Characterizations of Conditional Mutual Independence: Equivalence and Implication
This provides theoretical tools for probability theory, but it is incremental as it builds on existing notions of conditional independence.
The paper tackled the fundamental problems of determining equivalence and implication between two conditional mutual independencies (CMIs) on discrete random variables, obtaining necessary and sufficient conditions for both using a canonical form.
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.