General moment closure for the neutral two-locus Wright-Fisher dynamics

The Wright-Fisher diffusion and its dual, the coalescent process, are at the core of many results and methods in population genetics. Approaches have been developed to study the dynamics of its moments under genetic drift, mutation, and recombination using ordinary differential equations. The dynamics of these moments can be used to study population genetic processes and are key building blocks of efficient methods to infer population genetic parameters, like demographic histories or fine-scale recombination rates. However, the system of equations does not close under recombination; that is, computing moments of a certain order requires knowledge of moments of higher order. By applying a coordinate transformation to the diffusion generator, we show that the canonical moments in these alternative coordinates yield a closed system, enabling more accurate numerical computations. Compared to previous approaches in the literature, we believe that this approach can be more readily extended to general scenarios. Through simulations, we verify that the derived system of differential equations can accurately capture the dynamics of the moments, and can be used to efficiently compute expected diversity and linkage statistics in population genetic samples.