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Extreme probability distributions of random sets, fuzzy sets and p-boxes

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  • A. Bernardini
  • F. Tonon

Abstract

The uncertain information given by a random set on a finite space of singletons determines a set of probability distributions defined by the convex hull of a finite set of extreme distributions. After placing random sets in the context of the theory of imprecise probabilities, algorithms are given to calculate these extreme distributions, and hence exact upper/lower bounds on the expectation of functions of the uncertain variable. Detailed applications are given to consonant random sets (or their equivalent fuzzy sets) and to p-boxes (non-consonant random sets). A procedure is presented to calculate the random set equivalent to a p-box and hence to derive extreme distributions from a p-box. A hierarchy of non-consonant (and eventually consonant) random sets ordered by the inclusion of the corresponding sets of probability distributions can yield the same upper and lower cumulative distribution functions of the p-box. Simple numerical examples illustrate the presented concepts and algorithms.

Suggested Citation

  • A. Bernardini & F. Tonon, 2009. "Extreme probability distributions of random sets, fuzzy sets and p-boxes," International Journal of Reliability and Safety, Inderscience Enterprises Ltd, vol. 3(1/2/3), pages 57-78.
    • Handle: RePEc:ids:ijrsaf:v:3:y:2009:i:1/2/3:p:57-78
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    Cited by:

    1. Xiaomin You & Fulvio Tonon, 2012. "Event‐Tree Analysis with Imprecise Probabilities," Risk Analysis, John Wiley & Sons, vol. 32(2), pages 330-344, February.

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