  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.
