A procedure for Bayes nonparametric estimation from a Markov renewal process is developed. It is based on a conjugate class of a priori distributions on the parameter space of semi-Markov transition distributions. The class is characterized by a Dirichlet family of distributions for random Markov matrices and a Beta family of Levy processes for random cumulative hazard functions. The main result is the derivation of the posterior law from an observation of the Markov renewal process over a period of time.
"Bayes Estimation from a Markov Renewal Process." Ann. Statist. 18 (2) 603 - 616, June, 1990. https://doi.org/10.1214/aos/1176347618