Hierarchical Bayes Modeling for Large-Scale Inference

Bayesian modeling is now ubiquitous in problems of large-scale inference even when frequentist criteria are in mind for evaluating the performance of a procedure. By far most popular in literature of the past decade and a half are empirical Bayes methods, that have shown in practice to improve significantly over strictly-frequentist competitors in many different problems. As an alternative to empirical Bayes methods, in this paper we propose hierarchical Bayes modeling for large-scale problems, and address two separate points that, in our opinion, deserve more attention. The first is nonparametric "deconvolution" methods that are applicable also outside the sequence model. The second point is the adequacy of Bayesian modeling for situations where the parameters are by assumption deterministic. We provide partial answers to both: first, we demonstrate how our methodology applies in the analysis of a logistic regression model. Second, we appeal to Robbins's compound decision theory and provide an extension, to give formal justification for the Bayesian approach in the sequence case.