On the Intersection and Composition properties of conditional independence
For researchers in probabilistic reasoning and graphical models, this work clarifies the conditions under which these properties hold, but it is primarily a survey with incremental theoretical contributions.
This paper surveys the Intersection and Composition properties of conditional independence in compositional graphoids, providing systematic constructions of examples and counterexamples, necessary and sufficient conditions, and novel sufficient conditions for discrete random variables via information-theoretic tools.
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.