# Mathematical Biology Seminar

### Stewart Ethier

Department of Mathematics, University of Utah

#### A population genetics interpretation of the two-parameter Poisson-Dirichlet distribution

#### Friday, April 15 , 2016, at 3:00pm

LCB 219

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.)