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Probability-Possibility Transformations, Triangular Fuzzy Sets, and Probabilistic Inequalities

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Reliable Computing

Abstract

A possibility measure can encode a family of probability measures. This fact is the basis for a transformation of a probability distribution into a possibility distribution that generalises the notion of best interval substitute to a probability distribution with prescribed confidence. This paper describes new properties of this transformation, by relating it with the well-known probability inequalities of Bienaymé-Chebychev and Camp-Meidel. The paper also provides a justification of symmetric triangular fuzzy numbers in the spirit of such inequalities. It shows that the cuts of such a triangular fuzzy number contains the “confidence intervals” of any symmetric probability distribution with the same mode and support. This result is also the basis of a fuzzy approach to the representation of uncertainty in measurement. It consists in representing measurements by a family of nested intervals with various confidence levels. From the operational point of view, the proposed representation is compatible with the recommendations of the ISO Guide for the expression of uncertainty in physical measurement.

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Authors and Affiliations

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

    Didier Dubois & Henri Prade

  2. Laboratoire d'Informatique, Systemes, Traitement de l'Information et de la Connaissance, LISTIC, Universite de Savoie, BP 806, 74016, Annecy, France

    Laurent Foulloy & Gilles Mauris

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  1. Didier Dubois
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  2. Laurent Foulloy
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  3. Gilles Mauris
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  4. Henri Prade
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Dubois, D., Foulloy, L., Mauris, G. et al. Probability-Possibility Transformations, Triangular Fuzzy Sets, and Probabilistic Inequalities. Reliable Computing 10, 273–297 (2004). https://doi.org/10.1023/B:REOM.0000032115.22510.b5

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