Bayesian case-deletion model complexity and information criterion

We establish a connection between Bayesian case influence measures for assessing the influence of individual observations and Bayesian predictive methods for evaluating the predictive performance of a model and comparing different models fit to the same dataset. Based on such a connection, we formally propose a new set of Bayesian case-deletion model complexity (BCMC) measures for quantifying the effective number of parameters in a given statistical model and its properties in linear models are explored. Adding certain functions of BCMC to a conditional deviance function leads to a Bayesian case-deletion information criterion (BCIC) for comparing models. We systematically investigate some properties of BCIC and its connections with other information criteria, such as the Deviance Information Criterion (DIC). We illustrate the proposed methodology for the linear mixed model with simulations and a real data example.