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Griffiths, Robert, Jenkins, Paul and Spanò, Dario (2018) Wright-Fisher diffusion bridges. Theoretical Population Biology, 122 . pp. 67-77. doi:10.1016/j.tpb.2017.09.005
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WRAP-Wright-Fisher-diffusion-bridges-Jenkins-2017.pdf - Accepted Version - Requires a PDF viewer. Available under License Creative Commons Attribution Non-commercial No Derivatives 4.0. Download (741Kb) | Preview |
Official URL: https://doi.org/10.1016/j.tpb.2017.09.005
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 $t\in (0,T)$. 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 $t\in (0,T)$. 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 P\'olya 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.
This paper is dedicated to the memory of Paul Joyce.
| Item Type: | Journal Article | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| Subjects: | Q Science > QH Natural history | ||||||||
| Divisions: | Faculty of Science > Statistics | ||||||||
| Library of Congress Subject Headings (LCSH): | Population biology -- Mathematical models | ||||||||
| Journal or Publication Title: | Theoretical Population Biology | ||||||||
| Publisher: | Academic Press | ||||||||
| ISSN: | 0040-5809 | ||||||||
| Official Date: | July 2018 | ||||||||
| Dates: |
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| Volume: | 122 | ||||||||
| Page Range: | pp. 67-77 | ||||||||
| DOI: | 10.1016/j.tpb.2017.09.005 | ||||||||
| Status: | Peer Reviewed | ||||||||
| Publication Status: | Published | ||||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||||
| Open Access Version: |
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