ABSTRACT
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Statistical modeling of a finite collection of spatial random variables is often done through a Markov random field (MRF). An MRF is specified through the set of conditional distributions of one component given all the others. This enables one to focus on a single random variable at a time and leads to simple computational procedures for simulating MRFs, in particular for Bayesian inference via Markov chain Monte Carlo (MCMC). The main purpose of this chapter is to give a thorough introduction to the Gaussian case, so-called Gaussian MRFs (GMRFs), with a focus toward general properties and efficient computations. Examples and applications appear in Chapters 13 and 14. At the end, we will discuss the general case where the joint distribution is not Gaussian, and, in particular, the famous HammersleyClifford theorem. A modern and general reference to GMRFs is the monograph by Rue and Held (2005), while for MRFs in general, one can consult Guyon (1995) and Lauritzen (1996) for the methodology background and Li (2001) for spatial applications in image analysis. The seminal papers by J. Besag (1974, 1975) are still worth reading.
