Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Damjan Škulj

TL;DR

This work studies a complete characterization of extreme points for $p$-boxes on finite domains via normal cones. It develops a complete normal-simplicial fan analysis, classifying all possible normal cones (MESCs) and detailing their adjacency structure, clarifying how extreme points relate to supporting hyperplanes. In the $p$-box setting, normal cones are generated by indicator gambles and constrained by the bounds $\underline{F}$ and $\overline{F}$; the authors derive a new explicit condition: for an extreme $F$, each $F(x_i)$ must lie in the bounds or neighboring values while minimizing the linear functional $P_F(h)$, connecting extremes to supporting hyperplanes. Collectively, the results yield a constructive way to enumerate all extreme points and offer a graph-based view via cone adjacencies, enabling extension to generalized or multivariate $p$-boxes.

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.

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

TL;DR

This work studies a complete characterization of extreme points for -boxes on finite domains via normal cones. It develops a complete normal-simplicial fan analysis, classifying all possible normal cones (MESCs) and detailing their adjacency structure, clarifying how extreme points relate to supporting hyperplanes. In the -box setting, normal cones are generated by indicator gambles and constrained by the bounds and ; the authors derive a new explicit condition: for an extreme , each must lie in the bounds or neighboring values while minimizing the linear functional , connecting extremes to supporting hyperplanes. Collectively, the results yield a constructive way to enumerate all extreme points and offer a graph-based view via cone adjacencies, enabling extension to generalized or multivariate -boxes.

Abstract

Probability boxes, also known as -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 -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 -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.
Paper Structure (12 sections, 12 theorems, 16 equations, 6 figures)

This paper contains 12 sections, 12 theorems, 16 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Normal cones
  3. Normal cones and normal fans of convex polyhedra
  4. Adjacent cones and graph theoretical properties of maximal elementary simplicial cones
  5. Normal cones of lower probabilities
  6. Normal cones of $p$-boxes
  7. $p$-boxes and their credal sets
  8. General structure of normal cones of $p$-boxes
  9. Extreme points of $p$-boxes
  10. Relating extreme points to their normal cones
  11. Adjacency structure of the normal cones
  12. Conclusions

Key Result

Proposition 1

Let $\mathcal{C}$ be a convex polyhedron of the form eq-convex-polyhedron and $N(\mathcal{C}, x)$ its normal cone in $x$. The following propositions hold.

Figures (6)

  • Figure 1: Extreme points corresponding to Example \ref{['ex-cones-cases']}--1 (left) and --2 (right).
  • Figure 2: Extreme distribution corresponding to Example \ref{['ex-cones-cases']}--3.
  • Figure 3: Extreme points corresponding to Example \ref{['ex-cones-cases']}--4, with the excluded singletons $\{2\}$ (left), $\{3\}$ (middle) and $\{4\}$ (right).
  • Figure 4: $p$-boxes with the adjacent extreme points from Example \ref{['ex-adjacent1']}--1 depicted.
  • Figure 5: $p$-boxes with the adjacent extreme points from Example \ref{['ex-adjacent1']}--2 depicted.
  • ...and 1 more figures

Theorems & Definitions (22)

  • Proposition 1: (see gruber:07CDG, Proposition 14.1)
  • Proposition 2
  • Definition 1
  • Definition 2
  • Lemma 1
  • Corollary 1
  • Proposition 3
  • Proposition 4: Normal cone additivity
  • Proposition 5
  • Corollary 2: of Proposition \ref{['prop-lprob-triangulation']}
  • ...and 12 more