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Recurrence Equations for the Probability Distribution of Sample Configurations in Exact Population Genetics Models

Part of: Combinatorial probability

Published online by Cambridge University Press:  14 July 2016

Sabin Lessard
Sabin Lessard*
Affiliation:
Université de Montréal
*
∗Postal address: Département de Mathématiques et de Statistique, Université de Montréal, Montréal, Québec H3C 3J7, Canada. Email address: lessards@dms.umontreal.ca
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Abstract

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Recurrence equations for the number of types and the frequency of each type in a random sample drawn from a finite population undergoing discrete, nonoverlapping generations and reproducing according to the Cannings exchangeable model are deduced under the assumption of a mutation scheme with infinitely many types. The case of overlapping generations in discrete time is also considered. The equations are developed for the Wright-Fisher model and the Moran model, and extended to the case of the limit coalescent with nonrecurrent mutation as the population size goes to ∞ and the mutation rate to 0. Computations of the total variation distance for the distribution of the number of types in the sample suggest that the exact Moran model provides a better approximation for the sampling formula under the exact Wright-Fisher model than the Ewens sampling formula in the limit of the Kingman coalescent with nonrecurrent mutation. On the other hand, this model seems to provide a good approximation for a Λ-coalescent with nonrecurrent mutation as long as the probability of multiple mergers and the mutation rate are small enough.

Keywords

Ewens sampling formulacoalescent theoryCannings modelWright-Fisher modelMoran modelexchangeable population genetics modelΛ-coalescent

MSC classification

Secondary: 60C05: Combinatorial probability 92D15: Problems related to evolution
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 3 , September 2010 , pp. 732 - 751
Copyright
Copyright © Applied Probability Trust 2010 

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