Mutual Conditional Independence and its Applications to Inference in Markov Networks

Abstract

The fundamental concepts underlying in Markov networks are the conditional independence and the set of rules called Markov properties that translates conditional independence constraints into graphs. In this article we introduce the concept of mutual conditional independence relationship among elements of an independent set of a Markov network. We first prove that the mutual conditional independence property holds within the elements of a maximal independent set afterwards we prove equivalence between the set of mutual conditional independence relations encoded by all the maximal independent sets and the three Markov properties(pair-wise, local and the global) under certain regularity conditions. The proof employs diverse methods involving graphoid axioms, factorization of the joint probability density functions and the graph theory. We present inference methods for decomposable and non-decomposable graphical models exploiting newly revealed mutual conditional independence property.