The stationary distribution of a sample from the Wright-Fisher diffusion model with general small mutation rates

TL;DR

This paper derives an approximate stationary distribution for samples from the Wright-Fisher diffusion model with small mutation rates using a coalescent approach, providing insights into allele configurations.

Contribution

It introduces a novel coalescent-based approximation for the stationary distribution under small mutation rates, differing from previous flux-based methods.

Findings

Approximate stationary distribution characterized for small mutation rates

Explicit solution for pure birth process trees with rate lambda

Connection established with allele configurations in exchangeable coalescent trees

Abstract

The stationary distribution of a sample taken from a Wright-Fisher diffusion with general small mutation rates is found using a coalescent approach. The approximation is equivalent to having at most one mutation in the coalescent tree to the first order in the rates. The sample probabilities characterize an approximation for the stationary distribution from the Wright-Fisher diffusion. The approach is different from Burden and Tang (2016,2017) who use a probability flux argument to obtain the same results from a forward diffusion generator equation. The solution has interest because the solution is not known when rates are not small. An analogous solution is found for the configuration of alleles in a general exchangeable binary coalescent tree. In particular an explicit solution is found for a pure birth process tree when individuals reproduce at rate lambda.