Fitting Gaussian Markov random fields to Gaussian fields

Håvard Rue*, Håkon Tjelmeland

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

163 Scopus citations


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 neighhourhood. 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.

Original languageEnglish (US)
Pages (from-to)31-49
Number of pages19
JournalScandinavian Journal of Statistics
Issue number1
StatePublished - Mar 2002
Externally publishedYes


  • Conditional autoregressive models
  • Empirical Bayes
  • Gaussian Markov random field
  • Gaussian fields
  • Kullback-Leibler discrepancy
  • Markov chain Monte Carlo
  • Maximum likelihood

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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