  In this paper, we adopt a nonparametric Bayesian approach and investigate the
asymptotic behavior of the posterior distribution in continuous time and
general state space semi-Markov processes. In particular, we obtain posterior
concentration rates for semi-Markov kernels. For the purposes of this study, we
construct robust statistical tests between Hellinger balls around semi-Markov
kernels and present some specifications to particular cases, including
discrete-time semi-Markov processes and finite state space Markov processes.
The objective of this paper is to provide sufficient conditions on priors and
semi-Markov kernels that enable us to establish posterior concentration rates.
