Statistical Science

Is Bayes Posterior just Quick and Dirty Confidence?

D. A. S. Fraser

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Abstract

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.

Article information

Source
Statist. Sci. Volume 26, Number 3 (2011), 299-316.

Dates
First available: 31 October 2011

Permanent link to this document
http://projecteuclid.org/euclid.ss/1320066918

Digital Object Identifier
doi:10.1214/11-STS352

Mathematical Reviews number (MathSciNet)
MR2918001

Zentralblatt MATH identifier
06075172

Citation

Fraser, D. A. S. Is Bayes Posterior just Quick and Dirty Confidence?. Statistical Science 26 (2011), no. 3, 299--316. doi:10.1214/11-STS352. http://projecteuclid.org/euclid.ss/1320066918.


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See also

  • Discussion of: Is Bayes Posterior just Quick and Dirty Confidence? by D. A. S. Fraser.
  • Discussion of: Is Bayes Posterior just Quick and Dirty Confidence? by D. A. S. Fraser.
  • Discussion of: Is Bayes Posterior just Quick and Dirty Confidence? by D. A. S. Fraser.
  • Discussion of: Is Bayes Posterior just Quick and Dirty Confidence? by D. A. S. Fraser.
  • Rejoinder: Is Bayes Posterior just Quick and Dirty Confidence?.