Advertisement

Consonant Belief Function Induced by a Confidence Set of Pignistic Probabilities

  • Astride Aregui
  • Thierry Denoeux
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4724)

Abstract

A new method is proposed for building a predictive belief function from statistical data in the Transferable Belief Model framework. The starting point of this method is the assumption that, if the probability distribution ℙ X of a random variable X is known, then the belief function quantifying our belief regarding a future realization of X should have its pignistic probability distribution equal to ℙ X . When PX is unknown but a random sample of X is available, it is possible to build a set \(\mathcal{P}\) of probability distributions containing ℙ X with some confidence level. Following the Least Commitment Principle, we then look for a belief function less committed than all belief functions with pignistic probability distribution in \(\mathcal{P}\). Our method selects the most committed consonant belief function verifying this property. This general principle is applied to the case of the normal distribution.

Keywords

Dempster-Shafer theory Evidence theory Transferable BeliefModel possibility distribution statistical data 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aregui, A., Denœux, T.: Constructing predictive belief functions from continuous sample data using confidence bands. In: De Cooman, G., Vejnarová, J., Zaffalon, M. (eds.) Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications (ISIPTA 2007), Czech Republic, pp. 11–20 (July 2007)Google Scholar
  2. 2.
    Aregui, A., Denœux, T.: Fusion of one-class classifiers in the belief function framework. In: Proceedings of the 10th Int. Conf. on Information Fusion, Quebec, Canada (July 2007)Google Scholar
  3. 3.
    Arnold, B.C., Shavelle, R.M.: Joint confidence sets for the mean and variance of a normal distribution. The American Statistician 52(2), 133–140 (1998)MathSciNetGoogle Scholar
  4. 4.
    Dempster, A.P.: Upper and lower probabilities generated by a random closed interval. Annals of Mathematical Statistics 39(3), 957–966 (1968)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Denœux, T.: Constructing belief functions from sample data using multinomial confidence regions. International Journal of Approximate Reasoning 42(3), 228–252 (2006)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Denœux, T., Smets, P.: Classification using belief functions: the relationship between the case-based and model-based approaches. IEEE Transactions on Systems, Man and Cybernetics B 36(6), 1395–1406 (2006)CrossRefGoogle Scholar
  7. 7.
    Dubois, D., Prade, H.: On several representations of an uncertainty body of evidence. In: Gupta, M.M., Sanchez, E. (eds.) Fuzzy Information and Decision Processes, pp. 167–181. North-Holland, Amsterdam (1982)Google Scholar
  8. 8.
    Dubois, D., Prade, H.: A set-theoretic view of belief functions: logical operations and approximations by fuzzy sets. International Journal of General Systems 12(3), 193–226 (1986)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dubois, D., Prade, H., Smets, P.: New semantics for quantitative possibility theory. In: Benferhat, S., Besnard, P. (eds.) ECSQARU 2001. LNCS (LNAI), vol. 2143, pp. 410–421. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  10. 10.
    Dubois, D., Prade, H., Smets, P.: A definition of subjective possibility. International Journal of Approximate Reasoning (in press, 2007)Google Scholar
  11. 11.
    Hacking, I.: Logic of Statistical Inference. Cambridge University Press, Cambridge (1965)CrossRefMATHGoogle Scholar
  12. 12.
    Masson, M.-H., Denœux, T.: Inferring a possibility distribution from empirical data. Fuzzy Sets and Systems 157(3), 319–340 (2006)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Ristic, B., Smets, P.: Target classification approach based on the belief function theory. IEEE Transactions on Aerospace and Electronic Systems 41(2), 574–583 (2005)CrossRefGoogle Scholar
  14. 14.
    Shafer, G.: A mathematical theory of evidence. Princeton University Press, Princeton (1976)MATHGoogle Scholar
  15. 15.
    Smets, P.: Un modèle mathématico-statistique simulant le processus du diagnostic médical. PhD thesis, Université Libre de Bruxelles, Brussels, Belgium (in French) (1978)Google Scholar
  16. 16.
    Smets, P.: Belief functions: the disjunctive rule of combination and the generalized Bayesian theorem. International Journal of Approximate Reasoning 9, 1–35 (1993)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Smets, P.: Belief induced by the partial knowledge of the probabilities. In: Heckerman, D., et al. (eds.) Uncertainty in AI 1994, pp. 523–530. Morgan Kaufmann, San Mateo (1994)CrossRefGoogle Scholar
  18. 18.
    Smets, P.: Practical uses of belief functions. In: Laskey, K.B., Prade, H. (eds.) Uncertainty in Artificial Intelligence 15 (UAI 1999), Stockholm, Sweden, pp. 612–621 (1999)Google Scholar
  19. 19.
    Smets, P.: Belief functions on real numbers. International Journal of Approximate Reasoning 40(3), 181–223 (2005)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Smets, P.: Decision making in the TBM: the necessity of the pignistic transformation. International Journal of Approximate Reasoning 38, 133–147 (2005)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Smets, P., Kennes, R.: The Transferable Belief Model. Artificial Intelligence 66, 191–243 (1994)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Yager, R.R.: The entailment principle for Dempster-Shafer granules. Int. J. of Intelligent Systems 1, 247–262 (1986)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Astride Aregui
    • 1
    • 2
  • Thierry Denoeux
    • 1
  1. 1.HEUDIASYC, UTC, CNRS Centre de Recherche de RoyallieuCompiègneFrance
  2. 2.CIRSEE, Suez EnvironnementLe PecqFrance

Personalised recommendations