Advertisement

A Mathematical Formulation of Dempster-Shafer’s Belief Functions

  • Chun-Hung Tzeng
Conference paper
Part of the Informatik-Fachberichte book series (INFORMATIK, volume 287)

Abstract

This paper introduces a unified mathematical formulation for both Bayesian approach and Dempster-Shafer’s approach in handling uncertainty in artificial intelligence. Each body of uncertain information in this formulation is an information quadruplet, consisting of a code space, a message space, an interpretation function, and an evidence space. Each information quadruplet contains prior information as well as possible new evidence which may appear later. Bayes rule is used to update the prior information. This paper also introduces an idea of independent information and its combination, in which a combination formula is derived. A conventional Bayesian prior probability measure is the prior information of a special information quadruplet; Bayesian conditioning is the combination of special independent information. A Dempster’s belief function is the belief function of a different information quadruplet; the Dempster combination rule is the combination rule of such independent quadruplets. This paper shows that both the conventional Bayesian approach and Dempster-Shafer’s approach originate from the same mathematical theory.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Dempster, A P (1967) Upper and lower probabilities induced by a multivalued mapping, Annals of Mathematical Statistics (38) 325–339MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    Dempster, A P (1968) A generalization of Bayesian inference, Journal of the Royal Statistical Society (38) 205–247MathSciNetGoogle Scholar
  3. [3]
    Hummel, R A and Landy, M S (1988) A Statistical Viewpoint on the Theory of Evidence, IEEE Transactions on Pattern Analysis and Machine Intelligence 10, no. 2, 235–247zbMATHCrossRefGoogle Scholar
  4. [4]
    Kyburg, II E Jr (1987) Bayesian and non-Bayesian evidential updating, Artificial Intelligence 31, 271–293MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    Laskey, K B (1987) Belief in belief functions: an examination of Shafer’s canonical examples, in Uncertainty in Artificial Intelligence, Third Workshop on Uncertainty in Artificial Intelligence, Seattle, Washington, 39–46Google Scholar
  6. [6]
    Pear, J (1988) Probabilistic Reasoning in Intelligent System: Networks of Plausible Inference, Morgan Kaufmann Publisher, San Mateo, California.Google Scholar
  7. [7]
    Ruspini, E II (1987) Epistemic logics, probability, and the calculus of evidence, in Proceedings of IJCAI-87, 924–931Google Scholar
  8. [8]
    Shafer, G (1976) A mathematical theory of evidence, Princeton University Press, Princeton, New JerseyzbMATHGoogle Scholar
  9. [9]
    Shafer, G (1981) Constructive probability, Synihese 48, 1–60MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    Shafer, G (1987) Probability Judgment in Artificial Intelligence and Expert Systems, Statistical Science 2 No. 1, 3–44MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    Tzeng, C II (1988a) A Theory of Heuristic Information in Game-Tree Search, Springer-Verlag, BerlinzbMATHCrossRefGoogle Scholar
  12. [12]
    Tzeng, C II (1988b) A Study of Dempster-Shafer’s Theory through Shafer’s Canonical Examples, Proceedings of International Computer Science Conference ’88, Artificial Intelligence: Theory and Applications, Ilong Kong, December 19–21, 1988, 69–76Google Scholar
  13. [13]
    Tzeng, C II (1990) A Mathematical Formulation of Uncertain Information, to appear in Annals of Mathematics and Artificial Intelligence. Google Scholar
  14. [14]
    William, P M (1982) Discussion of “belief functions and parametric models” by Glenn Shafer, Journal of the Royal Statistical Society, Ser. B, 44(3), 341–343MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Chun-Hung Tzeng
    • 1
  1. 1.Computer Science DepartmentBall State UniversityMuncieUSA

Personalised recommendations