Posterior and Cross-validatory Predictive Checks: A Comparison of MCMC and INLA

Model criticism and comparison of Bayesian hierarchical models is often based on posterior or leave-one-out cross-validatory predictive checks. Cross-validatory checks are usually preferred because posterior predictive checks are difficult to assess and tend to be too conservative. However, techniques for statistical inference in such models often try to avoid full (manual) leave-one-out cross-validation, since it is very time-consuming. In this paper we will compare two approaches for estimating Bayesian hierarchical models: Markov chain Monte Carlo (MCMC) and integrated nested Laplace approximations (INLA). We review how both approaches allow for the computation of leave-one-out cross-validatory checks without re-running the model for each observation in turn. We then empirically compare the two approaches in an extensive case study analysing the spatial distribution of bovine viral diarrhoe (BVD) among cows in Switzerland.