Known results on the moments of the distribution generated by the two-locus Wright–Fisher diffusion model, and the duality between the diffusion process and the ancestral process with recombination are briefly summarized. A numerical method for computing moments using a Markov chain Monte Carlo simulation and a method to compute closed-form expressions of the moments are presented. By applying the duality argument, the properties of the ancestral recombination graph are studied in terms of the moments.

Gene conversion is a genetic mechanism by which one gene is ‘copied and pasted’ onto another gene, where the direction can be biased between the different types. In this paper, a stochastic model for biased gene conversion within a d-unlinked multigene family and its diffusion approximation are developed for a finite Moran population. A connection with a d-island model is made. A formula for the fixation probability in the absence of mutation is given. A two-timescale argument is applied in the case of the strong conversion limit. The dual process is generally shown to be a biased voter model, which generates an ancestral bias graph for a given sample. An importance sampling algorithm for computing the likelihood of the sample is deduced.