Extreme probability distributions of random sets, fuzzy sets and p-boxes. (29th June 2009) - Bernardini, A.; Tonon, F.
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.