An information-borrowing Bayesian method for inferring high-dimensional population-level brain connectivity from small samples

Estimating population-level dependence between brain regions is challenging when the number of regions greatly exceeds the number of individuals. We introduce an information-borrowing generalized Bayesian method for high-dimensional covariance estimation from region-level brain measurements. A divide-and-conquer parameterization separates marginal variances from correlation calibrations: an empirical \emph{pilot} correlation matrix--by default, obtained by standardizing a sample covariance matrix (SCM)--provides a reference structure that is adaptively calibrated through region-specific parameters shared across pairwise correlations, reducing the number of free parameters from $\mathcal{O}(d^2)$ to $\mathcal{O}(d)$ ($d$ is the number of regions) while guaranteeing a positive-definite covariance through the inclusion of a ridge term. Horseshoe priors on the calibration parameters enable selective borrowing and shrinkage of pilot information. Under suitable conditions on pilot accuracy, we establish theoretical guarantees for covariance estimation while retaining scalability in ultra-high-dimensional settings via a low-rank likelihood representation and a Woodbury-accelerated Hamiltonian Monte Carlo algorithm. Simulations showed lower estimation error than the SCM, bagging and iteratively refined principal orthogonal complement thresholding (POETRY). Applications to a semi-synthetic cortical-thickness and ADHD resting-state fMRI data revealed significant group differences in the default-mode--visual network and somatomotor network--limbic network blocks after Benjamini--Hochberg correction.