M. Studeny and R. R. Bouckaert: On chain graph models for description of conditional independence structures. The Annals of Statistics 26 (1998), n. 4, pp. 1434-1495.

A chain graph is a graph admitting both directed and undirected edges with forbidden directed cycles. It generalizes both the concept of undirected graph and the concept of directed acyclic graph. 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 independency knowledge of 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 chain graph is defined and it is shown that the classic moralization criterion for chain graphs embraces exactly the graphoid closure of the input list. A new direct separation criterion for reading independency statements from a chain graph is introduced and shown to be equivalent to the moralization criterion. Using this new criterion, it is proved that for every chain graph there exists a strictly positive discrete probability distribution that embodies exactly the independency statements displayed by the graph. In particular, both criteria are shown to be complete and the use of chain graphs as tools for description of conditional independence structures is justified.

AMS classification 62H99, 62H05, 68R10

chain graph
conditional independence
Markovian distribution
input list
moralization criterion
c-separation criterion
strong completeness

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