Choice of priors for hierarchical models: Admissibility and computation

Hierarchical Bayesian analysis is extensively utilized in statistical practice. Surprisingly, however, little is known about optimal choice of the prior distribution for the hyperparameters at the highest level of the hierarchical model. The standard choice for such a hyper prior is simply the constant prior. We will see, however, that this choice is usually not optimal in the sense of admissibility. Furthermore, the constant hyper prior can even lead to improper posterior distributions. We consider the block multivariate normal mean estimation problem (sometimes called the ‘matrix of means’ problem), with unknown covariance matrix at the highest level of the hierarchical model. For a variety of hyper priors, we first give necessary and sufficient conditions under which the posterior distribution is proper. Next, admissibility and inadmissibility of the resulting Bayes estimators is studied, under quadratic loss. Finally, recommended prior distributions for the model are presented. Recommended prior distributions for the covariance matrix do not result in closed form estimates, so that simulation from the resulting posterior distributions is necessary. We study various Metropolis-Hastings schemes for implementing the simulation, and recommend a particular scheme as being efficient and very easy to implement. Finally, a few comparisons of the risk of Bayesian estimators for covariance matrices are presented