Belief Functions: The Disjunctive Rule of Combination and the Generalized Bayesian Theorem

  • Philippe Smets
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 219)


We generalize the Bayes’ theorem within the transferable belief model framework. The Generalized Bayesian Theorem (GBT) allows us to compute the belief over a space Θ given an observation x⊆ X when one knows only the beliefs over X for every θi ∈ Θ. We also discuss the Disjunctive Rule of Combination (DRC) for distinct pieces of evidence. This rule allows us to compute the belief over X from the beliefs induced by two distinct pieces of evidence when one knows only that one of the pieces of evidence holds. The properties of the DRC and GBT and their uses for belief propagation in directed belief networks are analysed. The use of the discounting factors is justfied. The application of these rules is illustrated by an example of medical diagnosis.


Belief functions Bayes’ theorem Disjunctive rule of combination 


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  1. Cohen, M.S., Laskey, K.B. and Ulvila, J.W. (1987) The management of uncertainty in intelligence data: a self-reconciling evidential database. Falls Church, VA: Decision Science Consortium, Inc.Google Scholar
  2. Delgado M. and Moral S. (1987) On the concept of possibility-probabilty consistency. Fuzzy Sets and Systems 21: 311–3018.MATHCrossRefMathSciNetGoogle Scholar
  3. Dubois D. and Prade H. (1985) Théorie des possibilités. Masson, Paris.Google Scholar
  4. Dubois D. and Prade H. (1986a) A set theoretical view of belief functions. Int. J. Gen. Systems, 12: 193–226.CrossRefMathSciNetGoogle Scholar
  5. Dubois D. and Prade H. (1986b) On the unicity of Dempster rule of combination. Int. J. Intelligent Systems, 1: 133–142.MATHCrossRefGoogle Scholar
  6. Dubois D. and Prade H. (1987) The principle of minimum specificity as a basis for evidential reasoning. in: Uncertainty in knowledge-based systems, Bouchon B. and Yager R. eds, Springer Verlag, Berlin, p. 75–84.Google Scholar
  7. Dubois D. and Prade H. (1988) Representation and combination of uncertainty with belief functions and possibility measures. Computational Intelligence, 4: 244–264.CrossRefGoogle Scholar
  8. Edwards A.W.F. (1972) Likelihood. Cambridge University Press, Cambridge, UK.MATHGoogle Scholar
  9. Gebhardt F. and Kruse R. (1993) The context model: an integrating view of vagueness and uncertainty. Int. J. Approx. Reas. 9(3), 283–314.MATHCrossRefMathSciNetGoogle Scholar
  10. Hacking I. (1965) Logic of statistical inference. Cambridge University Press, Cambridge, U.K.MATHGoogle Scholar
  11. Halpern J.Y. and Fagin R. (1990) Two views of belief: Belief as Generlaized Probability and Belief as Evidence. Proc. Eighth National Conf. on AI, 112–119.Google Scholar
  12. Hsia Y.-T. (1991) Characterizing Belief with Minimum Commitment. IJCAI-91: 1184–1189.Google Scholar
  13. Kennes R. and Smets Ph. (1990) Computational Aspects of the Mšbius Transform. Procs of the 6th Conf. on Uncertainty in AI, Cambridge, USA.Google Scholar
  14. KLAWONN F. and SCHWECKE E. (1992) On the axiomatic justification of Dempster’s rule of combination. Int. J. Intel. Systems 7: 469–478.MATHCrossRefGoogle Scholar
  15. KLAWONN F. and SMETS Ph. (1992) The dynamic of belief in the transferable belief model and specialization-generalization matrices. in Dubois D., Wellman M.P., d’Ambrosio B. and Smets P. Uncertainty in AI 92. Morgan Kaufmann, San Mateo, Ca, USA, 1992, p. 130–137.Google Scholar
  16. Kruse R. and Schwecke E. (1990) Specialization: a new concept for uncertainty handling with belief functions. Int. J. Gen. Systems 18: 49–60.MATHCrossRefGoogle Scholar
  17. Kohlas J. and Monney P. A. (1990) Modeling and reasoning with hints. Technical Report. Inst. Automation and OR. Univ. Fribourg.Google Scholar
  18. Moral S. (1985) Informaciòn difusa. Relationes entre probabilidad y possibilidad. Tesis Doctoral, Universidad de Granada.Google Scholar
  19. Nguyen T. H. and Smets Ph. (1993) On Dynamics of Cautious Belief and Conditional Objects. Int. J. Approx. Reas. 8(2), 89–104.MATHCrossRefMathSciNetGoogle Scholar
  20. Pearl J. (1988) Probabilistic reasoning in intelligent systems: networks of plausible inference. Morgan Kaufmann Pub. San Mateo, Ca, USA.Google Scholar
  21. Pearl J. (1990) Reasoning with Belief Functions: an Analysis of Compatibility. Intern. J. Approx. Reasoning, 4: 363–390.CrossRefMathSciNetGoogle Scholar
  22. Shafer G. (1976) A mathematical theory of evidence. Princeton Univ. Press. Princeton, NJ.MATHGoogle Scholar
  23. Shafer G. (1982) Belief functions and parametric models. J. Roy. Statist. Soc. B44 322-352.MathSciNetGoogle Scholar
  24. Shafer G., Shenoy P.P. and Mellouli K. (1987) Propagating belief functions in qualitative Markov trees. Int. J. Approx. Reasoning, 1: 349–400.MATHCrossRefMathSciNetGoogle Scholar
  25. Smets Ph. (1978) Un modèle mathématico-statistique simulant le processus du diagnostic médical. Doctoral dissertation, Université Libre de Bruxelles, Bruxelles, (Available through University Microfilm International, 30–32 Mortimer Street, London W1N 7RA, thesis 80-70,003)Google Scholar
  26. Smets Ph. (1981) Medical Diagnosis : Fuzzy Sets and Degrees of Belief. Fuzzy Sets and systems, 5 : 259–266.MATHCrossRefMathSciNetGoogle Scholar
  27. Smets P. (1982) Possibilistic Inference from Statistical Data. In : Second World Conference on Mathematics at the Service of Man. A. Ballester, D. Cardus and E. Trillas eds. Universidad Politecnica de Las Palmas, pp. 611–613.Google Scholar
  28. Smets Ph. (1986) Bayes’ theorem generalized for belief functions. Proc. ECAI-86, vol. II. 169–171, 1986.Google Scholar
  29. Smets Ph. (1988) Belief functions. in Smets Ph, Mamdani A. , Dubois D. and Prade H. ed. Non standard logics for automated reasoning. Academic Press, London p. 253–286.Google Scholar
  30. Smets Ph. (1990) The combination of evidence in the transferable belief model. IEEE Trans. Pattern analysis and Machine Intelligence, 12: 447–458.CrossRefGoogle Scholar
  31. Smets Ph. (1991) The Transferable Belief Model and Other Interpretations of Dempster-Shafer’s Model. in Bonissone P.P., Henrion M., Kanal L.N. and Lemmer J.F. eds. Uncertainty in Artificial Intelligence 6, North Holland, Amsteram, 375–384.Google Scholar
  32. Smets Ph. (1992a) Resolving misunderstandings about belief functions: A response to the many criticisms raised by J. Pearl. Int. J. Approximate Reasoning. 6: 321–344.MATHCrossRefGoogle Scholar
  33. Smets Ph. (1992b) The nature of the unnormalized beliefs encountered in the transferable belief model. in Dubois D., Wellman M.P., d’Ambrosio B. and Smets P. Uncertainty in AI 92. Morgan Kaufmann, San Mateo, Ca, USA, 1992, p. 292–297.Google Scholar
  34. Smets Ph. (1992c) The concept of distinct evidence., IPMU 92 Proceedings, p. 789–794.Google Scholar
  35. Smets P. and Kennes R. (1994) The transferable belief model. Artificial Intelligence, 66(2), 191–234.MATHCrossRefMathSciNetGoogle Scholar
  36. Yager R. (1986) The entailment principle for Dempster-Shafer granules. Int. J. Intell. Systems, 1: 247–262MATHCrossRefGoogle Scholar
  37. Zadeh L.A. (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems. 1: 3–28.MATHCrossRefMathSciNetGoogle Scholar

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  • Philippe Smets

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