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), designed for models with latent Gaussian structures. In this study we compare the approximations obtained with INLA to alternative approaches across a progression of models of varying complexity. In particular, we evaluate a simple linear latent model with exact analytical truth, a Bayesian linear regression model, probit and logit Bayesian regression models, a longitudinal Poisson generalized linear mixed model, and two extra simulation scenarios designed to probe the limits of the latent Gaussian approximation.