Information-theoretic characterizations of Markov random fields and subfields

Let X _i , i ϵ V form a Markov random field (MRF) represented by an undirected graph G = (V, E), and V' be a subset of V. We determine the smallest graph that can always represent the subfield X _i , i ϵ V' as an MRF. Based on this result, we obtain a necessary and sufficient condition for a subfield of a Markov tree to be also a Markov tree. When G is a path so that X _i , i ϵ V form a Markov chain, it is known that the I - Measure is always nonnegative [3]. We prove that Markov chain is essentially the only MRF that possesses this property. Our work is built on the set-theoretic characterization of an MRF in [4]. Unlike most works in the literature, we do not make the standard assumption that the underlying probability distribution is factorizable with respect to the graph representing the MRF.