Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations

• Published in: Journal of the Royal Statistical Society. Series B, Statistical methodology Vol. 71; no. 2; pp. 319 - 392
• Main Authors: Rue, Håvard, Martino, Sara, Chopin, Nicolas
• Format: Journal Article
• Language: English
• Published: Oxford, UK Blackwell Publishing Ltd 01-04-2009; Blackwell Publishing; Royal Statistical Society; Oxford University Press
• Series: Journal of the Royal Statistical Society Series B
• Subjects: Algorithms; Approximate Bayesian inference; Approximation; Bayesian analysis; Bayesian inference; Gaussian distributions; Gaussian Markov random fields; Generalized additive mixed models; Inference; Laplace approximation; Linear models; Markov analysis; Markov chains; Markovian processes; Modeling; Monte Carlo simulation; Non Gaussianity; Parallel computing; Regression analysis; Sparse matrices; Spatial models; Statistical discrepancies; Statistical methods; Statistical models; Statistics; Stochastic models; Structured additive regression models; Studies
• ISSN: 1369-7412; 1467-9868
• DOI: 10.1111/j.1467-9868.2008.00700.x

Structured additive regression models are perhaps the most commonly used class of models in statistical applications. It includes, among others, (generalized) linear models, (generalized) additive models, smoothing spline models, state space models, semiparametric regression, spatial and spatiotemporal models, log-Gaussian Cox processes and geostatistical and geoadditive models. We consider approximate Bayesian inference in a popular subset of structured additive regression models, latent Gaussian models, where the latent field is Gaussian, controlled by a few hyperparameters and with non-Gaussian response variables. The posterior marginals are not available in closed form owing to the non-Gaussian response variables. For such models, Markov chain Monte Carlo methods can be implemented, but they are not without problems, in terms of both convergence and computational time. In some practical applications, the extent of these problems is such that Markov chain Monte Carlo sampling is simply not an appropriate tool for routine analysis. We show that, by using an integrated nested Laplace approximation and its simplified version, we can directly compute very accurate approximations to the posterior marginals. The main benefit of these approximations is computational: where Markov chain Monte Carlo algorithms need hours or days to run, our approximations provide more precise estimates in seconds or minutes. Another advantage with our approach is its generality, which makes it possible to perform Bayesian analysis in an automatic, streamlined way, and to compute model comparison criteria and various predictive measures so that models can be compared and the model under study can be challenged.