Variations on Undirected Graphical Models and their Relationships

We compare alternative definitions of undirected graphical models for discrete, finite variables. Lauritzen (1996) provides several definitions of such models and describes their relationships. He shows that the definitions agree only when joint distributions represented by the models are limited to strictly positive distributions. Heckerman et al. (2000), in their paper on dependency networks, describe another definition of undirected graphical models for strictly positive distributions. They show that this definition agrees with those of Lauritzen again when distributions are strictly positive. In this paper, we extend the definition of Heckerman et al. to arbitrary distributions and show how this definition relates to those of Lauritzen in the general case.