Extreme probability distributions of random sets, fuzzy sets and p-boxes. (29th June 2009)

• Record Type: Journal Article
• Authors: Bernardini, A.; Tonon, F.
• Is Part Of: International journal of reliability and safety. Volume 3:Number 1/2/3(2009)
• Page Start: 57
• Page End: 78
• Publication Date: 2009-06-29
• Subjects: random sets -- fuzzy sets -- p-boxes -- extreme probability distributions -- imprecise probabilities -- uncertainty; Reliability (Engineering) -- Periodicals; Industrial safety -- Periodicals; 620.00452
• Languages: English
• ISSNs: 1479-389X

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.