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

Theoretical Population Biology
P Donnelly

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.

References

Mar 31, 1977·Nature·J H Gillespie
Apr 1, 1977·Theoretical Population Biology·G A Watterson, H A Guess
Apr 1, 1977·Theoretical Population Biology·J F Kingman
Mar 1, 1972·Theoretical Population Biology·S Karlin, J McGregor
Mar 1, 1972·Theoretical Population Biology·W J Ewens
Feb 1, 1980·Theoretical Population Biology·R C Griffiths

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Citations

Mar 17, 2009·Journal of Mathematical Biology·Joseph C Watkins
Nov 7, 2006·Bulletin of Mathematical Biology·Jukka CoranderTimo Koski
Dec 1, 1992·Genetical Research·G A Watterson, P Donnelly
Nov 28, 2007·PLoS Computational Biology·Hailiang HuangJoel S Bader
Jan 19, 2008·Genetics·Anders ErikssonSerik Sagitov
Nov 4, 2008·Statistical Applications in Genetics and Molecular Biology·Asger HobolthCarsten Wiuf
Apr 18, 2009·Theoretical Population Biology·Ganeshkumar Ganapathy, Marcy K Uyenoyama
Oct 25, 2007·Journal of Evolutionary Biology·E G Leigh
Jan 1, 1995·Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics·P G Higgs
Jan 1, 1987·Journal of Mathematical Biology·P Donnelly, S Tavaré
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Feb 10, 2017·Journal of the Royal Society, Interface·N ZoroaJ Casas
Mar 8, 2020·Theoretical Population Biology·John Wakeley
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May 4, 2000·Theoretical Population Biology·M Stephens
Jul 29, 1998·Theoretical Population Biology·Y X Fu
Mar 20, 2001·American Journal of Human Genetics·M StephensP Donnelly

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