Bayesian Interpretation of Some Empirical Bayes Procedures in Hierarchical Models

In modern statistics and its applications, in particular in econometrics, random parameters or latent variables are widely used, and their estimation or prediction is of interest. Under some prior assumptions, Bayes formula can be used to obtain their posterior distribution. However, on the sampling-theory grounds, the unknown constants appearing in the prior distribution are estimated using the data being actually modelled. We call such approaches quasi-Bayesian; empirical Bayes procedures give important examples. In this paper we propose theoretical framework that enables Bayesian validation (or interpretation) of quasi-Bayesian inference techniques. Our framework amounts to establishing a formal Bayesian model that justifies the quasi-Bayesian "posterior" as a valid posterior distribution. From the Bayesian model validating the quasi-posterior, i.e. from the joint distribution of observations and other quantities, one can deduce the true sampling model, that is the conditional distribution of observations, and the true prior (or marginal) distribution of the remaining quantities. We illustrate our approach not only by simple examples, but also by the complicated Bayesian model validating one of the basic empirical Bayes estimators of the multivariate normal mean. This model is in fact a non-standard joint measure that separates two subsets of the Cartesian product of the observation space and the parameter space.