Binary Markov Mesh Models and Symmetric Markov Random Fields: Some Results on their Equivalence

In this article, we focus on statistical models for binary data on a regular two-dimensional lattice. We study two classes of models, the Markov mesh models (MMMs) based on causal-like, asymmetric spatial dependence, and symmetric Markov random fields (SMFs) based on noncausal-like, symmetric spatial dependence. Building on results of Enting (1977), we give sufficient conditions for the asymmetrically defined binary MMMs (of third order) to be equivalent to a symmetrically defined binary SMF. Although not every binary SMF can be written as a binary MMM, our results show that many can. For such SMFs, their joint distribution can be written in closed form and their realizations can be simulated with just one pass through the lattice. An important consequence of the latter observation is that there are nontrivial spatial processes for which exact probabilities can be used to benchmark the performance of Markov-chain-Monte-Carlo and other algorithms.