English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

We provide a general theorem bounding the error in the approximation of arandom measure of interest--for example, the empirical population measure oftypes in a Wright-Fisher model--and a Dirichlet process, which is a measurehaving Poisson-Dirichlet distributed atoms with i.i.d. labels from a diffusedistribution. The implicit metric of the approximation theorem captures thesizes and locations of the masses, and so also yields bounds on theapproximation between the masses of the measure of interest and thePoisson-Dirichlet distribution. We apply the result to bound the error in theapproximation of the stationary distribution of types in the finiteWright-Fisher model with infinite-alleles mutation structure (not necessarilyparent independent) by the Poisson-Dirichlet distribution. An importantconsequence of our result is an explicit upper bound on the total variationdistance between the random partition generated by sampling from a finiteWright-Fisher stationary distribution, and the Ewens Sampling Formula. Thebound 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 analysisrequires a result of separate interest, giving an explicit bound on the secondmoment of the number of types of a finite Wright-Fisher stationarydistribution. The general approximation result follows from a new developmentof Stein's method for the Dirichlet process, which follows by viewing theDirichlet process as the stationary distribution of a Fleming-Viot process, andthen applying Barbour's generator approach.

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