Choice of hierarchical priors: admissibility in estimation of normal means

Abstract

In hierarchical Bayesian modeling of normal means, it is common to complete the prior specification by choosing a constant prior density for unmodeled hyperparameters (e.g., variances and highest-level means). This common practice often results in an inadequate overall prior, inadequate in the sense that estimators resulting from its use can be inadmissible under quadratic loss. In this paper, hierarchical priors for normal means are categorized in terms of admissibility and inadmissibility of resulting estimators for a quite general scenario. The Jeffreys prior for the hyper-variance and a shrinkage prior for the hypermeans are recommended as admissible alternatives. Incidental to this analysis is presentation of the conditions under which the (generally improper) priors result in proper posteriors.