  Inferring dependencies between complex biological traits while accounting for
evolutionary relationships between specimens is of great scientific interest
yet remains infeasible when trait and specimen counts grow large. The
state-of-the-art approach uses a phylogenetic multivariate probit model to
accommodate binary and continuous traits via a latent variable framework, and
utilizes an efficient bouncy particle sampler (BPS) to tackle the computational
bottleneck -- integrating many latent variables from a high-dimensional
truncated normal distribution. This approach breaks down as the number of
specimens grows and fails to reliably characterize conditional dependencies
between traits. Here, we propose an inference pipeline for phylogenetic probit
models that greatly outperforms BPS. The novelty lies in 1) a combination of
the recent Zigzag Hamiltonian Monte Carlo (Zigzag-HMC) with linear-time
gradient evaluations and 2) a joint sampling scheme for highly correlated
latent variables and correlation matrix elements. In an application exploring
HIV-1 evolution from 535 viruses, the inference requires joint sampling from an
11,235-dimensional truncated normal and a 24-dimensional covariance matrix. Our
method yields a 5-fold speedup compared to BPS and makes it possible to learn
partial correlations between candidate viral mutations and virulence.
Computational speedup now enables us to tackle even larger problems: we study
the evolution of influenza H1N1 glycosylations on around 900 viruses. For
broader applicability, we extend the phylogenetic probit model to incorporate
categorical traits, and demonstrate its use to study Aquilegia flower and
pollinator co-evolution.
