Fitting Gaussian Markov Random Fields to Gaussian Fields

TLDR

Bayesian spatial modeling often requires constructing a homogeneous Gaussian Markov random field on a lattice that captures observed or prior correlation structures, and the Markov property is crucial for efficient MCMC algorithms. The authors aim to reformulate the construction of such GMRFs as fitting them to a prescribed stationary Gaussian field when both local and global properties matter, and to introduce a new criterion for this task. They propose a criterion based on the Kullback–Leibler discrepancy that overcomes its shortcomings, and show that GMRFs with small neighbourhoods can approximate Gaussian fields with long correlation lengths. The study finds that the standard Kullback–Leibler approach fails, whereas the new criterion enables small‑neighbourhood GMRFs to closely match Gaussian fields, with implications for likelihood‑based inference of general Markov random fields.

Abstract

This paper discusses the following task often encountered in building Bayesian spatial models: construct a homogeneous Gaussian Markov random field (GMRF) on a lattice with correlation properties either as present in some observed data, or consistent with prior knowledge. The Markov property is essential in designing computationally efficient Markov chain Monte Carlo algorithms to analyse such models. We argue that we can restate both tasks as that of fitting a GMRF to a prescribed stationary Gaussian field on a lattice when both local and global properties are important. We demonstrate that using the Kullback–Leibler discrepancy often fails for this task, giving severely undesirable behaviour of the correlation function for lags outside the neighbourhood. We propose a new criterion that resolves this difficulty, and demonstrate that GMRFs with small neighbourhoods can approximate Gaussian fields surprisingly well even with long correlation lengths. Finally, we discuss implications of our findings for likelihood based inference for general Markov random fields when global properties are also important.