693},[199,200],[23,71695,71697,71711],{"className":71696},[204],[23,71698,71700],{"className":71699,"style":1008},[208],[23,71701,71702,71705],{"style":70256},[23,71703],{"className":71704,"style":1015},[216],[23,71706,71708],{"className":71707},[1019,1020,1021,1022],[23,71709,20684],{"className":71710},[153,1022],[23,71712,297],{"className":71713},[296],[23,71715,71717],{"className":71716},[204],[23,71718,71720],{"className":71719,"style":1057},[208],[23,71721],{},[23,71723,71725,71728],{"className":71724},[135],[23,71726,20684],{"className":71727},[135],[23,71729,71731],{"className":71730},[998],[23,71732,71734,71768],{"className":71733},[199,200],[23,71735,71737,71765],{"className":71736},[204],[23,71738,71740,71754],{"className":71739,"style":11006},[208],[23,71741,71742,71745],{"style":19349},[23,71743],{"className":71744,"style":1015},[216],[23,71746,71748],{"className":71747},[1019,1020,1021,1022],[23,71749,71751],{"className":71750},[135,1022],[23,71752,26253],{"className":71753,"style":45028},[135,412,1022],[23,71755,71756,71759],{"style":4989},[23,71757],{"className":71758,"style":1015},[216],[23,71760,71762],{"className":71761},[1019,1020,1021,1022],[23,71763,9956],{"className":71764},[135,1022],[23,71766,297],{"className":71767},[296],[23,71769,71771],{"className":71770},[204],[23,71772,71774],{"className":71773,"style":19379},[208],[23,71775],{},[23,71777],{"className":71778,"style":392},[148],[23,71780,5106],{"className":71781},[612],[23,71783],{"className":71784,"style":392},[148],[23,71786,71788,71791,71794,71834,71888,71891,71894],{"className":71787},[126],[23,71789],{"className":71790,"style":26503},[130],[23,71792,20684],{"className":71793},[135],[23,71795,71797,71800],{"className":71796},[135],[23,71798,8340],{"className":71799,"style":11402},[135,136],[23,71801,71803],{"className":71802},[998],[23,71804,71806,71826],{"className":71805},[199,200],[23,71807,71809,71823],{"className":71808},[204],[23,71810,71812],{"className":71811,"style":34196},[208],[23,71813,71814,71817],{"style":29550},[23,71815],{"className":71816,"style":1015},[216],[23,71818,71820],{"className":71819},[1019,1020,1021,1022],[23,71821,70164],{"className":71822},[153,1022],[23,71824,297],{"className":71825},[296],[23,71827,71829],{"className":71828},[204],[23,71830,71832],{"className":71831,"style":2048},[208],[23,71833],{},[23,71835,71837,71840],{"className":71836},[135],[23,71838,20684],{"className":71839},[135],[23,71841,71843],{"className":71842},[998],[23,71844,71846,71880],{"className":71845},[199,200],[23,71847,71849,71877],{"className":71848},[204],[23,71850,71852,71866],{"className":71851,"style":11006},[208],[23,71853,71854,71857],{"style":19349},[23,71855],{"className":71856,"style":1015},[216],[23,71858,71860],{"className":71859},[1019,1020,1021,1022],[23,71861,71863],{"className":71862},[135,1022],[23,71864,26253],{"className":71865,"style":45028},[135,412,1022],[23,71867,71868,71871],{"style":4989},[23,71869],{"className":71870,"style":1015},[216],[23,71872,71874],{"className":71873},[1019,1020,1021,1022],[23,71875,9956],{"className":71876},[135,1022],[23,71878,297],{"className":71879},[296],[23,71881,71883],{"className":71882},[204],[23,71884,71886],{"className":71885,"style":19379},[208],[23,71887],{},[23,71889],{"className":71890,"style":149},[148],[23,71892,7382],{"className":71893},[153],[23,71895],{"className":71896,"style":149},[148],[23,71898,71900,71903,71906,71946,72000],{"className":71899},[126],[23,71901],{"className":71902,"style":71678},[130],[23,71904,20684],{"className":71905},[135],[23,71907,71909,71912],{"className":71908},[135],[23,71910,8340],{"className":71911,"style":11402},[135,136],[23,71913,71915],{"className":71914},[998],[23,71916,71918,71938],{"className":71917},[199,200],[23,71919,71921,71935],{"className":71920},[204],[23,71922,71924],{"className":71923,"style":1008},[208],[23,71925,71926,71929],{"style":70256},[23,71927],{"className":71928,"style":1015},[216],[23,71930,71932],{"className":71931},[1019,1020,1021,1022],[23,71933,20684],{"className":71934},[153,1022],[23,71936,297],{"className":71937},[296],[23,71939,71941],{"className":71940},[204],[23,71942,71944],{"className":71943,"style":1057},[208],[23,71945],{},[23,71947,71949,71952],{"className":71948},[135],[23,71950,20684],{"className":71951},[135],[23,71953,71955],{"className":71954},[998],[23,71956,71958,71992],{"className":71957},[199,200],[23,71959,71961,71989],{"className":71960},[204],[23,71962,71964,71978],{"className":71963,"style":11006},[208],[23,71965,71966,71969],{"style":19349},[23,71967],{"className":71968,"style":1015},[216],[23,71970,71972],{"className":71971},[1019,1020,1021,1022],[23,71973,71975],{"className":71974},[135,1022],[23,71976,26253],{"className":71977,"style":45028},[135,412,1022],[23,71979,71980,71983],{"style":4989},[23,71981],{"className":71982,"style":1015},[216],[23,71984,71986],{"className":71985},[1019,1020,1021,1022],[23,71987,9956],{"className":71988},[135,1022],[23,71990,297],{"className":71991},[296],[23,71993,71995],{"className":71994},[204],[23,71996,71998],{"className":71997,"style":19379},[208],[23,71999],{},[23,72001,112],{"className":72002},[135],[11,72004,72005,72006,72034],{},"因为 ",[23,72007,72009,72022],{"className":72008,"translate":27},[31],[23,72010,72012],{"className":72011},[35],[37,72013,72014],{"xmlns":39},[42,72015,72016,72020],{},[45,72017,72018],{},[48,72019,68146],{"mathvariant":111},[114,72021,69104],{"encoding":116},[23,72023,72025],{"className":72024,"ariaHidden":122},[121],[23,72026,72028,72031],{"className":72027},[126],[23,72029],{"className":72030,"style":470},[130],[23,72032,68146],{"className":72033},[135]," 非减，",[23,72036,72038],{"className":72037,"translate":27},[26],[23,72039,72041,72097],{"className":72040,"translate":27},[31],[23,72042,72044],{"className":72043},[35],[37,72045,72046],{"xmlns":39,"display":40},[42,72047,72048,72094],{},[45,72049,72050,72052,72054,72056,72058,72066,72068,72070,72072,72074,72076,72082,72090,72092],{},[48,72051,68146],{"mathvariant":111},[51,72053,54],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[48,72055,20684],{"mathvariant":111},[48,72057,8340],{},[19141,72059,72060,72062,72064],{},[48,72061,20684],{"mathvariant":111},[48,72063,26253],{"mathvariant":375},[2581,72065,9956],{},[51,72067,66],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[51,72069,7382],{},[48,72071,68146],{"mathvariant":111},[51,72073,54],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[48,72075,20684],{"mathvariant":111},[898,72077,72078,72080],{},[48,72079,8340],{},[51,72081,20684],{},[19141,72083,72084,72086,72088],{},[48,72085,20684],{"mathvariant":111},[48,72087,26253],{"mathvariant":375},[2581,72089,9956],{},[51,72091,66],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[48,72093,112],{"mathvariant":111},[114,72095,72096],{"encoding":116},"\\Omega\\bigl(\\|f\\|_{\\mathcal H}^2\\bigr)\\ge  \\Omega\\bigl(\\|f_\\parallel\\|_{\\mathcal H}^2\\bigr).",[23,72098,72100,72190],{"className":72099,"ariaHidden":122},[121],[23,72101,72103,72106,72109,72115,72118,72121,72175,72181,72184,72187],{"className":72102},[126],[23,72104],{"className":72105,"style":17160},[130],[23,72107,68146],{"className":72108},[135],[23,72110,72112],{"className":72111},[140],[23,72113,54],{"className":72114},[1066,1067],[23,72116,20684],{"className":72117},[135],[23,72119,8340],{"className":72120,"style":11402},[135,136],[23,72122,72124,72127],{"className":72123},[135],[23,72125,20684],{"className":72126},[135],[23,72128,72130],{"className":72129},[998],[23,72131,72133,72167],{"className":72132},[199,200],[23,72134,72136,72164],{"className":72135},[204],[23,72137,72139,72153],{"className":72138,"style":11006},[208],[23,72140,72141,72144],{"style":19349},[23,72142],{"className":72143,"style":1015},[216],[23,72145,72147],{"className":72146},[1019,1020,1021,1022],[23,72148,72150],{"className":72149},[135,1022],[23,72151,26253],{"className":72152,"style":45028},[135,412,1022],[23,72154,72155,72158],{"style":4989},[23,72156],{"className":72157,"style":1015},[216],[23,72159,72161],{"className":72160},[1019,1020,1021,1022],[23,72162,9956],{"className":72163},[135,1022],[23,72165,297],{"className":72166},[296],[23,72168,72170],{"className":72169},[204],[23,72171,72173],{"className":72172,"style":19379},[208],[23,72174],{},[23,72176,72178],{"className":72177},[169],[23,72179,66],{"className":72180},[1066,1067],[23,72182],{"className":72183,"style":149},[148],[23,72185,7382],{"className":72186},[153],[23,72188],{"className":72189,"style":149},[148],[23,72191,72193,72196,72199,72205,72208,72248,72302,72308],{"className":72192},[126],[23,72194],{"className":72195,"style":71678},[130],[23,72197,68146],{"className":72198},[135],[23,72200,72202],{"className":72201},[140],[23,72203,54],{"className":72204},[1066,1067],[23,72206,20684],{"className":72207},[135],[23,72209,72211,72214],{"className":72210},[135],[23,72212,8340],{"className":72213,"style":11402},[135,136],[23,72215,72217],{"className":72216},[998],[23,72218,72220,72240],{"className":72219},[199,200],[23,72221,72223,72237],{"className":72222},[204],[23,72224,72226],{"className":72225,"style":1008},[208],[23,72227,72228,72231],{"style":70256},[23,72229],{"className":72230,"style":1015},[216],[23,72232,72234],{"className":72233},[1019,1020,1021,1022],[23,72235,20684],{"className":72236},[153,1022],[23,72238,297],{"className":72239},[296],[23,72241,72243],{"className":72242},[204],[23,72244,72246],{"className":72245,"style":1057},[208],[23,72247],{},[23,72249,72251,72254],{"className":72250},[135],[23,72252,20684],{"className":72253},[135],[23,72255,72257],{"className":72256},[998],[23,72258,72260,72294],{"className":72259},[199,200],[23,72261,72263,72291],{"className":72262},[204],[23,72264,72266,72280],{"className":72265,"style":11006},[208],[23,72267,72268,72271],{"style":19349},[23,72269],{"className":72270,"style":1015},[216],[23,72272,72274],{"className":72273},[1019,1020,1021,1022],[23,72275,72277],{"className":72276},[135,1022],[23,72278,26253],{"className":72279,"style":45028},[135,412,1022],[23,72281,72282,72285],{"style":4989},[23,72283],{"className":72284,"style":1015},[216],[23,72286,72288],{"className":72287},[1019,1020,1021,1022],[23,72289,9956],{"className":72290},[135,1022],[23,72292,297],{"className":72293},[296],[23,72295,72297],{"className":72296},[204],[23,72298,72300],{"className":72299,"style":19379},[208],[23,72301],{},[23,72303,72305],{"className":72304},[169],[23,72306,66],{"className":72307},[1066,1067],[23,72309,112],{"className":72310},[135],[11,72312,72313],{},"把两件事合起来：",[23,72315,72317],{"className":72316,"translate":27},[26],[23,72318,72320,72458],{"className":72319,"translate":27},[31],[23,72321,72323],{"className":72322},[35],[37,72324,72325],{"xmlns":39,"display":40},[42,72326,72327,72455],{},[45,72328,72329,72331,72333,72335,72337,72343,72345,72347,72349,72351,72353,72355,72361,72363,72365,72367,72369,72371,72373,72375,72381,72383,72385,72387,72389,72391,72397,72399,72405,72407,72409,72411,72413,72419,72421,72427,72429,72431,72433,72435,72437,72439,72445,72451,72453],{},[48,72330,555],{},[51,72332,54],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[48,72334,8340],{},[51,72336,54],{"stretchy":53},[898,72338,72339,72341],{},[48,72340,753],{},[2581,72342,4435],{},[51,72344,66],{"stretchy":53},[51,72346,747],{"separator":122},[51,72348,62563],{},[51,72350,747],{"separator":122},[48,72352,8340],{},[51,72354,54],{"stretchy":53},[898,72356,72357,72359],{},[48,72358,753],{},[48,72360,4806],{},[51,72362,66],{"stretchy":53},[51,72364,66],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[51,72366,5106],{},[48,72368,68146],{"mathvariant":111},[51,72370,54],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[48,72372,20684],{"mathvariant":111},[48,72374,8340],{},[4518,72376,72377,72379],{},[48,72378,20684],{"mathvariant":111},[2581,72380,9956],{},[51,72382,66],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[51,72384,7382],{},[148,72386],{"linebreak":29629},[48,72388,555],{},[51,72390,54],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[898,72392,72393,72395],{},[48,72394,8340],{},[51,72396,20684],{},[51,72398,54],{"stretchy":53},[898,72400,72401,72403],{},[48,72402,753],{},[2581,72404,4435],{},[51,72406,66],{"stretchy":53},[51,72408,747],{"separator":122},[51,72410,62563],{},[51,72412,747],{"separator":122},[898,72414,72415,72417],{},[48,72416,8340],{},[51,72418,20684],{},[51,72420,54],{"stretchy":53},[898,72422,72423,72425],{},[48,72424,753],{},[48,72426,4806],{},[51,72428,66],{"stretchy":53},[51,72430,66],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[51,72432,5106],{},[48,72434,68146],{"mathvariant":111},[51,72436,54],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[48,72438,20684],{"mathvariant":111},[898,72440,72441,72443],{},[48,72442,8340],{},[51,72444,20684],{},[4518,72446,72447,72449],{},[48,72448,20684],{"mathvariant":111},[2581,72450,9956],{},[51,72452,66],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[48,72454,112],{"mathvariant":111},[114,72456,72457],{"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).",[23,72459,72461,72607,72669,72672,72892],{"className":72460,"ariaHidden":122},[121],[23,72462,72464,72467,72470,72476,72479,72482,72522,72525,72528,72531,72534,72537,72540,72543,72546,72549,72589,72592,72598,72601,72604],{"className":72463},[126],[23,72465],{"className":72466,"style":2213},[130],[23,72468,555],{"className":72469},[135,136],[23,72471,72473],{"className":72472},[140],[23,72474,54],{"className":72475},[1066,1067],[23,72477,8340],{"className":72478,"style":11402},[135,136],[23,72480,54],{"className":72481},[140],[23,72483,72485,72488],{"className":72484},[135],[23,72486,753],{"className":72487},[135,136],[23,72489,72491],{"className":72490},[998],[23,72492,72494,72514],{"className":72493},[199,200],[23,72495,72497,72511],{"className":72496},[204],[23,72498,72500],{"className":72499,"style":18651},[208],[23,72501,72502,72505],{"style":2029},[23,72503],{"className":72504,"style":1015},[216],[23,72506,72508],{"className":72507},[1019,1020,1021,1022],[23,72509,4435],{"className":72510},[135,1022],[23,72512,297],{"className":72513},[296],[23,72515,72517],{"className":72516},[204],[23,72518,72520],{"className":72519,"style":2048},[208],[23,72521],{},[23,72523,66],{"className":72524},[169],[23,72526,747],{"className":72527},[824],[23,72529],{"className":72530,"style":280},[148],[23,72532,62563],{"className":72533},[1839],[23,72535],{"className":72536,"style":280},[148],[23,72538,747],{"className":72539},[824],[23,72541],{"className":72542,"style":280},[148],[23,72544,8340],{"className":72545,"style":11402},[135,136],[23,72547,54],{"className":72548},[140],[23,72550,72552,72555],{"className":72551},[135],[23,72553,753],{"className":72554},[135,136],[23,72556,72558],{"className":72557},[998],[23,72559,72561,72581],{"className":72560},[199,200],[23,72562,72564,72578],{"className":72563},[204],[23,72565,72567],{"className":72566,"style":6499},[208],[23,72568,72569,72572],{"style":2029},[23,72570],{"className":72571,"style":1015},[216],[23,72573,72575],{"className":72574},[1019,1020,1021,1022],[23,72576,4806],{"className":72577},[135,136,1022],[23,72579,297],{"className":72580},[296],[23,72582,72584],{"className":72583},[204],[23,72585,72587],{"className":72586,"style":2048},[208],[23,72588],{},[23,72590,66],{"className":72591},[169],[23,72593,72595],{"className":72594},[169],[23,72596,66],{"className":72597},[1066,1067],[23,72599],{"className":72600,"style":392},[148],[23,72602,5106],{"className":72603},[612],[23,72605],{"className":72606,"style":392},[148],[23,72608,72610,72613,72616,72622,72625,72628,72657,72663,72666],{"className":72609},[126],[23,72611],{"className":72612,"style":17160},[130],[23,72614,68146],{"className":72615},[135],[23,72617,72619],{"className":72618},[140],[23,72620,54],{"className":72621},[1066,1067],[23,72623,20684],{"className":72624},[135],[23,72626,8340],{"className":72627,"style":11402},[135,136],[23,72629,72631,72634],{"className":72630},[135],[23,72632,20684],{"className":72633},[135],[23,72635,72637],{"className":72636},[998],[23,72638,72640],{"className":72639},[199],[23,72641,72643],{"className":72642},[204],[23,72644,72646],{"className":72645,"style":11006},[208],[23,72647,72648,72651],{"style":4989},[23,72649],{"className":72650,"style":1015},[216],[23,72652,72654],{"className":72653},[1019,1020,1021,1022],[23,72655,9956],{"className":72656},[135,1022],[23,72658,72660],{"className":72659},[169],[23,72661,66],{"className":72662},[1066,1067],[23,72664],{"className":72665,"style":149},[148],[23,72667,7382],{"className":72668},[153],[23,72670],{"className":72671},[148,29629],[23,72673,72675,72678,72681,72687,72727,72730,72770,72773,72776,72779,72782,72785,72788,72791,72831,72834,72874,72877,72883,72886,72889],{"className":72674},[126],[23,72676],{"className":72677,"style":988},[130],[23,72679,555],{"className":72680},[135,136],[23,72682,72684],{"className":72683},[140],[23,72685,54],{"className":72686},[1066,1067],[23,72688,72690,72693],{"className":72689},[135],[23,72691,8340],{"className":72692,"style":11402},[135,136],[23,72694,72696],{"className":72695},[998],[23,72697,72699,72719],{"className":72698},[199,200],[23,72700,72702,72716],{"className":72701},[204],[23,72703,72705],{"className":72704,"style":1008},[208],[23,72706,72707,72710],{"style":70256},[23,72708],{"className":72709,"style":1015},[216],[23,72711,72713],{"className":72712},[1019,1020,1021,1022],[23,72714,20684],{"className":72715},[153,1022],[23,72717,297],{"className":72718},[296],[23,72720,72722],{"className":72721},[204],[23,72723,72725],{"className":72724,"style":1057},[208],[23,72726],{},[23,72728,54],{"className":72729},[140],[23,72731,72733,72736],{"className":72732},[135],[23,72734,753],{"className":72735},[135,136],[23,72737,72739],{"className":72738},[998],[23,72740,72742,72762],{"className":72741},[199,200],[23,72743,72745,72759],{"className":72744},[204],[23,72746,72748],{"className":72747,"style":18651},[208],[23,72749,72750,72753],{"style":2029},[23,72751],{"className":72752,"style":1015},[216],[23,72754,72756],{"className":72755},[1019,1020,1021,1022],[23,72757,4435],{"className":72758},[135,1022],[23,72760,297],{"className":72761},[296],[23,72763,72765],{"className":72764},[204],[23,72766,72768],{"className":72767,"style":2048},[208],[23,72769],{},[23,72771,66],{"className":72772},[169],[23,72774,747],{"className":72775},[824],[23,72777],{"className":72778,"style":280},[148],[23,72780,62563],{"className":72781},[1839],[23,72783],{"className":72784,"style":280},[148],[23,72786,747],{"className":72787},[824],[23,72789],{"className":72790,"style":280},[148],[23,72792,72794,72797],{"className":72793},[135],[23,72795,8340],{"className":72796,"style":11402},[135,136],[23,72798,72800],{"className":72799},[998],[23,72801,72803,72823],{"className":72802},[199,200],[23,72804,72806,72820],{"className":72805},[204],[23,72807,72809],{"className":72808,"style":1008},[208],[23,72810,72811,72814],{"style":70256},[23,72812],{"className":72813,"style":1015},[216],[23,72815,72817],{"className":72816},[1019,1020,1021,1022],[23,72818,20684],{"className":72819},[153,1022],[23,72821,297],{"className":72822},[296],[23,72824,72826],{"className":72825},[204],[23,72827,72829],{"className":72828,"style":1057},[208],[23,72830],{},[23,72832,54],{"className":72833},[140],[23,72835,72837,72840],{"className":72836},[135],[23,72838,753],{"className":72839},[135,136],[23,72841,72843],{"className":72842},[998],[23,72844,72846,72866],{"className":72845},[199,200],[23,72847,72849,72863],{"className":72848},[204],[23,72850,72852],{"className":72851,"style":6499},[208],[23,72853,72854,72857],{"style":2029},[23,72855],{"className":72856,"style":1015},[216],[23,72858,72860],{"className":72859},[1019,1020,1021,1022],[23,72861,4806],{"className":72862},[135,136,1022],[23,72864,297],{"className":72865},[296],[23,72867,72869],{"className":72868},[204],[23,72870,72872],{"className":72871,"style":2048},[208],[23,72873],{},[23,72875,66],{"className":72876},[169],[23,72878,72880],{"className":72879},[169],[23,72881,66],{"className":72882},[1066,1067],[23,72884],{"className":72885,"style":392},[148],[23,72887,5106],{"className":72888},[612],[23,72890],{"className":72891,"style":392},[148],[23,72893,72895,72898,72901,72907,72910,72950,72979,72985],{"className":72894},[126],[23,72896],{"className":72897,"style":71678},[130],[23,72899,68146],{"className":72900},[135],[23,72902,72904],{"className":72903},[140],[23,72905,54],{"className":72906},[1066,1067],[23,72908,20684],{"className":72909},[135],[23,72911,72913,72916],{"className":72912},[135],[23,72914,8340],{"className":72915,"style":11402},[135,136],[23,72917,72919],{"className":72918},[998],[23,72920,72922,72942],{"className":72921},[199,200],[23,72923,72925,72939],{"className":72924},[204],[23,72926,72928],{"className":72927,"style":1008},[208],[23,72929,72930,72933],{"style":70256},[23,72931],{"className":72932,"style":1015},[216],[23,72934,72936],{"className":72935},[1019,1020,1021,1022],[23,72937,20684],{"className":72938},[153,1022],[23,72940,297],{"className":72941},[296],[23,72943,72945],{"className":72944},[204],[23,72946,72948],{"className":72947,"style":1057},[208],[23,72949],{},[23,72951,72953,72956],{"className":72952},[135],[23,72954,20684],{"className":72955},[135],[23,72957,72959],{"className":72958},[998],[23,72960,72962],{"className":72961},[199],[23,72963,72965],{"className":72964},[204],[23,72966,72968],{"className":72967,"style":11006},[208],[23,72969,72970,72973],{"style":4989},[23,72971],{"className":72972,"style":1015},[216],[23,72974,72976],{"className":72975},[1019,1020,1021,1022],[23,72977,9956],{"className":72978},[135,1022],[23,72980,72982],{"className":72981},[169],[23,72983,66],{"className":72984},[1066,1067],[23,72986,112],{"className":72987},[135],[11,72989,72990,72991,73019,73020,73090,73091,73119,73120,73148,73149,73218,73219,73352],{},"所以，对任意 ",[23,72992,72994,73007],{"className":72993,"translate":27},[31],[23,72995,72997],{"className":72996},[35],[37,72998,72999],{"xmlns":39},[42,73000,73001,73005],{},[45,73002,73003],{},[48,73004,8340],{},[114,73006,8340],{"encoding":116},[23,73008,73010],{"className":73009,"ariaHidden":122},[121],[23,73011,73013,73016],{"className":73012},[126],[23,73014],{"className":73015,"style":5414},[130],[23,73017,8340],{"className":73018,"style":11402},[135,136],"，它的投影 ",[23,73021,73023,73041],{"className":73022,"translate":27},[31],[23,73024,73026],{"className":73025},[35],[37,73027,73028],{"xmlns":39},[42,73029,73030,73038],{},[45,73031,73032],{},[898,73033,73034,73036],{},[48,73035,8340],{},[51,73037,20684],{},[114,73039,73040],{"encoding":116},"f_\\parallel",[23,73042,73044],{"className":73043,"ariaHidden":122},[121],[23,73045,73047,73050],{"className":73046},[126],[23,73048],{"className":73049,"style":70235},[130],[23,73051,73053,73056],{"className":73052},[135],[23,73054,8340],{"className":73055,"style":11402},[135,136],[23,73057,73059],{"className":73058},[998],[23,73060,73062,73082],{"className":73061},[199,200],[23,73063,73065,73079],{"className":73064},[204],[23,73066,73068],{"className":73067,"style":1008},[208],[23,73069,73070,73073],{"style":70256},[23,73071],{"className":73072,"style":1015},[216],[23,73074,73076],{"className":73075},[1019,1020,1021,1022],[23,73077,20684],{"className":73078},[153,1022],[23,73080,297],{"className":73081},[296],[23,73083,73085],{"className":73084},[204],[23,73086,73088],{"className":73087,"style":1057},[208],[23,73089],{}," 的目标值都不比 ",[23,73092,73094,73107],{"className":73093,"translate":27},[31],[23,73095,73097],{"className":73096},[35],[37,73098,73099],{"xmlns":39},[42,73100,73101,73105],{},[45,73102,73103],{},[48,73104,8340],{},[114,73106,8340],{"encoding":116},[23,73108,73110],{"className":73109,"ariaHidden":122},[121],[23,73111,73113,73116],{"className":73112},[126],[23,73114],{"className":73115,"style":5414},[130],[23,73117,8340],{"className":73118,"style":11402},[135,136]," 差。若 ",[23,73121,73123,73136],{"className":73122,"translate":27},[31],[23,73124,73126],{"className":73125},[35],[37,73127,73128],{"xmlns":39},[42,73129,73130,73134],{},[45,73131,73132],{},[48,73133,8340],{},[114,73135,8340],{"encoding":116},[23,73137,73139],{"className":73138,"ariaHidden":122},[121],[23,73140,73142,73145],{"className":73141},[126],[23,73143],{"className":73144,"style":5414},[130],[23,73146,8340],{"className":73147,"style":11402},[135,136]," 是最优解，则 ",[23,73150,73152,73169],{"className":73151,"translate":27},[31],[23,73153,73155],{"className":73154},[35],[37,73156,73157],{"xmlns":39},[42,73158,73159,73167],{},[45,73160,73161],{},[898,73162,73163,73165],{},[48,73164,8340],{},[51,73166,20684],{},[114,73168,73040],{"encoding":116},[23,73170,73172],{"className":73171,"ariaHidden":122},[121],[23,73173,73175,73178],{"className":73174},[126],[23,73176],{"className":73177,"style":70235},[130],[23,73179,73181,73184],{"className":73180},[135],[23,73182,8340],{"className":73183,"style":11402},[135,136],[23,73185,73187],{"className":73186},[998],[23,73188,73190,73210],{"className":73189},[199,200],[23,73191,73193,73207],{"className":73192},[204],[23,73194,73196],{"className":73195,"style":1008},[208],[23,73197,73198,73201],{"style":70256},[23,73199],{"className":73200,"style":1015},[216],[23,73202,73204],{"className":73203},[1019,1020,1021,1022],[23,73205,20684],{"className":73206},[153,1022],[23,73208,297],{"className":73209},[296],[23,73211,73213],{"className":73212},[204],[23,73214,73216],{"className":73215,"style":1057},[208],[23,73217],{}," 也是最优解，且 ",[23,73220,73222,73248],{"className":73221,"translate":27},[31],[23,73223,73225],{"className":73224},[35],[37,73226,73227],{"xmlns":39},[42,73228,73229,73245],{},[45,73230,73231,73237,73239],{},[898,73232,73233,73235],{},[48,73234,8340],{},[51,73236,20684],{},[51,73238,372],{},[898,73240,73241,73243],{},[48,73242,26253],{"mathvariant":375},[2581,73244,1532],{},[114,73246,73247],{"encoding":116},"f_\\parallel\\in\\mathcal H_0",[23,73249,73251,73306],{"className":73250,"ariaHidden":122},[121],[23,73252,73254,73257,73297,73300,73303],{"className":73253},[126],[23,73255],{"className":73256,"style":70235},[130],[23,73258,73260,73263],{"className":73259},[135],[23,73261,8340],{"className":73262,"style":11402},[135,136],[23,73264,73266],{"className":73265},[998],[23,73267,73269,73289],{"className":73268},[199,200],[23,73270,73272,73286],{"className":73271},[204],[23,73273,73275],{"className":73274,"style":1008},[208],[23,73276,73277,73280],{"style":70256},[23,73278],{"className":73279,"style":1015},[216],[23,73281,73283],{"className":73282},[1019,1020,1021,1022],[23,73284,20684],{"className":73285},[153,1022],[23,73287,297],{"className":73288},[296],[23,73290,73292],{"className":73291},[204],[23,73293,73295],{"className":73294,"style":1057},[208],[23,73296],{},[23,73298],{"className":73299,"style":149},[148],[23,73301,372],{"className":73302},[153],[23,73304],{"className":73305,"style":149},[148],[23,73307,73309,73312],{"className":73308},[126],[23,73310],{"className":73311,"style":42831},[130],[23,73313,73315,73318],{"className":73314},[135],[23,73316,26253],{"className":73317,"style":45028},[135,412],[23,73319,73321],{"className":73320},[998],[23,73322,73324,73344],{"className":73323},[199,200],[23,73325,73327,73341],{"className":73326},[204],[23,73328,73330],{"className":73329,"style":18651},[208],[23,73331,73332,73335],{"style":61635},[23,73333],{"className":73334,"style":1015},[216],[23,73336,73338],{"className":73337},[1019,1020,1021,1022],[23,73339,1532],{"className":73340},[135,1022],[23,73342,297],{"className":73343},[296],[23,73345,73347],{"className":73346},[204],[23,73348,73350],{"className":73349,"style":2048},[208],[23,73351],{},"。这就证明了第 1 条。",[11,73354,73355,73356,73384,73385,46547,73510,73683,73684,73712,73713,73805,73806,73876,73877],{},"若 ",[23,73357,73359,73372],{"className":73358,"translate":27},[31],[23,73360,73362],{"className":73361},[35],[37,73363,73364],{"xmlns":39},[42,73365,73366,73370],{},[45,73367,73368],{},[48,73369,68146],{"mathvariant":111},[114,73371,69104],{"encoding":116},[23,73373,73375],{"className":73374,"ariaHidden":122},[121],[23,73376,73378,73381],{"className":73377},[126],[23,73379],{"className":73380,"style":470},[130],[23,73382,68146],{"className":73383},[135]," 严格递增且 ",[23,73386,73388,73410],{"className":73387,"translate":27},[31],[23,73389,73391],{"className":73390},[35],[37,73392,73393],{"xmlns":39},[42,73394,73395,73407],{},[45,73396,73397,73403,73405],{},[898,73398,73399,73401],{},[48,73400,8340],{},[51,73402,70164],{},[51,73404,9019],{"mathvariant":111},[2581,73406,1532],{},[114,73408,73409],{"encoding":116},"f_\\perp\\neq 0",[23,73411,73413,73501],{"className":73412,"ariaHidden":122},[121],[23,73414,73416,73419,73459,73462,73498],{"className":73415},[126],[23,73417],{"className":73418,"style":5414},[130],[23,73420,73422,73425],{"className":73421},[135],[23,73423,8340],{"className":73424,"style":11402},[135,136],[23,73426,73428],{"className":73427},[998],[23,73429,73431,73451],{"className":73430},[199,200],[23,73432,73434,73448],{"className":73433},[204],[23,73435,73437],{"className":73436,"style":34196},[208],[23,73438,73439,73442],{"style":29550},[23,73440],{"className":73441,"style":1015},[216],[23,73443,73445],{"className":73444},[1019,1020,1021,1022],[23,73446,70164],{"className":73447},[153,1022],[23,73449,297],{"className":73450},[296],[23,73452,73454],{"className":73453},[204],[23,73455,73457],{"className":73456,"style":2048},[208],[23,73458],{},[23,73460],{"className":73461,"style":149},[148],[23,73463,73465,73492,73495],{"className":73464},[153],[23,73466,73468],{"className":73467},[153],[23,73469,73471],{"className":73470},[135,9097],[23,73472,73474],{"className":73473},[9101],[23,73475,73477,73480,73489],{"className":73476},[9105],[23,73478],{"className":73479,"style":5414},[130],[23,73481,73483],{"className":73482},[9112],[23,73484,73486],{"className":73485},[135],[23,73487,9119],{"className":73488},[153],[23,73490],{"className":73491},[9123],[23,73493],{"className":73494},[148,9127],[23,73496,69],{"className":73497},[153],[23,73499],{"className":73500,"style":149},[148],[23,73502,73504,73507],{"className":73503},[126],[23,73505],{"className":73506,"style":2770},[130],[23,73508,1532],{"className":73509},[135],[23,73511,73513,73551],{"className":73512,"translate":27},[31],[23,73514,73516],{"className":73515},[35],[37,73517,73518],{"xmlns":39},[42,73519,73520,73548],{},[45,73521,73522,73524,73526,73532,73534,73536,73542],{},[48,73523,20684],{"mathvariant":111},[48,73525,8340],{},[4518,73527,73528,73530],{},[48,73529,20684],{"mathvariant":111},[2581,73531,9956],{},[51,73533,2579],{},[48,73535,20684],{"mathvariant":111},[898,73537,73538,73540],{},[48,73539,8340],{},[51,73541,20684],{},[4518,73543,73544,73546],{},[48,73545,20684],{"mathvariant":111},[2581,73547,9956],{},[114,73549,73550],{"encoding":116},"\\|f\\|^2>\\|f_\\parallel\\|^2",[23,73552,73554,73604],{"className":73553,"ariaHidden":122},[121],[23,73555,73557,73560,73563,73566,73595,73598,73601],{"className":73556},[126],[23,73558],{"className":73559,"style":10665},[130],[23,73561,20684],{"className":73562},[135],[23,73564,8340],{"className":73565,"style":11402},[135,136],[23,73567,73569,73572],{"className":73568},[135],[23,73570,20684],{"className":73571},[135],[23,73573,73575],{"className":73574},[998],[23,73576,73578],{"className":73577},[199],[23,73579,73581],{"className":73580},[204],[23,73582,73584],{"className":73583,"style":4693},[208],[23,73585,73586,73589],{"style":4696},[23,73587],{"className":73588,"style":1015},[216],[23,73590,73592],{"className":73591},[1019,1020,1021,1022],[23,73593,9956],{"className":73594},[135,1022],[23,73596],{"className":73597,"style":149},[148],[23,73599,2579],{"className":73600},[153],[23,73602],{"className":73603,"style":149},[148],[23,73605,73607,73611,73614,73654],{"className":73606},[126],[23,73608],{"className":73609,"style":73610},[130],"height:1.1693em;vertical-align:-0.3552em;",[23,73612,20684],{"className":73613},[135],[23,73615,73617,73620],{"className":73616},[135],[23,73618,8340],{"className":73619,"style":11402},[135,136],[23,73621,73623],{"className":73622},[998],[23,73624,73626,73646],{"className":73625},[199,200],[23,73627,73629,73643],{"className":73628},[204],[23,73630,73632],{"className":73631,"style":1008},[208],[23,73633,73634,73637],{"style":70256},[23,73635],{"className":73636,"style":1015},[216],[23,73638,73640],{"className":73639},[1019,1020,1021,1022],[23,73641,20684],{"className":73642},[153,1022],[23,73644,297],{"className":73645},[296],[23,73647,73649],{"className":73648},[204],[23,73650,73652],{"className":73651,"style":1057},[208],[23,73653],{},[23,73655,73657,73660],{"className":73656},[135],[23,73658,20684],{"className":73659},[135],[23,73661,73663],{"className":73662},[998],[23,73664,73666],{"className":73665},[199],[23,73667,73669],{"className":73668},[204],[23,73670,73672],{"className":73671,"style":4693},[208],[23,73673,73674,73677],{"style":4696},[23,73675],{"className":73676,"style":1015},[216],[23,73678,73680],{"className":73679},[1019,1020,1021,1022],[23,73681,9956],{"className":73682},[135,1022],"，上面的不等式是严格的，",[23,73685,73687,73700],{"className":73686,"translate":27},[31],[23,73688,73690],{"className":73689},[35],[37,73691,73692],{"xmlns":39},[42,73693,73694,73698],{},[45,73695,73696],{},[48,73697,8340],{},[114,73699,8340],{"encoding":116},[23,73701,73703],{"className":73702,"ariaHidden":122},[121],[23,73704,73706,73709],{"className":73705},[126],[23,73707],{"className":73708,"style":5414},[130],[23,73710,8340],{"className":73711,"style":11402},[135,136]," 不可能是最优解。因此每个最优解都必须满足 ",[23,73714,73716,73738],{"className":73715,"translate":27},[31],[23,73717,73719],{"className":73718},[35],[37,73720,73721],{"xmlns":39},[42,73722,73723,73735],{},[45,73724,73725,73731,73733],{},[898,73726,73727,73729],{},[48,73728,8340],{},[51,73730,70164],{},[51,73732,69],{},[2581,73734,1532],{},[114,73736,73737],{"encoding":116},"f_\\perp=0",[23,73739,73741,73796],{"className":73740,"ariaHidden":122},[121],[23,73742,73744,73747,73787,73790,73793],{"className":73743},[126],[23,73745],{"className":73746,"style":5414},[130],[23,73748,73750,73753],{"className":73749},[135],[23,73751,8340],{"className":73752,"style":11402},[135,136],[23,73754,73756],{"className":73755},[998],[23,73757,73759,73779],{"className":73758},[199,200],[23,73760,73762,73776],{"className":73761},[204],[23,73763,73765],{"className":73764,"style":34196},[208],[23,73766,73767,73770],{"style":29550},[23,73768],{"className":73769,"style":1015},[216],[23,73771,73773],{"className":73772},[1019,1020,1021,1022],[23,73774,70164],{"className":73775},[153,1022],[23,73777,297],{"className":73778},[296],[23,73780,73782],{"className":73781},[204],[23,73783,73785],{"className":73784,"style":2048},[208],[23,73786],{},[23,73788],{"className":73789,"style":149},[148],[23,73791,69],{"className":73792},[153],[23,73794],{"className":73795,"style":149},[148],[23,73797,73799,73802],{"className":73798},[126],[23,73800],{"className":73801,"style":2770},[130],[23,73803,1532],{"className":73804},[135],"，即落在 ",[23,73807,73809,73827],{"className":73808,"translate":27},[31],[23,73810,73812],{"className":73811},[35],[37,73813,73814],{"xmlns":39},[42,73815,73816,73824],{},[45,73817,73818],{},[898,73819,73820,73822],{},[48,73821,26253],{"mathvariant":375},[2581,73823,1532],{},[114,73825,73826],{"encoding":116},"\\mathcal H_0",[23,73828,73830],{"className":73829,"ariaHidden":122},[121],[23,73831,73833,73836],{"className":73832},[126],[23,73834],{"className":73835,"style":42831},[130],[23,73837,73839,73842],{"className":73838},[135],[23,73840,26253],{"className":73841,"style":45028},[135,412],[23,73843,73845],{"className":73844},[998],[23,73846,73848,73868],{"className":73847},[199,200],[23,73849,73851,73865],{"className":73850},[204],[23,73852,73854],{"className":73853,"style":18651},[208],[23,73855,73856,73859],{"style":61635},[23,73857],{"className":73858,"style":1015},[216],[23,73860,73862],{"className":73861},[1019,1020,1021,1022],[23,73863,1532],{"className":73864},[135,1022],[23,73866,297],{"className":73867},[296],[23,73869,73871],{"className":73870},[204],[23,73872,73874],{"className":73873,"style":2048},[208],[23,73875],{}," 中。第 2 条得证。",[23,73878,73880,73893],{"className":73879,"translate":27},[31],[23,73881,73883],{"className":73882},[35],[37,73884,73885],{"xmlns":39},[42,73886,73887,73891],{},[45,73888,73889],{},[48,73890,8251],{"mathvariant":111},[114,73892,8254],{"encoding":116},[23,73894,73896],{"className":73895,"ariaHidden":122},[121],[23,73897,73899,73902],{"className":73898},[126],[23,73900],{"className":73901,"style":8264},[130],[23,73903,8251],{"className":73904},[135,8268],[4079,73906,73907],{},[4082,73908],{"height":73909,"width":73910,"style":4086,"data-d2-theme":4087,"alt":73911,"src":73912},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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代回目标函数。对每个训练点，",[23,74230,74232],{"className":74231,"translate":27},[26],[23,74233,74235,74319],{"className":74234,"translate":27},[31],[23,74236,74238],{"className":74237},[35],[37,74239,74240],{"xmlns":39,"display":40},[42,74241,74242,74316],{},[45,74243,74244,74246,74248,74254,74256,74258,74272,74278,74280,74282,74284,74290,74292,74298,74300,74302,74304,74306,74308,74314],{},[48,74245,8340],{},[51,74247,54],{"stretchy":53},[898,74249,74250,74252],{},[48,74251,753],{},[48,74253,33497],{},[51,74255,66],{"stretchy":53},[51,74257,69],{},[61717,74259,74260,74262,74270],{},[51,74261,33855],{},[45,74263,74264,74266,74268],{},[48,74265,35522],{},[51,74267,69],{},[2581,74269,4435],{},[48,74271,4806],{},[898,74273,74274,74276],{},[48,74275,4526],{},[48,74277,35522],{},[88,74279,90],{},[48,74281,4836],{},[51,74283,54],{"stretchy":53},[898,74285,74286,74288],{},[48,74287,753],{},[48,74289,33497],{},[51,74291,747],{"separator":122},[898,74293,74294,74296],{},[48,74295,753],{},[48,74297,35522],{},[51,74299,66],{"stretchy":53},[51,74301,69],{},[51,74303,54],{"stretchy":53},[48,74305,26922],{},[48,74307,4526],{},[898,74309,74310,74312],{},[51,74311,66],{"stretchy":53},[48,74313,33497],{},[51,74315,747],{"separator":122},[114,74317,74318],{"encoding":116},"f(x_i)=\\sum_{j=1}^n\\alpha_j\\,k(x_i,x_j)=(K\\alpha)_i,",[23,74320,74322,74386,74606],{"className":74321,"ariaHidden":122},[121],[23,74323,74325,74328,74331,74334,74374,74377,74380,74383],{"className":74324},[126],[23,74326],{"className":74327,"style":131},[130],[23,74329,8340],{"className":74330,"style":11402},[135,136],[23,74332,54],{"className":74333},[140],[23,74335,74337,74340],{"className":74336},[135],[23,74338,753],{"className":74339},[135,136],[23,74341,74343],{"className":74342},[998],[23,74344,74346,74366],{"className":74345},[199,200],[23,74347,74349,74363],{"className":74348},[204],[23,74350,74352],{"className":74351,"style":33651},[208],[23,74353,74354,74357],{"style":2029},[23,74355],{"className":74356,"style":1015},[216],[23,74358,74360],{"className":74359},[1019,1020,1021,1022],[23,74361,33497],{"className":74362},[135,136,1022],[23,74364,297],{"className":74365},[296],[23,74367,74369],{"className":74368},[204],[23,74370,74372],{"className":74371,"style":2048},[208],[23,74373],{},[23,74375,66],{"className":74376},[169],[23,74378],{"className":74379,"style":149},[148],[23,74381,69],{"className":74382},[153],[23,74384],{"className":74385,"style":149},[148],[23,74387,74389,74392,74456,74459,74499,74502,74505,74508,74548,74551,74554,74594,74597,74600,74603],{"className":74388},[126],[23,74390],{"className":74391,"style":62961},[130],[23,74393,74395],{"className":74394},[1786,7098],[23,74396,74398,74448],{"className":74397},[199,200],[23,74399,74401,74445],{"className":74400},[204],[23,74402,74404,74424,74434],{"className":74403,"style":61928},[208],[23,74405,74406,74409],{"style":34079},[23,74407],{"className":74408,"style":34083},[216],[23,74410,74412],{"className":74411},[1019,1020,1021,1022],[23,74413,74415,74418,74421],{"className":74414},[135,1022],[23,74416,35522],{"className":74417,"style":35607},[135,136,1022],[23,74419,69],{"className":74420},[153,1022],[23,74422,4435],{"className":74423},[135,1022],[23,74425,74426,74429],{"style":34092},[23,74427],{"className":74428,"style":34083},[216],[23,74430,74431],{},[23,74432,33855],{"className":74433},[1786,1790,1791],[23,74435,74436,74439],{"style":61961},[23,74437],{"className":74438,"style":34083},[216],[23,74440,74442],{"className":74441},[1019,1020,1021,1022],[23,74443,4806],{"className":74444},[135,136,1022],[23,74446,297],{"className":74447},[296],[23,74449,74451],{"className":74450},[204],[23,74452,74454],{"className":74453,"style":63091},[208],[23,74455],{},[23,74457],{"className":74458,"style":280},[148],[23,74460,74462,74465],{"className":74461},[135],[23,74463,4526],{"className":74464,"style":4645},[135,136],[23,74466,74468],{"className":74467},[998],[23,74469,74471,74491],{"className":74470},[199,200],[23,74472,74474,74488],{"className":74473},[204],[23,74475,74477],{"className":74476,"style":33651},[208],[23,74478,74479,74482],{"style":68976},[23,74480],{"className":74481,"style":1015},[216],[23,74483,74485],{"className":74484},[1019,1020,1021,1022],[23,74486,35522],{"className":74487,"style":35607},[135,136,1022],[23,74489,297],{"className":74490},[296],[23,74492,74494],{"className":74493},[204],[23,74495,74497],{"className":74496,"style":14296},[208],[23,74498],{},[23,74500],{"className":74501,"style":280},[148],[23,74503,4836],{"className":74504,"style":4851},[135,136],[23,74506,54],{"className":74507},[140],[23,74509,74511,74514],{"className":74510},[135],[23,74512,753],{"className":74513},[135,136],[23,74515,74517],{"className":74516},[998],[23,74518,74520,74540],{"className":74519},[199,200],[23,74521,74523,74537],{"className":74522},[204],[23,74524,74526],{"className":74525,"style":33651},[208],[23,74527,74528,74531],{"style":2029},[23,74529],{"className":74530,"style":1015},[216],[2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是核矩阵，",[23,74765,74767,74810],{"className":74766,"translate":27},[31],[23,74768,74770],{"className":74769},[35],[37,74771,74772],{"xmlns":39},[42,74773,74774,74808],{},[45,74775,74776,74786,74788,74790,74792,74798,74800,74806],{},[898,74777,74778,74780],{},[48,74779,26922],{},[45,74781,74782,74784],{},[48,74783,33497],{},[48,74785,35522],{},[51,74787,69],{},[48,74789,4836],{},[51,74791,54],{"stretchy":53},[898,74793,74794,74796],{},[48,74795,753],{},[48,74797,33497],{},[51,74799,747],{"separator":122},[898,74801,74802,74804],{},[48,74803,753],{},[48,74805,35522],{},[51,74807,66],{"stretchy":53},[114,74809,63342],{"encoding":116},[23,74811,74813,74871],{"className":74812,"ariaHidden":122},[121],[23,74814,74816,74819,74862,74865,74868],{"className":74815},[126],[23,74817],{"className":74818,"style":63352},[130],[23,74820,74822,74825],{"className":74821},[135],[23,74823,26922],{"className":74824,"style":37145},[135,136],[23,74826,74828],{"className":74827},[998],[23,74829,74831,74854],{"className":74830},[199,200],[23,74832,74834,74851],{"className":74833},[204],[23,74835,74837],{"className":74836,"style":33651},[208],[23,74838,74839,74842],{"style":63373},[23,74840],{"className":74841,"style":1015},[216],[23,74843,74845],{"className":74844},[1019,1020,1021,1022],[23,74846,74848],{"className":74847},[135,1022],[23,74849,63386],{"className":74850,"style":35607},[135,136,1022],[23,74852,297],{"className":74853},[296],[23,74855,74857],{"className":74856},[204],[23,74858,74860],{"className":74859,"style":14296},[208],[23,74861],{},[23,74863],{"className":74864,"style":149},[148],[23,74866,69],{"className":74867},[153],[23,74869],{"className":74870,"style":149},[148],[23,74872,74874,74877,74880,74883,74923,74926,74929,74969],{"className":74873},[126],[23,74875],{"className":74876,"style":14257},[130],[23,74878,4836],{"className":74879,"style":4851},[135,136],[23,74881,54],{"className":74882},[140],[23,74884,74886,74889],{"className":74885},[135],[23,74887,753],{"className":74888},[135,136],[23,74890,74892],{"className":74891},[998],[23,74893,74895,74915],{"className":74894},[199,200],[23,74896,74898,74912],{"className":74897},[204],[23,74899,74901],{"className":74900,"style":33651},[208],[23,74902,74903,74906],{"style":2029},[23,74904],{"className":74905,"style":1015},[216],[23,74907,74909],{"className":74908},[1019,1020,1021,1022],[23,74910,33497],{"className":74911},[135,136,1022],[23,74913,297],{"className":74914},[296],[23,74916,74918],{"className":74917},[204],[23,74919,74921],{"className":74920,"style":2048},[208],[23,74922],{},[23,74924,747],{"className":74925},[824],[23,74927],{"className":74928,"style":280},[148],[23,74930,74932,74935],{"className":74931},[135],[23,74933,753],{"className":74934},[135,136],[23,74936,74938],{"className":74937},[998],[23,74939,74941,74961],{"className":74940},[199,200],[23,74942,74944,74958],{"className":74943},[204],[23,74945,74947],{"className":74946,"style":33651},[208],[23,74948,74949,74952],{"style":2029},[23,74950],{"className":74951,"style":1015},[216],[23,74953,74955],{"className":74954},[1019,1020,1021,1022],[23,74956,35522],{"className":74957,"style":35607},[135,136,1022],[23,74959,297],{"className":74960},[296],[23,74962,74964],{"className":74963},[204],[23,74965,74967],{"className":74966,"style":14296},[208],[23,74968],{},[23,74970,66],{"className":74971},[169],"。正则项则是",[23,74974,74976],{"className":74975,"translate":27},[26],[23,74977,74979,75131],{"className":74978,"translate":27},[31],[23,74980,74982],{"className":74981},[35],[37,74983,74984],{"xmlns":39,"display":40},[42,74985,74986,75128],{},[45,74987,74988,74990,74992,75000,75002,75004,75010,75016,75018,75020,75022,75024,75030,75032,75034,75040,75046,75048,75050,75052,75054,75060,75062,75064,75066,75068,75080,75086,75092,75094,75096,75098,75104,75106,75112,75114,75116,75122,75124,75126],{},[48,74989,20684],{"mathvariant":111},[48,74991,8340],{},[19141,74993,74994,74996,74998],{},[48,74995,20684],{"mathvariant":111},[48,74997,26253],{"mathvariant":375},[2581,74999,9956],{},[51,75001,69],{},[51,75003,61189],{"fence":122,"stretchy":122,"minsize":7407,"maxsize":7407},[6963,75005,75006,75008],{},[51,75007,33855],{},[48,75009,33497],{},[898,75011,75012,75014],{},[48,75013,4526],{},[48,75015,33497],{},[48,75017,4836],{},[51,75019,54],{"stretchy":53},[51,75021,5758],{},[51,75023,747],{"separator":122},[898,75025,75026,75028],{},[48,75027,753],{},[48,75029,33497],{},[51,75031,66],{"stretchy":53},[51,75033,747],{"separator":122},[6963,75035,75036,75038],{},[51,75037,33855],{},[48,75039,35522],{},[898,75041,75042,75044],{},[48,75043,4526],{},[48,75045,35522],{},[48,75047,4836],{},[51,75049,54],{"stretchy":53},[51,75051,5758],{},[51,75053,747],{"separator":122},[898,75055,75056,75058],{},[48,75057,753],{},[48,75059,35522],{},[51,75061,66],{"stretchy":53},[51,75063,61214],{"fence":122,"stretchy":122,"minsize":7407,"maxsize":7407},[148,75065],{"linebreak":29629},[51,75067,69],{},[6963,75069,75070,75072],{},[51,75071,33855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R^n}\\left\\{L\\bigl(K\\alpha\\bigr)+\\Omega\\bigl(\\alpha^\\top K\\alpha\\bigr)\\right\\}.\n\\tag{4}",[23,76006,76008,76202],{"className":76007,"ariaHidden":122},[121],[23,76009,76011,76015,76098,76101,76196,76199],{"className":76010},[126],[23,76012],{"className":76013,"style":76014},[130],"height:1.6747em;vertical-align:-0.7756em;",[23,76016,76018],{"className":76017},[1786,7098],[23,76019,76021,76089],{"className":76020},[199,200],[23,76022,76024,76086],{"className":76023},[204],[23,76025,76027,76076],{"className":76026,"style":7108},[208],[23,76028,76030,76033],{"style":76029},"top:-2.3518em;margin-left:0em;",[23,76031],{"className":76032,"style":217},[216],[23,76034,76036],{"className":76035},[1019,1020,1021,1022],[23,76037,76039,76042,76045],{"className":76038},[135,1022],[23,76040,4526],{"className":76041,"style":4645},[135,136,1022],[23,76043,372],{"className":76044},[153,1022],[23,76046,76048,76051],{"className":76047},[135,1022],[23,76049,572],{"className":76050},[135,644,1022],[23,76052,76054],{"className":76053},[998],[23,76055,76057],{"className":76056},[199],[23,76058,76060],{"className":76059},[204],[23,76061,76064],{"className":76062,"style":76063},[208],"height:0.5935em;",[23,76065,76067,76070],{"style":76066},"top:-2.786em;margin-right:0.0714em;",[23,76068],{"className":76069,"style":24885},[216],[23,76071,76073],{"className":76072},[1019,24889,1067,1022],[23,76074,4806],{"className":76075},[135,136,1022],[23,76077,76078,76081],{"style":7132},[23,76079],{"className":76080,"style":217},[216],[23,76082,76083],{},[23,76084,6969],{"className":76085},[1786],[23,76087,297],{"className":76088},[296],[23,76090,76092],{"className":76091},[204],[23,76093,76096],{"className":76094,"style":76095},[208],"height:0.7756em;",[23,76097],{},[23,76099],{"className":76100,"style":280},[148],[23,76102,76104,76110,76113,76119,76122,76125,76131,76134,76137,76140,76143,76149,76178,76181,76184,76190],{"className":76103},[1839],[23,76105,76107],{"className":76106,"style":1844},[140,1843],[23,76108,9014],{"className":76109},[1066,1067],[23,76111,555],{"className":76112},[135,136],[23,76114,76116],{"className":76115},[140],[23,76117,54],{"className":76118},[1066,1067],[23,76120,26922],{"className":76121,"style":37145},[135,136],[23,76123,4526],{"className":76124,"style":4645},[135,136],[23,76126,76128],{"className":76127},[169],[23,76129,66],{"className":76130},[1066,1067],[23,76132],{"className":76133,"style":392},[148],[23,76135,5106],{"className":76136},[612],[23,76138],{"className":76139,"style":392},[148],[23,76141,68146],{"className":76142},[135],[23,76144,76146],{"className":76145},[140],[23,76147,54],{"className":76148},[1066,1067],[23,76150,76152,76155],{"className":76151},[135],[23,76153,4526],{"className":76154,"style":4645},[135,136],[23,76156,76158],{"className":76157},[998],[23,76159,76161],{"className":76160},[199],[23,76162,76164],{"className":76163},[204],[23,76165,76167],{"className":76166,"style":4986},[208],[23,76168,76169,76172],{"style":4989},[23,76170],{"className":76171,"style":1015},[216],[23,76173,76175],{"className":76174},[1019,1020,1021,1022],[23,76176,22475],{"className":76177},[135,1022],[23,76179,26922],{"className":76180,"style":37145},[135,136],[23,76182,4526],{"className":76183,"style":4645},[135,136],[23,76185,76187],{"className":76186},[169],[23,76188,66],{"className":76189},[1066,1067],[23,76191,76193],{"className":76192,"style":1844},[169,1843],[23,76194,9024],{"className":76195},[1066,1067],[23,76197],{"className":76198,"style":280},[148],[23,76200,112],{"className":76201},[135],[23,76203,76205,76208],{"className":76204},[62182],[23,76206],{"className":76207,"style":76014},[130],[23,76209,76211,76214,76220],{"className":76210},[135,6514],[23,76212,54],{"className":76213},[135],[23,76215,76217],{"className":76216},[135],[23,76218,24934],{"className":76219},[135],[23,76221,66],{"className":76222},[135],[11,76224,76225,76226,76254,76255,76283,76284,76312,76313,76341,76342,76393,76394,76422,76423,76452,76453,76481],{},"这就是表示定理的全部实用价值：原本在无穷维空间里找 ",[23,76227,76229,76242],{"className":76228,"translate":27},[31],[23,76230,76232],{"className":76231},[35],[37,76233,76234],{"xmlns":39},[42,76235,76236,76240],{},[45,76237,76238],{},[48,76239,8340],{},[114,76241,8340],{"encoding":116},[23,76243,76245],{"className":76244,"ariaHidden":122},[121],[23,76246,76248,76251],{"className":76247},[126],[23,76249],{"className":76250,"style":5414},[130],[23,76252,8340],{"className":76253,"style":11402},[135,136],"，现在只需找 ",[23,76256,76258,76271],{"className":76257,"translate":27},[31],[23,76259,76261],{"className":76260},[35],[37,76262,76263],{"xmlns":39},[42,76264,76265,76269],{},[45,76266,76267],{},[48,76268,4806],{},[114,76270,4806],{"encoding":116},[23,76272,76274],{"className":76273,"ariaHidden":122},[121],[23,76275,76277,76280],{"className":76276},[126],[23,76278],{"className":76279,"style":703},[130],[23,76281,4806],{"className":76282},[135,136]," 个系数。核矩阵 ",[23,76285,76287,76300],{"className":76286,"translate":27},[31],[23,76288,76290],{"className":76289},[35],[37,76291,76292],{"xmlns":39},[42,76293,76294,76298],{},[45,76295,76296],{},[48,76297,26922],{},[114,76299,26922],{"encoding":116},[23,76301,76303],{"className":76302,"ariaHidden":122},[121],[23,76304,76306,76309],{"className":76305},[126],[23,76307],{"className":76308,"style":470},[130],[23,76310,26922],{"className":76311,"style":37145},[135,136]," 是半正定的；若 ",[23,76314,76316,76329],{"className":76315,"translate":27},[31],[23,76317,76319],{"className":76318},[35],[37,76320,76321],{"xmlns":39},[42,76322,76323,76327],{},[45,76324,76325],{},[48,76326,26922],{},[114,76328,26922],{"encoding":116},[23,76330,76332],{"className":76331,"ariaHidden":122},[121],[23,76333,76335,76338],{"className":76334},[126],[23,76336],{"className":76337,"style":470},[130],[23,76339,26922],{"className":76340,"style":37145},[135,136]," 可逆，则 ",[23,76343,76345,76363],{"className":76344,"translate":27},[31],[23,76346,76348],{"className":76347},[35],[37,76349,76350],{"xmlns":39},[42,76351,76352,76360],{},[45,76353,76354,76356,76358],{},[48,76355,4526],{},[51,76357,756],{},[48,76359,8340],{},[114,76361,76362],{"encoding":116},"\\alpha\\mapsto f",[23,76364,76366,76384],{"className":76365,"ariaHidden":122},[121],[23,76367,76369,76372,76375,76378,76381],{"className":76368},[126],[23,76370],{"className":76371,"style":59984},[130],[23,76373,4526],{"className":76374,"style":4645},[135,136],[23,76376],{"className":76377,"style":149},[148],[23,76379,756],{"className":76380},[153],[23,76382],{"className":76383,"style":149},[148],[23,76385,76387,76390],{"className":76386},[126],[23,76388],{"className":76389,"style":5414},[130],[23,76391,8340],{"className":76392,"style":11402},[135,136]," 是一一对应；若 ",[23,76395,76397,76410],{"className":76396,"translate":27},[31],[23,76398,76400],{"className":76399},[35],[37,76401,76402],{"xmlns":39},[42,76403,76404,76408],{},[45,76405,76406],{},[48,76407,26922],{},[114,76409,26922],{"encoding":116},[23,76411,76413],{"className":76412,"ariaHidden":122},[121],[23,76414,76416,76419],{"className":76415},[126],[23,76417],{"className":76418,"style":470},[130],[23,76420,26922],{"className":76421,"style":37145},[135,136]," 奇异，不同的 ",[23,76424,76426,76440],{"className":76425,"translate":27},[31],[23,76427,76429],{"className":76428},[35],[37,76430,76431],{"xmlns":39},[42,76432,76433,76437],{},[45,76434,76435],{},[48,76436,4526],{},[114,76438,76439],{"encoding":116},"\\alpha",[23,76441,76443],{"className":76442,"ariaHidden":122},[121],[23,76444,76446,76449],{"className":76445},[126],[23,76447],{"className":76448,"style":703},[130],[23,76450,4526],{"className":76451,"style":4645},[135,136]," 可能对应同一个函数（相差一个 ",[23,76454,76456,76469],{"className":76455,"translate":27},[31],[23,76457,76459],{"className":76458},[35],[37,76460,76461],{"xmlns":39},[42,76462,76463,76467],{},[45,76464,76465],{},[48,76466,26922],{},[114,76468,26922],{"encoding":116},[23,76470,76472],{"className":76471,"ariaHidden":122},[121],[23,76473,76475,76478],{"className":76474},[126],[23,76476],{"className":76477,"style":470},[130],[23,76479,26922],{"className":76480,"style":37145},[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。",[2501,76483,76485],{"id":76484},"_34-唯一性系数与偏置项","3.4 唯一性、系数与偏置项",[11,76487,76488,76489,76492,76493,76496],{},"表示定理给出的是",[18,76490,76491],{},"解的形式","，不是",[18,76494,76495],{},"解的唯一性","。以下几点容易混淆：",[346,76498,76499,76924,76989],{},[349,76500,76501,69087,76504,76532,76533,76561,76562,76621,76622,76672,76673,76701,76702,76760,76761,76894,76895,76923],{},[18,76502,76503],{},"函数唯一 vs 系数唯一。",[23,76505,76507,76520],{"className":76506,"translate":27},[31],[23,76508,76510],{"className":76509},[35],[37,76511,76512],{"xmlns":39},[42,76513,76514,76518],{},[45,76515,76516],{},[48,76517,26922],{},[114,76519,26922],{"encoding":116},[23,76521,76523],{"className":76522,"ariaHidden":122},[121],[23,76524,76526,76529],{"className":76525},[126],[23,76527],{"className":76528,"style":470},[130],[23,76530,26922],{"className":76531,"style":37145},[135,136]," 奇异，",[23,76534,76536,76549],{"className":76535,"translate":27},[31],[23,76537,76539],{"className":76538},[35],[37,76540,76541],{"xmlns":39},[42,76542,76543,76547],{},[45,76544,76545],{},[48,76546,4526],{},[114,76548,76439],{"encoding":116},[23,76550,76552],{"className":76551,"ariaHidden":122},[121],[23,76553,76555,76558],{"className":76554},[126],[23,76556],{"className":76557,"style":703},[130],[23,76559,4526],{"className":76560,"style":4645},[135,136]," 不唯一，但 ",[23,76563,76565,76583],{"className":76564,"translate":27},[31],[23,76566,76568],{"className":76567},[35],[37,76569,76570],{"xmlns":39},[42,76571,76572,76580],{},[45,76573,76574],{},[4518,76575,76576,76578],{},[48,76577,8340],{},[51,76579,6883],{},[114,76581,76582],{"encoding":116},"f^*",[23,76584,76586],{"className":76585,"ariaHidden":122},[121],[23,76587,76589,76592],{"className":76588},[126],[23,76590],{"className":76591,"style":5414},[130],[23,76593,76595,76598],{"className":76594},[135],[23,76596,8340],{"className":76597,"style":11402},[135,136],[23,76599,76601],{"className":76600},[998],[23,76602,76604],{"className":76603},[199],[23,76605,76607],{"className":76606},[204],[23,76608,76610],{"className":76609,"style":6914},[208],[23,76611,76612,76615],{"style":4696},[23,76613],{"className":76614,"style":1015},[216],[23,76616,76618],{"className":76617},[1019,1020,1021,1022],[23,76619,6883],{"className":76620},[612,1022]," 可能唯一。核岭回归在 ",[23,76623,76625,76642],{"className":76624,"translate":27},[31],[23,76626,76628],{"className":76627},[35],[37,76629,76630],{"xmlns":39},[42,76631,76632,76640],{},[45,76633,76634,76636,76638],{},[48,76635,61761],{},[51,76637,2579],{},[2581,76639,1532],{},[114,76641,69685],{"encoding":116},[23,76643,76645,76663],{"className":76644,"ariaHidden":122},[121],[23,76646,76648,76651,76654,76657,76660],{"className":76647},[126],[23,76649],{"className":76650,"style":4450},[130],[23,76652,61761],{"className":76653},[135,136],[23,76655],{"className":76656,"style":149},[148],[23,76658,2579],{"className":76659},[153],[23,76661],{"className":76662,"style":149},[148],[23,76664,76666,76669],{"className":76665},[126],[23,76667],{"className":76668,"style":2770},[130],[23,76670,1532],{"className":76671},[135]," 时就是如此：目标函数关于 ",[23,76674,76676,76689],{"className":76675,"translate":27},[31],[23,76677,76679],{"className":76678},[35],[37,76680,76681],{"xmlns":39},[42,76682,76683,76687],{},[45,76684,76685],{},[48,76686,8340],{},[114,76688,8340],{"encoding":116},[23,76690,76692],{"className":76691,"ariaHidden":122},[121],[23,76693,76695,76698],{"className":76694},[126],[23,76696],{"className":76697,"style":5414},[130],[23,76699,8340],{"className":76700,"style":11402},[135,136]," 严格凸，",[23,76703,76705,76722],{"className":76704,"translate":27},[31],[23,76706,76708],{"className":76707},[35],[37,76709,76710],{"xmlns":39},[42,76711,76712,76720],{},[45,76713,76714],{},[4518,76715,76716,76718],{},[48,76717,8340],{},[51,76719,6883],{},[114,76721,76582],{"encoding":116},[23,76723,76725],{"className":76724,"ariaHidden":122},[121],[23,76726,76728,76731],{"className":76727},[126],[23,76729],{"className":76730,"style":5414},[130],[23,76732,76734,76737],{"className":76733},[135],[23,76735,8340],{"className":76736,"style":11402},[135,136],[23,76738,76740],{"className":76739},[998],[23,76741,76743],{"className":76742},[199],[23,76744,76746],{"className":76745},[204],[23,76747,76749],{"className":76748,"style":6914},[208],[23,76750,76751,76754],{"style":4696},[23,76752],{"className":76753,"style":1015},[216],[23,76755,76757],{"className":76756},[1019,1020,1021,1022],[23,76758,6883],{"className":76759},[612,1022]," 唯一；而 ",[23,76762,76764,76802],{"className":76763,"translate":27},[31],[23,76765,76767],{"className":76766},[35],[37,76768,76769],{"xmlns":39},[42,76770,76771,76799],{},[45,76772,76773,76775,76777,76779,76781,76783,76785,76787,76797],{},[48,76774,4526],{},[51,76776,69],{},[51,76778,54],{"stretchy":53},[48,76780,26922],{},[51,76782,5106],{},[48,76784,61761],{},[48,76786,34357],{},[4518,76788,76789,76791],{},[51,76790,66],{"stretchy":53},[45,76792,76793,76795],{},[51,76794,4529],{},[2581,76796,4435],{},[48,76798,660],{},[114,76800,76801],{"encoding":116},"\\alpha=(K+\\lambda I)^{-1}y",[23,76803,76805,76823,76844],{"className":76804,"ariaHidden":122},[121],[23,76806,76808,76811,76814,76817,76820],{"className":76807},[126],[23,76809],{"className":76810,"style":703},[130],[23,76812,4526],{"className":76813,"style":4645},[135,136],[23,76815],{"className":76816,"style":149},[148],[23,76818,69],{"className":76819},[153],[23,76821],{"className":76822,"style":149},[148],[23,76824,76826,76829,76832,76835,76838,76841],{"className":76825},[126],[23,76827],{"className":76828,"style":131},[130],[23,76830,54],{"className":76831},[140],[23,76833,26922],{"className":76834,"style":37145},[135,136],[23,76836],{"className":76837,"style":392},[148],[23,76839,5106],{"className":76840},[612],[23,76842],{"className":76843,"style":392},[148],[23,76845,76847,76850,76853,76856,76891],{"className":76846},[126],[23,76848],{"className":76849,"style":10665},[130],[23,76851,61761],{"className":76852},[135,136],[23,76854,34357],{"className":76855,"style":454},[135,136],[23,76857,76859,76862],{"className":76858},[169],[23,76860,66],{"className":76861},[169],[23,76863,76865],{"className":76864},[998],[23,76866,76868],{"className":76867},[199],[23,76869,76871],{"className":76870},[204],[23,76872,76874],{"className":76873,"style":4693},[208],[23,76875,76876,76879],{"style":4696},[23,76877],{"className":76878,"style":1015},[216],[23,76880,76882],{"className":76881},[1019,1020,1021,1022],[23,76883,76885,76888],{"className":76884},[135,1022],[23,76886,4529],{"className":76887},[135,1022],[23,76889,4435],{"className":76890},[135,1022],[23,76892,660],{"className":76893,"style":676},[135,136]," 只是其中一个系数表示，所有相差 ",[23,76896,76898,76911],{"className":76897,"translate":27},[31],[23,76899,76901],{"className":76900},[35],[37,76902,76903],{"xmlns":39},[42,76904,76905,76909],{},[45,76906,76907],{},[48,76908,26922],{},[114,76910,26922],{"encoding":116},[23,76912,76914],{"className":76913,"ariaHidden":122},[121],[23,76915,76917,76920],{"className":76916},[126],[23,76918],{"className":76919,"style":470},[130],[23,76921,26922],{"className":76922,"style":37145},[135,136]," 零空间的系数给出同一个函数。",[349,76925,76926,76988],{},[18,76927,76928,76929,76987],{},"损失不严格凸时，",[23,76930,76932,76949],{"className":76931,"translate":27},[31],[23,76933,76935],{"className":76934},[35],[37,76936,76937],{"xmlns":39},[42,76938,76939,76947],{},[45,76940,76941],{},[4518,76942,76943,76945],{},[48,76944,8340],{},[51,76946,6883],{},[114,76948,76582],{"encoding":116},[23,76950,76952],{"className":76951,"ariaHidden":122},[121],[23,76953,76955,76958],{"className":76954},[126],[23,76956],{"className":76957,"style":5414},[130],[23,76959,76961,76964],{"className":76960},[135],[23,76962,8340],{"className":76963,"style":11402},[135,136],[23,76965,76967],{"className":76966},[998],[23,76968,76970],{"className":76969},[199],[23,76971,76973],{"className":76972},[204],[23,76974,76976],{"className":76975,"style":6914},[208],[23,76977,76978,76981],{"style":4696},[23,76979],{"className":76980,"style":1015},[216],[23,76982,76984],{"className":76983},[1019,1020,1021,1022],[23,76985,6883],{"className":76986},[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。",[349,76990,76991,76994,76995,77061,77062,77090,77091,77119,77120,77365,77366,77394,77395,77398],{},[18,76992,76993],{},"偏置项要单独处理。"," 如果模型是 ",[23,76996,76998,77022],{"className":76997,"translate":27},[31],[23,76999,77001],{"className":77000},[35],[37,77002,77003],{"xmlns":39},[42,77004,77005,77019],{},[45,77006,77007,77009,77011,77013,77015,77017],{},[48,77008,8340],{},[51,77010,54],{"stretchy":53},[48,77012,753],{},[51,77014,66],{"stretchy":53},[51,77016,5106],{},[48,77018,35537],{},[114,77020,77021],{"encoding":116},"f(x)+b",[23,77023,77025,77052],{"className":77024,"ariaHidden":122},[121],[23,77026,77028,77031,77034,77037,77040,77043,77046,77049],{"className":77027},[126],[23,77029],{"className":77030,"style":131},[130],[23,77032,8340],{"className":77033,"style":11402},[135,136],[23,77035,54],{"className":77036},[140],[23,77038,753],{"className":77039},[135,136],[23,77041,66],{"className":77042},[169],[23,77044],{"className":77045,"style":392},[148],[23,77047,5106],{"className":77048},[612],[23,77050],{"className":77051,"style":392},[148],[23,77053,77055,77058],{"className":77054},[126],[23,77056],{"className":77057,"style":779},[130],[23,77059,35537],{"className":77060},[135,136],"，而 ",[23,77063,77065,77078],{"className":77064,"translate":27},[31],[23,77066,77068],{"className":77067},[35],[37,77069,77070],{"xmlns":39},[42,77071,77072,77076],{},[45,77073,77074],{},[48,77075,35537],{},[114,77077,35537],{"encoding":116},[23,77079,77081],{"className":77080,"ariaHidden":122},[121],[23,77082,77084,77087],{"className":77083},[126],[23,77085],{"className":77086,"style":779},[130],[23,77088,35537],{"className":77089},[135,136]," 不受正则项约束，那么表示定理直接作用于 ",[23,77092,77094,77107],{"className":77093,"translate":27},[31],[23,77095,77097],{"className":77096},[35],[37,77098,77099],{"xmlns":39},[42,77100,77101,77105],{},[45,77102,77103],{},[48,77104,8340],{},[114,77106,8340],{"encoding":116},[23,77108,77110],{"className":77109,"ariaHidden":122},[121],[23,77111,77113,77116],{"className":77112},[126],[23,77114],{"className":77115,"style":5414},[130],[23,77117,8340],{"className":77118,"style":11402},[135,136]," 的部分仍然给出 ",[23,77121,77123,77171],{"className":77122,"translate":27},[31],[23,77124,77126],{"className":77125},[35],[37,77127,77128],{"xmlns":39},[42,77129,77130,77168],{},[45,77131,77132,77138,77140,77146,77152,77154,77156,77158,77160,77166],{},[4518,77133,77134,77136],{},[48,77135,8340],{},[51,77137,6883],{},[51,77139,69],{},[898,77141,77142,77144],{},[51,77143,33855],{},[48,77145,33497],{},[898,77147,77148,77150],{},[48,77149,4526],{},[48,77151,33497],{},[48,77153,4836],{},[51,77155,54],{"stretchy":53},[51,77157,5758],{},[51,77159,747],{"separator":122},[898,77161,77162,77164],{},[48,77163,753],{},[48,77165,33497],{},[51,77167,66],{"stretchy":53},[114,77169,77170],{"encoding":116},"f^*=\\sum_i\\alpha_i k(\\cdot,x_i)",[23,77172,77174,77218],{"className":77173,"ariaHidden":122},[121],[23,77175,77177,77180,77209,77212,77215],{"className":77176},[126],[23,77178],{"className":77179,"style":5414},[130],[23,77181,77183,77186],{"className":77182},[135],[23,77184,8340],{"className":77185,"style":11402},[135,136],[23,77187,77189],{"className":77188},[998],[23,77190,77192],{"className":77191},[199],[23,77193,77195],{"className":77194},[204],[23,77196,77198],{"className":77197,"style":6914},[208],[23,77199,77200,77203],{"style":4696},[23,77201],{"className":77202,"style":1015},[216],[23,77204,77206],{"className":77205},[1019,1020,1021,1022],[23,77207,6883],{"className":77208},[612,1022],[23,77210],{"className":77211,"style":149},[148],[23,77213,69],{"className":77214},[153],[23,77216],{"className":77217,"style":149},[148],[23,77219,77221,77224,77264,77267,77307,77310,77313,77316,77319,77322,77362],{"className":77220},[126],[23,77222],{"className":77223,"style":33579},[130],[23,77225,77227,77230],{"className":77226},[1786],[23,77228,33855],{"className":77229,"style":33586},[1786,1790,20458],[23,77231,77233],{"className":77232},[998],[23,77234,77236,77256],{"className":77235},[199,200],[23,77237,77239,77253],{"className":77238},[204],[23,77240,77242],{"className":77241,"style":33599},[208],[23,77243,77244,77247],{"style":33602},[23,77245],{"className":77246,"style":1015},[216],[23,77248,77250],{"className":77249},[1019,1020,1021,1022],[23,77251,33497],{"className":77252},[135,136,1022],[23,77254,297],{"className":77255},[296],[23,77257,77259],{"className":77258},[204],[23,77260,77262],{"className":77261,"style":33621},[208],[23,77263],{},[23,77265],{"className":77266,"style":280},[148],[23,77268,77270,77273],{"className":77269},[135],[23,77271,4526],{"className":77272,"style":4645},[135,136],[23,77274,77276],{"className":77275},[998],[23,77277,77279,77299],{"className":77278},[199,200],[23,77280,77282,77296],{"className":77281},[204],[23,77283,77285],{"className":77284,"style":33651},[208],[23,77286,77287,77290],{"style":68976},[23,77288],{"className":77289,"style":1015},[216],[23,77291,77293],{"className":77292},[1019,1020,1021,1022],[23,77294,33497],{"className":77295},[135,136,1022],[23,77297,297],{"className":77298},[296],[23,77300,77302],{"className":77301},[204],[23,77303,77305],{"className":77304,"style":2048},[208],[23,77306],{},[23,77308,4836],{"className":77309,"style":4851},[135,136],[23,77311,54],{"className":77312},[140],[23,77314,5758],{"className":77315},[135],[23,77317,747],{"className":77318},[824],[23,77320],{"className":77321,"style":280},[148],[23,77323,77325,77328],{"className":77324},[135],[23,77326,753],{"className":77327},[135,136],[23,77329,77331],{"className":77330},[998],[23,77332,77334,77354],{"className":77333},[199,200],[23,77335,77337,77351],{"className":77336},[204],[23,77338,77340],{"className":77339,"style":33651},[208],[23,77341,77342,77345],{"style":2029},[23,77343],{"className":77344,"style":1015},[216],[23,77346,77348],{"className":77347},[1019,1020,1021,1022],[23,77349,33497],{"className":77350},[135,136,1022],[23,77352,297],{"className":77353},[296],[23,77355,77357],{"className":77356},[204],[23,77358,77360],{"className":77359,"style":2048},[208],[23,77361],{},[23,77363,66],{"className":77364},[169],"，但 ",[23,77367,77369,77382],{"className":77368,"translate":27},[31],[23,77370,77372],{"className":77371},[35],[37,77373,77374],{"xmlns":39},[42,77375,77376,77380],{},[45,77377,77378],{},[48,77379,35537],{},[114,77381,35537],{"encoding":116},[23,77383,77385],{"className":77384,"ariaHidden":122},[121],[23,77386,77388,77391],{"className":77387},[126],[23,77389],{"className":77390,"style":779},[130],[23,77392,35537],{"className":77393},[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",[18,77396,77397],{},"半参数表示定理","（semi-parametric representer theorem）：若模型为",[23,77400,77402],{"className":77401,"translate":27},[26],[23,77403,77405,77481],{"className":77404,"translate":27},[31],[23,77406,77408],{"className":77407},[35],[37,77409,77410],{"xmlns":39,"display":40},[42,77411,77412,77478],{},[45,77413,77414,77416,77418,77420,77422,77424,77438,77444,77450,77452,77454,77456,77458,77460,77462,77464,77466,77468,77470,77472,77474,77476],{},[48,77415,8340],{},[51,77417,54],{"stretchy":53},[48,77419,753],{},[51,77421,66],{"stretchy":53},[51,77423,69],{},[61717,77425,77426,77428,77436],{},[51,77427,33855],{},[45,77429,77430,77432,77434],{},[48,77431,35522],{},[51,77433,69],{},[2581,77435,4435],{},[48,77437,11],{},[898,77439,77440,77442],{},[48,77441,4548],{},[48,77443,35522],{},[898,77445,77446,77448],{},[48,77447,18367],{},[48,77449,35522],{},[51,77451,54],{"stretchy":53},[48,77453,753],{},[51,77455,66],{"stretchy":53},[51,77457,5106],{},[48,77459,7091],{},[51,77461,54],{"stretchy":53},[48,77463,753],{},[51,77465,66],{"stretchy":53},[51,77467,747],{"separator":122},[148,77469],{"width":750},[48,77471,7091],{},[51,77473,372],{},[48,77475,26253],{"mathvariant":375},[51,77477,747],{"separator":122},[114,77479,77480],{"encoding":116},"f(x)=\\sum_{j=1}^p\\beta_j\\phi_j(x)+g(x),\\qquad g\\in\\mathcal H,",[23,77482,77484,77511,77686,77725],{"className":77483,"ariaHidden":122},[121],[23,77485,77487,77490,77493,77496,77499,77502,77505,77508],{"className":77486},[126],[23,77488],{"className":77489,"style":131},[130],[23,77491,8340],{"className":77492,"style":11402},[135,136],[23,77494,54],{"className":77495},[140],[23,77497,753],{"className":77498},[135,136],[23,77500,66],{"className":77501},[169],[23,77503],{"className":77504,"style":149},[148],[23,77506,69],{"className":77507},[153],[23,77509],{"className":77510,"style":149},[148],[23,77512,77514,77518,77584,77587,77628,77668,77671,77674,77677,77680,77683],{"className":77513},[126],[23,77515],{"className":77516,"style":77517},[130],"height:3.1123em;vertical-align:-1.4138em;",[23,77519,77521],{"className":77520},[1786,7098],[23,77522,77524,77576],{"className":77523},[199,200],[23,77525,77527,77573],{"className":77526},[204],[23,77528,77531,77551,77561],{"className":77529,"style":77530},[208],"height:1.6985em;",[23,77532,77533,77536],{"style":34079},[23,77534],{"className":77535,"style":34083},[216],[23,77537,77539],{"className":77538},[1019,1020,1021,1022],[23,77540,77542,77545,77548],{"className":77541},[135,1022],[23,77543,35522],{"className":77544,"style":35607},[135,136,1022],[23,77546,69],{"className":77547},[153,1022],[23,77549,4435],{"className":77550},[135,1022],[23,77552,77553,77556],{"style":34092},[23,77554],{"className":77555,"style":34083},[216],[23,77557,77558],{},[23,77559,33855],{"className":77560},[1786,1790,1791],[23,77562,77564,77567],{"style":77563},"top:-4.3471em;margin-left:0em;",[23,77565],{"className":77566,"style":34083},[216],[23,77568,77570],{"className":77569},[1019,1020,1021,1022],[23,77571,11],{"className":77572},[135,136,1022],[23,77574,297],{"className":77575},[296],[23,77577,77579],{"className":77578},[204],[23,77580,77582],{"className":77581,"style":63091},[208],[23,77583],{},[23,77585],{"className":77586,"style":280},[148],[23,77588,77590,77593],{"className":77589},[135],[23,77591,4548],{"className":77592,"style":4655},[135,136],[23,77594,77596],{"className":77595},[998],[23,77597,77599,77620],{"className":77598},[199,200],[23,77600,77602,77617],{"className":77601},[204],[23,77603,77605],{"className":77604,"style":33651},[208],[23,77606,77608,77611],{"style":77607},"top:-2.55em;margin-left:-0.0528em;margin-right:0.05em;",[23,77609],{"className":77610,"style":1015},[216],[23,77612,77614],{"className":77613},[1019,1020,1021,1022],[23,77615,35522],{"className":77616,"style":35607},[135,136,1022],[23,77618,297],{"className":77619},[296],[23,77621,77623],{"className":77622},[204],[23,77624,77626],{"className":77625,"style":14296},[208],[23,77627],{},[23,77629,77631,77634],{"className":77630},[135],[23,77632,18367],{"className":77633},[135,136],[23,77635,77637],{"className":77636},[998],[23,77638,77640,77660],{"className":77639},[199,200],[23,77641,77643,77657],{"className":77642},[204],[23,77644,77646],{"className":77645,"style":33651},[208],[23,77647,77648,77651],{"style":2029},[23,77649],{"className":77650,"style":1015},[216],[23,77652,77654],{"className":77653},[1019,1020,1021,1022],[23,77655,35522],{"className":77656,"style":35607},[135,136,1022],[23,77658,297],{"className":77659},[296],[23,77661,77663],{"className":77662},[204],[23,77664,77666],{"className":77665,"style":14296},[208],[23,77667],{},[23,77669,54],{"className":77670},[140],[23,77672,753],{"className":77673},[135,136],[23,77675,66],{"className":77676},[169],[23,77678],{"className":77679,"style":392},[148],[23,77681,5106],{"className":77682},[612],[23,77684],{"className":77685,"style":392},[148],[23,77687,77689,77692,77695,77698,77701,77704,77707,77710,77713,77716,77719,77722],{"className":77688},[126],[23,77690],{"className":77691,"style":131},[130],[23,77693,7091],{"className":77694,"style":676},[135,136],[23,77696,54],{"className":77697},[140],[23,77699,753],{"className":77700},[135,136],[23,77702,66],{"className":77703},[169],[23,77705,747],{"className":77706},[824],[23,77708],{"className":77709,"style":828},[148],[23,77711],{"className":77712,"style":280},[148],[23,77714,7091],{"className":77715,"style":676},[135,136],[23,77717],{"className":77718,"style":149},[148],[23,77720,372],{"className":77721},[153],[23,77723],{"className":77724,"style":149},[148],[23,77726,77728,77731,77734],{"className":77727},[126],[23,77729],{"className":77730,"style":817},[130],[23,77732,26253],{"className":77733,"style":45028},[135,412],[23,77735,747],{"className":77736},[824],[11,77738,24433,77739,77880,77881,77980,77981,78009,78010,78039],{},[23,77740,77742,77772],{"className":77741,"translate":27},[31],[23,77743,77745],{"className":77744},[35],[37,77746,77747],{"xmlns":39},[42,77748,77749,77769],{},[45,77750,77751,77757,77759,77761,77763],{},[898,77752,77753,77755],{},[48,77754,18367],{},[2581,77756,4435],{},[51,77758,747],{"separator":122},[51,77760,62563],{},[51,77762,747],{"separator":122},[898,77764,77765,77767],{},[48,77766,18367],{},[48,77768,11],{},[114,77770,77771],{"encoding":116},"\\phi_1,\\dots,\\phi_p",[23,77773,77775],{"className":77774,"ariaHidden":122},[121],[23,77776,77778,77782,77822,77825,77828,77831,77834,77837,77840],{"className":77777},[126],[23,77779],{"className":77780,"style":77781},[130],"height:0.9805em;vertical-align:-0.2861em;",[23,77783,77785,77788],{"className":77784},[135],[23,77786,18367],{"className":77787},[135,136],[23,77789,77791],{"className":77790},[998],[23,77792,77794,77814],{"className":77793},[199,200],[23,77795,77797,77811],{"className":77796},[204],[23,77798,77800],{"className":77799,"style":18651},[208],[23,77801,77802,77805],{"style":2029},[23,77803],{"className":77804,"style":1015},[216],[23,77806,77808],{"className":77807},[1019,1020,1021,1022],[23,77809,4435],{"className":77810},[135,1022],[23,77812,297],{"className":77813},[296],[23,77815,77817],{"className":77816},[204],[23,77818,77820],{"className":77819,"style":2048},[208],[23,77821],{},[23,77823,747],{"className":77824},[824],[23,77826],{"className":77827,"style":280},[148],[23,77829,62563],{"className":77830},[1839],[23,77832],{"className":77833,"style":280},[148],[23,77835,747],{"className":77836},[824],[23,77838],{"className":77839,"style":280},[148],[23,77841,77843,77846],{"className":77842},[135],[23,77844,18367],{"className":77845},[135,136],[23,77847,77849],{"className":77848},[998],[23,77850,77852,77872],{"className":77851},[199,200],[23,77853,77855,77869],{"className":77854},[204],[23,77856,77858],{"className":77857,"style":6499},[208],[23,77859,77860,77863],{"style":2029},[23,77861],{"className":77862,"style":1015},[216],[23,77864,77866],{"className":77865},[1019,1020,1021,1022],[23,77867,11],{"className":77868},[135,136,1022],[23,77870,297],{"className":77871},[296],[23,77873,77875],{"className":77874},[204],[23,77876,77878],{"className":77877,"style":14296},[208],[23,77879],{}," 张成一个有限维空间（例如常数、线性函数），正则项只惩罚 ",[23,77882,77884,77908],{"className":77883,"translate":27},[31],[23,77885,77887],{"className":77886},[35],[37,77888,77889],{"xmlns":39},[42,77890,77891,77905],{},[45,77892,77893,77895,77897],{},[48,77894,20684],{"mathvariant":111},[48,77896,7091],{},[19141,77898,77899,77901,77903],{},[48,77900,20684],{"mathvariant":111},[48,77902,26253],{"mathvariant":375},[2581,77904,9956],{},[114,77906,77907],{"encoding":116},"\\|g\\|_{\\mathcal H}^2",[23,77909,77911],{"className":77910,"ariaHidden":122},[121],[23,77912,77914,77918,77921,77924],{"className":77913},[126],[23,77915],{"className":77916,"style":77917},[130],"height:1.0894em;vertical-align:-0.2753em;",[23,77919,20684],{"className":77920},[135],[23,77922,7091],{"className":77923,"style":676},[135,136],[23,77925,77927,77930],{"className":77926},[135],[23,77928,20684],{"className":77929},[135],[23,77931,77933],{"className":77932},[998],[23,77934,77936,77971],{"className":77935},[199,200],[23,77937,77939,77968],{"className":77938},[204],[23,77940,77942,77957],{"className":77941,"style":4693},[208],[23,77943,77945,77948],{"style":77944},"top:-2.4247em;margin-left:0em;margin-right:0.05em;",[23,77946],{"className":77947,"style":1015},[216],[23,77949,77951],{"className":77950},[1019,1020,1021,1022],[23,77952,77954],{"className":77953},[135,1022],[23,77955,26253],{"className":77956,"style":45028},[135,412,1022],[23,77958,77959,77962],{"style":4696},[23,77960],{"className":77961,"style":1015},[216],[23,77963,77965],{"className":77964},[1019,1020,1021,1022],[23,77966,9956],{"className":77967},[135,1022],[23,77969,297],{"className":77970},[296],[23,77972,77974],{"className":77973},[204],[23,77975,77978],{"className":77976,"style":77977},[208],"height:0.2753em;",[23,77979],{},"，则最优的 ",[23,77982,77984,77997],{"className":77983,"translate":27},[31],[23,77985,77987],{"className":77986},[35],[37,77988,77989],{"xmlns":39},[42,77990,77991,77995],{},[45,77992,77993],{},[48,77994,7091],{},[114,77996,7091],{"encoding":116},[23,77998,78000],{"className":77999,"ariaHidden":122},[121],[23,78001,78003,78006],{"className":78002},[126],[23,78004],{"className":78005,"style":672},[130],[23,78007,7091],{"className":78008,"style":676},[135,136]," 仍在训练点的张成空间中，而 ",[23,78011,78013,78027],{"className":78012,"translate":27},[31],[23,78014,78016],{"className":78015},[35],[37,78017,78018],{"xmlns":39},[42,78019,78020,78024],{},[45,78021,78022],{},[48,78023,4548],{},[114,78025,78026],{"encoding":116},"\\beta",[23,78028,78030],{"className":78029,"ariaHidden":122},[121],[23,78031,78033,78036],{"className":78032},[126],[23,78034],{"className":78035,"style":5414},[130],[23,78037,4548],{"className":78038,"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。",[338,78041,78043],{"id":78042},"四为什么是范数的非减函数","四、为什么是「范数的非减函数」",[2501,78045,78047],{"id":78046},"_41-充分性","4.1 充分性",[11,78049,78050,78051,78079,78080,78149,78150,78219,78220,63915,78392,78420],{},"第三节的证明已经说明：只要 ",[23,78052,78054,78067],{"className":78053,"translate":27},[31],[23,78055,78057],{"className":78056},[35],[37,78058,78059],{"xmlns":39},[42,78060,78061,78065],{},[45,78062,78063],{},[48,78064,68146],{"mathvariant":111},[114,78066,69104],{"encoding":116},[23,78068,78070],{"className":78069,"ariaHidden":122},[121],[23,78071,78073,78076],{"className":78072},[126],[23,78074],{"className":78075,"style":470},[130],[23,78077,68146],{"className":78078},[135]," 非减，投影 ",[23,78081,78083,78100],{"className":78082,"translate":27},[31],[23,78084,78086],{"className":78085},[35],[37,78087,78088],{"xmlns":39},[42,78089,78090,78098],{},[45,78091,78092],{},[898,78093,78094,78096],{},[48,78095,8340],{},[51,78097,20684],{},[114,78099,73040],{"encoding":116},[23,78101,78103],{"className":78102,"ariaHidden":122},[121],[23,78104,78106,78109],{"className":78105},[126],[23,78107],{"className":78108,"style":70235},[130],[23,78110,78112,78115],{"className":78111},[135],[23,78113,8340],{"className":78114,"style":11402},[135,136],[23,78116,78118],{"className":78117},[998],[23,78119,78121,78141],{"className":78120},[199,200],[23,78122,78124,78138],{"className":78123},[204],[23,78125,78127],{"className":78126,"style":1008},[208],[23,78128,78129,78132],{"style":70256},[23,78130],{"className":78131,"style":1015},[216],[23,78133,78135],{"className":78134},[1019,1020,1021,1022],[23,78136,20684],{"className":78137},[153,1022],[23,78139,297],{"className":78140},[296],[23,78142,78144],{"className":78143},[204],[23,78145,78147],{"className":78146,"style":1057},[208],[23,78148],{}," 就不会让目标变差，因此总有一个最优解落在 ",[23,78151,78153,78170],{"className":78152,"translate":27},[31],[23,78154,78156],{"className":78155},[35],[37,78157,78158],{"xmlns":39},[42,78159,78160,78168],{},[45,78161,78162],{},[898,78163,78164,78166],{},[48,78165,26253],{"mathvariant":375},[2581,78167,1532],{},[114,78169,73826],{"encoding":116},[23,78171,78173],{"className":78172,"ariaHidden":122},[121],[23,78174,78176,78179],{"className":78175},[126],[23,78177],{"className":78178,"style":42831},[130],[23,78180,78182,78185],{"className":78181},[135],[23,78183,26253],{"className":78184,"style":45028},[135,412],[23,78186,78188],{"className":78187},[998],[23,78189,78191,78211],{"className":78190},[199,200],[23,78192,78194,78208],{"className":78193},[204],[23,78195,78197],{"className":78196,"style":18651},[208],[23,78198,78199,78202],{"style":61635},[23,78200],{"className":78201,"style":1015},[216],[23,78203,78205],{"className":78204},[1019,1020,1021,1022],[23,78206,1532],{"className":78207},[135,1022],[23,78209,297],{"className":78210},[296],[23,78212,78214],{"className":78213},[204],[23,78215,78217],{"className":78216,"style":2048},[208],[23,78218],{}," 中。这一步只用了 ",[23,78221,78223,78261],{"className":78222,"translate":27},[31],[23,78224,78226],{"className":78225},[35],[37,78227,78228],{"xmlns":39},[42,78229,78230,78258],{},[45,78231,78232,78234,78236,78242,78244,78246,78252],{},[48,78233,20684],{"mathvariant":111},[48,78235,8340],{},[4518,78237,78238,78240],{},[48,78239,20684],{"mathvariant":111},[2581,78241,9956],{},[51,78243,7382],{},[48,78245,20684],{"mathvariant":111},[898,78247,78248,78250],{},[48,78249,8340],{},[51,78251,20684],{},[4518,78253,78254,78256],{},[48,78255,20684],{"mathvariant":111},[2581,78257,9956],{},[114,78259,78260],{"encoding":116},"\\|f\\|^2\\ge\\|f_\\parallel\\|^2",[23,78262,78264,78314],{"className":78263,"ariaHidden":122},[121],[23,78265,78267,78270,78273,78276,78305,78308,78311],{"className":78266},[126],[23,78268],{"className":78269,"style":10665},[130],[23,78271,20684],{"className":78272},[135],[23,78274,8340],{"className":78275,"style":11402},[135,136],[23,78277,78279,78282],{"className":78278},[135],[23,78280,20684],{"className":78281},[135],[23,78283,78285],{"className":78284},[998],[23,78286,78288],{"className":78287},[199],[23,78289,78291],{"className":78290},[204],[23,78292,78294],{"className":78293,"style":4693},[208],[23,78295,78296,78299],{"style":4696},[23,78297],{"className":78298,"style":1015},[216],[23,78300,78302],{"className":78301},[1019,1020,1021,1022],[23,78303,9956],{"className":78304},[135,1022],[23,78306],{"className":78307,"style":149},[148],[23,78309,7382],{"className":78310},[153],[23,78312],{"className":78313,"style":149},[148],[23,78315,78317,78320,78323,78363],{"className":78316},[126],[23,78318],{"className":78319,"style":73610},[130],[23,78321,20684],{"className":78322},[135],[23,78324,78326,78329],{"className":78325},[135],[23,78327,8340],{"className":78328,"style":11402},[135,136],[23,78330,78332],{"className":78331},[998],[23,78333,78335,78355],{"className":78334},[199,200],[23,78336,78338,78352],{"className":78337},[204],[23,78339,78341],{"className":78340,"style":1008},[208],[23,78342,78343,78346],{"style":70256},[23,78344],{"className":78345,"style":1015},[216],[23,78347,78349],{"className":78348},[1019,1020,1021,1022],[23,78350,20684],{"className":78351},[153,1022],[23,78353,297],{"className":78354},[296],[23,78356,78358],{"className":78357},[204],[23,78359,78361],{"className":78360,"style":1057},[208],[23,78362],{},[23,78364,78366,78369],{"className":78365},[135],[23,78367,20684],{"className":78368},[135],[23,78370,78372],{"className":78371},[998],[23,78373,78375],{"className":78374},[199],[23,78376,78378],{"className":78377},[204],[23,78379,78381],{"className":78380,"style":4693},[208],[23,78382,78383,78386],{"style":4696},[23,78384],{"className":78385,"style":1015},[216],[23,78387,78389],{"className":78388},[1019,1020,1021,1022],[23,78390,9956],{"className":78391},[135,1022],[23,78393,78395,78408],{"className":78394,"translate":27},[31],[23,78396,78398],{"className":78397},[35],[37,78399,78400],{"xmlns":39},[42,78401,78402,78406],{},[45,78403,78404],{},[48,78405,68146],{"mathvariant":111},[114,78407,69104],{"encoding":116},[23,78409,78411],{"className":78410,"ariaHidden":122},[121],[23,78412,78414,78417],{"className":78413},[126],[23,78415],{"className":78416,"style":470},[130],[23,78418,68146],{"className":78419},[135]," 的单调性，没有别的东西。",[2501,78422,78424],{"id":78423},"_42-反例非单调正则项","4.2 反例：非单调正则项",[11,78426,78427,78428,78456],{},"反过来，如果 ",[23,78429,78431,78444],{"className":78430,"translate":27},[31],[23,78432,78434],{"className":78433},[35],[37,78435,78436],{"xmlns":39},[42,78437,78438,78442],{},[45,78439,78440],{},[48,78441,68146],{"mathvariant":111},[114,78443,69104],{"encoding":116},[23,78445,78447],{"className":78446,"ariaHidden":122},[121],[23,78448,78450,78453],{"className":78449},[126],[23,78451],{"className":78452,"style":470},[130],[23,78454,68146],{"className":78455},[135]," 不是非减的，表示定理就可能失效。考虑一个最小的例子。",[11,78458,78459,78460,78541,78542,78710,78711,78792,78793,78908,78909,79001],{},"取 ",[23,78461,78463,78485],{"className":78462,"translate":27},[31],[23,78464,78466],{"className":78465},[35],[37,78467,78468],{"xmlns":39},[42,78469,78470,78482],{},[45,78471,78472,78474,78476],{},[48,78473,434],{"mathvariant":375},[51,78475,69],{},[4518,78477,78478,78480],{},[48,78479,572],{"mathvariant":571},[2581,78481,9956],{},[114,78483,78484],{"encoding":116},"\\mathcal X=\\mathbb R^2",[23,78486,78488,78506],{"className":78487,"ariaHidden":122},[121],[23,78489,78491,78494,78497,78500,78503],{"className":78490},[126],[23,78492],{"className":78493,"style":470},[130],[23,78495,434],{"className":78496,"style":474},[135,412],[23,78498],{"className":78499,"style":149},[148],[23,78501,69],{"className":78502},[153],[23,78504],{"className":78505,"style":149},[148],[23,78507,78509,78512],{"className":78508},[126],[23,78510],{"className":78511,"style":4693},[130],[23,78513,78515,78518],{"className":78514},[135],[23,78516,572],{"className":78517},[135,644],[23,78519,78521],{"className":78520},[998],[23,78522,78524],{"className":78523},[199],[23,78525,78527],{"className":78526},[204],[23,78528,78530],{"className":78529,"style":4693},[208],[23,78531,78532,78535],{"style":4696},[23,78533],{"className":78534,"style":1015},[216],[23,78536,78538],{"className":78537},[1019,1020,1021,1022],[23,78539,9956],{"className":78540},[135,1022],"，线性核 ",[23,78543,78545,78589],{"className":78544,"translate":27},[31],[23,78546,78548],{"className":78547},[35],[37,78549,78550],{"xmlns":39},[42,78551,78552,78586],{},[45,78553,78554,78556,78558,78560,78562,78568,78570,78572,78574,78576,78578,78584],{},[48,78555,4836],{},[51,78557,54],{"stretchy":53},[48,78559,753],{},[51,78561,747],{"separator":122},[4518,78563,78564,78566],{},[48,78565,753],{},[51,78567,38295],{"mathvariant":111,"lspace":1522,"rspace":1522},[51,78569,66],{"stretchy":53},[51,78571,69],{},[51,78573,61189],{"stretchy":53},[48,78575,753],{},[51,78577,747],{"separator":122},[4518,78579,78580,78582],{},[48,78581,753],{},[51,78583,38295],{"mathvariant":111,"lspace":1522,"rspace":1522},[51,78585,61214],{"stretchy":53},[114,78587,78588],{"encoding":116},"k(x,x')=\\langle x,x'\\rangle",[23,78590,78592,78657],{"className":78591,"ariaHidden":122},[121],[23,78593,78595,78598,78601,78604,78607,78610,78613,78645,78648,78651,78654],{"className":78594},[126],[23,78596],{"className":78597,"style":66382},[130],[23,78599,4836],{"className":78600,"style":4851},[135,136],[23,78602,54],{"className":78603},[140],[23,78605,753],{"className":78606},[135,136],[23,78608,747],{"className":78609},[824],[23,78611],{"className":78612,"style":280},[148],[23,78614,78616,78619],{"className":78615},[135],[23,78617,753],{"className":78618},[135,136],[23,78620,78622],{"className":78621},[998],[23,78623,78625],{"className":78624},[199],[23,78626,78628],{"className":78627},[204],[23,78629,78631],{"className":78630,"style":38337},[208],[23,78632,78633,78636],{"style":4696},[23,78634],{"className":78635,"style":1015},[216],[23,78637,78639],{"className":78638},[1019,1020,1021,1022],[23,78640,78642],{"className":78641},[135,1022],[23,78643,38295],{"className":78644},[135,1022],[23,78646,66],{"className":78647},[169],[23,78649],{"className":78650,"style":149},[148],[23,78652,69],{"className":78653},[153],[23,78655],{"className":78656,"style":149},[148],[23,78658,78660,78663,78666,78669,78672,78675,78707],{"className":78659},[126],[23,78661],{"className":78662,"style":66382},[130],[23,78664,61189],{"className":78665},[140],[23,78667,753],{"className":78668},[135,136],[23,78670,747],{"className":78671},[824],[23,78673],{"className":78674,"style":280},[148],[23,78676,78678,78681],{"className":78677},[135],[23,78679,753],{"className":78680},[135,136],[23,78682,78684],{"className":78683},[998],[23,78685,78687],{"className":78686},[199],[23,78688,78690],{"className":78689},[204],[23,78691,78693],{"className":78692,"style":38337},[208],[23,78694,78695,78698],{"style":4696},[23,78696],{"className":78697,"style":1015},[216],[23,78699,78701],{"className":78700},[1019,1020,1021,1022],[23,78702,78704],{"className":78703},[135,1022],[23,78705,38295],{"className":78706},[135,1022],[23,78708,61214],{"className":78709},[169],"，此时 ",[23,78712,78714,78736],{"className":78713,"translate":27},[31],[23,78715,78717],{"className":78716},[35],[37,78718,78719],{"xmlns":39},[42,78720,78721,78733],{},[45,78722,78723,78725,78727],{},[48,78724,26253],{"mathvariant":375},[51,78726,69],{},[4518,78728,78729,78731],{},[48,78730,572],{"mathvariant":571},[2581,78732,9956],{},[114,78734,78735],{"encoding":116},"\\mathcal H=\\mathbb R^2",[23,78737,78739,78757],{"className":78738,"ariaHidden":122},[121],[23,78740,78742,78745,78748,78751,78754],{"className":78741},[126],[23,78743],{"className":78744,"style":470},[130],[23,78746,26253],{"className":78747,"style":45028},[135,412],[23,78749],{"className":78750,"style":149},[148],[23,78752,69],{"className":78753},[153],[23,78755],{"className":78756,"style":149},[148],[23,78758,78760,78763],{"className":78759},[126],[23,78761],{"className":78762,"style":4693},[130],[23,78764,78766,78769],{"className":78765},[135],[23,78767,572],{"className":78768},[135,644],[23,78770,78772],{"className":78771},[998],[23,78773,78775],{"className":78774},[199],[23,78776,78778],{"className":78777},[204],[23,78779,78781],{"className":78780,"style":4693},[208],[23,78782,78783,78786],{"style":4696},[23,78784],{"className":78785,"style":1015},[216],[23,78787,78789],{"className":78788},[1019,1020,1021,1022],[23,78790,9956],{"className":78791},[135,1022],"。只有一个训练点 ",[23,78794,78796,78826],{"className":78795,"translate":27},[31],[23,78797,78799],{"className":78798},[35],[37,78800,78801],{"xmlns":39},[42,78802,78803,78823],{},[45,78804,78805,78811,78813,78815,78817,78819,78821],{},[898,78806,78807,78809],{},[48,78808,753],{},[2581,78810,4435],{},[51,78812,69],{},[51,78814,54],{"stretchy":53},[2581,78816,4435],{},[51,78818,747],{"separator":122},[2581,78820,1532],{},[51,78822,66],{"stretchy":53},[114,78824,78825],{"encoding":116},"x_1=(1,0)",[23,78827,78829,78884],{"className":78828,"ariaHidden":122},[121],[23,78830,78832,78835,78875,78878,78881],{"className":78831},[126],[23,78833],{"className":78834,"style":12734},[130],[23,78836,78838,78841],{"className":78837},[135],[23,78839,753],{"className":78840},[135,136],[23,78842,78844],{"className":78843},[998],[23,78845,78847,78867],{"className":78846},[199,200],[23,78848,78850,78864],{"className":78849},[204],[23,78851,78853],{"className":78852,"style":18651},[208],[23,78854,78855,78858],{"style":2029},[23,78856],{"className":78857,"style":1015},[216],[23,78859,78861],{"className":78860},[1019,1020,1021,1022],[23,78862,4435],{"className":78863},[135,1022],[23,78865,297],{"className":78866},[296],[23,78868,78870],{"className":78869},[204],[23,78871,78873],{"className":78872,"style":2048},[208],[23,78874],{},[23,78876],{"className":78877,"style":149},[148],[23,78879,69],{"className":78880},[153],[23,78882],{"className":78883,"style":149},[148],[23,78885,78887,78890,78893,78896,78899,78902,78905],{"className":78886},[126],[23,78888],{"className":78889,"style":131},[130],[23,78891,54],{"className":78892},[140],[23,78894,4435],{"className":78895},[135],[23,78897,747],{"className":78898},[824],[23,78900],{"className":78901,"style":280},[148],[23,78903,1532],{"className":78904},[135],[23,78906,66],{"className":78907},[169],"，标签 ",[23,78910,78912,78934],{"className":78911,"translate":27},[31],[23,78913,78915],{"className":78914},[35],[37,78916,78917],{"xmlns":39},[42,78918,78919,78931],{},[45,78920,78921,78927,78929],{},[898,78922,78923,78925],{},[48,78924,660],{},[2581,78926,4435],{},[51,78928,69],{},[2581,78930,1532],{},[114,78932,78933],{"encoding":116},"y_1=0",[23,78935,78937,78992],{"className":78936,"ariaHidden":122},[121],[23,78938,78940,78943,78983,78986,78989],{"className":78939},[126],[23,78941],{"className":78942,"style":672},[130],[23,78944,78946,78949],{"className":78945},[135],[23,78947,660],{"className":78948,"style":676},[135,136],[23,78950,78952],{"className":78951},[998],[23,78953,78955,78975],{"className":78954},[199,200],[23,78956,78958,78972],{"className":78957},[204],[23,78959,78961],{"className":78960,"style":18651},[208],[23,78962,78963,78966],{"style":21854},[23,78964],{"className":78965,"style":1015},[216],[23,78967,78969],{"className":78968},[1019,1020,1021,1022],[23,78970,4435],{"className":78971},[135,1022],[23,78973,297],{"className":78974},[296],[23,78976,78978],{"className":78977},[204],[23,78979,78981],{"className":78980,"style":2048},[208],[23,78982],{},[23,78984],{"className":78985,"style":149},[148],[23,78987,69],{"className":78988},[153],[23,78990],{"className":78991,"style":149},[148],[23,78993,78995,78998],{"className":78994},[126],[23,78996],{"className":78997,"style":2770},[130],[23,78999,1532],{"className":79000},[135],"。损失取平方损失",[23,79003,79005],{"className":79004,"translate":27},[26],[23,79006,79008,79058],{"className":79007,"translate":27},[31],[23,79009,79011],{"className":79010},[35],[37,79012,79013],{"xmlns":39,"display":40},[42,79014,79015,79055],{},[45,79016,79017,79019,79021,79023,79025,79031,79033,79035,79037,79039,79041,79047,79053],{},[48,79018,555],{},[51,79020,54],{"stretchy":53},[48,79022,8340],{},[51,79024,54],{"stretchy":53},[898,79026,79027,79029],{},[48,79028,753],{},[2581,79030,4435],{},[51,79032,66],{"stretchy":53},[51,79034,66],{"stretchy":53},[51,79036,69],{},[48,79038,8340],{},[51,79040,54],{"stretchy":53},[898,79042,79043,79045],{},[48,79044,753],{},[2581,79046,4435],{},[4518,79048,79049,79051],{},[51,79050,66],{"stretchy":53},[2581,79052,9956],{},[51,79054,747],{"separator":122},[114,79056,79057],{"encoding":116},"L(f(x_1))=f(x_1)^2,",[23,79059,79061,79131],{"className":79060,"ariaHidden":122},[121],[23,79062,79064,79067,79070,79073,79076,79079,79119,79122,79125,79128],{"className":79063},[126],[23,79065],{"className":79066,"style":131},[130],[23,79068,555],{"className":79069},[135,136],[23,79071,54],{"className":79072},[140],[23,79074,8340],{"className":79075,"style":11402},[135,136],[23,79077,54],{"className":79078},[140],[23,79080,79082,79085],{"className":79081},[135],[23,79083,753],{"className":79084},[135,136],[23,79086,79088],{"className":79087},[998],[23,79089,79091,79111],{"className":79090},[199,200],[23,79092,79094,79108],{"className":79093},[204],[23,79095,79097],{"className":79096,"style":18651},[208],[23,79098,79099,79102],{"style":2029},[23,79100],{"className":79101,"style":1015},[216],[23,79103,79105],{"className":79104},[1019,1020,1021,1022],[23,79106,4435],{"className":79107},[135,1022],[23,79109,297],{"className":79110},[296],[23,79112,79114],{"className":79113},[204],[23,79115,79117],{"className":79116,"style":2048},[208],[23,79118],{},[23,79120,1095],{"className":79121},[169],[23,79123],{"className":79124,"style":149},[148],[23,79126,69],{"className":79127},[153],[23,79129],{"className":79130,"style":149},[148],[23,79132,79134,79137,79140,79143,79183,79212],{"className":79133},[126],[23,79135],{"className":79136,"style":26503},[130],[23,79138,8340],{"className":79139,"style":11402},[135,136],[23,79141,54],{"className":79142},[140],[23,79144,79146,79149],{"className":79145},[135],[23,79147,753],{"className":79148},[135,136],[23,79150,79152],{"className":79151},[998],[23,79153,79155,79175],{"className":79154},[199,200],[23,79156,79158,79172],{"className":79157},[204],[23,79159,79161],{"className":79160,"style":18651},[208],[23,79162,79163,79166],{"style":2029},[23,79164],{"className":79165,"style":1015},[216],[23,79167,79169],{"className":79168},[1019,1020,1021,1022],[23,79170,4435],{"className":79171},[135,1022],[23,79173,297],{"className":79174},[296],[23,79176,79178],{"className":79177},[204],[23,79179,79181],{"className":79180,"style":2048},[208],[23,79182],{},[23,79184,79186,79189],{"className":79185},[169],[23,79187,66],{"className":79188},[169],[23,79190,79192],{"className":79191},[998],[23,79193,79195],{"className":79194},[199],[23,79196,79198],{"className":79197},[204],[23,79199,79201],{"className":79200,"style":11006},[208],[23,79202,79203,79206],{"style":4989},[23,79204],{"className":79205,"style":1015},[216],[23,79207,79209],{"className":79208},[1019,1020,1021,1022],[23,79210,9956],{"className":79211},[135,1022],[23,79213,747],{"className":79214},[824],[11,79216,79217],{},"正则项取",[23,79219,79221],{"className":79220,"translate":27},[26],[23,79222,79224,79266],{"className":79223,"translate":27},[31],[23,79225,79227],{"className":79226},[35],[37,79228,79229],{"xmlns":39,"display":40},[42,79230,79231,79263],{},[45,79232,79233,79235,79237,79239,79241,79243,79245,79247,79249,79251,79253,79255,79261],{},[48,79234,68146],{"mathvariant":111},[51,79236,54],{"stretchy":53},[48,79238,8340],{},[51,79240,66],{"stretchy":53},[51,79242,69],{},[51,79244,54],{"stretchy":53},[48,79246,20684],{"mathvariant":111},[48,79248,8340],{},[48,79250,20684],{"mathvariant":111},[51,79252,4529],{},[2581,79254,4435],{},[4518,79256,79257,79259],{},[51,79258,66],{"stretchy":53},[2581,79260,9956],{},[48,79262,112],{"mathvariant":111},[114,79264,79265],{"encoding":116},"\\Omega(f)=(\\|f\\|-1)^2.",[23,79267,79269,79296,79323],{"className":79268,"ariaHidden":122},[121],[23,79270,79272,79275,79278,79281,79284,79287,79290,79293],{"className":79271},[126],[23,79273],{"className":79274,"style":131},[130],[23,79276,68146],{"className":79277},[135],[23,79279,54],{"className":79280},[140],[23,79282,8340],{"className":79283,"style":11402},[135,136],[23,79285,66],{"className":79286},[169],[23,79288],{"className":79289,"style":149},[148],[23,79291,69],{"className":79292},[153],[23,79294],{"className":79295,"style":149},[148],[23,79297,79299,79302,79305,79308,79311,79314,79317,79320],{"className":79298},[126],[23,79300],{"className":79301,"style":131},[130],[23,79303,54],{"className":79304},[140],[23,79306,20684],{"className":79307},[135],[23,79309,8340],{"className":79310,"style":11402},[135,136],[23,79312,20684],{"className":79313},[135],[23,79315],{"className":79316,"style":392},[148],[23,79318,4529],{"className":79319},[612],[23,79321],{"className":79322,"style":392},[148],[23,79324,79326,79329,79332,79361],{"className":79325},[126],[23,79327],{"className":79328,"style":26503},[130],[23,79330,4435],{"className":79331},[135],[23,79333,79335,79338],{"className":79334},[169],[23,79336,66],{"className":79337},[169],[23,79339,79341],{"className":79340},[998],[23,79342,79344],{"className":79343},[199],[23,79345,79347],{"className":79346},[204],[23,79348,79350],{"className":79349,"style":11006},[208],[23,79351,79352,79355],{"style":4989},[23,79353],{"className":79354,"style":1015},[216],[23,79356,79358],{"className":79357},[1019,1020,1021,1022],[23,79359,9956],{"className":79360},[135,1022],[23,79362,112],{"className":79363},[135],[11,79365,79366,79367,79418],{},"这是一个非负的、只依赖范数的正则项，但它在 ",[23,79368,79370,79391],{"className":79369,"translate":27},[31],[23,79371,79373],{"className":79372},[35],[37,79374,79375],{"xmlns":39},[42,79376,79377,79389],{},[45,79378,79379,79381,79383,79385,79387],{},[51,79380,924],{"stretchy":53},[2581,79382,1532],{},[51,79384,747],{"separator":122},[2581,79386,4435],{},[51,79388,945],{"stretchy":53},[114,79390,46872],{"encoding":116},[23,79392,79394],{"className":79393,"ariaHidden":122},[121],[23,79395,79397,79400,79403,79406,79409,79412,79415],{"className":79396},[126],[23,79398],{"className":79399,"style":131},[130],[23,79401,924],{"className":79402},[140],[23,79404,1532],{"className":79405},[135],[23,79407,747],{"className":79408},[824],[23,79410],{"className":79411,"style":280},[148],[23,79413,4435],{"className":79414},[135],[23,79416,945],{"className":79417},[169]," 上递减，不是非减函数。",[11,79420,79421],{},"训练点张成的子空间是",[23,79423,79425],{"className":79424,"translate":27},[26],[23,79426,79428,79498],{"className":79427,"translate":27},[31],[23,79429,79431],{"className":79430},[35],[37,79432,79433],{"xmlns":39,"display":40},[42,79434,79435,79495],{},[45,79436,79437,79443,79445,79447,79449,79451,79453,79455,79457,79459,79465,79467,79469,79471,79473,79475,79477,79479,79481,79483,79485,79487,79489,79491,79493],{},[898,79438,79439,79441],{},[48,79440,26253],{"mathvariant":375},[2581,79442,1532],{},[51,79444,69],{},[48,79446,23],{"mathvariant":111},[51,79448,6961],{},[51,79450,9014],{"stretchy":53},[48,79452,4836],{},[51,79454,54],{"stretchy":53},[51,79456,5758],{},[51,79458,747],{"separator":122},[898,79460,79461,79463],{},[48,79462,753],{},[2581,79464,4435],{},[51,79466,66],{"stretchy":53},[51,79468,9024],{"stretchy":53},[51,79470,69],{},[51,79472,9014],{"stretchy":53},[51,79474,54],{"stretchy":53},[48,79476,495],{},[51,79478,747],{"separator":122},[2581,79480,1532],{},[51,79482,66],{"stretchy":53},[51,79484,558],{},[48,79486,495],{},[51,79488,372],{},[48,79490,572],{"mathvariant":571},[51,79492,9024],{"stretchy":53},[48,79494,112],{"mathvariant":111},[114,79496,79497],{"encoding":116},"\\mathcal H_0=\\operatorname{span}\\{k(\\cdot,x_1)\\}=\\{(a,0):a\\in\\mathbb R\\}.",[23,79499,79501,79556,79638,79671,79689],{"className":79500,"ariaHidden":122},[121],[23,79502,79504,79507,79547,79550,79553],{"className":79503},[126],[23,79505],{"className":79506,"style":42831},[130],[23,79508,79510,79513],{"className":79509},[135],[23,79511,26253],{"className":79512,"style":45028},[135,412],[23,79514,79516],{"className":79515},[998],[23,79517,79519,79539],{"className":79518},[199,200],[23,79520,79522,79536],{"className":79521},[204],[23,79523,79525],{"className":79524,"style":18651},[208],[23,79526,79527,79530],{"style":61635},[23,79528],{"className":79529,"style":1015},[216],[23,79531,79533],{"className":79532},[1019,1020,1021,1022],[23,79534,1532],{"className":79535},[135,1022],[23,79537,297],{"className":79538},[296],[23,79540,79542],{"className":79541},[204],[23,79543,79545],{"className":79544,"style":2048},[208],[23,79546],{},[23,79548],{"className":79549,"style":149},[148],[23,79551,69],{"className":79552},[153],[23,79554],{"className":79555,"style":149},[148],[23,79557,79559,79562,79568,79571,79574,79577,79580,79583,79586,79626,79629,79632,79635],{"className":79558},[126],[23,79560],{"className":79561,"style":131},[130],[23,79563,79565],{"className":79564},[1786],[23,79566,23],{"className":79567},[135,5391],[23,79569,9014],{"className":79570},[140],[23,79572,4836],{"className":79573,"style":4851},[135,136],[23,79575,54],{"className":79576},[140],[23,79578,5758],{"className":79579},[135],[23,79581,747],{"className":79582},[824],[23,79584],{"className":79585,"style":280},[148],[23,79587,79589,79592],{"className":79588},[135],[23,79590,753],{"className":79591},[135,136],[23,79593,79595],{"className":79594},[998],[23,79596,79598,79618],{"className":79597},[199,200],[23,79599,79601,79615],{"className":79600},[204],[23,79602,79604],{"className":79603,"style":18651},[208],[23,79605,79606,79609],{"style":2029},[23,79607],{"className":79608,"style":1015},[216],[23,79610,79612],{"className":79611},[1019,1020,1021,1022],[23,79613,4435],{"className":79614},[135,1022],[23,79616,297],{"className":79617},[296],[23,79619,79621],{"className":79620},[204],[23,79622,79624],{"className":79623,"style":2048},[208],[23,79625],{},[23,79627,15224],{"className":79628},[169],[23,79630],{"className":79631,"style":149},[148],[23,79633,69],{"className":79634},[153],[23,79636],{"className":79637,"style":149},[148],[23,79639,79641,79644,79647,79650,79653,79656,79659,79662,79665,79668],{"className":79640},[126],[23,79642],{"className":79643,"style":131},[130],[23,79645,50680],{"className":79646},[140],[23,79648,495],{"className":79649},[135,136],[23,79651,747],{"className":79652},[824],[23,79654],{"className":79655,"style":280},[148],[23,79657,1532],{"className":79658},[135],[23,79660,66],{"className":79661},[169],[23,79663],{"className":79664,"style":149},[148],[23,79666,558],{"className":79667},[153],[23,79669],{"className":79670,"style":149},[148],[23,79672,79674,79677,79680,79683,79686],{"className":79673},[126],[23,79675],{"className":79676,"style":513},[130],[23,79678,495],{"className":79679},[135,136],[23,79681],{"className":79682,"style":149},[148],[23,79684,372],{"className":79685},[153],[23,79687],{"className":79688,"style":149},[148],[23,79690,79692,79695,79698,79701],{"className":79691},[126],[23,79693],{"className":79694,"style":131},[130],[23,79696,572],{"className":79697},[135,644],[23,79699,9024],{"className":79700},[169],[23,79702,112],{"className":79703},[135],[11,79705,59356,79706,79780,79781,79888,79889,77365,79917,79978,79979,80007,80008,80036,80037,79780,80111,80217,80218,77061,80246,79978,80307,80007,80335,80363,80364,80416,80417,80486],{},[23,79707,79709,79735],{"className":79708,"translate":27},[31],[23,79710,79712],{"className":79711},[35],[37,79713,79714],{"xmlns":39},[42,79715,79716,79732],{},[45,79717,79718,79720,79722,79724,79726,79728,79730],{},[48,79719,8340],{},[51,79721,69],{},[51,79723,54],{"stretchy":53},[2581,79725,1532],{},[51,79727,747],{"separator":122},[2581,79729,1532],{},[51,79731,66],{"stretchy":53},[114,79733,79734],{"encoding":116},"f=(0,0)",[23,79736,79738,79756],{"className":79737,"ariaHidden":122},[121],[23,79739,79741,79744,79747,79750,79753],{"className":79740},[126],[23,79742],{"className":79743,"style":5414},[130],[23,79745,8340],{"className":79746,"style":11402},[135,136],[23,79748],{"className":79749,"style":149},[148],[23,79751,69],{"className":79752},[153],[23,79754],{"className":79755,"style":149},[148],[23,79757,79759,79762,79765,79768,79771,79774,79777],{"className":79758},[126],[23,79760],{"className":79761,"style":131},[130],[23,79763,54],{"className":79764},[140],[23,79766,1532],{"className":79767},[135],[23,79769,747],{"className":79770},[824],[23,79772],{"className":79773,"style":280},[148],[23,79775,1532],{"className":79776},[135],[23,79778,66],{"className":79779},[169]," 处，",[23,79782,79784,79812],{"className":79783,"translate":27},[31],[23,79785,79787],{"className":79786},[35],[37,79788,79789],{"xmlns":39},[42,79790,79791,79809],{},[45,79792,79793,79795,79797,79803,79805,79807],{},[48,79794,8340],{},[51,79796,54],{"stretchy":53},[898,79798,79799,79801],{},[48,79800,753],{},[2581,79802,4435],{},[51,79804,66],{"stretchy":53},[51,79806,69],{},[2581,79808,1532],{},[114,79810,79811],{"encoding":116},"f(x_1)=0",[23,79813,79815,79879],{"className":79814,"ariaHidden":122},[121],[23,79816,79818,79821,79824,79827,79867,79870,79873,79876],{"className":79817},[126],[23,79819],{"className":79820,"style":131},[130],[23,79822,8340],{"className":79823,"style":11402},[135,136],[23,79825,54],{"className":79826},[140],[23,79828,79830,79833],{"className":79829},[135],[23,79831,753],{"className":79832},[135,136],[23,79834,79836],{"className":79835},[998],[23,79837,79839,79859],{"className":79838},[199,200],[23,79840,79842,79856],{"className":79841},[204],[23,79843,79845],{"className":79844,"style":18651},[208],[23,79846,79847,79850],{"style":2029},[23,79848],{"className":79849,"style":1015},[216],[23,79851,79853],{"className":79852},[1019,1020,1021,1022],[23,79854,4435],{"className":79855},[135,1022],[23,79857,297],{"className":79858},[296],[23,79860,79862],{"className":79861},[204],[23,79863,79865],{"className":79864,"style":2048},[208],[23,79866],{},[23,79868,66],{"className":79869},[169],[23,79871],{"className":79872,"style":149},[148],[23,79874,69],{"className":79875},[153],[23,79877],{"className":79878,"style":149},[148],[23,79880,79882,79885],{"className":79881},[126],[23,79883],{"className":79884,"style":2770},[130],[23,79886,1532],{"className":79887},[135],"，损失为 ",[23,79890,79892,79905],{"className":79891,"translate":27},[31],[23,79893,79895],{"className":79894},[35],[37,79896,79897],{"xmlns":39},[42,79898,79899,79903],{},[45,79900,79901],{},[2581,79902,1532],{},[114,79904,1532],{"encoding":116},[23,79906,79908],{"className":79907,"ariaHidden":122},[121],[23,79909,79911,79914],{"className":79910},[126],[23,79912],{"className":79913,"style":2770},[130],[23,79915,1532],{"className":79916},[135],[23,79918,79920,79942],{"className":79919,"translate":27},[31],[23,79921,79923],{"className":79922},[35],[37,79924,79925],{"xmlns":39},[42,79926,79927,79939],{},[45,79928,79929,79931,79933,79935,79937],{},[48,79930,20684],{"mathvariant":111},[48,79932,8340],{},[48,79934,20684],{"mathvariant":111},[51,79936,69],{},[2581,79938,1532],{},[114,79940,79941],{"encoding":116},"\\|f\\|=0",[23,79943,79945,79969],{"className":79944,"ariaHidden":122},[121],[23,79946,79948,79951,79954,79957,79960,79963,79966],{"className":79947},[126],[23,79949],{"className":79950,"style":131},[130],[23,79952,20684],{"className":79953},[135],[23,79955,8340],{"className":79956,"style":11402},[135,136],[23,79958,20684],{"className":79959},[135],[23,79961],{"className":79962,"style":149},[148],[23,79964,69],{"className":79965},[153],[23,79967],{"className":79968,"style":149},[148],[23,79970,79972,79975],{"className":79971},[126],[23,79973],{"className":79974,"style":2770},[130],[23,79976,1532],{"className":79977},[135],"，正则项为 ",[23,79980,79982,79995],{"className":79981,"translate":27},[31],[23,79983,79985],{"className":79984},[35],[37,79986,79987],{"xmlns":39},[42,79988,79989,79993],{},[45,79990,79991],{},[2581,79992,4435],{},[114,79994,4435],{"encoding":116},[23,79996,79998],{"className":79997,"ariaHidden":122},[121],[23,79999,80001,80004],{"className":80000},[126],[23,80002],{"className":80003,"style":2770},[130],[23,80005,4435],{"className":80006},[135],"，目标值为 ",[23,80009,80011,80024],{"className":80010,"translate":27},[31],[23,80012,80014],{"className":80013},[35],[37,80015,80016],{"xmlns":39},[42,80017,80018,80022],{},[45,80019,80020],{},[2581,80021,4435],{},[114,80023,4435],{"encoding":116},[23,80025,80027],{"className":80026,"ariaHidden":122},[121],[23,80028,80030,80033],{"className":80029},[126],[23,80031],{"className":80032,"style":2770},[130],[23,80034,4435],{"className":80035},[135],"。在 ",[23,80038,80040,80066],{"className":80039,"translate":27},[31],[23,80041,80043],{"className":80042},[35],[37,80044,80045],{"xmlns":39},[42,80046,80047,80063],{},[45,80048,80049,80051,80053,80055,80057,80059,80061],{},[48,80050,8340],{},[51,80052,69],{},[51,80054,54],{"stretchy":53},[2581,80056,1532],{},[51,80058,747],{"separator":122},[2581,80060,4435],{},[51,80062,66],{"stretchy":53},[114,80064,80065],{"encoding":116},"f=(0,1)",[23,80067,80069,80087],{"className":80068,"ariaHidden":122},[121],[23,80070,80072,80075,80078,80081,80084],{"className":80071},[126],[23,80073],{"className":80074,"style":5414},[130],[23,80076,8340],{"className":80077,"style":11402},[135,136],[23,80079],{"className":80080,"style":149},[148],[23,80082,69],{"className":80083},[153],[23,80085],{"className":80086,"style":149},[148],[23,80088,80090,80093,80096,80099,80102,80105,80108],{"className":80089},[126],[23,80091],{"className":80092,"style":131},[130],[23,80094,54],{"className":80095},[140],[23,80097,1532],{"className":80098},[135],[23,80100,747],{"className":80101},[824],[23,80103],{"className":80104,"style":280},[148],[23,80106,4435],{"className":80107},[135],[23,80109,66],{"className":80110},[169],[23,80112,80114,80141],{"className":80113,"translate":27},[31],[23,80115,80117],{"className":80116},[35],[37,80118,80119],{"xmlns":39},[42,80120,80121,80139],{},[45,80122,80123,80125,80127,80133,80135,80137],{},[48,80124,8340],{},[51,80126,54],{"stretchy":53},[898,80128,80129,80131],{},[48,80130,753],{},[2581,80132,4435],{},[51,80134,66],{"stretchy":53},[51,80136,69],{},[2581,80138,1532],{},[114,80140,79811],{"encoding":116},[23,80142,80144,80208],{"className":80143,"ariaHidden":122},[121],[23,80145,80147,80150,80153,80156,80196,80199,80202,80205],{"className":80146},[126],[23,80148],{"className":80149,"style":131},[130],[23,80151,8340],{"className":80152,"style":11402},[135,136],[23,80154,54],{"className":80155},[140],[23,80157,80159,80162],{"className":80158},[135],[23,80160,753],{"className":80161},[135,136],[23,80163,80165],{"className":80164},[998],[23,80166,80168,80188],{"className":80167},[199,200],[23,80169,80171,80185],{"className":80170},[204],[23,80172,80174],{"className":80173,"style":18651},[208],[23,80175,80176,80179],{"style":2029},[23,80177],{"className":80178,"style":1015},[216],[23,80180,80182],{"className":80181},[1019,1020,1021,1022],[23,80183,4435],{"className":80184},[135,1022],[23,80186,297],{"className":80187},[296],[23,80189,80191],{"className":80190},[204],[23,80192,80194],{"className":80193,"style":2048},[208],[23,80195],{},[23,80197,66],{"className":80198},[169],[23,80200],{"className":80201,"style":149},[148],[23,80203,69],{"className":80204},[153],[23,80206],{"className":80207,"style":149},[148],[23,80209,80211,80214],{"className":80210},[126],[23,80212],{"className":80213,"style":2770},[130],[23,80215,1532],{"className":80216},[135],"，损失仍为 ",[23,80219,80221,80234],{"className":80220,"translate":27},[31],[23,80222,80224],{"className":80223},[35],[37,80225,80226],{"xmlns":39},[42,80227,80228,80232],{},[45,80229,80230],{},[2581,80231,1532],{},[114,80233,1532],{"encoding":116},[23,80235,80237],{"className":80236,"ariaHidden":122},[121],[23,80238,80240,80243],{"className":80239},[126],[23,80241],{"className":80242,"style":2770},[130],[23,80244,1532],{"className":80245},[135],[23,80247,80249,80271],{"className":80248,"translate":27},[31],[23,80250,80252],{"className":80251},[35],[37,80253,80254],{"xmlns":39},[42,80255,80256,80268],{},[45,80257,80258,80260,80262,80264,80266],{},[48,80259,20684],{"mathvariant":111},[48,80261,8340],{},[48,80263,20684],{"mathvariant":111},[51,80265,69],{},[2581,80267,4435],{},[114,80269,80270],{"encoding":116},"\\|f\\|=1",[23,80272,80274,80298],{"className":80273,"ariaHidden":122},[121],[23,80275,80277,80280,80283,80286,80289,80292,80295],{"className":80276},[126],[23,80278],{"className":80279,"style":131},[130],[23,80281,20684],{"className":80282},[135],[23,80284,8340],{"className":80285,"style":11402},[135,136],[23,80287,20684],{"className":80288},[135],[23,80290],{"className":80291,"style":149},[148],[23,80293,69],{"className":80294},[153],[23,80296],{"className":80297,"style":149},[148],[23,80299,80301,80304],{"className":80300},[126],[23,80302],{"className":80303,"style":2770},[130],[23,80305,4435],{"className":80306},[135],[23,80308,80310,80323],{"className":80309,"translate":27},[31],[23,80311,80313],{"className":80312},[35],[37,80314,80315],{"xmlns":39},[42,80316,80317,80321],{},[45,80318,80319],{},[2581,80320,1532],{},[114,80322,1532],{"encoding":116},[23,80324,80326],{"className":80325,"ariaHidden":122},[121],[23,80327,80329,80332],{"className":80328},[126],[23,80330],{"className":80331,"style":2770},[130],[23,80333,1532],{"className":80334},[135],[23,80336,80338,80351],{"className":80337,"translate":27},[31],[23,80339,80341],{"className":80340},[35],[37,80342,80343],{"xmlns":39},[42,80344,80345,80349],{},[45,80346,80347],{},[2581,80348,1532],{},[114,80350,1532],{"encoding":116},[23,80352,80354],{"className":80353,"ariaHidden":122},[121],[23,80355,80357,80360],{"className":80356},[126],[23,80358],{"className":80359,"style":2770},[130],[23,80361,1532],{"className":80362},[135],"。所以最优解是 ",[23,80365,80367,80389],{"className":80366,"translate":27},[31],[23,80368,80370],{"className":80369},[35],[37,80371,80372],{"xmlns":39},[42,80373,80374,80386],{},[45,80375,80376,80378,80380,80382,80384],{},[51,80377,54],{"stretchy":53},[2581,80379,1532],{},[51,80381,747],{"separator":122},[2581,80383,4435],{},[51,80385,66],{"stretchy":53},[114,80387,80388],{"encoding":116},"(0,1)",[23,80390,80392],{"className":80391,"ariaHidden":122},[121],[23,80393,80395,80398,80401,80404,80407,80410,80413],{"className":80394},[126],[23,80396],{"className":80397,"style":131},[130],[23,80399,54],{"className":80400},[140],[23,80402,1532],{"className":80403},[135],[23,80405,747],{"className":80406},[824],[23,80408],{"className":80409,"style":280},[148],[23,80411,4435],{"className":80412},[135],[23,80414,66],{"className":80415},[169],"，它落在 ",[23,80418,80420,80437],{"className":80419,"translate":27},[31],[23,80421,80423],{"className":80422},[35],[37,80424,80425],{"xmlns":39},[42,80426,80427,80435],{},[45,80428,80429],{},[898,80430,80431,80433],{},[48,80432,26253],{"mathvariant":375},[2581,80434,1532],{},[114,80436,73826],{"encoding":116},[23,80438,80440],{"className":80439,"ariaHidden":122},[121],[23,80441,80443,80446],{"className":80442},[126],[23,80444],{"className":80445,"style":42831},[130],[23,80447,80449,80452],{"className":80448},[135],[23,80450,26253],{"className":80451,"style":45028},[135,412],[23,80453,80455],{"className":80454},[998],[23,80456,80458,80478],{"className":80457},[199,200],[23,80459,80461,80475],{"className":80460},[204],[23,80462,80464],{"className":80463,"style":18651},[208],[23,80465,80466,80469],{"style":61635},[23,80467],{"className":80468,"style":1015},[216],[23,80470,80472],{"className":80471},[1019,1020,1021,1022],[23,80473,1532],{"className":80474},[135,1022],[23,80476,297],{"className":80477},[296],[23,80479,80481],{"className":80480},[204],[23,80482,80484],{"className":80483,"style":2048},[208],[23,80485],{}," 的正交补里，而不是训练点的张成空间中。",[11,80488,80489,80490,80493],{},"这个反例说明：",[18,80491,80492],{},"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。",[2501,80495,80497],{"id":80496},"_43-充要条件","4.3 充要条件",[11,80499,80500],{},"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，",[23,80502,80504],{"className":80503,"translate":27},[26],[23,80505,80507,80545],{"className":80506,"translate":27},[31],[23,80508,80510],{"className":80509},[35],[37,80511,80512],{"xmlns":39,"display":40},[42,80513,80514,80542],{},[45,80515,80516,80518,80520,80522,80524,80526,80528,80530,80532,80534,80540],{},[48,80517,68146],{"mathvariant":111},[51,80519,54],{"stretchy":53},[48,80521,8340],{},[51,80523,66],{"stretchy":53},[51,80525,69],{},[48,80527,18372],{},[51,80529,54],{"stretchy":53},[48,80531,20684],{"mathvariant":111},[48,80533,8340],{},[898,80535,80536,80538],{},[48,80537,20684],{"mathvariant":111},[48,80539,26253],{"mathvariant":375},[51,80541,66],{"stretchy":53},[114,80543,80544],{"encoding":116},"\\Omega(f)=h(\\|f\\|_{\\mathcal H})",[23,80546,80548,80575],{"className":80547,"ariaHidden":122},[121],[23,80549,80551,80554,80557,80560,80563,80566,80569,80572],{"className":80550},[126],[23,80552],{"className":80553,"style":131},[130],[23,80555,68146],{"className":80556},[135],[23,80558,54],{"className":80559},[140],[23,80561,8340],{"className":80562,"style":11402},[135,136],[23,80564,66],{"className":80565},[169],[23,80567],{"className":80568,"style":149},[148],[23,80570,69],{"className":80571},[153],[23,80573],{"className":80574,"style":149},[148],[23,80576,80578,80581,80584,80587,80590,80593,80636],{"className":80577},[126],[23,80579],{"className":80580,"style":131},[130],[23,80582,18372],{"className":80583},[135,136],[23,80585,54],{"className":80586},[140],[23,80588,20684],{"className":80589},[135],[23,80591,8340],{"className":80592,"style":11402},[135,136],[23,80594,80596,80599],{"className":80595},[135],[23,80597,20684],{"className":80598},[135],[23,80600,80602],{"className":80601},[998],[23,80603,80605,80628],{"className":80604},[199,200],[23,80606,80608,80625],{"className":80607},[204],[23,80609,80611],{"className":80610,"style":2026},[208],[23,80612,80613,80616],{"style":2029},[23,80614],{"className":80615,"style":1015},[216],[23,80617,80619],{"className":80618},[1019,1020,1021,1022],[23,80620,80622],{"className":80621},[135,1022],[23,80623,26253],{"className":80624,"style":45028},[135,412,1022],[23,80626,297],{"className":80627},[296],[23,80629,80631],{"className":80630},[204],[23,80632,80634],{"className":80633,"style":2048},[208],[23,80635],{},[23,80637,66],{"className":80638},[169],[11,80640,80641,80642,80738],{},"对某个非减的 ",[23,80643,80645,80675],{"className":80644,"translate":27},[31],[23,80646,80648],{"className":80647},[35],[37,80649,80650],{"xmlns":39},[42,80651,80652,80672],{},[45,80653,80654,80656,80658,80660,80662,80664,80666,80668,80670],{},[48,80655,18372],{},[51,80657,558],{},[51,80659,924],{"stretchy":53},[2581,80661,1532],{},[51,80663,747],{"separator":122},[48,80665,46451],{"mathvariant":111},[51,80667,66],{"stretchy":53},[51,80669,568],{},[48,80671,572],{"mathvariant":571},[114,80673,80674],{"encoding":116},"h:[0,\\infty)\\to\\mathbb R",[23,80676,80678,80696,80729],{"className":80677,"ariaHidden":122},[121],[23,80679,80681,80684,80687,80690,80693],{"className":80680},[126],[23,80682],{"className":80683,"style":779},[130],[23,80685,18372],{"className":80686},[135,136],[23,80688],{"className":80689,"style":149},[148],[23,80691,558],{"className":80692},[153],[23,80694],{"className":80695,"style":149},[148],[23,80697,80699,80702,80705,80708,80711,80714,80717,80720,80723,80726],{"className":80698},[126],[23,80700],{"className":80701,"style":131},[130],[23,80703,924],{"className":80704},[140],[23,80706,1532],{"className":80707},[135],[23,80709,747],{"className":80710},[824],[23,80712],{"className":80713,"style":280},[148],[23,80715,46451],{"className":80716},[135],[23,80718,66],{"className":80719},[169],[23,80721],{"className":80722,"style":149},[148],[23,80724,568],{"className":80725},[153],[23,80727],{"className":80728,"style":149},[148],[23,80730,80732,80735],{"className":80731},[126],[23,80733],{"className":80734,"style":640},[130],[23,80736,572],{"className":80737},[135,644]," 成立，是表示定理的充要条件。",[11,80740,80741],{},"这里还要区分两个层次：",[346,80743,80744,80820],{},[349,80745,80746,80749,80750,80819],{},[18,80747,80748],{},"非减","：保证存在一个落在 ",[23,80751,80753,80770],{"className":80752,"translate":27},[31],[23,80754,80756],{"className":80755},[35],[37,80757,80758],{"xmlns":39},[42,80759,80760,80768],{},[45,80761,80762],{},[898,80763,80764,80766],{},[48,80765,26253],{"mathvariant":375},[2581,80767,1532],{},[114,80769,73826],{"encoding":116},[23,80771,80773],{"className":80772,"ariaHidden":122},[121],[23,80774,80776,80779],{"className":80775},[126],[23,80777],{"className":80778,"style":42831},[130],[23,80780,80782,80785],{"className":80781},[135],[23,80783,26253],{"className":80784,"style":45028},[135,412],[23,80786,80788],{"className":80787},[998],[23,80789,80791,80811],{"className":80790},[199,200],[23,80792,80794,80808],{"className":80793},[204],[23,80795,80797],{"className":80796,"style":18651},[208],[23,80798,80799,80802],{"style":61635},[23,80800],{"className":80801,"style":1015},[216],[23,80803,80805],{"className":80804},[1019,1020,1021,1022],[23,80806,1532],{"className":80807},[135,1022],[23,80809,297],{"className":80810},[296],[23,80812,80814],{"className":80813},[204],[23,80815,80817],{"className":80816,"style":2048},[208],[23,80818],{}," 中的最优解；",[349,80821,80822,80825,80826,80895],{},[18,80823,80824],{},"严格递增","：保证每个最优解都落在 ",[23,80827,80829,80846],{"className":80828,"translate":27},[31],[23,80830,80832],{"className":80831},[35],[37,80833,80834],{"xmlns":39},[42,80835,80836,80844],{},[45,80837,80838],{},[898,80839,80840,80842],{},[48,80841,26253],{"mathvariant":375},[2581,80843,1532],{},[114,80845,73826],{"encoding":116},[23,80847,80849],{"className":80848,"ariaHidden":122},[121],[23,80850,80852,80855],{"className":80851},[126],[23,80853],{"className":80854,"style":42831},[130],[23,80856,80858,80861],{"className":80857},[135],[23,80859,26253],{"className":80860,"style":45028},[135,412],[23,80862,80864],{"className":80863},[998],[23,80865,80867,80887],{"className":80866},[199,200],[23,80868,80870,80884],{"className":80869},[204],[23,80871,80873],{"className":80872,"style":18651},[208],[23,80874,80875,80878],{"style":61635},[23,80876],{"className":80877,"style":1015},[216],[23,80879,80881],{"className":80880},[1019,1020,1021,1022],[23,80882,1532],{"className":80883},[135,1022],[23,80885,297],{"className":80886},[296],[23,80888,80890],{"className":80889},[204],[23,80891,80893],{"className":80892,"style":2048},[208],[23,80894],{}," 中。",[11,80897,80898,80899,80951,80952,80980,80981,81050,81051,81120],{},"如果 ",[23,80900,80902,80921],{"className":80901,"translate":27},[31],[23,80903,80905],{"className":80904},[35],[37,80906,80907],{"xmlns":39},[42,80908,80909,80918],{},[45,80910,80911,80913,80916],{},[48,80912,68146],{"mathvariant":111},[51,80914,80915],{},"≡",[2581,80917,1532],{},[114,80919,80920],{"encoding":116},"\\Omega\\equiv 0",[23,80922,80924,80942],{"className":80923,"ariaHidden":122},[121],[23,80925,80927,80930,80933,80936,80939],{"className":80926},[126],[23,80928],{"className":80929,"style":470},[130],[23,80931,68146],{"className":80932},[135],[23,80934],{"className":80935,"style":149},[148],[23,80937,80915],{"className":80938},[153],[23,80940],{"className":80941,"style":149},[148],[23,80943,80945,80948],{"className":80944},[126],[23,80946],{"className":80947,"style":2770},[130],[23,80949,1532],{"className":80950},[135],"，目标函数只依赖 ",[23,80953,80955,80968],{"className":80954,"translate":27},[31],[23,80956,80958],{"className":80957},[35],[37,80959,80960],{"xmlns":39},[42,80961,80962,80966],{},[45,80963,80964],{},[48,80965,8340],{},[114,80967,8340],{"encoding":116},[23,80969,80971],{"className":80970,"ariaHidden":122},[121],[23,80972,80974,80977],{"className":80973},[126],[23,80975],{"className":80976,"style":5414},[130],[23,80978,8340],{"className":80979,"style":11402},[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 ",[23,80982,80984,81001],{"className":80983,"translate":27},[31],[23,80985,80987],{"className":80986},[35],[37,80988,80989],{"xmlns":39},[42,80990,80991,80999],{},[45,80992,80993],{},[898,80994,80995,80997],{},[48,80996,26253],{"mathvariant":375},[2581,80998,1532],{},[114,81000,73826],{"encoding":116},[23,81002,81004],{"className":81003,"ariaHidden":122},[121],[23,81005,81007,81010],{"className":81006},[126],[23,81008],{"className":81009,"style":42831},[130],[23,81011,81013,81016],{"className":81012},[135],[23,81014,26253],{"className":81015,"style":45028},[135,412],[23,81017,81019],{"className":81018},[998],[23,81020,81022,81042],{"className":81021},[199,200],[23,81023,81025,81039],{"className":81024},[204],[23,81026,81028],{"className":81027,"style":18651},[208],[23,81029,81030,81033],{"style":61635},[23,81031],{"className":81032,"style":1015},[216],[23,81034,81036],{"className":81035},[1019,1020,1021,1022],[23,81037,1532],{"className":81038},[135,1022],[23,81040,297],{"className":81041},[296],[23,81043,81045],{"className":81044},[204],[23,81046,81048],{"className":81047,"style":2048},[208],[23,81049],{}," 里。表示定理的「存在」版本仍然成立（取 ",[23,81052,81054,81071],{"className":81053,"translate":27},[31],[23,81055,81057],{"className":81056},[35],[37,81058,81059],{"xmlns":39},[42,81060,81061,81069],{},[45,81062,81063],{},[898,81064,81065,81067],{},[48,81066,8340],{},[51,81068,20684],{},[114,81070,73040],{"encoding":116},[23,81072,81074],{"className":81073,"ariaHidden":122},[121],[23,81075,81077,81080],{"className":81076},[126],[23,81078],{"className":81079,"style":70235},[130],[23,81081,81083,81086],{"className":81082},[135],[23,81084,8340],{"className":81085,"style":11402},[135,136],[23,81087,81089],{"className":81088},[998],[23,81090,81092,81112],{"className":81091},[199,200],[23,81093,81095,81109],{"className":81094},[204],[23,81096,81098],{"className":81097,"style":1008},[208],[23,81099,81100,81103],{"style":70256},[23,81101],{"className":81102,"style":1015},[216],[23,81104,81106],{"className":81105},[1019,1020,1021,1022],[23,81107,20684],{"className":81108},[153,1022],[23,81110,297],{"className":81111},[296],[23,81113,81115],{"className":81114},[204],[23,81116,81118],{"className":81117,"style":1057},[208],[23,81119],{}," 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。",[338,81122,81124],{"id":81123},"五五个经典例子","五、五个经典例子",[2501,81126,81128],{"id":81127},"_51-核岭回归","5.1 核岭回归",[11,81130,81131,81132,69664,81202,81252],{},"取平方损失和 ",[23,81133,81135,81160],{"className":81134,"translate":27},[31],[23,81136,81138],{"className":81137},[35],[37,81139,81140],{"xmlns":39},[42,81141,81142,81158],{},[45,81143,81144,81146,81148,81150,81152,81154,81156],{},[48,81145,68146],{"mathvariant":111},[51,81147,54],{"stretchy":53},[48,81149,5345],{},[51,81151,66],{"stretchy":53},[51,81153,69],{},[48,81155,61761],{},[48,81157,5345],{},[114,81159,69621],{"encoding":116},[23,81161,81163,81190],{"className":81162,"ariaHidden":122},[121],[23,81164,81166,81169,81172,81175,81178,81181,81184,81187],{"className":81165},[126],[23,81167],{"className":81168,"style":131},[130],[23,81170,68146],{"className":81171},[135],[23,81173,54],{"className":81174},[140],[23,81176,5345],{"className":81177},[135,136],[23,81179,66],{"className":81180},[169],[23,81182],{"className":81183,"style":149},[148],[23,81185,69],{"className":81186},[153],[23,81188],{"className":81189,"style":149},[148],[23,81191,81193,81196,81199],{"className":81192},[126],[23,81194],{"className":81195,"style":779},[130],[23,81197,61761],{"className":81198},[135,136],[23,81200,5345],{"className":81201},[135,136],[23,81203,81205,81222],{"className":81204,"translate":27},[31],[23,81206,81208],{"className":81207},[35],[37,81209,81210],{"xmlns":39},[42,81211,81212,81220],{},[45,81213,81214,81216,81218],{},[48,81215,61761],{},[51,81217,2579],{},[2581,81219,1532],{},[114,81221,69685],{"encoding":116},[23,81223,81225,81243],{"className":81224,"ariaHidden":122},[121],[23,81226,81228,81231,81234,81237,81240],{"className":81227},[126],[23,81229],{"className":81230,"style":4450},[130],[23,81232,61761],{"className":81233},[135,136],[23,81235],{"className":81236,"style":149},[148],[23,81238,2579],{"className":81239},[153],[23,81241],{"className":81242,"style":149},[148],[23,81244,81246,81249],{"className":81245},[126],[23,81247],{"className":81248,"style":2770},[130],[23,81250,1532],{"className":81251},[135],"）：",[23,81254,81256],{"className":81255,"translate":27},[26],[23,81257,81259,81353],{"className":81258,"translate":27},[31],[23,81260,81262],{"className":81261},[35],[37,81263,81264],{"xmlns":39,"display":40},[42,81265,81266,81350],{},[45,81267,81268,81284,81348],{},[6963,81269,81270,81276],{},[45,81271,81272,81274],{},[48,81273,6969],{},[51,81275,6961],{},[45,81277,81278,81280,81282],{},[48,81279,8340],{},[51,81281,372],{},[48,81283,26253],{"mathvariant":375},[45,81285,81286,81288,81302,81304,81310,81312,81314,81316,81322,81324,81330,81332,81334,81336,81338,81346],{},[51,81287,9014],{"fence":122},[61717,81289,81290,81292,81300],{},[51,81291,33855],{},[45,81293,81294,81296,81298],{},[48,81295,33497],{},[51,81297,69],{},[2581,81299,4435],{},[48,81301,4806],{},[51,81303,54],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[898,81305,81306,81308],{},[48,81307,660],{},[48,81309,33497],{},[51,81311,4529],{},[48,81313,8340],{},[51,81315,54],{"stretchy":53},[898,81317,81318,81320],{},[48,81319,753],{},[48,81321,33497],{},[51,81323,66],{"stretchy":53},[4518,81325,81326,81328],{},[51,81327,66],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[2581,81329,9956],{},[51,81331,5106],{},[48,81333,61761],{},[48,81335,20684],{"mathvariant":111},[48,81337,8340],{},[19141,81339,81340,81342,81344],{},[48,81341,20684],{"mathvariant":111},[48,81343,26253],{"mathvariant":375},[2581,81345,9956],{},[51,81347,9024],{"fence":122},[48,81349,112],{"mathvariant":111},[114,81351,81352],{"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\\}.",[23,81354,81356],{"className":81355,"ariaHidden":122},[121],[23,81357,81359,81362,81415,81418,81705,81708],{"className":81358},[126],[23,81360],{"className":81361,"style":61841},[130],[23,81363,81365],{"className":81364},[1786,7098],[23,81366,81368,81407],{"className":81367},[199,200],[23,81369,81371,81404],{"className":81370},[204],[23,81372,81374,81394],{"className":81373,"style":7108},[208],[23,81375,81376,81379],{"style":14742},[23,81377],{"className":81378,"style":217},[216],[23,81380,81382],{"className":81381},[1019,1020,1021,1022],[23,81383,81385,81388,81391],{"className":81384},[135,1022],[23,81386,8340],{"className":81387,"style":11402},[135,136,1022],[23,81389,372],{"className":81390},[153,1022],[23,81392,26253],{"className":81393,"style":45028},[135,412,1022],[23,81395,81396,81399],{"style":7132},[23,81397],{"className":81398,"style":217},[216],[23,81400,81401],{},[23,81402,6969],{"className":81403},[1786],[23,81405,297],{"className":81406},[296],[23,81408,81410],{"className":81409},[204],[23,81411,81413],{"className":81412,"style":61901},[208],[23,81414],{},[23,81416],{"className":81417,"style":280},[148],[23,81419,81421,81427,81491,81497,81537,81540,81543,81546,81549,81552,81592,81595,81627,81630,81633,81636,81639,81642,81645,81699],{"className":81420},[1839],[23,81422,81424],{"className":81423,"style":1844},[140,1843],[23,81425,9014],{"className":81426},[1066,10073],[23,81428,81430],{"className":81429},[1786,7098],[23,81431,81433,81483],{"className":81432},[199,200],[23,81434,81436,81480],{"className":81435},[204],[23,81437,81439,81459,81469],{"className":81438,"style":61928},[208],[23,81440,81441,81444],{"style":34079},[23,81442],{"className":81443,"style":34083},[216],[23,81445,81447],{"className":81446},[1019,1020,1021,1022],[23,81448,81450,81453,81456],{"className":81449},[135,1022],[23,81451,33497],{"className":81452},[135,136,1022],[23,81454,69],{"className":81455},[153,1022],[23,81457,4435],{"className":81458},[135,1022],[23,81460,81461,81464],{"style":34092},[23,81462],{"className":81463,"style":34083},[216],[23,81465,81466],{},[23,81467,33855],{"className":81468},[1786,1790,1791],[23,81470,81471,81474],{"style":61961},[23,81472],{"className":81473,"style":34083},[216],[23,81475,81477],{"className":81476},[1019,1020,1021,1022],[23,81478,4806],{"className":81479},[135,136,1022],[23,81481,297],{"className":81482},[296],[23,81484,81486],{"className":81485},[204],[23,81487,81489],{"className":81488,"style":34110},[208],[23,81490],{},[23,81492,81494],{"className":81493},[140],[23,81495,54],{"className":81496},[1066,1067],[23,81498,81500,81503],{"className":81499},[135],[23,81501,660],{"className":81502,"style":676},[135,136],[23,81504,81506],{"className":81505},[998],[23,81507,81509,81529],{"className":81508},[199,200],[23,81510,81512,81526],{"className":81511},[204],[23,81513,81515],{"className":81514,"style":33651},[208],[23,81516,81517,81520],{"style":21854},[23,81518],{"className":81519,"style":1015},[216],[23,81521,81523],{"className":81522},[1019,1020,1021,1022],[23,81524,33497],{"className":81525},[135,136,1022],[23,81527,297],{"className":81528},[296],[23,81530,81532],{"className":81531},[204],[23,81533,81535],{"className":81534,"style":2048},[208],[23,81536],{},[23,81538],{"className":81539,"style":392},[148],[23,81541,4529],{"className":81542},[612],[23,81544],{"className":81545,"style":392},[148],[23,81547,8340],{"className":81548,"style":11402},[135,136],[23,81550,54],{"className":81551},[140],[23,81553,81555,81558],{"className":81554},[135],[23,81556,753],{"className":81557},[135,136],[23,81559,81561],{"className":81560},[998],[23,81562,81564,81584],{"className":81563},[199,200],[23,81565,81567,81581],{"className":81566},[204],[23,81568,81570],{"className":81569,"style":33651},[208],[23,81571,81572,81575],{"style":2029},[23,81573],{"className":81574,"style":1015},[216],[23,81576,81578],{"className":81577},[1019,1020,1021,1022],[23,81579,33497],{"className":81580},[135,136,1022],[23,81582,297],{"className":81583},[296],[23,81585,81587],{"className":81586},[204],[23,81588,81590],{"className":81589,"style":2048},[208],[23,81591],{},[23,81593,66],{"className":81594},[169],[23,81596,81598,81604],{"className":81597},[169],[23,81599,81601],{"className":81600},[169],[23,81602,66],{"className":81603},[1066,1067],[23,81605,81607],{"className":81606},[998],[23,81608,81610],{"className":81609},[199],[23,81611,81613],{"className":81612},[204],[23,81614,81616],{"className":81615,"style":11192},[208],[23,81617,81618,81621],{"style":11195},[23,81619],{"className":81620,"style":1015},[216],[23,81622,81624],{"className":81623},[1019,1020,1021,1022],[23,81625,9956],{"className":81626},[135,1022],[23,81628],{"className":81629,"style":392},[148],[23,81631,5106],{"className":81632},[612],[23,81634],{"className":81635,"style":392},[148],[23,81637,61761],{"className":81638},[135,136],[23,81640,20684],{"className":81641},[135],[23,81643,8340],{"className":81644,"style":11402},[135,136],[23,81646,81648,81651],{"className":81647},[135],[23,81649,20684],{"className":81650},[135],[23,81652,81654],{"className":81653},[998],[23,81655,81657,81691],{"className":81656},[199,200],[23,81658,81660,81688],{"className":81659},[204],[23,81661,81663,81677],{"className":81662,"style":11006},[208],[23,81664,81665,81668],{"style":19349},[23,81666],{"className":81667,"style":1015},[216],[23,81669,81671],{"className":81670},[1019,1020,1021,1022],[23,81672,81674],{"className":81673},[135,1022],[23,81675,26253],{"className":81676,"style":45028},[135,412,1022],[23,81678,81679,81682],{"style":4989},[23,81680],{"className":81681,"style":1015},[216],[23,81683,81685],{"className":81684},[1019,1020,1021,1022],[23,81686,9956],{"className":81687},[135,1022],[23,81689,297],{"className":81690},[296],[23,81692,81694],{"className":81693},[204],[23,81695,81697],{"className":81696,"style":19379},[208],[23,81698],{},[23,81700,81702],{"className":81701,"style":1844},[169,1843],[23,81703,9024],{"className":81704},[1066,10073],[23,81706],{"className":81707,"style":280},[148],[23,81709,112],{"className":81710},[135],[11,81712,81713,81714,81958],{},"由表示定理，",[23,81715,81717,81764],{"className":81716,"translate":27},[31],[23,81718,81720],{"className":81719},[35],[37,81721,81722],{"xmlns":39},[42,81723,81724,81762],{},[45,81725,81726,81732,81734,81740,81746,81748,81750,81752,81754,81760],{},[4518,81727,81728,81730],{},[48,81729,8340],{},[51,81731,6883],{},[51,81733,69],{},[898,81735,81736,81738],{},[51,81737,33855],{},[48,81739,33497],{},[898,81741,81742,81744],{},[48,81743,4526],{},[48,81745,33497],{},[48,81747,4836],{},[51,81749,54],{"stretchy":53},[51,81751,5758],{},[51,81753,747],{"separator":122},[898,81755,81756,81758],{},[48,81757,753],{},[48,81759,33497],{},[51,81761,66],{"stretchy":53},[114,81763,77170],{"encoding":116},[23,81765,81767,81811],{"className":81766,"ariaHidden":122},[121],[23,81768,81770,81773,81802,81805,81808],{"className":81769},[126],[23,81771],{"className":81772,"style":5414},[130],[23,81774,81776,81779],{"className":81775},[135],[23,81777,8340],{"className":81778,"style":11402},[135,136],[23,81780,81782],{"className":81781},[998],[23,81783,81785],{"className":81784},[199],[23,81786,81788],{"className":81787},[204],[23,81789,81791],{"className":81790,"style":6914},[208],[23,81792,81793,81796],{"style":4696},[23,81794],{"className":81795,"style":1015},[216],[23,81797,81799],{"className":81798},[1019,1020,1021,1022],[23,81800,6883],{"className":81801},[612,1022],[23,81803],{"className":81804,"style":149},[148],[23,81806,69],{"className":81807},[153],[23,81809],{"className":81810,"style":149},[148],[23,81812,81814,81817,81857,81860,81900,81903,81906,81909,81912,81915,81955],{"className":81813},[126],[23,81815],{"className":81816,"style":33579},[130],[23,81818,81820,81823],{"className":81819},[1786],[23,81821,33855],{"className":81822,"style":33586},[1786,1790,20458],[23,81824,81826],{"className":81825},[998],[23,81827,81829,81849],{"className":81828},[199,200],[23,81830,81832,81846],{"className":81831},[204],[23,81833,81835],{"className":81834,"style":33599},[208],[23,81836,81837,81840],{"style":33602},[23,81838],{"className":81839,"style":1015},[216],[23,81841,81843],{"className":81842},[1019,1020,1021,1022],[23,81844,33497],{"className":81845},[135,136,1022],[23,81847,297],{"className":81848},[296],[23,81850,81852],{"className":81851},[204],[23,81853,81855],{"className":81854,"style":33621},[208],[23,81856],{},[23,81858],{"className":81859,"style":280},[148],[23,81861,81863,81866],{"className":81862},[135],[23,81864,4526],{"className":81865,"style":4645},[135,136],[23,81867,81869],{"className":81868},[998],[23,81870,81872,81892],{"className":81871},[199,200],[23,81873,81875,81889],{"className":81874},[204],[23,81876,81878],{"className":81877,"style":33651},[208],[23,81879,81880,81883],{"style":68976},[23,81881],{"className":81882,"style":1015},[216],[23,81884,81886],{"className":81885},[1019,1020,1021,1022],[23,81887,33497],{"className":81888},[135,136,1022],[23,81890,297],{"className":81891},[296],[23,81893,81895],{"className":81894},[204],[23,81896,81898],{"className":81897,"style":2048},[208],[23,81899],{},[23,81901,4836],{"className":81902,"style":4851},[135,136],[23,81904,54],{"className":81905},[140],[23,81907,5758],{"className":81908},[135],[23,81910,747],{"className":81911},[824],[23,81913],{"className":81914,"style":280},[148],[23,81916,81918,81921],{"className":81917},[135],[23,81919,753],{"className":81920},[135,136],[23,81922,81924],{"className":81923},[998],[23,81925,81927,81947],{"className":81926},[199,200],[23,81928,81930,81944],{"className":81929},[204],[23,81931,81933],{"className":81932,"style":33651},[208],[23,81934,81935,81938],{"style":2029},[23,81936],{"className":81937,"style":1015},[216],[23,81939,81941],{"className":81940},[1019,1020,1021,1022],[23,81942,33497],{"className":81943},[135,136,1022],[23,81945,297],{"className":81946},[296],[23,81948,81950],{"className":81949},[204],[23,81951,81953],{"className":81952,"style":2048},[208],[23,81954],{},[23,81956,66],{"className":81957},[169],"。代入式 (4)：",[23,81960,81962],{"className":81961,"translate":27},[26],[23,81963,81965,82011],{"className":81964,"translate":27},[31],[23,81966,81968],{"className":81967},[35],[37,81969,81970],{"xmlns":39,"display":40},[42,81971,81972,82008],{},[45,81973,81974,81976,81978,81980,81982,81984,81990,81992,81994,81996,82002,82004,82006],{},[48,81975,20684],{"mathvariant":111},[48,81977,660],{},[51,81979,4529],{},[48,81981,26922],{},[48,81983,4526],{},[4518,81985,81986,81988],{},[48,81987,20684],{"mathvariant":111},[2581,81989,9956],{},[51,81991,5106],{},[48,81993,61761],{},[88,81995,90],{},[4518,81997,81998,82000],{},[48,81999,4526],{},[48,82001,22475],{"mathvariant":111},[48,82003,26922],{},[48,82005,4526],{},[48,82007,112],{"mathvariant":111},[114,82009,82010],{"encoding":116},"\\|y-K\\alpha\\|^2+\\lambda\\,\\alpha^\\top K\\alpha.",[23,82012,82014,82035,82085],{"className":82013,"ariaHidden":122},[121],[23,82015,82017,82020,82023,82026,82029,82032],{"className":82016},[126],[23,82018],{"className":82019,"style":131},[130],[23,82021,20684],{"className":82022},[135],[23,82024,660],{"className":82025,"style":676},[135,136],[23,82027],{"className":82028,"style":392},[148],[23,82030,4529],{"className":82031},[612],[23,82033],{"className":82034,"style":392},[148],[23,82036,82038,82041,82044,82047,82076,82079,82082],{"className":82037},[126],[23,82039],{"className":82040,"style":26503},[130],[23,82042,26922],{"className":82043,"style":37145},[135,136],[23,82045,4526],{"className":82046,"style":4645},[135,136],[23,82048,82050,82053],{"className":82049},[135],[23,82051,20684],{"className":82052},[135],[23,82054,82056],{"className":82055},[998],[23,82057,82059],{"className":82058},[199],[23,82060,82062],{"className":82061},[204],[23,82063,82065],{"className":82064,"style":11006},[208],[23,82066,82067,82070],{"style":4989},[23,82068],{"className":82069,"style":1015},[216],[23,82071,82073],{"className":82072},[1019,1020,1021,1022],[23,82074,9956],{"className":82075},[135,1022],[23,82077],{"className":82078,"style":392},[148],[23,82080,5106],{"className":82081},[612],[23,82083],{"className":82084,"style":392},[148],[23,82086,82088,82091,82094,82097,82126,82129,82132],{"className":82087},[126],[23,82089],{"className":82090,"style":4986},[130],[23,82092,61761],{"className":82093},[135,136],[23,82095],{"className":82096,"style":280},[148],[23,82098,82100,82103],{"className":82099},[135],[23,82101,4526],{"className":82102,"style":4645},[135,136],[23,82104,82106],{"className":82105},[998],[23,82107,82109],{"className":82108},[199],[23,82110,82112],{"className":82111},[204],[23,82113,82115],{"className":82114,"style":4986},[208],[23,82116,82117,82120],{"style":4989},[23,82118],{"className":82119,"style":1015},[216],[23,82121,82123],{"className":82122},[1019,1020,1021,1022],[23,82124,22475],{"className":82125},[135,1022],[23,82127,26922],{"className":82128,"style":37145},[135,136],[23,82130,4526],{"className":82131,"style":4645},[135,136],[23,82133,112],{"className":82134},[135],[11,82136,16379,82137,82165],{},[23,82138,82140,82153],{"className":82139,"translate":27},[31],[23,82141,82143],{"className":82142},[35],[37,82144,82145],{"xmlns":39},[42,82146,82147,82151],{},[45,82148,82149],{},[48,82150,4526],{},[114,82152,76439],{"encoding":116},[23,82154,82156],{"className":82155,"ariaHidden":122},[121],[23,82157,82159,82162],{"className":82158},[126],[23,82160],{"className":82161,"style":703},[130],[23,82163,4526],{"className":82164,"style":4645},[135,136]," 求梯度：",[23,82167,82169],{"className":82168,"translate":27},[26],[23,82170,82172,82242],{"className":82171,"translate":27},[31],[23,82173,82175],{"className":82174},[35],[37,82176,82177],{"xmlns":39,"display":40},[42,82178,82179,82239],{},[45,82180,82181,82183,82185,82187,82189,82191,82193,82195,82197,82199,82201,82203,82205,82207,82209,82211,82213,82215,82217,82219,82221,82223,82225,82227,82229,82231,82233,82235,82237],{},[51,82182,4529],{},[2581,82184,9956],{},[48,82186,26922],{},[51,82188,54],{"stretchy":53},[48,82190,660],{},[51,82192,4529],{},[48,82194,26922],{},[48,82196,4526],{},[51,82198,66],{"stretchy":53},[51,82200,5106],{},[2581,82202,9956],{},[48,82204,61761],{},[48,82206,26922],{},[48,82208,4526],{},[51,82210,69],{},[2581,82212,9956],{},[48,82214,26922],{},[51,82216,54],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[51,82218,54],{"stretchy":53},[48,82220,26922],{},[51,82222,5106],{},[48,82224,61761],{},[48,82226,34357],{},[51,82228,66],{"stretchy":53},[48,82230,4526],{},[51,82232,4529],{},[48,82234,660],{},[51,82236,66],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[48,82238,112],{"mathvariant":111},[114,82240,82241],{"encoding":116},"-2K(y-K\\alpha)+2\\lambda K\\alpha\n=2K\\bigl((K+\\lambda I)\\alpha-y\\bigr).",[23,82243,82245,82275,82299,82326,82359,82386],{"className":82244,"ariaHidden":122},[121],[23,82246,82248,82251,82254,82257,82260,82263,82266,82269,82272],{"className":82247},[126],[23,82249],{"className":82250,"style":131},[130],[23,82252,4529],{"className":82253},[135],[23,82255,9956],{"className":82256},[135],[23,82258,26922],{"className":82259,"style":37145},[135,136],[23,82261,54],{"className":82262},[140],[23,82264,660],{"className":82265,"style":676},[135,136],[23,82267],{"className":82268,"style":392},[148],[23,82270,4529],{"className":82271},[612],[23,82273],{"className":82274,"style":392},[148],[23,82276,82278,82281,82284,82287,82290,82293,82296],{"className":82277},[126],[23,82279],{"className":82280,"style":131},[130],[23,82282,26922],{"className":82283,"style":37145},[135,136],[23,82285,4526],{"className":82286,"style":4645},[135,136],[23,82288,66],{"className":82289},[169],[23,82291],{"className":82292,"style":392},[148],[23,82294,5106],{"className":82295},[612],[23,82297],{"className":82298,"style":392},[148],[23,82300,82302,82305,82308,82311,82314,82317,82320,82323],{"className":82301},[126],[23,82303],{"className":82304,"style":779},[130],[23,82306,9956],{"className":82307},[135],[23,82309,61761],{"className":82310},[135,136],[23,82312,26922],{"className":82313,"style":37145},[135,136],[23,82315,4526],{"className":82316,"style":4645},[135,136],[23,82318],{"className":82319,"style":149},[148],[23,82321,69],{"className":82322},[153],[23,82324],{"className":82325,"style":149},[148],[23,82327,82329,82332,82335,82338,82344,82347,82350,82353,82356],{"className":82328},[126],[23,82330],{"className":82331,"style":2213},[130],[23,82333,9956],{"className":82334},[135],[23,82336,26922],{"className":82337,"style":37145},[135,136],[23,82339,82341],{"className":82340},[140],[23,82342,54],{"className":82343},[1066,1067],[23,82345,54],{"className":82346},[140],[23,82348,26922],{"className":82349,"style":37145},[135,136],[23,82351],{"className":82352,"style":392},[148],[23,82354,5106],{"className":82355},[612],[23,82357],{"className":82358,"style":392},[148],[23,82360,82362,82365,82368,82371,82374,82377,82380,82383],{"className":82361},[126],[23,82363],{"className":82364,"style":131},[130],[23,82366,61761],{"className":82367},[135,136],[23,82369,34357],{"className":82370,"style":454},[135,136],[23,82372,66],{"className":82373},[169],[23,82375,4526],{"className":82376,"style":4645},[135,136],[23,82378],{"className":82379,"style":392},[148],[23,82381,4529],{"className":82382},[612],[23,82384],{"className":82385,"style":392},[148],[23,82387,82389,82392,82395,82401],{"className":82388},[126],[23,82390],{"className":82391,"style":2213},[130],[23,82393,660],{"className":82394,"style":676},[135,136],[23,82396,82398],{"className":82397},[169],[23,82399,66],{"className":82400},[1066,1067],[23,82402,112],{"className":82403},[135],[11,82405,82406,82407,82510,82511,82561,82562,82618],{},"令梯度为零，得到正规方程 ",[23,82408,82410,82444],{"className":82409,"translate":27},[31],[23,82411,82413],{"className":82412},[35],[37,82414,82415],{"xmlns":39},[42,82416,82417,82441],{},[45,82418,82419,82421,82423,82425,82427,82429,82431,82433,82435,82437,82439],{},[48,82420,26922],{},[51,82422,54],{"stretchy":53},[48,82424,26922],{},[51,82426,5106],{},[48,82428,61761],{},[48,82430,34357],{},[51,82432,66],{"stretchy":53},[48,82434,4526],{},[51,82436,69],{},[48,82438,26922],{},[48,82440,660],{},[114,82442,82443],{"encoding":116},"K(K+\\lambda I)\\alpha=Ky",[23,82445,82447,82471,82498],{"className":82446,"ariaHidden":122},[121],[23,82448,82450,82453,82456,82459,82462,82465,82468],{"className":82449},[126],[23,82451],{"className":82452,"style":131},[130],[23,82454,26922],{"className":82455,"style":37145},[135,136],[23,82457,54],{"className":82458},[140],[23,82460,26922],{"className":82461,"style":37145},[135,136],[23,82463],{"className":82464,"style":392},[148],[23,82466,5106],{"className":82467},[612],[23,82469],{"className":82470,"style":392},[148],[23,82472,82474,82477,82480,82483,82486,82489,82492,82495],{"className":82473},[126],[23,82475],{"className":82476,"style":131},[130],[23,82478,61761],{"className":82479},[135,136],[23,82481,34357],{"className":82482,"style":454},[135,136],[23,82484,66],{"className":82485},[169],[23,82487,4526],{"className":82488,"style":4645},[135,136],[23,82490],{"className":82491,"style":149},[148],[23,82493,69],{"className":82494},[153],[23,82496],{"className":82497,"style":149},[148],[23,82499,82501,82504,82507],{"className":82500},[126],[23,82502],{"className":82503,"style":817},[130],[23,82505,26922],{"className":82506,"style":37145},[135,136],[23,82508,660],{"className":82509,"style":676},[135,136],"。由于 ",[23,82512,82514,82531],{"className":82513,"translate":27},[31],[23,82515,82517],{"className":82516},[35],[37,82518,82519],{"xmlns":39},[42,82520,82521,82529],{},[45,82522,82523,82525,82527],{},[48,82524,61761],{},[51,82526,2579],{},[2581,82528,1532],{},[114,82530,69685],{"encoding":116},[23,82532,82534,82552],{"className":82533,"ariaHidden":122},[121],[23,82535,82537,82540,82543,82546,82549],{"className":82536},[126],[23,82538],{"className":82539,"style":4450},[130],[23,82541,61761],{"className":82542},[135,136],[23,82544],{"className":82545,"style":149},[148],[23,82547,2579],{"className":82548},[153],[23,82550],{"className":82551,"style":149},[148],[23,82553,82555,82558],{"className":82554},[126],[23,82556],{"className":82557,"style":2770},[130],[23,82559,1532],{"className":82560},[135]," 时 ",[23,82563,82565,82585],{"className":82564,"translate":27},[31],[23,82566,82568],{"className":82567},[35],[37,82569,82570],{"xmlns":39},[42,82571,82572,82582],{},[45,82573,82574,82576,82578,82580],{},[48,82575,26922],{},[51,82577,5106],{},[48,82579,61761],{},[48,82581,34357],{},[114,82583,82584],{"encoding":116},"K+\\lambda I",[23,82586,82588,82606],{"className":82587,"ariaHidden":122},[121],[23,82589,82591,82594,82597,82600,82603],{"className":82590},[126],[23,82592],{"className":82593,"style":62468},[130],[23,82595,26922],{"className":82596,"style":37145},[135,136],[23,82598],{"className":82599,"style":392},[148],[23,82601,5106],{"className":82602},[612],[23,82604],{"className":82605,"style":392},[148],[23,82607,82609,82612,82615],{"className":82608},[126],[23,82610],{"className":82611,"style":779},[130],[23,82613,61761],{"className":82614},[135,136],[23,82616,34357],{"className":82617,"style":454},[135,136]," 可逆，",[23,82620,82622],{"className":82621,"translate":27},[26],[23,82623,82625,82662],{"className":82624,"translate":27},[31],[23,82626,82628],{"className":82627},[35],[37,82629,82630],{"xmlns":39,"display":40},[42,82631,82632,82660],{},[45,82633,82634,82636,82638,82640,82642,82644,82646,82648,82658],{},[48,82635,4526],{},[51,82637,69],{},[51,82639,54],{"stretchy":53},[48,82641,26922],{},[51,82643,5106],{},[48,82645,61761],{},[48,82647,34357],{},[4518,82649,82650,82652],{},[51,82651,66],{"stretchy":53},[45,82653,82654,82656],{},[51,82655,4529],{},[2581,82657,4435],{},[48,82659,660],{},[114,82661,76801],{"encoding":116},[23,82663,82665,82683,82704],{"className":82664,"ariaHidden":122},[121],[23,82666,82668,82671,82674,82677,82680],{"className":82667},[126],[23,82669],{"className":82670,"style":703},[130],[23,82672,4526],{"className":82673,"style":4645},[135,136],[23,82675],{"className":82676,"style":149},[148],[23,82678,69],{"className":82679},[153],[23,82681],{"className":82682,"style":149},[148],[23,82684,82686,82689,82692,82695,82698,82701],{"className":82685},[126],[23,82687],{"className":82688,"style":131},[130],[23,82690,54],{"className":82691},[140],[23,82693,26922],{"className":82694,"style":37145},[135,136],[23,82696],{"className":82697,"style":392},[148],[23,82699,5106],{"className":82700},[612],[23,82702],{"className":82703,"style":392},[148],[23,82705,82707,82710,82713,82716,82751],{"className":82706},[126],[23,82708],{"className":82709,"style":26503},[130],[23,82711,61761],{"className":82712},[135,136],[23,82714,34357],{"className":82715,"style":454},[135,136],[23,82717,82719,82722],{"className":82718},[169],[23,82720,66],{"className":82721},[169],[23,82723,82725],{"className":82724},[998],[23,82726,82728],{"className":82727},[199],[23,82729,82731],{"className":82730},[204],[23,82732,82734],{"className":82733,"style":11006},[208],[23,82735,82736,82739],{"style":4989},[23,82737],{"className":82738,"style":1015},[216],[23,82740,82742],{"className":82741},[1019,1020,1021,1022],[23,82743,82745,82748],{"className":82744},[135,1022],[23,82746,4529],{"className":82747},[135,1022],[23,82749,4435],{"className":82750},[135,1022],[23,82752,660],{"className":82753,"style":676},[135,136],[11,82755,82756,82757,82785,82786,82814],{},"是一个解（即使 ",[23,82758,82760,82773],{"className":82759,"translate":27},[31],[23,82761,82763],{"className":82762},[35],[37,82764,82765],{"xmlns":39},[42,82766,82767,82771],{},[45,82768,82769],{},[48,82770,26922],{},[114,82772,26922],{"encoding":116},[23,82774,82776],{"className":82775,"ariaHidden":122},[121],[23,82777,82779,82782],{"className":82778},[126],[23,82780],{"className":82781,"style":470},[130],[23,82783,26922],{"className":82784,"style":37145},[135,136]," 奇异，这个 ",[23,82787,82789,82802],{"className":82788,"translate":27},[31],[23,82790,82792],{"className":82791},[35],[37,82793,82794],{"xmlns":39},[42,82795,82796,82800],{},[45,82797,82798],{},[48,82799,4526],{},[114,82801,76439],{"encoding":116},[23,82803,82805],{"className":82804,"ariaHidden":122},[121],[23,82806,82808,82811],{"className":82807},[126],[23,82809],{"className":82810,"style":703},[130],[23,82812,4526],{"className":82813,"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",[23,82816,82818],{"className":82817,"translate":27},[26],[23,82819,82821,82943],{"className":82820,"translate":27},[31],[23,82822,82824],{"className":82823},[35],[37,82825,82826],{"xmlns":39,"display":40},[42,82827,82828,82940],{},[45,82829,82830,82836,82838,82840,82842,82844,82846,82848,82850,82856,82858,82860,82862,82864,82866,82876,82878,82880,82882,82884,82886,82888,82890,82892,82894,82896,82898,82900,82902,82908,82910,82912,82914,82916,82918,82920,82922,82924,82930,82932,82938],{},[4518,82831,82832,82834],{},[48,82833,8340],{},[51,82835,6883],{},[51,82837,54],{"stretchy":53},[48,82839,753],{},[51,82841,66],{"stretchy":53},[51,82843,69],{},[48,82845,4836],{},[51,82847,54],{"stretchy":53},[48,82849,753],{},[4518,82851,82852,82854],{},[51,82853,66],{"stretchy":53},[48,82855,22475],{"mathvariant":111},[51,82857,54],{"stretchy":53},[48,82859,26922],{},[51,82861,5106],{},[48,82863,61761],{},[48,82865,34357],{},[4518,82867,82868,82870],{},[51,82869,66],{"stretchy":53},[45,82871,82872,82874],{},[51,82873,4529],{},[2581,82875,4435],{},[48,82877,660],{},[51,82879,747],{"separator":122},[148,82881],{"linebreak":29629},[48,82883,4836],{},[51,82885,54],{"stretchy":53},[48,82887,753],{},[51,82889,66],{"stretchy":53},[51,82891,69],{},[51,82893,54],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[48,82895,4836],{},[51,82897,54],{"stretchy":53},[48,82899,753],{},[51,82901,747],{"separator":122},[898,82903,82904,82906],{},[48,82905,753],{},[2581,82907,4435],{},[51,82909,66],{"stretchy":53},[51,82911,747],{"separator":122},[51,82913,62563],{},[51,82915,747],{"separator":122},[48,82917,4836],{},[51,82919,54],{"stretchy":53},[48,82921,753],{},[51,82923,747],{"separator":122},[898,82925,82926,82928],{},[48,82927,753],{},[48,82929,4806],{},[51,82931,66],{"stretchy":53},[4518,82933,82934,82936],{},[51,82935,66],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[48,82937,22475],{"mathvariant":111},[48,82939,112],{"mathvariant":111},[114,82941,82942],{"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.",[23,82944,82946,82999,83058,83111,83114,83141],{"className":82945,"ariaHidden":122},[121],[23,82947,82949,82952,82981,82984,82987,82990,82993,82996],{"className":82948},[126],[23,82950],{"className":82951,"style":131},[130],[23,82953,82955,82958],{"className":82954},[135],[23,82956,8340],{"className":82957,"style":11402},[135,136],[23,82959,82961],{"className":82960},[998],[23,82962,82964],{"className":82963},[199],[23,82965,82967],{"className":82966},[204],[23,82968,82970],{"className":82969,"style":7047},[208],[23,82971,82972,82975],{"style":4989},[23,82973],{"className":82974,"style":1015},[216],[23,82976,82978],{"className":82977},[1019,1020,1021,1022],[23,82979,6883],{"className":82980},[612,1022],[23,82982,54],{"className":82983},[140],[23,82985,753],{"className":82986},[135,136],[23,82988,66],{"className":82989},[169],[23,82991],{"className":82992,"style":149},[148],[23,82994,69],{"className":82995},[153],[23,82997],{"className":82998,"style":149},[148],[23,83000,83002,83005,83008,83011,83014,83043,83046,83049,83052,83055],{"className":83001},[126],[23,83003],{"className":83004,"style":4967},[130],[23,83006,4836],{"className":83007,"style":4851},[135,136],[23,83009,54],{"className":83010},[140],[23,83012,753],{"className":83013},[135,136],[23,83015,83017,83020],{"className":83016},[169],[23,83018,66],{"className":83019},[169],[23,83021,83023],{"className":83022},[998],[23,83024,83026],{"className":83025},[199],[23,83027,83029],{"className":83028},[204],[23,83030,83032],{"className":83031,"style":4986},[208],[23,83033,83034,83037],{"style":4989},[23,83035],{"className":83036,"style":1015},[216],[23,83038,83040],{"className":83039},[1019,1020,1021,1022],[23,83041,22475],{"className":83042},[135,1022],[23,83044,54],{"className":83045},[140],[23,83047,26922],{"className":83048,"style":37145},[135,136],[23,83050],{"className":83051,"style":392},[148],[23,83053,5106],{"className":83054},[612],[23,83056],{"className":83057,"style":392},[148],[23,83059,83061,83064,83067,83070,83105,83108],{"className":83060},[126],[23,83062],{"className":83063,"style":26503},[130],[23,83065,61761],{"className":83066},[135,136],[23,83068,34357],{"className":83069,"style":454},[135,136],[23,83071,83073,83076],{"className":83072},[169],[23,83074,66],{"className":83075},[169],[23,83077,83079],{"className":83078},[998],[23,83080,83082],{"className":83081},[199],[23,83083,83085],{"className":83084},[204],[23,83086,83088],{"className":83087,"style":11006},[208],[23,83089,83090,83093],{"style":4989},[23,83091],{"className":83092,"style":1015},[216],[23,83094,83096],{"className":83095},[1019,1020,1021,1022],[23,83097,83099,83102],{"className":83098},[135,1022],[23,83100,4529],{"className":83101},[135,1022],[23,83103,4435],{"className":83104},[135,1022],[23,83106,660],{"className":83107,"style":676},[135,136],[23,83109,747],{"className":83110},[824],[23,83112],{"className":83113},[148,29629],[23,83115,83117,83120,83123,83126,83129,83132,83135,83138],{"className":83116},[126],[23,83118],{"className":83119,"style":131},[130],[23,83121,4836],{"className":83122,"style":4851},[135,136],[23,83124,54],{"className":83125},[140],[23,83127,753],{"className":83128},[135,136],[23,83130,66],{"className":83131},[169],[23,83133],{"className":83134,"style":149},[148],[23,83136,69],{"className":83137},[153],[23,83139],{"className":83140,"style":149},[148],[23,83142,83144,83148,83154,83157,83160,83163,83166,83169,83209,83212,83215,83218,83221,83224,83227,83230,83233,83236,83239,83242,83245,83285,83288,83321],{"className":83143},[126],[23,83145],{"className":83146,"style":83147},[130],"height:1.439em;vertical-align:-0.35em;",[23,83149,83151],{"className":83150},[140],[23,83152,54],{"className":83153},[1066,1067],[23,83155,4836],{"className":83156,"style":4851},[135,136],[23,83158,54],{"className":83159},[140],[23,83161,753],{"className":83162},[135,136],[23,83164,747],{"className":83165},[824],[23,83167],{"className":83168,"style":280},[148],[23,83170,83172,83175],{"className":83171},[135],[23,83173,753],{"className":83174},[135,136],[23,83176,83178],{"className":83177},[998],[23,83179,83181,83201],{"className":83180},[199,200],[23,83182,83184,83198],{"className":83183},[204],[23,83185,83187],{"className":83186,"style":18651},[208],[23,83188,83189,83192],{"style":2029},[23,83190],{"className":83191,"style":1015},[216],[23,83193,83195],{"className":83194},[1019,1020,1021,1022],[23,83196,4435],{"className":83197},[135,1022],[23,83199,297],{"className":83200},[296],[23,83202,83204],{"className":83203},[204],[23,83205,83207],{"className":83206,"style":2048},[208],[23,83208],{},[23,83210,66],{"className":83211},[169],[23,83213,747],{"className":83214},[824],[23,83216],{"className":83217,"style":280},[148],[23,83219,62563],{"className":83220},[1839],[23,83222],{"className":83223,"style":280},[148],[23,83225,747],{"className":83226},[824],[23,83228],{"className":83229,"style":280},[148],[23,8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",[23,83328,83330,83354],{"className":83329,"translate":27},[31],[23,83331,83333],{"className":83332},[35],[37,83334,83335],{"xmlns":39},[42,83336,83337,83351],{},[45,83338,83339,83341,83343,83349],{},[48,83340,42344],{},[51,83342,54],{"stretchy":53},[4518,83344,83345,83347],{},[48,83346,4806],{},[2581,83348,24923],{},[51,83350,66],{"stretchy":53},[114,83352,83353],{"encoding":116},"O(n^3)",[23,83355,83357],{"className":83356,"ariaHidden":122},[121],[23,83358,83360,83363,83366,83369,83398],{"className":83359},[126],[23,83361],{"className":83362,"style":10665},[130],[23,83364,42344],{"className":83365,"style":144},[135,136],[23,83367,54],{"className":83368},[140],[23,83370,83372,83375],{"className":83371},[135],[23,83373,4806],{"className":83374},[135,136],[23,83376,83378],{"className":83377},[998],[23,83379,83381],{"className":83380},[199],[23,83382,83384],{"className":83383},[204],[23,83385,83387],{"className":83386,"style":4693},[208],[23,83388,83389,83392],{"style":4696},[23,83390],{"className":83391,"style":1015},[216],[23,83393,83395],{"className":83394},[1019,1020,1021,1022],[23,83396,24923],{"className":83397},[135,1022],[23,83399,66],{"className":83400},[169]," 时间、",[23,83403,83405,83429],{"className":83404,"translate":27},[31],[23,83406,83408],{"className":83407},[35],[37,83409,83410],{"xmlns":39},[42,83411,83412,83426],{},[45,83413,83414,83416,83418,83424],{},[48,83415,42344],{},[51,83417,54],{"stretchy":53},[4518,83419,83420,83422],{},[48,83421,4806],{},[2581,83423,9956],{},[51,83425,66],{"stretchy":53},[114,83427,83428],{"encoding":116},"O(n^2)",[23,83430,83432],{"className":83431,"ariaHidden":122},[121],[23,83433,83435,83438,83441,83444,83473],{"className":83434},[126],[23,83436],{"className":83437,"style":10665},[130],[23,83439,42344],{"className":83440,"style":144},[135,136],[23,83442,54],{"className":83443},[140],[23,83445,83447,83450],{"className":83446},[135],[23,83448,4806],{"className":83449},[135,136],[23,83451,83453],{"className":83452},[998],[23,83454,83456],{"className":83455},[199],[23,83457,83459],{"className":83458},[204],[23,83460,83462],{"className":83461,"style":4693},[208],[23,83463,83464,83467],{"style":4696},[23,83465],{"className":83466,"style":1015},[216],[23,83468,83470],{"className":83469},[1019,1020,1021,1022],[23,83471,9956],{"className":83472},[135,1022],[23,83474,66],{"className":83475},[169]," 内存；预测一个点需要 ",[23,83478,83480,83500],{"className":83479,"translate":27},[31],[23,83481,83483],{"className":83482},[35],[37,83484,83485],{"xmlns":39},[42,83486,83487,83497],{},[45,83488,83489,83491,83493,83495],{},[48,83490,42344],{},[51,83492,54],{"stretchy":53},[48,83494,4806],{},[51,83496,66],{"stretchy":53},[114,83498,83499],{"encoding":116},"O(n)",[23,83501,83503],{"className":83502,"ariaHidden":122},[121],[23,83504,83506,83509,83512,83515,83518],{"className":83505},[126],[23,83507],{"className":83508,"style":131},[130],[23,83510,42344],{"className":83511,"style":144},[135,136],[23,83513,54],{"className":83514},[140],[23,83516,4806],{"className":83517},[135,136],[23,83519,66],{"className":83520},[169]," 次核函数求值。",[2501,83523,83525],{"id":83524},"_52-支持向量机","5.2 支持向量机",[11,83527,83528],{},"软间隔 SVM 在 RKHS 中的 primal 问题是",[23,83530,83532],{"className":83531,"translate":27},[26],[23,83533,83535,83659],{"className":83534,"translate":27},[31],[23,83536,83538],{"className":83537},[35],[37,83539,83540],{"xmlns":39,"display":40},[42,83541,83542,83656],{},[45,83543,83544,83570,83654],{},[6963,83545,83546,83552],{},[45,83547,83548,83550],{},[48,83549,6969],{},[51,83551,6961],{},[45,83553,83554,83556,83558,83560,83562,83564,83566,83568],{},[48,83555,8340],{},[51,83557,372],{},[48,83559,26253],{"mathvariant":375},[51,83561,747],{"separator":122},[88,83563,90],{},[48,83565,35537],{},[51,83567,372],{},[48,83569,572],{"mathvariant":571},[45,83571,83572,83574,83580,83582,83584,83592,83594,83596,83610,83612,83614,83616,83618,83620,83622,83624,83630,83632,83634,83636,83642,83644,83646,83648,83650,83652],{},[51,83573,9014],{"fence":122},[71,83575,83576,83578],{},[2581,83577,4435],{},[2581,83579,9956],{},[48,83581,20684],{"mathvariant":111},[48,83583,8340],{},[19141,83585,83586,83588,83590],{},[48,83587,20684],{"mathvariant":111},[48,83589,26253],{"mathvariant":375},[2581,83591,9956],{},[51,83593,5106],{},[48,83595,24983],{},[61717,83597,83598,83600,83608],{},[51,83599,33855],{},[45,83601,83602,83604,83606],{},[48,83603,33497],{},[51,83605,69],{},[2581,83607,4435],{},[48,83609,4806],{},[48,83611,9555],{},[51,83613,6961],{},[51,83615,54],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[2581,83617,1532],{},[51,83619,747],{"separator":122},[2581,83621,4435],{},[51,83623,4529],{},[898,83625,83626,83628],{},[48,83627,660],{},[48,83629,33497],{},[51,83631,54],{"stretchy":53},[48,83633,8340],{},[51,83635,54],{"stretchy":53},[898,83637,83638,83640],{},[48,83639,753],{},[48,83641,33497],{},[51,83643,66],{"stretchy":53},[51,83645,5106],{},[48,83647,35537],{},[51,83649,66],{"stretchy":53},[51,83651,66],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[51,83653,9024],{"fence":122},[48,83655,112],{"mathvariant":111},[114,83657,83658],{"encoding":116},"\\min_{f\\in\\mathcal 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条件给出：",[346,85776,85777,86056,86335],{},[349,85778,73355,85779,46547,85964,63820],{},[23,85780,85782,85824],{"className":85781,"translate":27},[31],[23,85783,85785],{"className":85784},[35],[37,85786,85787],{"xmlns":39},[42,85788,85789,85821],{},[45,85790,85791,85797,85799,85801,85803,85809,85811,85813,85815,85817,85819],{},[898,85792,85793,85795],{},[48,85794,660],{},[48,85796,33497],{},[51,85798,54],{"stretchy":53},[48,85800,8340],{},[51,85802,54],{"stretchy":53},[898,85804,85805,85807],{},[48,85806,753],{},[48,85808,33497],{},[51,85810,66],{"stretchy":53},[51,85812,5106],{},[48,85814,35537],{},[51,85816,66],{"stretchy":53},[51,85818,2579],{},[2581,85820,4435],{},[114,85822,85823],{"encoding":116},"y_i(f(x_i)+b)>1",[23,85825,85827,85934,85955],{"className":85826,"ariaHidden":122},[121],[23,85828,85830,85833,85873,85876,85879,85882,85922,85925,85928,85931],{"className":85829},[126],[23,85831],{"className":85832,"style":131},[130],[23,85834,85836,85839],{"className":85835},[135],[23,85837,660],{"className":85838,"style":676},[135,136],[23,85840,85842],{"className":85841},[998],[23,85843,85845,85865],{"className":85844},[199,200],[23,85846,85848,85862],{"className":85847},[204],[23,85849,85851],{"className":85850,"style":33651},[208],[23,85852,85853,85856],{"style":21854},[23,85854],{"className":85855,"style":1015},[216],[23,85857,85859],{"className":85858},[1019,1020,1021,1022],[23,85860,33497],{"className":85861},[135,136,1022],[23,85863,297],{"className":85864},[296],[23,85866,85868],{"className":85867},[204],[23,85869,85871],{"className":85870,"style":2048},[208],[23,85872],{},[23,85874,54],{"className":85875},[140],[23,85877,8340],{"className":85878,"style":11402},[135,136],[23,85880,54],{"className":85881},[140],[23,85883,85885,85888],{"className":85884},[135],[23,85886,753],{"className":85887},[135,136],[23,85889,85891],{"className":85890},[998],[23,85892,85894,85914],{"className":85893},[199,200],[23,85895,85897,85911],{"className":85896},[204],[23,85898,85900],{"className":85899,"style":33651},[208],[23,85901,85902,85905],{"style":2029},[23,85903],{"className":85904,"style":1015},[216],[23,85906,85908],{"className":85907},[1019,1020,1021,1022],[23,85909,33497],{"className":85910},[135,136,1022],[23,85912,297],{"className":85913},[296],[23,85915,85917],{"className":85916},[204],[23,85918,85920],{"className":85919,"style":2048},[208],[23,85921],{},[23,85923,66],{"className":85924},[169],[23,85926],{"className":85927,"style":392},[148],[23,85929,5106],{"className":85930},[612],[23,85932],{"className":85933,"style":392},[148],[23,85935,85937,85940,85943,85946,85949,85952],{"className":85936},[126],[23,85938],{"className":85939,"style":131},[130],[23,85941,35537],{"className":85942},[135,136],[23,85944,66],{"className":85945},[169],[23,85947],{"className":85948,"style":149},[148],[23,85950,2579],{"className":85951},[153],[23,85953],{"className":85954,"style":149},[148],[23,85956,85958,85961],{"className":85957},[126],[23,85959],{"className":85960,"style":2770},[130],[23,85962,4435],{"className":85963},[135],[23,85965,85967,85989],{"className":85966,"translate":27},[31],[23,85968,85970],{"className":85969},[35],[37,85971,85972],{"xmlns":39},[42,85973,85974,85986],{},[45,85975,85976,85982,85984],{},[898,85977,85978,85980],{},[48,85979,4526],{},[48,85981,33497],{},[51,85983,69],{},[2581,85985,1532],{},[114,85987,85988],{"encoding":116},"\\alpha_i=0",[23,85990,85992,86047],{"className":85991,"ariaHidden":122},[121],[23,85993,85995,85998,86038,86041,86044],{"className":85994},[126],[23,85996],{"className":85997,"style":12734},[130],[23,85999,86001,86004],{"className":86000},[135],[23,86002,4526],{"className":86003,"style":4645},[135,136],[23,86005,86007],{"className":86006},[998],[23,86008,86010,86030],{"className":86009},[199,200],[23,86011,86013,86027],{"className":86012},[204],[23,86014,86016],{"className":86015,"style":33651},[208],[23,86017,86018,86021],{"style":68976},[23,86019],{"className":86020,"style":1015},[216],[23,86022,86024],{"className":86023},[1019,1020,1021,1022],[23,86025,33497],{"className":86026},[135,136,1022],[23,86028,297],{"className":86029},[296],[23,86031,86033],{"className":86032},[204],[23,86034,86036],{"className":86035,"style":2048},[208],[23,86037],{},[23,86039],{"className":86040,"style":149},[148],[23,86042,69],{"className":86043},[153],[23,86045],{"className":86046,"style":149},[148],[23,86048,86050,86053],{"className":86049},[126],[23,86051],{"className":86052,"style":2770},[130],[23,86054,1532],{"className":86055},[135],[349,86057,73355,86058,46547,86243,63820],{},[23,86059,86061,86103],{"className":86060,"translate":27},[31],[23,86062,86064],{"className":86063},[35],[37,86065,86066],{"xmlns":39},[42,86067,86068,86100],{},[45,86069,86070,86076,86078,86080,86082,86088,86090,86092,86094,86096,86098],{},[898,86071,86072,86074],{},[48,86073,660],{},[48,86075,33497],{},[51,86077,54],{"stretchy":53},[48,86079,8340],{},[51,86081,54],{"stretchy":53},[898,86083,86084,86086],{},[48,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C",[23,86552,86554,86573,86629],{"className":86553,"ariaHidden":122},[121],[23,86555,86557,86561,86564,86567,86570],{"className":86556},[126],[23,86558],{"className":86559,"style":86560},[130],"height:0.7804em;vertical-align:-0.136em;",[23,86562,1532],{"className":86563},[135],[23,86565],{"className":86566,"style":149},[148],[23,86568,12272],{"className":86569},[153],[23,86571],{"className":86572,"style":149},[148],[23,86574,86576,86580,86620,86623,86626],{"className":86575},[126],[23,86577],{"className":86578,"style":86579},[130],"height:0.786em;vertical-align:-0.15em;",[23,86581,86583,86586],{"className":86582},[135],[23,86584,4526],{"className":86585,"style":4645},[135,136],[23,86587,86589],{"className":86588},[998],[23,86590,86592,86612],{"className":86591},[199,200],[23,86593,86595,86609],{"className":86594},[204],[23,86596,86598],{"className":86597,"style":33651},[208],[23,86599,86600,86603],{"style":68976},[23,86601],{"className":86602,"style":1015},[216],[23,86604,86606],{"className":86605},[1019,1020,1021,1022],[23,86607,33497],{"className":86608},[135,136,1022],[23,86610,297],{"className":86611},[296],[23,86613,86615],{"className":86614},[204],[23,86616,86618],{"className":86617,"style":2048},[208],[23,86619],{},[23,86621],{"className":86622,"style":149},[148],[23,86624,12272],{"className":86625},[153],[23,86627],{"className":86628,"style":149},[148],[23,86630,86632,86635],{"className":86631},[126],[23,86633],{"className":86634,"style":470},[130],[23,86636,24983],{"className":86637,"style":37145},[135,136],[11,86639,86640,86733,86734,86737],{},[23,86641,86643,86665],{"className":86642,"translate":27},[31],[23,86644,86646],{"className":86645},[35],[37,86647,86648],{"xmlns":39},[42,86649,86650,86662],{},[45,86651,86652,86658,86660],{},[898,86653,86654,86656],{},[48,86655,4526],{},[48,86657,33497],{},[51,86659,2579],{},[2581,86661,1532],{},[114,86663,86664],{"encoding":116},"\\alpha_i>0",[23,86666,86668,86724],{"className":86667,"ariaHidden":122},[121],[23,86669,86671,86675,86715,86718,86721],{"className":86670},[126],[23,86672],{"className":86673,"style":86674},[130],"height:0.6891em;vertical-align:-0.15em;",[23,86676,86678,86681],{"className":86677},[135],[23,86679,4526],{"className":86680,"style":4645},[135,136],[23,86682,86684],{"className":86683},[998],[23,86685,86687,86707],{"className":86686},[199,200],[23,86688,86690,86704],{"className":86689},[204],[23,86691,86693],{"className":86692,"style":33651},[208],[23,86694,86695,86698],{"style":68976},[23,86696],{"className":86697,"style":1015},[216],[23,86699,86701],{"className":86700},[1019,1020,1021,1022],[23,86702,33497],{"className":86703},[135,136,1022],[23,86705,297],{"className":86706},[296],[23,86708,86710],{"className":86709},[204],[23,86711,86713],{"className":86712,"style":2048},[208],[23,86714],{},[23,86716],{"className":86717,"style":149},[148],[23,86719,2579],{"className":86720},[153],[23,86722],{"className":86723,"style":149},[148],[23,86725,86727,86730],{"className":86726},[126],[23,86728],{"className":86729,"style":2770},[130],[23,86731,1532],{"className":86732},[135]," 的点就是",[18,86735,86736],{},"支持向量","。决策函数为",[23,86739,86741],{"className":86740,"translate":27},[26],[23,86742,86744,86817],{"className":86743,"translate":27},[31],[23,86745,86747],{"className":86746},[35],[37,86748,86749],{"xmlns":39,"display":40},[42,86750,86751,86814],{},[45,86752,86753,86756,86758,86812],{},[48,86754,86755],{"mathvariant":111},"sign",[51,86757,6961],{},[45,86759,86760,86762,86776,86782,86788,86790,86792,86794,86796,86798,86804,86806,86808,86810],{},[51,86761,54],{"fence":122},[61717,86763,86764,86766,86774],{},[51,86765,33855],{},[45,86767,86768,86770,86772],{},[48,86769,33497],{},[51,86771,69],{},[2581,86773,4435],{},[48,86775,4806],{},[898,86777,86778,86780],{},[48,86779,4526],{},[48,86781,33497],{},[898,86783,86784,86786],{},[48,86785,660],{},[48,86787,33497],{},[88,86789,90],{},[48,86791,4836],{},[51,86793,54],{"stretchy":53},[48,86795,753],{},[51,86797,747],{"separator":122},[898,86799,86800,86802],{},[48,86801,753],{},[48,86803,33497],{},[51,86805,66],{"stretchy":53},[51,86807,5106],{},[48,86809,35537],{},[51,86811,66],{"fence":122},[48,86813,112],{"mathvariant":111},[114,86815,86816],{"encoding":116},"\\operatorname{sign}\\left(\\sum_{i=1}^n\\alpha_i y_i\\,k(x,x_i)+b\\right).",[23,86818,86820],{"className":86819,"ariaHidden":122},[121],[23,86821,86823,86826,86832,86835,87070,87073],{"className":86822},[126],[23,86824],{"className":86825,"style":61841},[130],[23,86827,86829],{"className":86828},[1786],[23,86830,86755],{"className":86831},[135,5391],[23,86833],{"className":86834,"style":280},[148],[23,86836,86838,86844,86908,86911,86951,86991,86994,86997,87000,87003,87006,87009,87049,87052,87055,87058,87061,87064],{"className":86837},[1839],[23,86839,86841],{"className":86840,"style":1844},[140,1843],[23,86842,54],{"className":86843},[1066,10073],[23,86845,86847],{"className":86846},[1786,7098],[23,86848,86850,86900],{"className":86849},[199,200],[23,86851,86853,86897],{"className":86852},[204],[23,86854,86856,86876,86886],{"className":86855,"style":61928},[208],[23,86857,86858,86861],{"style":34079},[23,86859],{"className":86860,"style":34083},[216],[23,86862,86864],{"className":86863},[1019,1020,1021,1022],[23,86865,86867,86870,86873],{"className":86866},[135,1022],[23,86868,33497],{"className":86869},[135,136,1022],[23,86871,69],{"className":86872},[153,1022],[23,86874,4435],{"className":86875},[135,1022],[23,86877,86878,86881],{"style":34092},[23,86879],{"className":86880,"style":34083},[216],[23,86882,86883],{},[23,86884,33855],{"className":86885},[1786,1790,1791],[23,86887,86888,86891],{"style":61961},[23,86889],{"className":86890,"style":34083},[216],[23,86892,86894],{"className":86893},[1019,1020,1021,1022],[23,86895,4806],{"className":86896},[135,136,1022],[23,86898,297],{"className":86899},[296],[23,86901,86903],{"className":86902},[204],[23,86904,86906],{"className":86905,"style":34110},[208],[23,86907],{},[23,86909],{"className":86910,"style":280},[148],[23,86912,86914,86917],{"className":86913},[135],[23,86915,4526],{"className":86916,"style":4645},[135,136],[23,86918,86920],{"className":86919},[998],[23,86921,86923,86943],{"className":86922},[199,200],[23,86924,86926,86940],{"className":86925},[204],[23,86927,86929],{"className":86928,"style":33651},[208],[23,86930,86931,86934],{"style":68976},[23,86932],{"className":86933,"style":1015},[216],[23,86935,86937],{"className":86936},[1019,1020,1021,1022],[23,86938,33497],{"className":86939},[135,136,1022],[23,86941,297],{"className":86942},[296],[23,86944,86946],{"className":86945},[204],[23,86947,86949],{"className":86948,"style":2048},[208],[23,86950],{},[23,86952,86954,86957],{"className":86953},[135],[23,86955,660],{"className":86956,"style":676},[135,136],[23,86958,86960],{"className":86959},[998],[23,86961,86963,86983],{"className":86962},[199,200],[23,86964,86966,86980],{"className":86965},[204],[23,86967,86969],{"className":86968,"style":33651},[208],[23,86970,86971,86974],{"style":21854},[23,86972],{"className":86973,"style":1015},[216],[23,86975,86977],{"className":86976},[1019,1020,1021,1022],[23,86978,33497],{"className":86979},[135,136,1022],[23,86981,297],{"className":86982},[296],[23,86984,86986],{"className":86985},[204],[23,86987,86989],{"className":86988,"style":2048},[208],[23,86990],{},[23,86992],{"className":86993,"style":280},[148],[23,86995,4836],{"className":86996,"style":4851},[135,136],[23,86998,54],{"className":86999},[140],[23,87001,753],{"className":87002},[135,136],[23,87004,747],{"className":87005},[824],[23,87007],{"className":87008,"style":280},[148],[23,87010,87012,87015],{"className":87011},[135],[23,87013,753],{"className":87014},[135,136],[23,87016,87018],{"className":87017},[998],[23,87019,87021,87041],{"className":87020},[199,200],[23,87022,87024,87038],{"className":87023},[204],[23,87025,87027],{"className":87026,"style":33651},[208],[23,87028,87029,87032],{"style":2029},[23,87030],{"className":87031,"style":1015},[216],[23,87033,87035],{"className":87034},[1019,1020,1021,1022],[23,87036,33497],{"className":87037},[135,136,1022],[23,87039,297],{"className":87040},[296],[23,87042,87044],{"className":87043},[204],[23,87045,87047],{"className":87046,"style":2048},[208],[23,87048],{},[23,87050,66],{"className":87051},[169],[23,87053],{"className":87054,"style":392},[148],[23,87056,5106],{"className":87057},[612],[23,87059],{"className":87060,"style":392},[148],[23,87062,35537],{"className":87063},[135,136],[23,87065,87067],{"className":87066,"style":1844},[169,1843],[23,87068,66],{"className":87069},[1066,10073],[23,87071],{"className":87072,"style":280},[148],[23,87074,112],{"className":87075},[135],[11,87077,87078,87079,87082,87083,87141,87142,87170,87171,87240,87241,87310,87311,87339],{},"这里要特别澄清一个常见误解：",[18,87080,87081],{},"SVM 的稀疏性不是表示定理给的。"," 表示定理只说 ",[23,87084,87086,87103],{"className":87085,"translate":27},[31],[23,87087,87089],{"className":87088},[35],[37,87090,87091],{"xmlns":39},[42,87092,87093,87101],{},[45,87094,87095],{},[4518,87096,87097,87099],{},[48,87098,8340],{},[51,87100,6883],{},[114,87102,76582],{"encoding":116},[23,87104,87106],{"className":87105,"ariaHidden":122},[121],[23,87107,87109,87112],{"className":87108},[126],[23,87110],{"className":87111,"style":5414},[130],[23,87113,87115,87118],{"className":87114},[135],[23,87116,8340],{"className":87117,"style":11402},[135,136],[23,87119,87121],{"className":87120},[998],[23,87122,87124],{"className":87123},[199],[23,87125,87127],{"className":87126},[204],[23,87128,87130],{"className":87129,"style":6914},[208],[23,87131,87132,87135],{"style":4696},[23,87133],{"className":87134,"style":1015},[216],[23,87136,87138],{"className":87137},[1019,1020,1021,1022],[23,87139,6883],{"className":87140},[612,1022]," 是 ",[23,87143,87145,87158],{"className":87144,"translate":27},[31],[23,87146,87148],{"className":87147},[35],[37,87149,87150],{"xmlns":39},[42,87151,87152,87156],{},[45,87153,87154],{},[48,87155,4806],{},[114,87157,4806],{"encoding":116},[23,87159,87161],{"className":87160,"ariaHidden":122},[121],[23,87162,87164,87167],{"className":87163},[126],[23,87165],{"className":87166,"style":703},[130],[23,87168,4806],{"className":87169},[135,136]," 个核函数的线性组合，",[23,87172,87174,87191],{"className":87173,"translate":27},[31],[23,87175,87177],{"className":87176},[35],[37,87178,87179],{"xmlns":39},[42,87180,87181,87189],{},[45,87182,87183],{},[898,87184,87185,87187],{},[48,87186,4526],{},[48,87188,33497],{},[114,87190,85542],{"encoding":116},[23,87192,87194],{"className":87193,"ariaHidden":122},[121],[23,87195,87197,87200],{"className":87196},[126],[23,87198],{"className":87199,"style":12734},[130],[23,87201,87203,87206],{"className":87202},[135],[23,87204,4526],{"className":87205,"style":4645},[135,136],[23,87207,87209],{"className":87208},[998],[23,87210,87212,87232],{"className":87211},[199,200],[23,87213,87215,87229],{"className":87214},[204],[23,87216,87218],{"className":87217,"style":33651},[208],[23,87219,87220,87223],{"style":68976},[23,87221],{"className":87222,"style":1015},[216],[23,87224,87226],{"className":87225},[1019,1020,1021,1022],[23,87227,33497],{"className":87228},[135,136,1022],[23,87230,297],{"className":87231},[296],[23,87233,87235],{"className":87234},[204],[23,87236,87238],{"className":87237,"style":2048},[208],[23,87239],{}," 完全可以全部非零。真正让大部分 ",[23,87242,87244,87261],{"className":87243,"translate":27},[31],[23,87245,87247],{"className":87246},[35],[37,87248,87249],{"xmlns":39},[42,87250,87251,87259],{},[45,87252,87253],{},[898,87254,87255,87257],{},[48,87256,4526],{},[48,87258,33497],{},[114,87260,85542],{"encoding":116},[23,87262,87264],{"className":87263,"ariaHidden":122},[121],[23,87265,87267,87270],{"className":87266},[126],[23,87268],{"className":87269,"style":12734},[130],[23,87271,87273,87276],{"className":87272},[135],[23,87274,4526],{"className":87275,"style":4645},[135,136],[23,87277,87279],{"className":87278},[998],[23,87280,87282,87302],{"className":87281},[199,200],[23,87283,87285,87299],{"className":87284},[204],[23,87286,87288],{"className":87287,"style":33651},[208],[23,87289,87290,87293],{"style":68976},[23,87291],{"className":87292,"style":1015},[216],[23,87294,87296],{"className":87295},[1019,1020,1021,1022],[23,87297,33497],{"className":87298},[135,136,1022],[23,87300,297],{"className":87301},[296],[23,87303,87305],{"className":87304},[204],[23,87306,87308],{"className":87307,"style":2048},[208],[23,87309],{}," 变成零的，是 hinge loss 的 KKT 条件：间隔大于 ",[23,87312,87314,87327],{"className":87313,"translate":27},[31],[23,87315,87317],{"className":87316},[35],[37,87318,87319],{"xmlns":39},[42,87320,87321,87325],{},[45,87322,87323],{},[2581,87324,4435],{},[114,87326,4435],{"encoding":116},[23,87328,87330],{"className":87329,"ariaHidden":122},[121],[23,87331,87333,87336],{"className":87332},[126],[23,87334],{"className":87335,"style":2770},[130],[23,87337,4435],{"className":87338},[135]," 的点对损失没有贡献，对应的乘子为零。",[2501,87341,87343],{"id":87342},"_53-核逻辑回归","5.3 核逻辑回归",[11,87345,87346],{},"取逻辑损失：",[23,87348,87350],{"className":87349,"translate":27},[26],[23,87351,87353,87463],{"className":87352,"translate":27},[31],[23,87354,87356],{"className":87355},[35],[37,87357,87358],{"xmlns":39,"display":40},[42,87359,87360,87460],{},[45,87361,87362,87378,87458],{},[6963,87363,87364,87370],{},[45,87365,87366,87368],{},[48,87367,6969],{},[51,87369,6961],{},[45,87371,87372,87374,87376],{},[48,87373,8340],{},[51,87375,372],{},[48,87377,26253],{"mathvariant":375},[45,87379,87380,87382,87396,87398,87400,87402,87404,87406,87408,87410,87412,87414,87420,87422,87424,87430,87432,87434,87436,87438,87444,87446,87448,87456],{},[51,87381,9014],{"fence":122},[61717,87383,87384,87386,87394],{},[51,87385,33855],{},[45,87387,87388,87390,87392],{},[48,87389,33497],{},[51,87391,69],{},[2581,87393,4435],{},[48,87395,4806],{},[48,87397,22422],{},[51,87399,6961],{},[51,87401,54],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[2581,87403,4435],{},[51,87405,5106],{},[48,87407,25012],{},[51,87409,6961],{},[51,87411,54],{"stretchy":53},[51,87413,4529],{},[898,87415,87416,87418],{},[48,87417,660],{},[48,87419,33497],{},[48,87421,8340],{},[51,87423,54],{"stretchy":53},[898,87425,87426,87428],{},[48,87427,753],{},[48,87429,33497],{},[51,87431,66],{"stretchy":53},[51,87433,66],{"stretchy":53},[51,87435,66],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[51,87437,5106],{},[71,87439,87440,87442],{},[48,87441,61761],{},[2581,87443,9956],{},[48,87445,20684],{"mathvariant":111},[48,87447,8340],{},[19141,87449,87450,87452,87454],{},[48,87451,20684],{"mathvariant":111},[48,87453,26253],{"mathvariant":375},[2581,87455,9956],{},[51,87457,9024],{"fence":122},[48,87459,112],{"mathvariant":111},[114,87461,87462],{"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\\}.",[23,87464,87466],{"className":87465,"ariaHidden":122},[121],[23,87467,87469,87472,87525,87528,87868,87871],{"className":87468},[126],[23,87470],{"className":87471,"style":61841},[130],[23,87473,87475],{"className":87474},[1786,7098],[23,87476,87478,87517],{"className":87477},[199,200],[23,87479,87481,87514],{"className":87480},[204],[23,87482,87484,87504],{"className":87483,"style":7108},[208],[23,87485,87486,87489],{"style":14742},[23,87487],{"className":87488,"style":217},[216],[23,87490,87492],{"className":87491},[1019,1020,1021,1022],[23,87493,87495,87498,87501],{"className":87494},[135,1022],[23,87496,8340],{"className":87497,"style":11402},[135,136,1022],[23,87499,372],{"className":87500},[153,1022],[23,87502,26253],{"className":87503,"style":45028},[135,412,1022],[23,87505,87506,87509],{"style":7132},[23,87507],{"className":87508,"style":217},[216],[23,87510,87511],{},[23,87512,6969],{"className":87513},[1786],[23,87515,297],{"className":87516},[296],[23,87518,87520],{"className":87519},[204],[23,87521,87523],{"className":87522,"style":61901},[208],[23,87524],{},[23,87526],{"className":87527,"style":280},[148],[23,87529,87531,87537,87601,87604,87609,87615,87618,87621,87624,87627,87630,87633,87636,87676,87679,87682,87722,87725,87731,87734,87737,87740,87802,87805,87808,87862],{"className":87530},[1839],[23,87532,87534],{"className":87533,"style":1844},[140,1843],[23,87535,9014],{"className":87536},[1066,10073],[23,87538,87540],{"className":87539},[1786,7098],[23,87541,87543,87593],{"className":87542},[199,200],[23,87544,87546,87590],{"className":87545},[204],[23,87547,87549,87569,87579],{"className":87548,"style":61928},[208],[23,87550,87551,87554],{"style":34079},[23,87552],{"className":87553,"style":34083},[216],[23,87555,87557],{"className":87556},[1019,1020,1021,1022],[23,87558,87560,87563,87566],{"className":87559},[135,1022],[23,87561,33497],{"className":87562},[135,136,1022],[23,87564,69],{"className":87565},[153,1022],[23,87567,4435],{"className":87568},[135,1022],[23,87570,87571,87574],{"style":34092},[23,87572],{"className":87573,"style":34083},[216],[23,87575,87576],{},[23,87577,33855],{"className":87578},[1786,1790,1791],[23,87580,87581,87584],{"style":61961},[23,87582],{"className":87583,"style":34083},[216],[23,87585,87587],{"className":87586},[1019,1020,1021,1022],[23,87588,4806],{"className":87589},[135,136,1022],[23,87591,297],{"className":87592},[296],[23,87594,87596],{"className":87595},[204],[23,87597,87599],{"className":87598,"style":34110},[208],[23,87600],{},[23,87602],{"className":87603,"style":280},[148],[23,87605,22579,87607],{"className":87606},[1786],[23,87608,7091],{"style":7090},[23,87610,87612],{"className":87611},[140],[23,87613,54],{"className":87614},[1066,1067],[23,87616,4435],{"className":87617},[135],[23,87619],{"className":87620,"style":392},[148],[23,87622,5106],{"className":87623},[612],[23,87625],{"className":87626,"style":392},[148],[23,87628,25012],{"className":87629},[1786],[23,87631,54],{"className":87632},[140],[23,87634,4529],{"className":87635},[135],[23,87637,87639,87642],{"className":87638},[135],[23,87640,660],{"className":87641,"style":676},[135,136],[23,87643,87645],{"className":87644},[998],[23,87646,87648,87668],{"className":87647},[199,200],[23,87649,87651,87665],{"className":87650},[204],[23,87652,87654],{"className":87653,"style":33651},[208],[23,87655,87656,87659],{"style":21854},[23,87657],{"className":87658,"style":1015},[216],[23,87660,87662],{"className":87661},[1019,1020,1021,1022],[23,87663,33497],{"className":87664},[135,136,1022],[23,87666,297],{"className":87667},[296],[23,87669,87671],{"className":87670},[204],[23,87672,87674],{"className":87673,"style":2048},[208],[23,87675],{},[23,87677,8340],{"className":87678,"style":11402},[135,136],[23,87680,54],{"className":87681},[140],[23,87683,87685,87688],{"className":87684},[135],[23,87686,753],{"className":87687},[135,136],[23,87689,87691],{"className":87690},[998],[23,87692,87694,87714],{"className":87693},[199,200],[23,87695,87697,87711],{"className":87696},[204],[23,87698,87700],{"className":87699,"style":33651},[208],[23,87701,87702,87705],{"style":2029},[23,87703],{"className":87704,"style":1015},[216],[23,87706,87708],{"className":87707},[1019,1020,1021,1022],[23,87709,33497],{"className":87710},[135,136,1022],[23,87712,297],{"className":87713},[296],[23,87715,87717],{"className":87716},[204],[23,87718,87720],{"className":87719,"style":2048},[208],[23,87721],{},[23,87723,1095],{"className":87724},[169],[23,87726,87728],{"className":87727},[169],[23,87729,66],{"className":87730},[1066,1067],[23,87732],{"className":87733,"style":392},[148],[23,87735,5106],{"className":87736},[612],[23,87738],{"className":87739,"style":392},[148],[23,87741,87743,87746,87799],{"className":87742},[135],[23,87744],{"className":87745},[140,192],[23,87747,87749],{"className":87748},[71],[23,87750,87752,87791],{"className":87751},[199,200],[23,87753,87755,87788],{"className":87754},[204],[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K\\alpha\n\\right\\}.",[23,88241,88243],{"className":88242,"ariaHidden":122},[121],[23,88244,88246,88249,88328,88331,88649,88652],{"className":88245},[126],[23,88247],{"className":88248,"style":61841},[130],[23,88250,88252],{"className":88251},[1786,7098],[23,88253,88255,88320],{"className":88254},[199,200],[23,88256,88258,88317],{"className":88257},[204],[23,88259,88261,88307],{"className":88260,"style":7108},[208],[23,88262,88263,88266],{"style":76029},[23,88264],{"className":88265,"style":217},[216],[23,88267,88269],{"className":88268},[1019,1020,1021,1022],[23,88270,88272,88275,88278],{"className":88271},[135,1022],[23,88273,4526],{"className":88274,"style":4645},[135,136,1022],[23,88276,372],{"className":88277},[153,1022],[23,88279,88281,88284],{"className":88280},[135,1022],[23,88282,572],{"className":88283},[135,644,1022],[23,88285,88287],{"className":88286},[998],[23,88288,88290],{"className":88289},[199],[23,88291,88293],{"className":88292},[204],[23,88294,88296],{"className":88295,"style":76063},[208],[23,88297,88298,88301],{"style":76066},[23,88299],{"className":88300,"style":24885},[216],[23,88302,88304],{"className":88303},[1019,24889,1067,1022],[23,88305,4806],{"className":88306},[135,136,1022],[23,88308,88309,88312],{"style":7132},[23,88310],{"className":88311,"style":217},[216],[23,88313,88314],{},[23,88315,6969],{"className":88316},[1786],[23,88318,297],{"className":88319},[296],[23,88321,88323],{"className":88322},[204],[23,88324,88326],{"className":88325,"style":76095},[208],[23,88327],{},[23,88329],{"className":88330,"style":280},[148],[23,88332,88334,88340,88404,88407,88412,88418,88421,88424,88427,88430,88433,88436,88439,88479,88482,88485,88488,88528,88531,88537,88540,88543,88546,88608,88637,88640,88643],{"className":88333},[1839],[23,88335,88337],{"className":88336,"style":1844},[140,1843],[23,88338,9014],{"className":88339},[1066,10073],[23,88341,88343],{"className":88342},[1786,7098],[23,88344,88346,88396],{"className":88345},[199,200],[23,88347,88349,88393],{"className":88348},[204],[23,88350,88352,88372,88382],{"className":88351,"style":61928},[208],[23,88353,88354,88357],{"style":34079},[23,88355],{"className":88356,"style":34083},[216],[23,88358,88360],{"className":88359},[1019,1020,1021,1022],[23,88361,88363,88366,88369],{"className":88362},[135,1022],[23,88364,33497],{"className":88365},[135,136,1022],[23,88367,69],{"className":88368},[153,1022],[23,88370,4435],{"className":88371},[135,1022],[23,88373,88374,88377],{"style":34092},[23,88375],{"className":88376,"style":34083},[216],[23,88378,88379],{},[23,88380,33855],{"className":88381},[1786,1790,1791],[23,88383,88384,88387],{"style":61961},[23,88385],{"className":88386,"style":34083},[216],[23,88388,88390],{"className":88389},[1019,1020,1021,1022],[23,88391,4806],{"className":88392},[135,136,1022],[23,88394,297],{"className":88395},[296],[23,88397,88399],{"className":88398},[204],[23,88400,88402],{"className":88401,"style":34110},[208],[23,88403],{},[23,88405],{"className":88406,"style":280},[148],[23,88408,22579,88410],{"className":88409},[1786],[23,88411,7091],{"style":7090},[23,88413,88415],{"className":88414},[140],[23,88416,54],{"className":88417},[1066,1067],[23,88419,4435],{"className":88420},[135],[23,88422],{"className":88423,"style":392},[148],[23,88425,5106],{"className":88426},[612],[23,88428],{"className":88429,"style":392},[148],[23,88431,25012],{"className":88432},[1786],[23,88434,54],{"className":88435},[140],[23,88437,4529],{"className":88438},[135],[23,88440,88442,88445],{"className":88441},[135],[23,88443,660],{"className":88444,"style":676},[135,136],[23,88446,88448],{"className":88447},[998],[23,88449,88451,88471],{"className":88450},[199,200],[23,88452,88454,88468],{"className":88453},[204],[23,88455,88457],{"className":88456,"style":33651},[208],[23,88458,88459,88462],{"style":21854},[23,88460],{"className":88461,"style":1015},[216],[23,88463,88465],{"className":88464},[1019,1020,1021,1022],[23,88466,33497],{"className":88467},[135,136,1022],[23,88469,297],{"className":88470},[296],[23,88472,88474],{"className":88473},[204],[23,88475,88477],{"className":88476,"style":2048},[208],[23,88478],{},[23,88480,54],{"className":88481},[140],[23,88483,26922],{"className":88484,"style":37145},[135,136],[23,88486,4526],{"className":88487,"style":4645},[135,136],[23,88489,88491,88494],{"className":88490},[169],[23,88492,66],{"className":88493},[169],[23,88495,88497],{"className":88496},[998],[23,88498,88500,88520],{"className":88499},[199,200],[23,88501,88503,88517],{"className":88502},[204],[23,88504,88506],{"className":88505,"style":33651},[208],[23,88507,88508,88511],{"style":2029},[23,88509],{"className":88510,"style":1015},[216],[23,88512,88514],{"className":88513},[1019,1020,1021,1022],[23,88515,33497],{"className":88516},[135,136,1022],[23,88518,297],{"className":88519},[296],[23,88521,88523],{"className":88522},[204],[23,88524,88526],{"className":88525,"style":2048},[208],[23,88527],{},[23,88529,66],{"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，",[23,88659,88661,88674],{"className":88660,"translate":27},[31],[23,88662,88664],{"className":88663},[35],[37,88665,88666],{"xmlns":39},[42,88667,88668,88672],{},[45,88669,88670],{},[48,88671,4526],{},[114,88673,76439],{"encoding":116},[23,88675,88677],{"className":88676,"ariaHidden":122},[121],[23,88678,88680,88683],{"className":88679},[126],[23,88681],{"className":88682,"style":703},[130],[23,88684,4526],{"className":88685,"style":4645},[135,136]," 一般是稠密的。",[2501,88688,88690],{"id":88689},"_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",[11,89893,89894,89895,8059,89969,87141,89997,90025],{},"表示定理最早的重要应用之一是平滑样条。取 ",[23,89896,89898,89924],{"className":89897,"translate":27},[31],[23,89899,89901],{"className":89900},[35],[37,89902,89903],{"xmlns":39},[42,89904,89905,89921],{},[45,89906,89907,89909,89911,89913,89915,89917,89919],{},[48,89908,434],{"mathvariant":375},[51,89910,69],{},[51,89912,924],{"stretchy":53},[2581,89914,1532],{},[51,89916,747],{"separator":122},[2581,89918,4435],{},[51,89920,945],{"stretchy":53},[114,89922,89923],{"encoding":116},"\\mathcal X=[0,1]",[23,89925,89927,89945],{"className":89926,"ariaHidden":122},[121],[23,89928,89930,89933,89936,89939,89942],{"className":89929},[126],[23,89931],{"className":89932,"style":470},[130],[23,89934,434],{"className":89935,"style":474},[135,412],[23,89937],{"className":89938,"style":149},[148],[23,89940,69],{"className":89941},[153],[23,89943],{"className":89944,"style":149},[148],[23,89946,89948,89951,89954,89957,89960,89963,89966],{"className":89947},[126],[23,89949],{"className":89950,"style":131},[130],[23,89952,924],{"className":89953},[140],[23,89955,1532],{"className":89956},[135],[23,89958,747],{"className":89959},[824],[23,89961],{"className":89962,"style":280},[148],[23,89964,4435],{"className":89965},[135],[23,89967,945],{"className":89968},[169],[23,89970,89972,89985],{"className":89971,"translate":27},[31],[23,89973,89975],{"className":89974},[35],[37,89976,89977],{"xmlns":39},[42,89978,89979,89983],{},[45,89980,89981],{},[48,89982,26253],{"mathvariant":375},[114,89984,45015],{"encoding":116},[23,89986,89988],{"className":89987,"ariaHidden":122},[121],[23,89989,89991,89994],{"className":89990},[126],[23,89992],{"className":89993,"style":470},[130],[23,89995,26253],{"className":89996,"style":45028},[135,412],[23,89998,90000,90013],{"className":89999,"translate":27},[31],[23,90001,90003],{"className":90002},[35],[37,90004,90005],{"xmlns":39},[42,90006,90007,90011],{},[45,90008,90009],{},[48,90010,50967],{},[114,90012,50967],{"encoding":116},[23,90014,90016],{"className":90015,"ariaHidden":122},[121],[23,90017,90019,90022],{"className":90018},[126],[23,90020],{"className":90021,"style":703},[130],[23,90023,50967],{"className":90024},[135,136]," 阶 Sobolev 空间，正则项为",[23,90027,90029],{"className":90028,"translate":27},[26],[23,90030,90032,90088],{"className":90031,"translate":27},[31],[23,90033,90035],{"className":90034},[35],[37,90036,90037],{"xmlns":39,"display":40},[42,90038,90039,90085],{},[45,90040,90041,90043,90051,90053,90065,90067,90069,90071,90077,90079,90081,90083],{},[48,90042,61761],{},[19141,90044,90045,90047,90049],{},[51,90046,1557],{},[2581,90048,1532],{},[2581,90050,4435],{},[51,90052,54],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[4518,90054,90055,90057],{},[48,90056,8340],{},[45,90058,90059,90061,90063],{},[51,90060,54],{"stretchy":53},[48,90062,50967],{},[51,90064,66],{"stretchy":53},[51,90066,54],{"stretchy":53},[48,90068,753],{},[51,90070,66],{"stretchy":53},[4518,90072,90073,90075],{},[51,90074,66],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[2581,90076,9956],{},[88,90078,90],{},[48,90080,1606],{},[48,90082,753],{},[48,90084,112],{"mathvariant":111},[114,90086,90087],{"encoding":116},"\\lambda\\int_0^1\\bigl(f^{(m)}(x)\\bigr)^2\\,dx.",[23,90089,90091],{"className":90090,"ariaHidden":122},[121],[23,90092,90094,90098,90101,90104,90157,90163,90201,90204,90207,90210,90242,90245,90248,90251],{"className":90093},[126],[23,90095],{"className":90096,"style":90097},[130],"height:2.476em;vertical-align:-0.9119em;",[23,90099,61761],{"className":90100},[135,136],[23,90102],{"className":90103,"style":280},[148],[23,90105,90107,90110],{"className":90106},[1786],[23,90108,1557],{"className":90109,"style":1792},[1786,1790,1791],[23,90111,90113],{"className":90112},[998],[23,90114,90116,90149],{"className":90115},[199,200],[23,90117,90119,90146],{"className":90118},[204],[23,90120,90123,90134],{"className":90121,"style":90122},[208],"height:1.564em;",[23,90124,90125,90128],{"style":1808},[23,90126],{"className":90127,"style":1015},[216],[23,90129,90131],{"className":90130},[1019,1020,1021,1022],[23,90132,1532],{"className":90133},[135,1022],[23,90135,90137,90140],{"style":90136},"top:-3.8129em;margin-right:0.05em;",[23,90138],{"className":90139,"style":1015},[216],[23,90141,90143],{"className":90142},[1019,1020,1021,1022],[23,90144,4435],{"className":90145},[135,1022],[23,90147,297],{"className":90148},[296],[23,90150,90152],{"className":90151},[204],[23,90153,90155],{"className":90154,"style":1830},[208],[23,90156],{},[23,90158,90160],{"className":90159},[140],[23,90161,54],{"className":90162},[1066,1067],[23,90164,90166,90169],{"className":90165},[135],[23,90167,8340],{"className":90168,"style":11402},[135,136],[23,90170,90172],{"className":90171},[998],[23,90173,90175],{"className":90174},[199],[23,90176,90178],{"className":90177},[204],[23,90179,90181],{"className":90180,"style":44186},[208],[23,90182,90183,90186],{"style":4989},[23,90184],{"className":90185,"style":1015},[216],[23,90187,90189],{"className":90188},[1019,1020,1021,1022],[23,90190,90192,90195,90198],{"className":90191},[135,1022],[23,90193,54],{"className":90194},[140,1022],[23,90196,50967],{"className":90197},[135,136,1022],[23,90199,66],{"className":90200},[169,1022],[23,90202,54],{"className":90203},[140],[23,90205,753],{"className":90206},[135,136],[23,90208,66],{"className":90209},[169],[23,90211,90213,90219],{"className":90212},[169],[23,90214,90216],{"className":90215},[169],[23,90217,66],{"className":90218},[1066,1067],[23,90220,90222],{"className":90221},[998],[23,90223,90225],{"className":90224},[199],[23,90226,90228],{"className":90227},[204],[23,90229,90231],{"className":90230,"style":11192},[208],[23,90232,90233,90236],{"style":11195},[23,90234],{"className":90235,"style":1015},[216],[23,90237,90239],{"className":90238},[1019,1020,1021,1022],[23,90240,9956],{"className":90241},[135,1022],[23,90243],{"className":90244,"style":280},[148],[23,90246,1606],{"className":90247},[135,136],[23,90249,753],{"className":90250},[135,136],[23,90252,112],{"className":90253},[135],[11,90255,90256],{},"最小化",[23,90258,90260],{"className":90259,"translate":27},[26],[23,90261,90263,90361],{"className":90262,"translate":27},[31],[23,90264,90266],{"className":90265},[35],[37,90267,90268],{"xmlns":39,"display":40},[42,90269,90270,90358],{},[45,90271,90272,90286,90288,90294,90296,90298,90300,90306,90308,90314,90316,90318,90326,90328,90340,90342,90344,90346,90352,90354,90356],{},[61717,90273,90274,90276,90284],{},[51,90275,33855],{},[45,90277,90278,90280,90282],{},[48,90279,33497],{},[51,90281,69],{},[2581,90283,4435],{},[48,90285,4806],{},[51,90287,54],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[898,90289,90290,90292],{},[48,90291,660],{},[48,90293,33497],{},[51,90295,4529],{},[48,90297,8340],{},[51,90299,54],{"stretchy":53},[898,90301,90302,90304],{},[48,90303,753],{},[48,90305,33497],{},[51,90307,66],{"stretchy":53},[4518,90309,90310,90312],{},[51,90311,66],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[2581,90313,9956],{},[51,90315,5106],{},[48,90317,61761],{},[19141,90319,90320,90322,90324],{},[51,90321,1557],{},[2581,90323,1532],{},[2581,903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",[23,90746,90748,90768],{"className":90747,"translate":27},[31],[23,90749,90751],{"className":90750},[35],[37,90752,90753],{"xmlns":39},[42,90754,90755,90765],{},[45,90756,90757,90759,90761,90763],{},[2581,90758,9956],{},[48,90760,50967],{},[51,90762,4529],{},[2581,90764,4435],{},[114,90766,90767],{"encoding":116},"2m-1",[23,90769,90771,90792],{"className":90770,"ariaHidden":122},[121],[23,90772,90774,90777,90780,90783,90786,90789],{"className":90773},[126],[23,90775],{"className":90776,"style":9404},[130],[23,90778,9956],{"className":90779},[135],[23,90781,50967],{"className":90782},[135,136],[23,90784],{"className":90785,"style":392},[148],[23,90787,4529],{"className":90788},[612],[23,90790],{"className":90791,"style":392},[148],[23,90793,90795,90798],{"className":90794},[126],[23,90796],{"className":90797,"style":2770},[130],[23,90799,4435],{"className":90800},[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（",[23,90803,90805,90818],{"className":90804,"translate":27},[31],[23,90806,90808],{"className":90807},[35],[37,90809,90810],{"xmlns":39},[42,90811,90812,90816],{},[45,90813,90814],{},[48,90815,50967],{},[114,90817,50967],{"encoding":116},[23,90819,90821],{"className":90820,"ariaHidden":122},[121],[23,90822,90824,90827],{"className":90823},[126],[23,90825],{"className":90826,"style":703},[130],[23,90828,50967],{"className":90829},[135,136]," 维）则由半参数版本处理。",[23,90832,90834,90852],{"className":90833,"translate":27},[31],[23,90835,90837],{"className":90836},[35],[37,90838,90839],{"xmlns":39},[42,90840,90841,90849],{},[45,90842,90843,90845,90847],{},[48,90844,50967],{},[51,90846,69],{},[2581,90848,9956],{},[114,90850,90851],{"encoding":116},"m=2",[23,90853,90855,90873],{"className":90854,"ariaHidden":122},[121],[23,90856,90858,90861,90864,90867,90870],{"className":90857},[126],[23,90859],{"className":90860,"style":703},[130],[23,90862,50967],{"className":90863},[135,136],[23,90865],{"className":90866,"style":149},[148],[23,90868,69],{"className":90869},[153],[23,90871],{"className":90872,"style":149},[148],[23,90874,90876,90879],{"className":90875},[126],[23,90877],{"className":90878,"style":2770},[130],[23,90880,9956],{"className":90881},[135]," 时就是常见的自然三次样条。",[11,90884,90885],{},"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。",[2501,90887,90889],{"id":90888},"_56-一张表","5.6 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I)^{-1}y",[23,91035,91037,91058],{"className":91036,"ariaHidden":122},[121],[23,91038,91040,91043,91046,91049,91052,91055],{"className":91039},[126],[23,91041],{"className":91042,"style":131},[130],[23,91044,54],{"className":91045},[140],[23,91047,26922],{"className":91048,"style":37145},[135,136],[23,91050],{"className":91051,"style":392},[148],[23,91053,5106],{"className":91054},[612],[23,91056],{"className":91057,"style":392},[148],[23,91059,91061,91064,91067,91070,91105],{"className":91060},[126],[23,91062],{"className":91063,"style":10665},[130],[23,91065,61761],{"className":91066},[135,136],[23,91068,34357],{"className":91069,"style":454},[135,136],[23,91071,91073,91076],{"className":91072},[169],[23,91074,66],{"className":91075},[169],[23,91077,91079],{"className":91078},[998],[23,91080,91082],{"className":91081},[199],[23,91083,91085],{"className":91084},[204],[23,91086,91088],{"className":91087,"style":4693},[208],[23,91089,91090,91093],{"style":4696},[23,91091],{"className":91092,"style":1015},[216],[23,91094,91096],{"className":91095},[1019,1020,1021,1022],[23,91097,91099,91102],{"className":91098},[135,1022],[23,91100,4529],{"className":91101},[135,1022],[23,91103,4435],{"className":91104},[135,1022],[23,91106,660],{"className":91107,"style":676},[135,136],[3788,91109,91110],{},"一般稠密",[3770,91112,91113,91116,91119,91264,91267],{},[3788,91114,91115],{},"SVM",[3788,91117,91118],{},"hinge",[3788,91120,91121],{},[23,91122,91124,91152],{"className":91123,"translate":27},[31],[23,91125,91127],{"className":91126},[35],[37,91128,91129],{"xmlns":39},[42,91130,91131,91149],{},[45,91132,91133,91139,91141,91143],{},[71,91134,91135,91137],{},[2581,91136,4435],{},[2581,91138,9956],{},[48,91140,20684],{"mathvariant":111},[48,91142,8340],{},[4518,91144,91145,91147],{},[48,91146,20684],{"mathvariant":111},[2581,91148,9956],{},[114,91150,91151],{"encoding":116},"\\frac12\\|f\\|^2",[23,91153,91155],{"className":91154,"ariaHidden":122},[121],[23,91156,91158,91161,91229,91232,91235],{"className":91157},[126],[23,91159],{"className":91160,"style":27420},[130],[23,91162,91164,91167,91226],{"className":91163},[135],[23,91165],{"className":91166},[140,192],[23,91168,91170],{"className":91169},[71],[23,91171,91173,91218],{"className":91172},[199,200],[23,91174,91176,91215],{"className":91175},[204],[23,91177,91179,91193,91201],{"className":91178,"style":10200},[208],[23,91180,91181,91184],{"style":10203},[23,91182],{"className":91183,"style":217},[216],[23,91185,91187],{"className":91186},[1019,1020,1021,1022],[23,91188,91190],{"className":91189},[135,1022],[23,91191,9956],{"className":91192},[135,1022],[23,91194,91195,91198],{"style":235},[23,91196],{"className":91197,"style":217},[216],[23,91199],{"className":91200,"style":243},[242],[23,91202,91203,91206],{"style":10226},[23,91204],{"className":91205,"style":217},[216],[23,91207,91209],{"className":91208},[1019,1020,1021,1022],[23,91210,91212],{"className":91211},[135,1022],[23,91213,4435],{"className":91214},[135,1022],[23,91216,297],{"className":91217},[296],[23,91219,91221],{"className":91220},[204],[23,91222,91224],{"className":91223,"style":10248},[208],[23,91225],{},[23,91227],{"className":91228},[169,192],[23,91230,20684],{"className":91231},[135],[23,91233,8340],{"className":91234,"style":11402},[135,136],[23,91236,91238,91241],{"className":91237},[135],[23,91239,20684],{"className":91240},[135],[23,91242,91244],{"className":91243},[998],[23,91245,91247],{"className":91246},[199],[23,91248,91250],{"className":91249},[204],[23,91251,91253],{"className":91252,"style":4693},[208],[23,91254,91255,91258],{"style":4696},[23,91256],{"className":91257,"style":1015},[216],[23,91259,91261],{"className":91260},[1019,1020,1021,1022],[23,91262,9956],{"className":91263},[135,1022],[3788,91265,91266],{},"对偶二次规划",[3788,91268,91269],{},"支持向量稀疏",[3770,91271,91272,91275,91278,91425,91428],{},[3788,91273,91274],{},"核逻辑回归",[3788,91276,91277],{},"logistic",[3788,91279,91280],{},[23,91281,91283,91311],{"className":91282,"translate":27},[31],[23,91284,91286],{"className":91285},[35],[37,91287,91288],{"xmlns":39},[42,91289,91290,91308],{},[45,91291,91292,91298,91300,91302],{},[71,91293,91294,91296],{},[48,91295,61761],{},[2581,91297,9956],{},[48,91299,20684],{"mathvariant":111},[48,91301,8340],{},[4518,91303,91304,91306],{},[48,91305,20684],{"mathvariant":111},[2581,91307,9956],{},[114,91309,91310],{"encoding":116},"\\frac{\\lambda}{2}\\|f\\|^2",[23,91312,91314],{"className":91313,"ariaHidden":122},[121],[23,91315,91317,91321,91390,91393,91396],{"className":91316},[126],[23,91318],{"className":91319,"style":91320},[130],"height:1.2251em;vertical-align:-0.345em;",[23,91322,91324,91327,91387],{"className":91323},[135],[23,91325],{"className":91326},[140,192],[23,91328,91330],{"className":91329},[71],[23,91331,91333,91379],{"className":91332},[199,200],[23,91334,91336,91376],{"className":91335},[204],[23,91337,91340,91354,91362],{"className":91338,"style":91339},[208],"height:0.8801em;",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先验",[3788,91442,91443],{},[23,91444,91446,91484],{"className":91445,"translate":27},[31],[23,91447,91449],{"className":91448},[35],[37,91450,91451],{"xmlns":39},[42,91452,91453,91481],{},[45,91454,91455,91457,91459,91461,91467,91469,91479],{},[51,91456,54],{"stretchy":53},[48,91458,26922],{},[51,91460,5106],{},[4518,91462,91463,91465],{},[48,91464,29638],{},[2581,91466,9956],{},[48,91468,34357],{},[4518,91470,91471,91473],{},[51,91472,66],{"stretchy":53},[45,91474,91475,91477],{},[51,91476,4529],{},[2581,91478,4435],{},[48,91480,660],{},[114,91482,91483],{"encoding":116},"(K+\\sigma^2 I)^{-1}y",[23,91485,91487,91508],{"className":91486,"ariaHidden":122},[121],[23,91488,91490,91493,91496,91499,91502,91505],{"className":91489},[126],[23,91491],{"className":91492,"style":131},[130],[23,91494,54],{"className":91495},[140],[23,91497,26922],{"className":91498,"style":37145},[135,136],[23,91500],{"className":91501,"style":392},[148],[23,91503,5106],{"className":91504},[612],[23,91506],{"className":91507,"style":392},[148],[23,91509,91511,91514,91543,91546,91581],{"className":91510},[126],[23,91512],{"className":91513,"style":10665},[130],[23,91515,91517,91520],{"className":91516},[135],[23,91518,29638],{"className":91519,"style":676},[135,136],[23,91521,91523],{"className":91522},[998],[23,91524,91526],{"className":91525},[199],[23,91527,91529],{"className":91528},[204],[23,91530,91532],{"className":91531,"style":4693},[208],[23,91533,91534,91537],{"style":4696},[23,91535],{"className":91536,"style":1015},[216],[23,91538,91540],{"className":91539},[1019,1020,1021,1022],[23,91541,9956],{"className":91542},[135,1022],[23,91544,34357],{"className":91545,"style":454},[135,136],[23,91547,91549,91552],{"className":91548},[169],[23,91550,66],{"className":91551},[169],[23,91553,91555],{"className":91554},[998],[23,91556,91558],{"className":91557},[199],[23,91559,91561],{"className":91560},[204],[23,91562,91564],{"className":91563,"style":4693},[208],[23,91565,91566,91569],{"style":4696},[23,91567],{"className":91568,"style":1015},[216],[23,91570,91572],{"className":91571},[1019,1020,1021,1022],[23,91573,91575,91578],{"className":91574},[135,1022],[23,91576,4529],{"className":91577},[135,1022],[23,91579,4435],{"className":91580},[135,1022],[23,91582,660],{"className":91583,"style":676},[135,136],[3788,91585,91110],{},[3770,91587,91588,91591,91593,91715,91718],{},[3788,91589,91590],{},"平滑样条",[3788,91592,90919],{},[3788,91594,91595],{},[23,91596,91598,91632],{"className":91597,"translate":27},[31],[23,91599,91601],{"className":91600},[35],[37,91602,91603],{"xmlns":39},[42,91604,91605,91629],{},[45,91606,91607,91609,91611,91623],{},[51,91608,1557],{},[51,91610,54],{"stretchy":53},[4518,91612,91613,91615],{},[48,91614,8340],{},[45,91616,91617,91619,91621],{},[51,91618,54],{"stretchy":53},[48,91620,50967],{},[51,91622,66],{"stretchy":53},[4518,91624,91625,91627],{},[51,91626,66],{"stretchy":53},[2581,91628,9956],{},[114,91630,91631],{"encoding":116},"\\int(f^{(m)})^2",[23,91633,91635],{"className":91634,"ariaHidden":122},[121],[23,91636,91638,91642,91645,91648,91686],{"className":91637},[126],[23,91639],{"className":91640,"style":91641},[130],"height:1.1941em;vertical-align:-0.3061em;",[23,91643,1557],{"className":91644,"style":20459},[1786,1790,20458],[23,91646,54],{"className":91647},[140],[23,91649,91651,91654],{"className":91650},[135],[23,91652,8340],{"className":91653,"style":11402},[135,136],[23,91655,91657],{"className":91656},[998],[23,91658,91660],{"className":91659},[199],[23,91661,91663],{"className":91662},[204],[23,91664,91666],{"className":91665,"style":35727},[208],[23,91667,91668,91671],{"style":4696},[23,91669],{"className":91670,"style":1015},[216],[23,91672,91674],{"className":91673},[1019,1020,1021,1022],[23,91675,91677,91680,91683],{"className":91676},[135,1022],[23,91678,54],{"className":91679},[140,1022],[23,91681,50967],{"className":91682},[135,136,1022],[23,91684,66],{"className":91685},[169,1022],[23,91687,91689,91692],{"className":91688},[169],[23,91690,66],{"className":91691},[169],[23,91693,91695],{"className":91694},[998],[23,91696,91698],{"className":91697},[199],[23,91699,91701],{"className":91700},[204],[23,91702,91704],{"className":91703,"style":4693},[208],[23,91705,91706,91709],{"style":4696},[23,91707],{"className":91708,"style":1015},[216],[23,91710,91712],{"className":91711},[1019,1020,1021,1022],[23,91713,9956],{"className":91714},[135,1022],[3788,91716,91717],{},"样条系数",[3788,91719,91110],{},[338,91721,91723],{"id":91722},"六计算代价与大规模近似","六、计算代价与大规模近似",[11,91725,91726,91727,91755,91756,91784,91785,91813,91814,91887,91888,91961,91962,92005,92006,8059,92103,92183],{},"表示定理把无穷维问题降到了 ",[23,91728,91730,91743],{"className":91729,"translate":27},[31],[23,91731,91733],{"className":91732},[35],[37,91734,91735],{"xmlns":39},[42,91736,91737,91741],{},[45,91738,91739],{},[48,91740,4806],{},[114,91742,4806],{"encoding":116},[23,91744,91746],{"className":91745,"ariaHidden":122},[121],[23,91747,91749,91752],{"className":91748},[126],[23,91750],{"className":91751,"style":703},[130],[23,91753,4806],{"className":91754},[135,136]," 维，但 ",[23,91757,91759,91772],{"className":91758,"translate":27},[31],[23,91760,91762],{"className":91761},[35],[37,91763,91764],{"xmlns":39},[42,91765,91766,91770],{},[45,91767,91768],{},[48,91769,4806],{},[114,91771,4806],{"encoding":116},[23,91773,91775],{"className":91774,"ariaHidden":122},[121],[23,91776,91778,91781],{"className":91777},[126],[23,91779],{"className":91780,"style":703},[130],[23,91782,4806],{"className":91783},[135,136]," 是训练样本数，可能很大。核矩阵 ",[23,91786,91788,91801],{"className":91787,"translate":27},[31],[23,91789,91791],{"className":91790},[35],[37,91792,91793],{"xmlns":39},[42,91794,91795,91799],{},[45,91796,91797],{},[48,91798,26922],{},[114,91800,26922],{"encoding":116},[23,91802,91804],{"className":91803,"ariaHidden":122},[121],[23,91805,91807,91810],{"className":91806},[126],[23,91808],{"className":91809,"style":470},[130],[23,91811,26922],{"className":91812,"style":37145},[135,136]," 需要 ",[23,91815,91817,91840],{"className":91816,"translate":27},[31],[23,91818,91820],{"className":91819},[35],[37,91821,91822],{"xmlns":39},[42,91823,91824,91838],{},[45,91825,91826,91828,91830,91836],{},[48,91827,42344],{},[51,91829,54],{"stretchy":53},[4518,91831,91832,91834],{},[48,91833,4806],{},[2581,91835,9956],{},[51,91837,66],{"stretchy":53},[114,91839,83428],{"encoding":116},[23,91841,91843],{"className":91842,"ariaHidden":122},[121],[23,91844,91846,91849,91852,91855,91884],{"className":91845},[126],[23,91847],{"className":91848,"style":10665},[130],[23,91850,42344],{"className":91851,"style":144},[135,136],[23,91853,54],{"className":91854},[140],[23,91856,91858,91861],{"className":91857},[135],[23,91859,4806],{"className":91860},[135,136],[23,91862,91864],{"className":91863},[998],[23,91865,91867],{"className":91866},[199],[23,91868,91870],{"className":91869},[204],[23,91871,91873],{"className":91872,"style":4693},[208],[23,91874,91875,91878],{"style":4696},[23,91876],{"className":91877,"style":1015},[216],[23,91879,91881],{"className":91880},[1019,1020,1021,1022],[23,91882,9956],{"className":91883},[135,1022],[23,91885,66],{"className":91886},[169]," 内存；核岭回归的 Cholesky 分解需要 ",[23,91889,91891,91914],{"className":91890,"translate":27},[31],[23,91892,91894],{"className":91893},[35],[37,91895,91896],{"xmlns":39},[42,91897,91898,91912],{},[45,91899,91900,91902,91904,91910],{},[48,91901,42344],{},[51,91903,54],{"stretchy":53},[4518,91905,91906,91908],{},[48,91907,4806],{},[2581,91909,24923],{},[51,91911,66],{"stretchy":53},[114,91913,83353],{"encoding":116},[23,91915,91917],{"className":91916,"ariaHidden":122},[121],[23,91918,91920,91923,91926,91929,91958],{"className":91919},[126],[23,91921],{"className":91922,"style":10665},[130],[23,91924,42344],{"className":91925,"style":144},[135,136],[23,91927,54],{"className":91928},[140],[23,91930,91932,91935],{"className":91931},[135],[23,91933,4806],{"className":91934},[135,136],[23,91936,91938],{"className":91937},[998],[23,91939,91941],{"className":91940},[199],[23,91942,91944],{"className":91943},[204],[23,91945,91947],{"className":91946,"style":4693},[208],[23,91948,91949,91952],{"style":4696},[23,91950],{"className":91951,"style":1015},[216],[23,91953,91955],{"className":91954},[1019,1020,1021,1022],[23,91956,24923],{"className":91957},[135,1022],[23,91959,66],{"className":91960},[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 ",[23,91963,91965,91984],{"className":91964,"translate":27},[31],[23,91966,91968],{"className":91967},[35],[37,91969,91970],{"xmlns":39},[42,91971,91972,91982],{},[45,91973,91974,91976,91978,91980],{},[48,91975,42344],{},[51,91977,54],{"stretchy":53},[48,91979,4806],{},[51,91981,66],{"stretchy":53},[114,91983,83499],{"encoding":116},[23,91985,91987],{"className":91986,"ariaHidden":122},[121],[23,91988,91990,91993,91996,91999,92002],{"className":91989},[126],[23,91991],{"className":91992,"style":131},[130],[23,91994,42344],{"className":91995,"style":144},[135,136],[23,91997,54],{"className":91998},[140],[23,92000,4806],{"className":92001},[135,136],[23,92003,66],{"className":92004},[169],"（SVM 中是 ",[23,92007,92009,92038],{"className":92008,"translate":27},[31],[23,92010,92012],{"className":92011},[35],[37,92013,92014],{"xmlns":39},[42,92015,92016,92035],{},[45,92017,92018,92020,92022,92033],{},[48,92019,42344],{},[51,92021,54],{"stretchy":53},[898,92023,92024,92026],{},[48,92025,4806],{},[45,92027,92028,92031],{},[48,92029,92030],{"mathvariant":111},"S",[48,92032,10794],{"mathvariant":111},[51,92034,66],{"stretchy":53},[114,92036,92037],{"encoding":116},"O(n_{\\mathrm{SV}})",[23,92039,92041],{"className":92040,"ariaHidden":122},[121],[23,92042,92044,92047,92050,92053,92100],{"className":92043},[126],[23,92045],{"className":92046,"style":131},[130],[23,92048,42344],{"className":92049,"style":144},[135,136],[23,92051,54],{"className":92052},[140],[23,92054,92056,92059],{"className":92055},[135],[23,92057,4806],{"className":92058},[135,136],[23,92060,92062],{"className":92061},[998],[23,92063,92065,92092],{"className":92064},[199,200],[23,92066,92068,92089],{"className":92067},[204],[23,92069,92071],{"className":92070,"style":2026},[208],[23,92072,92073,92076],{"style":2029},[23,92074],{"className":92075,"style":1015},[216],[23,92077,92079],{"className":92078},[1019,1020,1021,1022],[23,92080,92082],{"className":92081},[135,1022],[23,92083,92085],{"className":92084},[135,1022],[23,92086,92088],{"className":92087,"style":7090},[135,5391,1022],"SV",[23,92090,297],{"className":92091},[296],[23,92093,92095],{"className":92094},[204],[23,92096,92098],{"className":92097,"style":2048},[208],[23,92099],{},[23,92101,66],{"className":92102},[169],[23,92104,92106,92128],{"className":92105,"translate":27},[31],[23,92107,92109],{"className":92108},[35],[37,92110,92111],{"xmlns":39},[42,92112,92113,92125],{},[45,92114,92115],{},[898,92116,92117,92119],{},[48,92118,4806],{},[45,92120,92121,92123],{},[48,92122,92030],{"mathvariant":111},[48,92124,10794],{"mathvariant":111},[114,92126,92127],{"encoding":116},"n_{\\mathrm{SV}}",[23,92129,92131],{"className":92130,"ariaHidden":122},[121],[23,92132,92134,92137],{"className":92133},[126],[23,92135],{"className":92136,"style":12734},[130],[23,92138,92140,92143],{"className":92139},[135],[23,92141,4806],{"className":92142},[135,136],[23,92144,92146],{"className":92145},[998],[23,92147,92149,92175],{"className":92148},[199,200],[23,92150,92152,92172],{"className":92151},[204],[23,92153,92155],{"className":92154,"style":2026},[208],[23,92156,92157,92160],{"style":2029},[23,92158],{"className":92159,"style":1015},[216],[23,92161,92163],{"className":92162},[1019,1020,1021,1022],[23,92164,92166],{"className":92165},[135,1022],[23,92167,92169],{"className":92168},[135,1022],[23,92170,92088],{"className":92171,"style":7090},[135,5391,1022],[23,92173,297],{"className":92174},[296],[23,92176,92178],{"className":92177},[204],[23,92179,92181],{"className":92180,"style":2048},[208],[23,92182],{}," 为支持向量数）。",[11,92185,92186,92187,92215],{},"当 ",[23,92188,92190,92203],{"className":92189,"translate":27},[31],[23,92191,92193],{"className":92192},[35],[37,92194,92195],{"xmlns":39},[42,92196,92197,92201],{},[45,92198,92199],{},[48,92200,4806],{},[114,92202,4806],{"encoding":116},[23,92204,92206],{"className":92205,"ariaHidden":122},[121],[23,92207,92209,92212],{"className":92208},[126],[23,92210],{"className":92211,"style":703},[130],[23,92213,4806],{"className":92214},[135,136]," 很大时，常用的近似方法有：",[346,92217,92218,92737,92932,92989],{},[349,92219,92220,92223,92224,92276,92277,92532,92533,67589,92606,63820],{},[18,92221,92222],{},"Nyström 近似","：选 ",[23,92225,92227,92246],{"className":92226,"translate":27},[31],[23,92228,92230],{"className":92229},[35],[37,92231,92232],{"xmlns":39},[42,92233,92234,92243],{},[45,92235,92236,92238,92241],{},[48,92237,50967],{},[51,92239,92240],{},"≪",[48,92242,4806],{},[114,92244,92245],{"encoding":116},"m\\ll n",[23,92247,92249,92267],{"className":92248,"ariaHidden":122},[121],[23,92250,92252,92255,92258,92261,92264],{"className":92251},[126],[23,92253],{"className":92254,"style":513},[130],[23,92256,50967],{"className":92257},[135,136],[23,92259],{"className":92260,"style":149},[148],[23,92262,92240],{"className":92263},[153],[23,92265],{"className":92266,"style":149},[148],[23,92268,92270,92273],{"className":92269},[126],[23,92271],{"className":92272,"style":703},[130],[23,92274,4806],{"className":92275},[135,136]," 个「地标点」，用 ",[23,92278,92280,92338],{"className":92279,"translate":27},[31],[23,92281,92283],{"className":92282},[35],[37,92284,92285],{"xmlns":39},[42,92286,92287,92335],{},[45,92288,92289,92291,92293,92305,92323],{},[48,92290,26922],{},[51,92292,29070],{},[898,92294,92295,92297],{},[48,92296,26922],{},[45,92298,92299,92301,92303],{},[48,92300,4806],{},[51,92302,747],{"separator":122},[48,92304,50967],{},[19141,92306,92307,92309,92317],{},[48,92308,26922],{},[45,92310,92311,92313,92315],{},[48,92312,50967],{},[51,92314,747],{"separator":122},[48,92316,50967],{},[45,92318,92319,92321],{},[51,92320,4529],{},[2581,92322,4435],{},[898,92324,92325,92327],{},[48,92326,26922],{},[45,92328,92329,92331,92333],{},[48,92330,50967],{},[51,92332,747],{"separator":122},[48,92334,4806],{},[114,92336,92337],{"encoding":116},"K\\approx K_{n,m}K_{m,m}^{-1}K_{m,n}",[23,92339,92341,92359],{"className":92340,"ariaHidden":122},[121],[23,92342,92344,92347,92350,92353,92356],{"className":92343},[126],[23,92345],{"className":92346,"style":470},[130],[23,92348,26922],{"className":92349,"style":37145},[135,136],[23,92351],{"className":92352,"style":149},[148],[23,92354,29070],{"className":92355},[153],[23,92357],{"className":92358,"style":149},[148],[23,92360,92362,92366,92415,92483],{"className":92361},[126],[23,92363],{"className":92364,"style":92365},[130],"height:1.1972em;vertical-align:-0.3831em;",[23,92367,92369,92372],{"className":92368},[135],[23,92370,26922],{"className":92371,"style":37145},[135,136],[23,92373,92375],{"className":92374},[998],[23,92376,92378,92407],{"className":92377},[199,200],[23,92379,92381,92404],{"className":92380},[204],[23,92382,92384],{"className":92383,"style":6499},[208],[23,92385,92386,92389],{"style":63373},[23,92387],{"className":92388,"style":1015},[216],[23,92390,92392],{"className":92391},[1019,1020,1021,1022],[23,92393,92395,92398,92401],{"className":92394},[135,1022],[23,92396,4806],{"className":92397},[135,136,1022],[23,92399,747],{"className":92400},[824,1022],[23,92402,50967],{"className":92403},[135,136,1022],[23,92405,297],{"className":92406},[296],[23,92408,92410],{"className":92409},[204],[23,92411,92413],{"className":92412,"style":14296},[208],[23,92414],{},[23,92416,92418,92421],{"className":92417},[135],[23,92419,26922],{"className":92420,"style":37145},[135,136],[23,92422,92424],{"className":92423},[998],[23,92425,92427,92474],{"className":92426},[199,200],[23,92428,92430,92471],{"className":92429},[204],[23,92431,92433,92454],{"className":92432,"style":4693},[208],[23,92434,92436,92439],{"style":92435},"top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;",[23,92437],{"className":92438,"style":1015},[216],[23,92440,92442],{"className":92441},[1019,1020,1021,1022],[23,92443,92445,92448,92451],{"className":92444},[135,1022],[23,92446,50967],{"className":92447},[135,136,1022],[23,92449,747],{"className":92450},[824,1022],[23,92452,50967],{"className":92453},[135,136,1022],[23,92455,92456,92459],{"style":4696},[23,92457],{"className":92458,"style":1015},[216],[23,92460,92462],{"className":92461},[1019,1020,1021,1022],[23,92463,92465,92468],{"className":92464},[135,1022],[23,92466,4529],{"className":92467},[135,1022],[23,92469,4435],{"className":92470},[135,1022],[23,92472,297],{"className":92473},[296],[23,92475,92477],{"className":92476},[204],[23,92478,92481],{"className":92479,"style":92480},[208],"height:0.3831em;",[23,92482],{},[23,92484,92486,92489],{"className":92485},[135],[23,92487,26922],{"className":92488,"style":37145},[135,136],[23,92490,92492],{"className":92491},[998],[23,92493,92495,92524],{"className":92494},[199,200],[23,92496,92498,92521],{"className":92497},[204],[23,92499,92501],{"className":92500,"style":6499},[208],[23,92502,92503,92506],{"style":63373},[23,92504],{"className":92505,"style":1015},[216],[23,92507,92509],{"className":92508},[1019,1020,1021,1022],[23,92510,92512,92515,92518],{"className":92511},[135,1022],[23,92513,50967],{"className":92514},[135,136,1022],[23,92516,747],{"className":92517},[824,1022],[23,92519,4806],{"className":92520},[135,136,1022],[23,92522,297],{"className":92523},[296],[23,92525,92527],{"className":92526},[204],[23,92528,92530],{"className":92529,"style":14296},[208],[23,92531],{}," 逼近核矩阵，代价从 ",[23,92534,92536,92559],{"className":92535,"translate":27},[31],[23,92537,92539],{"className":92538},[35],[37,92540,92541],{"xmlns":39},[42,92542,92543,92557],{},[45,92544,92545,92547,92549,92555],{},[48,92546,42344],{},[51,92548,54],{"stretchy":53},[4518,92550,92551,92553],{},[48,92552,4806],{},[2581,92554,24923],{},[51,92556,66],{"stretchy":53},[114,92558,83353],{"encoding":116},[23,92560,92562],{"className":92561,"ariaHidden":122},[121],[23,92563,92565,92568,92571,92574,92603],{"className":92564},[126],[23,92566],{"className":92567,"style":10665},[130],[23,92569,42344],{"className":92570,"style":144},[135,136],[23,92572,54],{"className":92573},[140],[23,92575,92577,92580],{"className":92576},[135],[23,92578,4806],{"className":92579},[135,136],[23,92581,92583],{"className":92582},[998],[23,92584,92586],{"className":92585},[199],[23,92587,92589],{"className":92588},[204],[23,92590,92592],{"className":92591,"style":4693},[208],[23,92593,92594,92597],{"style":4696},[23,92595],{"className":92596,"style":1015},[216],[23,92598,92600],{"className":92599},[1019,1020,1021,1022],[23,92601,24923],{"className":92602},[135,1022],[23,92604,66],{"className":92605},[169],[23,92607,92609,92643],{"className":92608,"translate":27},[31],[23,92610,92612],{"className":92611},[35],[37,92613,92614],{"xmlns":39},[42,92615,92616,92640],{},[45,92617,92618,92620,92622,92624,92630,92632,92638],{},[48,92619,42344],{},[51,92621,54],{"stretchy":53},[48,92623,4806],{},[4518,92625,92626,92628],{},[48,92627,50967],{},[2581,92629,9956],{},[51,92631,5106],{},[4518,92633,92634,92636],{},[48,92635,50967],{},[2581,92637,24923],{},[51,92639,66],{"stretchy":53},[114,92641,92642],{"encoding":116},"O(nm^2+m^3)",[23,92644,92646,92699],{"className":92645,"ariaHidden":122},[121],[23,92647,92649,92652,92655,92658,92661,92690,92693,92696],{"className":92648},[126],[23,92650],{"className":92651,"style":10665},[130],[23,92653,42344],{"className":92654,"style":144},[135,136],[23,92656,54],{"className":92657},[140],[23,92659,4806],{"className":92660},[135,136],[23,92662,92664,92667],{"className":92663},[135],[23,92665,50967],{"className":92666},[135,136],[23,92668,92670],{"className":92669},[998],[23,92671,92673],{"className":92672},[199],[23,92674,92676],{"className":92675},[204],[23,92677,92679],{"className":92678,"style":4693},[208],[23,92680,92681,92684],{"style":4696},[23,92682],{"className":92683,"style":1015},[216],[23,92685,92687],{"className":92686},[1019,1020,1021,1022],[23,92688,9956],{"className":92689},[135,1022],[23,92691],{"className":92692,"style":392},[148],[23,92694,5106],{"className":92695},[612],[23,92697],{"className":92698,"style":392},[148],[23,92700,92702,92705,92734],{"className":92701},[126],[23,92703],{"className":92704,"style":10665},[130],[23,92706,92708,92711],{"className":92707},[135],[23,92709,50967],{"className":92710},[135,136],[23,92712,92714],{"className":92713},[998],[23,92715,92717],{"className":92716},[199],[23,92718,92720],{"className":92719},[204],[23,92721,92723],{"className":92722,"style":4693},[208],[23,92724,92725,92728],{"style":4696},[23,92726],{"className":92727,"style":1015},[216],[23,92729,92731],{"className":92730},[1019,1020,1021,1022],[23,92732,24923],{"className":92733},[135,1022],[23,92735,66],{"className":92736},[169],[349,92738,92739,92742,92743,92771,92772,92800,92801,63820],{},[18,92740,92741],{},"随机傅里叶特征","：用 ",[23,92744,92746,92759],{"className":92745,"translate":27},[31],[23,92747,92749],{"className":92748},[35],[37,92750,92751],{"xmlns":39},[42,92752,92753,92757],{},[45,92754,92755],{},[48,92756,63],{},[114,92758,63],{"encoding":116},[23,92760,92762],{"className":92761,"ariaHidden":122},[121],[23,92763,92765,92768],{"className":92764},[126],[23,92766],{"className":92767,"style":470},[130],[23,92769,63],{"className":92770,"style":144},[135,136]," 个随机特征近似平移不变核，把核方法转成 ",[23,92773,92775,92788],{"className":92774,"translate":27},[31],[23,92776,92778],{"className":92777},[35],[37,92779,92780],{"xmlns":39},[42,92781,92782,92786],{},[45,92783,92784],{},[48,92785,63],{},[114,92787,63],{"encoding":116},[23,92789,92791],{"className":92790,"ariaHidden":122},[121],[23,92792,92794,92797],{"className":92793},[126],[23,92795],{"className":92796,"style":470},[130],[23,92798,63],{"className":92799,"style":144},[135,136]," 维的线性方法，代价约为 ",[23,92802,92804,92838],{"className":92803,"translate":27},[31],[23,92805,92807],{"className":92806},[35],[37,92808,92809],{"xmlns":39},[42,92810,92811,92835],{},[45,92812,92813,92815,92817,92819,92825,92827,92833],{},[48,92814,42344],{},[51,92816,54],{"stretchy":53},[48,92818,4806],{},[4518,92820,92821,92823],{},[48,92822,63],{},[2581,92824,9956],{},[51,92826,5106],{},[4518,92828,92829,92831],{},[48,92830,63],{},[2581,92832,24923],{},[51,92834,66],{"stretchy":53},[114,92836,92837],{"encoding":116},"O(nD^2+D^3)",[23,92839,92841,92894],{"className":92840,"ariaHidden":122},[121],[23,92842,92844,92847,92850,92853,92856,92885,92888,92891],{"className":92843},[126],[23,92845],{"className":92846,"style":10665},[130],[23,92848,42344],{"className":92849,"style":144},[135,136],[23,92851,54],{"className":92852},[140],[23,92854,4806],{"className":92855},[135,136],[23,92857,92859,92862],{"className":92858},[135],[23,92860,63],{"className":92861,"style":144},[135,136],[23,92863,92865],{"className":92864},[998],[23,92866,92868],{"className":92867},[199],[23,92869,92871],{"className":92870},[204],[23,92872,92874],{"className":92873,"style":4693},[208],[23,92875,92876,92879],{"style":4696},[23,92877],{"className":92878,"style":1015},[216],[23,92880,92882],{"className":92881},[1019,1020,1021,1022],[23,92883,9956],{"className":92884},[135,1022],[23,92886],{"className":92887,"style":392},[148],[23,92889,5106],{"className":92890},[612],[23,92892],{"className":92893,"style":392},[148],[23,92895,92897,92900,92929],{"className":92896},[126],[23,92898],{"className":92899,"style":10665},[130],[23,92901,92903,92906],{"className":92902},[135],[23,92904,63],{"className":92905,"style":144},[135,136],[23,92907,92909],{"className":92908},[998],[23,92910,92912],{"className":92911},[199],[23,92913,92915],{"className":92914},[204],[23,92916,92918],{"className":92917,"style":4693},[208],[23,92919,92920,92923],{"style":4696},[23,92921],{"className":92922,"style":1015},[216],[23,92924,92926],{"className":92925},[1019,1020,1021,1022],[23,92927,24923],{"className":92928},[135,1022],[23,92930,66],{"className":92931},[169],[349,92933,92934,92937,92938,92988],{},[18,92935,92936],{},"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 ",[23,92939,92941,92958],{"className":92940,"translate":27},[31],[23,92942,92944],{"className":92943},[35],[37,92945,92946],{"xmlns":39},[42,92947,92948,92956],{},[45,92949,92950,92952,92954],{},[48,92951,4806],{},[51,92953,563],{},[48,92955,4806],{},[114,92957,60659],{"encoding":116},[23,92959,92961,92979],{"className":92960,"ariaHidden":122},[121],[23,92962,92964,92967,92970,92973,92976],{"className":92963},[126],[23,92965],{"className":92966,"style":5445},[130],[23,92968,4806],{"className":92969},[135,136],[23,92971],{"className":92972,"style":392},[148],[23,92974,563],{"className":92975},[612],[23,92977],{"className":92978,"style":392},[148],[23,92980,92982,92985],{"className":92981},[126],[23,92983],{"className":92984,"style":703},[130],[23,92986,4806],{"className":92987},[135,136]," 矩阵；",[349,92990,92991,92994],{},[18,92992,92993],{},"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。",[11,92996,92997,92998,93026],{},"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 ",[23,92999,93001,93014],{"className":93000,"translate":27},[31],[23,93002,93004],{"className":93003},[35],[37,93005,93006],{"xmlns":39},[42,93007,93008,93012],{},[45,93009,93010],{},[48,93011,4806],{},[114,93013,4806],{"encoding":116},[23,93015,93017],{"className":93016,"ariaHidden":122},[121],[23,93018,93020,93023],{"className":93019},[126],[23,93021],{"className":93022,"style":703},[130],[23,93024,4806],{"className":93025},[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。",[338,93028,93030],{"id":93029},"七表示定理与深度学习","七、表示定理与深度学习",[11,93032,93033,93034,93037],{},"表示定理并没有随着深度学习的兴起而过时。相反，它在",[18,93035,93036],{},"神经正切核","（Neural Tangent Kernel, NTK）的框架里重新出现。",[11,93039,93040],{},"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：",[23,93042,93044],{"className":93043,"translate":27},[26],[23,93045,93047,93125],{"className":93046,"translate":27},[31],[23,93048,93050],{"className":93049},[35],[37,93051,93052],{"xmlns":39,"display":40},[42,93053,93054,93122],{},[45,93055,93056,93058,93060,93062,93064,93066,93078,93080,93082,93084,93086,93088,93090,93102,93104,93106,93108,93118,93120],{},[48,93057,8340],{},[51,93059,54],{"stretchy":53},[48,93061,753],{},[51,93063,66],{"stretchy":53},[51,93065,69],{},[898,93067,93068,93070],{},[48,93069,4836],{},[45,93071,93072,93074,93076],{},[48,93073,12805],{"mathvariant":111},[48,93075,20707],{"mathvariant":111},[48,93077,26922],{"mathvariant":111},[51,93079,54],{"stretchy":53},[48,93081,753],{},[51,93083,747],{"separator":122},[48,93085,434],{},[51,93087,66],{"stretchy":53},[51,93089,54],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[898,93091,93092,93094],{},[48,93093,26922],{},[45,93095,93096,93098,93100],{},[48,93097,12805],{"mathvariant":111},[48,93099,20707],{"mathvariant":111},[48,93101,26922],{"mathvariant":111},[51,93103,5106],{},[48,93105,61761],{},[48,93107,34357],{},[4518,93109,93110,93112],{},[51,93111,66],{"fence":122,"stretchy":122,"minsize":923,"maxsize":923},[45,93113,93114,93116],{},[51,93115,4529],{},[2581,93117,4435],{},[48,93119,660],{},[51,93121,747],{"separator":122},[114,93123,93124],{"encoding":116},"f(x)=k_{\\mathrm{NTK}}(x,X)\\bigl(K_{\\mathrm{NTK}}+\\lambda I\\bigr)^{-1}y,",[23,93126,93128,93155,93288],{"className":93127,"ariaHidden":122},[121],[23,93129,93131,93134,93137,93140,93143,93146,93149,93152],{"className":93130},[126],[23,93132],{"className":93133,"style":131},[130],[23,93135,8340],{"className":93136,"style":11402},[135,136],[23,93138,54],{"className":93139},[140],[23,93141,753],{"className":93142},[135,136],[23,93144,66],{"className":93145},[169],[23,93147],{"className":93148,"style":149},[148],[23,93150,69],{"className":93151},[153],[23,93153],{"className":93154,"style":149},[148],[23,93156,93158,93161,93209,93212,93215,93218,93221,93224,93227,93233,93279,93282,93285],{"className":93157},[126],[23,93159],{"className":93160,"style":2213},[130],[23,93162,93164,93167],{"className":93163},[135],[23,93165,4836],{"className":93166,"style":4851},[135,136],[23,93168,93170],{"className":93169},[998],[23,93171,93173,93201],{"className":93172},[199,200],[23,93174,93176,93198],{"className":93175},[204],[23,93177,93179],{"className":93178,"style":2026},[208],[23,93180,93182,93185],{"style":93181},"top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;",[23,93183],{"className":93184,"style":1015},[216],[23,93186,93188],{"className":93187},[1019,1020,1021,1022],[23,93189,93191],{"className":93190},[135,1022],[23,93192,93194],{"className":93193},[135,1022],[23,93195,93197],{"className":93196},[135,5391,1022],"NTK",[23,93199,297],{"className":93200},[296],[23,93202,93204],{"className":93203},[204],[23,93205,93207],{"className":93206,"style":2048},[208],[23,93208],{},[23,93210,54],{"className":93211},[140],[23,93213,753],{"className":93214},[135,136],[23,93216,747],{"className":93217},[824],[23,93219],{"className":93220,"style":280},[148],[23,93222,434],{"className":93223,"style":454},[135,136],[23,93225,66],{"className":93226},[169],[23,93228,93230],{"className":93229},[140],[23,93231,54],{"className":93232},[1066,1067],[23,93234,93236,93239],{"className":93235},[135],[23,93237,26922],{"className":93238,"style":37145},[135,136],[23,93240,93242],{"className":93241},[998],[23,93243,93245,93271],{"className":93244},[199,200],[23,93246,93248,93268],{"className":93247},[204],[23,93249,93251],{"className":93250,"style":2026},[208],[23,93252,93253,93256],{"style":63373},[23,93254],{"className":93255,"style":1015},[216],[23,93257,93259],{"className":93258},[1019,1020,1021,1022],[23,93260,93262],{"className":93261},[135,1022],[23,93263,93265],{"className":93264},[135,1022],[23,93266,93197],{"className":93267},[135,5391,1022],[23,93269,297],{"className":93270},[296],[23,93272,93274],{"className":93273},[204],[23,93275,93277],{"className":93276,"style":2048},[208],[23,93278],{},[23,93280],{"className":93281,"style":392},[148],[23,93283,5106],{"className":93284},[612],[23,93286],{"className":93287,"style":392},[148],[23,93289,93291,93294,93297,93300,93338,93341],{"className":93290},[126],[23,93292],{"className":93293,"style":34177},[130],[23,93295,61761],{"className":93296},[135,136],[23,93298,34357],{"className":93299,"style":454},[135,136],[23,93301,93303,93309],{"className":93302},[169],[23,93304,93306],{"className":93305},[169],[23,93307,66],{"className":93308},[1066,1067],[23,93310,93312],{"className":93311},[998],[23,93313,93315],{"className":93314},[199],[23,93316,93318],{"className":93317},[204],[23,93319,93321],{"className":93320,"style":11192},[208],[23,93322,93323,93326],{"style":11195},[23,93324],{"className":93325,"style":1015},[216],[23,93327,93329],{"className":93328},[1019,1020,1021,1022],[23,93330,93332,93335],{"className":93331},[135,1022],[23,93333,4529],{"className":93334},[135,1022],[23,93336,4435],{"className":93337},[135,1022],[23,93339,660],{"className":93340,"style":676},[135,136],[23,93342,747],{"className":93343},[824],[11,93345,24433,93346,93428,93429,93511],{},[23,93347,93349,93373],{"className":93348,"translate":27},[31],[23,93350,93352],{"className":93351},[35],[37,93353,93354],{"xmlns":39},[42,93355,93356,93370],{},[45,93357,93358],{},[898,93359,93360,93362],{},[48,93361,4836],{},[45,93363,93364,93366,93368],{},[48,93365,12805],{"mathvariant":111},[48,93367,20707],{"mathvariant":111},[48,93369,26922],{"mathvariant":111},[114,93371,93372],{"encoding":116},"k_{\\mathrm{NTK}}",[23,93374,93376],{"className":93375,"ariaHidden":122},[121],[23,93377,93379,93382],{"className":93378},[126],[23,93380],{"className":93381,"style":46086},[130],[23,93383,93385,93388],{"className":93384},[135],[23,93386,4836],{"className":93387,"style":4851},[135,136],[23,93389,93391],{"className":93390},[998],[23,93392,93394,93420],{"className":93393},[199,200],[23,93395,93397,93417],{"className":93396},[204],[23,93398,93400],{"className":93399,"style":2026},[208],[23,93401,93402,93405],{"style":93181},[23,93403],{"className":93404,"style":1015},[216],[23,93406,93408],{"className":93407},[1019,1020,1021,1022],[23,93409,93411],{"className":93410},[135,1022],[23,93412,93414],{"className":93413},[135,1022],[23,93415,93197],{"className":93416},[135,5391,1022],[23,93418,297],{"className":93419},[296],[23,93421,93423],{"className":93422},[204],[23,93424,93426],{"className":93425,"style":2048},[208],[23,93427],{}," 是网络对应的神经正切核，",[23,93430,93432,93456],{"className":93431,"translate":27},[31],[23,93433,93435],{"className":93434},[35],[37,93436,93437],{"xmlns":39},[42,93438,93439,93453],{},[45,93440,93441],{},[898,93442,93443,93445],{},[48,93444,26922],{},[45,93446,93447,93449,93451],{},[48,93448,12805],{"mathvariant":111},[48,93450,20707],{"mathvariant":111},[48,93452,26922],{"mathvariant":111},[114,93454,93455],{"encoding":116},"K_{\\mathrm{NTK}}",[23,93457,93459],{"className":93458,"ariaHidden":122},[121],[23,93460,93462,93465],{"className":93461},[126],[23,93463],{"className":93464,"style":42831},[130],[23,93466,93468,93471],{"className":93467},[135],[23,93469,26922],{"className":93470,"style":37145},[135,136],[23,93472,93474],{"className":93473},[998],[23,93475,93477,93503],{"className":93476},[199,200],[23,93478,93480,93500],{"className":93479},[204],[23,93481,93483],{"className":93482,"style":2026},[208],[23,93484,93485,93488],{"style":63373},[23,93486],{"className":93487,"style":1015},[216],[23,93489,93491],{"className":93490},[1019,1020,1021,1022],[23,93492,93494],{"className":93493},[135,1022],[23,93495,93497],{"className":93496},[135,1022],[23,93498,93197],{"className":93499},[135,5391,1022],[23,93501,297],{"className":93502},[296],[23,93504,93506],{"className":93505},[204],[23,93507,93509],{"className":93508,"style":2048},[208],[23,93510],{}," 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。",[11,93513,93514],{},"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。",[338,93516,93518],{"id":93517},"九结语","九、结语",[11,93520,93521,93522,93525],{},"表示定理的核心是一句几何话：",[18,93523,93524],{},"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。",[11,93527,93528,93529,93643],{},"这个论证只用了两件事：再生性 ",[23,93530,93532,93574],{"className":93531,"translate":27},[31],[23,93533,93535],{"className":93534},[35],[37,93536,93537],{"xmlns":39},[42,93538,93539,93571],{},[45,93540,93541,93543,93545,93547,93549,93551,93553,93555,93557,93559,93561,93563,93565,93567,93569],{},[48,93542,8340],{},[51,93544,54],{"stretchy":53},[48,93546,753],{},[51,93548,66],{"stretchy":53},[51,93550,69],{},[51,93552,61189],{"stretchy":53},[48,93554,8340],{},[51,93556,747],{"separator":122},[48,93558,4836],{},[51,93560,54],{"stretchy":53},[51,93562,5758],{},[51,93564,747],{"separator":122},[48,93566,753],{},[51,93568,66],{"stretchy":53},[51,93570,61214],{"stretchy":53},[114,93572,93573],{"encoding":116},"f(x)=\\langle f,k(\\cdot,x)\\rangle",[23,93575,93577,93604],{"className":93576,"ariaHidden":122},[121],[23,93578,93580,93583,93586,93589,93592,93595,93598,93601],{"className":93579},[126],[23,93581],{"className":93582,"style":131},[130],[23,93584,8340],{"className":93585,"style":11402},[135,136],[23,93587,54],{"className":93588},[140],[23,93590,753],{"className":93591},[135,136],[23,93593,66],{"className":93594},[169],[23,93596],{"className":93597,"style":149},[148],[23,93599,69],{"className":93600},[153],[23,93602],{"className":93603,"style":149},[148],[23,93605,93607,93610,93613,93616,93619,93622,93625,93628,93631,93634,93637,93640],{"className":93606},[126],[23,93608],{"className":93609,"style":131},[130],[23,93611,61189],{"className":93612},[140],[23,93614,8340],{"className":93615,"style":11402},[135,136],[23,93617,747],{"className":93618},[824],[23,93620],{"className":93621,"style":280},[148],[23,93623,4836],{"className":93624,"style":4851},[135,136],[23,93626,54],{"className":93627},[140],[23,93629,5758],{"className":93630},[135],[23,93632,747],{"className":93633},[824],[23,93635],{"className":93636,"style":280},[148],[23,93638,753],{"className":93639},[135,136],[23,93641,61354],{"className":93642},[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。",[11,93645,93646,93647],{},"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。",[18,93648,93649,93650,93678,93679,93707],{},"无穷维并没有消失，它只是被 ",[23,93651,93653,93666],{"className":93652,"translate":27},[31],[23,93654,93656],{"className":93655},[35],[37,93657,93658],{"xmlns":39},[42,93659,93660,93664],{},[45,93661,93662],{},[48,93663,4806],{},[114,93665,4806],{"encoding":116},[23,93667,93669],{"className":93668,"ariaHidden":122},[121],[23,93670,93672,93675],{"className":93671},[126],[23,93673],{"className":93674,"style":703},[130],[23,93676,4806],{"className":93677},[135,136]," 个训练点压成了 ",[23,93680,93682,93695],{"className":93681,"translate":27},[31],[23,93683,93685],{"className":93684},[35],[37,93686,93687],{"xmlns":39},[42,93688,93689,93693],{},[45,93690,93691],{},[48,93692,4806],{},[114,93694,4806],{"encoding":116},[23,93696,93698],{"className":93697,"ariaHidden":122},[121],[23,93699,93701,93704],{"className":93700},[126],[23,93702],{"className":93703,"style":703},[130],[23,93705,4806],{"className":93706},[135,136]," 维。",{"title":60557,"searchDepth":60558,"depth":60558,"links":93709},[93710,93711,93716,93722,93727,93735,93736,93737],{"id":60696,"depth":60558,"text":60697},{"id":62374,"depth":60558,"text":62375,"children":93712},[93713,93714,93715],{"id":62378,"depth":60564,"text":62379},{"id":65733,"depth":60564,"text":65734},{"id":66571,"depth":60564,"text":66572},{"id":67651,"depth":60558,"text":67652,"children":93717},[93718,93719,93720,93721],{"id":67655,"depth":60564,"text":67656},{"id":69755,"depth":60564,"text":69756},{"id":73976,"depth":60564,"text":73977},{"id":76484,"depth":60564,"text":76485},{"id":78042,"depth":60558,"text":78043,"children":93723},[93724,93725,93726],{"id":78046,"depth":60564,"text":78047},{"id":78423,"depth":60564,"text":78424},{"id":80496,"depth":60564,"text":80497},{"id":81123,"depth":60558,"text":81124,"children":93728},[93729,93730,93731,93732,93733,93734],{"id":81127,"depth":60564,"text":81128},{"id":83524,"depth":60564,"text":83525},{"id":87342,"depth":60564,"text":87343},{"id":88689,"depth":60564,"text":88690},{"id":89890,"depth":60564,"text":89891},{"id":90888,"depth":60564,"text":90889},{"id":91722,"depth":60558,"text":91723},{"id":93029,"depth":60558,"text":93030},{"id":93517,"depth":60558,"text":93518},"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":60633,"description":93738},"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。",[60627],"2025-09-11T21:00:12+08:00","wFRdXtfB52qEpuspNQ6_5BNMWT6yuyp37_5ikKAOpvU",{"id":93748,"title":93749,"body":93750,"description":60557,"draft":60617,"enableComment":60618,"extension":60619,"image":60620,"important":60617,"location":60620,"meta":93827,"navigation":60618,"ogImage":60620,"path":93828,"seo":93829,"stem":93830,"summary":93831,"tags":93832,"time":93834,"__hash__":93835},"blog\u002Fblog\u002F2025\u002F2025-08-17-the-lamentable-programmers.md","可叹的程序员们",{"type":8,"value":93751,"toc":93825},[93752,93757,93760,93763,93766,93769,93772,93775,93778,93781,93784,93787,93790,93793,93796,93799,93802,93805,93808,93811],[6857,93753,93754],{},[11,93755,93756],{},"个人奋斗固然重要，但也必须顺应历史的进程。",[11,93758,93759],{},"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。",[11,93761,93762],{},"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。",[11,93764,93765],{},"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。",[11,93767,93768],{},"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。",[11,93770,93771],{},"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。",[11,93773,93774],{},"历史终将证明，繁荣终究是短暂的。",[11,93776,93777],{},"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。",[11,93779,93780],{},"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。」",[11,93782,93783],{},"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。",[11,93785,93786],{},"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？",[11,93788,93789],{},"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。",[11,93791,93792],{},"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？",[11,93794,93795],{},"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。",[11,93797,93798],{},"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。",[11,93800,93801],{},"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。",[11,93803,93804],{},"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。",[11,93806,93807],{},"属于程序员的时代如今已渐近尾声，我们该何去何从？😔",[11,93809,93810],{},"最后，以下是我分享的一点心得：",[346,93812,93813,93816,93819,93822],{},[349,93814,93815],{},"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。",[349,93817,93818],{},"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。",[349,93820,93821],{},"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。",[349,93823,93824],{},"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":60557,"searchDepth":60558,"depth":60558,"links":93826},[],{},"\u002Fblog\u002F2025\u002F2025-08-17-the-lamentable-programmers",{"title":93749,"description":60557},"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。",[93833],"thoughts","2025-08-17T00:00:00+08:00","4uHo_S5Roo5bILXrENJLvY3R7QenxS0G4TWTljLUdBg",1792504628991]