Dual process in the two-parameter Poisson-Dirichlet diffusion

TL;DR

This paper identifies a dual process for the two-parameter Poisson-Dirichlet diffusion, linking it to Kingman's coalescent with mutation, and uses this duality to derive the diffusion's transition density through a novel probabilistic approach.

Contribution

It introduces a dual process for the two-parameter Poisson-Dirichlet diffusion and employs it to derive the transition density using an extended Polya urn scheme.

Findings

Dual process given by Kingman's coalescent with mutation

Transition density derived via probabilistic proof

Dual depends on parameters through test functions only

Abstract

The two-parameter Poisson-Dirichlet diffusion takes values in the infinite ordered simplex and extends the celebrated infinitely-many-neutral-alleles model, having a two-parameter Poisson-Dirichlet stationary distribution. Here we identify a dual process for this diffusion and obtain its transition probabilities. The dual is shown to be given by Kingman's coalescent with mutation, conditional on a given configuration of leaves. Interestingly, the dual depends on the additional parameter of the stationary distribution only through the test functions and not through the transition rates. After discussing the sampling probabilities of a two-parameter Poisson-Dirichlet partition drawn conditionally on another partition, we use these notions together with the dual process to derive the transition density of the diffusion. Our derivation provides a new probabilistic proof of this result, leveraging on an extension of Pitman's Polya urn scheme, whereby the urn is split after a finite number of steps and two urns are run independently onwards. The proof strategy exemplifies the power of duality and could be exported to other models where a dual is available.