  We consider the modeling of data generated by a latent continuous-time Markov
jump process with a state space of finite but unknown dimensions. Typically in
such models, the number of states has to be pre-specified, and Bayesian
inference for a fixed number of states has not been studied until recently. In
addition, although approaches to address the problem for discrete-time models
have been developed, no method has been successfully implemented for the
continuous-time case. We focus on reversible jump Markov chain Monte Carlo
which allows the trans-dimensional move among different numbers of states in
order to perform Bayesian inference for the unknown number of states.
Specifically, we propose an efficient split-combine move which can facilitate
the exploration of the parameter space, and demonstrate that it can be
implemented effectively at scale. Subsequently, we extend this algorithm to the
context of model-based clustering, allowing numbers of states and clusters both
determined during the analysis. The model formulation, inference methodology,
and associated algorithm are illustrated by simulation studies. Finally, We
apply this method to real data from a Canadian healthcare system in Quebec.
