Normal cones corresponding to credal sets of lower probabilities

TL;DR

This paper investigates the geometric structure of credal sets, specifically normal cones, in the context of lower probabilities, providing detailed descriptions for key classes and their relation to extreme points.

Contribution

It offers a detailed analysis of the normal cone structures for credal sets of coherent lower probabilities, especially 2-monotone lower probabilities and probability intervals.

Findings

Normal cones form polyhedral complexes called normal fans.

Normal fan structures relate to the extreme points of credal sets.

General results on triangulated normal fans are provided.

Abstract

Credal sets are one of the most important models for describing probabilistic uncertainty. They usually arise as convex sets of probabilistic models compatible with judgments provided in terms of coherent lower previsions or more specific models such as coherent lower probabilities or probability intervals. In finite spaces, credal sets usually take the form of convex polytopes. Many properties of convex polytopes can be derived from their normal cones, which form polyhedral complexes called normal fans. We analyze the properties of normal cones corresponding to credal sets of coherent lower probabilities. For two important classes of coherent lower probabilities, 2-monotone lower probabilities and probability intervals, we provide a detailed description of the normal fan structure. These structures are related to the structure of the extreme points of the credal sets. To arrive at our main results, we provide some general results on triangulated normal fans of convex polyhedra and their adjacency structure.