The one-parameter Poisson-Dirichlet distribution PD(θ) (θ > 0) on the infinite-dimensional ordered simplex was introduced by Kingman (1975). It has applications in number theory, combinatorics, Bayesian statistics, and population genetics. In particular, it is closely related to the celebrated Ewens sampling formula. Moreover, PD(θ) is the unique stationary distribution of the infinitely many neutral alleles diffusion model with mutation parameter θ.
The two-parameter Poisson-Dirichlet distribution PD(θ, α) (0 ≤ α < 1, θ > -α) on the same simplex was introduced by Pitman and Yor (1997). It has applications in numerous fields but, until now, no connection with population genetics has been found. We argue that PD(θ, α) is the unique stationary distribution of the infinitely many neutral alleles diffusion model with mutation parameter θ+α and immigration parameter α, assuming a certain state-dependent immigration kernel.
(Joint work with Cristina Costantini, Pierpaolo De Blasi, Matteo Ruggiero, and Dario Spanò.)