ON THE RELATION BETWEEN CONDITIONAL-INDEPENDENCE MODELS DETERMINED BY FINITE DISTRIBUTIVE LATTICES AND BY DIRECTED ACYCLIC GRAPHS

The relations among the classes of multivariate conditional independence models determined by directed acyclic graphs (DAG), undirected graphs (UDG), decomposable graphs (DEC), and finite distributive lattices (LCI) are investigated, First, LCI models that admit positive joint densities are characterized in terms of an appropriate factorization of the density. This factorization is then recognized as a particular form of the recursive factorization that characterizes DAG models, thereby establishing that the LCI models comprise a subclass of the class of DAG models. Precisely, the class of LCI models coincides with the subclass of transitive DAG models. Furthermore, the class of LCI models has nontrivial intersection with the class of DEC models. A series of examples illustrating these relations are presented.