Multivariate Dirichlet Moments and a Polychromatic Ewens Sampling Formula

TL;DR

This paper derives a simple formula for multivariate Dirichlet moments, extends it to Dirichlet-Ferguson and Gamma measures, and introduces a polychromatic Ewens sampling formula linked to urn models and partition processes.

Contribution

It provides a non-recursive formula for multivariate Dirichlet moments and introduces a novel polychromatic Ewens sampling formula with theoretical properties.

Findings

Elementary non-recursive Dirichlet moments formula

Extension to Dirichlet-Ferguson and Gamma measures

Polychromatic Ewens sampling formula with consistency

Abstract

We present an elementary non-recursive formula for the multivariate moments of the Dirichlet distribution on the standard simplex, in terms of the pattern inventory of the moments' exponents. We obtain analog formulas for the multivariate moments of the Dirichlet-Ferguson and Gamma measures. We further introduce a polychromatic analogue of Ewens sampling formula on colored integer partitions, discuss its relation with suitable extensions of Hoppe's urn model and of the Chinese restaurant process, and prove that it satisfies an adapted notion of consistency in the sense of Kingman.