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Licensed Unlicensed Requires Authentication Published by De Gruyter June 6, 2018

Bayesian inference of selection in the Wright-Fisher diffusion model

Jeffrey J. Gory, Radu Herbei and Laura S. Kubatko

Abstract

The increasing availability of population-level allele frequency data across one or more related populations necessitates the development of methods that can efficiently estimate population genetics parameters, such as the strength of selection acting on the population(s), from such data. Existing methods for this problem in the setting of the Wright-Fisher diffusion model are primarily likelihood-based, and rely on numerical approximation for likelihood computation and on bootstrapping for assessment of variability in the resulting estimates, requiring extensive computation. Recent work has provided a method for obtaining exact samples from general Wright-Fisher diffusion processes, enabling the development of methods for Bayesian estimation in this setting. We develop and implement a Bayesian method for estimating the strength of selection based on the Wright-Fisher diffusion for data sampled at a single time point. The method utilizes the latest algorithms for exact sampling to devise a Markov chain Monte Carlo procedure to draw samples from the joint posterior distribution of the selection coefficient and the allele frequencies. We demonstrate that when assumptions about the initial allele frequencies are accurate the method performs well for both simulated data and for an empirical data set on hypoxia in flies, where we find evidence for strong positive selection in a region of chromosome 2L previously identified. We discuss possible extensions of our method to the more general settings commonly encountered in practice, highlighting the advantages of Bayesian approaches to inference in this setting.

Funding source: Division of Mathematical Sciences

Award Identifier / Grant number: 1209142

Award Identifier / Grant number: 1407604

Award Identifier / Grant number: 1106706

Funding statement: This work was supported by National Science Foundation grants DMS-1209142 and DMS-1407604 (R.H.) and Funder Id: 10.13039/100000121, DMS-1106706 (L.S.K.).

Acknowledgement

We thank Joshua Schraiber and an anonymous reviewer for helpful comments on an earlier draft.

Appendix

Exact simulation of the neutral Wright-Fisher diffusion

Consider a neutral WF diffusion process Qt

(11){dQt=β0(Qt)dt+Qt(1Qt)dBt,Q0=q0,0tT

where the drift function is β0(x) = (1/2)(θ1(1 − x) − θ2x), for 0 ≤ x ≤ 1 and θ1, θ2 > 0 (note the absence of the selection parameter). The necessary details for simulating a variate QT from the model (11) for a fixed T > 0, given that Q0 = q0 ∈ (0, 1) are given in Jenkins and Spanó (2015) and we briefly review the main ideas. It is well-known (Griffiths, 1979; Tavaré, 1984; Ethier & Griffiths, 1993; Griffiths & Spanò, 2013) that, conditional on Q0 = q0, the variate QT has a probability density function given by

(12)f(x|q0)=m=0qmθ(T)l=0mBin(l;m,q0)Beta(x;θ1+l,θ2+ml)

where θ = θ1 + θ2, and in(⋅) and eta(⋅) are the Binomial and Beta density functions, given by

Bin(l;m,q0)=(ml)q0l(1q0)mlBeta(x;θ1,θ2)xθ11(1x)θ21.

The mixture weights qmθ(T) in (12) have an alternating series expansion

(13)qmθ(T)=k=m(1)kmakmθek(k+θ1)T/2

where

akmθ=(θ+2k1)(θ+m)(k1)m!(km)!

with the convention that a(x) ≡ Γ(a + x)/Γ(a) for a > 0 and x ≥ −1. Sampling from the density (12) is achieved via Algorithm 2 described below (see Jenkins & Spanó, 2015).

Algorithm 2.

Algorithm 2. Exact Sampling for the neutral WF diffusion:

1: Draw 𝖬 from the discrete distribution {qmθ(T),m=0,1,};

2: Draw LBin(M,q0);

3: Draw QBeta(θ1+L,θ2+ML);

4: Return 𝖰.

The computational burden in Algorithm 2 is in Step 1, since the weights qmθ(T) do not have a closed form expression and (13) suggests an infinite amount of computation. However, given the alternating series representation in (13), one can use ideas described in Chapter 5 of Devroye (1986) to devise an exact procedure for simulating the variate 𝖬, which only requires finite computing time. Details of such a procedure are provided in Section 3.2 of Jenkins and Spanó (2015).

Exact simulation of general WF-diffusion models

As we mention above, our approach relies on being able to obtain exact draws from a general Wright-Fisher diffusion (1). We briefly review the general setup for sampling an SDE. Assume that {Xt, 0 ≤ tT} is a stochastic process described via

(14)dXt=β(Xt)dt+σ(Xt)dBt,X(0)=x0,0tT

where the functions β(⋅) and σ(⋅) are assumed to be smooth enough such that (14) has a unique weak solution. Such conditions can be found in Karatzas and Shreve (1998), for example. Our goal is to simulate exact draws from the distribution of Xt, for some fixed t > 0, conditional on X0 = x0. In general, the transition density for the process Xt is not available in closed form, not even in an infinite series representation as is the case with the neutral Wright-Fisher diffusion. Recently, in a series of papers (Beskos & Roberts, 2005; Beskos, Papaspiliopoulos & Roberts 2006b; 2008), the authors have developed a rejection sampling approach, which yields an exact (distribution-wise) skeleton of the full path (Xt, 0 ≤ tT) for a very large class of diffusions. Briefly, let Ω = 𝒞([0, T]) and let ω be a typical element of Ω. Let ℚ be the probability measure induced by {Xt, 0 ≤ tT} onto Ω and ℤ be another probability measure on Ω which is user-selected in an appropriate way. Under regularity conditions (see Beskos & Roberts, 2005) one can use the Girsanov formula to derive and expression for the Radon-Nikodym derivative

dQdZ(ω)G(ω)

where, in principle, one can arrange that 𝖦(⋅) ∈ (0, 1). Thus, a rejection sampling strategy will be appropriate:

Algorithm 3.

Algorithm 3. Exact sampling for general diffusions

1: Sample ω ∼ ℤ;

2: Accept ω as a draw from ℚ with probability 𝖦(ω) ∈ (0, 1).

The proposal distribution ℤ has to be selected in a way that Step 1 of Algorithm 3 can be done efficiently, i.e. a biased Brownian Bridge measure. We refer the reader to Beskos and Roberts (2005) for a detailed description of all of the conditions and the general setup.

General Wright-Fisher diffusions. If the diffusion coefficient in (14) takes the form

σ(x)=x(1x),

then the SDE (14) is a general WF diffusion. As Jenkins and Spanó (2015) suggest, in this case, it is efficient to select the proposal measure ℤ to be the law induced by the neutral WF diffusion (11).

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Published Online: 2018-6-6

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