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August 2011 Is Bayes Posterior just Quick and Dirty Confidence?
D. A. S. Fraser
Statist. Sci. 26(3): 299-316 (August 2011). DOI: 10.1214/11-STS352


Bayes [Philos. Trans. R. Soc. Lond. 53 (1763) 370–418; 54 296–325] 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: Fisher [Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 222 (1922) 309–368] for likelihood and Neyman [Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 237 (1937) 333–380] for confidence. Lindley [J. Roy. Statist. Soc. Ser. B 20 (1958) 102–107] 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 approach 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.


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D. A. S. Fraser. "Is Bayes Posterior just Quick and Dirty Confidence?." Statist. Sci. 26 (3) 299 - 316, August 2011.


Published: August 2011
First available in Project Euclid: 31 October 2011

zbMATH: 1246.62040
MathSciNet: MR2918001
Digital Object Identifier: 10.1214/11-STS352

Keywords: Bayes , Bayes error rate , Confidence , default prior , evaluating a prior , nonlinear parameter , Posterior , prior

Rights: Copyright © 2011 Institute of Mathematical Statistics

Vol.26 • No. 3 • August 2011
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