Reinforced random processes in continuous time

We introduce a stochastic process based on nonhomogeneous Poisson processes and urn processes which can be reinforced to produce a mixture of semi-Markov processes. By working with the notion of exchangeable blocks within the process, we present a Bayesian nonparametric framework for handling data which arises in the form of a semi-Markov process. That is, if units provide information as a semi-Markov process and units are regarded as being exchangeable then we show how to construct the sequence of predictive distributions without explicit reference to the de Finetti measure, or prior.