Exact simulation of the Wright-Fisher diffusion

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.

36 pages, 2 figure, 2 tables. This version corrects minor errors in the statements of Propositions 6 and 7

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• Frequent asymmetric migrations suppress natural selection in spatially structured populations
• Mutant fate in spatially structured populations on graphs: connecting models to experiments
• Smoothing distributions for conditional Fleming-Viot and Dawson-Watanabe diffusions
• Two level natural selection with a quasi-stationarity approach
• EWF : simulating exact paths of the Wright--Fisher diffusion
• Approximate filtering via discrete dual processes
• Simulating bridges using confluent diffusions
• An R package for nonparametric inference on dynamic populations with infinitely many types
• Efficient Bernoulli factory MCMC for intractable posteriors
• Sampling probabilities, diffusions, ancestral graphs, and duality under strong selection
• Discrete mixture representations of spherical distributions
• The Rescaled Polya Urn and the Wright-Fisher process with mutation
• An algorithm for simulating Brownian increments on a sphere
• Diffusion Limits at Small Times for Coalescent Processes with Mutation and Selection