Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-28T04:36:13.664Z Has data issue: false hasContentIssue false
  • English
  • Français

On equivalence of Markov properties over undirected graphs

Part of: Stochastic processes Graph theory Discrete mathematics in relation to computer science

Published online by Cambridge University Press:  14 July 2016

F. Matúš
F. Matúš*
Affiliation:
Institute of Information Theory and Automation, Prague
*
∗Postal address: Institute of Information Theory and Automation, Czechoslovak Academy of Sciences, Pod vodárenskou véží 4, 182 08 Prague, Czechoslovakia.
Get access Rights & Permissions [Opens in a new window]

Abstract

The dependence of coincidence of the global, local and pairwise Markov properties on the underlying undirected graph is examined. The pairs of these properties are found to be equivalent for graphs with some small excluded subgraphs. Probabilistic representations of the corresponding conditional independence structures are discussed.

Keywords

MARKOV RANDOM FIELDSCONDITIONAL INDEPENDENCE

MSC classification

Primary: 68R10: Graph theory (including graph drawing)
Secondary: 05C80: Random graphs 60G60: Random fields
Type
Short Communications
Information
Journal of Applied Probability , Volume 29 , Issue 3 , September 1992 , pp. 745 - 749
Copyright
Copyright © Applied Probability Trust 1992 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This research was supported partially by Internal Grant 27510 of Czechoslovak Academy of Sciences.

References

Dawid, A. P. (1979) Conditional independence in statistical theory (with discussion). J. Roy. Statist. Soc. B41, 1–31.Google Scholar
Frydenberg, M. (1990) Marginalization and collapsibility in graphical interaction models. Ann. Statist. 18, 790–805.CrossRefGoogle Scholar
Lauritzen, S. L. (1989) Lectures on Contingency Tables, 3rd edn. University of Aalborg Press.Google Scholar
Lauritzen, S. L., Dawid, A. P., Larsen, B. N. and Leimer, H. G. (1990) Independence properties of directed Markov fields, Networks 20, 579–605.CrossRefGoogle Scholar
Matúš, F. (1992) Ascending and descending conditional independence relations. In Transactions of the Eleventh Prague Conference on Information Theory, Statistical Decision Functions and Random Processes, A, 189–200, Academia, Prague.Google Scholar
Mouchart, M. and Rolin, J. M. (1984) A note on conditional independence with statistical applications. Statistica 44, 557–584.Google Scholar
Pearl, J. (1988) Probabilistic Reasoning in Intelligent Systems. Morgan Kaufman, San Mateo, California.Google Scholar
Speed, T. P. (1979) A note on nearest-neighbour Gibbs and Markov probabilities. Sankkya A41, 184–197.Google Scholar
Studený, M. (1989) Multiinformation and the problem of characterization of conditional independence relations. Probi. Control Inf. Theory 18, 3–16.Google Scholar
Studený, M. (1992) Conditional independence relations have no finite complete characterization. In Transactions of the Eleventh Prague Conference on Information Theory, Statistical Decision Functions and Random Processes , B, 377–396, Academia, Prague.Google Scholar
Whittaker, J. (1990) Graphical Models in Applied Multivariate Statistics. Wiley, Chichester.Google Scholar