  Phylogenetic comparative methods correct for shared evolutionary history
among a set of non-independent organisms by modeling sample traits as arising
from a diffusion process along on the branches of a possibly unknown history.
To incorporate such uncertainty, we present a scalable Bayesian inference
framework under a general Gaussian trait evolution model that exploits
Hamiltonian Monte Carlo (HMC). HMC enables efficient sampling of the
constrained model parameters and takes advantage of the tree structure for fast
likelihood and gradient computations, yielding algorithmic complexity linear in
the number of observations. This approach encompasses a wide family of
stochastic processes, including the general Ornstein-Uhlenbeck (OU) process,
with possible missing data and measurement errors. We implement inference tools
for a biologically relevant subset of all these models into the BEAST
phylogenetic software package and develop model comparison through marginal
likelihood estimation. We apply our approach to study the morphological
evolution in the superfamilly of Musteloidea (including weasels and allies) as
well as the heritability of HIV virulence. This second problem furnishes a new
measure of evolutionary heritability that demonstrates its utility through a
targeted simulation study.
