New models for Markov random fields

The Hammersley-Clifford theorem gives the form that the joint probability density (or mass) function of a Markov random field must take. Its exponent must be a sum of functions of variables, where each function in the summand involves only those variables whose sites form a clique. From a statistical modeling point of view, it is important to establish the converse result, namely, to give the conditional probability specifications that yield a Markov random field. Besag addressed this question by developing a one-parameter exponential family of conditional probability models. In this article the authors develop new models for Markov random fields by establishing sufficient conditions for the conditional probability specifications to yield a Markov random field.