Bayesian Robustness with Mixture Classes of Priors

Abstract

Uncertainty in specification of the prior distribution is a common concern with Bayesian analysis. The robust Bayesian approach is to work with a class of prior distributions, which model uncertainty about the prior, instead of a single distribution. One is interested in the range of the posterior expectations of certain parametric functions as the prior varies over the class being considered--if this range is small, the analysis is robust to misspecification of the prior. Relatively little research has dealt with robustness with respect to priors on several parameters, especially the problem of imposing shape and smoothness constraints on the priors in the class. To address this problem, we consider neighborhood classes of mixture priors. Results are presented for two kinds of "mixture classes," which yield different types of neighborhoods. The problem of finding suprema and infima of posterior expectations of parametric functions is seen to reduce to numerical maximization and minimization. In the applications we consider mixtures of uniform densities on variously shaped sets. This allows one to model symmetry and unimodality of different types in more than one dimension. Numerical examples are provided.