An objective prior for hyperparameters in normal hierarchical models

Hierarchical models are the workhorse of much of Bayesian analysis, yet there is uncertainty as to which priors to use for hyperparameters. Formal approaches to objective Bayesian analysis, such as the Jeffreys-rule approach or reference prior approach, are only implementable in simple hierarchical settings. It is thus common to use less formal approaches, such as utilizing formal priors from non-hierarchical models in hierarchical settings. This can be fraught with danger, however. For instance, non-hierarchical Jeffreys-rule priors for variances or covariance matrices result in improper posterior distributions if they are used at higher levels of a hierarchical model. Berger et al. (2005) approached the question of choice of hyperpriors in normal hierarchical models by looking at the frequentist notion of admissibility of resulting estimators. Hyperpriors that are ‘on the boundary of admissibility’ are sensible choices for objective priors, being as diffuse as possible without resulting in inadmissible procedures. The admissibility (and propriety) properties of a number of priors were considered in the paper, but no overall conclusion was reached as to a specific prior. In this paper, we complete the story and propose a particular objective prior for use in all normal hierarchical models, based on considerations of admissibility, ease of implementation and performance.