  Local clock models propose that the rate of molecular evolution is constant
within phylogenetic sub-trees. Current local clock inference procedures scale
poorly to large taxa problems, impose model misspecification, or require a
priori knowledge of the existence and location of clocks. To overcome these
challenges, we present an autocorrelated, Bayesian model of heritable clock
rate evolution that leverages heavy-tailed priors with mean zero to shrink
increments of change between branch-specific clocks. We further develop an
efficient Hamiltonian Monte Carlo sampler that exploits closed form gradient
computations to scale our model to large trees. Inference under our
shrinkage-clock exhibits an over 3-fold speed increase compared to the popular
random local clock when estimating branch-specific clock rates on a simulated
dataset. We further show our shrinkage-clock recovers known local clocks within
a rodent and mammalian phylogeny. Finally, in a problem that once appeared
computationally impractical, we investigate the heritable clock structure of
various surface glycoproteins of influenza A virus in the absence of prior
knowledge about clock placement.
