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.

翻译：条件独立性，以及更一般的条件互独立性，是概率论中的核心概念。在其一般形式下，它们将函数依赖作为特例包含在内。本文探讨了关于条件互独立性的两个基本问题。设 $K$ 和 $K'$ 是定义在有限离散随机变量集上的两个条件互独立性（CMI）。我们获得了以下条件的充分必要条件：i) $K$ 等价于 $K'$；ii) $K$ 蕴含 $K'$。这些表征基于为条件互独立性引入的一种规范形式。