Approximate Bayesian Inference in Spatial Generalized Linear Mixed Models

In this paper we propose fast approximate methods for computing posterior marginals in spatial generalized linear mixed models. We consider the common geostatistical special case with a high dimensional latent spatial variable and observations at only a few known registration sites. Our methods of inference are deterministic, using no random sampling. We present two methods of approximate inference. The first is very fast to compute and via examples we find that this approximation is ’practically sufficient’. By this expression we mean that the results obtained by this approximate method do not show any bias or dispersion effects that might affect decision making. The other approximation is an improved version of the first one, and via examples we demonstrate that the inferred posterior approximations of this improved version are ’practically exact’. By this expression we mean that one would have to run Markov chain Monte Carlo simulations for longer than is typically done to detect any indications of bias or dispersion error effects in the approximate results. The two methods of approximate inference can help to expand the scope of geostatistical models, for instance in the context of model choice, model assessment, and sampling design. The approximations take seconds of CPU time, in sharp contrast to overnight Markov chain Monte Carlo runs for solving these types of problems. Our approach to approximate inference could easily be part of standard softwares.