An Alternative Markov Property for Chain Graphs

Graphical Markov models use graphs, either undirected, directed, or mixed, to represent possible dependences among statistical variables. Applications of undirected graphs (UDGs) include models for spatial dependence and image analysis, while acyclic directed graphs (ADGs), which are especially convenient for statistical analysis, arise in such fields as genetics and psychometrics and as models for expert systems and Bayesian belief networks. Lauritzen, Wermuth and Frydenberg (LWF) introduced a Markov property for chain graphs, which are mixed graphs that can be used to represent simultaneously both causal and associative dependencies and which include both UDGs and ADGs as special cases. In this paper an alternative Markov property (AMP) for chain graphs is introduced, which in some ways is a more direct extension of the ADG Markov property than is the LWF property for chain graph.

Enumerating Markov Equivalence Classes of Acyclic Digraph Models

Graphical Markov models determined by acyclic digraphs (ADGs), also called directed acyclic graphs (DAGs), are widely studied in statistics, computer science (as Bayesian networks), operations research (as influence diagrams), and many related fields. Because different ADGs may determine the same Markov equivalence class, it long has been of interest to determine the efficiency gained in model specification and search by working directly with Markov equivalence classes of ADGs rather than with ADGs themselves. A computer program was written to enumerate the equivalence classes of ADG models as specified by Pearl & Verma's equivalence criterion. The program counted equivalence classes for models up to and including 10 vertices. The ratio of number of classes to ADGs appears to approach an asymptote of about 0.267. Classes were analyzed according to number of edges and class size. By edges, the distribution of number of classes approaches a Gaussian shape. By class size, classes of size 1 are most common, with the proportions for larger sizes initially decreasing but then following a more irregular pattern. The maximum number of classes generated by any undirected graph was found to increase approximately factorially. The program also includes a new variation of orderly algorithm for generating undirected graphs.