  The observed sequence variation at a locus informs about the evolutionary
history of the sample and past population size dynamics. The Kingman coalescent
is used in a generative model of molecular sequence variation to infer
evolutionary parameters. However, it is well understood that inference under
this model does not scale well with sample size. Here, we build on recent work
based on a lower resolution coalescent process, the Tajima coalescent, to model
longitudinal samples. While the Kingman coalescent models the ancestry of
labeled individuals, the heterochronous Tajima coalescent models the ancestry
of individuals labeled by their sampling time. We propose a new inference
scheme for the reconstruction of effective population size trajectories based
on this model with the potential to improve computational efficiency. Modeling
of longitudinal samples is necessary for applications (e.g. ancient DNA and RNA
from rapidly evolving pathogens like viruses) and statistically desirable
(variance reduction and parameter identifiability). We propose an efficient
algorithm to calculate the likelihood and employ a Bayesian nonparametric
procedure to infer the population size trajectory. We provide a new MCMC
sampler to explore the space of heterochronous Tajima's genealogies and model
parameters. We compare our procedure with state-of-the-art methodologies in
simulations and applications.
