Uniformly Generating Distribution Functions for Discrete Random Variables

TL;DR

This work addresses the problem of uniformly generating distribution functions for an $n$-valued discrete random variable by recasting the task as uniform sampling from the $(n-1)$-simplex $S^{n-1}$. It proposes a constructive, optimally efficient approach that sequentially samples the first $n-1$ coordinates using their marginal densities $\psi$, with inverse-transform sampling guided by $\Psi^{-1}$. The key contributions are closed-form expressions for the marginals $\psi$ and their CDFs $\Psi$, plus a practical algorithm that uses exactly $n-1$ Uniform$[0,1]$ samples and $n-2$ constant-time inverse-CDF evaluations to produce unbiased samples. This method enables unbiased generation of distribution functions for probabilistic inference and validation, avoiding inefficient rejection or rescaling techniques as $n$ grows.

Abstract

An algorithm is presented which, with optimal efficiency, solves the problem of uniform random generation of distribution functions for an n-valued random variable.

Figures (1)

• Figure 1: (a) The 3-dimensional case: for any outcome of $x_{1}$ rescaling of the 1-simplex is determined. The marginal distribution of $x_{1}$ is therefore proportional to the 1-volume of $S^{1}$. (b) 5000 samples of $S^{2}$ as obtained by applying the proposed algorithm.