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.


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


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