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

TL;DR

This paper provides a comprehensive mathematical characterization of the structure of $p$-boxes, including normal cones and extreme points, which are crucial for understanding bounds on expectations in imprecise probability models.

Contribution

It offers a complete characterization of normal cones and extreme points for $p$-boxes on finite domains, linking geometric structures to probabilistic bounds.

Findings

Characterization of all normal cones for $p$-boxes.

Relation between normal cones and extreme points.

Identification of adjacency structure among normal cones.

Abstract

Probability boxes, also known as $p$-boxes, correspond to sets of probability distributions bounded by a pair of distribution functions. They fall into the class of models known as imprecise probabilities. One of the central questions related to imprecise probabilities are the intervals of values corresponding to expectations of random variables, and especially the interval bounds. In general, those are attained in extremal points of credal sets, which denote convex sets of compatible probabilistic models. The aim of this paper is a characterization and identification of extreme points corresponding to $p$-boxes on finite domains. To accomplish this, we utilize the concept of normal cones. In the settings of imprecise probabilities, those correspond to sets of random variables whose extremal expectations are attained in a common extreme point. Our main results include a characterization all possible normal cones of $p$-boxes, their relation with extreme points, and the identification of adjacency structure on the collection of normal cones, closely related to the adjacency structure in the set of extreme points.