Exact simulation of the Wright-Fisher diffusion

TL;DR

This paper introduces a method for exact simulation of Wright-Fisher diffusion processes, including those with natural selection, using eigenfunction expansions and retrospective algorithms, overcoming previous simulation challenges.

Contribution

It presents the first exact simulation algorithms for a broad class of Wright-Fisher diffusions and their bridges, including models with natural selection.

Findings

Exact simulation of Wright-Fisher diffusions achieved

Applicable to processes with natural selection

Provides methods for simulating ancestral processes

Abstract

The Wright-Fisher family of diffusion processes is a widely used class of evolutionary models. However, simulation is difficult because there is no known closed-form formula for its transition function. In this article we demonstrate that it is in fact possible to simulate exactly from a broad class of Wright-Fisher diffusion processes and their bridges. For those diffusions corresponding to reversible, neutral evolution, our key idea is to exploit an eigenfunction expansion of the transition function; this approach even applies to its infinite-dimensional analogue, the Fleming-Viot process. We then develop an exact rejection algorithm for processes with more general drift functions, including those modelling natural selection, using ideas from retrospective simulation. Our approach also yields methods for exact simulation of the moment dual of the Wright-Fisher diffusion, the ancestral process of an infinite-leaf Kingman coalescent tree. We believe our new perspective on diffusion simulation holds promise for other models admitting a transition eigenfunction expansion.