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。在其一般形式中，它们将函数依赖关系作为一种特例包含在内。本文探讨了与条件互独立性相关的两个基本问题。设 $K$ 和 $K'$ 为定义在一组有限离散随机变量上的两个条件互独立性（CMIs）。我们已获得以下情况的充分必要条件：i) $K$ 等价于 $K'$；ii) $K$ 蕴含 $K'$。这些刻画是基于为条件互独立性引入的一种规范形式给出的。