An Efficient Bayesian Phylogenetic Approach for Joint Inference of Continuous and Discrete Trait Evolution under a State‐dependent Ornstein–Uhlenbeck Model

Abstract Macroevolutionary adaptation of a continuous trait to different discrete states across species can be modelled using a phylogenetic state‐dependent Ornstein–Uhlenbeck (OU) process. Existing inference methods face two challenges. First, although efficient likelihood algorithms for continuous trait evolution models exist, none have been specifically described for a state‐dependent OU model. Second, the commonly adopted sequential inference approach, where the state‐dependent OU model parameters are inferred conditionally on a discrete character history, treats the character history as known and does not allow the continuous trait to inform character history estimates. We present two mathematically equivalent approaches to compute the likelihood under a state‐dependent OU process: a variance–covariance approach and a pruning approach. We demonstrate the computational efficiency of our pruning algorithm for likelihood calculation of a state‐dependent OU model implemented in RevBayes. Coupled with a data augmentation approach to sample character histories of a discrete character, our model can jointly infer continuous and discrete trait evolution using Markov chain Monte Carlo. We validate our derivation and implementation through simulation tests. Using a case study of tooth crown evolution in ruminants, we compare joint and sequential inference approaches. We show that the root state estimates and some OU parameter estimates are qualitatively different between joint and sequential approaches. This indicates that the continuous trait can be informative for estimating the character history, which in turn impacts the OU parameter estimates. Our state‐dependent OU process and pruning algorithm enable studies using large phylogenies. Together with data augmentation, our joint inference approach provides a more robust and flexible method of macroevolutionary adaptation of continuous traits to underlying discrete states.