2019/10/11 by Han L. Gan, Gan, Han L., Nathan T. Ross +1 · 1 citation
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1910.04976
openalex publication_date 2019/10/11 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28
We provide a general theorem bounding the error in the approximation of a\nrandom measure of interest--for example, the empirical population measure of\ntypes in a Wright-Fisher model--and a Dirichlet process, which is a measure\nhaving Poisson-Dirichlet distributed atoms with i.i.d. labels from a diffuse\ndistribution. The implicit metric of the approximation theorem captures the\nsizes and locations of the masses, and so also yields bounds on the\napproximation between the masses of the measure of interest and the\nPoisson-Dirichlet distribution. We apply the result to bound the error in the\napproximation of the stationary distribution of types in the finite\nWright-Fisher model with infinite-alleles mutation structure (not necessarily\nparent independent) by the Poisson-Dirichlet distribution. An important\nconsequence of our result is an explicit upper bound on the total variation\ndistance between the random partition generated by sampling from a finite\nWright-Fisher stationary distribution, and the Ewens Sampling Formula. The\nbound is small if the sample size n is much smaller than\nN1/6\log(N)-1/2, where N is the total population size. Our analysis\nrequires a result of separate interest, giving an explicit bound on the second\nmoment of the number of types of a finite Wright-Fisher stationary\ndistribution. The general approximation result follows from a new development\nof Stein's method for the Dirichlet process, which follows by viewing the\nDirichlet process as the stationary distribution of a Fleming-Viot process, and\nthen applying Barbour's generator approach.\n