Penalized Pseudolikelihood Inference in Spatial Interaction Models with Covariates

Given spatially located observed random variables (x, z = {(xi, zi)}i, we propose a new method for non-parametric estimation of the potential functions of a Markov random field p(x|z), based on a roughness penalty approach. The new estimator maximizes the penalized log-pseudolikelihood function and is a natural cubic spline. The calculations involved do not rely on Monte Carlo simulation. We suggest the use of B-splines to stabilize the numerical procedure. An application in Bayesian image reconstruction is described.