  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.
