Credible rectangles for high-dimensional posterior comparison

2026-05-28

AI Summary

High-dimensional brain connectome analyses often neglect estimation uncertainty, impeding reliable individualized inference and longitudinal comparisons. This work proposes the first framework enabling direct Bayesian comparison of two brain scans from a single patient. By leveraging an inverse-Wishart prior, the method constructs posterior credible hyperrectangles for high-dimensional correlation matrices, integrating scalable algorithms with Bernstein–von Mises theory to rigorously quantify uncertainty while controlling the Bayesian family-wise error rate. Evaluated on synthetic data, the approach outperforms conventional multiple testing procedures, and when applied to real fMRI data, it substantially enhances result interpretability. The framework is broadly applicable to comparative analysis of general high-dimensional dependent data.

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.

Problem

• high-dimensional posterior comparison
• uncertainty quantification
• brain connectivity
• credible rectangles
• correlation matrices

Innovation

• credible hyperrectangles
• Bayesian uncertainty quantification
• high-dimensional posterior comparison
• brain connectivity
• inverse-Wishart model