  Rao and Teh (2012, 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. A similar result is also proved for a special version of the algorithm,
used for probabilistic inference in Continuous Time Bayesian Networks.
