Hybrid Probabilistic-Possibilistic Mixtures and Utility Functions

  • Didier Dubois
  • Endre Pap
  • Henri Prade
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 51)


A basic building block in the standard mathematics of decision under uncertainty is the notion of probabilistic mixture. In order to generalize decision theory to non probabilistic uncertainty, one approach is to generalize mixture sets. In the recent past it has been proved that generalized mixtures can be non trivially defined, and they have been instrumental in the development of possibilistic utility theory. This paper characterizes the families of operations involved in generalized mixtures, due to a previous result on the characterization of the pairs of continuous t-norm and t-conorm such that the former is conditionally distributive over the latter. What is obtained is a family of mixtures that combine probabilistic and possibilistic mixtures via a threshold. It is based on a restricted family of t-conorm/ t-norm pairs which are very special ordinal sums. Any practically useful theory of pseudo-additive measures must use such special pairs of operations in order to extend the additivity property, and the notion of probabilistic independence.


Binary Tree Probabilistic Mixture Possibility Measure Triangular Norm Probabilistic Independence 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Didier Dubois
    • 1
  • Endre Pap
    • 2
  • Henri Prade
    • 1
  1. 1.I.R.I.T.Université Paul SabatierToulouse Cedex 4France
  2. 2.Institute of MathematicsUniversity of Novi SadNovi SadYugoslavia

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