Characterizations of Conditional Mutual Independence: Equivalence and Implication

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 $K^{\mathrm{\prime }}$ 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 $K^{\mathrm{\prime }}$; ii) $K$ implies $K^{\mathrm{\prime }}$. These characterizations are in terms of a canonical form introduced for conditional mutual independence.