  Inferring concerted changes among biological traits along an evolutionary
history remains an important yet challenging problem. Besides adjusting for
spurious correlation induced from the shared history, the task also requires
sufficient flexibility and computational efficiency to incorporate multiple
continuous and discrete traits as data size increases. To accomplish this, we
jointly model mixed-type traits by assuming latent parameters for binary
outcome dimensions at the tips of an unknown tree informed by molecular
sequences. This gives rise to a phylogenetic multivariate probit model. With
large sample sizes, posterior computation under this model is problematic, as
it requires repeated sampling from a high-dimensional truncated normal
distribution. Current best practices employ multiple-try rejection sampling
that suffers from slow-mixing and a computational cost that scales
quadratically in sample size. We develop a new inference approach that exploits
1) the bouncy particle sampler (BPS) based on piecewise deterministic Markov
processes to simultaneously sample all truncated normal dimensions, and 2)
novel dynamic programming that reduces the cost of likelihood and gradient
evaluations for BPS to linear in sample size. In an application with 535 HIV
viruses and 24 traits that necessitates sampling from a 12,840-dimensional
truncated normal, our method makes it possible to estimate the across-trait
correlation and detect factors that affect the pathogen's capacity to cause
disease. This inference framework is also applicable to a broader class of
covariance structures beyond comparative biology.
