Scalable Bayesian divergence time estimation with ratio transformations

Divergence time estimation is crucial to provide temporal signals for
dating biologically important events, from species divergence to viral
transmissions in space and time. With the advent of high-throughput
sequencing, recent Bayesian phylogenetic studies have analyzed hundreds to
thousands of sequences. Such large-scale analyses challenge
divergence time reconstruction by requiring inference on
highly-correlated internal node heights that often become
computationally infeasible. To overcome this limitation, we
explore a ratio transformation that maps the original N - 1
internal node heights into a space of one height parameter and N
- 2 ratio parameters. To make the analyses scalable, we develop a
collection of linear-time algorithms to compute the gradient and
Jacobian-associated terms of the log-likelihood with respect to these
ratios. We then apply Hamiltonian Monte Carlo sampling with the ratio
transform in a Bayesian framework to learn the divergence times in four
pathogenic viruses (West Nile virus, rabies virus, Lassa virus
and Ebola virus) and the coralline red algae.  Our
method both resolves a mixing issue in the West Nile virus example and
improves  inference efficiency by at least 5-fold for
the Lassa and rabies virus examples as well  as for the
algae example. Our method now also makes it computationally feasible
to  incorporate mixed-effects molecular clock models for
the Ebola virus example, confirms  the findings from the
original study and reveals clearer multimodal distributions of
the  divergence times of some clades of interest.