  Kingman's coalescent is one of the most popular models in population
genetics. It describes the genealogy of a population whose genetic composition
evolves in time according to the Wright-Fisher model, or suitable
approximations of it belonging to the broad class of Fleming-Viot processes.
Ancestral inference under Kingman's coalescent has had much attention in the
literature, both in practical data analysis, and from a theoretical and
methodological point of view. Given a sample of individuals taken from the
population at time $t>0$, most contributions have aimed at making frequentist
or Bayesian parametric inference on quantities related to the genealogy of the
sample. In this paper we propose a Bayesian nonparametric predictive approach
to ancestral inference. That is, under the prior assumption that the
composition of the population evolves in time according to a neutral
Fleming-Viot process, and given the information contained in an initial sample
of $m$ individuals taken from the population at time $t>0$, we estimate
quantities related to the genealogy of an additional unobservable sample of
size $m^{\prime}\geq1$. As a by-product of our analysis we introduce a class of
Bayesian nonparametric estimators (predictors) which can be thought of as
Good-Turing type estimators for ancestral inference. The proposed approach is
illustrated through an application to genetic data.
