On chain graph models for description of conditional independence structures

A chain graph (CG) is a graph admitting both directed and undirected edges with (partially) directed cycles forbidden. It generalizes both the concept of undirected graph (UG) and the concept of directed acyclic graph (DAG). A chain graph can be used to describe efficiently the conditional independence structure of a multidimensional discrete probability distribution in the form of a graphoid, that is, in the form of a list of statements X is independent of Y given Z obeying a set of five properties (axioms). An input list of independency statements for every CG is defined and it is shown that the classic moralization criterion for CGs embraces exactly the graphoid closure of the input list. A new direct separation criterion for reading independency statements from a CG is introduced and shown to be equivalent to the moralization criterion. Using this new criterion, it is proved that for every CG, there exists a strictly positive discrete probability distribution that embodies exactly the independency statements displayed by the graph. Thus, both criteria are shown to be complete and the use of CGs as tools for description of conditional independence structures is justified.