The Annals of Applied Probability

Padé approximants and exact two-locus sampling distributions

Paul A. Jenkins and Yun S. Song

Full-text: Open access


For population genetics models with recombination, obtaining an exact, analytic sampling distribution has remained a challenging open problem for several decades. Recently, a new perspective based on asymptotic series has been introduced to make progress on this problem. Specifically, closed-form expressions have been derived for the first few terms in an asymptotic expansion of the two-locus sampling distribution when the recombination rate ρ is moderate to large. In this paper, a new computational technique is developed for finding the asymptotic expansion to an arbitrary order. Computation in this new approach can be automated easily. Furthermore, it is proved here that only a finite number of terms in the asymptotic expansion is needed to recover (via the method of Padé approximants) the exact two-locus sampling distribution as an analytic function of ρ; this function is exact for all values of ρ ∈ [0, ∞). It is also shown that the new computational framework presented here is flexible enough to incorporate natural selection.

Article information

Ann. Appl. Probab., Volume 22, Number 2 (2012), 576-607.

First available in Project Euclid: 2 April 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}

Population genetics recombination sampling distribution asymptotic expansion Padé approximants


Jenkins, Paul A.; Song, Yun S. Padé approximants and exact two-locus sampling distributions. Ann. Appl. Probab. 22 (2012), no. 2, 576--607. doi:10.1214/11-AAP780.

Export citation


  • Baker, G. A. and Graves-Morris, P. (1996). Padé Approximants, 2nd ed. Encyclopedia of Mathematics and its Applications 59. Cambridge Univ. Press, Cambridge.
  • Buzbas, E. O., Joyce, P. and Abdo, Z. (2009). Estimation of selection intensity under overdominance by Bayesian methods. Stat. Appl. Genet. Mol. Biol. 8 Art. 32, 24.
  • Dingle, R. B. (1973). Asymptotic Expansions: Their Derivation and Interpretation. Academic Press, New York.
  • Ethier, S. N. (1979). A limit theorem for two-locus diffusion models in population genetics. J. Appl. Probab. 16 402–408.
  • Ethier, S. N. and Griffiths, R. C. (1990). On the two-locus sampling distribution. J. Math. Biol. 29 131–159.
  • Ethier, S. N. and Nagylaki, T. (1989). Diffusion approximations of the two-locus Wright–Fisher model. J. Math. Biol. 27 17–28.
  • Ewens, W. J. (1972). The sampling theory of selectively neutral alleles. Theoretical Population Biology 3 87–112. Erratum, ibid. 3 (1972) 240; erratum, ibid. 3 (1972) 376.
  • Fearnhead, P. (2003). Haplotypes: The joint distribution of alleles at linked loci. J. Appl. Probab. 40 505–512.
  • Golding, G. B. (1984). The sampling distribution of linkage disequilibrium. Genetics 108 257–274.
  • Griffiths, R. C. and Tavaré, S. (1994). Simulating probability distributions in the coalescent. Theoretical Population Biology 46 131–159.
  • Grote, M. N. and Speed, T. P. (2002). Approximate Ewens formulae for symmetric overdominance selection. Ann. Appl. Probab. 12 637–663.
  • Handa, K. (2005). Sampling formulae for symmetric selection. Electron. Commun. Probab. 10 223–234.
  • Hudson, R. R. (2001). Two-locus sampling distributions and their application. Genetics 159 1805–1817.
  • Huillet, T. (2007). Ewens sampling formulae with and without selection. J. Comput. Appl. Math. 206 755–773.
  • 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. Probab. 20 1005–1028.
  • 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., Myers, S. R., Hunt, S., Deloukas, P., Bentley, D. R. and Donnelly, P. (2004). The fine-scale structure of recombination rate variation in the human genome. Science 304 581–584.
  • Ohta, T. and Kimura, M. (1969a). Linkage disequilibrium at steady state determined by random genetic drift and recurrent mutations. Genetics 63 229–238.
  • Ohta, T. and Kimura, M. (1969b). Linkage disequilibrium due to random genetic drift. Genetical Research 13 47–55.
  • Song, Y. S. and Song, J. S. (2007). Analytic computation of the expectation of the linkage disequilibrium coefficient r2. Theoretical Population Biology 71 49–60.
  • Wright, S. (1949). Adaptation and selection. In Genetics, Paleontology and Evolution (G. L. Jepson, E. Mayr and G. G. Simpson, eds.) 365–389. Princeton Univ. Press, Princeton, NJ.