  We propose a Bayesian hidden Markov model for analyzing time series and
sequential data where a special structure of the transition probability matrix
is embedded to model explicit-duration semi-Markovian dynamics. Our formulation
allows for the development of highly flexible and interpretable models that can
integrate available prior information on state durations while keeping a
moderate computational cost to perform efficient posterior inference. We show
the benefits of choosing a Bayesian approach for HSMM estimation over its
frequentist counterpart, in terms of model selection and out-of-sample
forecasting, also highlighting the computational feasibility of our inference
procedure whilst incurring negligible statistical error. The use of our
methodology is illustrated in an application relevant to e-Health, where we
investigate rest-activity rhythms using telemetric activity data collected via
a wearable sensing device. This analysis considers for the first time Bayesian
model selection for the form of the explicit state dwell distribution. We
further investigate the inclusion of a circadian covariate into the emission
density and estimate this in a data-driven manner.
