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Computation 2014, 2(4), 199-220; doi:10.3390/computation2040199

Computation of the Likelihood in Biallelic Diffusion Models Using Orthogonal Polynomials

Institute of Animal Breeding and Genetics, University of Veterinary Medicine, Vienna, Veterinarplatz 1, 1210 Vienna, Austria
Received: 14 July 2014 / Revised: 12 October 2014 / Accepted: 16 October 2014 / Published: 14 November 2014
(This article belongs to the Special Issue Genomes and Evolution: Computational Approaches)
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Abstract

In population genetics, parameters describing forces such as mutation, migration and drift are generally inferred from molecular data. Lately, approximate methods based on simulations and summary statistics have been widely applied for such inference, even though these methods waste information. In contrast, probabilistic methods of inference can be shown to be optimal, if their assumptions are met. In genomic regions where recombination rates are high relative to mutation rates, polymorphic nucleotide sites can be assumed to evolve independently from each other. The distribution of allele frequencies at a large number of such sites has been called “allele-frequency spectrum” or “site-frequency spectrum” (SFS). Conditional on the allelic proportions, the likelihoods of such data can be modeled as binomial. A simple model representing the evolution of allelic proportions is the biallelic mutation-drift or mutation-directional selection-drift diffusion model. With series of orthogonal polynomials, specifically Jacobi and Gegenbauer polynomials, or the related spheroidal wave function, the diffusion equations can be solved efficiently. In the neutral case, the product of the binomial likelihoods with the sum of such polynomials leads to finite series of polynomials, i.e., relatively simple equations, from which the exact likelihoods can be calculated. In this article, the use of orthogonal polynomials for inferring population genetic parameters is investigated. View Full-Text
Keywords: site frequency spectrum; mutation; drift; biallelic diffusion; directional selection; orthogonal polynomials; inference site frequency spectrum; mutation; drift; biallelic diffusion; directional selection; orthogonal polynomials; inference
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Vogl, C. Computation of the Likelihood in Biallelic Diffusion Models Using Orthogonal Polynomials. Computation 2014, 2, 199-220.

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