  We propose a sparse vector autoregressive (VAR) hidden semi-Markov model
(HSMM) for modeling temporal and contemporaneous (e.g. spatial) dependencies in
multivariate nonstationary time series. The HSMM's generic state distribution
is embedded in a special transition matrix structure, facilitating efficient
likelihood evaluations and arbitrary approximation accuracy. To promote
sparsity of the VAR coefficients, we deploy an $l_1$-ball projection prior,
which combines differentiability with a positive probability of obtaining exact
zeros, achieving variable selection within each switching state. This also
facilitates posterior estimation via Hamiltonian Monte Carlo (HMC). We further
place non-local priors on the parameters of the HSMM dwell distribution
improving the ability of Bayesian model selection to distinguish whether the
data is better supported by the simpler hidden Markov model (HMM), or the more
flexible HSMM. Our proposed methodology is illustrated via an application to
human gesture phase segmentation based on sensor data, where we successfully
identify and characterize the periods of rest and active gesturing, as well as
the dynamical patterns involved in the gesture movements associated with each
of these states.
