Bayesian optimality and intervals for Stein‐type estimates

We provide a novel Bayesian decision‐theoretic motivation for Stein‐type estimates, producing them as an adaptive choice between standard point estimation and estimation that rewards proximity to the origin. Unlike conventional approaches, our arguments provide shrunken estimates under any sampling model or prior. They also lead naturally to a form of credible interval, describing uncertainty about the underlying parameters yet focusing on the shrunken estimate. One specific method for constructing intervals provides a close Bayesian analogue of Samworth (2005)'s approximate confidence intervals around Stein estimates. Several examples are given, showing how shrinkage's focus on regions remote to the high‐support areas of the posterior can lead to substantially larger credible sets.