Extensions of Undirected and Acyclic, Directed Graphical Models
Proceedings of the Sixth International Workshop on Artificial Intelligence and Statistics, PMLR R1:407-420, 1997.
The use of acyclic, directed graphs (often called ’DAG’s) to simultaneously represent causal hypotheses and to encode independence and conditional independence constraints associated with those hypotheses has proved fruitful in the construction of expert systems, in the development of efficient updating algorithms (Pearl, 1988, Lauritzen et al. 1988), and in inferring causal structure (Pearl and Verma, 1991; Cooper and Herskovits 1992; Spirtes, Glymour and Scheines, 1993). In section 1 I will survey a number of extensions of the DAG framework based on directed graphs and chain graphs (Lauritzen and Wermuth 1989; Frydenberg 1990; Koster 1996; Andersson, Madigan and Perlman 1996). Those based on directed graphs include models based on directed cyclic and acyclic graphs, possibly including latent variables and/or selection bias (Pearl, 1988; Spirtes, Glymour and Scheines 1993; Spirtes 1995; Spirtes, Meek, and Richardson 1995; Richardson 1996a, 1996b; Koster 1996; Pearl and Dechter 1996; Cox and Wermuth, 1996). In section 2 I state two properties, motivated by causal and spatial intuitions, that the set of conditional independencies entailed by a graphical model might satisfy. I proceed to show that the sets of independencies entailed by (i) an undirected graph via separation, and (ii) a (cyclic or acyclic) directed graph (possibly with latent and/or selection variables) via ct-separation, satisfy both properties. By contrast neither of these properties, in general, will hold in a chain graph under the Lauritzen-Wermuth-Frydenberg (LWF) interpretation. One property holds for chain graphs under the Andersson-Madigan-Perlman (AMP) interpretation, the other does not. The examination of these properties and others like them may provide insight into the current vigorous debate concerning the applicability of chain graphs under different global Markov properties.