Configured spaces and their M\"obius polynomials

TL;DR

This paper explores the relationship between hypergraph configurations and probability spaces, introducing the concept of relative M"obius polynomials to analyze event dependence structures with a focus on optimal event probabilities.

Contribution

It introduces the notion of configurations and their associated M"obius polynomials, establishing existence and uniqueness of an optimal event probability in this framework.

Findings

Existence and uniqueness of an optimal probability t for configurations.

Development of a new formula for the derivative of the M"obius polynomial.

Extension of previous graph-based results to general hypergraph configurations.

Abstract

An arbitrary dependence structure between a finite family of events of a probability space defines a hypergraph structure. We study the converse operation, starting from a hypergraph structure, to determine a canonical probability space with events indexed by the vertices and that satisfy the prescribed dependence structure; we also require that events that are not mutually independent must be mutually exclusive. Such a hypergraph structure, we call a configuration. Requiring furthermore that all events are given the same positive probability $t$, converts this problem to the study of a family of M\"obius polynomials, which we call the relative M\"obius polynomials (or independence polynomial) of the configuration. This study was very much already complete in the case where the configuration actually derives from a graph structure, but remained open in the general case. We show the existence and uniqueness of an optimal real $t$, optimal in the sense that the events cover a subset of maximal probability. We base our study on a new formula for the derivative of the M\"obius polynomial.