Wright–Fisher diffusion bridges

Abstract

The trajectory of the frequency of an allele which begins at x at time 0 and is known to have frequency z at time T can be modelled by the bridge process of the Wright–Fisher diffusion. Bridges when x=z=0 are particularly interesting because they model the trajectory of the frequency of an allele which appears at a time, then is lost by random drift or mutation after a time T. The coalescent genealogy back in time of a population in a neutral Wright–Fisher diffusion process is well understood. In this paper we obtain a new interpretation of the coalescent genealogy of the population in a bridge from a time t∈(0,T). In a bridge with allele frequencies of 0 at times 0 and T the coalescence structure is that the population coalesces in two directions from t to 0 and t to T such that there is just one lineage of the allele under consideration at times 0 and T. The genealogy in Wright–Fisher diffusion bridges with selection is more complex than in the neutral model, but still with the property of the population branching and coalescing in two directions from time t∈(0,T). The density of the frequency of an allele at time t is expressed in a way that shows coalescence in the two directions. A new algorithm for exact simulation of a neutral Wright–Fisher bridge is derived. This follows from knowing the density of the frequency in a bridge and exact simulation from the Wright–Fisher diffusion. The genealogy of the neutral Wright–Fisher bridge is also modelled by branching Pólya urns, extending a representation in a Wright–Fisher diffusion. This is a new very interesting representation that relates Wright–Fisher bridges to classical urn models in a Bayesian setting.