The Annals of Statistics

A characterization of Markov equivalence classes for acyclic digraphs

Steen A. Andersson, David Madigan, and Michael D. Perlman

Full-text: Open access

Abstract

Undirected graphs and acyclic digraphs (ADG's), as well as their mutual extension to chain graphs, are widely used to describe dependencies among variables in multiviarate distributions. In particular, the likelihood functions of ADG models admit convenient recursive factorizations that often allow explicit maximum likelihood estimates and that are well suited to building Bayesian networks for expert systems. Whereas the undirected graph associated with a dependence model is uniquely determined, there may be many ADG's that determine the same dependence (i.e., Markov) model. Thus, the family of all ADG's with a given set of vertices is naturally partitioned into Markov-equivalence classes, each class being associated with a unique statistical model. Statistical procedures, such as model selection of model averaging, that fail to take into account these equivalence classes may incur substantial computational or other inefficiences. Here it is show that each Markov-equivalence class is uniquely determined by a single chain graph, the essential graph, that is itself simultaneously Markov equivalent to all ADG's in the equivalence class. Essential graphs are characterized, a polynomial-time algorithm for their construction is given, and their applications to model selection and other statistical questions are described.

Article information

Source
Ann. Statist. Volume 25, Number 2 (1997), 505-541.

Dates
First available in Project Euclid: 12 September 2002

Permanent link to this document
http://projecteuclid.org/euclid.aos/1031833662

Digital Object Identifier
doi:10.1214/aos/1031833662

Mathematical Reviews number (MathSciNet)
MR1439312

Zentralblatt MATH identifier
0876.60095

Subjects
Primary: 60K99: None of the above, but in this section 62H05: Characterization and structure theory 62M99: None of the above, but in this section 68R10: Graph theory (including graph drawing) [See also 05Cxx, 90B10, 90B35, 90C35] 68T30: Knowledge representation 94C15: Applications of graph theory [See also 05Cxx, 68R10]

Keywords
Graphical Markov model acyclic digraph Markov equivalence essential graph model selection

Citation

Andersson, Steen A.; Madigan, David; Perlman, Michael D. A characterization of Markov equivalence classes for acyclic digraphs. Ann. Statist. 25 (1997), no. 2, 505--541. doi:10.1214/aos/1031833662. http://projecteuclid.org/euclid.aos/1031833662.


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