A structural characterization of DAG-isomorphic dependency models

Abstract

Graphical models have been extensively used in probabilistic reasoning for representing conditionali ndependency (CI) information. Among them two of the well known models are undirected graphs (UGs), and directed acyclic graphs (DAGs). Given a set of CIs, it woul d be desirable to know whether this set can be perfectly represented by a UG or DAG. A necessary and sufficient condition using axioms has been found for a set of CIs that can be perfectly represented by a UG; while negative result has been shown for DAGs, i.e., there does not exist a finite set of axioms which can characterize a set of CIs having a perfect DAG. However, this does not exclude other possible ways for such a characterization. In this paper, by studying the relationship between CIs and factorizations of a joint probability distribution, we show that there does exist such a characterization for DAGs in terms of the structure of the given set of CIs. More precisely, we demonstrate that if the given set of CIs satisfies certain constraints, then it has a perfect DAG representation.