On Characterizations of Markov Random Fields and Subfields.

arXiv: Discrete Mathematics(2016)

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摘要
Let $X_i, i V$ form a Markov random field (MRF) represented by an undirected graph $G = (V,E)$, and $Vu0027$ be a subset of $V$. determine the smallest graph that can always represent the subfield $X_i, i Vu0027$ 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 $X_i, i V$ form a Markov chain, it is known that the $I$-Measure is always nonnegative [10]. 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 [13]. 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, which is possible by the Hammersley-Clifford theorem provided that the underlying distribution is strictly positive. As such, our results apply even when such a factorization does not exist.
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