Looking down in the ancestral selection graph: A probabilistic approach to the common ancestor type distribution

TL;DR

This paper introduces a novel probabilistic construction combining ancestral selection graph and lookdown methods to derive and interpret the series representation of the probability that a beneficial type is the ancestor in a Wright-Fisher model with selection and mutation.

Contribution

It develops a new probabilistic construction that unifies ancestral selection graph and lookdown approaches, enabling transparent derivation and interpretation of the series coefficients for ancestor type probability.

Findings

Provides a probabilistic interpretation of series coefficients.

Enables transparent derivation of ancestor type probabilities.

Integrates ancestral selection graph with lookdown construction.

Abstract

In a (two-type) Wright-Fisher diffusion with directional selection and two-way mutation, let $x$ denote today's frequency of the beneficial type, and given $x$, let $h(x)$ be the probability that, among all individuals of today's population, the individual whose progeny will eventually take over in the population is of the beneficial type. Fearnhead [Fearnhead, P., 2002. The common ancestor at a nonneutral locus. J. Appl. Probab. 39, 38-54] and Taylor [Taylor, J. E., 2007. The common ancestor process for a Wright-Fisher diffusion. Electron. J. Probab. 12, 808-847] obtained a series representation for $h(x)$. We develop a construction that contains elements of both the ancestral selection graph and the lookdown construction and includes pruning of certain lines upon mutation. Besides being interesting in its own right, this construction allows a transparent derivation of the series coefficients of $h(x)$ and gives them a probabilistic meaning.