×

Is Bayes posterior just quick and dirty confidence? (English) Zbl 1246.62040

Summary: Th. Bayes [Philos. Trans. R. Soc. Lond. 53 370–418 (1763; Zbl 1250.60007); ibid. 54, 296–325 (1763; doi:10.1098/rstl.1764.0050)] introduced the observed likelihood function to statistical inference and provided a weight function to calibrate the parameter; he also introduced a confidence distribution on the parameter space but did not provide present justifications. Of course the names likelihood and confidence did not appear until much later: R.A. Fisher [Lond. Phil. Trans. (A) 222, 309–368 (1922; JFM 48.1280.02)] for likelihood and J. Neyman [Philos. Trans. Roy. Soc. London, Ser. A 236, 333–380 (1937; Zbl 0017.12403; JFM 63.0515.02)] for confidence. D.V. Lindley [J. R. Stat. Soc., Ser. B 20, 102–107 (1958; Zbl 0085.35503)] showed that the Bayes and the confidence results were different when the model was not location.
This paper examines the occurrence of true statements from the Bayes and from the confidence approach, and shows that the proportion of true statements in the Bayes case depends critically on the presence of linearity in the model; and with departure from this linearity the Bayes approach can be a poor approximation and be seriously misleading. Bayesian integration of weighted likelihood thus provides a first-order linear approximation to confidence, but without linearity can give substantially incorrect results.

MSC:

62F15 Bayesian inference
62A01 Foundations and philosophical topics in statistics
62F25 Parametric tolerance and confidence regions
PDF BibTeX XML Cite
Full Text: DOI arXiv Euclid

References:

[1] Abebe, F., Cakmak, S., Cheah, P. K., Fraser, D. A. S., Kuhn, J., McDunnough, P., Reid, N. and Tapia, A. (1995). Third order asymptotic model: Exponential and location type approximations. Parisankhyan Samikkha 2 25-33.
[2] Andrews, D. F., Fraser, D. A. S. and Wong, A. C. M. (2005). Computation of distribution functions from likelihood information near observed data. J. Statist. Plann. Inference 134 180-193. · Zbl 1066.62021
[3] Bayes, T. (1763). An essay towards solving a problem in the doctrine of chances. Philos. Trans. R. Soc. Lond. 53 370-418; 54 296-325. Reprinted in Biometrika 45 (1958) 293-315. · Zbl 1250.60007
[4] Bédard, M., Fraser, D. A. S. and Wong, A. (2007). Higher accuracy for Bayesian and frequentist inference: Large sample theory for small sample likelihood. Statist. Sci. 22 301-321. · Zbl 1246.62027
[5] Bernardo, J.-M. (1979). Reference posterior distributions for Bayesian inference (with discussion). J. Roy. Statist. Soc. Ser. B 41 113-147. · Zbl 0428.62004
[6] Bernardo, J.-M. and Smith, A. F. M. (1994). Bayesian Theory . Wiley, Chichester. · Zbl 0796.62002
[7] Box, G. E. P. and Cox, D. R. (1964). An analysis of transformations (with discussion). J. Roy. Statist. Soc. Ser. B 26 211-252. · Zbl 0156.40104
[8] Cakmak, S., Fraser, D. A. S., McDunnough, P., Reid, N. and Yuan, X. (1998). Likelihood centered asymptotic model: Exponential and location model versions. J. Statist. Plann. Inference 66 211-222. · Zbl 0953.62017
[9] Cox, D. R. (1958). Some problems connected with statistical inference. Ann. Math. Statist. 29 357-372. · Zbl 0088.11702
[10] Dawid, A. P., Stone, M. and Zidek, J. V. (1973). Marginalization paradoxes in Bayesian and structural inference (with discussion). J. Roy. Statist. Soc. Ser. B 35 189-233. · Zbl 0271.62009
[11] Fisher, R. A. (1922). On the mathematical foundations of theoretical statistics. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 222 309-368. · JFM 48.1280.02
[12] Fisher, R. A. (1930). Inverse probability. Proc. Camb. Phil. Soc. 26 528-535. · JFM 56.1083.05
[13] Fisher, R. A. (1935). The fiducial argument in statistical inference. Ann. Eugenics B 391-398. · JFM 62.1345.02
[14] Fraser, A. M. Fraser, D. A. S. and Fraser, M. J. (2010a). Parameter curvature revisited and the Bayesian frequentist divergence. J. Statist. Res. 44 335-346. · Zbl 1204.58009
[15] Fraser, A. M., Fraser, D. A. S. and Staicu, A. M. (2010c). Second order ancillary: A differential view with continuity. Bernoulli 16 1208-1223. · Zbl 1207.62041
[16] Fraser, D. A. S. and McDunnough, P. (1980). Some remarks on conditional and unconditional inference for location-scale models. Statist. Hefte (N.F.) 21 224-231. · Zbl 0444.62045
[17] Fraser, D. A. S., Reid, N. and Wong, A. (2004). Setting confidence intervals for bounded parameters: A different perspective. Phys. Rev. D 69 033002.
[18] Fraser, D. A. S., Reid, N., Marras, E. and Yi, G. Y. (2010b). Default prior for Bayesian and frequentist inference. J. Roy. Statist. Soc. Ser. B 75 631-654.
[19] Heinrich, J. (2006). The Bayesian approach to setting limits: What to avoid? In Statistical Problems in Particle Physics , Astrophysics and Cosmology (L. Lyons and Ü. M. Karagöz, eds.) 98-102. Imperial College Press, London.
[20] Jeffreys, H. (1939). Theory of Probability , 3rd ed. Oxford Univ. Press, Oxford. · Zbl 0023.14501
[21] Jeffreys, H. (1946). An invariant form for the prior probability in estimation problems. Proc. Roy. Soc. London. Ser. A 186 453-461. · Zbl 0063.03050
[22] Lindley, D. V. (1958). Fiducial distributions and Bayes’ theorem. J. Roy. Statist. Soc. Ser. B 20 102-107. · Zbl 0085.35503
[23] Mandelkern, M. (2002). Setting confidence intervals for bounded parameters. Statist. Sci. 17 149-172. · Zbl 1013.62028
[24] Neyman, J. (1937). Outline of a theory of statistical estimation based on the classical theory of probability. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 237 333-380. · Zbl 0017.12403
[25] Reid, N. and Fraser, D. A. S. (2003). Likelihood inference in the presence of nuisance parameters. In Proceedings of PHYSTAT2003 (L. Lyons, R. Mount and R. Reitmeyer, eds.) 265-271. SLAC E-Conf C030908.
[26] Stainforth, D. A., Allen, M. R., Tredger, E. R. and Smith, L. A. (2007). Confidence, uncertainty and decision-support relevance in climate predictions. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 365 2145-2162. See also: Gambling on tomorrow. Modelling the Earth’s climate mathematically is hard already. Now a new difficulty is emerging. Economist August 18 (2007) 69.
[27] Wasserman, L. (2000). Asymptotic inference for mixture models using data-dependent priors. J. R. Stat. Soc. Ser. B Stat. Methodol. 62 159-180. · Zbl 0976.62028
[28] Woodroofe, M. and Wang, H. (2000). The problem of low counts in a signal plus noise model. Ann. Statist. 28 1561-1569. · Zbl 1105.62300
[29] Zhang, T. and Woodroofe, M. (2003). Credible and confidence sets for restricted parameter spaces. J. Statist. Plann. Inference 115 479-490. · Zbl 1030.62019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.