Padé approximants and exact two-locus sampling distributions

Abstract

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 $\rho$ 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 $\rho$; this function is exact for all values of $\rho \in [0,\infty )$. It is also shown that the new computational framework presented here is flexible enough to incorporate natural selection.

Published in at http://dx.doi.org/10.1214/11-AAP780 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• An asymptotic sampling formula for the coalescent with Recombination

Cited by in corpus (5)

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• Tractable diffusion and coalescent processes for weakly correlated loci
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• Large-sample analysis of cost functionals for inference under the coalescent