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Propagation of Gaussian belief functions

  • Liping Liu
Part of the Lecture Notes in Statistics book series (LNS, volume 112)

Abstract

A Gaussian belief function (GBF) can be intuitively described as a Gaussian distribution over a hyperplane, whose parallel sub-hyperplanes are the focal elements. It includes as special cases non-probabilistic linear equations, statistical observations, multivariate Gaussian distributions, and vacuous belief functions. The notion of GBFs was proposed in [Dem 90b], formalized in [Sha 92] and [Liu 95a], and successfully applied in combining independent statistical models in [Liu 95b]. In this paper, we propose a join-tree computation scheme for GBFs. We first represent Dempster’s rule for combining GBFs obtained in [Liu 95a] equivalently in terms of matrix sweepings. We then show that the operations of GBFs follow the axioms of Shenoy and Shafer [ShSh 90] and justify the possibility of a join-tree computation scheme for GBFs. The result enriches theory of local computation by extending its application to the combination of statistical models and the integration of knowledge bases. An example is carried out to illustrate how combined inference can be made in accordance with multiple statistical models using graphical belief function models.

Keywords

Variable Space Sample Space Local Computation Multivariate Gaussian Distribution Belief Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [BoTi 73]
    Box, G. E. P. and Tiao, G. C. (1973)Bayesian Inference in Statistical Analysis. Addison-Wesley, Reading.zbMATHGoogle Scholar
  2. [HuSc 90]
    Hunter, J. E. and Schmidt, F. L. (1990)Methods of Meta-Analysis: Correcting Error and Bias in Research Findings. Sage, Newbury Park.Google Scholar
  3. [Dem 90a]
    Dempster, A. P. (1990a) “Construction and Local Computation Aspects of Network Belief Functions,” InInfluence Diagrams Belief Nets and Decision AnalysisR. M. Oliver, J. Q. Smith (eds.), John Wiley and Sons, Chichester.Google Scholar
  4. [Dem 90b]
    Dempster, A. P. (1990b) “Normal Belief Functions and the Kalmam Filter,” Research Report, Department of Statistics, Harvard University, Cambridge, MA.Google Scholar
  5. [JLO 90]
    Jensen, F. V., Lauritzen, S. L., and Olesen, K. G. (1990) “Bayesian Updating in Causal Probabilistic Networks by Local Computations,”Computational Statistics Quarterly 4269–282.MathSciNetGoogle Scholar
  6. [LaSp 88]
    Lauritzen, S. L. and Spiegelhalter, D. J. (1988) “Local Computation with Probabilities on Graphical Structures and Their Application to Expert Systems (with discussion),”Journal of the Royal Statistical SocietySeries B50157–224.MathSciNetzbMATHGoogle Scholar
  7. [LaWe 89]
    Lauritzen, S. L. and Wermuth, N. (1989) “Graphical Models for Associations between Variables, Some of Which are Qualitative and Some Quantitative,”The Annals of Statistics 1731–57.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [LaWe 89]
    Lauritzen, S. L. and Wermuth, N. (1989) “Graphical Models for Associations between Variables, Some of Which are Qualitative and Some Quantitative,”The Annals of Statistics 1731–57.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [Liu 95b]
    Liu, L. (1995) “Model Combination Using Gaussian Belief Functions,” Research Report, School of Business, University of Kansas, Lawrence, KS.Google Scholar
  10. [Liu 95c]
    Liu, L. (1995) “Local Computation of Gaussian Belief Functions,” Research Report, School of Business, University of Kansas, Lawrence, KS.Google Scholar
  11. [Kong 86]
    Kong, A. (1986) “Multivariate Belief Functions and Graphical Models,” Ph.D. Thesis, Department of Statistics, Harvard University, Cambridge, MA.Google Scholar
  12. [Mai 83]
    Maier, D. (1983)The Theory of Relational Databases. Computer Science Press, Rockville.Google Scholar
  13. [Mel 87]
    Mellouli, K. (1987) “On the Propagation of Beliefs in Networks using the Dempster-Shafer Theory of Evidence,” Ph.D. Thesis, School of Business, University of Kansas, Lawrence, KS.Google Scholar
  14. [Sha 76]
    Shafer, G. (1976)A Mathematical Theory of Evidence. Princeton University Press, Princeton.zbMATHGoogle Scholar
  15. [Sha 92]
    Shafer, G. (1992) “A Note on Dempster’s Gaussian Belief Functions,” Working paper, School of Business, University of Kansas, Lawrence, KS.Google Scholar
  16. [SSM 87]
    Shafer, G., Shenoy, P. P., and Mellouli, K. (1987) “Propagating Belief Functions in Qualitative Markov Trees,”International Journal of Approximate Reasoning 3383–411.MathSciNetGoogle Scholar
  17. [ShSh 90]
    Shenoy, P. P. and Shafer, G. (1990) “Axioms for Probability and Belief-Function Propagation,”Uncertainty in Artificial Intelligence 4169–198.MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1996

Authors and Affiliations

  • Liping Liu
    • 1
  1. 1.School of BusinessAlbany State CollegeAlbanyUSA

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