Computing marginals from the marginal representation in Markov trees

Part of the Lecture Notes in Computer Science book series (LNCS, volume 945)


Local computational techniques have been proposed to compute marginals for the variables in belief networks or valuation networks, based on the secondary structures called clique trees or Markov trees. However, these techniques only compute the marginal on the subset of variables contained in one node of the secondary structure. This paper presents a method for computing the marginal on the subset that may not be contained in one node. The proposed method allows us to change the structure of the Markov tree without changing any information contained in the nodes, thus avoids the possible repeated computations. Moreover, it can compute the marginal on any subset from the marginal representation already obtained. An efficient implementation of this method is also proposed.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    F. V. Jensen, K. G. Olesen and K. Anderson: An Algebra of Bayesian belief Universes for Knowledge-based Systems, Networks, 20, (1990) pp. 637–659Google Scholar
  2. 2.
    A. Kong: Multivariate Belief Functions & Graphical Models, Ph.D dissertation, Department of Statistics, Harvard University, Cambridge, MA (1986).Google Scholar
  3. 3.
    S. L. Lauritzen and D. J. Spiegelhalter: Local Computation with Probabilities on Graphical Structures and Their Application to Expert Systems, Journal of the Royal Statistical Society, Series B, 50, No 2, (1988) pp. 157–224.Google Scholar
  4. 4.
    J. Pearl: Probabilistic Reasoning in Intelligence Systems: Networks of Plausible Inference, Morgan Kaufmann, Los Altos, CA (1988).Google Scholar
  5. 5.
    R. D. Shachter: Evaluating Influence Diagrams, Operations Research 34, (1986) pp. 871–882Google Scholar
  6. 6.
    R. D. Shachter, S. K. Andersen and P. Szolovits: The Equivalence of Exact Methods for Probabilistic Inference in Belief Networks, Department of Engineering-Economic Systems, Stanford University (1992).Google Scholar
  7. 7.
    G. Shafer: Probabilistic Expert Systems Society for Industrial and Applied Mathematics, Philadelphia, PA, (1993) to appear.Google Scholar
  8. 8.
    G. Shafer and P. P. Shenoy: Local Computation in Hypertrees, Working Paper No. 201, School of Business, University of Kansas, Lawrence, KS (1988).Google Scholar
  9. 9.
    P. P. Shenoy: A Valuation-Based Language for Expert Systems, International Journal of Approximate Reasoning, 3 (1989) pp. 383–411.Google Scholar
  10. 10.
    P. P. Shenoy: Valuation-Based Systems: A framework for managing uncertainty in expert systems, Fuzzy logic for the Management of Uncertainty edited by Zadeh, L. A. and Kacprzyk J., John Wiley & Sons, New York. (1992) pp. 83–104.Google Scholar
  11. 11.
    P. P. Shenoy: Conditional Independence in Uncertainty Theories, in Proc. 8th Uncertainty in AI edited by Dubois D., Wellman M. P., D'Ambrosio B. D. and Smets Ph., San Mateo, Calif.: Morgan Kaufmann, (1992) pp. 284–291.Google Scholar
  12. 12.
    P. P. Shenoy: Valuation Networks and Conditional Independence, in Proc. 9th Uncertainty in AI edited by Wellman M. P., Mamdani A. and Heckerman D., San Mateo, Calif.: Morgan Kaufmann (1993).Google Scholar
  13. 13.
    Ph. Smets: Belief Functions: the Disjunctive Rule of Combination and the Generalized Bayesian Theorem, International Journal of Approximate Reasoning, Vol. 9, No. 1 (1993) pp1–35.Google Scholar
  14. 14.
    R. E. Tarjan and M. Yannakakis: Simple Linear-time Algorithms to Test Chordality of Graphs, Test Acyclicity of Hypergraphs, and Selectivity Reduce Acyclic Hypergraphs, SIAM J. Comput., 13, (1984) pp. 566–579.Google Scholar
  15. 15.
    H. Xu: Computing Marginals from the Marginal Representation in Markov Trees, Technical Report TR/IRIDIA/93-17 (1993).Google Scholar
  16. 16.
    L. Zhang: Studies on Finding Hypertree Covers of Hypergraphs, Working Paper No. 198, School of Business, University of Kansas, Lawrence, KS, (1988).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Hong Xu
    • 1
  1. 1.IRIDIA and Service d'AutomatiqueUniversité libre de BruxellesBruxellesBelgique

Personalised recommendations