Mutual Conditional Independence and its Applications to Inference in Markov Networks

TL;DR

This paper introduces the concept of mutual conditional independence in Markov networks, proving its equivalence to existing Markov properties and developing inference methods based on this new property.

Contribution

It defines mutual conditional independence among elements of maximal independent sets and establishes its equivalence to traditional Markov properties under certain conditions.

Findings

Mutual conditional independence holds within maximal independent sets.

Equivalence between mutual conditional independence and Markov properties is proven.

New inference methods are developed exploiting this property.

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.