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Aggregation of Imprecise Probabilities

  • Serafín Moral
  • José del Sagrado
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 12)

Abstract

Methods to aggregate convex sets of probabilities are proposed. Source reliability is taken into account by transforming the given information and making it less precise. An important property of the aggregation will be that the precision of the result will depend on the initial compatibility of sources. Special attention will be paid to the particular case of probability intervals giving adaptations of aggregation procedures.

Keywords

Convex Hull Aggregation Function Equivalence Relationship Belief Function Single Probability 
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.

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References

  1. 1.
    D.A. Bell, J.W. Guan, S.K. Lee: Generalized union and project operations for pooling uncertain and imprecise information, Data & Knowledge Engineering, 18, 89–117, 1996.MATHCrossRefGoogle Scholar
  2. 2.
    L.M. Campos, J.F. Huete, S. Moral: Probability intervals: a tool for uncertain reasoning, International Journal of Uncertainty, Fuzziness and Kowledge-Based Systems, 2, 167–196, 1994.MATHCrossRefGoogle Scholar
  3. 3.
    J.E. Cano, S. Moral, J.F. Verdegay-Lopez: Partial inconsistency of probability envelopes, Fuzzy Sets and Systems, 52, 201–216, 1992.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    R.M. Cooke, T. Bedford: Expert opinion, Cambridge Program for Industry, 1996.Google Scholar
  5. 5.
    A.P. Dempster: Upper and lower probabilities induced by a multivalued mapping. Ann. Math. Statist. 38, 325–339, 1967.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    H. Edelsbruner: Algorithms in Combinatorial Geometry, Springer-Verlag, Berlin, 1987.CrossRefGoogle Scholar
  7. 7.
    S. French: Group consensus probability distribution: A critical survey, Bayesian Statistics 2, J.M. Bernardo et al. (Eds.) Elsevier, 183–202, 1985.Google Scholar
  8. 8.
    C. Genest, J.V. Zidek: Combining probability distributions: a critique and an annotated bibliography. Statistical Science 1, 114–146, 1986.MathSciNetCrossRefGoogle Scholar
  9. 9.
    J.L. Imbert, P. Van Hentenryck: Redundancy elimination with a lexicographic solved form. Annals of Mathematics and Artificial Intelligence, 17, 85–106, 1996MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    R. Hummel, M. Landy: Evidence as opinions of experts. In: Uncertainty in Artificial Intelligence 2, J.F. Lemmer and L.N. Kanal, eds., North-Holland, 43–53, 1988.Google Scholar
  11. 11.
    R. Hummel, L. Manevitz: Combining bodies of dependent information. Proc. Tenth Int. Conf. Artificial Intelligence, Milan, 1015–1017, 1987.Google Scholar
  12. 12.
    R. Hummel, L. Manevitz: Combination calculi for uncertain reasoning. Representing uncertainty using distributions. Annals of Mathematics and Artificial Intelligence, 1996.Google Scholar
  13. 13.
    R. Hummel, L. Manevitz: A statistical approach to the representation of uncertainty in beliefs using spread of opinions. IEEE Transactions on Systems, Man, and Cybernetics — Part A: Systems and Humans, 26, 378–384, 1996.CrossRefGoogle Scholar
  14. 14.
    R. Laddaga: Lehrer and the consensus proposal. Synthese 36, 473–477, 1977.MATHCrossRefGoogle Scholar
  15. 15.
    C. Lassez, J.L. Lassez: Quantifier elimination for conjunctions of linear constraints via a convex hull algorithm. In: Symbolic and Numerical Computation for Artificial Intelligence, B.R. Donald et al. (eds.) Academic Press, London, 103–119, 1992.Google Scholar
  16. 16.
    T.H. Mattheis, D.S. Rubin: A survey and comparison of methods for finding all vertices of convex polyhedral sets. Technical Report N. 77–14, Department of Operations Research and Systems Analysis, University of Carolina at Chapel Hill, 1977.Google Scholar
  17. 17.
    K.J. McConway: Marginalization and linear opinion pools. J. Amer. Statist. Assoc. 76, 410–414, 1981.MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    K.C. Ng, B. Abramson: Consensus diagnosis: a simulation study, IEEE Transactions on Systems, Man and Cybernetics, Vol. 22, 1992.Google Scholar
  19. 19.
    F.P. Preparata, M.I. Shamos: Computational Geometry. An Introduction. Springer Verlag, New York, 1985.CrossRefGoogle Scholar
  20. 20.
    G. Shafer: A Mathematical Theory of Evidence, Princeton University Press, Princeton, 1976.MATHGoogle Scholar
  21. 21.
    Ph. Smets: Belief functions, Non-Standard Logics for Automated Reasoning, Ph. Smets, E.H. Mamdani, D. Dubois, H. Prade (eds.) Academic Press, London, 253–286, 1988.Google Scholar
  22. 22.
    M. Stone: The opinion pool: Ann. Math. Stat., 32, 1961, 1339–1342.MATHCrossRefGoogle Scholar
  23. 23.
    E. Trillas, L. Valverde: On implication and indistinguishability in the setting of fuzzy logic, Management Decision Support Systems using Fuzzy Sets and Possibility Theory, J. Kacprzyk and R. Yager, eds., TUV Rheinland, Köln, 198–212, 1985.Google Scholar
  24. 24.
    P. Walley: The elicitation and aggregation of beliefs. Statistics Research Report. University of Warwick, 1982.Google Scholar
  25. 25.
    P. Walley: Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London, 1991.MATHGoogle Scholar
  26. 26.
    P. Walley: Measures of uncertainty in expert systems, Artificial Intelligence, 31, 1–58, 1996.MathSciNetCrossRefGoogle Scholar
  27. 27.
    R.L. Winkler: The consensus of subjective probability distributions. Management Science, 15, B61 - B75, 1968.CrossRefGoogle Scholar
  28. 28.
    R.R. Yager: On the aggregation of prioritized belief structures. IEEE Transactions of Systems, Man, and Cybernetics, 26, 708–717, 1996.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Serafín Moral
    • 1
  • José del Sagrado
    • 2
  1. 1.Dpto. Ciencias de la Computación e Inteligencia ArtificialUniversidad de GranadaGranadaSpain
  2. 2.Dpto. Lenguajes y ComputaciónUniversidad de AlmeríaAlmeríaSpain

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