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

The marginal likelihood is a well established model selection criterion in Bayesian statistics. It also allows to efficiently calculate the marginal posterior model probabilities that can be used for Bayesian model averaging of quantities of interest. For many complex models, including latent modeling approaches, marginal likelihoods are however difficult to compute. One recent promising approach for approximating the marginal likelihood is Integrated Nested Laplace Approximation (INLA), design for models with latent Gaussian structures. In this study we compare the approximations obtained with INLA to some alternative approaches on a number of examples of different complexity. In particular we address a simple linear latent model, a Bayesian linear regression model, logistic Bayesian regression models with probit and logit links, and a Poisson longitudinal generalized linear mixed model.