[Paper Review] Enumerating Markov Equivalence Classes of Acyclic Digraph Models

Steven B. Gillispie, Perlman, Michael D.|arXiv (Cornell University)|Jan 10, 2013

Bayesian Modeling and Causal InferenceComputer Science31 references73 citations

TL;DR

This paper presents a computational enumeration of Markov equivalence classes of acyclic digraphs (DAGs) using Pearl and Verma's equivalence criterion, employing an optimized orderly algorithm to generate undirected graphs and classify DAGs into equivalence classes. The key finding is that the ratio of equivalence classes to DAGs asymptotically approaches approximately 0.267, with class size and edge distribution showing Gaussian and irregular patterns, respectively.

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.

Motivation & Objective

• To systematically enumerate Markov equivalence classes of acyclic digraph models to improve model specification and search efficiency.
• To analyze the distribution of equivalence classes by number of edges and class size for DAGs up to 10 vertices.
• To develop and apply a new variation of the orderly algorithm for generating undirected graphs as a foundation for equivalence class enumeration.
• To determine the maximum number of equivalence classes generated by any single undirected graph, observing factorial growth.

Proposed method

• Implementation of a computer program based on Pearl and Verma's Markov equivalence criterion to classify DAGs into equivalence classes.
• Adaptation of the orderly algorithm for generating undirected graphs, with a novel variation to support efficient equivalence class enumeration.
• Enumeration of all DAGs up to 10 vertices, followed by grouping into Markov equivalence classes using the equivalence criterion.
• Statistical analysis of equivalence classes by edge count and class size, including distribution patterns and asymptotic behavior.
• Computation of the maximum number of equivalence classes per undirected graph, revealing approximately factorial growth.

Experimental results

Research questions

• RQ1 What is the asymptotic ratio of Markov equivalence classes to DAGs as the number of vertices increases?
• RQ2 How are equivalence classes distributed by the number of edges in the DAGs?
• RQ3 What is the frequency distribution of class sizes, and does it follow a predictable pattern?
• RQ4 What is the maximum number of equivalence classes that can be generated from a single undirected graph, and how does it scale with graph size?
• RQ5 How does the performance and structure of the new orderly algorithm variant compare to standard methods in generating equivalence classes?

Key findings

• The ratio of the number of Markov equivalence classes to the number of DAGs asymptotically approaches approximately 0.267 as the number of vertices increases.
• The distribution of equivalence classes by number of edges approaches a Gaussian (normal) shape, indicating a central tendency in edge counts across classes.
• Classes of size 1 are the most frequent, with proportions for larger class sizes initially decreasing before following a more irregular pattern.
• The maximum number of equivalence classes generated by any single undirected graph increases approximately factorially with the number of vertices.
• The new variation of the orderly algorithm successfully supports the enumeration of equivalence classes and enables scalable computation up to 10 vertices.