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The Bernoulli structure of discrete distributions

Roberto Fontana, Patrizia Semeraro

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 $μ_s$ on $\mathcal{D}_d$ that is Hausdorff absolutely continuous. We also prove that the measure $μ_s$ 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.

The Bernoulli structure of discrete distributions

Abstract

Any discrete distribution with support on can be constructed as the distribution of sums of Bernoulli variables. We prove that the class of -dimensional Bernoulli variables whose sums have the same distribution is a convex polytope and we analytically find its extremal points. Our main result is to prove that the Hausdorff measure of the polytopes is a continuous function over and it is the density of a finite measure on that is Hausdorff absolutely continuous. We also prove that the measure normalized over the simplex belongs to the class of Dirichlet distributions. We observe that the symmetric binomial distribution is the mean of the Dirichlet distribution on and that when increases it converges to the mode.

Paper Structure

This paper contains 9 sections, 14 theorems, 68 equations, 4 tables.

Table of Contents

  1. Introduction
  2. The convex polytope $\mathcal{P}(p)$
  3. Moments
  4. Entropy
  5. The induced measure on $\mathcal{D}_d$
  6. Measuring a neighbourhood of $\mathcal{P}(p)$
  7. The binomial distribution
  8. Moments and correlation
  9. Entropy

Key Result

Theorem 2.1

For any $p\in \mathcal{D}_d$ the class $\mathcal{P}(p)=\{\boldsymbol{f}\in \mathcal{F}_d: \, p_{\boldsymbol{f}}=p\}$ is the convex polytope $\mathcal{P}(p)=\prod_{k=0}^d\Delta_{n_k, \sqrt{2}p_k}$, where $n_k=\binom{d}{k}-1$ and its extremal points are where $\sigma=(\sigma_0, \ldots, \sigma_k, \ldots, \sigma_d)$, $\sigma_k=1,\ldots, \binom{d}{k}$, $k=0,\ldots, d$.

Theorems & Definitions (35)

  • Theorem 2.1
  • proof
  • Corollary 2.1
  • proof
  • Example 1
  • Remark 1
  • Proposition 2.1
  • proof
  • Remark 2
  • Example 2
  • ...and 25 more