268:
36:
1945:
1233:
with respect to the above partial order. An inadmissible rule is not preferred (except for reasons of simplicity or computational efficiency), since by definition there is some other rule that will achieve equal or lower risk for
998:
1562:
741:
2811:
technique with respect to a mean-squared-error loss function. Thus least squares estimation is not an admissible estimation procedure in this context. Some others of the standard estimates associated with the
618:
1164:
689:
323:
such that there is no other rule that is always "better" than it (or at least sometimes better and never worse), in the precise sense of "better" defined below. This concept is analogous to
2599:
873:
2692:
1755:
506:
1805:
571:
2339:
2064:
543:
2391:
2239:
At first, this may appear rather different from the Bayes rule approach of the previous section, not a generalization. However, notice that the Bayes risk already averages over
1624:
827:
771:
648:
797:
2753:
2633:
2311:
2203:
2027:
1661:
1470:
1377:
1074:
2544:
2491:
2283:
2234:
2168:
474:
450:
405:
381:
2775:
2460:
2438:
2361:
2259:
2115:
1974:
1683:
1587:
1439:
1413:
1327:
1305:
1279:
1257:
1216:
1186:
1101:
1045:
1023:
426:
357:
2714:
2655:
2569:
2513:
2416:
2137:
2093:
1996:
1827:
1777:
1719:
2515:
for which a finite-expected-loss action does exist. In addition, a generalized Bayes rule may be desirable because it must choose a minimum-expected-loss action
1840:
2396:
Then why is the notion of generalized Bayes rule an improvement? It is indeed equivalent to the notion of Bayes rule when a Bayes rule exists and all
897:
2724:
According to the complete class theorems, under mild conditions every admissible rule is a (generalized) Bayes rule (with respect to some prior
1478:
298:
702:
89:
2787:
Conversely, while Bayes rules with respect to proper priors are virtually always admissible, generalized Bayes rules corresponding to
583:
1106:
171:
2363:
minimizes this expectation of expected loss (i.e., is a Bayes rule) if and only if it minimizes the expected loss for each
653:
2958:
2939:
2917:
2898:
2848:
291:
254:
1281:
is admissible does not mean it is a good rule to use. Being admissible means there is no other single rule that is
181:
207:
84:
1389:. That is, it is our believed probability distribution on the states of nature, prior to observing data. For a
2574:
832:
145:
1694:
2660:
1728:
479:
1782:
548:
2977:
284:
176:
114:
2635:. In this case, the Bayes risk is not even well-defined, nor is there any well-defined distribution over
2316:
2032:
511:
2366:
1592:
802:
746:
623:
312:
776:
2982:
2727:
2607:
2288:
2177:
2001:
1635:
1444:
1351:
166:
135:
2804:
2788:
1050:
228:
109:
2518:
2465:
2264:
2208:
2142:
455:
431:
386:
362:
1193:
1077:
249:
161:
1307:
that occur in practice. (The Bayes risk discussed below is a way of explicitly considering which
2807:
is a nonlinear estimator of the mean of
Gaussian random vectors and can be shown to dominate the
2758:
2443:
2421:
2418:
have positive probability. However, no Bayes rule exists if the Bayes risk is infinite (for all
2344:
2242:
2098:
1957:
1666:
1570:
1422:
1396:
1310:
1288:
1262:
1240:
1199:
1169:
1084:
1028:
1006:
2808:
1338:
1225:(with respect to the loss function) if and only if no other rule dominates it; otherwise it is
578:
410:
341:
140:
2205:. There may be more than one generalized Bayes rule, since there may be multiple choices of
1390:
43:
2697:
2638:
2552:
2496:
2399:
2120:
2076:
1979:
1831:
1810:
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1702:
1380:
875:. (It is possible though unconventional to recast the following definitions in terms of a
223:
104:
74:
8:
2813:
55:
47:
27:
2792:
1940:{\displaystyle \rho (\pi ,\delta \mid x)=\operatorname {E} _{\pi (\theta \mid x)}.\,\!}
1385:
272:
197:
99:
69:
1663:. There may be more than one such Bayes rule. If the Bayes risk is infinite for all
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79:
51:
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94:
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in
Bayesian fashion, and the Bayes risk may be recovered as the expectation over
1628:
1285:
as good or better – but other admissible rules might achieve lower risk for most
1230:
130:
2927:
1725:. Whereas the frequentist approach (i.e., risk) averages over possible samples
889:
993:{\displaystyle R(\theta ,\delta )=\operatorname {E} _{F(x\mid \theta )}}.\,\!}
2971:
2836:
2571:, whereas a Bayes rule would be allowed to deviate from this policy on a set
884:
696:
320:
244:
2778:
2440:). In this case it is still useful to define a generalized Bayes rule
1557:{\displaystyle r(\pi ,\delta )=\operatorname {E} _{\pi (\theta )}.\,\!}
736:{\displaystyle L:\Theta \times {\mathcal {A}}\rightarrow \mathbb {R} }
2604:
More important, it is sometimes convenient to use an improper prior
2716:, so that it is still possible to define a generalized Bayes rule.
613:{\displaystyle \delta :{\mathcal {X}}\rightarrow {\mathcal {A}}}
1807:. Thus, the Bayesian approach is to consider for our observed
1379:
be a probability distribution on the states of nature. From a
2784:
it is sufficient to consider only (generalized) Bayes rules.
2719:
1159:{\displaystyle R(\theta ,\delta ^{*})\leq R(\theta ,\delta )}
35:
743:, which specifies the loss we would incur by taking action
2694:—and hence the expected loss—may be well-defined for each
1699:
In the
Bayesian approach to decision theory, the observed
1332:
545:
and therefore provides evidence about the state of nature
2462:, which at least chooses a minimum-expected-loss action
799:. Usually we will take this action after observing data
2170:
that minimizes the expected loss. This is known as a
2073:
Having made explicit the expected loss for each given
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2211:
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2145:
2123:
2101:
2079:
2035:
2004:
1982:
1960:
1843:
1813:
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1763:
1731:
1705:
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1638:
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1573:
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389:
365:
344:
2948:
2755:—possibly an improper one—that favors distributions
2820:when the population mean and variance are unknown.
1695:
Bayes estimator § Generalized Bayes estimators
16:
Type of "good" decision rule in
Bayesian statistics
2769:
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821:
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420:
399:
375:
351:
2910:Statistical Decision Theory and Bayesian Analysis
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2705:
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2407:
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2225:
2194:
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2084:
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2018:
1987:
1965:
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1818:
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1615:
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1553:
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1404:
1368:
1318:
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1270:
1248:
1207:
1177:
1092:
1065:
1036:
1025:has low risk depends on the true state of nature
1014:
989:
864:
680:
562:
534:
497:
476:the actions that may be taken. An observation of
2969:
2393:separately (i.e., is a generalized Bayes rule).
684:{\displaystyle \delta (x)\in {\mathcal {A}}\,\!}
2926:
2601:of measure 0 without affecting the Bayes risk.
2777:where that rule achieves low risk). Thus, in
1757:, the Bayesian would fix the observed sample
292:
2888:
2872:
2860:
2841:The Oxford Dictionary of Statistical Terms
2720:Admissibility of (generalized) Bayes rules
2095:separately, we can define a decision rule
1688:
299:
285:
2765:
2743:
2704:
2682:
2645:
2623:
2594:{\displaystyle X\subseteq {\mathcal {X}}}
2559:
2534:
2503:
2482:
2450:
2428:
2406:
2351:
2329:
2301:
2249:
2224:
2193:
2158:
2127:
2105:
2083:
2054:
2017:
1986:
1964:
1935:
1817:
1795:
1767:
1745:
1709:
1673:
1651:
1614:
1577:
1552:
1460:
1429:
1415:with no such special interpretation. The
1403:
1367:
1317:
1295:
1269:
1247:
1206:
1176:
1091:
1064:
1035:
1013:
988:
868:{\displaystyle L(\theta ,\delta (x))\,\!}
863:
729:
679:
561:
533:
496:
417:
348:
2851:(entry for admissible decision function)
2816:are also inadmissible: for example, the
1229:. Thus an admissible decision rule is a
172:Integrated nested Laplace approximations
2687:{\displaystyle \pi (\theta \mid x)\,\!}
1750:{\displaystyle x\in {\mathcal {X}}\,\!}
1383:point of view, we would regard it as a
1333:Bayes rules and generalized Bayes rules
501:{\displaystyle x\in {\mathcal {X}}\,\!}
2970:
2907:
2791:need not yield admissible procedures.
1800:{\displaystyle \theta \in \Theta \,\!}
879:, which is the negative of the loss.)
566:{\displaystyle \theta \in \Theta \,\!}
2236:that achieve the same expected loss.
2334:{\displaystyle \theta \sim \pi \,\!}
2059:{\displaystyle F(x\mid \theta )\,\!}
1339:Bayes estimator § Admissibility
538:{\displaystyle F(x\mid \theta )\,\!}
2889:Cox, D. R.; Hinkley, D. V. (1974).
2386:{\displaystyle x\in {\mathcal {X}}}
1619:{\displaystyle r(\pi ,\delta )\,\!}
822:{\displaystyle x\in {\mathcal {X}}}
766:{\displaystyle a\in {\mathcal {A}}}
643:{\displaystyle x\in {\mathcal {X}}}
13:
2586:
2378:
2270:
2246:
1950:where the expectation is over the
1872:
1792:
1740:
1504:
1400:
923:
814:
792:{\displaystyle \theta \in \Theta }
786:
758:
720:
712:
674:
635:
605:
595:
558:
491:
461:
437:
414:
392:
368:
345:
14:
2994:
2912:(2nd ed.). Springer-Verlag.
2748:{\displaystyle \pi (\theta )\,\!}
2628:{\displaystyle \pi (\theta )\,\!}
2306:{\displaystyle x\sim \theta \,\!}
2198:{\displaystyle \pi (\theta )\,\!}
2022:{\displaystyle \pi (\theta )\,\!}
1685:, then no Bayes rule is defined.
1656:{\displaystyle \pi (\theta )\,\!}
1465:{\displaystyle \pi (\theta )\,\!}
1372:{\displaystyle \pi (\theta )\,\!}
773:when the true state of nature is
266:
182:Approximate Bayesian computation
34:
2818:sample estimate of the variance
1069:{\displaystyle \delta ^{*}\,\!}
452:the possible observations, and
208:Maximum a posteriori estimation
2866:
2854:
2830:
2795:is one such famous situation.
2740:
2734:
2679:
2667:
2620:
2614:
2539:{\displaystyle \delta (x)\,\!}
2531:
2525:
2486:{\displaystyle \delta (x)\!\,}
2478:
2472:
2278:{\displaystyle {\mathcal {X}}}
2229:{\displaystyle \delta (x)\,\!}
2221:
2215:
2190:
2184:
2163:{\displaystyle \delta (x)\,\!}
2155:
2149:
2051:
2039:
2014:
2008:
1929:
1926:
1923:
1917:
1905:
1899:
1891:
1879:
1865:
1847:
1648:
1642:
1611:
1599:
1546:
1543:
1531:
1525:
1517:
1511:
1497:
1485:
1457:
1451:
1364:
1358:
1343:
1153:
1141:
1132:
1113:
981:
978:
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957:
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942:
930:
916:
904:
860:
857:
851:
839:
725:
666:
660:
600:
530:
518:
469:{\displaystyle {\mathcal {A}}}
445:{\displaystyle {\mathcal {X}}}
400:{\displaystyle {\mathcal {A}}}
376:{\displaystyle {\mathcal {X}}}
1:
2949:Robert, Christian P. (1994).
2932:Optimal Statistical Decisions
2882:
1393:, it is merely a function on
330:
2285:of the expected loss (where
1779:and average over hypotheses
115:Principle of maximum entropy
7:
2798:
2770:{\displaystyle \theta \,\!}
2455:{\displaystyle \delta \,\!}
2433:{\displaystyle \delta \,\!}
2356:{\displaystyle \delta \,\!}
2254:{\displaystyle \Theta \,\!}
2110:{\displaystyle \delta \,\!}
1969:{\displaystyle \theta \,\!}
1678:{\displaystyle \delta \,\!}
1582:{\displaystyle \delta \,\!}
1434:{\displaystyle \delta \,\!}
1408:{\displaystyle \Theta \,\!}
1322:{\displaystyle \theta \,\!}
1300:{\displaystyle \theta \,\!}
1274:{\displaystyle \delta \,\!}
1259:. But just because a rule
1252:{\displaystyle \theta \,\!}
1211:{\displaystyle \theta \,\!}
1181:{\displaystyle \theta \,\!}
1096:{\displaystyle \delta \,\!}
1040:{\displaystyle \theta \,\!}
1018:{\displaystyle \delta \,\!}
829:, so that the loss will be
650:, we choose to take action
313:statistical decision theory
85:Bernstein–von Mises theorem
10:
2999:
2934:. Wiley Classics Library.
2657:. However, the posterior
1692:
1336:
428:are the states of nature,
321:rule for making a decision
2908:Berger, James O. (1980).
421:{\displaystyle \Theta \,}
352:{\displaystyle \Theta \,}
110:Principle of indifference
2823:
1003:Whether a decision rule
317:admissible decision rule
162:Markov chain Monte Carlo
2117:by specifying for each
1689:Generalized Bayes rules
620:, where upon observing
167:Laplace's approximation
154:Posterior approximation
2891:Theoretical Statistics
2873:Cox & Hinkley 1974
2861:Cox & Hinkley 1974
2809:ordinary least squares
2771:
2749:
2710:
2688:
2651:
2629:
2595:
2565:
2540:
2509:
2487:
2456:
2434:
2412:
2387:
2357:
2341:). Roughly speaking,
2335:
2307:
2279:
2255:
2230:
2199:
2172:generalized Bayes rule
2164:
2133:
2111:
2089:
2060:
2023:
1992:
1970:
1941:
1823:
1801:
1773:
1751:
1715:
1679:
1657:
1620:
1583:
1558:
1466:
1435:
1409:
1373:
1323:
1301:
1275:
1253:
1212:
1182:
1160:
1097:
1070:
1041:
1019:
994:
869:
823:
793:
767:
737:
685:
644:
614:
567:
539:
502:
470:
446:
422:
401:
377:
353:
273:Mathematics portal
216:Evidence approximation
2805:James–Stein estimator
2772:
2750:
2711:
2709:{\displaystyle x\,\!}
2689:
2652:
2650:{\displaystyle x\,\!}
2630:
2596:
2566:
2564:{\displaystyle x\,\!}
2541:
2510:
2508:{\displaystyle x\,\!}
2488:
2457:
2435:
2413:
2411:{\displaystyle x\,\!}
2388:
2358:
2336:
2308:
2280:
2256:
2231:
2200:
2165:
2134:
2132:{\displaystyle x\,\!}
2112:
2090:
2088:{\displaystyle x\,\!}
2061:
2024:
1993:
1991:{\displaystyle x\,\!}
1971:
1942:
1824:
1822:{\displaystyle x\,\!}
1802:
1774:
1772:{\displaystyle x\,\!}
1752:
1716:
1714:{\displaystyle x\,\!}
1680:
1658:
1621:
1584:
1559:
1467:
1436:
1419:of the decision rule
1410:
1374:
1324:
1302:
1276:
1254:
1213:
1183:
1161:
1098:
1071:
1042:
1020:
995:
870:
824:
794:
768:
738:
686:
645:
615:
568:
540:
503:
471:
447:
423:
402:
378:
354:
177:Variational inference
2759:
2728:
2698:
2661:
2639:
2608:
2575:
2553:
2519:
2497:
2466:
2444:
2422:
2400:
2367:
2345:
2317:
2289:
2265:
2243:
2209:
2178:
2143:
2121:
2099:
2077:
2033:
2002:
1980:
1958:
1841:
1811:
1783:
1761:
1729:
1703:
1667:
1636:
1593:
1571:
1479:
1445:
1423:
1397:
1352:
1329:occur in practice.)
1311:
1289:
1263:
1241:
1200:
1170:
1107:
1085:
1051:
1029:
1007:
898:
833:
803:
777:
747:
703:
654:
624:
584:
549:
512:
480:
456:
432:
411:
387:
363:
342:
255:Posterior predictive
224:Evidence lower bound
105:Likelihood principle
75:Bayesian probability
2978:Bayesian statistics
2953:. Springer-Verlag.
2951:The Bayesian Choice
2814:normal distribution
1472:is the expectation
1221:A decision rule is
1047:. A decision rule
28:Bayesian statistics
22:Part of a series on
2767:
2745:
2706:
2684:
2647:
2625:
2591:
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2536:
2505:
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2452:
2430:
2408:
2383:
2353:
2331:
2303:
2275:
2251:
2226:
2195:
2160:
2129:
2107:
2085:
2056:
2019:
1988:
1966:
1937:
1819:
1797:
1769:
1747:
1711:
1675:
1653:
1616:
1579:
1554:
1462:
1431:
1405:
1386:prior distribution
1369:
1319:
1297:
1271:
1249:
1208:
1192:the inequality is
1178:
1156:
1093:
1066:
1037:
1015:
990:
865:
819:
789:
763:
733:
681:
640:
610:
563:
535:
508:is distributed as
498:
466:
442:
418:
397:
373:
349:
198:Bayesian estimator
146:Hierarchical model
70:Bayesian inference
2983:Optimal decisions
1632:with respect to
325:Pareto efficiency
309:
308:
203:Credible interval
136:Linear regression
2990:
2964:
2945:
2923:
2904:
2876:
2870:
2864:
2858:
2852:
2834:
2776:
2774:
2773:
2768:
2754:
2752:
2751:
2746:
2715:
2713:
2712:
2707:
2693:
2691:
2690:
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2653:
2648:
2634:
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2600:
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2589:
2570:
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2542:
2537:
2514:
2512:
2511:
2506:
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2489:
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2436:
2431:
2417:
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2409:
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2332:
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2284:
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2273:
2260:
2258:
2257:
2252:
2235:
2233:
2232:
2227:
2204:
2202:
2201:
2196:
2174:with respect to
2169:
2167:
2166:
2161:
2138:
2136:
2135:
2130:
2116:
2114:
2113:
2108:
2094:
2092:
2091:
2086:
2065:
2063:
2062:
2057:
2028:
2026:
2025:
2020:
1997:
1995:
1994:
1989:
1975:
1973:
1972:
1967:
1946:
1944:
1943:
1938:
1895:
1894:
1828:
1826:
1825:
1820:
1806:
1804:
1803:
1798:
1778:
1776:
1775:
1770:
1756:
1754:
1753:
1748:
1744:
1743:
1720:
1718:
1717:
1712:
1684:
1682:
1681:
1676:
1662:
1660:
1659:
1654:
1625:
1623:
1622:
1617:
1588:
1586:
1585:
1580:
1567:A decision rule
1563:
1561:
1560:
1555:
1521:
1520:
1471:
1469:
1468:
1463:
1441:with respect to
1440:
1438:
1437:
1432:
1414:
1412:
1411:
1406:
1378:
1376:
1375:
1370:
1328:
1326:
1325:
1320:
1306:
1304:
1303:
1298:
1280:
1278:
1277:
1272:
1258:
1256:
1255:
1250:
1217:
1215:
1214:
1209:
1187:
1185:
1184:
1179:
1165:
1163:
1162:
1157:
1131:
1130:
1102:
1100:
1099:
1094:
1081:a decision rule
1075:
1073:
1072:
1067:
1063:
1062:
1046:
1044:
1043:
1038:
1024:
1022:
1021:
1016:
999:
997:
996:
991:
984:
946:
945:
877:utility function
874:
872:
871:
866:
828:
826:
825:
820:
818:
817:
798:
796:
795:
790:
772:
770:
769:
764:
762:
761:
742:
740:
739:
734:
732:
724:
723:
690:
688:
687:
682:
678:
677:
649:
647:
646:
641:
639:
638:
619:
617:
616:
611:
609:
608:
599:
598:
572:
570:
569:
564:
544:
542:
541:
536:
507:
505:
504:
499:
495:
494:
475:
473:
472:
467:
465:
464:
451:
449:
448:
443:
441:
440:
427:
425:
424:
419:
406:
404:
403:
398:
396:
395:
382:
380:
379:
374:
372:
371:
358:
356:
355:
350:
301:
294:
287:
271:
270:
237:Model evaluation
38:
19:
18:
2998:
2997:
2993:
2992:
2991:
2989:
2988:
2987:
2968:
2967:
2961:
2942:
2928:DeGroot, Morris
2920:
2901:
2885:
2880:
2879:
2875:, Exercise 11.7
2871:
2867:
2859:
2855:
2835:
2831:
2826:
2801:
2793:Stein's example
2789:improper priors
2782:decision theory
2760:
2757:
2756:
2729:
2726:
2725:
2722:
2699:
2696:
2695:
2662:
2659:
2658:
2640:
2637:
2636:
2609:
2606:
2605:
2585:
2584:
2576:
2573:
2572:
2554:
2551:
2550:
2520:
2517:
2516:
2498:
2495:
2494:
2467:
2464:
2463:
2445:
2442:
2441:
2423:
2420:
2419:
2401:
2398:
2397:
2377:
2376:
2368:
2365:
2364:
2346:
2343:
2342:
2318:
2315:
2314:
2290:
2287:
2286:
2269:
2268:
2266:
2263:
2262:
2244:
2241:
2240:
2210:
2207:
2206:
2179:
2176:
2175:
2144:
2141:
2140:
2122:
2119:
2118:
2100:
2097:
2096:
2078:
2075:
2074:
2034:
2031:
2030:
2003:
2000:
1999:
1998:(obtained from
1981:
1978:
1977:
1959:
1956:
1955:
1875:
1871:
1842:
1839:
1838:
1812:
1809:
1808:
1784:
1781:
1780:
1762:
1759:
1758:
1739:
1738:
1730:
1727:
1726:
1704:
1701:
1700:
1697:
1691:
1668:
1665:
1664:
1637:
1634:
1633:
1594:
1591:
1590:
1589:that minimizes
1572:
1569:
1568:
1507:
1503:
1480:
1477:
1476:
1446:
1443:
1442:
1424:
1421:
1420:
1398:
1395:
1394:
1353:
1350:
1349:
1346:
1341:
1335:
1312:
1309:
1308:
1290:
1287:
1286:
1264:
1261:
1260:
1242:
1239:
1238:
1231:maximal element
1201:
1198:
1197:
1171:
1168:
1167:
1126:
1122:
1108:
1105:
1104:
1103:if and only if
1086:
1083:
1082:
1058:
1054:
1052:
1049:
1048:
1030:
1027:
1026:
1008:
1005:
1004:
953:
926:
922:
899:
896:
895:
834:
831:
830:
813:
812:
804:
801:
800:
778:
775:
774:
757:
756:
748:
745:
744:
728:
719:
718:
704:
701:
700:
673:
672:
655:
652:
651:
634:
633:
625:
622:
621:
604:
603:
594:
593:
585:
582:
581:
550:
547:
546:
513:
510:
509:
490:
489:
481:
478:
477:
460:
459:
457:
454:
453:
436:
435:
433:
430:
429:
412:
409:
408:
391:
390:
388:
385:
384:
367:
366:
364:
361:
360:
343:
340:
339:
333:
305:
265:
250:Model averaging
229:Nested sampling
141:Empirical Bayes
131:Conjugate prior
100:Cromwell's rule
17:
12:
11:
5:
2996:
2986:
2985:
2980:
2966:
2965:
2959:
2946:
2940:
2924:
2918:
2905:
2899:
2884:
2881:
2878:
2877:
2865:
2863:, Section 11.8
2853:
2828:
2827:
2825:
2822:
2800:
2797:
2764:
2742:
2739:
2736:
2733:
2721:
2718:
2703:
2681:
2678:
2675:
2672:
2669:
2666:
2644:
2622:
2619:
2616:
2613:
2588:
2583:
2580:
2558:
2533:
2530:
2527:
2524:
2502:
2480:
2477:
2474:
2471:
2449:
2427:
2405:
2380:
2375:
2372:
2350:
2328:
2325:
2322:
2300:
2297:
2294:
2272:
2248:
2223:
2220:
2217:
2214:
2192:
2189:
2186:
2183:
2157:
2154:
2151:
2148:
2126:
2104:
2082:
2068:Bayes' theorem
2053:
2050:
2047:
2044:
2041:
2038:
2016:
2013:
2010:
2007:
1985:
1963:
1948:
1947:
1934:
1931:
1928:
1925:
1922:
1919:
1916:
1913:
1910:
1907:
1904:
1901:
1898:
1893:
1890:
1887:
1884:
1881:
1878:
1874:
1870:
1867:
1864:
1861:
1858:
1855:
1852:
1849:
1846:
1816:
1794:
1791:
1788:
1766:
1742:
1737:
1734:
1721:is considered
1708:
1690:
1687:
1672:
1650:
1647:
1644:
1641:
1613:
1610:
1607:
1604:
1601:
1598:
1576:
1565:
1564:
1551:
1548:
1545:
1542:
1539:
1536:
1533:
1530:
1527:
1524:
1519:
1516:
1513:
1510:
1506:
1502:
1499:
1496:
1493:
1490:
1487:
1484:
1459:
1456:
1453:
1450:
1428:
1402:
1366:
1363:
1360:
1357:
1345:
1342:
1334:
1331:
1316:
1294:
1268:
1246:
1205:
1175:
1155:
1152:
1149:
1146:
1143:
1140:
1137:
1134:
1129:
1125:
1121:
1118:
1115:
1112:
1090:
1061:
1057:
1034:
1012:
1001:
1000:
987:
983:
980:
977:
974:
971:
968:
965:
962:
959:
956:
952:
949:
944:
941:
938:
935:
932:
929:
925:
921:
918:
915:
912:
909:
906:
903:
862:
859:
856:
853:
850:
847:
844:
841:
838:
816:
811:
808:
788:
785:
782:
760:
755:
752:
731:
727:
722:
717:
714:
711:
708:
694:Also define a
676:
671:
668:
665:
662:
659:
637:
632:
629:
607:
602:
597:
592:
589:
560:
557:
554:
532:
529:
526:
523:
520:
517:
493:
488:
485:
463:
439:
416:
394:
370:
347:
332:
329:
307:
306:
304:
303:
296:
289:
281:
278:
277:
276:
275:
260:
259:
258:
257:
252:
247:
239:
238:
234:
233:
232:
231:
226:
218:
217:
213:
212:
211:
210:
205:
200:
192:
191:
187:
186:
185:
184:
179:
174:
169:
164:
156:
155:
151:
150:
149:
148:
143:
138:
133:
125:
124:
123:Model building
120:
119:
118:
117:
112:
107:
102:
97:
92:
87:
82:
80:Bayes' theorem
77:
72:
64:
63:
59:
58:
40:
39:
31:
30:
24:
23:
15:
9:
6:
4:
3:
2:
2995:
2984:
2981:
2979:
2976:
2975:
2973:
2962:
2960:3-540-94296-3
2956:
2952:
2947:
2943:
2941:0-471-68029-X
2937:
2933:
2929:
2925:
2921:
2919:0-387-96098-8
2915:
2911:
2906:
2902:
2900:0-412-12420-3
2896:
2892:
2887:
2886:
2874:
2869:
2862:
2857:
2850:
2849:0-19-920613-9
2846:
2842:
2838:
2833:
2829:
2821:
2819:
2815:
2810:
2806:
2796:
2794:
2790:
2785:
2783:
2780:
2762:
2737:
2731:
2717:
2701:
2676:
2673:
2670:
2664:
2642:
2617:
2611:
2602:
2581:
2578:
2556:
2549:
2528:
2522:
2500:
2475:
2469:
2447:
2425:
2403:
2394:
2373:
2370:
2348:
2326:
2323:
2320:
2298:
2295:
2292:
2237:
2218:
2212:
2187:
2181:
2173:
2152:
2146:
2124:
2102:
2080:
2071:
2069:
2048:
2045:
2042:
2036:
2011:
2005:
1983:
1961:
1953:
1932:
1920:
1914:
1911:
1908:
1902:
1896:
1888:
1885:
1882:
1876:
1868:
1862:
1859:
1856:
1853:
1850:
1844:
1837:
1836:
1835:
1834:
1833:
1832:expected loss
1814:
1789:
1786:
1764:
1735:
1732:
1724:
1706:
1696:
1686:
1670:
1645:
1639:
1631:
1630:
1608:
1605:
1602:
1596:
1574:
1549:
1540:
1537:
1534:
1528:
1522:
1514:
1508:
1500:
1494:
1491:
1488:
1482:
1475:
1474:
1473:
1454:
1448:
1426:
1418:
1392:
1388:
1387:
1382:
1361:
1355:
1340:
1330:
1314:
1292:
1284:
1266:
1244:
1237:
1232:
1228:
1224:
1219:
1203:
1195:
1191:
1173:
1150:
1147:
1144:
1138:
1135:
1127:
1123:
1119:
1116:
1110:
1088:
1080:
1079:
1059:
1055:
1032:
1010:
985:
972:
966:
963:
960:
954:
947:
939:
936:
933:
927:
919:
913:
910:
907:
901:
894:
893:
892:
891:
887:
886:
885:risk function
880:
878:
854:
848:
845:
842:
836:
809:
806:
783:
780:
753:
750:
715:
709:
706:
699:
698:
697:loss function
692:
669:
663:
657:
630:
627:
590:
587:
580:
576:
575:decision rule
555:
552:
527:
524:
521:
515:
486:
483:
338:
328:
326:
322:
318:
314:
302:
297:
295:
290:
288:
283:
282:
280:
279:
274:
269:
264:
263:
262:
261:
256:
253:
251:
248:
246:
243:
242:
241:
240:
236:
235:
230:
227:
225:
222:
221:
220:
219:
215:
214:
209:
206:
204:
201:
199:
196:
195:
194:
193:
189:
188:
183:
180:
178:
175:
173:
170:
168:
165:
163:
160:
159:
158:
157:
153:
152:
147:
144:
142:
139:
137:
134:
132:
129:
128:
127:
126:
122:
121:
116:
113:
111:
108:
106:
103:
101:
98:
96:
95:Cox's theorem
93:
91:
88:
86:
83:
81:
78:
76:
73:
71:
68:
67:
66:
65:
61:
60:
57:
53:
49:
45:
42:
41:
37:
33:
32:
29:
26:
25:
21:
20:
2950:
2931:
2909:
2890:
2868:
2856:
2840:
2832:
2802:
2786:
2723:
2603:
2547:
2395:
2238:
2171:
2072:
1951:
1949:
1830:
1722:
1698:
1627:
1626:is called a
1566:
1416:
1384:
1347:
1282:
1235:
1227:inadmissible
1226:
1222:
1220:
1189:
1076:
1002:
883:
881:
695:
693:
574:
334:
316:
310:
245:Bayes factor
2779:frequentist
1391:frequentist
1344:Bayes rules
890:expectation
882:Define the
2972:Categories
2883:References
2493:for those
2139:an action
1693:See also:
1629:Bayes rule
1417:Bayes risk
1337:See also:
1223:admissible
331:Definition
190:Estimators
62:Background
48:Likelihood
2930:(2004) .
2893:. Wiley.
2837:Dodge, Y.
2763:θ
2738:θ
2732:π
2674:∣
2671:θ
2665:π
2618:θ
2612:π
2582:⊆
2523:δ
2470:δ
2448:δ
2426:δ
2374:∈
2349:δ
2327:π
2324:∼
2321:θ
2299:θ
2296:∼
2247:Θ
2213:δ
2188:θ
2182:π
2147:δ
2103:δ
2049:θ
2046:∣
2012:θ
2006:π
1962:θ
1952:posterior
1915:δ
1909:θ
1897:
1886:∣
1883:θ
1877:π
1860:∣
1857:δ
1851:π
1845:ρ
1793:Θ
1790:∈
1787:θ
1736:∈
1671:δ
1646:θ
1640:π
1609:δ
1603:π
1575:δ
1541:δ
1535:θ
1523:
1515:θ
1509:π
1495:δ
1489:π
1455:θ
1449:π
1427:δ
1401:Θ
1362:θ
1356:π
1315:θ
1293:θ
1267:δ
1245:θ
1204:θ
1196:for some
1174:θ
1151:δ
1145:θ
1136:≤
1128:∗
1124:δ
1117:θ
1089:δ
1078:dominates
1060:∗
1056:δ
1033:θ
1011:δ
967:δ
961:θ
948:
940:θ
937:∣
914:δ
908:θ
849:δ
843:θ
810:∈
787:Θ
784:∈
781:θ
754:∈
726:→
716:×
713:Θ
670:∈
658:δ
631:∈
601:→
588:δ
559:Θ
556:∈
553:θ
528:θ
525:∣
487:∈
415:Θ
346:Θ
90:Coherence
44:Posterior
2799:Examples
1381:Bayesian
1166:for all
579:function
407:, where
56:Evidence
2843:. OUP.
2839:(2003)
888:as the
335:Define
2957:
2938:
2916:
2897:
2847:
2066:using
1976:given
1283:always
1194:strict
2824:Notes
2548:every
1723:fixed
577:is a
573:. A
319:is a
315:, an
52:Prior
2955:ISBN
2936:ISBN
2914:ISBN
2895:ISBN
2845:ISBN
2803:The
2546:for
2313:and
2029:and
1829:the
1348:Let
383:and
337:sets
2070:).
1954:of
1236:all
1190:and
311:In
2974::
1218:.
1188:,
691:.
359:,
327:.
54:Ă·
50:Ă—
46:=
2963:.
2944:.
2922:.
2903:.
2741:)
2735:(
2702:x
2680:)
2677:x
2668:(
2643:x
2621:)
2615:(
2587:X
2579:X
2557:x
2532:)
2529:x
2526:(
2501:x
2479:)
2476:x
2473:(
2404:x
2379:X
2371:x
2293:x
2271:X
2222:)
2219:x
2216:(
2191:)
2185:(
2156:)
2153:x
2150:(
2125:x
2081:x
2052:)
2043:x
2040:(
2037:F
2015:)
2009:(
1984:x
1933:.
1930:]
1927:)
1924:)
1921:x
1918:(
1912:,
1906:(
1903:L
1900:[
1892:)
1889:x
1880:(
1873:E
1869:=
1866:)
1863:x
1854:,
1848:(
1815:x
1765:x
1741:X
1733:x
1707:x
1649:)
1643:(
1612:)
1606:,
1600:(
1597:r
1550:.
1547:]
1544:)
1538:,
1532:(
1529:R
1526:[
1518:)
1512:(
1505:E
1501:=
1498:)
1492:,
1486:(
1483:r
1458:)
1452:(
1365:)
1359:(
1154:)
1148:,
1142:(
1139:R
1133:)
1120:,
1114:(
1111:R
986:.
982:]
979:)
976:)
973:x
970:(
964:,
958:(
955:L
951:[
943:)
934:x
931:(
928:F
924:E
920:=
917:)
911:,
905:(
902:R
861:)
858:)
855:x
852:(
846:,
840:(
837:L
815:X
807:x
759:A
751:a
730:R
721:A
710::
707:L
675:A
667:)
664:x
661:(
636:X
628:x
606:A
596:X
591::
531:)
522:x
519:(
516:F
492:X
484:x
462:A
438:X
393:A
369:X
300:e
293:t
286:v
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