Simulation-Free Variational Inference for Latent Continuous-Time Markov Chains

Latent continuous-time Markov chains (CTMCs) are a widely-used model class in systems biology, epidemiology, and spatial statistics, but learning them from noisy observations requires inference over unobserved continuous-time dynamics and can become prohibitively expensive in large state spaces. We introduce CTMC Matching, a simulation-free variational inference method for latent CTMCs that directly parameterizes the posterior's time marginals. Given a marginal curve, the KL-optimal path measure realizing those marginals has an explicit structure: its transition rates are an exponential tilt of the prior rates by a potential that solves a concave dual problem. We prove that this projection is exact: at the exact smoothing marginals it recovers the smoothing posterior. We develop efficient algorithms that couple this inner optimization with end-to-end learning of both the variational posterior, amortized with a neural network, and the model parameters. For structured CTMCs such as interacting particle systems and chemical reaction networks, we exploit model structure to obtain scalable implementations without enumerating the full joint state space. Across a range of systems, CTMC Matching achieves competitive inference and parameter-learning performance while avoiding latent-trajectory simulation during optimization.