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Inference of population genetic parameters with a biallelic mutation drift model using the coalescent, diffusion with orthogonal polynomials, and the Moran model

In population genetics, extant samples are usually used for inference of past population genetic forces. With the Kingman coalescent and the backward diffusion equation, inference of the marginal likelihood proceeds from an extant sample backward in time. Conditional on an extant sample, the Moran model can also be used backward in time with identical results, up to a scaling of time. In particular, all three approaches -- the coalescent, the backward diffusion, and the Moran model -- lead to the identical marginal likelihood of the sample. If probabilities of ancestral states are also inferred, either of discrete ancestral allele particle configurations, as in the coalescent, or of ancestral population allele proportions, as in the backward diffusion, the backward algorithm needs to be combined with the corresponding forward algorithm to the forward-backward algorithm. Generally orthogonal polynomials, solving the diffusion equation, are numerically simpler than the other approaches: they implicitly sum over many intermediate ancestral particle configurations; furthermore, while the Moran model requires iterative matrix multiplication with a transition matrix of a dimension of the population size squared, expansion of the polynomials is only necessary up to the sample size. For discrete samples, forward-in-time moving pure birth processes similar to the Polya- or Hoppe-urn models complement the backward-looking coalescent. Because, the sample size is a random variable forward in time, pure-birth processes are unsuited to model population demography given extant samples. With orthogonal polynomials, however, not only ancestral allele proportions but also probabilities of ancestral particle configurations can be calculated easily. Assuming only mutation and drift, the use of orthogonal polynomials is numerically advantageous over alternative strategies.