Bayesian predictive model comparison via parallel sampling

Methods of model comparison and checking, and associated criteria, are proposed based on parallel sampling of two or more models subsequent to convergence. These complement Bayesian predictive criteria already proposed (e.g. error sum of squares and deviance based) but are on a scale that may be compared across applications. Penalised criteria for model comparison based on the AIC are also investigated, together with AIC model weights and evidence ratios. Parallel sampling enables posterior summaries to be obtained for continuous comparison measures (e.g. likelihood and evidence ratios). A forward selection procedure for regression is suggested as one possible extension, as well as procedures for model averaging and posterior predictive checking. Comparisons with the DIC are made together with implications of parallel sampling for assessing the density of the DIC. Three worked examples illustrate the working of the procedures in practice.