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The fundamental concepts underlying in Markov networks are the conditionalindependence and the set of rules called Markov properties that translatesconditional independence constraints into graphs. In this article we introducethe concept of mutual conditional independence relationship among elements ofan independent set of a Markov network. We first prove that the mutualconditional independence property holds within the elements of a maximalindependent set afterwards we prove equivalence between the set of mutualconditional independence relations encoded by all the maximal independent setsand the three Markov properties(pair-wise, local and the global) under certainregularity conditions. The proof employs diverse methods involving graphoidaxioms, factorization of the joint probability density functions and the graphtheory. We present inference methods for decomposable and non-decomposablegraphical models exploiting newly revealed mutual conditional independenceproperty.

Mutual Conditional Independence and its Applications to Inference in Markov Networks