Introduction and the Shrinkage Argument

Abstract

The study of priors ensuring, up to the desired order of asymptotics, the approximate frequentist validity of posterior credible sets has received significant attention in recent years and a considerable interest is still continuing in this field. Bayesian credible sets based on these priors have approximately correct frequentist coverage as well. Such priors are generically known as probability matching priors, or matching priors in short. As noted by Tibshirani (1989) among others, study in this direction has several important practical implications with appeal to both Bayesians and frequentists: (a) First, the ensuing matching priors are, in a sense, noninformative. The approximate agreement between the Bayesian and frequentist coverage probabilities of the associated credible sets provides an external validation for these priors. They can form the basis of an objective Bayesian analysis and are potentially useful for comparative purposes in subjective Bayesian analyses as well. (b) Second, Bayesian credible sets given by matching priors can also be interpreted as accurate frequentist confidence sets because of their approximately correct frequentist coverage. Thus the exploration of matching priors provides a route for obtaining accurate frequentist confidence sets which are meaningful also to a Bayesian. (c) In addition, research in this area has led to the development of a powerful and transparent Bayesian route, via a shrinkage argument, for higher order asymptotic frequentist computations.