Expressing Regret: A Unified View of Credible Intervals

Abstract

Posterior uncertainty is typically summarized as a credible interval, an interval in the parameter space that contains a fixed proportion—usually 95%—of the posterior’s support. For multivariate parameters, credible sets perform the same role. There are of course many potential 95% intervals from which to choose, yet even standard choices are rarely justified in any formal way. In this article we give a general method, focusing on the loss function that motivates an estimate—the Bayes rule—around which we construct a credible set. The set contains all points which, as estimates, would have minimally-worse expected loss than the Bayes rule: we call this excess expected loss “regret.” The approach can be used for any model and prior, and we show how it justifies all widely used choices of credible interval/set. Further examples show how it provides insights into more complex estimation problems. Supplementary materials for this article are available online.