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Bayesian hierarchical models with latent Gaussian layers have proven veryflexible in capturing complex stochastic behavior and hierarchical structuresin high-dimensional spatial and spatio-temporal data. Whereas simulation-basedBayesian inference through Markov Chain Monte Carlo may be hampered by slowconvergence and numerical instabilities, the inferential framework ofIntegrated Nested Laplace Approximation (INLA) is capable to provide accurateand relatively fast analytical approximations to posterior quantities ofinterest. It heavily relies on the use of Gauss-Markov dependence structures toavoid the numerical bottleneck of high-dimensional nonsparse matrixcomputations. With a view towards space-time applications, we here review theprincipal theoretical concepts, model classes and inference tools within theINLA framework. Important elements to construct space-time models are certainspatial Mat\'ern-like Gauss-Markov random fields, obtained as approximatesolutions to a stochastic partial differential equation. Efficientimplementation of statistical inference tools for a large variety of models isavailable through the INLA package of the R software. To showcase the practicaluse of R-INLA and to illustrate its principal commands and syntax, acomprehensive simulation experiment is presented using simulated non Gaussianspace-time count data with a first-order autoregressive dependence structure intime.

Latent Gaussian modeling and INLA: A review with focus on space-time applications