Advances in Applied Probability

Closed-form asymptotic sampling distributions under the coalescent with recombination for an arbitrary number of loci

Anand Bhaskar and Yun S. Song

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Obtaining a closed-form sampling distribution for the coalescent with recombination is a challenging problem. In the case of two loci, a new framework based on an asymptotic series has recently been developed to derive closed-form results when the recombination rate is moderate to large. In this paper, an arbitrary number of loci is considered and combinatorial approaches are employed to find closed-form expressions for the first couple of terms in an asymptotic expansion of the multi-locus sampling distribution. These expressions are universal in the sense that their functional form in terms of the marginal one-locus distributions applies to all finite- and infinite-alleles models of mutation.

Article information

Adv. in Appl. Probab., Volume 44, Number 2 (2012), 391-407.

First available in Project Euclid: 16 June 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 92D15: Problems related to evolution
Secondary: 65C50: Other computational problems in probability 92D10: Genetics {For genetic algebras, see 17D92}

Coalescent theory recombination asymptotic expansion sampling distribution


Bhaskar, Anand; Song, Yun S. Closed-form asymptotic sampling distributions under the coalescent with recombination for an arbitrary number of loci. Adv. in Appl. Probab. 44 (2012), no. 2, 391--407. doi:10.1239/aap/1339878717.

Export citation


  • Ethier, S. N. (1979). A limit theorem for two-locus diffusion models in population genetics. J. Appl. Prob. 16, 402–408.
  • Ethier, S. N. and Griffiths, R. C. (1990). On the two-locus sampling distribution. J. Math. Biol. 29, 131–159.
  • Ewens, W. J. (1972). The sampling theory of selectively neutral alleles. Theoret. Pop. Biol. 3, 87–112.
  • Fearnhead, P. and Donnelly, P. (2001). Estimating recombination rates from population genetic data. Genetics 159, 1299–1318.
  • Golding, G. B. (1984). The sampling distribution of linkage disequilibrium. Genetics 108, 257–274.
  • Griffiths, R. C. (1981). Neutral two-locus multiple allele models with recombination. Theoret. Pop. Biol. 19, 169–186.
  • Griffiths, R. C. and Marjoram, P. (1996). Ancestral inference from samples of DNA sequences with recombination. J. Comput. Biol. 3, 479–502.
  • Griffiths, R. C., Jenkins, P. A. and Song, Y. S. (2008). Importance sampling and the two-locus model with subdivided population structure. Adv. Appl. Prob. 40, 473–500.
  • Hudson, R. R. (1985). The sampling distribution of linkage disequilibrium under an infinite allele model without selection. Genetics 109, 611–631.
  • Hudson, R. R. (2001). Two-locus sampling distributions and their application. Genetics 159, 1805–1817.
  • Jenkins, P. A. and Song, Y. S. (2009). Closed-form two-locus sampling distributions: accuracy and universality. Genetics 183, 1087–1103.
  • Jenkins, P. A. and Song, Y. S. (2010). An asymptotic sampling formula for the coalescent with recombination. Ann. Appl. Prob. 20, 1005–1028.
  • Jenkins, P. A. and Song, Y. S. (2012). Padé approximants and exact two-locus sampling distributions. Ann. Appl. Prob. 22, 576–607.
  • Kingman, J. F. C. (1982). The coalescent. Stoch. Process. Appl. 13, 235–248.
  • Kingman, J. F. C. (1982). On the genealogy of large populations. In Essays in Statistical Science (J. Appl. Prob. Spec. Vol. 19A), eds J. Gani and E. J. Hannan, Applied Probability Trust, Sheffield, pp. 27–43.
  • Kuhner, M. K., Yamato, J. and Felsenstein, J. (2000). Maximum likelihood estimation of recombination rates from population data. Genetics 156, 1393–1401.
  • McVean, G., Awadalla, P. and Fearnhead, P. (2002). A coalescent-based method for detecting and estimating recombination from gene sequences. Genetics 160, 1231–1241.
  • McVean, G. A. T. et al. (2004). The fine-scale structure of recombination rate variation in the human genome. Science 304, 581–584.
  • Nielsen, R. (2000). Estimation of population parameters and recombination rates from single nucleotide polymorphisms. Genetics 154, 931–942.
  • Stephens, M. and Donnelly, P. (2000). Inference in molecular population genetics. J. Roy. Statist. Soc. B 62, 605–655.
  • Wang, Y. and Rannala, B. (2008). Bayesian inference of fine-scale recombination rates using population genomic data. Phil. Trans. R. Soc. B 363, 3921–3930.
  • Wright, S. (1949). Adaptation and selection. In Genetics, Paleontology and Evolution, eds G. L. Jepson, G. G. Simpson and E. Mayr, Princeton University Press, pp. 365–389.