Generalizations of the Relative Belief Transform

Part of the Advances in Intelligent and Soft Computing book series (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. However, the relative belief transform only exists when some mass is assigned to singletons. We show here that relative belief is only a member of a class of “relative mass” mappings, which can be interpreted as lowcost proxies for both plausibility and pignistic transforms.

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References

  1. 1.
    Burger, T.: Defining new approximations of belief functions by means of Dempster’s combination. In: Proc. of BELIEF 2010 (2010)Google Scholar
  2. 2.
    Cobb, B.R., Shenoy, P.P.: A Comparison of Methods for Transforming Belief Function Models to Probability Models. In: Nielsen, T.D., Zhang, N.L. (eds.) ECSQARU 2003. LNCS (LNAI), vol. 2711, pp. 255–266. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  3. 3.
    Cobb, B., Shenoy, P.: A comparison of Bayesian and belief function reasoning. Information Systems Frontiers 5(4), 345–358 (2003)CrossRefGoogle Scholar
  4. 4.
    Cuzzolin, F.: Two new Bayesian approximations of belief functions based on convex geometry. IEEE Transactions on Systems, Man, and Cybernetics - Part B 37(4), 993–1008 (2007)CrossRefGoogle Scholar
  5. 5.
    Cuzzolin, F.: A geometric approach to the theory of evidence. IEEE Transactions on Systems, Man, and Cybernetics - Part C 38(4), 522–534 (2008)CrossRefGoogle Scholar
  6. 6.
    Cuzzolin, F.: Dual properties of the relative belief of singletons. In: Proc. of the Pacific Rim International Conference on AI, pp. 78–90 (2008)Google Scholar
  7. 7.
    Cuzzolin, F.: Geometry of relative plausibility and relative belief of singletons. Annals of Mathematics and Artificial Intelligence, 1–33 (2010)Google Scholar
  8. 8.
    Cuzzolin, F.: Semantics of the relative belief of singletons. In: Workshop on Uncertainty and Logic, Kanazawa, Japan (2008)Google Scholar
  9. 9.
    Daniel, M.: On transformations of belief functions to probabilities. Int. J. of Intelligent Systems 21(3), 261–282 (2006)MATHCrossRefGoogle Scholar
  10. 10.
    Dempster, A.P.: A generalization of Bayesian inference. Journal of the Royal Statistical Society, Series B 30, 205–247 (1968)MathSciNetGoogle Scholar
  11. 11.
    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
  12. 12.
    Haenni, R.: Aggregating referee scores: an algebraic approach. In: 2nd International Workshop on Computational Social Choice, COMSOC 2008, pp. 277–288 (2008)Google Scholar
  13. 13.
    Lowrance, J., Garvey, T., Strat, T.: A framework for evidential-reasoning systems. In: Proc. of the National Conference on Artificial Intelligence, pp. 896–903 (1986)Google Scholar
  14. 14.
    Shafer, G.: A mathematical theory of evidence. Princeton University Press (1976)Google Scholar
  15. 15.
    Smets, P., Kennes, R.: The transferable belief model. Artificial Intelligence 66(2), 191–234 (1994)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Smets, P.: Decision making in the TBM: the necessity of the pignistic transformation. IJAR 38(2), 133–147 (2005)MathSciNetMATHGoogle Scholar
  17. 17.
    Sudano, J.: Equivalence between belief theories and nave Bayesian fusion for systems with independent evidential data. In: Proc. of the 6th International Conference of Information Fusion, vol. 2, pp. 1239–1243 (2003)Google Scholar
  18. 18.
    Tessem, B.: Approximations for efficient computation in the theory of evidence. Artificial Intelligence 61(2), 315–329 (1993)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Voorbraak, F.: A computationally efficient approximation of Dempster-Shafer theory. International Journal on Man-Machine Studies 30, 525–536 (1989)MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Oxford Brookes UniversityOxfordUK

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