Penalized pseudolikelihood inference in spatial interaction models with covariates

Given spatially located observed random variables ((x) under bar, (z) under bar) = {(x(i), z(i))}(i), we propose a new method for non-parametric estim ation of the potential functions of a Markov random field p((x) under bar\( z) under bar), based on a roughness penalty approach. The new estimator max imizes the penalized log-pseudolikelihood function and is a natural cubic s pline. The calculations Involved do not rely on Monte Carlo simulation. We suggest the use of B-splines to stabilize the numerical procedure. An appli cation in Bayesian image reconstruction is described.