Nonparametric Modeling of Continuous-Time Markov Chains

Inferring the infinitesimal rates of continuous-time Markov chains (CTMCs) is a central challenge in many scientific domains. This task is hindered by three factors: quadratic growth in the number of rates as the CTMC state space expands, strong dependencies among rates, and incomplete information for many transitions. We introduce a new Bayesian framework that flexibly models the CTMC rates by incorporating covariates through Gaussian processes (GPs). This approach improves inference by integrating new information and contributes to the understanding of the CTMC stochastic behavior by shedding light on potential external drivers. Unlike previous approaches limited to linear covariate effects, our method captures complex non-linear relationships, enabling fuller use of covariate information and more accurate characterization of their influence. To perform efficient inference, we employ a scalable Hamiltonian Monte Carlo (HMC) sampler. We address the prohibitive cost of computing the exact likelihood gradient by integrating the HMC trajectories with a scalable gradient approximation, reducing the computational complexity from $O(K^5)$ to $O(K^2)$, where $K$ is the number of CTMC states. Finally, we demonstrate our method on Bayesian phylogeography inference -- a domain where CTMCs are central -- showing effectiveness on both synthetic and real datasets.