Gibbs and Markov properties of graphs

• Authors: Jan T. A. Koster
• Venue: Annals of Mathematics and Artificial Intelligence
• Year: 1997

Directed acyclic graphs (DAG’s) and, more generally, chain graphs have in recent years been widely used for statistical modelling. Their Gibbs and Markov properties are now well understood and are exploited, e.g., in reducing the complexity encountered in estimating the joint distribution of many random variables. The scope of the models has been restricted to acyclic or recursive processes and this restriction was long considered imperative, due to the supposed fundamentally different nature of processes involving reciprocal interactions between variables. Recently however it was shown independently by Spirtes (Spirtes, 1995) and Koster (Koster, 1996) that graphs containing directed cycles may be given a proper Markov interpretation. This paper further generalizes the scope of graphical models. It studies a class of conditional independence (CI) probability models determined by a general graph which may have directed and undirected edges, and may contain directed cycles. This class of graphical models strictly includes the well‐known class of graphical chain models studied by Frydenberg et al., and the class of probability models determined by a directed cyclic graph or a reciprocal graph, studied recently by Spirtes and Koster. It is shown that the Markov property determined by a graph is equivalent to the existence of a Gibbs‐factorization of the density (assumed positive). To better understand the structural aspects of the Gibbs and Markov properties embodied by graphs the notion of lattice conditional independence (LCI), introduced by Andersson and Perlman (Andersson and Perlman, 1993), is needed. The Gibbs‐factorization has an outer ‘skeleton’ which is determined by the ring of all anterior sets of the graph.