Is Bayes Posterior just Quick and Dirty Confidence?

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.

Published in at http://dx.doi.org/10.1214/11-STS352 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (3)

• Likelihood inference in the presence of nuisance parameters
• Second order ancillary: A differential view from continuity
• Higher Accuracy for Bayesian and Frequentist Inference: Large Sample Theory for Small Sample Likelihood

Cited by in corpus (17)

• Inferential models: A framework for prior-free posterior probabilistic inference
• Marginal inferential models: prior-free probabilistic inference on interest parameters
• Conditional inferential models: combining information for prior-free probabilistic inference
• False confidence, non-additive beliefs, and valid statistical inference
• A comparison of learning rate selection methods in generalized Bayesian inference
• On an inferential model construction using generalized associations
• Posterior Model Consistency in Variable Selection as the Model Dimension Grows
• Model-free posterior inference on the area under the receiver operating characteristic curve
• Gibbs posterior inference on multivariate quantiles
• Frameworks for prior-free posterior probabilistic inference
• Controlling the degree of caution in statistical inference with the Bayesian and frequentist approaches as opposite extremes
• Direct and approximately valid probabilistic inference on a class of statistical functionals
• Discussion: Foundations of Statistical Inference, Revisited
• Asymptotic efficiency of inferential models and a possibilistic Bernstein--von Mises theorem
• Improved inference on risk measures for univariate extremes
• Possibilistic inferential models: a review
• Discussion of "On the Birnbaum Argument for the Strong Likelihood Principle"