Mode Poset Probability Polytopes

TL;DR

This paper investigates the geometric structure of probability polytopes defined by mode and strong mode conditions, analyzing their vertices, facets, and volume depending on mode sets and vicinity structures.

Contribution

It introduces a detailed geometric analysis of mode and strong mode probability polytopes, including their vertices, facets, and volume characteristics.

Findings

Characterization of vertices and facets of mode polytopes

Volume calculations for polytopes based on mode sets

Structural insights into strong mode probability polytopes

Abstract

A mode of a probability vector is a local maximum with respect to some vicinity structure on the set of elementary events. The mode inequalities cut out a polytope from the simplex of probability vectors. Related to this is the concept of strong modes. A strong mode of a distribution is an elementary event that has more probability mass than all its direct neighbors together. The set of probability distributions with a given set of strong modes is again a polytope. We study the vertices, the facets, and the volume of such polytopes depending on the sets of (strong) modes and the vicinity structures.