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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 ISSN 0040-5809.
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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 | ||||||||
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Subjects: | Q Science > QH Natural history | ||||||||
Divisions: | Faculty of Science, Engineering and Medicine > 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 | ||||||||
Date of first compliant deposit: | 17 October 2017 | ||||||||
Date of first compliant Open Access: | 6 October 2018 | ||||||||
Open Access Version: |
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