Methods for exact Bayesian inference under a uniform diffuse prior are set forth for the continuous time homogeneous Markov chain model. It is shown how the exact posterior distribution of any function of interest may be computed using Monte Carlo integration. The solution handles the problems of embeddability in a very natural way, and provides (to our knowledge) the only solution that systematically takes this problem into account. The methods are illustrated using several sets of data.
Exact Inference for Continuous Time Markov Chain Models
Geweke, J., Marshall, R. C., & Zarkin, G. A. (1986). Exact Inference for Continuous Time Markov Chain Models. Review of Economic Studies, 53(4), 653-669. https://doi.org/10.2307/2297610