Likelihood Estimation and Inference for the Autologistic Model

The autologistic model is commonly used to model spatial binary data on the lattice. However, if the lattice size is too large, then exact calculation of its normalizing constant poses a major difficulty. Various different methods for estimation of model parameters, such as pseudo-likelihood, have been proposed to overcome this problem. This article presents a method to estimate the normalizing constant in an efficient manner. In particular, this allows tasks such as maximum likelihood estimation and inference for model parameters. We also consider the true likelihood approximated by the product of likelihoods for which the normalizing constant can be found by an analytic computational method by wrapping the lattice on the cylinder. This gives a simulation-free method of inference. We compare estimates of model parameters based on our new methods with the commonly used pseudo-likelihood approach. Although we have not considered Bayesian inferences here, the method can be straightforwardly extended to find posterior distributions. We apply our methods to the well-known endive data and to simulated data and find that our methods give substantially increased accuracy of estimation of model parameters.