Skip to main content
SpringerLink
Log in
Book cover

Belief Functions: Theory and Applications pp 101–108Cite as

Game-Theoretical Semantics of Epistemic Probability Transformations

  • Conference paper

  • 1148 Accesses

Part of the book series: Advances in Intelligent and Soft Computing ((AINSC,volume 164))

Abstract

Probability transformation of belief functions can be classified into different families, according to the operator they commute with. In particular, as they commute with Dempster’s rule, relative plausibility and belief transforms form one such “epistemic” family, and possess natural rationales within Shafer’s formulation of the theory of evidence, while they are not consistent with the credal or probability-bound semantic of belief functions. We prove here, however, that these transforms can be given in this latter case an interesting rationale in terms of optimal strategies in a non-cooperative game.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bogler, P.: Shafer-Dempster reasoning with applications to multisensor target identification systems. IEEE Trans. on Systems, Man and Cybernetics 17(6), 968–977 (1987)

    Google Scholar 

  2. Bowles, S.: Microeconomics: Behavior, institutions, and evolution. Princeton University Press (2004)

    Google Scholar 

  3. Burger, T.: Defining new approximations of belief functions by means of Dempster’s combination. In: Proc. of BELIEF, Brest, France (2010)

    Google Scholar 

  4. Chateauneuf, A., Jaffray, J.: Some characterizations of lower probabilities and other monotone capacities through the use of Möbius inversion. Mathematical Social Sciences 17, 263–283 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cobb, B., Shenoy, P.: A comparison of Bayesian and belief function reasoning. Information Systems Frontiers 5(4), 345–358 (2003)

    Article  Google Scholar 

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

    Article  Google Scholar 

  7. Cuzzolin, F.: Two new Bayesian approximations of belief functions based on convex geometry. IEEE Trans. on Systems, Man, and Cybernetics B 37(4), 993–1008 (2007)

    Article  Google Scholar 

  8. Cuzzolin, F.: A geometric approach to the theory of evidence. IEEE Trans. on Systems, Man, and Cybernetics C 38(4), 522–534 (2008)

    Article  Google Scholar 

  9. Cuzzolin, F.: Dual properties of the relative belief of singletons. In: Proc. of PRICAI, Hanoi, Vietnam, pp. 78–90 (2008)

    Google Scholar 

  10. Cuzzolin, F.: Semantics of the relative belief of singletons. In: Workshop on Uncertainty and Logic, Kanazawa, Japan (2008)

    Google Scholar 

  11. Cuzzolin, F.: The geometry of consonant belief functions: simplicial complexes of necessity measures. Fuzzy Sets and Systems 161(10), 1459–1479 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cuzzolin, F.: On the relative belief transform. International Journal of Approximate Reasoning (in press, 2012)

    Google Scholar 

  13. Daniel, M.: On transformations of belief functions to probabilities. Int. J. of Intelligent Systems 21(3), 261–282 (2006)

    Article  MATH  Google Scholar 

  14. Dempster, A.P.: Lindley’s paradox: Comment. Journal of the American Statistical Association 77(378), 339–341 (1982)

    Google Scholar 

  15. Dempster, A.P.: A generalization of Bayesian inference. In: Classic Works of the Dempster-Shafer Theory of Belief Functions, pp. 73–104 (2008)

    Google Scholar 

  16. Dezert, J., Smarandache, F.: A new probabilistic transformation of belief mass assignment. In: Proc. of the 11th International Conference of Information Fusion, pp. 1–8 (2008)

    Google Scholar 

  17. Fagin, R., Halpern, J.: Uncertainty, belief and probability. In: Proc. of IJCAI, pp. 1161–1167 (1989)

    Google Scholar 

  18. Kohlas, J., Monney, P.-A.: A Mathematical Theory of Hints. An Approach to Dempster-Shafer Theory of Evidence. Springer (1995)

    Google Scholar 

  19. Nguyen, H.: On random sets and belief functions. Journal of Mathematical Analysis and Applications 65, 531–542 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  20. Schubert, J.: On ‘rho’ in a decision-theoretic apparatus of Dempster-Shafer theory. International Journal of Approximate Reasoning 13, 185–200 (1995)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  22. Shafer, G.: Constructive probability. Synthese 48, 309–370 (1981)

    Article  MathSciNet  Google Scholar 

  23. Shenoy, P.: No double counting semantics for conditional independence. Working Paper No. 307. School of Business, University of Kansas (2005)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  25. Smets, P., Kennes, R.: The transferable belief model. Artificial Intelligence 66(2), 191–234 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  26. Strat, T.M.: Decision analysis using belief functions. International Journal of Approximate Reasoning 4, 391–417 (1990)

    Article  MATH  Google Scholar 

  27. Sudano, J.: Equivalence between belief theories and nave Bayesian fusion for systems with independent evidential data. In: Proc. of ICIF, vol. 2, pp. 1239–1243 (2003)

    Google Scholar 

  28. von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press (1944)

    Google Scholar 

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

    Article  MATH  Google Scholar 

  30. Wald, A.: Statistical decision functions. Wiley, New York (1950)

    MATH  Google Scholar 

  31. Walley, P.: Belief function representations of statistical evidence. The Annals of Statistics 15, 1439–1465 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  32. Xu, H., Hsia, Y.-T., Smets, P.: The transferable belief model for decision making in the valuation-based systems. IEEE Trans. on Systems, Man, and Cybernetics 26, 698–707 (1996)

    Article  Google Scholar 

  33. Zadeh, L.: A simple view of the Dempster-Shafer theory of evidence and its implications for the rule of combination. AI Magazine 7(2), 85–90 (1986)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Oxford Brookes University, Oxford, UK

    Fabio Cuzzolin

Authors
  1. Fabio Cuzzolin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fabio Cuzzolin .

Editor information

Editors and Affiliations

  1. , Centre de Recherches de Royallieu, Université de Technologie de Compiègne, Compiègne, 60205, France

    Thierry Denoeux

  2. Université de Picardie Jules Verne, Compiègne, France

    Marie-Hélène Masson

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Cuzzolin, F. (2012). Game-Theoretical Semantics of Epistemic Probability Transformations. In: Denoeux, T., Masson, MH. (eds) Belief Functions: Theory and Applications. Advances in Intelligent and Soft Computing, vol 164. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29461-7_12

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-29461-7_12

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-29460-0

  • Online ISBN: 978-3-642-29461-7

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation