  We propose a flexible Bayesian approach for sparse Gaussian graphical
modeling of multivariate time series. We account for temporal correlation in
the data by assuming that observations are characterized by an underlying and
unobserved hidden discrete autoregressive process. We assume multivariate
Gaussian emission distributions and capture spatial dependencies by modeling
the state-specific precision matrices via graphical horseshoe priors. We
characterize the mixing probabilities of the hidden process via a cumulative
shrinkage prior that accommodates zero-inflated parameters for non-active
components, and further incorporate a sparsity-inducing Dirichlet prior to
estimate the effective number of states from the data. For posterior inference,
we develop a sampling procedure that allows estimation of the number of
discrete autoregressive lags and the number of states, and that cleverly avoids
having to deal with the changing dimensions of the parameter space. We
thoroughly investigate performance of our proposed methodology through several
simulation studies. We further illustrate the use of our approach for the
estimation of dynamic brain connectivity based on fMRI data collected on a
subject performing a task-based experiment on latent learning
