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On Chain Graph Models For Description Of Conditional Independence Structures
 Ann. Statist
, 1998
"... This paper deals with chain graphs (CGs) which allow both directed and undirected edges. This class of graphs, introduced by Lauritzen and Wermuth [15], generalizes both UGs and DAGs. To establish the semantics of CGs one should associate an independency model to every CG. Some steps were already ma ..."
Abstract

Cited by 19 (3 self)
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This paper deals with chain graphs (CGs) which allow both directed and undirected edges. This class of graphs, introduced by Lauritzen and Wermuth [15], generalizes both UGs and DAGs. To establish the semantics of CGs one should associate an independency model to every CG. Some steps were already made. Lauritzen and Wermuth [16] intended to use CGs to describe independency models for strictly positive probability distributions and introduced the concept of the chain Markov property which is analogous to the concept of causal input list for DAGs. Lauritzen and Frydenberg [17, 9] generalized the concept of moral graph and introduced a moralization criterion for reading independency statements from a CG. Frydenberg [9] characterized CGs with the same Markov ON CHAIN GRAPH MODELS 3 property (that is producing the same CGmodel) and Andersson, Madigan and Perlman [3] used special CGs to represent uniquely classes of Markov equivalent DAGs. Whittaker [31] in his book gave several examples of the use of CGs, and other recent works also deal with them [6, 20, 23, 30], the most comprehensive account is provided by the book [19]. Several results proved here were already presented (without proof) in our previous conference contribution [5]. An alternative approach to the generalization of UGs and DAGs was started by Cox and Wermuth [7] who introduced a wider class of jointresponse chain graphs which allow also 'dashed' directed and undirected edges in addition to the classic 'solid' directed and undirected edges treated in this paper. Andersson, Madigan and Perlman [1] introduced an alternative Markov property to give an interpretation to those jointresponse CGs which combine dashed directed edges with solid undirected edges (of course, another independency model is associated...
Chain Graphs: Semantics and Expressiveness
, 1995
"... . A chain graph (CG) is a graph admitting both directed and undirected edges with forbidden directed cycles. It generalizes both the concept of undirected graph (UG) and the concept of directed acyclic graph (DAG). CGs can be used efficiently to store graphoids, that is, independency knowledge of th ..."
Abstract

Cited by 8 (2 self)
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. A chain graph (CG) is a graph admitting both directed and undirected edges with forbidden directed cycles. It generalizes both the concept of undirected graph (UG) and the concept of directed acyclic graph (DAG). CGs can be used efficiently to store graphoids, that is, independency knowledge of the form "X is independent of Y given Z" obeying a set of five properties (axioms). Two equivalent criteria for reading independencies from a CG are formulated, namely the moralization criterion and the separation criterion. These criteria give exactly the graphoid closure of the input list for the CG. Moreover, a construction of a CG from a graphoid (through an input list), which produces a minimal Imap of that graphoid, is given. 1 Introduction Using graphs to describe independency structure arising among variables has a long and rich tradition. One can distinguish two classic approaches (for details see the book [11]): using undirected graphs (UGs), called also Markov networks, or using...