  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.
