NORGES TEKNISK-NATURVITENSKAPELIGE UNIVERSITET Approximate Inference for Hierarchical Gaussian Markov Random Fields Models

Many commonly used models in statistics can be formulated as Hierarchical Gaussian Markov random field (GMRF) models. These are characterised by assuming a (often large) GMRF as the second stage in the hierarchical model and a few hyperparameters at the third stage. Markov chain Monte Carlo is the common approach to do inference from such models. The variance of the Monte Carlo estimates is Op(M) where M is the number of samples in the chain so, in order to obtain precise estimates of marginal densities, say, we need M to be very large. Inspired by the fact that often one-block and independence samplers can be constructed for hierarchical GMRF models, we will in this work investigate whether MCMC is really needed to estimate marginal densities, which often is the goal of the analysis. By making use of GMRFapproximations, we show by typical examples that marginal densities can indeed be very precisely estimated by deterministic schemes. The methodological and practical consequence of these findings are indeed positive; We conjecture that for most hierarchical GMRF-models there is really no need for MCMC based inference to estimate marginal densities. Further, by making use of numerical methods for sparse matrices the computational costs of these deterministic schemes are nearly instant compared to the MCMC alternative.