Bayesian tests and model diagnostics in conditionally independent hierarchical models

Consider the conditionally independent hierarchical model (CIHM) in which observations y_i are independently distributed from f(y_i/Θ_i)the parameters Θ_i are independently distributed from distributions g(Θ|λ), and the hyperparameters λ are distributed according to a distribution h(λ). The posterior distribution of all parameters of the CIHM can be efficiently simulated by Markov chain Monte Carlo (MCMC) algorithms. Although these simulation algorithms have facilitated the application of CIHMs, they generally have not addressed the problem of computing quantities useful in model selection. This article explores how MCMC simulation algorithms and other related computational algorithms can be used to compute Bayes factors that are useful in criticizing a particular CIHM. In the case where the CIHM models a belief that the parameters are exchangeable or lie on a regression surface, the Bayes factor can measure the consistency of the data with the structural prior belief. Bayes factors can also be used to judge the suitability of particular assumptions in CIHMs, including the choice of link function, the nonexistence or existence of outliers, and the prior belief in exchangeability. The methods are illustrated in the situation in which a CIHM is used to model structural prior information about a set of binomial probabilities.