The Annals of Statistics

Separation and Completeness Properties for Amp Chain Graph Markov Models

Michael Levitz, David Madigan, and Michael D. Perlman

Full-text: Open access

Abstract

Pearl ’s well-known $d$-separation criterion for an acyclic directed graph (ADG) is a pathwise separation criterion that can be used to efficiently identify all valid conditional independence relations in the Markov model determined by the graph. This paper introduces $p$-separation, a pathwise separation criterion that efficiently identifies all valid conditional independences under the Andersson–Madigan–Perlman (AMP) alternative Markov property for chain graphs ( = adicyclic graphs), which include both ADGs and undirected graphs as special cases. The equivalence of p-separation to the augmentation criterion occurring in the AMP global Markov property is established, and $p$-separation is applied to prove completeness of the global Markov propertyfor AMP chain graph models. Strong completeness of the AMP Markov property is established, that is, the existence of Markov perfect distributions that satisfy those and only those conditional independences implied by the AMP property (equivalently, by $p$-separation). A linear-time algorithm for determining $p$-separation is presented.

Article information

Source
Ann. Statist., Volume 29, Number 6 (2001), 1751-1784.

Dates
First available in Project Euclid: 5 March 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1015345961

Digital Object Identifier
doi:10.1214/aos/1015345961

Mathematical Reviews number (MathSciNet)
MR1891745

Zentralblatt MATH identifier
1043.62080

Subjects
Primary: 62M45: Neural nets and related approaches 60K99: None of the above, but in this section
Secondary: 68R10: Graph theory (including graph drawing) [See also 05Cxx, 90B10, 90B35, 90C35] 68T30: Knowledge representation

Keywords
Graphical Markov model acyclic directed graph Bayesian network $d$-separation chain graph AMP model $p$-separation completeness efficient algorithm

Citation

Levitz, Michael; Perlman, Michael D.; Madigan, David. Separation and Completeness Properties for Amp Chain Graph Markov Models. Ann. Statist. 29 (2001), no. 6, 1751--1784. doi:10.1214/aos/1015345961. https://projecteuclid.org/euclid.aos/1015345961


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References

  • Andersson, S. A., Madigan, D. and Perlman, M. D. (1996). An alternative Markov property for chain graphs. In Uncertainty in Artificial Intelligence: Proceedings of the Twelfth Conference (F. Jensen and E. Horvitz, eds.) 40-48. Morgan Kaufmann, San Francisco. Andersson, S. A., Madigan, D. and Perlman, M. D. (1997a). On the Markov equivalence of chain graphs, undirected graphs and acyclic digraphs. Scand. J. Statist. 24 81-102. Andersson, S. A., Madigan, D. and Perlman, M. D. (1997b). A characterization of Markov equivalence classes for acyclic digraphs. Ann. Statist. 25 505-541.
  • Andersson, S. A., Madigan, D. and Perlman, M. D. (2001). Alternative Markov properties for chain graphs. Scand. J. Statist. 28 33-85.
  • Andersson, S. A. and Perlman, M. D. (1998). Normal linear regression models with recursive graphical Markov structure. J. Multivariate Anal. 66 133-187.
  • Bouckaert, R. R. and Studen´y, M. (1995). Chain graphs: semantics and expressiveness. In Symbolic and Quantitative Approaches to Reasoning and Uncertainty. Lecture Notes in AI 946 67-76. Springer, New York.
  • Buntine, W. L. (1995). Chain graphs for learning. In Uncertainty in Artificial Intelligence: Proceedings of the Eleventh Conference (P. Besnard and S. Hanks, eds.) 87-98. Morgan Kaufmann, San Francisco.
  • Cowell, R. G., Dawid, A. P., Lauritzen, S. L. and Spiegelhalter, D. J. (1999). Probabilistic Networks and Expert Systems. Springer, New York.
  • Cox, D. R. (1999). Graphical Markov models: statistical motivation. Paper presented at the Workshop on Conditional Independence Structures and Graphical Models, Fields Institute for Research in Mathematical Sciences, Toronto, Canada.
  • Cox, D. R. and Wermuth, N. (1996). Multivariate Dependencies: Models, Analysis and Interpretation. Chapman and Hall, London. Frydenberg, M. (1990a). The chain graph Markov property. Scand. J. Statist. 17 333-353. Frydenberg, M. (1990b). Marginalization and collapsibilityin graphical interaction models. Ann. Statist. 18 790-805.
  • Geiger, D. and Pearl, J. (1988). On the logic of causal models. In Proceedings of the Fourth Workshop on Uncertainty in AI (R. Shachter, T. Levitt, L. Kanal and J. Lemmer, eds.) 136-147. North-Holland, Amsterdam.
  • Geiger, D., Verma, T. and Pearl, J. (1990). Identifying independence in Bayesian networks. Networks 20 507-534.
  • Heckerman, D., Geiger, D. and Chickering, D. M. (1995). Learning Bayesian networks: the combination of knowledge and statistical data. Machine Learning 20 197-243.
  • Højsgaard, S. and Thiesson, B. (1995). BIFROST:Block recursive models induced from relevant knowledge, observations, and statistical techniques. Comput. Statist. Data Anal. 19 155-175.
  • Koster, J. T. A. (1999). Linear Structural Equations and Graphical Models. The Fields Institute, Toronto, Canada.
  • Lauritzen, S. L. (1996). Graphical Models. Oxford Univ. Press.
  • Lauritzen, S. L., Dawid, A. P., Larsen, B. N. and Leimer, H. G. (1990). Independence properties of directed Markov fields. Networks 20 491-505.
  • Lauritzen, S. L. and Wermuth, N. (1989). Graphical models for association between variables, some of which are qualitative and some quantitative. Ann. Statist. 17 31-57.
  • Madigan, D., Andersson, S. A., Perlman, M. D. and Volinsky, C. M. (1996). Bayesian model averaging and model selection for Markov equivalence classes of acyclic diagraphs. Comm. Statist. Theory Methods Ser. A 25 2493-2519.
  • Meek, C. (1995). Strong completeness and faithfulness in Bayesian networks. In Uncertainty in Artificial Intelligence: Proceedings of the Eleventh Conference (P. Besnard and S. Hanks, eds.) 403-410. Morgan Kaufmann, San Mateo, CA.
  • Okamoto, M. (1973). Distinctness of the eigenvalues of a quadratic form in a multivariate sample. Ann. Statist. 1 763-765.
  • Pearl, J. (1988). Probabilistic Reasoning in Intelligent Systems. Morgan Kaufmann, San Mateo, CA.
  • Spiegelhalter, D. J., Dawid, A. P., Lauritzen, S. L. and Cowell, R. G. (1993). Bayesian analysis in expert systems (with discussion). Statist. Sci. 8 219-283.
  • Spirtes, P., Glymour, C. and Scheines, R. (1993). Causation, Prediction, and Search. Lecture Notes in Statist. 81 Springer, New York.
  • Studen´y, M. (1996). On separation criterion and recoveryalgorithm for chain graphs. In Uncertainty in Artificial Intelligence: Proceedings of the Twelfth Conference (F. Jensen and E. Horvitz, eds.) 509-516. Morgan Kaufmann, San Francisco.
  • Studen´y, M. (1997). A recoveryalgorithm for chain graphs. Internat. J. Approx. Reason. 17 265-293.
  • Studen´y, M. (1998). Bayesian networks from the point of view of chain graphs. In Uncertainty in Artificial Intelligence: Proceedings of the Fourteenth Conference (G. F. Cooper and S. Moral, eds.) 496-503. Morgan Kaufmann, San Francisco.
  • Studen´y, M. and Bouckaert, R. R. (1998). On chain graph models for description of conditional independence structure. Ann. Statist. 26 1434-1495.
  • Wermuth, N. and Lauritzen, S. L. (1990). On substantive research hypotheses, conditional independence graphs, and graphical chain models (with discussion). J. Roy. Statist. Soc. Ser. B 52 21-72.
  • Whittaker, J. (1990). Graphical Models in Applied Multivariate Statistics. Wiley, New York.