pith. machine review for the scientific record. sign in

arxiv: 2604.04964 · v2 · submitted 2026-04-03 · 📊 stat.ME

Recognition: no theorem link

Bayesian Global-Local Shrinkage with Univariate Guidance for Ultra-High-Dimensional Regression

Priyam Das
Authors on Pith no claims yet

Pith reviewed 2026-05-13 17:49 UTC · model grok-4.3

classification 📊 stat.ME
keywords Bayesian sparse regressionglobal-local shrinkagehorseshoe priorunivariate guidancehigh-dimensional regressionposterior contractionactive-set MCMC
0
0 comments X

The pith

BUGS modulates the regularized horseshoe prior with univariate marginal associations to achieve adaptive shrinkage and posterior contraction in ultra-high dimensions.

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

The paper develops BUGS, a Bayesian global-local shrinkage method for ultra-high-dimensional regression. It continuously adjusts each predictor's local shrinkage variance using that predictor's marginal association with the outcome, embedding this guidance inside the nonlinear structure of a regularized horseshoe prior. This produces stronger shrinkage on noise variables while protecting signals. The authors prove that the resulting prior concentrates on sparse vectors, the posterior contracts at the minimax rate, and guided shrinkage separates from unguided cases. BUGS-Active restricts MCMC updates to a data-driven active set of size much smaller than p, lowering per-iteration cost to linear in the active set while retaining sure screening and contraction. Simulations and a DNA methylation analysis with roughly 850,000 predictors show improved signal recovery and false-discovery control compared with standard shrinkage approaches.

Core claim

BUGS embeds univariate guidance within the nonlinear variance structure of a regularized horseshoe prior, inducing adaptive shrinkage that enhances signal-noise separation. The framework establishes prior concentration, posterior contraction, and guidance-induced shrinkage separation. BUGS-Active, an active-set MCMC approximation, reduces per-iteration complexity from O(p) to O(|A_n|) while preserving sure screening and contraction.

What carries the argument

Continuous modulation of local shrinkage variance parameters by univariate marginal associations, placed inside the regularized horseshoe prior.

If this is right

  • Prior concentration around sparse vectors and posterior contraction at the minimax rate continue to hold under the guided prior.
  • Guidance-induced shrinkage separation improves recovery of true signals while tightening control of false discoveries.
  • BUGS-Active preserves sure screening and contraction rates at O(|A_n|) per-iteration cost, enabling scaling to p=1,000,000.
  • Performance remains robust when the supplied univariate guidance carries little or no information about the true signals.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Marginal screening statistics can be fused directly into the prior rather than applied only as a pre-filter.
  • The same continuous-modulation idea may transfer to other global-local shrinkage families beyond the horseshoe.
  • In genomic applications the resulting sparse models may yield more interpretable selections because marginal relevance is explicitly respected.

Load-bearing premise

The continuous modulation of local shrinkage variance by univariate marginal associations produces the claimed separation and contraction even when the marginal information is noisy or only partially correlated with the true signals, and the active-set restriction does not materially alter the posterior geometry.

What would settle it

A simulation study in which marginal associations are generated independently of the true nonzero coefficients, testing whether posterior contraction at the claimed rate still occurs and whether BUGS-Active continues to recover the correct support.

Figures

Figures reproduced from arXiv: 2604.04964 by Priyam Das.

Figure 1
Figure 1. Figure 1: Illustration of marginal guidance in a sparse regression example with [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Schematic representation of the guided regularized horseshoe prior. The effective scale [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Effect of marginal guidance on the prior. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Summary of the top 10 CpGs identified by the BUGS framework. [PITH_FULL_IMAGE:figures/full_fig_p027_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Effect of the guidance budget on posterior estimation in BUGS-Active under Scenario 1 [PITH_FULL_IMAGE:figures/full_fig_p029_5.png] view at source ↗
read the original abstract

We propose Bayesian Univariate-Guided Sparse Regression (BUGS), a novel global-local shrinkage framework that incorporates marginal association information directly into the prior through a continuous modulation of shrinkage. Unlike existing approaches that treat predictors symmetrically or rely on post hoc screening, BUGS embeds univariate guidance within the nonlinear variance structure of a regularized horseshoe prior, inducing adaptive shrinkage that enhances signal-noise separation. We establish theoretical guarantees including prior concentration, posterior contraction, and guidance-induced shrinkage separation, while demonstrating robustness under uninformative guidance. To enable scalability in ultra-high dimensions, we develop BUGS-Active, an active-set MCMC approximation that restricts local updates to a data-adaptive subset A_n, reducing per-iteration complexity from O(p) to O(|A_n|) while preserving key theoretical properties such as sure screening and contraction. Empirically, the proposed framework achieves strong signal recovery together with substantially improved control of false discovery rates relative to existing methods. BUGS-Active scales to dimensions up to p = 1,000,000, and is applied to a DNA methylation study with n=1051 subjects and approximately 850,000 CpG sites, yielding accurate prediction and interpretable sparse selection. These results establish marginally guided shrinkage as a powerful and scalable paradigm for high-dimensional Bayesian inference.

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.

Circularity Check

1 steps flagged

Minor circularity in presenting 'guidance-induced shrinkage separation' as derived result

specific steps
  1. self definitional [Abstract]
    "BUGS embeds univariate guidance within the nonlinear variance structure of a regularized horseshoe prior, inducing adaptive shrinkage that enhances signal-noise separation. We establish theoretical guarantees including prior concentration, posterior contraction, and guidance-induced shrinkage separation"

    The separation is asserted as a theoretical guarantee, yet it is produced by construction when the univariate marginal associations (the 'guidance') are inserted into the local shrinkage variances. No additional derivation step is required beyond the prior definition.

full rationale

The core construction modulates the horseshoe prior variance directly with univariate marginal associations computed from the same data. The abstract and theoretical claims then list 'guidance-induced shrinkage separation' among the established guarantees. This separation follows immediately from the prior definition once the marginals are plugged in, rather than emerging as an independent consequence. Standard posterior contraction arguments remain non-circular, but the separation claim reduces to the model specification itself. No load-bearing self-citation chain or fitted-parameter renaming is evident from the provided text.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard posterior contraction theory for shrinkage priors plus the novel modulation mechanism; no new physical entities are introduced.

free parameters (1)
  • local shrinkage modulation parameters
    Parameters controlling how strongly univariate marginals affect local variance are introduced and must be chosen or estimated.
axioms (1)
  • standard math Standard results on posterior contraction for global-local shrinkage priors hold under the modulated variance structure
    Invoked to establish contraction and sure screening.

pith-pipeline@v0.9.0 · 5524 in / 1439 out tokens · 88893 ms · 2026-05-13T17:49:04.118173+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

  1. [1]

    , " * write output.state after.block = add.period write newline

    ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type url volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence := ...

    work page

  2. [2]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

    work page

  3. [3]

    The horseshoe+ estimator of ultra-sparse signals

    Bhadra et al. The horseshoe+ estimator of ultra-sparse signals. Bayesian Analysis, 12 0 (4): 0 1105--1131, 2017

    work page 2017

  4. [4]

    Bhattacharya, A

    A. Bhattacharya, A. Chakraborty, and B. Mallick. Fast sampling with gaussian scale mixture priors in high-dimensional regression. Biometrika, 103 0 (4): 0 985--991, 2016. doi:10.1093/biomet/asw042

    work page doi:10.1093/biomet/asw042 2016

  5. [5]

    Dirichlet--laplace priors for optimal shrinkage

    Bhattacharya et al. Dirichlet--laplace priors for optimal shrinkage. Journal of the American Statistical Association, 110 0 (512): 0 1479--1490, 2015

    work page 2015

  6. [6]

    Global patterns of 16s rrna diversity at a depth of millions of sequences per sample

    Caporaso et al. Global patterns of 16s rrna diversity at a depth of millions of sequences per sample. Proc. Natl. Acad. Sci., 108 0 (1): 0 4516--4522, 2011. doi:10.1073/pnas.1000080107

    work page doi:10.1073/pnas.1000080107 2011

  7. [7]

    Carvalho, N

    C. Carvalho, N. Polson, and J. Scott. The horseshoe estimator for sparse signals. Biometrika, 97 0 (2): 0 465--480, 2010

    work page 2010

  8. [8]

    Chatterjee, T

    S. Chatterjee, T. Hastie, and R. Tibshirani. Univariate-guided sparse regression. Harvard Data Science Review, 7 0 (3), 2025

    work page 2025

  9. [9]

    B. Efron. Large-Scale Inference: Empirical Bayes Methods for Estimation, Testing, and Prediction. Cambridge University Press, Cambridge, 2010

    work page 2010

  10. [10]

    Fan and J

    J. Fan and J. Lv. Sure independence screening for ultrahigh dimensional feature space. Journal of the Royal Statistical Society: Series B, 70 0 (5): 0 849--911, 2008

    work page 2008

  11. [11]

    Regularization paths for generalized linear models via coordinate descent

    Jerome Friedman, Trevor Hastie, and Robert Tibshirani. Regularization paths for generalized linear models via coordinate descent. Journal of Statistical Software, 33 0 (1): 0 1--22, 2010

    work page 2010

  12. [12]

    Ghosal, J.K

    S. Ghosal, J.K. Ghosh, and A. van der Vaart. Convergence rates of posterior distributions. The Annals of Statistics, 28 0 (2): 0 500--531, 2000. doi:10.1214/aos/1016218228

    work page doi:10.1214/aos/1016218228 2000

  13. [13]

    Johndrow, P

    J. Johndrow, P. Orenstein, and A. Bhattacharya. Scalable approximate mcmc algorithms for the horseshoe prior. Journal of Machine Learning Research, 21 0 (73): 0 1--61, 2020

    work page 2020

  14. [14]

    Kang and K

    M. Kang and K. Lee. Mhorseshoe: Approximate Algorithm for Horseshoe Prior, 2025. R package version 0.1.5

    work page 2025

  15. [15]

    Longitudinal prediction of dna methylation to forecast epigenetic outcomes

    Leroy et al. Longitudinal prediction of dna methylation to forecast epigenetic outcomes. eBioMedicine, 115: 0 105709, 2025

    work page 2025

  16. [16]

    Microbiome, metagenomics, and high-dimensional compositional data analysis

    Hongzhe Li. Microbiome, metagenomics, and high-dimensional compositional data analysis. Annual Review of Statistics and Its Application, 2: 0 73--94, 2015. doi:10.1146/annurev-statistics-010814-020351

    work page doi:10.1146/annurev-statistics-010814-020351 2015

  17. [17]

    Enes Makalic and Daniel F. Schmidt. BayesReg: Flexible Bayesian Penalized Regression Modelling . https://www.mathworks.com/matlabcentral/fileexchange/60823-flexible-bayesian-penalized-regression-modelling, 2020

    work page 2020

  18. [18]

    R. Neal. Slice sampling. Annals of Statistics, 31 0 (3): 0 705--767, 2003

    work page 2003

  19. [19]

    Park and G

    T. Park and G. Casella. The bayesian lasso. Journal of the American Statistical Association, 103 0 (482): 0 681--686, 2008. doi:10.1198/016214508000000337

    work page doi:10.1198/016214508000000337 2008

  20. [20]

    Sparsity information and regularization in the horseshoe and other shrinkage priors

    Juho Piironen and Aki Vehtari. Sparsity information and regularization in the horseshoe and other shrinkage priors. Electronic Journal of Statistics, 11 0 (2): 0 5018--5051, 2017

    work page 2017

  21. [21]

    Polson and J

    N. Polson and J. Scott. Shrink globally, act locally: Sparse bayesian regularization and prediction. In J. M. Bernardo, M. J. Bayarri, J. O. Berger, A. P. Dawid, D. Heckerman, A. F. M. Smith, and M. West, editors, Bayesian Statistics 9, pages 501--538. Oxford University Press, 2011

    work page 2011

  22. [22]

    H. Robbins. An empirical bayes approach to statistics. Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1: 0 157--163, 1956

    work page 1956

  23. [23]

    Rockova and E

    V. Rockova and E. George. The spike-and-slab LASSO . Journal of the American Statistical Association, 113 0 (521): 0 431--444, 2018. doi:10.1080/01621459.2016.1260469

    work page doi:10.1080/01621459.2016.1260469 2018

  24. [24]

    Next-generation sequencing technologies for environmental dna research

    Shokralla et al. Next-generation sequencing technologies for environmental dna research. Molecular Ecology, 21 0 (8): 0 1794--1805, 2012. doi:10.1111/j.1365-294X.2012.05538.x

    work page doi:10.1111/j.1365-294x.2012.05538.x 2012

  25. [25]

    Tibshirani

    R. Tibshirani. Regression shrinkage and selection via the L asso. Journal of the Royal Statistical Society: Series B, 58 0 (1): 0 267--288, 1996

    work page 1996

  26. [26]

    van der Pas, B

    S. van der Pas, B. Kleijn, and A. van der Vaart. The horseshoe estimator: Posterior concentration around nearly black vectors. Electronic Journal of Statistics, 8 0 (2): 0 2585--2618, 2014

    work page 2014

  27. [27]

    van der Pas, B

    S. van der Pas, B. Szab \'o , and A. van der Vaart. Adaptive posterior contraction rates for the horseshoe. Electronic Journal of Statistics, 11 0 (2): 0 3196--3225, 2017

    work page 2017

  28. [28]

    Bayesian regression using a prior on the model fit: The R2-D2 shrinkage prior

    Zhang et al. Bayesian regression using a prior on the model fit: The R2-D2 shrinkage prior. Journal of the American Statistical Association, 117 0 (538): 0 862--874, 2022

    work page 2022

  29. [29]

    H. Zou. The adaptive lasso and its oracle properties. Journal of the American Statistical Association, 101 0 (476): 0 1418--1429, 2006. doi:10.1198/016214506000000735

    work page doi:10.1198/016214506000000735 2006

  30. [30]

    Journal of the Royal Statistical Society Series B: Statistical Methodology , author =

    H. Zou and T. Hastie. Regularization and variable selection via the elastic net. Journal of the Royal Statistical Society: Series B, 67 0 (2): 0 301--320, 2005. doi:10.1111/j.1467-9868.2005.00503.x

    work page doi:10.1111/j.1467-9868.2005.00503.x 2005