  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.
