Approximate Bayesian Inference in Conditionally Independent Hierarchical Models (Parametric Empirical Bayes Models)

Abstract

Abstract We consider two-stage models of the kind used in parametric empirical Bayes (PEB) methodology, calling them conditionally independent hierarchical models. We suppose that there are k “units,” which may be experimental subjects, cities, study centers, etcetera. At the first stage, the observation vectors Yi for units i = 1, …, k are independently distributed with densities p(yi | θi ), or more generally, p(yi | θi, λ). At the second stage, the unit-specific parameter vectors θi are iid with densities p(θi | λ). The PEB approach proceeds by regarding the second-stage distribution as a prior and noting that, if λ were known, inference about θ could be based on its posterior. Since λ is not known, the simplest PEB methods estimate the parameter λ by maximum likelihood or some variant, and then treat λ as if it were known to be equal to this estimate. Although this procedure is sometimes satisfactory, a well-known defect is that it neglects the uncertainty due to the estimation of λ. In this article w...