Partition structures, Polya urns, the Ewens sampling formula, and the ages of alleles

Abstract

It has recently been shown that the Ewens sampling formula may be generated by a Polya-like urn model. A genealogical proof of this result equates the labelling of balls in the urn to the partition by age of alleles in the sample. This urn construction is shown to be equivalent to the construction of Kingman ( Proc. Roy. Soc. London Ser. A 361 (1978), 1–20) using a Poisson-Dirichlet “paintbox” and as a consequence, the partition by ages is seen to be equivalent to the size biased permutation of the Poisson-Dirichlet distribution. This approach unifies and extends many results on ages of alleles, the Polya urn, and the Poisson-Dirichlet distribution. Furthermore the Ewens sampling formula is characterized as being the only partition structure which may be generated by an urn-like mechanism.