Credible rectangles for high-dimensional posterior comparison

Abstract

We propose a Bayesian framework for uncertainty quantification and comparison in brain connectivity graph analysis. Standard graph-based approaches typically rely on point estimates of correlation matrices, overlooking the uncertainty induced by high-dimensional estimation from limited data. Our methodology constructs and compares credible hyperrectangles derived from posterior distributions, providing interpretable tools for subject-level inference and longitudinal monitoring. We develop scalable algorithms for estimating these regions in high dimensions and establish theoretical guarantees in the inverse-Wishart model for resting-state fMRI data, including a Bernstein--von Mises theorem for correlation matrices and control of a Bayesian family-wise error rate. The proposed framework enables principled detection of significant connectivity differences both globally and locally while preserving joint dependency structures. While demonstrating competitive performance against multiple-testing procedures on synthetic datasets, our approach also facilitates the direct comparison of two distinct scans from a single patient, a capability currently absent from the literature. We leverage this novelty on real datasets to improve interpretability. Beyond fMRI data, the approach provides a general framework for comparison problems in high-dimensional dependent settings.