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Phylogenetic comparative methods correct for shared evolutionary historyamong a set of non-independent organisms by modeling sample traits as arisingfrom a diffusion process along on the branches of a possibly unknown history.To incorporate such uncertainty, we present a scalable Bayesian inferenceframework under a general Gaussian trait evolution model that exploitsHamiltonian Monte Carlo (HMC). HMC enables efficient sampling of theconstrained model parameters and takes advantage of the tree structure for fastlikelihood and gradient computations, yielding algorithmic complexity linear inthe number of observations. This approach encompasses a wide family ofstochastic processes, including the general Ornstein-Uhlenbeck (OU) process,with possible missing data and measurement errors. We implement inference toolsfor a biologically relevant subset of all these models into the BEASTphylogenetic software package and develop model comparison through marginallikelihood estimation. We apply our approach to study the morphologicalevolution in the superfamilly of Musteloidea (including weasels and allies) aswell as the heritability of HIV virulence. This second problem furnishes a newmeasure of evolutionary heritability that demonstrates its utility through atargeted simulation study.

Efficient Bayesian Inference of General Gaussian Models on Large Phylogenetic Trees