  Understanding the dynamics of functional brain connectivity patterns using
noninvasive neuroimaging techniques is an important focus in human
neuroscience. Vector autoregressive (VAR) processes and Granger causality
analysis (GCA) have been extensively used for this purpose. While
high-resolution multi-subject neuroimaging data are routinely collected
now-a-days, the statistics literature on VAR models has remained heavily
focused on small-to-moderate dimensional problems and single-subject data.
Motivated by these issues, we develop a novel Bayesian random effects panel VAR
model for multi-subject high-dimensional neuroimaging data. We begin with a
single-subject model that structures the VAR coefficients as a three-way
tensor, then reduces the dimensions by applying a Tucker tensor decomposition.
A novel sparsity-inducing shrinkage prior allows data-adaptive rank and lag
selection. We then extend the approach to a novel random effects model for
multi-subject data that carefully avoids the dimensions getting exploded with
the number of subjects but also flexibly accommodates subject-specific
heterogeneity. We design a Markov chain Monte Carlo algorithm for posterior
computation. Finally, GCA with posterior false discovery control is performed
on the posterior samples. The method shows excellent empirical performance in
simulation experiments. Applied to our motivating functional magnetic resonance
imaging study, the approach allows the directional connectivity of human brain
networks to be studied in fine detail, revealing meaningful but previously
unsubstantiated cortical connectivity patterns.
