Hierarchical models are increasingly used in many applications. Along with this increased use comes a desire to investigate whether the model is compatible with the observed data. Bayesian methods are well suited to eliminate the many (nuisance) parameters in these complicated models; in this paper we investigate Bayesian methods for model checking. Since we contemplate model checking as a preliminary, exploratory analysis, we concentrate on objective Bayesian methods in which careful specification of an informative prior distribution is avoided. Numerous examples are given and different proposals are investigated and critically compared.
"Bayesian Checking of the Second Levels of Hierarchical Models." Statist. Sci. 22 (3) 322 - 343, August 2007. https://doi.org/10.1214/07-STS235