Markov Chain Monte Carlo Simulation of the Wright-Fisher Diffusion

Abstract

In population genetics, the proportions of alleles at any given time are of interest. From generation to generation, these proportions vary and over a long time horizon the likelihoods for the proportions are given by a stationary distribution corresponding to the dynamics of the population. We investigate a diffusion approximation for the Wright-Fisher model and develop a Markov chain Monte Carlo simulation to approximate the evolution of the proportions of alleles in the population. Our aim is to estimate the stationary distribution, especially for parameters of the model for which no analytical formulas are known. We discretize the space of the diffusion process and construct a continuous time Markov chain which converges weakly to the diffusion.