The probability function of spatial statistical models involves, in general, an extremely awkward normalizing function of the parameters known as the partition function in statistical mechanics with the consequence that a direct approach to statistical inference through maximum likelihood (ML) is rarely possible. In order to avoid such intractability Besag (1975) introduced an alternative technique known as the method of maximum pseudo-likelihood (MPL) owing to its merit of being easy to implement. The maximum pseudo-likelihood estimator (MPLE) is the value of the parameter that maximizes the pseudo-likelihood defined as the direct product of conditional probabilities or conditional probability densities of the variable at each site. It has been mathematically demonstrated that, under suitable conditions, the MPLEs are strongly consistent and asymptotically normally distributed around the true parameter value for large samples of various spatial processes. On the other hand, the MPL method trades away efficiency for computational ease. It has been shown that in many situations the MPLE is not efficient in comparison with the ML estimator (MLE). According to these studies, the MPLEs are as good as the MLEs in the weak interaction case, but the difference between the two becomes substantial when spatial interactions are strong.
Huang and Ogata (2002) address the problem of improving the efficiency of MPLEs while still keeping the technique computationally feasible and proposed the maximum generalized pseudo-likelihood (MGPL) method for Markov random field (MRF) models on lattice. The MGPL estimator (MGPLE) is the value of the parameter that maximizes the generalized pseudo-likelihood function (GPL). This GPL is the multivariate version of Besag's pseudo-likelihood which is constructed first by defining a group of adjacent sites for each site in the lattice and then taking the product of the multivariate conditional probability distributions (MCPD) of the groups of random variables defined on each group of adjacent sites. Simulation results for an Ising and two auto-normal models on a region of square lattice showed better performance of the MGPLE than the MPLE, and the performance became better as the size of the groups of adjacent sites increased. On the other hand, it was observed that as the size of the groups of adjacent sites increased, the computing complexity for the MGPLE increased exponentially due to the presence of a normalizing integral (a sum in the case of discrete site variables) in the expression for each MCPD which has to be evaluated all over the support of the joint distribution for groups of site variables in each case. Because of this, for continuous MRFs other than auto-normal and discrete MRFs with site variables assuming more than two values, an enormous effort might be required making the implementation of the MGPL method practically unfeasible even for small square lattices. For example, in MRFs where each site variable, conditional on its neighbors, follows the distribution of a Winsorized Poisson random variable (Kaiser and Cressie (1997)) the computation of the normalizing integrals rapidly becomes prohibitive with the size of the groups of adjacent sites even for small square lattices, as the support of this distribution may be in the hundreds (or thousands).
In our research we propose a conditional pairwise pseudo-likelihood (CPPL) for parameter estimation in Markov random fields on lattice. The CPPL is defined as the direct product of conditional pairwise distributions corresponding to the pairs of random variables associated with the cliques of size two from the collection of spatial locations on a region of a lattice. Thus the CPPL is a modified version of Besag's pseudo-likelihood (PL) and Huang and Ogata's generalized pseudo-likelihood (GPL) in that it is not constructed based on defining a group of adjacent sites for each site in the lattice. We carry out calculations of the correspondingly defined maximum conditional pairwise pseudo-likelihood estimator (MCPPLE) for Markov random fields with Winsorized Poisson conditional distributions on the lattice. These simulation studies show that the MCPPLE has significantly better performance than Besag's maximum pseudo-likelihood estimator (MPLE), and its calculation is almost as easy to implement as the MPLE. Therefore, we suggest that for situations where each discrete local random variable conditional on its neighbors assumes more than two possible values, as in the Winsorized Poisson case, estimation based on the CPPL may be a computationally more feasible alternative than estimation based on Huang and Ogata's GPL.