  This paper presents a Bayesian reformulation of covariate-assisted principal
(CAP) regression of Zhao et al. (2021), which aims to identify components in
the covariance of response signal that are associated with covariates in a
regression framework. We introduce a geometric formulation and
reparameterization of individual covariance matrices in their tangent space. By
mapping the covariance matrices to the tangent space, we leverage Euclidean
geometry to perform posterior inference. This approach enables joint estimation
of all parameters and uncertainty quantification within a unified framework,
fusing dimension reduction for covariance matrices with regression model
estimation. We validate the proposed method through simulation studies and
apply it to analyze associations between covariates and brain functional
connectivity, utilizing data from the Human Connectome Project.
