On 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 (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).