Stein's method for the Poisson-Dirichlet distribution and the Ewens Sampling Formula, with applications to Wright-Fisher models

Abstract

We provide a general theorem bounding the error in the approximation of a random measure of interest--for example, the empirical population measure of types in a Wright-Fisher model--and a Dirichlet process, which is a measure having Poisson-Dirichlet distributed atoms with i.i.d. labels from a diffuse distribution. The implicit metric of the approximation theorem captures the sizes and locations of the masses, and so also yields bounds on the approximation between the masses of the measure of interest and the Poisson-Dirichlet distribution. We apply the result to bound the error in the approximation of the stationary distribution of types in the finite Wright-Fisher model with infinite-alleles mutation structure (not necessarily parent independent) by the Poisson-Dirichlet distribution. An important consequence of our result is an explicit upper bound on the total variation distance between the random partition generated by sampling from a finite Wright-Fisher stationary distribution, and the Ewens Sampling Formula. The bound is small if the sample size $n$ is much smaller than $N^{1/6}\log(N)^{-1/2}$, where $N$ is the total population size. Our analysis requires a result of separate interest, giving an explicit bound on the second moment of the number of types of a finite Wright-Fisher stationary distribution. The general approximation result follows from a new development of Stein's method for the Dirichlet process, which follows by viewing the Dirichlet process as the stationary distribution of a Fleming-Viot process, and then applying Barbour's generator approach.