Empirical Bayesian Analysis Through the Lens of a Particular Class of Constrained Bayesian Hierarchical Models

Empirical Bayesian analysis is a well-known approach that incorporates an estimator into a Bayesian analysis. In this article, we offer another approach, which has several useful properties. Our solution is based on the framework introduced by Yekutieli (2012) to account for the variability introduced by selecting parameters. Specifically, we assume that the unknown parameter is contained within a ball centered at an estimator, and the radius is given a prior distribution. We refer to our method as the auxiliary parameter constrained Bayesian hierarchical model (C-BHM). This general framework is particularly exciting as traditional empirical Bayesian analysis and parametric Bayesian analysis can be written as special cases. Hence, this C-BHM represents a unifying framework within the area of Bayesian statistics. Several technical results are provided. Furthermore, we show analytically that one can outperform both empirical and fully Bayesian analysis through the Bayes factor. We illustrate the C-BHM to extend the Fay-Herriot model, which is often used in the survey sampling setting. To demonstrate the usefulness of our method we provide simulations and an illustration to data obtained from the U.S. Census Bureau's Small Area Income and Poverty Estimates (SAIPE) program.