Markov random fields and Markov chains on trees

Abstract

We consider probability measures on a space S(^A) (where S and A are countable and the σ-field is the natural one) which are Markov random fields with respect to a given neighbour relation ~ on A. In particular, we study the set G(II) of Markov random fields corresponding to a given Markov specification II, i.e. to a consistent family of "Markov" conditional probability distributions associated with the finite subsets of A. First, we review the relation between II and G(II). We consider also the representation of II by a family of interaction functions associated with the simplices of the graph (A,~) , together with some related problems. The rest of the thesis is concerned with the case where (A,~) is a tree. We define Markov chains on and consider their relation to the wider class of Markov random fields. We then derive analytical methods for the study of the set M(II) of Markov chains in G(II). These results are applied to homogeneous Markov specifications on regular infinite trees. Finally, we consider Markov specifications which are either attractive or repulsive with respect to a total ordering on S. For these we obtain quite strong results, including an exact condition for G(II) to contain precisely one element. We thereby generalise results obtained by Preston and Spitzer for binary S.