Constrained Bayesian Hierarchical Models for Gaussian Data: A Model Selection Criterion Approach

Consider the setting where there are B>1 candidate statistical models, and one is interested in model selection. Two common approaches to solve this problem are to select a single model or to combine the candidate models through model averaging. Instead, we select a subset of the combined parameter space associated with the models. Specifically, a model averaging perspective is used to increase the parameter space, and a model selection criterion is used to select a subset of this expanded parameter space. We account for the variability of the criterion by adapting Yekutieli (2012)'s method to Bayesian model averaging (BMA). Yekutieli (2012)'s method treats model selection as a truncation problem. We truncate the joint support of the data and the parameter space to only include small values of the covariance penalized error (CPE) criterion. The CPE is a general expression that contains several information criteria as special cases. Simulation results show that as long as the truncated set does not have near zero probability, we tend to obtain lower mean squared error than BMA. Additional theoretical results are provided that provide the foundation for these observations. We apply our approach to a dataset consisting of American Community Survey (ACS) period estimates to illustrate that this perspective can lead to improvements of a single model.