On the Spatial Dependence Structure of Isotropic Pairwise Gaussian-Markov Random Field Models

Details On the Spatial Dependence Structure of Isotropic Pairwise Gaussian-Markov Random Field Models On the Spatial Dependence Structure of Isotropic Pairwise Gaussian-Markov Random Field Models

2011-08-25

arxiv.org/abs/1108.4973v4

Computer Science

Information Theory

10 pages

Scientific paper

Markov Random Field (MRF) models are powerful tools for contextual modeling. However, little is known about how the spatial dependence between their elements is encoded in terms of statistical information, more precisely, information-theoretic measures. In this paper, we enlight the connection between Fisher information, Shannon entropy and spatial properties of the random field in case of Gaussian random variables (a Gaussian Markov random field, or simply GMRF), by defining analytical expressions to compute local and global versions of these measures using Besag's pseudo-likelihood function (conditional independence assumption). Besides, we use the derived expressions to define an exact expression for the asymptotic variance of the maximum pseudo-likelihood estimator of the spatial dependence parameter, showing that, since information equality fails, it is not possible to define a lower bound (Cramer-Rao limit). Moreover, the obtained results indicate that accuracy on the estimation of the spatial dependence structure of GMRF's depends essentially on the massive presence of contextual patterns satisfying two intuitive conditions: high local log-likelihood value (minimization of type-I Fisher information), which means concentration of patterns that are likely to be observed, and high local log-likelihood curvature (maximization of type-II Fisher information), which means that small perturbations on data cannot cause abrupt changes on the spatial dependence structure.

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