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

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