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Inferring dependencies between complex biological traits while accounting forevolutionary relationships between specimens is of great scientific interestyet remains infeasible when trait and specimen counts grow large. Thestate-of-the-art approach uses a phylogenetic multivariate probit model toaccommodate binary and continuous traits via a latent variable framework, andutilizes an efficient bouncy particle sampler (BPS) to tackle the computationalbottleneck -- integrating many latent variables from a high-dimensionaltruncated normal distribution. This approach breaks down as the number ofspecimens grows and fails to reliably characterize conditional dependenciesbetween traits. Here, we propose an inference pipeline for phylogenetic probitmodels that greatly outperforms BPS. The novelty lies in 1) a combination ofthe recent Zigzag Hamiltonian Monte Carlo (Zigzag-HMC) with linear-timegradient evaluations and 2) a joint sampling scheme for highly correlatedlatent variables and correlation matrix elements. In an application exploringHIV-1 evolution from 535 viruses, the inference requires joint sampling from an11,235-dimensional truncated normal and a 24-dimensional covariance matrix. Ourmethod yields a 5-fold speedup compared to BPS and makes it possible to learnpartial correlations between candidate viral mutations and virulence.Computational speedup now enables us to tackle even larger problems: we studythe evolution of influenza H1N1 glycosylations on around 900 viruses. Forbroader applicability, we extend the phylogenetic probit model to incorporatecategorical traits, and demonstrate its use to study Aquilegia flower andpollinator co-evolution.

Accelerating Bayesian inference of dependency between complex biological traits