Improving the INLA approach for approximate Bayesian inference for latent Gaussian models

We introduce a new copula-based correction for generalized linear mixed models (GLMMs) within the integrated nested Laplace approximation (INLA) approach for approximate Bayesian inference for latent Gaussian models. GLMMs for Binomial and Poisson data with many zeroes and low counts have been a somewhat difficult case for INLA, and the case of binary data has been particularly problematic. Our new correction has been implemented as part of the R-INLA package, and adds only negligible computational cost. Empirical evaluations on both real and simulated data indicate that the method works well.