Simultaneous credible bands for latent Gaussian models

Deterministic Bayesian inference for latent Gaussian models has recently become available using integrated nested Laplace approximations (INLA). Applying the INLAmethodology, marginal estimates for elements of the latent field can be computed efficiently, providing relevant summary statistics like posterior means, variances and pointwise credible intervals. In this paper, we extend the use of INLA to joint inference and present an algorithm to derive analytical simultaneous credible bands for subsets of the latent field. The algorithm is based on approximating the joint distribution of the subsets by multivariate Gaussian mixtures. Additionally, we present a saddlepoint approximation to compute Bayesian contour probabilities, representing the posterior support of fixed parameter vectors of interest. The given methods are applied to various examples from the literature.