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. 14341495.
 Abstract
 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
 Keywords
 chain graph
 conditional independence
 Markovian distribution
 input list
 graphoid
 moralization criterion
 cseparation criterion
 strong completeness

 A
pdf copy of the paper (374kB) is already openaccess available.
The problem solved by the paper was formulated in the works:
 S. L. Lauritzen: Mixed graphical association models.
Scandinavian Journal of Statistics
16 (1989), pp. 273306.
 M. Frydenberg: The chain graph Markov property.
Scandinavian Journal of Statistics
17 (1990), n. 3, pp. 333353.
The paper also builds on the following works:
 S.L. Lauritzen, A.P. Dawid, B.N. Larsen, H.G. Leimer:
Independence properties of directed Markov fields.
Networks
20 (1990), n. 3, pp. 491505.
 D. Geiger and J. Pearl: On the logic of causal
models. In Uncertainty in Artificial
Intelligence 4 (R.D. Schachter, T.S. Lewitt, L.N. Kanal, J.F.
Lemmer eds.) NorthHolland 1990, pp. 314.
 D. Geiger and J. Pearl:
Logical and algorithmical properties of conditional independence
and graphical models. Annals of Statistics
21 (1993), n. 4, pp. 20012021.