MCMC implementation for Bayesian hidden semi-Markov models with illustrative applications

Hidden Markov models (HMMs) are flexible, well-established models useful in a diverse range of applications. However, one potential limitation of such models lies in their inability to explicitly structure the holding times of each hidden state. Hidden semi-Markov models (HSMMs) are more useful in the latter respect as they incorporate additional temporal structure by explicit modelling of the holding times. However, HSMMs have generally received less attention in the literature, mainly due to their intensive computational requirements. Here a Bayesian implementation of HSMMs is presented. Recursive algorithms are proposed in conjunction with Metropolis-Hastings in such a way as to avoid sampling from the distribution of the hidden state sequence in the MCMC sampler. This provides a computationally tractable estimation framework for HSMMs avoiding the limitations associated with the conventional EM algorithm regarding model flexibility. Performance of the proposed implementation is demonstrated through simulation experiments as well as an illustrative application relating to recurrent failures in a network of underground water pipes where random effects are also included into the HSMM to allow for pipe heterogeneity.