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Typical Bayesian methods for models with latent variables (or random effects)involve directly sampling the latent variables along with the model parameters.In high-level software code for model definitions (using, e.g., BUGS, JAGS,Stan), the likelihood is therefore specified as conditional on the latentvariables. This can lead researchers to perform model comparisons viaconditional likelihoods, where the latent variables are considered modelparameters. In other settings, however, typical model comparisons involvemarginal likelihoods where the latent variables are integrated out. Thisdistinction is often overlooked despite the fact that it can have a largeimpact on the comparisons of interest. In this paper, we clarify and illustratethese issues, focusing on the comparison of conditional and marginal DevianceInformation Criteria (DICs) and Watanabe-Akaike Information Criteria (WAICs) inpsychometric modeling. The conditional/marginal distinction corresponds towhether the model should be predictive for the clusters that are in the data orfor new clusters (where "clusters" typically correspond to higher-level unitslike people or schools). Correspondingly, we show that marginal WAICcorresponds to leave-one-cluster out (LOcO) cross-validation, whereasconditional WAIC corresponds to leave-one-unit out (LOuO). These results leadto recommendations on the general application of the criteria to models withlatent variables.

Bayesian comparison of latent variable models: Conditional vs marginal likelihoods