Compositional graphoids are fundamental discrete structures which appear in probabilistic reasoning, particularly in the area of graphical models. They are semigraphoids which satisfy the Intersection and Composition properties. These important properties, however, are not enjoyed by general probability distributions. This paper surveys what is known about them, providing systematic constructions of examples and counterexamples as well as necessary and sufficient conditions. Novel sufficient conditions for both properties are derived in the context of discrete random variables via information-theoretic tools.

翻译：组合图状是概率推理中出现的基本离散结构，特别是在图模型领域。它们是满足交集与合成性质的半图状。然而，一般概率分布并不具备这些重要性质。本文综述了关于这些性质的已知结果，提供了示例与反例的系统构造以及充分必要条件。通过信息论工具，在离散随机变量背景下，推导出了这两种性质的新颖充分条件。