Efficient parameter sampling for Markov jump processes

Markov jump processes (MJPs) are continuous-time stochastic processes widely used in a variety of applied disciplines. Inference for MJPs typically proceeds via Markov chain Monte Carlo, the state-of-the-art being a uniformization-based auxiliary variable Gibbs sampler. This was designed for situations where the MJP parameters are known, and Bayesian inference over unknown parameters is typically carried out by incorporating it into a larger Gibbs sampler. This strategy of sampling parameters given path, and path given parameters can result in poor Markov chain mixing. In this work, we propose a simple and elegant algorithm to address this problem. Our scheme brings Metropolis-Hastings approaches for discrete-time hidden Markov models to the continuous-time setting, resulting in a complete and clean recipe for parameter and path inference in MJPs. In our experiments, we demonstrate superior performance over Gibbs sampling, as well as another popular approach, particle MCMC. We also show our sampler inherits geometric mixing from an `ideal' sampler that operates without computational constraints.