  Analysis of structural and functional connectivity (FC) of human brains is of
pivotal importance for diagnosis of cognitive ability. The Human Connectome
Project (HCP) provides an excellent source of neural data across different
regions of interest (ROIs) of the living human brain. Individual specific data
were available from an existing analysis (Dai et al., 2017) in the form of time
varying covariance matrices representing the brain activity as the subjects
perform a specific task. As a preliminary objective of studying the
heterogeneity of brain connectomics across the population, we develop a
probabilistic model for a sample of covariance matrices using a scaled Wishart
distribution. We stress here that our data units are available in the form of
covariance matrices, and we use the Wishart distribution to create our
likelihood function rather than its more common usage as a prior on covariance
matrices. Based on empirical explorations suggesting the data matrices to have
low effective rank, we further model the center of the Wishart distribution
using an orthogonal factor model type decomposition. We encourage shrinkage
towards a low rank structure through a novel shrinkage prior and discuss
strategies to sample from the posterior distribution using a combination of
Gibbs and slice sampling. We extend our modeling framework to a dynamic setting
to detect change points. The efficacy of the approach is explored in various
simulation settings and exemplified on several case studies including our
motivating HCP data. We extend our modeling framework to a dynamic setting to
detect change points.
