2544:
279:
1199:
677:
1284:. Unlike a likelihood-ratio test, this Bayesian model comparison does not depend on any single set of parameters, as it integrates over all parameters in each model (with respect to the respective priors). An advantage of the use of Bayes factors is that it automatically, and quite naturally, includes a penalty for including too much model structure. It thus guards against
47:
1194:{\displaystyle K={\frac {\Pr(D|M_{1})}{\Pr(D|M_{2})}}={\frac {\int \Pr(\theta _{1}|M_{1})\Pr(D|\theta _{1},M_{1})\,d\theta _{1}}{\int \Pr(\theta _{2}|M_{2})\Pr(D|\theta _{2},M_{2})\,d\theta _{2}}}={\frac {\frac {\Pr(M_{1}|D)\Pr(D)}{\Pr(M_{1})}}{\frac {\Pr(M_{2}|D)\Pr(D)}{\Pr(M_{2})}}}={\frac {\Pr(M_{1}|D)}{\Pr(M_{2}|D)}}{\frac {\Pr(M_{2})}{\Pr(M_{1})}}.}
2070:
at the 5% significance level, the Bayes factor hardly considers this to be an extreme result. Note, however, that a non-uniform prior (for example one that reflects the fact that you expect the number of success and failures to be of the same order of magnitude) could result in a Bayes factor that is
362:
have been suggested. For certain special cases, simplified algebraic expressions can be derived; for instance, the Savage–Dickey density ratio in the case of a precise (equality constrained) hypothesis against an unrestricted alternative. Another approximation, derived by applying
357:
Although conceptually simple, the computation of the Bayes factor can be challenging depending on the complexity of the model and the hypotheses. Since closed-form expressions of the marginal likelihood are generally not available, numerical approximations based on
2268:
2005:
1847:
346:, although it uses the integrated (i.e., marginal) likelihood rather than the maximized likelihood. As such, both quantities only coincide under simple hypotheses (e.g., two specific parameter values). Also, in contrast with
2509:
2384:
2447:
542:
1731:
2132:
2059:
is 0.02, and as a two-tailed test of getting a figure as extreme as or more extreme than 115 is 0.04. Note that 115 is more than two standard deviations away from 100. Thus, whereas a
2140:
1865:
1260:
2319:
takes into account the number of free parameters in the models, unlike the classical likelihood ratio. The relative likelihood method could be applied as follows. Model
665:
638:
582:
432:
1748:
371:(BIC); in large data sets the Bayes factor will approach the BIC as the influence of the priors wanes. In small data sets, priors generally matter and must not be
3161:
309:
100:
2300:
because it has a free parameter which allows it to model the data more closely. The ability of Bayes factors to take this into account is a reason why
182:
3335:
Kadane, Joseph B.; Dickey, James M. (1980). "Bayesian
Decision Theory and the Simplification of Models". In Kmenta, Jan; Ramsey, James B. (eds.).
2459:
1292:
can be used for model selection in a
Bayesian framework, with the caveat that approximate-Bayesian estimates of Bayes factors are often biased.
2792:
2679:
1508:
a change in a weight of evidence of 1 deciban or 1/3 of a bit (i.e. a change in an odds ratio from evens to about 5:4) is about as finely as
2742:
Llorente, Fernando; et al. (2023). "Marginal likelihood computation for model selection and hypothesis testing: an extensive review".
347:
334:, and is used to quantify the support for one model over the other. The models in question can have a common set of parameters, such as a
3419:
3363:
3344:
3313:
3290:
3258:
3239:
3145:
2928:
2900:
2817:
1647:
302:
265:
2333:
2562:
2396:
1289:
192:
2045:
should be rejected at the 5% significance level, since the probability of getting 115 or more successes from a sample of 200 if
452:
95:
218:
2572:
2567:
2063:
2024:
368:
156:
3107:
Williams, Matt; BĂĄĂĄth, Rasmus; Philipp, Michael (2017). "Using Bayes
Factors to Test Hypotheses in Developmental Research".
1660:
1650:
on . We take a sample of 200, and find 115 successes and 85 failures. The likelihood can be calculated according to the
2089:
3178:(1979). "Studies in the History of Probability and Statistics. XXXVII A. M. Turing's statistical work in World War II".
2557:
2327:
295:
187:
125:
3480:
3470:
1314:
364:
177:
146:
3450:
1288:. For models where an explicit version of the likelihood is not available or too costly to evaluate numerically,
338:
and an alternative, but this is not necessary; for instance, it could also be a non-linear model compared to its
239:
120:
2263:{\displaystyle \textstyle P(X=115\mid M_{2})={{200 \choose 115}{\hat {q}}^{115}(1-{\hat {q}})^{85}}\approx 0.06}
3475:
2067:
2000:{\displaystyle P(X=115\mid M_{2})=\int _{0}^{1}{200 \choose 115}q^{115}(1-q)^{85}dq={1 \over 201}\approx 0.005}
359:
260:
172:
3375:"Efficiency Testing of Prediction Markets: Martingale Approach, Likelihood Ratio and Bayes Factor Analysis"
1207:
2582:
1307:
151:
3441:
3322:
2075:
1651:
1281:
395:
343:
54:
2543:
278:
1348:
gives one hypothesis (or model) preferred status (the 'null hypothesis'), and only considers evidence
3061:
2948:
2916:
2577:
1513:
339:
234:
115:
85:
375:
since the Bayes factor will be undefined if either of the two integrals in its ratio is not finite.
2316:
384:
331:
66:
58:
38:
2834:
2010:
The ratio is then 1.2, which is "barely worth mentioning" even if it points very slightly towards
3155:
3002:
2971:
2865:
2765:
2747:
2549:
2301:
2079:
1639:
1345:
1277:
643:
616:
283:
208:
80:
110:
1842:{\displaystyle P(X=115\mid M_{1})={200 \choose 115}\left({1 \over 2}\right)^{200}\approx 0.006}
3415:
3359:
3340:
3309:
3305:
3286:
3254:
3235:
3141:
3135:
3089:
3030:
2924:
2896:
2857:
2813:
2788:
2675:
1356:
and not just against a null hypothesis is one of the key advantages of this analysis method.
550:
443:
400:
388:
327:
213:
90:
62:
2783:
Congdon, Peter (2014). "Estimating model probabilities or marginal likelihoods in practice".
3396:
3386:
3271:
3189:
3116:
3079:
3069:
3020:
3012:
2963:
2888:
2849:
2757:
2722:
2712:
2667:
2642:
2305:
105:
3201:
3454:
3197:
2991:"Simulation-based model selection for dynamical systems in systems and population biology"
2587:
2035:
1603:
1359:
1300:
592:
335:
141:
584:
represents the probability that some data are produced under the assumption of the model
354:
of a null hypothesis, rather than only allowing the null to be rejected or not rejected.
17:
3065:
3084:
3049:
3025:
2990:
2835:"Bayesian hypothesis testing for psychologists: A tutorial on the Savage–Dickey method"
1493:
1280:
of the parameter for each statistical model is used, then the test becomes a classical
595:
problem in which one wishes to choose between two models on the basis of observed data
372:
3214:
3464:
2869:
2769:
2609:
3327:
3016:
2853:
2697:"The Bayesian Methodology of Sir Harold Jeffreys as a Practical Alternative to the
2309:
3120:
2883:
Ibrahim, Joseph G.; Chen, Ming-Hui; Sinha, Debajyoti (2001). "Model
Comparison".
2833:
Wagenmakers, Eric-Jan; Lodewyckx, Tom; Kuriyal, Himanshu; Grasman, Raoul (2010).
2892:
2631:"The philosophy of Bayes factors and the quantification of statistical evidence"
2060:
2021:
1285:
3435:
2717:
2696:
1276:. If instead of the Bayes factor integral, the likelihood corresponding to the
3193:
3180:
3175:
2671:
2647:
2630:
2604:
2539:
2304:
has been put forward as a theoretical justification for and generalisation of
1505:
3275:
2662:
Lesaffre, Emmanuel; Lawson, Andrew B. (2012). "Bayesian hypothesis testing".
1519:
An alternative table, widely cited, is provided by Kass and
Raftery (1995):
3074:
3281:
Duda, Richard O.; Hart, Peter E.; Stork, David G. (2000). "Section 9.6.5".
3093:
3034:
2861:
1262:, the Bayes factor is equal to the ratio of the posterior probabilities of
3438:—an R package for computing Bayes factors in common research designs
3249:
Denison, D. G. T.; Holmes, C. C.; Mallick, B. K.; Smith, A. F. M. (2002).
2504:{\displaystyle \exp \left({\frac {7.7297-10.2467}{2}}\right)\approx 0.284}
3391:
3374:
2592:
1606:
that produces either a success or a failure. We want to compare a model
3401:
2808:
Koop, Gary (2003). "Model
Comparison: The Savage–Dickey Density Ratio".
383:
The Bayes factor is the ratio of two marginal likelihoods; that is, the
2975:
2887:. Springer Series in Statistics. New York: Springer. pp. 246–254.
2761:
2727:
1497:
46:
3447:
2038:) would have produced a very different result. Such a test says that
3050:"Lack of confidence in approximate Bayesian computation model choice"
2967:
2752:
2629:
Morey, Richard D.; Romeijn, Jan-Willem; Rouder, Jeffrey N. (2016).
588:; evaluating it correctly is the key to Bayesian model comparison.
3007:
1509:
2921:
Bayesian
Methods : A Social and Behavioral Sciences Approach
1492:
The second column gives the corresponding weights of evidence in
342:. The Bayes factor can be thought of as a Bayesian analog to the
1337:
is more strongly supported by the data under consideration than
1501:
3048:
Robert, C.P.; J. Cornuet; J. Marin & N.S. Pillai (2011).
3251:
Bayesian
Methods for Nonlinear Classification and Regression
2919:(2002). "Bayesian Hypothesis Testing and the Bayes Factor".
2379:{\displaystyle 2\cdot 0-2\cdot \ln(0.005956)\approx 10.2467}
2277:). That gives a likelihood ratio of 0.1 and points towards
1303:, computing the expected value or cost of each model choice;
3457:—web-based version of much of the BayesFactor package
2442:{\displaystyle 2\cdot 1-2\cdot \ln(0.056991)\approx 7.7297}
3269:
Advances in
Methods and Practices in Psychological Science
3267:
Dienes, Z. (2019). How do I know what my theory predicts?
1204:
When the two models have equal prior probability, so that
537:{\displaystyle \Pr(M|D)={\frac {\Pr(D|M)\Pr(M)}{\Pr(D)}}.}
1504:
are added in the third column for clarity. According to
2071:
more in agreement with the frequentist hypothesis test.
27:
Statistical factor used to compare competing hypotheses
2144:
1352:
it. The fact that a Bayes factor can produce evidence
3300:
Gelman, A.; Carlin, J.; Stern, H.; Rubin, D. (1995).
2462:
2399:
2336:
2143:
2092:
1868:
1751:
1726:{\displaystyle {{200 \choose 115}q^{115}(1-q)^{85}}.}
1663:
1210:
680:
646:
619:
553:
455:
403:
3444:— Online calculator for informed Bayes factors
2812:. Somerset: John Wiley & Sons. pp. 69–71.
2666:. Somerset: John Wiley & Sons. pp. 72–78.
2127:{\displaystyle {\hat {q}}={\frac {115}{200}}=0.575}
2942:
2940:
2503:
2441:
2378:
2262:
2126:
1999:
1841:
1725:
1254:
1193:
659:
632:
576:
536:
426:
1934:
1921:
1802:
1789:
1681:
1668:
1233:
1211:
1166:
1145:
1112:
1083:
1054:
1040:
1013:
990:
976:
949:
885:
851:
792:
758:
719:
690:
554:
516:
502:
482:
456:
404:
3412:Introduction to Bayesian Inference and Decision
3054:Proceedings of the National Academy of Sciences
2956:Journal of the American Statistical Association
2947:Robert E. Kass & Adrian E. Raftery (1995).
599:, the plausibility of the two different models
367:to the integrated likelihoods, is known as the
350:, Bayes factors support evaluation of evidence
3339:. New York: Academic Press. pp. 245–268.
3140:(3rd ed.). Oxford, England. p. 432.
387:of two statistical models integrated over the
3215:Sharpening Ockham's Razor On a Bayesian Strop
2196:
2183:
303:
8:
3160:: CS1 maint: location missing publisher (
613:, parametrised by model parameter vectors
310:
296:
29:
3400:
3390:
3285:(2nd ed.). Wiley. pp. 487–489.
3083:
3073:
3024:
3006:
2751:
2726:
2716:
2646:
2473:
2461:
2398:
2393:has 1 parameter, and so its AIC value is
2335:
2273:(rather than averaging over all possible
2246:
2231:
2230:
2215:
2204:
2203:
2195:
2182:
2180:
2179:
2167:
2142:
2108:
2094:
2093:
2091:
1981:
1966:
1944:
1933:
1920:
1918:
1912:
1907:
1891:
1867:
1827:
1813:
1801:
1788:
1786:
1774:
1750:
1713:
1691:
1680:
1667:
1665:
1664:
1662:
1243:
1221:
1209:
1176:
1155:
1142:
1128:
1122:
1099:
1093:
1080:
1064:
1029:
1023:
1000:
965:
959:
945:
933:
925:
916:
903:
894:
876:
867:
861:
840:
832:
823:
810:
801:
783:
774:
768:
752:
737:
728:
708:
699:
687:
679:
651:
645:
624:
618:
563:
552:
491:
479:
465:
454:
413:
402:
3328:Probability Theory: the logic of science
2923:. Chapman & Hall. pp. 199–237.
2518:to minimize the information loss. Thus
2315:On the other hand, the modern method of
1521:
1368:
183:Integrated nested Laplace approximations
2787:(2nd ed.). Wiley. pp. 38–40.
2621:
247:
226:
200:
164:
133:
72:
37:
3230:Bernardo, J.; Smith, A. F. M. (1994).
3153:
1255:{\displaystyle \Pr(M_{1})=\Pr(M_{2})}
7:
3356:Bayesian Statistics: an introduction
1613:where the probability of success is
348:null hypothesis significance testing
2695:Ly, Alexander; et al. (2020).
1552:Not worth more than a bare mention
1362:gave a scale for interpretation of
3373:Richard, Mark; Vecer, Jan (2021).
2705:Computational Brain & Behavior
2635:Journal of Mathematical Psychology
2187:
1925:
1793:
1672:
667:, is assessed by the Bayes factor
25:
2989:Toni, T.; Stumpf, M.P.H. (2009).
1516:in a hypothesis in everyday use.
3337:Evaluation of Econometric Models
2563:Approximate Bayesian computation
2542:
1290:approximate Bayesian computation
277:
193:Approximate Bayesian computation
45:
3414:(2nd ed.). Probabilistic.
1299:to treat model comparison as a
219:Maximum a posteriori estimation
2854:10.1016/j.cogpsych.2009.12.001
2573:Deviance information criterion
2568:Bayesian information criterion
2430:
2424:
2367:
2361:
2243:
2236:
2221:
2209:
2173:
2148:
2099:
1963:
1950:
1897:
1872:
1780:
1755:
1710:
1697:
1512:can reasonably perceive their
1249:
1236:
1227:
1214:
1182:
1169:
1161:
1148:
1136:
1129:
1115:
1107:
1100:
1086:
1070:
1057:
1049:
1043:
1037:
1030:
1016:
1006:
993:
985:
979:
973:
966:
952:
922:
895:
888:
882:
868:
854:
829:
802:
795:
789:
775:
761:
743:
729:
722:
714:
700:
693:
571:
564:
557:
525:
519:
511:
505:
499:
492:
485:
473:
466:
459:
421:
414:
407:
369:Bayesian information criterion
1:
3121:10.1080/15427609.2017.1370964
3109:Research in Human Development
3017:10.1093/bioinformatics/btp619
2326:has 0 parameters, and so its
2293:is a more complex model than
2558:Akaike information criterion
2328:Akaike information criterion
547:The key data-dependent term
326:is a ratio of two competing
126:Principle of maximum entropy
2893:10.1007/978-1-4757-3447-8_6
2525:is slightly preferred, but
1278:maximum likelihood estimate
660:{\displaystyle \theta _{2}}
633:{\displaystyle \theta _{1}}
96:Bernstein–von Mises theorem
3497:
3134:Jeffreys, Harold (1998) .
2885:Bayesian Survival Analysis
2785:Applied Bayesian Modelling
2718:10.1007/s42113-019-00070-x
1315:minimum description length
3137:The Theory of Probability
2672:10.1002/9781119942412.ch3
2648:10.1016/j.jmp.2015.11.001
1638:is unknown and we take a
121:Principle of indifference
18:Bayesian model comparison
3448:Bayes Factor Calculators
3410:Winkler, Robert (2003).
3276:10.1177/2515245919876960
1422:Barely worth mentioning
577:{\displaystyle \Pr(D|M)}
427:{\displaystyle \Pr(M|D)}
173:Markov chain Monte Carlo
3442:Bayes factor calculator
3194:10.1093/biomet/66.2.393
3075:10.1073/pnas.1102900108
365:Laplace's approximation
178:Laplace's approximation
165:Posterior approximation
3302:Bayesian Data Analysis
3283:Pattern classification
2701:Value Hypothesis Test"
2664:Bayesian Biostatistics
2583:Minimum message length
2505:
2443:
2380:
2264:
2128:
2034:(here considered as a
2001:
1843:
1727:
1344:. Note that classical
1308:minimum message length
1295:Other approaches are:
1256:
1195:
661:
634:
578:
538:
428:
284:Mathematics portal
227:Evidence approximation
2810:Bayesian Econometrics
2511:times as probable as
2506:
2444:
2381:
2265:
2129:
2078:would have found the
2076:likelihood-ratio test
2002:
1844:
1728:
1652:binomial distribution
1539:Strength of evidence
1383:Strength of evidence
1282:likelihood-ratio test
1257:
1196:
662:
635:
579:
539:
429:
396:posterior probability
391:of their parameters.
344:likelihood-ratio test
330:represented by their
188:Variational inference
3392:10.3390/risks9020031
2842:Cognitive Psychology
2532:cannot be excluded.
2460:
2397:
2334:
2141:
2090:
1866:
1749:
1661:
1627:, and another model
1208:
678:
644:
617:
551:
453:
401:
340:linear approximation
266:Posterior predictive
235:Evidence lower bound
116:Likelihood principle
86:Bayesian probability
3354:Lee, P. M. (2012).
3066:2011PNAS..10815112R
3060:(37): 15112–15117.
2746:. to appear: 3–58.
2317:relative likelihood
2068:significant results
1917:
1399:Negative (supports
389:prior probabilities
39:Bayesian statistics
33:Part of a series on
3481:Statistical ratios
3471:Bayesian inference
3453:2015-05-07 at the
3306:Chapman & Hall
2762:10.1137/20M1310849
2599:Statistical ratios
2550:Mathematics portal
2501:
2439:
2376:
2302:Bayesian inference
2260:
2259:
2124:
2080:maximum likelihood
1997:
1903:
1839:
1723:
1640:prior distribution
1602:Suppose we have a
1346:hypothesis testing
1330:> 1 means that
1252:
1191:
657:
630:
574:
534:
424:
328:statistical models
209:Bayesian estimator
157:Hierarchical model
81:Bayesian inference
2794:978-1-119-95151-3
2681:978-0-470-01823-1
2578:Lindley's paradox
2489:
2239:
2212:
2194:
2116:
2102:
1989:
1932:
1821:
1800:
1736:Thus we have for
1679:
1595:
1594:
1490:
1489:
1186:
1140:
1075:
1074:
1010:
940:
747:
529:
320:
319:
214:Credible interval
147:Linear regression
16:(Redirected from
3488:
3425:
3406:
3404:
3394:
3369:
3350:
3319:
3296:
3264:
3245:
3217:
3212:
3206:
3205:
3172:
3166:
3165:
3159:
3151:
3131:
3125:
3124:
3104:
3098:
3097:
3087:
3077:
3045:
3039:
3038:
3028:
3010:
2986:
2980:
2979:
2953:
2944:
2935:
2934:
2913:
2907:
2906:
2880:
2874:
2873:
2839:
2830:
2824:
2823:
2805:
2799:
2798:
2780:
2774:
2773:
2755:
2739:
2733:
2732:
2730:
2720:
2692:
2686:
2685:
2659:
2653:
2652:
2650:
2626:
2552:
2547:
2546:
2510:
2508:
2507:
2502:
2494:
2490:
2485:
2474:
2448:
2446:
2445:
2440:
2385:
2383:
2382:
2377:
2269:
2267:
2266:
2261:
2252:
2251:
2250:
2241:
2240:
2232:
2220:
2219:
2214:
2213:
2205:
2201:
2200:
2199:
2186:
2172:
2171:
2133:
2131:
2130:
2125:
2117:
2109:
2104:
2103:
2095:
2058:
2057:
2053:
2006:
2004:
2003:
1998:
1990:
1982:
1971:
1970:
1949:
1948:
1939:
1938:
1937:
1924:
1916:
1911:
1896:
1895:
1848:
1846:
1845:
1840:
1832:
1831:
1826:
1822:
1814:
1807:
1806:
1805:
1792:
1779:
1778:
1732:
1730:
1729:
1724:
1719:
1718:
1717:
1696:
1695:
1686:
1685:
1684:
1671:
1626:
1625:
1621:
1522:
1514:degree of belief
1369:
1301:decision problem
1261:
1259:
1258:
1253:
1248:
1247:
1226:
1225:
1200:
1198:
1197:
1192:
1187:
1185:
1181:
1180:
1164:
1160:
1159:
1143:
1141:
1139:
1132:
1127:
1126:
1110:
1103:
1098:
1097:
1081:
1076:
1073:
1069:
1068:
1052:
1033:
1028:
1027:
1011:
1009:
1005:
1004:
988:
969:
964:
963:
947:
946:
941:
939:
938:
937:
921:
920:
908:
907:
898:
881:
880:
871:
866:
865:
846:
845:
844:
828:
827:
815:
814:
805:
788:
787:
778:
773:
772:
753:
748:
746:
742:
741:
732:
717:
713:
712:
703:
688:
666:
664:
663:
658:
656:
655:
639:
637:
636:
631:
629:
628:
583:
581:
580:
575:
567:
543:
541:
540:
535:
530:
528:
514:
495:
480:
469:
433:
431:
430:
425:
417:
312:
305:
298:
282:
281:
248:Model evaluation
49:
30:
21:
3496:
3495:
3491:
3490:
3489:
3487:
3486:
3485:
3476:Model selection
3461:
3460:
3455:Wayback Machine
3432:
3422:
3409:
3372:
3366:
3353:
3347:
3334:
3316:
3299:
3293:
3280:
3261:
3248:
3242:
3232:Bayesian Theory
3229:
3226:
3224:Further reading
3221:
3220:
3213:
3209:
3174:
3173:
3169:
3152:
3148:
3133:
3132:
3128:
3106:
3105:
3101:
3047:
3046:
3042:
2988:
2987:
2983:
2968:10.2307/2291091
2951:
2949:"Bayes Factors"
2946:
2945:
2938:
2931:
2915:
2914:
2910:
2903:
2882:
2881:
2877:
2837:
2832:
2831:
2827:
2820:
2807:
2806:
2802:
2795:
2782:
2781:
2777:
2741:
2740:
2736:
2694:
2693:
2689:
2682:
2661:
2660:
2656:
2628:
2627:
2623:
2618:
2588:Model selection
2548:
2541:
2538:
2531:
2524:
2517:
2475:
2469:
2458:
2457:
2455:
2395:
2394:
2392:
2332:
2331:
2330:(AIC) value is
2325:
2299:
2292:
2283:
2242:
2202:
2181:
2163:
2139:
2138:
2088:
2087:
2064:hypothesis test
2055:
2051:
2050:
2044:
2036:null hypothesis
2033:
2025:hypothesis test
2016:
1962:
1940:
1919:
1887:
1864:
1863:
1858:
1809:
1808:
1787:
1770:
1747:
1746:
1742:
1709:
1687:
1666:
1659:
1658:
1633:
1623:
1619:
1618:
1612:
1604:random variable
1600:
1528:
1496:(also known as
1405:
1360:Harold Jeffreys
1343:
1336:
1324:
1275:
1268:
1239:
1217:
1206:
1205:
1172:
1165:
1151:
1144:
1118:
1111:
1089:
1082:
1060:
1053:
1019:
1012:
996:
989:
955:
948:
929:
912:
899:
872:
857:
847:
836:
819:
806:
779:
764:
754:
733:
718:
704:
689:
676:
675:
647:
642:
641:
620:
615:
614:
612:
605:
593:model selection
549:
548:
515:
481:
451:
450:
399:
398:
381:
336:null hypothesis
316:
276:
261:Model averaging
240:Nested sampling
152:Empirical Bayes
142:Conjugate prior
111:Cromwell's rule
28:
23:
22:
15:
12:
11:
5:
3494:
3492:
3484:
3483:
3478:
3473:
3463:
3462:
3459:
3458:
3445:
3439:
3431:
3430:External links
3428:
3427:
3426:
3420:
3407:
3370:
3364:
3351:
3345:
3332:
3320:
3314:
3297:
3291:
3278:
3265:
3259:
3253:. John Wiley.
3246:
3240:
3234:. John Wiley.
3225:
3222:
3219:
3218:
3207:
3188:(2): 393–396.
3167:
3146:
3126:
3099:
3040:
2995:Bioinformatics
2981:
2936:
2929:
2908:
2901:
2875:
2848:(3): 158–189.
2825:
2818:
2800:
2793:
2775:
2734:
2711:(2): 153–161.
2687:
2680:
2654:
2620:
2619:
2617:
2614:
2613:
2612:
2607:
2601:
2600:
2596:
2595:
2590:
2585:
2580:
2575:
2570:
2565:
2560:
2554:
2553:
2537:
2534:
2529:
2522:
2515:
2500:
2497:
2493:
2488:
2484:
2481:
2478:
2472:
2468:
2465:
2453:
2438:
2435:
2432:
2429:
2426:
2423:
2420:
2417:
2414:
2411:
2408:
2405:
2402:
2390:
2375:
2372:
2369:
2366:
2363:
2360:
2357:
2354:
2351:
2348:
2345:
2342:
2339:
2323:
2297:
2290:
2281:
2271:
2270:
2258:
2255:
2249:
2245:
2238:
2235:
2229:
2226:
2223:
2218:
2211:
2208:
2198:
2193:
2190:
2185:
2178:
2175:
2170:
2166:
2162:
2159:
2156:
2153:
2150:
2147:
2123:
2120:
2115:
2112:
2107:
2101:
2098:
2042:
2031:
2014:
2008:
2007:
1996:
1993:
1988:
1985:
1980:
1977:
1974:
1969:
1965:
1961:
1958:
1955:
1952:
1947:
1943:
1936:
1931:
1928:
1923:
1915:
1910:
1906:
1902:
1899:
1894:
1890:
1886:
1883:
1880:
1877:
1874:
1871:
1856:
1850:
1849:
1838:
1835:
1830:
1825:
1820:
1817:
1812:
1804:
1799:
1796:
1791:
1785:
1782:
1777:
1773:
1769:
1766:
1763:
1760:
1757:
1754:
1740:
1734:
1733:
1722:
1716:
1712:
1708:
1705:
1702:
1699:
1694:
1690:
1683:
1678:
1675:
1670:
1631:
1610:
1599:
1596:
1593:
1592:
1589:
1586:
1580:
1579:
1576:
1573:
1567:
1566:
1563:
1560:
1554:
1553:
1550:
1547:
1541:
1540:
1537:
1532:
1526:
1488:
1487:
1484:
1481:
1478:
1472:
1471:
1468:
1465:
1462:
1456:
1455:
1452:
1449:
1446:
1440:
1439:
1436:
1433:
1430:
1424:
1423:
1420:
1417:
1414:
1408:
1407:
1403:
1397:
1394:
1391:
1385:
1384:
1381:
1378:
1375:
1341:
1334:
1323:
1322:Interpretation
1320:
1319:
1318:
1311:
1304:
1273:
1266:
1251:
1246:
1242:
1238:
1235:
1232:
1229:
1224:
1220:
1216:
1213:
1202:
1201:
1190:
1184:
1179:
1175:
1171:
1168:
1163:
1158:
1154:
1150:
1147:
1138:
1135:
1131:
1125:
1121:
1117:
1114:
1109:
1106:
1102:
1096:
1092:
1088:
1085:
1079:
1072:
1067:
1063:
1059:
1056:
1051:
1048:
1045:
1042:
1039:
1036:
1032:
1026:
1022:
1018:
1015:
1008:
1003:
999:
995:
992:
987:
984:
981:
978:
975:
972:
968:
962:
958:
954:
951:
944:
936:
932:
928:
924:
919:
915:
911:
906:
902:
897:
893:
890:
887:
884:
879:
875:
870:
864:
860:
856:
853:
850:
843:
839:
835:
831:
826:
822:
818:
813:
809:
804:
800:
797:
794:
791:
786:
782:
777:
771:
767:
763:
760:
757:
751:
745:
740:
736:
731:
727:
724:
721:
716:
711:
707:
702:
698:
695:
692:
686:
683:
654:
650:
627:
623:
610:
603:
573:
570:
566:
562:
559:
556:
545:
544:
533:
527:
524:
521:
518:
513:
510:
507:
504:
501:
498:
494:
490:
487:
484:
478:
475:
472:
468:
464:
461:
458:
444:Bayes' theorem
423:
420:
416:
412:
409:
406:
380:
377:
318:
317:
315:
314:
307:
300:
292:
289:
288:
287:
286:
271:
270:
269:
268:
263:
258:
250:
249:
245:
244:
243:
242:
237:
229:
228:
224:
223:
222:
221:
216:
211:
203:
202:
198:
197:
196:
195:
190:
185:
180:
175:
167:
166:
162:
161:
160:
159:
154:
149:
144:
136:
135:
134:Model building
131:
130:
129:
128:
123:
118:
113:
108:
103:
98:
93:
91:Bayes' theorem
88:
83:
75:
74:
70:
69:
51:
50:
42:
41:
35:
34:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3493:
3482:
3479:
3477:
3474:
3472:
3469:
3468:
3466:
3456:
3452:
3449:
3446:
3443:
3440:
3437:
3434:
3433:
3429:
3423:
3421:0-9647938-4-9
3417:
3413:
3408:
3403:
3398:
3393:
3388:
3384:
3380:
3376:
3371:
3367:
3365:9781118332573
3361:
3357:
3352:
3348:
3346:0-12-416550-8
3342:
3338:
3333:
3331:, chapter 24.
3330:
3329:
3324:
3323:Jaynes, E. T.
3321:
3317:
3315:0-412-03991-5
3311:
3307:
3303:
3298:
3294:
3292:0-471-05669-3
3288:
3284:
3279:
3277:
3273:
3270:
3266:
3262:
3260:0-471-49036-9
3256:
3252:
3247:
3243:
3241:0-471-92416-4
3237:
3233:
3228:
3227:
3223:
3216:
3211:
3208:
3203:
3199:
3195:
3191:
3187:
3183:
3182:
3177:
3171:
3168:
3163:
3157:
3149:
3147:9780191589676
3143:
3139:
3138:
3130:
3127:
3122:
3118:
3114:
3110:
3103:
3100:
3095:
3091:
3086:
3081:
3076:
3071:
3067:
3063:
3059:
3055:
3051:
3044:
3041:
3036:
3032:
3027:
3022:
3018:
3014:
3009:
3004:
3001:(1): 104–10.
3000:
2996:
2992:
2985:
2982:
2977:
2973:
2969:
2965:
2961:
2957:
2950:
2943:
2941:
2937:
2932:
2930:1-58488-288-3
2926:
2922:
2918:
2912:
2909:
2904:
2902:0-387-95277-2
2898:
2894:
2890:
2886:
2879:
2876:
2871:
2867:
2863:
2859:
2855:
2851:
2847:
2843:
2836:
2829:
2826:
2821:
2819:0-470-84567-8
2815:
2811:
2804:
2801:
2796:
2790:
2786:
2779:
2776:
2771:
2767:
2763:
2759:
2754:
2749:
2745:
2738:
2735:
2729:
2724:
2719:
2714:
2710:
2706:
2702:
2700:
2691:
2688:
2683:
2677:
2673:
2669:
2665:
2658:
2655:
2649:
2644:
2640:
2636:
2632:
2625:
2622:
2615:
2611:
2610:Relative risk
2608:
2606:
2603:
2602:
2598:
2597:
2594:
2591:
2589:
2586:
2584:
2581:
2579:
2576:
2574:
2571:
2569:
2566:
2564:
2561:
2559:
2556:
2555:
2551:
2545:
2540:
2535:
2533:
2528:
2521:
2514:
2498:
2495:
2491:
2486:
2482:
2479:
2476:
2470:
2466:
2463:
2452:
2436:
2433:
2427:
2421:
2418:
2415:
2412:
2409:
2406:
2403:
2400:
2389:
2373:
2370:
2364:
2358:
2355:
2352:
2349:
2346:
2343:
2340:
2337:
2329:
2322:
2318:
2313:
2311:
2310:Type I errors
2307:
2306:Occam's razor
2303:
2296:
2289:
2285:
2280:
2276:
2256:
2253:
2247:
2233:
2227:
2224:
2216:
2206:
2191:
2188:
2176:
2168:
2164:
2160:
2157:
2154:
2151:
2145:
2137:
2136:
2135:
2121:
2118:
2113:
2110:
2105:
2096:
2085:
2082:estimate for
2081:
2077:
2072:
2069:
2065:
2062:
2048:
2041:
2037:
2030:
2026:
2023:
2018:
2013:
1994:
1991:
1986:
1983:
1978:
1975:
1972:
1967:
1959:
1956:
1953:
1945:
1941:
1929:
1926:
1913:
1908:
1904:
1900:
1892:
1888:
1884:
1881:
1878:
1875:
1869:
1862:
1861:
1860:
1855:
1836:
1833:
1828:
1823:
1818:
1815:
1810:
1797:
1794:
1783:
1775:
1771:
1767:
1764:
1761:
1758:
1752:
1745:
1744:
1743:
1739:
1720:
1714:
1706:
1703:
1700:
1692:
1688:
1676:
1673:
1657:
1656:
1655:
1653:
1649:
1645:
1641:
1637:
1630:
1616:
1609:
1605:
1597:
1590:
1587:
1585:
1582:
1581:
1577:
1574:
1572:
1569:
1568:
1564:
1561:
1559:
1556:
1555:
1551:
1548:
1546:
1543:
1542:
1538:
1536:
1533:
1531:
1524:
1523:
1520:
1517:
1515:
1511:
1507:
1503:
1499:
1495:
1485:
1482:
1479:
1477:
1474:
1473:
1469:
1466:
1463:
1461:
1458:
1457:
1453:
1450:
1447:
1445:
1442:
1441:
1437:
1434:
1431:
1429:
1426:
1425:
1421:
1418:
1415:
1413:
1410:
1409:
1402:
1398:
1395:
1392:
1390:
1387:
1386:
1382:
1379:
1376:
1374:
1371:
1370:
1367:
1365:
1361:
1357:
1355:
1351:
1347:
1340:
1333:
1329:
1321:
1316:
1312:
1309:
1305:
1302:
1298:
1297:
1296:
1293:
1291:
1287:
1283:
1279:
1272:
1265:
1244:
1240:
1230:
1222:
1218:
1188:
1177:
1173:
1156:
1152:
1133:
1123:
1119:
1104:
1094:
1090:
1077:
1065:
1061:
1046:
1034:
1024:
1020:
1001:
997:
982:
970:
960:
956:
942:
934:
930:
926:
917:
913:
909:
904:
900:
891:
877:
873:
862:
858:
848:
841:
837:
833:
824:
820:
816:
811:
807:
798:
784:
780:
769:
765:
755:
749:
738:
734:
725:
709:
705:
696:
684:
681:
674:
673:
672:
670:
652:
648:
625:
621:
609:
602:
598:
594:
589:
587:
568:
560:
531:
522:
508:
496:
488:
476:
470:
462:
449:
448:
447:
445:
441:
437:
418:
410:
397:
392:
390:
386:
378:
376:
374:
370:
366:
361:
355:
353:
349:
345:
341:
337:
333:
329:
325:
313:
308:
306:
301:
299:
294:
293:
291:
290:
285:
280:
275:
274:
273:
272:
267:
264:
262:
259:
257:
254:
253:
252:
251:
246:
241:
238:
236:
233:
232:
231:
230:
225:
220:
217:
215:
212:
210:
207:
206:
205:
204:
199:
194:
191:
189:
186:
184:
181:
179:
176:
174:
171:
170:
169:
168:
163:
158:
155:
153:
150:
148:
145:
143:
140:
139:
138:
137:
132:
127:
124:
122:
119:
117:
114:
112:
109:
107:
106:Cox's theorem
104:
102:
99:
97:
94:
92:
89:
87:
84:
82:
79:
78:
77:
76:
71:
68:
64:
60:
56:
53:
52:
48:
44:
43:
40:
36:
32:
31:
19:
3411:
3402:10419/258120
3382:
3378:
3355:
3336:
3326:
3301:
3282:
3268:
3250:
3231:
3210:
3185:
3179:
3170:
3136:
3129:
3112:
3108:
3102:
3057:
3053:
3043:
2998:
2994:
2984:
2962:(430): 791.
2959:
2955:
2920:
2911:
2884:
2878:
2845:
2841:
2828:
2809:
2803:
2784:
2778:
2743:
2737:
2708:
2704:
2698:
2690:
2663:
2657:
2638:
2634:
2624:
2526:
2519:
2512:
2450:
2387:
2320:
2314:
2294:
2287:
2286:
2278:
2274:
2272:
2083:
2074:A classical
2073:
2066:would yield
2046:
2039:
2028:
2019:
2011:
2009:
1853:
1852:whereas for
1851:
1737:
1735:
1643:
1635:
1628:
1614:
1607:
1601:
1583:
1570:
1565:Substantial
1557:
1544:
1534:
1529:
1518:
1494:decihartleys
1491:
1475:
1470:Very strong
1459:
1443:
1438:Substantial
1427:
1411:
1400:
1388:
1372:
1363:
1358:
1353:
1349:
1338:
1331:
1327:
1325:
1294:
1270:
1263:
1203:
668:
607:
600:
596:
590:
585:
546:
442:is given by
439:
435:
393:
382:
360:MCMC samples
356:
351:
324:Bayes factor
323:
321:
256:Bayes factor
255:
3436:BayesFactor
3115:: 321–337.
2744:SIAM Review
2728:2066/226717
2308:, reducing
2061:frequentist
2022:frequentist
1464:31.6 to 100
1326:A value of
1286:overfitting
438:given data
434:of a model
385:likelihoods
3465:Categories
3304:. London:
3181:Biometrika
3176:Good, I.J.
2917:Gill, Jeff
2753:2005.08334
2616:References
2605:Odds ratio
2134:, whence
1506:I. J. Good
1467:5.0 to 6.6
1451:3.3 to 5.0
1448:10 to 31.6
1435:1.6 to 3.3
379:Definition
201:Estimators
73:Background
59:Likelihood
3385:(2): 31.
3358:. Wiley.
3156:cite book
3008:0911.1705
2870:206867662
2770:210156537
2496:≈
2480:−
2467:
2456:is about
2449:. Hence
2434:≈
2422:
2416:⋅
2410:−
2404:⋅
2386:. Model
2371:≈
2359:
2353:⋅
2347:−
2341:⋅
2254:≈
2237:^
2228:−
2210:^
2161:∣
2100:^
2086:, namely
1992:≈
1957:−
1905:∫
1885:∣
1834:≈
1768:∣
1704:−
1591:Decisive
1575:10 to 100
1562:3.2 to 10
1486:Decisive
1432:3.2 to 10
931:θ
901:θ
859:θ
849:∫
838:θ
808:θ
766:θ
756:∫
671:given by
649:θ
622:θ
101:Coherence
55:Posterior
3451:Archived
3325:(1994),
3094:21876135
3035:19880371
2862:20064637
2641:: 6–18.
2536:See also
2428:0.056991
2365:0.005956
1859:we have
1646:that is
1588:> 100
1558:1/2 to 1
1549:1 to 3.2
1545:0 to 1/2
1498:decibans
1483:> 6.6
1480:> 100
1460:10 to 10
1444:10 to 10
1428:10 to 10
1419:0 to 1.6
1416:1 to 3.2
1412:10 to 10
591:Given a
373:improper
352:in favor
332:evidence
67:Evidence
3202:0548210
3085:3174657
3062:Bibcode
3026:2796821
2976:2291091
2593:E-Value
2483:10.2467
2374:10.2467
2054:⁄
1648:uniform
1622:⁄
1598:Example
1578:Strong
1476:> 10
1454:Strong
1389:< 10
1350:against
1313:to use
1306:to use
3418:
3362:
3343:
3312:
3289:
3257:
3238:
3200:
3144:
3092:
3082:
3033:
3023:
2974:
2927:
2899:
2868:
2860:
2816:
2791:
2768:
2678:
2477:7.7297
2437:7.7297
1634:where
1584:> 2
1571:1 to 2
1510:humans
1396:< 0
1393:< 1
1317:(MDL).
1310:(MML).
3379:Risks
3003:arXiv
2972:JSTOR
2952:(PDF)
2866:S2CID
2838:(PDF)
2766:S2CID
2748:arXiv
2499:0.284
2122:0.575
1995:0.005
1837:0.006
1377:dHart
63:Prior
3416:ISBN
3360:ISBN
3341:ISBN
3310:ISBN
3287:ISBN
3255:ISBN
3236:ISBN
3162:link
3142:ISBN
3090:PMID
3031:PMID
2925:ISBN
2897:ISBN
2858:PMID
2814:ISBN
2789:ISBN
2676:ISBN
2257:0.06
1642:for
1502:bits
1380:bits
1269:and
640:and
606:and
394:The
322:The
3397:hdl
3387:doi
3272:doi
3190:doi
3117:doi
3080:PMC
3070:doi
3058:108
3021:PMC
3013:doi
2964:doi
2889:doi
2850:doi
2758:doi
2723:hdl
2713:doi
2668:doi
2643:doi
2464:exp
2217:115
2192:115
2189:200
2158:115
2114:200
2111:115
2027:of
1987:201
1946:115
1930:115
1927:200
1882:115
1829:200
1798:115
1795:200
1765:115
1693:115
1677:115
1674:200
1525:log
1500:);
1354:for
3467::
3395:.
3381:.
3377:.
3308:.
3198:MR
3196:.
3186:66
3184:.
3158:}}
3154:{{
3113:14
3111:.
3088:.
3078:.
3068:.
3056:.
3052:.
3029:.
3019:.
3011:.
2999:26
2997:.
2993:.
2970:.
2960:90
2958:.
2954:.
2939:^
2895:.
2864:.
2856:.
2846:60
2844:.
2840:.
2764:.
2756:.
2721:.
2707:.
2703:.
2674:.
2639:72
2637:.
2633:.
2419:ln
2356:ln
2312:.
2284:.
2248:85
2049:=
2020:A
2017:.
1968:85
1715:85
1654::
1617:=
1527:10
1406:)
1366::
1234:Pr
1212:Pr
1167:Pr
1146:Pr
1113:Pr
1084:Pr
1055:Pr
1041:Pr
1014:Pr
991:Pr
977:Pr
950:Pr
886:Pr
852:Pr
793:Pr
759:Pr
720:Pr
691:Pr
555:Pr
517:Pr
503:Pr
483:Pr
457:Pr
446::
405:Pr
65:Ă·
61:Ă—
57:=
3424:.
3405:.
3399::
3389::
3383:9
3368:.
3349:.
3318:.
3295:.
3274::
3263:.
3244:.
3204:.
3192::
3164:)
3150:.
3123:.
3119::
3096:.
3072::
3064::
3037:.
3015::
3005::
2978:.
2966::
2933:.
2905:.
2891::
2872:.
2852::
2822:.
2797:.
2772:.
2760::
2750::
2731:.
2725::
2715::
2709:3
2699:P
2684:.
2670::
2651:.
2645::
2530:1
2527:M
2523:2
2520:M
2516:2
2513:M
2492:)
2487:2
2471:(
2454:1
2451:M
2431:)
2425:(
2413:2
2407:1
2401:2
2391:2
2388:M
2368:)
2362:(
2350:2
2344:0
2338:2
2324:1
2321:M
2298:1
2295:M
2291:2
2288:M
2282:2
2279:M
2275:q
2244:)
2234:q
2225:1
2222:(
2207:q
2197:)
2184:(
2177:=
2174:)
2169:2
2165:M
2155:=
2152:X
2149:(
2146:P
2119:=
2106:=
2097:q
2084:q
2056:2
2052:1
2047:q
2043:1
2040:M
2032:1
2029:M
2015:1
2012:M
1984:1
1979:=
1976:q
1973:d
1964:)
1960:q
1954:1
1951:(
1942:q
1935:)
1922:(
1914:1
1909:0
1901:=
1898:)
1893:2
1889:M
1879:=
1876:X
1873:(
1870:P
1857:2
1854:M
1824:)
1819:2
1816:1
1811:(
1803:)
1790:(
1784:=
1781:)
1776:1
1772:M
1762:=
1759:X
1756:(
1753:P
1741:1
1738:M
1721:.
1711:)
1707:q
1701:1
1698:(
1689:q
1682:)
1669:(
1644:q
1636:q
1632:2
1629:M
1624:2
1620:1
1615:q
1611:1
1608:M
1535:K
1530:K
1404:2
1401:M
1373:K
1364:K
1342:2
1339:M
1335:1
1332:M
1328:K
1274:2
1271:M
1267:1
1264:M
1250:)
1245:2
1241:M
1237:(
1231:=
1228:)
1223:1
1219:M
1215:(
1189:.
1183:)
1178:1
1174:M
1170:(
1162:)
1157:2
1153:M
1149:(
1137:)
1134:D
1130:|
1124:2
1120:M
1116:(
1108:)
1105:D
1101:|
1095:1
1091:M
1087:(
1078:=
1071:)
1066:2
1062:M
1058:(
1050:)
1047:D
1044:(
1038:)
1035:D
1031:|
1025:2
1021:M
1017:(
1007:)
1002:1
998:M
994:(
986:)
983:D
980:(
974:)
971:D
967:|
961:1
957:M
953:(
943:=
935:2
927:d
923:)
918:2
914:M
910:,
905:2
896:|
892:D
889:(
883:)
878:2
874:M
869:|
863:2
855:(
842:1
834:d
830:)
825:1
821:M
817:,
812:1
803:|
799:D
796:(
790:)
785:1
781:M
776:|
770:1
762:(
750:=
744:)
739:2
735:M
730:|
726:D
723:(
715:)
710:1
706:M
701:|
697:D
694:(
685:=
682:K
669:K
653:2
626:1
611:2
608:M
604:1
601:M
597:D
586:M
572:)
569:M
565:|
561:D
558:(
532:.
526:)
523:D
520:(
512:)
509:M
506:(
500:)
497:M
493:|
489:D
486:(
477:=
474:)
471:D
467:|
463:M
460:(
440:D
436:M
422:)
419:D
415:|
411:M
408:(
311:e
304:t
297:v
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
