Characterizations of Conditional Mutual Independence: Equivalence and Implication

TL;DR

This paper provides a comprehensive characterization of conditional mutual independence, establishing necessary and sufficient conditions for equivalence and implication between different CMIs using a canonical form.

Contribution

It introduces a canonical form for CMIs and derives necessary and sufficient conditions for their equivalence and implication, advancing theoretical understanding.

Findings

Established a necessary and sufficient condition for CMI equivalence.

Provided a necessary and sufficient condition for CMI implication.

Introduced a canonical form for conditional mutual independence.

Abstract

Conditional independence, and more generally conditional mutual independence, are central notions in probability theory. In their general forms, they include functional dependence as a special case. In this paper, we tackle two fundamental problems related to conditional mutual independence. Let $K$ and $K'$ be two conditional mutual independncies (CMIs) defined on a finite set of discrete random variables. We have obtained a necessary and sufficient condition for i) $K$ is equivalent to $K'$; ii) $K$ implies $K'$. These characterizations are in terms of a canonical form introduced for conditional mutual independence.