Bernoulli factories and duality in Wright–Fisher and Allen–Cahn models of population genetics

Abstract

Mathematical models of genetic evolution often come in pairs, connected by a so-called duality relation. The most seminal example are the Wright–Fisher diffusion and the Kingman coalescent, where the former describes the stochastic evolution of neutral allele frequencies in a large population forwards in time, and the latter describes the genetic ancestry of randomly sampled individuals from the population backwards in time. As well as providing a richer description than either model in isolation, duality often yields equations satisfied by quantities of interest. We employ the so-called Bernoulli factory – a celebrated tool in simulation-based computing – to derive duality relations for broad classes of genetics models. As concrete examples, we present Wright–Fisher diffusions with general drift functions, and Allen–Cahn equations with general, nonlinear forcing terms. The drift and forcing functions can be interpreted as the action of frequency-dependent selection. To our knowledge, this work is the first time a connection has been drawn between Bernoulli factories and duality in models of population genetics.