Bayes/non-Bayes blended inference

5 October 2012

Updated with a new multiple comparison procedure and applications on 30 June 2012 and with slides for a presentation on 5 October 2012:

D. R. Bickel, “Blending Bayesian and frequentist methods according to the precision of prior information with applications to hypothesis testing,” Working Paper, University of Ottawa, deposited in uO Research at (2012)2012 preprint | 2011 preprint | Slides

This framework of statistical inference facilitates the development of new methodology to bridge the gap between the frequentist and Bayesian theories. As an example, a simple and practical method for combining p-values with a set of possible posterior probabilities is provided.

In this new approach to statistics, Bayesian inference is used when the prior distribution is known, frequentist inference is used when nothing is known about the prior, and both types of inference are blended according to game theory when the prior is known to be a member of some set. (The robust Bayes framework represents knowledge about a prior in terms of a set of possible priors.) If the benchmark posterior that corresponds to frequentist inference lies within the set of Bayesian posteriors derived from the set of priors, then the benchmark posterior is used for inference. Otherwise, the posterior within that set that is closest to the benchmark posterior is used for inference.