Coherent Conditional Probability as a Measure of Uncertainty of the Relevant Conditioning Events

  • Giulianella Coletti
  • Romano Scozzafava
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2711)


In previous papers, by resorting to the most effective concept of conditional probability, we have been able not only to define fuzzy subsets, but also to introduce in a very natural way the basic continuous T-norms and the relevant dual T-conorms, bound to the former by coherence. Moreover, we have given, as an interesting and fundamental by-product of our approach, a natural interpretation of possibility functions, both from a semantic and a syntactic point of view.

In this paper we study the properties of a coherent conditional probability looked on as a general non-additive uncertainty measure of the conditioning events, and we prove that this measure is a capacity if and only if it is a possibility.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bouchon-Meunier, B., Coletti, G., Marsala, C.: Conditional Possibility and Necessity. In: Bouchon-Meunier, B., Gutiérrez-Rios, J., Magdalena, L., Yager, R.R. (eds.) Technologies for Constructing Intelligent Systems, vol. 2, pp. 59–71. Springer, Berlin (2001)Google Scholar
  2. 2.
    Coletti, G., Scozzafava, R.: Characterization of Coherent Conditional Probabilities as a Tool for their Assessment and Extension. International Journal of Uncertainty, Fuzziness and Knowledge-Based System 4, 103–127 (1996)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Coletti, G., Scozzafava, R.: Zero probabilities in stochastic independence. In: Bouchon-Meunier, B., Yager, R.R., Zadeh, L.A. (eds.) Information, Uncertainty, Fusion, pp. 185–196. Kluwer, Dordrecht (2000) (Selected papers from IPMU 1998)Google Scholar
  4. 4.
    Coletti, G., Scozzafava, R.: Conditional Subjective Probability and Fuzzy Theory. In: Proc. of 18th NAFIPS International Conference, pp. 77–80. IEEE, New York (1999)Google Scholar
  5. 5.
    Coletti, G., Scozzafava, R.: Conditioning and Inference in Intelligent Systems. Soft Computing 3, 118–130 (1999)Google Scholar
  6. 6.
    Coletti, G., Scozzafava, R.: Fuzzy sets as conditional probabilities: which meaningful operations can be defined? In: Proc. of 20th NAFIPS International Conference, pp. 1892–1895. IEEE, Vancouver (2001)Google Scholar
  7. 7.
    Coletti, G., Scozzafava, R.: From conditional events to conditional measures: a new axiomatic approach. Annals of Mathematics and Artificial Intelligence 32, 373–392 (2001)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Coletti, G., Scozzafava, R.: Probabilistic logic in a coherent setting. Trends in Logic, (15) (2002)Google Scholar
  9. 9.
    Coletti, G., Scozzafava, R.: Conditional probability, fuzzy sets and possibility: a unifying view. Fuzzy Sets and Systems (2003) (to appear)Google Scholar
  10. 10.
    Coletti, G., Scozzafava, R.: Coherent conditional probability as a measure of information of the relevant conditioning events. In: Berthold, M.R., Lenz, H.-J., Bradley, E., Kruse, R., Borgelt, C. (eds.) IDA 2003. LNCS, vol. 2810, pp. 123–133. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  11. 11.
    Coletti, G., Scozzafava, R., Vantaggi, B.: Coherent Conditional Probability as a Tool for Default Reasoning. In: Proc. IPMU 2002, Annecy, France, pp. 1663–1670 (2002)Google Scholar
  12. 12.
    de Finetti, B.: Sull’impostazione assiomatica del calcolo delle probabilità. Annali Univ. Trieste 19, 3–55 (1949) (Engl. transl.: Probability, Induction, Statistics, ch. 5. Wiley, London, 1972) Google Scholar
  13. 13.
    Dubins, L.E.: Finitely Additive Conditional Probabilities, Conglomerability and Disintegration. Annals of Probability 3, 89–99 (1975)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Dubois, D., Prade, H.: Possibility Theory. Plenum Press, New York (1988)MATHGoogle Scholar
  15. 15.
    Koopman, B.O.: The Bases of Probability. Bulletin A.M.S. 46, 763–774 (1940)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Krauss, P.H.: Representation of Conditional Probability Measures on Boolean Algebras. Acta Math. Acad. Scient. Hungar. 19, 229–241 (1968)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Rényi, A.: On Conditional Probability Spaces Generated by a Dimensionally Ordered Set of Measures. Theory of Probability and its Applications 1, 61–71 (1956)MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Giulianella Coletti
    • 1
  • Romano Scozzafava
    • 2
  1. 1.Dipartimento di MatematicaUniversità di PerugiaPerugiaItaly
  2. 2.Dipartimento Metodi e Modelli MatematiciUniversità “La Sapienza”RomaItaly

Personalised recommendations