An Approach to Robust Bayesian Analysis for Multidimensional Parameter Spaces

Abstract

Abstract In some multiparameter Bayesian problems the prior is more naturally described as a joint density for all of the parameters and in others it is more naturally described as a product of marginal and conditional densities. Although ordinary quantile, ε-contamination, and density-ratio classes of priors may be sensible choices for robust Bayesian analyses in the former case, they may be less satisfactory in the latter. This article shows how to construct product classes of priors for use in the second case, using one class of priors for the marginal and another for each conditional. Then the article introduces density-bounded classes as an alternative to the more familiar classes for representing uncertainty about the prior. It is often of interest to find tight bounds on posterior expectations as the prior ranges over the class of priors—an apparently difficult nonlinear optimization problem regardless of which class of priors is used. This article describes a technique for turning the problem into a sequence of linear optimizations that can be much easier to handle. Then it shows how to solve the linear optimization problem for quantile, ε-contamination, density-ratio, and density-bounded classes. The ideas are illustrated with an example from a clinical trial.