Extreme points of the set of probabilities associated with a possibility measure

I. Montes, E. Miranda

Possibility measures, or supremum-preserving set functions, constitute one of the most prominent alternatives to probability theory in situations of imprecise information. Under a epistemic interpretation, where it is assumed the existence of a precise but unknown model, a possibility measure can be equivalently represented by means of the credal set it determines: the convex set of probability measures it dominates. This convex set can be characterized by its extreme points, which may be used to ease the computations.
The number of extreme points of such credal set has been shown to be upper bounded by 2^{n-1}, where n is the cardinality of the referential space. Here we improve upon this result by providing a formula for the number of extreme points in terms of the cardinalities of the focal elements. In addition, we determine in which cases the maximal number of extreme points is attained and investigate in some detail possibility measures associated with probability boxes.

Palabras clave: Possibility measure, credal set, extreme points, probability boxes

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