Characterization of Inclusion Neighbourhood in Terms of the Essential Graph: Upper Neighbours

  • Milan Studený
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2711)

Abstract

The problem of efficient characterization of inclusion neighbourhood is crucial for some methods of learning (equivalence classes of) Bayesian networks. In this paper, neighbouring equivalence classes of a given equivalence class of Bayesian networks are characterized efficiently by means of the respective essential graph. The characterization reveals hidded internal structure of the inclusion neighbourhood. More exactly, upper neighbours, that is, those neighbouring equivalence classes which describe more independencies, are completely characterized here. First, every upper neighbour is characterized by a pair ([a,b],C) where [a,b] is an edge in the essential graph and \(C \subseteq N \ \{a,b\}\) a disjoint set of nodes. Second, if [a,b] is fixed, the class of sets C which characterize the respective neighbours is a tuft/ of sets determined by its least set and the list of its maximal sets. These sets can be read directly from the essential graph. An analogous characterization of lower neighbours, which is more complex, is mentioned.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Andersson, S.A., Madigan, D., Perlman, M.D.: A characterization of Markov equivalence classes for acyclic digraphs. Annals of Statistics 25, 505–541 (1997)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Auvray, V., Wehenkel, L.: On the construction of the inclusion boundary neighbourhood for Markov equivalence clases of Bayesian network structures. In: Darwiche, A., Friedman, N. (eds.) Uncertainty in Artificial Intelligence 18, pp. 26–35. Morgan Kaufmann, San Francisco (2002)Google Scholar
  3. 3.
    Bouckaert, R.R.: Bayesian belief networks: from construction to inference. PhD thesis, University of Utrecht (1995)Google Scholar
  4. 4.
    Castelo, R.: The discrete acyclic digraph Markov model in data mining. PhD thesis, University of Utrecht (2002)Google Scholar
  5. 5.
    Chickering, D.M.: Optimal structure identification with greedy search. To appear. In: Journal of Machine Learning Research (2003)Google Scholar
  6. 6.
    Frydenberg, M.: The chain graph Markov property. Scandinavian Journal of Statistics 17, 333–353 (1990)MATHMathSciNetGoogle Scholar
  7. 7.
    Kočka, T., Castello, R.: Improved learning of Bayesian networks. In: Breese, J., Koller, D. (eds.) Uncertainty in Artificial Intelligence 17, pp. 269–276. Morgan Kaufmann, San Francisco (2001)Google Scholar
  8. 8.
    Lauritzen, S.L.: Graphical Models. Clarendon Press, Oxford (1996)Google Scholar
  9. 9.
    Meek, C.: Graphical models, selectin causal and statistical models. PhD thesis, Carnegie Mellon University (1997)Google Scholar
  10. 10.
    Pearl, J.: Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, San Mateo (1988)Google Scholar
  11. 11.
    Rovenato, A.: A unified approach to the characterisation of equivalence classes of DAGs, chain graphs with no flags and chain graphs. Technical report, Dipartimento di Scienze Sociali Cognitive e Quantitative, University of Modena and Reggio Emilia (2003) submitted to Annals of StatisticsGoogle Scholar
  12. 12.
    Studený, M.: A recovery algorithm for chain graphs. International Journal of Approximate Reasoning 17, 265–293 (1997)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Studený, M.: On probabilistic conditional independence structures. A research monograph, Institute of Information Theory and Automation, Prague, Czech Republic. Springer Heidelberg (2003)Google Scholar
  14. 14.
    Studený, M.: Characterization of essential graphs by means of an operation of legal merging of components. Submitted to International Journal of Uncertainty, Fuzziness and Knowledge-based SystemsGoogle Scholar
  15. 15.
    Studený, M.: Characterization of inclusion neighbourhood in terms of the essential graph: lower neighbours. In: preparation, to be submitted to Proceedings of WUPES (2003)Google Scholar
  16. 16.
    Verma, T., Pearl, J.: Equivalence and synthesis of causal models. In: Bonissone, P.P., Henrion, M., Kanal, L.N., Lemmer, J.F. (eds.) Uncertainty in Artificial Intelligence 6, pp. 220–227. Elsevier, New York (1991)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Milan Studený
    • 1
  1. 1.Institute of Information Theory and AutomationAcademy of Sciences of the Czech RepublicPragueCzech Republic

Personalised recommendations