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On Consistent Approximations of Belief Functions in the Mass Space

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Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU 2011)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 6717))

Abstract

In this paper we study the class of consistent belief functions, as counterparts of consistent knowledge bases in classical logic. We prove that such class can be defined univocally no matter our definition of proposition implied by a belief function. As consistency can be desirable in decision making, the problem of mapping an arbitrary belief function to a consistent one arises, and can be posed in a geometric setup. We analyze here all the consistent transformations induced by minimizing L p distances between belief functions, represented by the vectors of their basic probabilities.

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References

  1. Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press, Princeton (1976)

    MATH  Google Scholar 

  2. Dempster, A.P.: Upper and lower probabilities induced by a multivariate mapping. Annals of Mathematical Statistics 38, 325–339 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  3. Yager, R.R.: On the dempster-shafer framework and new combination rules. Information Sciences 41, 93–138 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  4. Smets, P.: The degree of belief in a fuzzy event. Inf. Sciences 25, 1–19 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ramer, A., Klir, G.J.: Measures of discord in the Dempster-Shafer theory. Information Sciences 67(1-2), 35–50 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  6. Liu, W.: Analyzing the degree of conflict among belief functions. Artif. Intell. 170(11), 909–924 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hunter, A., Liu, W.: Fusion rules for merging uncertain information. Information Fusion 7(1), 97–134 (2006)

    Article  Google Scholar 

  8. Lo, K.C.: Agreement and stochastic independence of belief functions. Mathematical Social Sciences 51(1), 1–22 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Paris, J.B., Picado-Muino, D., Rosefield, M.: Information from inconsistent knowledge: A probability logic approach. In: Advances in Soft Computing, vol. 46, pp. 291–307. Springer, Heidelberg (2008)

    Google Scholar 

  10. Haenni, R.: Towards a unifying theory of logical and probabilistic reasoning. In: Proceedings of ISIPTA 2005, pp. 193–202 (2005)

    Google Scholar 

  11. Priest, G., Routley, R., Norman, J.: Paraconsistent logic: Essays on the inconsistent. Philosophia Verlag (1989)

    Google Scholar 

  12. Batens, D., Mortensen, C., Priest, G.: Frontiers of paraconsistent logic. In: Studies in Logic and Computation, vol. 8. Research Studies Press (2000)

    Google Scholar 

  13. Daniel, M.: On transformations of belief functions to probabilities. International Journal of Intelligent Systems 21(6), 261–282 (2006)

    Article  MATH  Google Scholar 

  14. Cuzzolin, F.: Two new Bayesian approximations of belief functions based on convex geometry. IEEE Tr. SMC-B 37(4), 993–1008 (2007)

    Google Scholar 

  15. Dubois, D., Prade, H.: Consonant approximations of belief functions. International Journal of Approximate Reasoning 4, 419–449 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  16. Black, P.: An examination of belief functions and other monotone capacities. PhD dissertation, Department of Statistics, Carnegie Mellon University (1996)

    Google Scholar 

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

    Article  Google Scholar 

  18. Cuzzolin, F.: Consistent approximations of belief functions. In: Proceedings of ISIPTA 2009, pp. 139–148 (2009)

    Google Scholar 

  19. Saffiotti, A.: A belief-function logic. In: Universit Libre de Bruxelles, pp. 642–647. MIT Press, Cambridge

    Google Scholar 

  20. Mates, B.: Elementary Logic. Oxford University Press, Oxford (1972)

    MATH  Google Scholar 

  21. Cattaneo, M.E.G.V.: Combining belief functions issued from dependent sources. In: ISIPTA, pp. 133–147 (2003)

    Google Scholar 

  22. de Cooman, G.: Belief models: An order-theoretic investigation. Annals of Mathematics and Artificial Intelligence 45(1-2), 5–34 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Dubrovin, B.A., Novikov, S.P., Fomenko, A.T.: Sovremennaja geometrija. Metody i prilozenija, Nauka, Moscow (1986)

    Google Scholar 

  24. Cuzzolin, F.: An interpretation of consistent belief functions in terms of simplicial complexes. In: Proc. of ISAIM 2008 (2008)

    Google Scholar 

  25. Cuzzolin, F.: Geometric conditioning of belief functions. In: Proceedings of BELIEF 2010, Brest, France (2010)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Oxford Brookes University, Oxford, UK

    Fabio Cuzzolin

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  1. Fabio Cuzzolin
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Editor information

Editors and Affiliations

  1. School of Electronics, Electrical Engineering and Computer Science, Queen’s University Belfast, BT9 5BN, Belfast, UK

    Weiru Liu

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Cuzzolin, F. (2011). On Consistent Approximations of Belief Functions in the Mass Space. In: Liu, W. (eds) Symbolic and Quantitative Approaches to Reasoning with Uncertainty. ECSQARU 2011. Lecture Notes in Computer Science(), vol 6717. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22152-1_25

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  • DOI: https://doi.org/10.1007/978-3-642-22152-1_25

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-22151-4

  • Online ISBN: 978-3-642-22152-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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