The Bernoulli structure of discrete distributions

TL;DR

This paper explores the geometric structure of Bernoulli distributions with fixed sum, revealing their convex polytope form, extremal points, and a continuous measure related to Dirichlet distributions, advancing understanding of discrete distribution geometry.

Contribution

It analytically characterizes the convex polytopes of Bernoulli distributions with fixed sum and links their measure to Dirichlet distributions, providing new geometric insights.

Findings

The class of Bernoulli distributions with fixed sum forms a convex polytope.

The Hausdorff measure of these polytopes is a continuous function over the distribution simplex.

The measure normalized over the simplex belongs to the Dirichlet distribution class.

Abstract

Any discrete distribution with support on $\{0,\ldots, d\}$ can be constructed as the distribution of sums of Bernoulli variables. We prove that the class of $d$-dimensional Bernoulli variables $\boldsymbol{X}=(X_1,\ldots, X_d)$ whose sums $\sum_{i=1}^dX_i$ have the same distribution $p$ is a convex polytope $\mathcal{P}(p)$ and we analytically find its extremal points. Our main result is to prove that the Hausdorff measure of the polytopes $\mathcal{P}(p), p\in \mathcal{D}_d,$ is a continuous function $l(p)$ over $\mathcal{D}_d$ and it is the density of a finite measure $\mu_s$ on $\mathcal{D}_d$ that is Hausdorff absolutely continuous. We also prove that the measure $\mu_s$ normalized over the simplex $\mathcal{D}$ belongs to the class of Dirichlet distributions. We observe that the symmetric binomial distribution is the mean of the Dirichlet distribution on $\mathcal{D}$ and that when $d$ increases it converges to the mode.