Pith. sign in

REVIEW 3 major objections 5 minor 50 references

Bayesian Inference of Spatially Varying Correlations via the Thresholded Correlation Gaussian Process

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper proposes a thresholded correlation Gaussian process prior that makes spatially varying correlations between two imaging modalities identifiable and consistently recoverable, and demonstrates it on brain imaging data.

desk verdict The TCGP prior is a genuine new construction for spatially varying correlations with real computational machinery, but Theorem 3's exact-zero selection consistency is not proven: the proof assumes a limit interchange that need not commute. read the letter →

arxiv 2411.18012 v1 pith:5PMNIIVQ submitted 2024-11-27 stat.ME

Moyan Li , Lexin Li , Jian Kang This is my paper · ORCID
classification stat.ME MSC 62F1562G2062P10
keywords BayesiannonparametricsthresholdedGaussianprocessspatiallyvaryingcorrelationmultimodalneuroimagingposteriorconsistencyselectionGibbssamplingfunctionalMRI
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proposes a Bayesian nonparametric model for estimating how the correlation between two imaging modalities varies across space, and for locating the brain regions where that correlation is nonzero. The prior, called the thresholded correlation Gaussian process (TCGP), is built by passing a latent Gaussian process through a hard threshold, so the resulting correlation map is smooth within regions, sparse (exactly zero where the latent process stays below a threshold), and can jump at region boundaries. The paper establishes that the model is identifiable, that the prior has large support, and that the posterior is consistent, with posterior probability tending to one on the correct sign of the correlation at every location. If correct, this gives researchers a principled, uncertainty-quantified way to identify activation regions from multimodal imaging data, with performance that is maintained when the number of subjects is small or the signal is weak.

What carries the argument

The load-bearing object is the thresholded correlation Gaussian process prior (Definition 1, equations (2.2)–(2.6)). A latent Gaussian process $\xi(v)$ is thresholded at level $\omega$ to define $\sigma_+(v) = G_\omega\{\xi(v)\}$ and $\sigma_-(v) = G_\omega\{-\xi(v)\}$, which in turn scale two independent Gaussian processes whose sum and difference form the two modality means; after integrating those out, the correlation is the closed-form function in (2.6). The threshold enforces sparsity and identifiability, since $\sigma_+(v)\sigma_-(v)=0$ everywhere, and the Gaussian process kernel imposes spatial smoothness on the nonzero part. The equivalent model (2.9) uses $Y_{\pm,i}(v) = s\{\pm\rho(v)\}E_{\pm,i}(v)+\varepsilon_{\pm,i}(v)$ for a monotone function $s$, which turns the correlation into a coefficient multiplying a Gaussian process; this representation supports the posterior-consistency proof and yields full conditional distributions that are mixtures of truncated normals or uniforms, leading to a Gibbs sampler without gradient approximations.

What would settle it

Run the TCGP posterior on data generated from a true $\rho_0$ that is everywhere nonzero with $0<|\rho_0(v)|<\gamma$, so the null set is empty and no jump exists; the Theorem 3 premise is violated, and the estimated sign map and posterior inclusion probabilities will either declare spurious zero regions or fail to recover the true sign pattern.

Watch

Extended reading notes

Core claim

The central claim is that the correlation function $\rho(v)$ between two aligned image modalities can be represented by equation (2.6), driven by a single latent Gaussian process $\xi(v)$ through the thresholding function $G_\omega(x) = x I(x > \omega)$. Because $\sigma_+(v) = G_\omega\{\xi(v)\}$ and $\sigma_-(v) = G_\omega\{-\xi(v)\}$, at most one of the positive and negative variance components is active at any voxel; this forces exact zeros where $|\xi(v)| \le \omega$ and delivers piecewise smooth, jump-discontinuous correlation maps. The paper proves that this TCGP formulation is identifiable (Proposition 1), that the prior assigns positive probability to every small supremum-norm neighborhood of any true correlation function in the class (Theorem 1), and that the posterior concentrates and recovers the sign pattern $\operatorname{sgn}\{\rho(v)\}$ for all $v$ as the number of subjects and voxels grow (Theorems 2 and 3). A reparametrization in terms of average and contrast images $Y_+(v), Y_-(v)$ turns $\rho(v)$ into a mean parameter, which is what makes both the theory and the Gibbs sampler tractable.

Load-bearing premise

The load-bearing premise is that the true correlation is exactly zero throughout a whole region and bounded away from zero everywhere else; if real associations are merely small but never exactly zero, the thresholded prior is misspecified and the sign-consistency guarantee does not apply.

Editorial extensions

If this is right

  • Neuroscientists can report whole-brain maps of positive, negative, and null correlations between two modalities, with posterior inclusion probabilities as uncertainty measures, rather than a single test statistic per voxel.
  • The method is usable in small-sample imaging studies: simulations with $n=500$ and weak signals still show sensitivity around 0.89–0.90 and low false discovery rates, while voxel-wise tests have near-zero power.
  • The hybrid mini-batch MCMC reduces per-iteration cost from $O(m^2)$ to $O(m_s^2)$, so the procedure scales to the roughly 117,000 voxels of a typical whole-brain analysis.
  • The prior has large support over a wide class of sparse, piecewise smooth, jump-discontinuous correlation functions, so the model is not restricted to a parametric family.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The exact-zero assumption is a modeling commitment; in real data where all correlations are small but nonzero, the method should be read as detecting correlations above the threshold, and the sign-consistency theorem should not be invoked.
  • The adaptive choice of the prior range for $\omega$ from quantiles of $|\xi(v)|$ (Sections 4.2 and 6) lies outside the fixed-prior theory; a natural diagnostic is to repeat the analysis across quantile levels and check that declared regions are stable.
  • The same construction could be transplanted to temporally varying or group-level correlations, replacing the spatial domain with time or subject index; the proof structure would need new assumptions on the domain and smoothness, but the thresholding mechanism is domain-agnostic.
  • Because the transformed model (2.9) is a mean regression with a Gaussian process, the framework could in principle be extended to non-Gaussian imaging data through link functions, although the current theory and Gibbs sampler are tailored to Gaussian likelihoods.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proposes a Bayesian nonparametric model for spatially varying correlations between two imaging modalities, built on a thresholded correlation Gaussian process (TCGP) prior. The construction thresholds a latent Gaussian process so that at each spatial location the correlation is either zero or bounded away from zero, yielding piecewise smooth, sparse, and discontinuous correlation surfaces. The authors prove identifiability (Proposition 1), large prior support (Theorem 1), posterior consistency (Theorem 2), and sign/selection consistency (Theorem 3). They derive full conditional distributions and propose a Gibbs sampler and a hybrid mini-batch MCMC for computation. Simulations on 2D and 3D images and an analysis of Human Connectome Project fMRI data compare favorably with voxel-wise, region-wise, and integrated competitors. The main weaknesses are an apparently incomplete proof of exact-zero selection consistency in Theorem 3, a mismatch between the fixed prior analyzed in the theory and the data-adaptive prior used in the implementation, and the absence of an analysis of the Karhunen-Loève truncation error introduced in Section 4.1.

Significance. If the theoretical results hold, the paper is a substantial contribution to Bayesian nonparametric modeling of spatially varying correlations. The thresholded-correlation construction is novel, and the model has the desirable features of sparsity, piecewise smoothness, and jump discontinuities. The paper explicitly provides machine-checkable-style proofs, though they are not machine-checked, and the empirical results are strong: the proposed method substantially outperforms voxel-wise and region-wise baselines and the integrated method of Li et al. (2019) in the reported simulations, and the HCP analysis yields interpretable brain regions with plausible associations. The proposed Gibbs sampler with analytically derived full conditionals is an algorithmic contribution in its own right. However, the central theoretical claim of exact-zero selection consistency currently rests on a questionable double-limit interchange, and the implemented adaptive prior for the threshold parameter falls outside the scope of the theorems. If the proof gap is closed and the theory is aligned with the implementation, the paper would merit publication in a leading statistics journal.

major comments (3)
  1. [Appendix A.4, proof of Theorem 3] The proof of exact-zero selection consistency is incomplete because it interchanges limits in n and k without justification. The argument defines F_k(R0) = {∫_{R0}|ρ|dv < 1/k}, obtains pr(F_k(R0)|Y) → 1 for each fixed k from Theorem 2, and then uses monotone continuity to conclude pr({ρ=0 on R0}|Y) = lim_k pr(F_k(R0)|Y) = 1. For a fixed data set, monotone continuity indeed gives pr(∩_k F_k|Y) = lim_k pr(F_k|Y), but Theorem 2 only yields lim_n pr(F_k|Y) = 1 for each k. The double limit need not commute; a sequence of posteriors can concentrate on points just above zero on R0 at a rate that outpaces the convergence in n. Thus the proof as written does not establish that the posterior mass on the exact-zero set tends to one, which is the central claim of Theorem 3.
  2. [Sections 4.2 and 6; Theorems 2 and 3] The theoretical results assume a fixed prior specification for the threshold ω, but the implemented and recommended procedure uses an adaptive, data-dependent prior: Section 4.2 states that aω and bω are chosen as quantiles of {|ξ(v)|} and are 'adaptively changed based on the value of {|ξ(v)|}v∈B in each iteration,' and Section 6 sets aω to the 75% quantile and bω to the 100% quantile of the posterior draws. This empirical-Bayes-like prior is not covered by the proof of Theorem 2, which constructs sieves and tests under a fixed TCGP(ω0, κ, τ²) prior. Since ω controls which voxels are treated as exactly zero, the proof gap directly affects the sparsity and selection claims. The authors should either extend the theory to data-dependent priors, or modify the implementation to use a fixed prior whose range is justified independently of the data.
  3. [Section 4.1, Eq. (4.11); Theorems 2 and 3] The posterior consistency theorems are stated for the infinite-dimensional model (2.9)–(2.10), but all simulations and the HCP analysis use the finite-dimensional KL-truncated model (4.11) with a fixed truncation level L (L = 540 in the HCP analysis). No bound is given for the truncation error, and no rate condition linking L to n and m is stated. Consequently, the theoretical results do not apply to the model actually implemented, even if the proof in Appendix A.4 were correct. The authors should either extend the theory to the truncated model or demonstrate that the truncation error is asymptotically negligible under Assumption 3.
minor comments (5)
  1. [Appendix A.4, Lemma 5] There is a typo in the last line of the proof of Lemma 5: 'ccompletes' should read 'completes'.
  2. [Appendix A.4, Lemma 7] In the proof of Lemma 7, 'at lease' should read 'at least'.
  3. [Table 6] The caption of Table 6 contains the misspelling 'False Discorvery Rate'; it should be 'False Discovery Rate'.
  4. [Figures 1–3] The figure numbering appears inconsistent with the text: Section 2.1 refers to 'Figure 1' for the graphical model illustration, while Section 5 refers to 'Figure 1' for a simulation slice; later Section 6 refers to 'Figure 3' for the HCP activation maps. Please renumber the figures and correct all cross-references.
  5. [Section 5, first paragraph] The 3D simulation generates data using posterior means obtained from the same HCP data set analyzed in Section 6; this is a model-based truth and constitutes self-validation. It would be useful to state this limitation explicitly in the simulation section and to include at least one simulation with a truth not derived from the model's own posterior, for example with smoothly varying but nonzero correlations, to assess robustness.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: TCGP derivations are self-contained; mild empirical-Bayes choice and model-based simulation do not make the claims circular.

full rationale

The derivation chain is self-contained. The TCGP prior is defined explicitly in Definition 1 and equation (2.6), and the large-support, identifiability, posterior-consistency, and selection-consistency claims are proved from the prior and model assumptions using external classical results (Choi 2005; Ghosal & Roy 2006; Tokdar & Ghosh 2007; Choudhuri et al. 2004; Adler & Taylor 2009), not by invoking the conclusions. The thresholded-GP construction cites prior work by overlapping authors (Kang et al. 2018; Wu et al. 2024) only as motivation and for a computational KL-truncation rule, and the theoretical results do not depend on those citations. The data-dependent choice of the omega prior range (Section 4.2, Section 6) is an empirical-Bayes computational device and is not part of the theoretical claims, which assume a fixed prior; it does not make a fitted parameter be renamed as a prediction. The Section 5 simulation uses the model's own posterior means as truth, which is a self-consistency check rather than an independent benchmark, but it is not a step in the derivation of the method or its guarantees. The proof of Theorem 3 has a potential double-limit gap when passing from pr(F_k(R0)|Y) to 1 for each k to pr(Intersection_k F_k(R0)|Y) to 1, but that is a correctness concern, not a circularity: the target conclusion is not assumed as an input anywhere.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The model has several data-dependent parameters (ω prior range, KL truncation L, kernel hyperparameters) that are chosen in implementation but are not part of the theoretical guarantees. The structural assumptions about the true correlation function are explicit but strong.

free parameters (5)
  • threshold parameter ω = prior U(aω,bω) with aω=75% quantile of |ξ(v)|, bω=100% quantile in HCP analysis; min/max in simulations
    Controls sparsity; the prior range is chosen from the data (Section 4.2, Section 6), a tuning step not covered by the theory.
  • number of KL basis functions L = L=540 in HCP (variance percentage >60%)
    Truncates the infinite-dimensional GP; the choice is data-dependent and the approximation error is not analyzed (Section 4.1).
  • Matern kernel hyperparameters γ1, γ2 = not specified
    The correlation kernel shape is user-chosen and said to be determined by Bayes factor, but the values used in simulations and HCP are not reported.
  • mini-batch parameters ms and T0 = ms=m/16, T0=20
    Computational tuning; sensitivity analysis in Appendix D.3 shows stability.
  • PIP threshold = 0.5
    Used to declare a voxel as activated; standard but arbitrary.
assumptions (5)
  • domain assumption True correlation function ρ0 belongs to Θρ (Assumption 1)
    The true function must be exactly zero on a set R0 with nonempty interior and bounded away from zero elsewhere; otherwise selection consistency (Theorem 3) does not apply.
  • standard math Kernel κ factorizes as product of univariate densities and has derivatives up to order 2α+2 (Assumption 2)
    Used to prove large support via KL expansions and to control tail probabilities in Lemma 4.
  • domain assumption Number of spatial locations m grows polynomially in n (Assumption 3)
    Ensures posterior concentration of the spatially varying functions.
  • domain assumption Errors ε_k,i(v) are independent across k, i, v and normal with variance τ_k²(v) (model 2.1)
    Standard independent noise assumption for imaging data; used to derive likelihood.
  • domain assumption Latent processes E±,i are independent GPs with mean 0 and correlation kernel κ (equation 2.10)
    This specification yields the variance of Y± and the bivariate normal structure used in the proofs.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Bayesian Inference of Spatially Varying Correlations via the Thresholded Correlation Gaussian Process." pith.science (2026). https://pith.science/paper/5PMNIIVQ

@misc{pith2026241118012,
  author       = {Pith},
  title        = {Pith review of: Bayesian Inference of Spatially Varying Correlations via the Thresholded Correlation Gaussian Process},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5PMNIIVQ}},
  note         = {Machine review of arXiv:2411.18012}
}
read the original abstract

A central question in multimodal neuroimaging analysis is to understand the association between two imaging modalities and to identify brain regions where such an association is statistically significant. In this article, we propose a Bayesian nonparametric spatially varying correlation model to make inference of such regions. We build our model based on the thresholded correlation Gaussian process (TCGP). It ensures piecewise smoothness, sparsity, and jump discontinuity of spatially varying correlations, and is well applicable even when the number of subjects is limited or the signal-to-noise ratio is low. We study the identifiability of our model, establish the large support property, and derive the posterior consistency and selection consistency. We also develop a highly efficient Gibbs sampler and its variant to compute the posterior distribution. We illustrate the method with both simulations and an analysis of functional magnetic resonance imaging data from the Human Connectome Project.

Figures

Figures reproduced from arXiv: 2411.18012 by the authors.

Figure 1
Figure 1. Graphical illustration of the proposed Bayesian spatially varying correlation [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Simulation results of the 3D image example for [PITH_FULL_IMAGE:figures/full_fig_p030_2.png] view at source ↗
Figure 3
Figure 3. Results of Human Connectome Project data analysis. The sagittal slices of activa [PITH_FULL_IMAGE:figures/full_fig_p040_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Results of 2D image simulations. The first row is for a weak signal and the [PITH_FULL_IMAGE:figures/full_fig_p070_4.png]
Figure 5
Figure 5. Figure 5: Results of 2D image simulations. The posterior inclusion probability map of the [PITH_FULL_IMAGE:figures/full_fig_p071_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

50 extracted references · 48 canonical work pages

  1. [1]

    Acheson, D. J. and P. Hagoort (2013). Stimulating the brain's language network: syntactic ambiguity resolution after tms to the inferior frontal gyrus and middle temporal gyrus. Journal of cognitive neuroscience\/ 25\/ (10), 1664--1677

    work page 2013

  2. [2]

    Adler, R. and J. Taylor (2009). Random Fields and Geometry . Springer Monographs in Mathematics. Springer New York

    work page 2009

  3. [3]

    Korattikara, and M

    Ahn, S., A. Korattikara, and M. Welling (2012). Bayesian posterior sampling via stochastic gradient fisher scoring. arXiv preprint arXiv:1206.6380\/

    arXiv 2012

  4. [4]

    Barch, D. M., G. C. Burgess, M. P. Harms, S. E. Petersen, B. L. Schlaggar, M. Corbetta, M. F. Glasser, S. Curtiss, S. Dixit, C. Feldt, et al. (2013). Function in the human connectome: task-fmri and individual differences in behavior. Neuroimage\/ 80 , 169--189

    work page 2013

  5. [5]

    Bhattacharya, A. and D. B. Dunson (2011). Sparse bayesian infinite factor models. Biometrika\/ 98\/ (2), 291--306

    work page 2011

  6. [6]

    Kang, and T

    Cai, Q., J. Kang, and T. Yu (2020). Bayesian network marker selection via the thresholded graph laplacian gaussian prior. Bayesian Analysis\/ 15\/ (1), 79

    work page 2020

  7. [7]

    Calhoun, V. D. and T. Adali (2008). Feature-based fusion of medical imaging data. IEEE Transactions on Information Technology in Biomedicine\/ 13\/ (5), 711--720

    work page 2008

  8. [8]

    Kandeepan, M

    Cavaliere, C., S. Kandeepan, M. Aiello, D. Ribeiro de Paula, R. Marchitelli, S. Fiorenza, M. Orsini, L. Trojano, O. Masotta, K. St Lawrence, et al. (2018). Multimodal neuroimaging approach to variability of functional connectivity in disorders of consciousness: a pet/mri pilot study. Frontiers in Neurology\/ 9 , 861

    work page 2018

Show all 50 references
  1. [9]

    Chen, S., F. D. Bowman, and H. S. Mayberg (2016). A bayesian hierarchical framework for modeling brain connectivity for neuroimaging data. Biometrics\/ 72\/ (2), 596--605

    2016

  2. [10]

    Fox, and C

    Chen, T., E. Fox, and C. Guestrin (2014). Stochastic gradient hamiltonian monte carlo. In International conference on machine learning , pp.\ 1683--1691. PMLR

    2014

  3. [11]

    Diaz-Pinto, N

    Chen, X., A. Diaz-Pinto, N. Ravikumar, and A. F. Frangi (2021). Deep learning in medical image registration. Progress in Biomedical Engineering\/ 3\/ (1), 012003

    2021

  4. [12]

    Choi, T. (2005). Posterior consistency in nonparametric regression problems under Gaussian process priors . Ph.\ D. thesis, Carnegie Mellon University

    2005

  5. [13]

    Ghosal, and A

    Choudhuri, N., S. Ghosal, and A. Roy (2004). Bayesian estimation of the spectral density of a time series. Journal of the American Statistical Association\/ 99\/ (468), 1050--1059

    2004

  6. [14]

    Cohen, A. D., Z. Chen, O. Parker Jones, C. Niu, and Y. Wang (2020). Regression-based machine-learning approaches to predict task activation using resting-state fMRI . Human brain mapping\/ 41\/ (3), 815--826

    2020

  7. [15]

    Farrer, C., S. H. Frey, J. D. Van Horn, E. Tunik, D. Turk, S. Inati, and S. T. Grafton (2008). The angular gyrus computes action awareness representations. Cerebral cortex\/ 18\/ (2), 254--261

    2008

  8. [16]

    Kehtarnavaz, R

    Gholipour, A., N. Kehtarnavaz, R. Briggs, M. Devous, and K. Gopinath (2007). Brain functional localization: a survey of image registration techniques. IEEE transactions on medical imaging\/ 26\/ (4), 427--451

    2007

  9. [17]

    Ghosal, S. and A. Roy (2006). Posterior consistency of gaussian process prior for nonparametric binary regression. The Annals of Statistics\/ 34\/ (5), 2413--2429

    2006

  10. [18]

    Girolami, M. and B. Calderhead (2011). Riemann manifold langevin and hamiltonian monte carlo methods. Journal of the Royal Statistical Society: Series B (Statistical Methodology)\/ 73\/ (2), 123--214

    2011

  11. [19]

    Abend, J

    Harrewijn, A., R. Abend, J. Linke, M. A. Brotman, N. A. Fox, E. Leibenluft, A. M. Winkler, and D. S. Pine (2020). Combining fmri during resting state and an attention bias task in children. Neuroimage\/ 205 , 116301

    2020

  12. [20]

    Jones, O. P., N. Voets, J. Adcock, R. Stacey, and S. Jbabdi (2017). Resting connectivity predicts task activation in pre-surgical populations. NeuroImage: Clinical\/ 13 , 378--385

    2017

  13. [21]

    Kang, J., B. J. Reich, and A.-M. Staicu (2018). Scalar-on-image regression via the soft-thresholded gaussian process. Biometrika\/ 105\/ (1), 165--184

    2018

  14. [22]

    Koenigs, M., A. K. Barbey, B. R. Postle, and J. Grafman (2009). Superior parietal cortex is critical for the manipulation of information in working memory. Journal of Neuroscience\/ 29\/ (47), 14980--14986

    2009

  15. [23]

    Leshikar, E. D., A. Duarte, and C. Hertzog (2012). Task-selective memory effects for successfully implemented encoding strategies. PloS one\/ 7\/ (5), e38160

    2012

  16. [24]

    Li, D. and W. H. Wong (2017). Mini-batch tempered mcmc. arXiv preprint arXiv:1707.09705\/

    2017 arXiv

  17. [25]

    Huang, Z

    Li, J., C. Huang, Z. Hongtu, and for the Alzheimer’s Disease Neuroimaging Initiative (2017). A functional varying-coefficient single-index model for functional response data. Journal of the American Statistical Association\/ 112\/ (519), 1169--1181

    2017

  18. [26]

    Li, L., J. Kang, S. N. Lockhart, J. Adams, and W. J. Jagust (2019). Spatially adaptive varying correlation analysis for multimodal neuroimaging data. IEEE transactions on medical imaging\/ 38\/ (1), 113--123

    2019

  19. [27]

    Li, X., L. Wang, H. J. Wang, and A. D. N. Initiative (2021). Sparse learning and structure identification for ultrahigh-dimensional image-on-scalar regression. Journal of the American Statistical Association\/ 116\/ (536), 1994--2008

    2021

  20. [28]

    Liu, S., S. Liu, W. Cai, H. Che, S. Pujol, R. Kikinis, D. Feng, M. J. Fulham, et al. (2014). Multimodal neuroimaging feature learning for multiclass diagnosis of alzheimer's disease. IEEE transactions on biomedical engineering\/ 62\/ (4), 1132--1140

    2014

  21. [29]

    Kundu, G

    Lukemire, J., S. Kundu, G. Pagnoni, and Y. Guo (2021). Bayesian joint modeling of multiple brain functional networks. Journal of the American Statistical Association\/ 116\/ (534), 518--530

    2021

  22. [30]

    Mazziotta, J., A. Toga, A. Evans, P. Fox, J. Lancaster, K. Zilles, R. Woods, T. Paus, G. Simpson, B. Pike, et al. (2001). A probabilistic atlas and reference system for the human brain: International consortium for brain mapping (icbm). Philosophical Transactions of the Royal ...

    2001

  23. [31]

    Mercer, J. (1909). Xvi. functions of positive and negative type, and their connection the theory of integral equations. Philosophical transactions of the royal society of London. Series A, containing papers of a mathematical or physical character\/ 209\/ (441-458), 415--446

    1909

  24. [32]

    Nakajima, J. and M. West (2013). Bayesian analysis of latent threshold dynamic models. Journal of Business & Economic Statistics\/ 31\/ (2), 151--164

    2013

  25. [33]

    Ni, Y., F. C. Stingo, and V. Baladandayuthapani (2019). Bayesian graphical regression. Journal of the American Statistical Association\/ 114\/ (525), 184--197

    2019

  26. [34]

    Nishimura, A., D. B. Dunson, and J. Lu (2020). Discontinuous hamiltonian monte carlo for discrete parameters and discontinuous likelihoods. Biometrika\/ 107\/ (2), 365--380

    2020

  27. [35]

    Penny, W. D., K. J. Friston, J. T. Ashburner, S. J. Kiebel, and T. E. Nichols (2011). Statistical parametric mapping: the analysis of functional brain images . Elsevier

    2011

  28. [36]

    Roberts, G. O. and J. S. Rosenthal (1998). Optimal scaling of discrete approximations to langevin diffusions. Journal of the Royal Statistical Society: Series B (Statistical Methodology)\/ 60\/ (1), 255--268

    1998

  29. [37]

    Seghier, M. L. (2013). The angular gyrus: multiple functions and multiple subdivisions. The Neuroscientist\/ 19\/ (1), 43--61

    2013

  30. [38]

    Smith, S. M., C. F. Beckmann, J. Andersson, E. J. Auerbach, J. Bijsterbosch, G. Douaud, E. Duff, D. A. Feinberg, L. Griffanti, M. P. Harms, et al. (2013). Resting-state fMRI in the human connectome project. Neuroimage\/ 80 , 144--168

    2013

  31. [39]

    Tavor, I., O. P. Jones, R. B. Mars, S. Smith, T. Behrens, and S. Jbabdi (2016). Task-free mri predicts individual differences in brain activity during task performance. Science\/ 352\/ (6282), 216--220

    2016

  32. [40]

    Z \"o llei, and W

    Toews, M., L. Z \"o llei, and W. M. Wells (2013). Feature-based alignment of volumetric multi-modal images. In International Conference on Information Processing in Medical Imaging , pp.\ 25--36. Springer

    2013

  33. [41]

    Tokdar, S. T. and J. K. Ghosh (2007). Posterior consistency of logistic G aussian process priors in density estimation. Journal of Statistical Planning and Inference\/ 137\/ (1), 34--42

    2007

  34. [42]

    Uluda g , K. and A. Roebroeck (2014). General overview on the merits of multimodal neuroimaging data fusion. Neuroimage\/ 102 , 3--10

    2014

  35. [43]

    Wang, X., K. Shu, H. Kuang, S. Luo, R. Jin, and J. Liu (2021). The role of spatial alignment in multimodal medical image fusion using deep learning for diagnostic problems. In Proceedings of the 2021 International Conference on Intelligent Medicine and Health , pp.\ 40--46

    2021

  36. [44]

    Worsley, K. J., J. E. Taylor, F. Tomaiuolo, and J. Lerch (2004). Unified univariate and multivariate random field theory. NeuroImage\/ 23 , S189--S195

    2004

  37. [45]

    Guo, and J

    Wu, B., Y. Guo, and J. Kang (2024). Bayesian spatial blind source separation via the thresholded gaussian process. Journal of the American Statistical Association\/ 119\/ (545), 422--433

    2024

  38. [46]

    Rachel Wang, and W

    Wu, T.-Y., Y. Rachel Wang, and W. H. Wong (2022). Mini-batch metropolis--hastings with reversible sgld proposal. Journal of the American Statistical Association\/ 117\/ (537), 386--394

    2022

  39. [47]

    Zhang, X

    Zhu, D., T. Zhang, X. Jiang, X. Hu, H. Chen, N. Yang, J. Lv, J. Han, L. Guo, and T. Liu (2014). Fusing DTI and fMRI data: a survey of methods and applications. NeuroImage\/ 102 , 184--191

    2014

  40. [48]

    Fan, and L

    Zhu, H., J. Fan, and L. Kong (2014). Spatially varying coefficient model for neuroimaging data with jump discontinuities. Journal of the American Statistical Association\/ 109\/ (507), 1084--1098

    2014

  41. [49]

    Li, and B

    Zhu, H., T. Li, and B. Zhao (2023). Statistical learning methods for neuroimaging data analysis with applications. Annual Review of Biomedical Data Science\/ 6 , 73--104

    2023

  42. [50]

    Zou, Q.-H., C.-Z. Zhu, Y. Yang, X.-N. Zuo, X.-Y. Long, Q.-J. Cao, Y.-F. Wang, and Y.-F. Zang (2008). An improved approach to detection of amplitude of low-frequency fluctuation (alff) for resting-state fMRI : fractional ALFF . Journal of neuroscience methods\/ 172\/ (1), 137--141

    2008

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.