  Bayesian non-parametric methods based on Dirichlet process mixtures have seen
tremendous success in various domains and are appealing in being able to borrow
information by clustering samples that share identical parameters. However,
such methods can face hurdles in heterogeneous settings where objects are
expected to cluster only along a subset of axes or where clusters of samples
share only a subset of identical parameters. We overcome such limitations by
developing a novel class of product of Dirichlet process location-scale
mixtures that enable independent clustering at multiple scales, which result in
varying levels of information sharing across samples. First, we develop the
approach for independent multivariate data. Subsequently we generalize it to
multivariate time-series data under the framework of multi-subject Vector
Autoregressive (VAR) models that is our primary focus, which go beyond
parametric single-subject VAR models. We establish posterior consistency and
develop efficient posterior computation for implementation. Extensive numerical
studies involving VAR models show distinct advantages over competing methods,
in terms of estimation, clustering, and feature selection accuracy. Our resting
state fMRI analysis from the Human Connectome Project reveals biologically
interpretable connectivity differences between distinct intelligence groups,
while another air pollution application illustrates the superior forecasting
accuracy compared to alternate methods.
