Fast MCMC sampling for Markov jump processes and extensions

TL;DR

This paper introduces an exact, efficient MCMC sampling method for Markov jump processes using uniformization, improving computational performance for complex models like Bayesian networks.

Contribution

It presents a novel auxiliary variable Gibbs sampler based on uniformization that extends to various MJP-based models, offering significant computational advantages.

Findings

Exact sampling without time-discretization

Significant computational improvements over existing methods

Extension to complex models like Bayesian networks

Abstract

Markov jump processes (or continuous-time Markov chains) are a simple and important class of continuous-time dynamical systems. In this paper, we tackle the problem of simulating from the posterior distribution over paths in these models, given partial and noisy observations. Our approach is an auxiliary variable Gibbs sampler, and is based on the idea of uniformization. This sets up a Markov chain over paths by alternately sampling a finite set of virtual jump times given the current path and then sampling a new path given the set of extant and virtual jump times using a standard hidden Markov model forward filtering-backward sampling algorithm. Our method is exact and does not involve approximations like time-discretization. We demonstrate how our sampler extends naturally to MJP-based models like Markov-modulated Poisson processes and continuous-time Bayesian networks and show significant computational benefits over state-of-the-art MCMC samplers for these models.