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arxiv: 2605.30072 · v1 · pith:E56SKUGNnew · submitted 2026-05-28 · 📊 stat.ME

Credible rectangles for high-dimensional posterior comparison

Alice Chevaux , Julyan Arbel , Guillaume Kon Kam King , Sophie Achard This is my paper

Pith reviewed 2026-06-29 05:57 UTC · model grok-4.3

classification 📊 stat.ME
keywords Bayesian inferencecredible hyperrectanglescorrelation matricesbrain connectivityfMRI analysishigh-dimensional comparisonuncertainty quantificationposterior distributions
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The pith

Credible hyperrectangles from posterior distributions of correlation matrices enable direct comparison of two brain scans from the same patient.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a Bayesian approach to compare high-dimensional posterior distributions of brain connectivity graphs using credible hyperrectangles. This allows uncertainty quantification beyond point estimates and supports subject-level inference by comparing two scans from one individual. The method includes scalable estimation algorithms and theoretical results such as a Bernstein-von Mises theorem for correlation matrices under the inverse-Wishart model along with Bayesian family-wise error rate control. It shows competitive performance on synthetic data and applies to real fMRI datasets for better interpretability of connectivity differences.

Core claim

The paper claims that credible hyperrectangles derived from the posterior distributions provide interpretable tools for subject-level inference and longitudinal monitoring in brain connectivity analysis, enabling principled detection of significant connectivity differences both globally and locally while preserving joint dependency structures, with theoretical guarantees in the inverse-Wishart model including a Bernstein-von Mises theorem for correlation matrices and control of a Bayesian family-wise error rate.

What carries the argument

Credible hyperrectangles constructed from the posterior distributions of correlation matrices, which quantify uncertainty and allow comparison of high-dimensional dependent data.

Load-bearing premise

The inverse-Wishart model accurately captures the posterior distribution for correlation matrices in resting-state fMRI data.

What would settle it

Finding a real or simulated fMRI dataset where the credible hyperrectangles do not achieve the expected coverage or fail to control the Bayesian family-wise error rate at the stated level would challenge the theoretical guarantees.

Figures

Figures reproduced from arXiv: 2605.30072 by Alice Chevaux, Guillaume Kon Kam King, Julyan Arbel, Sophie Achard.

Figure 1
Figure 1. Figure 1: Projection of Naive (red), Bonferroni-type (blue) and Optimal (green) rectangles. [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Network representations of the gained and lost connections, respectively in blue and [PITH_FULL_IMAGE:figures/full_fig_p022_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Empirical correlation matrices and quantiles representation (lower and upper) of each [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Heatmaps of the mean length ml defined in Equation 5 as a function of the parameter ρ, number of variables p, and number of time points n. The mean length is computed for 95% credible rectangles derived for the distribution of correlation coefficients based on the distribution IWp(Σ(ρ), p + 2 + n) (defined in Equation 3). Each cell represents the average interval length over 100 simulations [PITH_FULL_IMA… view at source ↗
Figure 5
Figure 5. Figure 5: Ground-truth correlation matrices used for simulation study used in Section 5 for [PITH_FULL_IMAGE:figures/full_fig_p026_5.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper proposes a Bayesian framework for uncertainty quantification in high-dimensional correlation matrices arising from brain connectivity analysis. It constructs credible hyperrectangles from the posterior to enable global and local comparison of graphs, with scalable algorithms, theoretical results (Bernstein-von Mises theorem for correlation matrices and Bayesian family-wise error rate control) derived under the inverse-Wishart model, and an application to longitudinal subject-level inference on rs-fMRI data that is claimed to be novel.

Significance. If the inverse-Wishart posterior approximation is adequate and the hyperrectangles deliver the stated coverage and error-rate control, the method would provide a principled, dependency-preserving tool for direct within-subject scan comparison that is currently absent from the literature. The combination of high-dimensional theory and real-data demonstration could strengthen uncertainty-aware graphical modeling in neuroimaging and related fields.

major comments (2)
  1. [Abstract] Abstract and modeling section: the Bernstein-von Mises theorem and Bayesian family-wise error rate control are derived exclusively inside the inverse-Wishart model. No robustness analysis or diagnostic is supplied to assess whether the IW tails and dependence structure remain sufficiently accurate for the actual posterior of correlation matrices (or their differences) in resting-state fMRI; if they deviate materially, both rectangle coverage and error-rate control can fail even if the computational procedure is correct.
  2. [Abstract] The claim that the framework enables 'direct comparison of two distinct scans from a single patient' rests on the credible rectangles inheriting valid frequentist coverage from the BvM result. Because the BvM is model-specific, the manuscript must demonstrate that the IW posterior is close enough to the true sampling distribution of the sample correlation matrix for the coverage statement to transfer to real data; this step is load-bearing for the subject-level inference application.
minor comments (1)
  1. Notation for the credible hyperrectangle boundaries and the precise definition of the Bayesian family-wise error rate should be stated explicitly early in the manuscript to avoid ambiguity when the theoretical guarantees are invoked.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive major comments. We agree that the model-specific nature of the theoretical results requires additional support for the real-data claims and will strengthen the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract and modeling section: the Bernstein-von Mises theorem and Bayesian family-wise error rate control are derived exclusively inside the inverse-Wishart model. No robustness analysis or diagnostic is supplied to assess whether the IW tails and dependence structure remain sufficiently accurate for the actual posterior of correlation matrices (or their differences) in resting-state fMRI; if they deviate materially, both rectangle coverage and error-rate control can fail even if the computational procedure is correct.

    Authors: We agree that the BvM theorem and BFWE control are derived exclusively under the inverse-Wishart model and that the current manuscript provides no robustness diagnostics against deviations in tails or dependence that may occur in rs-fMRI data. In the revision we will add a dedicated simulation subsection that generates data from heavier-tailed or non-Wishart correlation models, recomputes the credible rectangles under the IW posterior, and reports empirical coverage and BFWE rates to quantify sensitivity. revision: yes

  2. Referee: [Abstract] The claim that the framework enables 'direct comparison of two distinct scans from a single patient' rests on the credible rectangles inheriting valid frequentist coverage from the BvM result. Because the BvM is model-specific, the manuscript must demonstrate that the IW posterior is close enough to the true sampling distribution of the sample correlation matrix for the coverage statement to transfer to real data; this step is load-bearing for the subject-level inference application.

    Authors: We concur that the subject-level comparison claim is load-bearing on the IW approximation being sufficiently close to the true sampling distribution. The revision will include Monte Carlo experiments that draw correlation matrices from non-IW distributions calibrated to fMRI characteristics, apply the IW-based credible rectangles, and tabulate attained coverage; we will also revise the abstract and discussion to state the guarantees as conditional on model adequacy while retaining the IW as a computationally convenient and standard working model. revision: yes

Circularity Check

0 steps flagged

No circularity; theoretical guarantees derived conditionally on explicit IW model assumption

full rationale

The manuscript states that it establishes a Bernstein-von Mises theorem for correlation matrices and Bayesian family-wise error rate control inside the inverse-Wishart model. This is an explicit modeling premise rather than a self-referential reduction. No equations, fitted parameters renamed as predictions, self-citations used as load-bearing uniqueness theorems, or ansatzes smuggled via prior work appear in the supplied text. The derivation chain therefore remains self-contained within the stated model assumptions and does not collapse to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so no free parameters, axioms, or invented entities can be identified from the provided text.

pith-pipeline@v0.9.1-grok · 5709 in / 1068 out tokens · 28230 ms · 2026-06-29T05:57:52.976487+00:00 · methodology

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Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages · 1 internal anchor

  1. [1]

    doi: 10.1111/jtsa.12170

    ISSN 1467-9892. doi: 10.1111/jtsa.12170. URLhttps://onlinelibrary.wiley.com/doi/abs/10.1111/jtsa. 12170. Julian Besag, Peter Green, David Higdon, and Kerrie Mengersen. Bayesian Computation and Stochastic Systems.Statistical Science, 10(1):3–41, Febru- ary

    work page doi:10.1111/jtsa.12170

  2. [2]

    doi: 10.1214/ss/1177010123

    ISSN 0883-4237, 2168-8745. doi: 10.1214/ss/1177010123. URL https://projecteuclid.org/journals/statistical-science/volume-10/issue-1/ Bayesian-Computation-and-Stochastic-Systems/10.1214/ss/1177010123.full. Jacob Bien and Robert J Tibshirani. Sparse estimation of a covariance matrix.Biometrika, 98 (4):807–820,

    work page doi:10.1214/ss/1177010123

  3. [3]

    Magnetic Resonance in Medicine34(4), 537–541 (1995)

    ISSN 1522-2594. doi: 10.1002/mrm.1910340409. eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/mrm.1910340409. Carlo Bonferroni. Teoria statistica delle classi e calcolo delle probabilita.Pubblicazioni del R istituto superiore di scienze economiche e commericiali di firenze, 8:3–62,

    work page doi:10.1002/mrm.1910340409

  4. [4]

    doi: 10.1016/j.jeconom.2018.03.020

    ISSN 0304-4076. doi: 10.1016/j.jeconom.2018.03.020. URLhttps://www.sciencedirect.com/ science/article/pii/S0304407618300782. Alice Chevaux, Ali Fahkar, K´ evin Polisano, Ir` ene Gannaz, and Sophie Achard. Benchmarking Brain Connectivity Graph Inference: A Novel Validation Approach. In33rd European Signal Processing Conference (EUSIPCO 2025), Palerme, Ital...

    work page doi:10.1016/j.jeconom.2018.03.020 2018

  5. [5]

    doi: 10.3758/s13428-017-0862-1

    ISSN 1554-3528. doi: 10.3758/s13428-017-0862-1. URLhttp://link.springer.com/10. 3758/s13428-017-0862-1. Christopher Ferrie. High posterior density ellipsoids of quantum states.New Journal of Physics, 16(2):023006, February

    work page doi:10.3758/s13428-017-0862-1

  6. [6]

    doi: 10.1088/1367-2630/16/2/023006

    ISSN 1367-2630. doi: 10.1088/1367-2630/16/2/023006. URL https://doi.org/10.1088/1367-2630/16/2/023006. Publisher: IOP Publishing. 31 Michael D. Fox and Marcus E. Raichle. Spontaneous fluctuations in brain activity observed with functional magnetic resonance imaging.Nature Reviews Neuroscience, 8(9):700–711, September

    work page doi:10.1088/1367-2630/16/2/023006

  7. [7]

    doi: 10.1038/nrn2201

    ISSN 1471-0048. doi: 10.1038/nrn2201. URLhttps://www.nature.com/ articles/nrn2201. Publisher: Nature Publishing Group. Andrew Gelman, John B Carlin, Hal S Stern, and Donald B Rubin.Bayesian data analysis. Chapman and Hall/CRC,

    work page doi:10.1038/nrn2201

  8. [8]

    doi: 10.1214/16-AOAS1015

    ISSN 1932-6157, 1941-7330. doi: 10.1214/16-AOAS1015. URL https://projecteuclid.org/journals/annals-of-applied-statistics/volume-11/ issue-2/Hypothesis-testing-for-network-data-in-functional-neuroimaging/10. 1214/16-AOAS1015.full. Matthew F. Glasser, Stamatios N. Sotiropoulos, J. Anthony Wilson, Timothy S. Coalson, Bruce Fischl, Jesper L. Andersson, Junqia...

    work page doi:10.1214/16-aoas1015 1932

  9. [9]

    doi: 10.1016/j.neuroimage.2013.04.127

    ISSN 1053-8119. doi: 10.1016/j.neuroimage.2013.04.127. URLhttps://www.sciencedirect.com/science/ article/pii/S1053811913005053. Jim Griffin. Expressing and Visualizing Model Uncertainty in Bayesian Variable Se- lection Using Cartesian Credible Sets.Bayesian Analysis, 20(4):1371–1397, De- cember

    work page doi:10.1016/j.neuroimage.2013.04.127 2013

  10. [10]

    doi: 10.1214/25-BA1568

    ISSN 1936-0975, 1931-6690. doi: 10.1214/25-BA1568. URL https://projecteuclid.org/journals/bayesian-analysis/volume-20/issue-4/ Expressing-and-Visualizing-Model-Uncertainty-in-Bayesian-Variable-Selection-Using/ 10.1214/25-BA1568.full. Publisher: International Society for Bayesian Analysis. L. R Haff. An identity for the Wishart distribution with applicatio...

    work page doi:10.1214/25-ba1568 1936

  11. [11]

    URL https://www.jstor.org/stable/1391142

    ISSN 1061-8600. URL https://www.jstor.org/stable/1391142. Sture Holm. A Simple Sequentially Rejective Multiple Test Procedure.Scandinavian Journal of Statistics, 6(2):65–70,

    work page arXiv

  12. [12]

    doi: 10.1016/j.ifacol.2024.10.004

    ISSN 2405-8963. doi: 10.1016/j.ifacol.2024.10.004. URLhttps://www. sciencedirect.com/science/article/pii/S2405896324017506. Yiye Jiang, Alice Chevaux, Wendy Meiring, Alex Petersen, Guillaume Kon Kam King, Julyan Arbel, and Sophie Achard. Prior elicitation for Bayesian estimation of single-subject con- nectivity networks, May

    work page doi:10.1016/j.ifacol.2024.10.004 2024

  13. [13]

    Prior elicitation for Bayesian estimation of single-subject connectivity networks

    URLhttp://arxiv.org/abs/2605.02587. arXiv:2605.02587 [stat.ME]. Ashwini Joshi, Terhi Ollila, Ilkka Immonen, and Sangita Kulathinal. Bayesian Simultaneous Credible Intervals for Effect Measures from Multiple Markers.Statistics in Biopharmaceutical Research, 15(3):638–653, July

    work page internal anchor Pith review Pith/arXiv arXiv

  14. [14]

    doi: 10.1080/19466315.2022.2110935

    ISSN 1946-6315. doi: 10.1080/19466315.2022.2110935. URLhttps://www.tandfonline.com/doi/full/10.1080/19466315.2022.2110935. 32 Junghi Kim, Jeffrey R. Wozniak, Bryon A. Mueller, Xiaotong Shen, and Wei Pan. Comparison of statistical tests for group differences in brain functional networks.NeuroImage, 101:681– 694, November

    work page doi:10.1080/19466315.2022.2110935 1946

  15. [15]

    doi: 10.1016/j.neuroimage.2014.07.031

    ISSN 1095-9572. doi: 10.1016/j.neuroimage.2014.07.031. Arnoˇ st Kom´ arek and Emmanuel Lesaffre. Bayesian semiparametric accelerated failure time model for paired doubly-interval-censored data.Statistical Modelling, 6(1):3–22,

    work page doi:10.1016/j.neuroimage.2014.07.031 2014

  16. [16]

    Olivier Ledoit and Michael Wolf

    doi: 10.1191/1471082X06st107oa. Olivier Ledoit and Michael Wolf. Honey, I shrunk the sample covariance matrix.UPF economics and business working paper, (691),

    work page doi:10.1191/1471082x06st107oa

  17. [17]

    doi: 10.1016/j.mri.2007.02.012

    ISSN 0730725X. doi: 10.1016/j.mri.2007.02.012. URLhttps://linkinghub.elsevier.com/retrieve/pii/ S0730725X07001415. Lydia Oujamaa, Chantal Delon-Martin, Chlo´ e Jaroszynski, Maite Termenon, Stein Silva, Jean- Fran¸ cois Payen, and Sophie Achard. Functional hub disruption emphasizes consciousness recovery in severe traumatic brain injury.Brain Communication...

    work page doi:10.1016/j.mri.2007.02.012 2007

  18. [18]

    doi: 10.1093/braincomms/fcad319

    ISSN 2632-1297. doi: 10.1093/braincomms/fcad319. URLhttps://academic.oup. com/braincomms/article/doi/10.1093/braincomms/fcad319/7441049. Edmund T. Rolls, Chu-Chung Huang, Ching-Po Lin, Jianfeng Feng, and Marc Joliot. Auto- mated anatomical labelling atlas 3.NeuroImage, 206:116189, February

    work page doi:10.1093/braincomms/fcad319

  19. [19]

    doi: 10.1016/j.neuroimage.2019.116189

    ISSN 1053-8119. doi: 10.1016/j.neuroimage.2019.116189. URLhttps://www.sciencedirect.com/science/ article/pii/S1053811919307803. Partha Sarkar, Kshitij Khare, Malay Ghosh, and Matt P. Wand. High-Dimensional Bernstein Von-Mises Theorems for Covariance and Precision Matrices, July

    work page doi:10.1016/j.neuroimage.2019.116189 2019

  20. [20]

    org/abs/2309.08556

    URLhttp://arxiv. org/abs/2309.08556. arXiv:2309.08556 [math]. Stephen M. Smith, Christian F. Beckmann, Jesper Andersson, Edward J. Auerbach, Ja- nine Bijsterbosch, Gwena¨ elle Douaud, Eugene Duff, David A. Feinberg, Ludovica Grif- fanti, Michael P. Harms, Michael Kelly, Timothy Laumann, Karla L. Miller, Steen Moeller, Steve Petersen, Jonathan Power, Ghola...

    work page arXiv

  21. [21]

    doi: 10.1016/j.neuroimage.2013.05.039

    ISSN 1095-9572. doi: 10.1016/j.neuroimage.2013.05.039. Niansheng Tang and Bin Yu. Simultaneous confidence interval for assessing non-inferiority with assay sensitivity in a three-arm trial with binary endpoints.Pharmaceutical Statistics, 19(5): 518–531, September

    work page doi:10.1016/j.neuroimage.2013.05.039 2013

  22. [22]

    doi: 10.1002/pst.2010

    ISSN 1539-1612. doi: 10.1002/pst.2010. Ma¨ ıt´ e Termenon, Assia Jaillard, Chantal Delon-Martin, and Sophie Achard. Reliability of graph analysis of resting state fMRI using test-retest dataset from the Human Connectome Project.NeuroImage, 142:172,

    work page doi:10.1002/pst.2010 2010

  23. [23]

    URLhttps: //inserm.hal.science/inserm-01330966

    doi: 10.1016/j.neuroimage.2016.05.062. URLhttps: //inserm.hal.science/inserm-01330966. N. Tzourio-Mazoyer, B. Landeau, D. Papathanassiou, F. Crivello, O. Etard, N. Delcroix, B. Ma- zoyer, and M. Joliot. Automated anatomical labeling of activations in SPM using a macro- scopic anatomical parcellation of the MNI MRI single-subject brain.NeuroImage, 15(1): 2...

    work page doi:10.1016/j.neuroimage.2016.05.062 2016

  24. [24]

    doi: 10.1006/nimg.2001.0978

    ISSN 1053-8119. doi: 10.1006/nimg.2001.0978. 33 Brandon Whitcher, Peter Guttorp, and Donald B. Percival. Wavelet analysis of covariance with application to atmospheric time series.Journal of Geophysical Research: Atmospheres, 105(D11):14941–14962,

    work page doi:10.1006/nimg.2001.0978 2001

  25. [25]

    doi: 10.1029/2000JD900110

    ISSN 2156-2202. doi: 10.1029/2000JD900110. URLhttps: //onlinelibrary.wiley.com/doi/abs/10.1029/2000JD900110. Andrew Zalesky, Alex Fornito, and Edward T. Bullmore. Network-based statistic: Identifying differences in brain networks.NeuroImage, 53(4):1197–1207, December

    work page doi:10.1029/2000jd900110

  26. [26]

    doi: 10.1016/j.neuroimage.2010.06.041

    ISSN 1053-8119. doi: 10.1016/j.neuroimage.2010.06.041. URLhttps://www.sciencedirect.com/science/ article/pii/S1053811910008852. 34

    work page doi:10.1016/j.neuroimage.2010.06.041 2010