High dimensional Bernoulli distributions: algebraic representation and applications

TL;DR

This paper introduces an algebraic framework for representing multivariate Bernoulli distributions with fixed means, enabling analytical generation, extremal point identification, and bounds in convex order in high dimensions.

Contribution

It provides a novel algebraic representation of multivariate Bernoulli classes, facilitating analytical generator derivation and applications in extremal and dependence structure problems.

Findings

Analytical generators for multivariate Bernoulli distributions derived

Method to find extremal points of high-dimensional Bernoulli polytopes

Lower bounds in convex order for sums with fixed margins determined

Abstract

The main contribution of this paper is to find a representation of the class $\mathcal{F}_d(p)$ of multivariate Bernoulli distributions with the same mean $p$ that allows us to find its generators analytically in any dimension. We map $\mathcal{F}_d(p)$ to an ideal of points and we prove that the class $\mathcal{F}_d(p)$ can be generated from a finite set of simple polynomials. We present two applications. Firstly, we show that polynomial generators help to find extremal points of the convex polytope $\mathcal{F}_d(p)$ in high dimensions. Secondly, we solve the problem of determining the lower bounds in the convex order for sums of multivariate Bernoulli distributions with given margins, but with an unspecified dependence structure.