Hostname: page-component-848d4c4894-2xdlg Total loading time: 0 Render date: 2024-06-21T17:32:20.607Z Has data issue: false hasContentIssue false
  • English
  • Français

Estimating the frequency of the oldest allele: a bayesian approach

Published online by Cambridge University Press:  01 July 2016

Paul Joyce
Paul Joyce*
Affiliation:
University of Southern California
*
∗Postal address: Department of Mathematics University of Southern California, Los Angeles, CA 90089-1113, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we calculate posterior distributions associated with a version of the Poisson–Dirichlet distribution called the GEM. The GEM has been shown (by several authors) to be the limiting stationary distribution for allele frequencies listed in age order associated with the neutral infinite alleles model. In view of this result, we use our posterior distributions to calculate Bayes estimators for the frequency of the oldest allele given a sample.

Keywords

GEM DISTRIBUTIONINFINITE ALLELESREVERSIBILITYSIZED-BIASEDAGE-ORDEREDGENERALIZED-DIRICHLETPOSTERIOR DISTRIBUTIONBAYES ESTIMATOR
Type
Research Article
Information
Advances in Applied Probability , Volume 23 , Issue 3 , September 1991 , pp. 456 - 475
Copyright
Copyright © Applied Probability Trust 1991 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Supported in part by NSF grant DMS 90-05833.

References

Abramowitz, M. and Stegun, L. A. (1972) Handbook of Mathematical Functions. Dover, New York.Google Scholar
Billingsley, P. (1977) Probability and Measure. Wiley, New York.Google Scholar
Connor, R. J. and Moismann, J. E. (1969) Concepts of independence for proportions with a generalization of the Dirichlet distribution. J. Amer. Statist. Assoc. 64, 194–206.Google Scholar
Donnelly, P. (1986) Partition structures, Polya urns, the Ewens sampling formula, and the ages of alleles. Theoret. Popn Biol. 30, 271–288.Google Scholar
Donnelly, P. and Joyce, P. (1989) Continuity and weak convergence of ranked size-biased permutations on the infinite simplex. Stoch. Proc. Appl. 31, 89–103.Google Scholar
Donnelly, P. and Joyce, P. (1991a) Consistent ordered sampling distributions: characterization and convergence. Adv. Appl. Prob. 23, 229–258.Google Scholar
Donnelly, P. and Joyce, P. (1991b) Weak convergence of population genealogical process to the coalescent with ages. Ann. Prob. Google Scholar
Donnelly, P. and Tavaré, S. (1986) The ages of alleles and a coalescent. Adv. Appl. Prob. 18, 1–19.Google Scholar
Donnelly, P. and Tavaré, S. (1987) The population genealogy of the infinitely-many neutral alleles model. J. Math. Biol. 251, 381–391.Google Scholar
Ethier, S. N. (1990a) The infinitely-many-neutral-alleles diffusion model with ages. Adv. Appl. Prob. 22, 1–24.Google Scholar
Ethier, S. N. (1990b) The distribution of the frequencies of age-ordered alleles in a diffusion model. Adv. Appl. Prob. 22, 519–532.CrossRefGoogle Scholar
Ewens, W. J. (1972) The sampling theory of selectively neutral alleles. Theoret. Popn Biol. 3, 87–112.CrossRefGoogle ScholarPubMed
Ewens, W. J. (1979) Mathematical Population Genetics. Springer-Verlag, Berlin.Google Scholar
Hoppe, F. M. (1984) Pólya-like urns and the Ewens sampling formula. J. Math. Biol. 20, 91–99.Google Scholar
Hoppe, F. M. (1986) Size-biased filtering of Poisson–Dirichlet samples with an application to partition structures in genetics. J. Appl. Prob. 23, 1008–1012.Google Scholar
Hoppe, F. M. (1987) The sampling theory of neutral alleles and an urn model in population genetics. J. Math. Biol. 25, 123–159.Google Scholar
Kelly, F. P. (1977) Exact results for the Moran neutral allele model. Adv. Appl. Prob. 9, 357–360.CrossRefGoogle Scholar
Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, New York.Google Scholar
Kingman, J. F. C. (1977) The population structure associated with the Ewens sampling formula. Theoret. Popn Biol. 11, 274–284.Google Scholar
Kingman, J. F. C. (1982) The coalescent. Stoch. Proc. Appl. 13, 235–248.Google Scholar
Patil, G. P. and Taillie, C. (1977) Diversity as a concept and its implications for random communities. Bull. Inst. Internat. Statist. Inst. 47, 497–515.Google Scholar
Watterson, G. A. (1976a) Reversibility and the age of an allele. I. Moran's infinitely-many neutral alleles model. Theoret. Popn Biol. 10, 239–253.Google Scholar
Watterson, G. A. (1976b) The stationary distribution of the infinitely-many neutral alleles diffusion model. J. Appl. Prob. 13, 639–651.CrossRefGoogle Scholar
Watterson, G. A. and Guess, H. A. (1977) Is the most frequent allele the oldest? Theoret. Popn Biol. 11, 141–160.Google Scholar