On Information-Theoretic Characterizations of Markov Random Fields and Subfields

Let X
_i
, i E 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 Xi, i E 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 Xi, i E V form a Markov chain, it is known that the I-Measure is always nonnegative (Kawabata and Yeung in 1992). We prove that Markov chain is essentially the only MRF such that the I-Measure is always nonnegative. By applying our characterization of the smallest graph representation of a subfield of an MRF, we develop a recursive approach for constructing information diagrams for MRFs. Our work is built on the set-theoretic characterization of an MRF (Yeung et al. in 2002).