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Adjusted Maximum Likelihood and Pseudo-Likelihood Estimation for Noisy Gaussian Markov Random Fields

The article is concerned with parameter estimation for Gaussian Markov random fields observed with additive independent identically distributed noise. In particular, we consider maximum likelihood and maximum pseudo-likelihood estimation for the noise-free case and make adjustments to the estimators in the presence of noise. The adjusted maximum likelihood estimator is computed in O(n2) time for a square image with n pixels. The estimation method is useful when only the moments of the noise are specified or when the exact maximum likelihood estimator is difficult to compute (e.g., for certain non-Gaussian noise distributions). The adjusted maximum pseudo-likelihood estimator is straightforward to calculate, is useful as a starting value in maximization routines, and is often a reasonable estimator in its own right. We discuss asymptotic properties of the adjusted estimators including consistency. We also consider constrained maximum pseudo-likelihood estimation and a Bayesian estimator. We compare the adjusted estimators with the exact Gaussian maximum likelihood estimator and toroidal boundary approximation Gaussian maximum likelihood estimator in a simulation study. The adjusted estimators are also robust to the specification of the noise distribution.