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

Authors: A. Bernardini, F. Tonon

Addresses: Dpt di Costruzioni e Trasporti, Universita degli Studi di Padova, Italy. ' Department of Civil Engineering, University of Texas, USA

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.

Keywords: random sets; fuzzy sets; p-boxes; extreme probability distributions; imprecise probabilities; uncertainty.

DOI: 10.1504/IJRS.2009.026835

International Journal of Reliability and Safety, 2009 Vol.3 No.1/2/3, pp.57 - 78

Published online: 27 Jun 2009 *

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