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 a significant improvement to the inference of natural selection from time series genetic data under a Wright-Fisher model. Existing approaches that can take genetic recombination effect and local linkage information into account are built upon either the diffusion approximation or the moment-based approximation of the Wright-Fisher model. However, such approximations are either limited to the increased computational cost like the diffusion approximation or suffer from the distribution support issue like the normal approximation, which can seriously affect computational efficiency and accuracy. In this work, we propose two novel moment-based approximations for the Wright-Fisher model of population dynamics subject to natural selection at a pair of linked loci. Our key innovation is that we extend existing approaches to calculate the mean and (co)variance of the two-locus Wright-Fisher model with selection and suggest a logistic normal distribution or a hierarchical beta distribution as a parametric continuous probability distribution to approximate the Wright-Fisher model by matching its first two moments to those of the Wright-Fisher model. Compared with the diffusion approximation, our approximations enable the computation of the transition probability distribution of the Wright-Fisher model at a far smaller computational cost and also allow us to avoid the distribution support issue found in the normal approximation.