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Mathematical models of genetic evolution often come in pairs, connected by aso-called duality relation. The most seminal example are the Wright-Fisherdiffusion and the Kingman coalescent, where the former describes the stochasticevolution of neutral allele frequencies in a large population forwards in time,and the latter describes the genetic ancestry of randomly sampled individualsfrom the population backwards in time. As well as providing a richerdescription than either model in isolation, duality often yields equationssatisfied by quantities of interest. We employ the so-called Bernoulli factory- a celebrated tool in simulation-based computing - to derive duality relationsfor broad classes of genetics models. As concrete examples, we presentWright-Fisher diffusions with general drift functions, and Allen-Cahn equationswith general, nonlinear forcing terms. The drift and forcing functions can beinterpreted as the action of frequency-dependent selection. To our knowledge,this work is the first time a connection has been drawn between Bernoullifactories and duality in models of population genetics.

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