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arXiv · 1608.03697

On Information-Theoretic Characterizations of Markov Random Fields and Subfields

Raymond W. Yeung·Ali Al-Bashabsheh·Chao Chen·Qi Chen·Pierre Moulin
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Abstract

Let $X_i, i \in V$ 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 $X_i, i \in V'$ 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 $X_i, i \in V$ form a Markov chain, it is known that the $I$-Measure is always nonnegative and the information diagram assumes a very special structure Kawabata and Yeung (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 in Yeung, Lee, and Ye (2002).

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Raymond W. Yeung, Ali Al-Bashabsheh, Chao Chen, Qi Chen, Pierre Moulin. 2016-08-12. On Information-Theoretic Characterizations of Markov Random Fields and Subfields. https://arxiv.org/abs/1608.03697

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