A stochastic diffusion process for the Dirichlet distribution

TL;DR

This paper introduces a nonlinear stochastic diffusion process that models the Dirichlet distribution as its equilibrium, enabling realistic simulation of constrained multivariate systems with conservation laws.

Contribution

It develops a novel coupled nonlinear diffusion process with multiplicative Wiener processes that ensures samples satisfy the Dirichlet constraint at all times.

Findings

Monte Carlo simulations confirm convergence to the Dirichlet distribution.

The process generalizes the Wright-Fisher model for multivariate cases.

Samples maintain the unit-sum constraint throughout evolution.

Abstract

The method of potential solutions of Fokker-Planck equations is used to develop a transport equation for the joint probability of N coupled stochastic variables with the Dirichlet distribution as its asymptotic solution. To ensure a bounded sample space, a coupled nonlinear diffusion process is required: the Wiener-processes in the equivalent system of stochastic differential equations are multiplicative with coefficients dependent on all the stochastic variables. Individual samples of a discrete ensemble, obtained from the stochastic process, satisfy a unit-sum constraint at all times. The process may be used to represent realizations of a fluctuating ensemble of N variables subject to a conservation principle. Similar to the multivariate Wright-Fisher process, whose invariant is also Dirichlet, the univariate case yields a process whose invariant is the beta distribution. As a test of the results, Monte-Carlo simulations are used to evolve numerical ensembles toward the invariant Dirichlet distribution.