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The marginal likelihood is a well established model selection criterion inBayesian statistics. It also allows to efficiently calculate the marginalposterior model probabilities that can be used for Bayesian model averaging ofquantities of interest. For many complex models, including latent modelingapproaches, marginal likelihoods are however difficult to compute. One recentpromising approach for approximating the marginal likelihood is IntegratedNested Laplace Approximation (INLA), design for models with latent Gaussianstructures. In this study we compare the approximations obtained with INLA tosome alternative approaches on a number of examples of different complexity. Inparticular we address a simple linear latent model, a Bayesian linearregression model, logistic Bayesian regression models with probit and logitlinks, and a Poisson longitudinal generalized linear mixed model.

Estimating the marginal likelihood with Integrated nested Laplace approximation (INLA)