Enumerating Markov Equivalence Classes of Acyclic Digraph Models

TL;DR

This paper presents a computational approach to enumerate and analyze Markov equivalence classes of acyclic digraph models, revealing their distribution and asymptotic properties, with implications for model selection in graphical models.

Contribution

The authors developed a program to enumerate Markov equivalence classes of ADGs up to 10 vertices, providing new insights into their distribution and asymptotic behavior.

Findings

Number of classes approaches 26.7% of ADGs asymptotically.

Distribution of classes by edges approximates a Gaussian.

Maximum number of classes per undirected graph increases factorially.

Abstract

Graphical Markov models determined by acyclic digraphs (ADGs), also called directed acyclic graphs (DAGs), are widely studied in statistics, computer science (as Bayesian networks), operations research (as influence diagrams), and many related fields. Because different ADGs may determine the same Markov equivalence class, it long has been of interest to determine the efficiency gained in model specification and search by working directly with Markov equivalence classes of ADGs rather than with ADGs themselves. A computer program was written to enumerate the equivalence classes of ADG models as specified by Pearl & Verma's equivalence criterion. The program counted equivalence classes for models up to and including 10 vertices. The ratio of number of classes to ADGs appears to approach an asymptote of about 0.267. Classes were analyzed according to number of edges and class size. By edges, the distribution of number of classes approaches a Gaussian shape. By class size, classes of size 1 are most common, with the proportions for larger sizes initially decreasing but then following a more irregular pattern. The maximum number of classes generated by any undirected graph was found to increase approximately factorially. The program also includes a new variation of orderly algorithm for generating undirected graphs.