Moment-based approximations for the Wright-Fisher model of population dynamics under natural selection at two linked loci

Properly modelling genetic recombination and local linkage has been shown to bring significant improvement to the inference of natural selection from time series data of allele frequencies under a Wright-Fisher model. Existing approaches that can account for genetic recombination and local linkage are built on either the diffusion approximation or a moment-based approximation of the Wright-Fisher model. However, methods based on the diffusion approximation are likely to require much higher computational cost, whereas moment-based approximations may suffer from the distribution support issue: for example, the normal approximation can seriously affect computational accuracy. In the present work, we introduce two novel moment-based approximations of the Wright-Fisher model on a pair of linked loci, both subject to natural selection. Our key innovation is to extend existing methods to account for both the mean and (co)variance of the two-locus Wright-Fisher model with selection. We devise two approximation schemes, using a logistic normal distribution and a hierarchical beta distribution, respectively, by matching the first two moments of the Wright-Fisher model and the approximating model. As compared with the diffusion approximation, our approximations enable the approximate computation of the transition probabilities of the Wright-Fisher model at a far smaller computational cost. We can also avoid the distribution support issue found in the normal approximation.