Jan 29

3:30 pm

## An Alternative Markov Property for Chain Graphs

### Michael Perlman

Seminar

University of Washington - Department of Statistics

Graphical Markov (=conditional independence) models use graphs, either directed, undirected, or mixed, to represent possible dependences among the variables of a multivariate statistical distribution. Applications include models for spatial dependence, image analysis, psychometrics, genetics, Bayesian belief networks, expert systems, and many others. The nodes of the graph represent the statistical variables, while the presence (absence) of an edge between two nodes indicates possible dependencies (independencies) between the two corresponding variables.

Graphical Markov models determined by acyclic directed graphs (ADGs) allow especially elegant statistical analysis. The likelihood function associated with an ADG model admits an convenient recursive factorization which, for categorical or multivariate normal data, yields explicit maximum likelihood estimates and likelihood ratio tests. ADG models allow efficient computational algorithms for exact probability calculations, as well as efficient updating algorithms for Bayesian analysis.

Lauritzen and Wermuth (1989) and Frydenberg (1990) generalized ADG models to chain graphs, which are "mixed" graphs (both directed and undirected edges) that contain no (partially) directed cycles. Loosely speaking, chain graph models provide a means to simultaneously represent dependencies some of which are causal and some associative.

In the present study we consider the same class of chain graphs as LWF, but propose an alternative Markov property (AMP) for these graphs. This is motivated by Gaussian chain graph models, where our interpretation is equivalent to imposing restrictions on regression coefficients, whereas the LWF interpretation is equivalent to imposing restrictions on the natural exponential parameters. In some ways, AMP chain graph models lie closer to ADG models than do LWF chain graph models.