A generalized maximum pseudo-likelihood estimator for noisy Markov fields

In this paper we present an asymptotic estimator, obtained by observing a noisy image, for the parameters of both a stationary Markov random field and an independent Bernoulli noise. We first estimate the parameter of the noise, by solving a polynomial equation of moderate degree (about 6-7 in the one-dimensional Ising model, and about 10-15 in the two-dimensional Ising model, for instance), and then apply the maximum pseudolikelihood method after removing the noise. Our method requires no extra simulation, and is likely to be applicable to any Markov random field, in any dimension. Here, we present the general theory and some examples in one dimension; more interesting examples in two dimensions will be discussed at length in a companion paper.