Integration within the Felsenstein equation for improved Markov chain Monte Carlo methods in population genetics

Abstract

In 1988, Felsenstein described a framework for assessing the likelihood of a genetic data set in which all of the possible genealogical histories of the data are considered, each in proportion to their probability. Although not analytically solvable, several approaches, including Markov chain Monte Carlo methods, have been developed to find approximate solutions. Here, we describe an approach in which Markov chain Monte Carlo simulations are used to integrate over the space of genealogies, whereas other parameters are integrated out analytically. The result is an approximation to the full joint posterior density of the model parameters. For many purposes, this function can be treated as a likelihood, thereby permitting likelihood-based analyses, including likelihood ratio tests of nested models. Several examples, including an application to the divergence of chimpanzee subspecies, are provided.speciation ͉ population structure ͉ divergence P opulation genetic and phylogenetic models that take a genealogical (i.e., gene tree) approach suffer two nested levels of ambiguity. First, the uncertainty of an estimate of a genealogy can be large and difficult to quantify, and second, it can be difficult to interpret a genealogy estimate explicitly in terms of an evolutionary or population genetics model. In his 1988 review, Felsenstein (1) conceptualized a way thru these uncertainties by positioning the genealogy as a nuisance variable in the definition of the likelihood of the parameters given the data (proportional to the sampling probability of the data):

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