A recovery algorithm for chain graphs

The class of chain graphs (CGs) involving both undirected graphs (=Markov networks) and directed acyclic graphs (= Bayesian networks) was introduced in middle eighties for description of probabilistic conditional independence structures. Every class of Markov equivalent CGs (that is, CGs describing the same conditional independence structure) has a natural representative, which is called the largest CG. The paper presents a recovery algorithm, which on the basis of the conditional independence structure given by a CG (in the form of a dependency model) finds the largest CG representing the corresponding class of Markov equivalent CGs. As a by-product a graphical characterization of graphs which are the largest CGs (for a class of Markov equivalent CGs) is obtained, and a simple algorithm changing every CG into the largest CG of the corresponding equivalence class is given. © 1997 Elsevier Science Inc. K E Y W O R D S : chain graph, dependency model, Markov equivalence, pattern, largest chain graph, recovery algorithm 1. I N T R O D U C T I O N Classical graphical approaches to the description of probabilistic conditional independence structures use either undirected graphs (UGs), also called Markov networks, or directed acyclic graphs (DAGs), known as Bayesian networks or (probabilistic) influence diagrams. In middle eighties Lauritzen and Wermuth [12] introduced the class of chain graphs (CGs), which includes both UGs and DAGs, but not only them. In CGs both undirected edges, called lines, and directed edges, called arrows, are Address correspondence to Milan Studenf,, Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, Pod voddrenskou vY~[ 4, 182 08 Prague, Czech Republic. E-mail: [email protected], cz. Received 1 December 1996; accepted 1 December 1996. International Journal of Approximate Reasoning 1997; 17:265-293 © 1997 Elsevier Science Inc. 0888-613X/97/$17.00 655 Avenue of the Americas, New York, NY 10010 PII S0888-613X(97)00018-2