A complete characterization of normal cones and extreme points for p-boxes

Abstract

Probability boxes, also called p-boxes, correspond to sets of probability distributions bounded by a pair of distribution functions. They belong to the class of models known as imprecise probabilities. One of the central issues related to imprecise probabilities is the intervals of values corresponding to the expectations of random variables, and in particular the interval bounds. In general, these are reached at the extreme points of credal sets, which denote convex sets of compatible probabilistic models. The goal of this paper is to characterize and identify extreme points corresponding to p-boxes on finite domains. To achieve this, we use the concept of normal cones. In the context of imprecise probabilities, these correspond to sets of random variables whose extreme expectations are reached at a common extreme point. Our main results include a characterization of all possible normal cones of p-boxes, their relation to extreme points, and the identification of an adjacency structure on the collection of normal cones that is closely related to the adjacency structure in the set of extreme points.