Game-Theoretical Semantics of Epistemic Probability Transformations

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


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 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. 2.
    Bowles, S.: Microeconomics: Behavior, institutions, and evolution. Princeton University Press (2004)Google Scholar
  3. 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. 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)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Cobb, B., Shenoy, P.: A comparison of Bayesian and belief function reasoning. Information Systems Frontiers 5(4), 345–358 (2003)CrossRefGoogle Scholar
  6. 6.
    Cuzzolin, F.: Geometry of Dempster’s rule of combination. IEEE Trans. on Systems, Man and Cybernetics B 34(2), 961–977 (2004)CrossRefGoogle Scholar
  7. 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)CrossRefGoogle Scholar
  8. 8.
    Cuzzolin, F.: A geometric approach to the theory of evidence. IEEE Trans. on Systems, Man, and Cybernetics C 38(4), 522–534 (2008)CrossRefGoogle Scholar
  9. 9.
    Cuzzolin, F.: Dual properties of the relative belief of singletons. In: Proc. of PRICAI, Hanoi, Vietnam, pp. 78–90 (2008)Google Scholar
  10. 10.
    Cuzzolin, F.: Semantics of the relative belief of singletons. In: Workshop on Uncertainty and Logic, Kanazawa, Japan (2008)Google Scholar
  11. 11.
    Cuzzolin, F.: The geometry of consonant belief functions: simplicial complexes of necessity measures. Fuzzy Sets and Systems 161(10), 1459–1479 (2010)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Cuzzolin, F.: On the relative belief transform. International Journal of Approximate Reasoning (in press, 2012)Google Scholar
  13. 13.
    Daniel, M.: On transformations of belief functions to probabilities. Int. J. of Intelligent Systems 21(3), 261–282 (2006)MATHCrossRefGoogle Scholar
  14. 14.
    Dempster, A.P.: Lindley’s paradox: Comment. Journal of the American Statistical Association 77(378), 339–341 (1982)Google Scholar
  15. 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. 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. 17.
    Fagin, R., Halpern, J.: Uncertainty, belief and probability. In: Proc. of IJCAI, pp. 1161–1167 (1989)Google Scholar
  18. 18.
    Kohlas, J., Monney, P.-A.: A Mathematical Theory of Hints. An Approach to Dempster-Shafer Theory of Evidence. Springer (1995)Google Scholar
  19. 19.
    Nguyen, H.: On random sets and belief functions. Journal of Mathematical Analysis and Applications 65, 531–542 (1978)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Schubert, J.: On ‘rho’ in a decision-theoretic apparatus of Dempster-Shafer theory. International Journal of Approximate Reasoning 13, 185–200 (1995)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Shafer, G.: A mathematical theory of evidence. Princeton University Press (1976)Google Scholar
  22. 22.
    Shafer, G.: Constructive probability. Synthese 48, 309–370 (1981)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Shenoy, P.: No double counting semantics for conditional independence. Working Paper No. 307. School of Business, University of Kansas (2005)Google Scholar
  24. 24.
    Smets, P.: Decision making in the TBM: the necessity of the pignistic transformation. International Journal of Approximate Reasoning 38(2), 133–147 (2005)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Smets, P., Kennes, R.: The transferable belief model. Artificial Intelligence 66(2), 191–234 (1994)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Strat, T.M.: Decision analysis using belief functions. International Journal of Approximate Reasoning 4, 391–417 (1990)MATHCrossRefGoogle Scholar
  27. 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. 28.
    von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press (1944)Google Scholar
  29. 29.
    Voorbraak, F.: A computationally efficient approximation of Dempster-Shafer theory. International Journal on Man-Machine Studies 30, 525–536 (1989)MATHCrossRefGoogle Scholar
  30. 30.
    Wald, A.: Statistical decision functions. Wiley, New York (1950)MATHGoogle Scholar
  31. 31.
    Walley, P.: Belief function representations of statistical evidence. The Annals of Statistics 15, 1439–1465 (1987)MathSciNetMATHCrossRefGoogle Scholar
  32. 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)CrossRefGoogle Scholar
  33. 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

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Oxford Brookes UniversityOxfordUK

Personalised recommendations