The extreme points of convex probability sets play an important practical role, especially for specific, easier to manipulate sets. Although this problem has been studied for many models (probability intervals, possibility distributions), it remains to be studied for imprecise cumulative distributions (a.k.a. p-boxes). This is what we do in this paper, where we characterize the maximal number of extreme points of a p-box, give a family of p-boxes that attains this number and show an algorithm that allows to compute the extreme points of a given p-box. To achieve all this, we also provide what we think to be a new characterization of extreme points of a belief function.
IntroductionImprecise probability theory [10] is a powerful unifying framework for uncertainty treatment, relying on convex sets of probabilities, or credal sets, to model the uncertainty. Formally, they encompass many existing models: belief functions, possibility distributions, probability intervals, . . . . To apply such models, it is important to study their practical aspects, among which is the characterization of their extreme points. Indeed, these extreme points can be used in many settings, such as graphical models or statistical learning.Extreme points of many models have already been studied. For instance, Dempster [3] shows that the maximal number of extreme points of a belief function on a n-element space is n!. It was later [6] proved that the maximal number of extreme points for possibility distributions in a n-element space is 2 n−1 , and in [8] an algorithm to extract them was provided. In [2], authors studied the extreme points of probability intervals. One practical and popular model for which extreme points have not been characterized are p-boxes [4]. They are special kinds of belief functions whose focal elements are ordered intervals [5,9,10], and are quite instrumental in applications such as risk and reliability analysis.In this paper, we investigate extreme point of p-boxes: we demonstrate that their maximal number is the Pell number, and give the family of p-boxes for which this bound is obtained. To do so, we introduce a new way to characterize the extreme points of a belief function. We also provide an algorithm to compute the extreme points of a given p-box. Section 2 introduces the new characterization, while Section 3.2 studies the extreme points of p-boxes. Due to space restrictions, proofs and side results have been removed.
Extreme points of belief functionsGiven a space X = {x 1 , . . . , x n }, a probability mass function is a function m : P(X ) → These two functions are conjugate since Bel(A) = 1 − Pl(A c ), and we can focus on one of them. A focal set of the belief function Bel is a set E such that m(E) > 0, and F will denote the set of focal sets. A belief function also induces a credal setBeing convex, the set M(Bel) can be characterized by its extreme points 1 , that we will denote Ext(Bel). It is known [1,3] that there is a correspondence between the extreme points of a belief function and the perm...
