Geometric ergodicity of Rao and Teh’s algorithm for homogeneous Markov jump processes

Rao and Teh (2013) introduced an efficient MCMC algorithm for sampling from the posterior distribution of a hidden Markov jump process. The algorithm is based on the idea of sampling virtual jumps. In the present paper we show that the Markov chain generated by Rao and Teh’s algorithm is geometrically ergodic. To this end we establish a geometric drift condition towards a small set. We work under the assumption that the parameters of the hidden process are known and the goal is to restore its trajectory.