Normal cones corresponding to credal sets of lower probabilities

Abstract

Credal sets are one of the most important models 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 lower probabilities or probability intervals. In finite spaces, credal sets most often have the form of convex polyhedra. Many properties of convex polyhedra can be derived from their normal cones that form polyhedral complexes called normal fans. We analyze the properties of normal cones corresponding to credal sets of lower probabilities. For two important classes of 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 extreme points of the credal sets. To obtain our main results, we provide some general results on triangulated normal fans of convex polyhedra and their adjacency structure.