Skip to main content

Dual Properties of the Relative Belief of Singletons

  • Conference paper

Part of the Lecture Notes in Computer Science book series (LNAI,volume 5351)

Abstract

In this paper we prove that a recent Bayesian approximation of belief functions, the relative belief of singletons, meets a number of properties with respect to Dempster’s rule of combination which mirrors those satisfied by the relative plausibility of singletons. In particular, its operator commutes with Dempster’s sum of plausibility functions, while perfectly representing a plausibility function when combined through Dempster’s rule. This suggests a classification of all Bayesian approximations into two families according to the operator they relate to.

Keywords

  • Bayesian Approximation
  • Belief Function
  • Dual Property
  • Approximate Reasoning
  • Operator Commute

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.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-540-89197-0_11
  • Chapter length: 13 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   169.00
Price excludes VAT (USA)
  • ISBN: 978-3-540-89197-0
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   219.00
Price excludes VAT (USA)

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Shafer, G.: A mathematical theory of evidence. Princeton University Press, Princeton (1976)

    MATH  Google Scholar 

  2. Daniel, M.: On transformations of belief functions to probabilities. International Journal of Intelligent Systems, special issue on Uncertainty Processing 21(3), 261–282 (2006)

    CrossRef  MATH  Google Scholar 

  3. Kramosil, I.: Approximations of believeability functions under incomplete identification of sets of compatible states. Kybernetika 31, 425–450 (1995)

    MathSciNet  MATH  Google Scholar 

  4. Yaghlane, A.B., Denœux, T., Mellouli, K.: Coarsening approximations of belief functions. In: Benferhat, S., Besnard, P. (eds.) ECSQARU 2001. LNCS (LNAI), vol. 2143, pp. 362–373. Springer, Heidelberg (2001)

    CrossRef  Google Scholar 

  5. Haenni, R., Lehmann, N.: Resource bounded and anytime approximation of belief function computations. International Journal of Approximate Reasoning 31(1-2), 103–154 (2002)

    MathSciNet  CrossRef  MATH  Google Scholar 

  6. Bauer, M.: Approximation algorithms and decision making in the Dempster-Shafer theory of evidence–an empirical study. International Journal of Approximate Reasoning 17, 217–237 (1997)

    MathSciNet  CrossRef  MATH  Google Scholar 

  7. Tessem, B.: Approximations for efficient computation in the theory of evidence. Artificial Intelligence 61(2), 315–329 (1993)

    MathSciNet  CrossRef  Google Scholar 

  8. Smets, P.: Belief functions versus probability functions. In: Bouchon, B., Saitta, L., Yager, R. (eds.) Uncertainty and Intelligent Systems, pp. 17–24. Springer, Berlin (1988)

    CrossRef  Google Scholar 

  9. Smets, P.: Decision making in the TBM: the necessity of the pignistic transformation. International Journal of Approximate Reasoning 38(2), 133–147 (2005)

    MathSciNet  CrossRef  MATH  Google Scholar 

  10. Cuzzolin, F.: Two new Bayesian approximations of belief functions based on convex geometry. IEEE Transactions on Systems, Man, and Cybernetics - Part B 37(4) (2007)

    Google Scholar 

  11. Voorbraak, F.: A computationally efficient approximation of Dempster-Shafer theory. International Journal on Man-Machine Studies 30, 525–536 (1989)

    CrossRef  MATH  Google Scholar 

  12. Dempster, A.: Upper and lower probabilities generated by a random closed interval. Annals of Mathematical Statistics 39, 957–966 (1968)

    MathSciNet  CrossRef  MATH  Google Scholar 

  13. Cobb, B., Shenoy, P.: On the plausibility transformation method for translating belief function models to probability models. Int. J. Approx. Reasoning 41(3), 314–330 (2006)

    MathSciNet  CrossRef  MATH  Google Scholar 

  14. Cuzzolin, F.: Semantics of the relative belief of singletons. In: Workshop on Uncertainty and Logic, Kanazawa, Japan, March 25-28 (2008)

    Google Scholar 

  15. Smets, P.: Belief functions: the disjunctive rule of combination and the generalized Bayesian theorem. International Journal of Approximate reasoning 9, 1–35 (1993)

    MathSciNet  CrossRef  MATH  Google Scholar 

  16. Aigner, M.: Combinatorial Theory. In: Classics in Mathematics. Springer, New York (1979)

    Google Scholar 

  17. Smets, P.: The canonical decomposition of a weighted belief. In: Proceedings of IJCAI 1995, Montréal, Canada, pp. 1896–1901 (1995)

    Google Scholar 

  18. Cuzzolin, F.: Geometry of upper probabilities. In: Proceedings of ISIPTA 2003 (2003)

    Google Scholar 

  19. Cuzzolin, F.: Geometry of Dempster’s rule of combination. IEEE Transactions on Systems, Man and Cybernetics part B 34(2), 961–977 (2004)

    CrossRef  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Cuzzolin, F. (2008). Dual Properties of the Relative Belief of Singletons. In: Ho, TB., Zhou, ZH. (eds) PRICAI 2008: Trends in Artificial Intelligence. PRICAI 2008. Lecture Notes in Computer Science(), vol 5351. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-89197-0_11

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-89197-0_11

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-89196-3

  • Online ISBN: 978-3-540-89197-0

  • eBook Packages: Computer ScienceComputer Science (R0)