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On the Combination of Evidence in Various Mathematical Frameworks

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Reliability Data Collection and Analysis

Part of the book series: Eurocourses ((EURR,volume 3))

Abstract

The problem of combining pieces of evidence issued from several sources of information turns out to be a very important issue in artificial intelligence. It is encountered in expert systems when several production rules conclude on the value of the same variable, but also in robotics when information coming from different sensors is to be aggregated. Solutions proposed in the literature so far have often been unsatisfactory because relying on a single theory of uncertainty, a unique mode of combination, or the absence of analysis of the reasons for uncertainty. Besides dependencies and redundancies between sources must be dealt with especially in knowledge bases, where sources correspond to production rules.

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References

  1. Berenstein, C., Kanal, L.N. and Lavine, P. (1986) Consensus rules. In L.N. Kanal and J.F. Lemmer (eds.) Uncertainty in Artificial Intelligence, Vol. 1, North-Holland, Amsterdam, pp. 27–32.

    Google Scholar 

  2. Buchanan, B.G. and Shortliffe, E.H. (1984) Rule-Based Expert Systems-The MYCIN Experiments of the Stanford Heuristic Programming Projects, Addison-Wesley, Reading, N.J..

    Google Scholar 

  3. Chamiak, E. (1983) The Bayesian basis of common sense medical diagnosis, Proc. 1983 American Assoc. Artificial Intelligence Conf., pp. 70–73.

    Google Scholar 

  4. Cheng, Y. and Kashyap, R.L. (1989) A study of associative evidential reasoning, IEEE Trans. on Pattern Analysis and Machine Intelligence 11, 623–631.

    Article  MATH  Google Scholar 

  5. Dempster, A.P.(1967) Upper and lower probabilities induced by a multivalued mapping, Ann. Math. Statistics, 325–339.

    Google Scholar 

  6. Domotor, Z. (1985) Probability kinematics conditionals and entropy principle, Synthese 63, 75–114.

    Article  MathSciNet  Google Scholar 

  7. Dubois, D. (1986) Generalized probabilistic independence, and its implications for utility, Operations Res. Letters 5, 255–260.

    Article  MATH  Google Scholar 

  8. Dubois, D. and Prade, H. (1982) On several representations of an uncertain body of evidence. In M.M. Gupta and E. Sanchez (eds.) Fuzzy Information and Decision Processes, North-Holland, Amsterdam, pp. 167–181.

    Google Scholar 

  9. Dubois, D. and Prade, H. (1985) (with the collaboration of Farreny, H., Martin-Clouaire, R. and Testemale, C.) Theorie des Possibilites-Applications a la Representation des Connaissances en Informatique, Masson, Paris. English version “Possibility Theory” published by Plenum Press, New York, 1988.

    Google Scholar 

  10. Dubois, D. and Prade, H. (1985) A review of fuzzy set aggregation connectives, Information Sciences 36, 85–121.

    Article  MathSciNet  MATH  Google Scholar 

  11. Dubois, D. and Prade, H. (1986) Weighted minimum and maximum operations in fuzzy set theory, Information Sciences 39, 205–210.

    Article  MathSciNet  MATH  Google Scholar 

  12. Dubois, D. and Prade, H. (1986) A set-theoretic view of belief functions-Logical operations and approximations by fuzzy sets, Int. Journal of General Systems 12, 193–226.

    Article  MathSciNet  Google Scholar 

  13. Dubois, D. and Prade, H. (1986) On the unicity of Dempster rule of combination, Int. J. Intelligent Systems 1, 133–142.

    Article  MATH  Google Scholar 

  14. Dubois, D. and Prade, H. (1988) Representation and combination of uncertainty with belief functions and possibility measures, Computational Intelligence 4(4), 244–264.

    Article  Google Scholar 

  15. Dubois, D. and Prade, H. (1988) Modelling uncertainty and inductive inference, Acta Psychologica 68, 53–78.

    Article  Google Scholar 

  16. D Dubois, D. and Prade, H. (1988) On the combination of uncertain or imprecise pieces of information in rule-based systems, Int. J. of Approximate Reasoning 2, 65–87.

    Article  MathSciNet  MATH  Google Scholar 

  17. Dubois, D. and Prade, H. (1988) Default reasoning and possibility theory, Artificial Intelligence 35, 243–257.

    Article  MathSciNet  MATH  Google Scholar 

  18. Dubois, D. and Prade, H. (1990) The logical view of conditioning and its application to possibility and evidence theories, Int. J. of Approximate Reasoning 4(1), 23–46.

    Article  MathSciNet  MATH  Google Scholar 

  19. Dubois, D. and Prade, H. (1990) Aggregation of possibility measures, to appear in J. Kacprzyk and M. Fedrizzi (eds.) Multiperson Decision-Making Under Fuzzy Sets and Possibility Theory, Kluwer Academic Pub., Dordrecht, The Netherlands.

    Google Scholar 

  20. Edwards, W.E (1972) Likelihood, Cambridge University Press, Cambridge, U.K..

    MATH  Google Scholar 

  21. Frank, M.J. (1979) On the simultaneous associativity of f(x,y) and x + y-f(x,y), Aequationes Math. 19, 194-226.

    Google Scholar 

  22. Fua, P. (1987) Using probability density functions in the framework of evidential reasoning In B. Bouchon and R.R. Yager (eds.) Uncertainty in Knowledge-Based Systems, Springer Verlag, pp. 103–110.

    Google Scholar 

  23. Goodman, L.R. and Nguyen, H.T. (1985) Uncertainty models for knowledgebased systems, North-Holland, Amsterdam.

    Google Scholar 

  24. Hajek, P. (1985) Combining functions for certainty degrees in consulting systems, Int. J. Man-Machine Studies 22, 59–76.

    Google Scholar 

  25. Higashi, M. and Klir, G. (1983) On the notion of distance representing information closeness: possibility and probability distributions, Int. J. of General Systems 9, 103–115.

    Article  MathSciNet  MATH  Google Scholar 

  26. Ishizuka, M., Fu, K.S. and Yao, J.T.P. (1982) Inference procedure with uncertainty for problem reduction method, Information Sciences 28, 179–206.

    Article  MathSciNet  MATH  Google Scholar 

  27. Jaynes, E.T. (1979) Where do we stand on maximum entropy, In Levine and Tribus (eds.) The Maximum Entropy Formalism, MIT Press, Cambridge, Mass..

    Google Scholar 

  28. Kyburg, H.E. (1974) The Logical Foundations of Statistical Inference, D. Reidel, Dordrecht, The Netherlands.

    Book  MATH  Google Scholar 

  29. Lindley, D.V. (1984) Reconciliation of probability distributions, Operations Research 32, 866–880.

    MathSciNet  Google Scholar 

  30. Loui, R.P. (1987) Computing reference classes In J.F. Lemmer and L.N. Kanal (eds.) Uncertainty in Artificial Intelligence, 2, North-Holland, Amsterdam, pp. 273–289.

    Google Scholar 

  31. Nguyen, H.T. (1978) On random sets and belief functions, J. Math. Anal. & Appl. 65, 531–542.

    Article  MathSciNet  MATH  Google Scholar 

  32. Oblow, E. (1987) O-theory-a hybrid uncertainty theory, Int. J. General Systems 13(2), 95–106.

    Article  MathSciNet  Google Scholar 

  33. Prade, H. (1985) A computational approach to approximate and plausible reasoning, with applications to expert systems, IEEE Trans. Pattern Analysis & Machine Intelligence 7, 260–283 (Corrections, 7, 747-748).

    Article  MATH  Google Scholar 

  34. Rauch, H.E. (1984) Probability concepts for an expert system used for data fusion, The AI Magazine 5(3), 55–60.

    Google Scholar 

  35. Rescher, N. (1969) Many-Valued Logic, Mc Graw-Hill, New York.

    MATH  Google Scholar 

  36. Sandri, S., Besi, A., Dubois, D., Mancini, G., Prade, H. and Testemale, C. (1989) Data fusion problems in an intelligent data bank interface, Proc. of the 6th EuReDatA Conference on Reliability, Data Collection and Use in Risk and Availability Assessment, Siena, Italy, (V. Colombari, ed.), Springer Verlag, Belin, 655–670.

    Google Scholar 

  37. Shafer, G. (1976) A Mathematical Theory of Evidence, Princeton University Press, N.J..

    MATH  Google Scholar 

  38. Shafer, G. (1986) The combination of evidence, Int. J Intelligent Systems 1, 155–180.

    Article  MathSciNet  MATH  Google Scholar 

  39. Shafer, G. (1987) Belief functions and possibility measures. In J.C. Bezdek (ed.) The Analysis of Fuzzy Information, Vol. 1, CRC Press, Boca Raton, Fl., pp. 51–84.

    Google Scholar 

  40. Sikorski, R. (1964) Boolean Algebras, Springer Verlag, Berlin.

    MATH  Google Scholar 

  41. Silvert, W. (1979) Symmetric summation: a class of operations on fuzzy sets, IEEE Trans. on Systems, Man and Cybernetics 9(10), 657–659.

    Article  MathSciNet  MATH  Google Scholar 

  42. Smets, P. (1982) Possibilistic inference from statistical data, Proc. of the 2nd World Conf. on Math. at the Service of Man, Las Palmas, Spain, June 28-July 3, pp. 611–613.

    Google Scholar 

  43. Wagner, C.G. (1989) Consensus for belief functions and related uncertainty measures, Theory and Decision 26, 295–304.

    Article  MathSciNet  MATH  Google Scholar 

  44. Walley, P. and Fine, T. (1982) Towards a frequentist theory of upper and lower probability, The Annals of Statistics 10, 741–761.

    Article  MathSciNet  MATH  Google Scholar 

  45. Wu, J.S., Apostolakis, G.E. and Okrent, D. (1990) Uncertainty in system analysis: probabilistic versus, non probabilistic theories, Reliability Eng. and Syst. Safety 30, 163–181.

    Article  Google Scholar 

  46. Yager, R.R. (1983) Hedging in the combination of evidence, Int. J. of Information and Optimization Science 4(1), 73–81.

    MathSciNet  MATH  Google Scholar 

  47. Yager, R.R. (1984) Approximate reasoning as a basis for rule-based expert systems, IEEE Trans. Systems, Man & Cybernetics 14, 636–643.

    Article  MathSciNet  MATH  Google Scholar 

  48. Yager, R.R. (1985) On the relationships of methods of aggregation evidence in expert systems, Cybernetics & Systems 16, 1–21.

    Article  MathSciNet  MATH  Google Scholar 

  49. Yager, R.R. (1987) Quasi associative operations in the combination of evidence, Kybernetes 16, 37–41.

    Article  MathSciNet  MATH  Google Scholar 

  50. Yager, R.R. (1988) Prioritized, non-pointwise, non-monotonic intersection and union for commonsense reasoning, In B. Bouchon, L. Saitta and R.R. Yager (eds.) Uncertainty and Intelligent Systems, Lectures Notes in Computer Sciences, n° 313, Springer Verlag, Berlin, pp. 359–365.

    Chapter  Google Scholar 

  51. Zadeh, L.A. (1965) Fuzzy sets, Information and Control 8, 338–353.

    Article  MathSciNet  MATH  Google Scholar 

  52. Zadeh, L.A. (1978) Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1(1), 3–28.

    Article  MathSciNet  MATH  Google Scholar 

  53. Zadeh, L.A. (1985) A simple view of the Dempster-Shafer theory of evidence and its implicatons for the rule of combination, The AI Magazine 7(2), 85–90.

    Google Scholar 

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Authors and Affiliations

  1. Institut de Recherche en Informatique de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062, Toulouse Cedex, France

    Didier Dubois & Henri Prade

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  1. Didier Dubois
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  2. Henri Prade
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Editors and Affiliations

  1. Commission of the European Communities, Joint Research Centre, Institute for Systems Engineering and Informatics, Ispra, Italy

    J. Flamm  & T. Luisi  & 

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© 1992 Springer Science+Business Media Dordrecht

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Dubois, D., Prade, H. (1992). On the Combination of Evidence in Various Mathematical Frameworks. In: Flamm, J., Luisi, T. (eds) Reliability Data Collection and Analysis. Eurocourses, vol 3. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-2438-6_13

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  • DOI: https://doi.org/10.1007/978-94-011-2438-6_13

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-5075-3

  • Online ISBN: 978-94-011-2438-6

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