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REVIEW 3 major objections 5 minor 82 references

CLT in high-dimensional Bayesian linear regression with low SNR

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper proves Gaussian limit theorems for posterior projections and posterior means in high-dimensional Bayesian linear regression under bounded signal-to-noise ratio, and uses them to build credible intervals with explicit average…

desk verdict A serious first stab at CLTs for non-contracting posteriors, but the general-design results have a real gap in the eigenvector error control; coverage claims need a fix. read the letter →

arxiv 2507.23285 v1 pith:NURHQVHN submitted 2025-07-31 math.ST math.PRstat.TH

Seunghyun Lee , Nabarun Deb , Sumit Mukherjee This is my paper · ORCID
classification math.STmath.PRstat.TH MSC 62F1262F1562F2562J0560F05
keywords high-dimensionalBayesianlinearregressioncentrallimittheoremmean-fieldapproximationBernstein-vonMisesuncertaintyquantificationlowsignal-to-noiseratiorandomfieldIsingmodelcredibleintervals
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 establishes central limit theorems for Bayesian linear regression in the bounded signal-to-noise regime, where the posterior does not contract and the prior keeps an asymptotically non-vanishing influence. For a delocalized unit direction $q$, the posterior projection $q^\top \beta$, centered by its mean-field approximation, converges in distribution to a Gaussian with an explicit variance, and the posterior mean also converges to a Gaussian centered at an explicit functional of the true coefficients. The limiting laws are prior-dependent rather than universal, and they hold without any sparsity assumption on the prior. As a consequence, credible intervals with correct nominal coverage (on average) become possible in settings where classical Bernstein-von Mises theory does not apply.

What carries the argument

The load-bearing object is the quadratic-interaction representation of the posterior, $\nu(d\beta) \propto \exp(\tfrac{1}{2}\beta^\top A_p \beta + c^\top \beta) \prod_i \mu_i(d\beta_i)$, which turns the linear projection $q^\top \beta$ into a linear statistic of a random-field Ising model. The mean-field fixed point $u_i = \psi'_i((A_p u)_i + c_i)$ supplies the centering; the finite-sample Berry-Esseen bound of Lee, Deb and Mukherjee (2025), reproduced as Lemmas A.1 and A.2, gives the Gaussian approximation with explicit error terms $R_{1p}, \dots, R_{4p}$; the high-temperature condition $\|A_p\|_4 \le \rho < 1$, the strong mean-field condition $\alpha_p = o(p^{-1/2})$, and homogeneous design make those error terms vanish. First- and second-order Poincar\'e inequalities control the fluctuations of the posterior mean around $q^\top u$.

What would settle it

Simulate the white-noise design at $p \approx c n^{2/3}$ with a compactly supported prior and a delocalized $q$, and compute the Kolmogorov-Smirnov distance between $q^\top \beta - q^\top u$ and $N(0, \upsilon_p)$ over repeated draws. If this distance stays bounded away from zero in any regime satisfying Assumptions 2.1-2.3, or if the error bound in Lemma A.1 fails while $\sqrt{p}\,\alpha_p = o(1)$, the claimed CLT would be refuted.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that non-contracting Bayesian linear regression still has a usable limit distribution theory: it is Gaussian, but the Gaussian is built around the mean-field posterior rather than around the least-squares estimator. Theorem 3.1 shows that under the white-noise design with $p \ll n^{2/3}$, a delocalized linear statistic of a posterior sample satisfies $q^\top \beta - q^\top u \to N(0, \upsilon)$, with $\upsilon$ an explicit limit computed from the prior and the true coefficients, and the posterior mean has its own Gaussian limit centered by $E[\psi'_0(\beta^*_i/\sigma^2 + W)]$ rather than by $q^\top \beta^*$. Theorem 3.4 extends the CLTs to general designs under an approximate delocalized eigenpair condition, with posterior variance $\upsilon/(1 - \lambda \upsilon)$ for the projection. Theorem 4.1 then computes the average coverage of the resulting credible intervals under any misspecified prior and shows that correct specification restores $1-\alpha$ coverage.

Load-bearing premise

The Gaussian limits stand on finite-sample Berry-Esseen bounds for random-field Ising models taken from the authors' companion paper; if those bounds fail, or require conditions beyond the high-temperature and strong mean-field assumptions used here, the central limit theorems do not follow.

Editorial extensions

If this is right

  • In the white-noise design, mean-field posterior credible intervals are asymptotically as valid as posterior intervals: $q^\top \beta$ under the mean-field approximation has the same Gaussian limit as under the true posterior.
  • For general designs the posterior variance $\upsilon/(1 - \lambda \upsilon)$ differs from the mean-field variance $\upsilon$, so mean-field intervals can over- or under-cover depending on the sign of the eigenparameter $\lambda$.
  • The coverage formula in Theorem 4.1 gives an explicit, computable coverage level for any misspecified prior, and correct prior specification recovers coverage $1-\alpha$ in an average sense.
  • For spike-and-slab priors, the posterior mean behaves as under the global null when the slab is symmetric or when $q$ is a contrast with $q_{\mathrm{tot}} = o(p^u)$; otherwise the limit has a nonzero center or diverges.
  • The results extend to $\gamma$-fractional posteriors by replacing $\sigma^2$ with $\sigma^2/\gamma$, with smaller $\gamma$ relaxing the high-temperature condition.

Reading between the lines

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

  • Editorial inference: the delocalization assumption $\|q\|_\infty \to 0$ is the uniform asymptotic negligibility condition, so per-coordinate posterior inference in this regime should generically be non-Gaussian rather than a mild technical restriction.
  • Editorial inference: the data-only pair $(\lambda_p, \upsilon_p)$ can serve as a cheap diagnostic for when mean-field uncertainty quantification fails, since multiplying the mean-field variance by $1/(1 - \lambda_p \upsilon_p)$ corrects the leading discrepancy.
  • Editorial inference: because the coverage guarantee in Theorem 4.1 averages over the random true coefficient $\beta^*$, conditional coverage for a realized $\beta^*$ should oscillate at $O(1)$; the paper's guarantee is not a conditional confidence statement.
  • Editorial inference: if the imported Berry-Esseen bounds are later sharpened, the $p \ll n^{2/3}$ threshold and the strong mean-field condition are the natural places to improve; conversely, if counterexamples exist near those boundaries, the high-temperature assumption is likely inherent to Gaussian limits in this setting.
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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 studies central limit theorems for one-dimensional projections of the posterior and for the posterior mean in high-dimensional Bayesian linear regression under a bounded-SNR, non-contracting regime. The posterior is written as a quadratic interaction model, and the paper shows that delocalized linear statistics, centered by the mean-field posterior mean q^T u or by explicit smooth transforms of the sufficient statistic, are asymptotically Gaussian with variance that can be consistently estimated. The results cover a white-noise random design under p << n^{2/3}, and a general (possibly deterministic) design under high-temperature and strong mean-field assumptions plus an approximate delocalized eigenpair condition. The paper also constructs credible intervals and derives their average coverage under misspecified priors, showing exact nominal coverage when the prior is correctly specified. The technical engine is a set of finite-sample Berry-Esseen bounds for random-field Ising models imported from the authors' companion paper, combined with Gaussian and second-order Poincaré inequalities.

Significance. If the main results are fully verified, this is a valuable contribution. The paper appears to be the first to derive limiting distribution theory for posterior projections in non-contracting Bayesian linear regression, where the prior remains influential asymptotically. The explicit Gaussian limits, the sample-based variance estimators, the sharpened dimension regime p << n^{2/3} for white-noise designs, and the coverage formulas under misspecified priors are concrete and falsifiable. The paper also carefully works out several examples (spike-and-slab, ANOVA, fractional posteriors) and gives finite-sample error bounds in the appendix rather than only asymptotic statements. The main caveats are that two central finite-sample Berry-Esseen lemmas are quoted from an unpublished companion manuscript without proof, and that one internal error-term estimate in the general-design proof does not close under the stated eigenpair assumption. Both issues are load-bearing but appear fixable, so a major revision is appropriate.

major comments (3)
  1. [Section A.4, proof of Theorem 3.4(a)(ii) and (b)] The proof does not establish that the eigenvector-error terms in Lemma A.2 vanish under assumption (3.6). After equations (A.32)-(A.33), the paper has R1p = O_P(α_p + ||ϵ||^2) and R2p = O_P(p α_p). Lemma A.2 contains the term sqrt(R2p)||ϵ||, which is O_P(sqrt(p α_p)||ϵ||). Assumption (3.6), namely (1 + sqrt(p)α_p)||ϵ|| = o(1), does not imply sqrt(p α_p)||ϵ|| = o(1). Under the strong mean-field condition α_p = o(p^{-1/2}), the factor sqrt(p)α_p tends to zero, so (3.6) reduces to ||ϵ|| = o(1); for instance α_p = p^{-0.6} gives sqrt(p α_p) = p^{0.2} → ∞, so the product need not vanish. The stronger condition (1 + sqrt(p α_p))||ϵ|| = o(1) is needed. This gap affects the modified-centering CLT (3.7), the posterior-mean CLT (3.8), and consequently the coverage conclusion in Theorem 4.1, which invokes part (b) of Theorem 3.4.
  2. [Appendix A.1, Lemmas A.1 and A.2] The two central finite-sample Berry-Esseen bounds are stated as Lemmas A.1 and A.2 but are not proved in this manuscript; they are quoted as Theorems 2.4 and 2.5 of Lee, Deb and Mukherjee (2025), a companion preprint by the same authors that is not yet published and does not appear to be accompanied by a complete proof or formal verification in the present submission. Since these lemmas are the main engine for both Theorems 3.1 and 3.4, the referee cannot fully verify the central claims as submitted. The authors should either include complete proofs of these lemmas, or provide a publicly available companion manuscript with full proofs, or state them with complete explicit hypotheses and derivations in an appendix.
  3. [Section 2.2 and Appendix A.2, dependency on companion results] Several other load-bearing ingredients are imported from the same companion paper without proof: Lemma 2.1 is stated as Lemma 2.2 of Lee, Deb and Mukherjee (2025), Proposition 2.3 as Theorem 2.3 there, and Lemma C.1 as Lemma 3.2(b) there. The verification that the white-noise design satisfies Assumptions 2.1-2.3 also relies on Lemmas 4.1 and 4.2 of that paper. This is more than routine citation: the structural properties of the mean-field optimizer and the moment bound of Lemma C.1 are used directly in the proofs of Propositions 2.3-2.4 and in the main theorems. Please clarify which results are new to this paper and make the companion proofs available for inspection.
minor comments (5)
  1. [Appendix A.1 and conditions after Lemma A.1] In the bullet list following Lemma A.1, the condition is written as “∥q∥ → 0”, but q is a unit vector; the intended condition is clearly ∥q∥∞ → 0. The same typo appears in the proof overview around the discussion of delocalized eigenvectors.
  2. [Example 2.2] The text repeatedly renders “ANOVA” as “ANOV A” in Example 2.2 and in Remark 3.2; please fix the formatting.
  3. [Theorem 3.4 statement] In the centering in (3.8), the conditional expectation is written as E[ψ'0(d0 β⋆_i + W0) | β⋆_i ], with an extra closing bracket before the comma; this should be E[ψ'0(d0 β⋆_i + W0) | β⋆_i] for consistency with the surrounding notation.
  4. [Proof of Theorem 3.4(b)] At the start of the proof of the first claim in (A.42), the text says “we assume E1 = o(1) for part (c)”, but the strengthened assumption appears in part (b) of Theorem 3.4; the reference should be corrected.
  5. [Lemma A.3 and surrounding display] In Lemma A.3 the statement says the law of δ(y,X) is conditional on X, β⋆, but the bounded-Lipschitz distance expression mixes conditional and unconditional notation; please make explicit that all expectations in the bound are conditional on (X, β⋆).

Circularity Check

1 steps flagged · score 5.0 of 10

Main CLTs inherit their Gaussian approximation from an unpublished, same-author companion paper; no fitted-input circularity, but the self-citation is load-bearing.

  1. self citation load bearing [Section 5 (proof overview) and Appendix A.1, Lemmas A.1 and A.2; also Lemma 2.1 and Proposition 2.3]
    "Our proof arguments build upon recently developed finite-sample CLTs for (i) Gibbs measures with quadratic interactions (Lee, Deb and Mukherjee, 2025) ... Mainly, we use two results from Lee, Deb and Mukherjee (2025) to understand the fluctuations of T (β). ... Lemma A.1 (Theorem 2.4 in Lee, Deb and Mukherjee (2025)). ... Lemma A.2 (Theorem 2.5 in Lee, Deb and Mukherjee (2025))."

    Theorems 3.1 and 3.4, and consequently the credible-interval coverage result Theorem 4.1, reduce the Gaussian limit for the posterior projection T(β) to finite-sample Berry-Esseen bounds stated as Lemmas A.1 and A.2. These bounds are not proved in this manuscript; they are quoted verbatim from a companion arXiv preprint by the same three authors. The paper does substantial independent work verifying that the error terms vanish, so the conclusion is not a tautology, but the central Gaussian approximation itself is imported from an unpublished same-author source. If that companion result is flawed or its assumptions are not met, the main CLTs do not follow.

full rationale

The paper's central Gaussian limits are not fitted-input predictions: the centers and variances are explicit functions of the prior, design, and data, and the coverage calculation in Theorem 4.1 computes the actual limiting probability rather than re-importing it from a fit. The main circularity concern is the reliance on the authors' own companion paper Lee, Deb and Mukherjee (2025) for the Random-Field-Ising Berry-Esseen inequalities (Lemmas A.1 and A.2), the uniqueness of the mean-field optimizer (Lemma 2.1), the second-moment bound (Proposition 2.3), and related random-matrix facts. These citations are load-bearing because Theorem 3.1(a), Theorem 3.4(a), and the posterior-mean CLTs are proved by checking that the error terms in those imported bounds vanish; the Gaussian approximation itself is not re-derived here. This is more than minor self-citation (score 4) because the unpublished companion underpins the main theorems, but it is not full circularity (score 8-10) because the present paper supplies substantial independent analysis of the dependent Gaussian field c, the variance limits, and the coverage formula, and the companion lemmas concern a broader Ising-model setting rather than the paper's exact conclusions. The skeptic's specific objection about condition (3.6) being too weak to control sqrt(R2p)||epsilon|| is a correctness gap in the error-term verification, not a circularity of the fit-or-definition type; it is therefore noted as a risk rather than added as a circular step. Overall score 5 reflects a moderate, load-bearing self-citation burden.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

The central claims rest on a set of design and prior assumptions, the DGP, and the existence of limiting variances. The most fragile are the Strong Mean-Field condition, the approximate-eigenvector condition, and the compact support of the prior. The paper does not introduce new physical or statistical entities.

assumptions (9)
  • domain assumption The prior µ is compactly supported on [-1,1] and nondegenerate.
    Assumed in Section 2 so that the log-mgf ψ and its derivatives are bounded and the posterior normalizing constant is finite. Excludes Gaussian or unbounded priors.
  • domain assumption High-temperature condition: ∥Ap∥4 ≤ ρ for some ρ ∈ (0,1).
    Assumption 2.1. Ensures uniqueness of the mean-field maximizer and controls the constants in the Berry-Esseen bounds.
  • domain assumption Strong Mean-Field condition: αp = max_i Σ_j Ap(i,j)^2 = o(p^{-1/2}).
    Assumption 2.2. Necessitated for the error terms in the CLT lemmas to vanish; without it the paper notes counterexamples to a law of large numbers.
  • domain assumption Homogeneous design: d_i ≥ d_min and Σ_i(d_i-d_0)^2 = O(1) (o(1) in Theorem 3.4(b)).
    Assumption 2.3. Allows replacing the per-coordinate prior tilts by a common limit µ0 and ψ0.
  • domain assumption Data generating process y = Xβ⋆ + ε⋆ with ε⋆ ~ N(0, σ²I).
    Assumption 2.4. Yields the Gaussian field c | X,β⋆ ~ N(Σpβ⋆, Σp), which is used throughout the proofs.
  • domain assumption Limits in (3.1) and (3.3) exist with υ>0, ς²>0.
    Theorem 3.1 assumes these limits; the examples show they hold for i.i.d. μ⋆ and spike-and-slab. If they fail, the CLT variances are not defined.
  • ad hoc to paper Approximate delocalized eigenpair: ∥Apq - λp q∥→0, λp→λ∈(-1,1), ∥q∥∞→0.
    Assumption (3.5) for Theorem 3.4. The eigenvector condition is a technical requirement inherited from the companion paper and limits the general design applicability.
  • domain assumption µ and µ⋆ are symmetric around 0; µ⋆ has finite fourth moments.
    Used in Theorem 3.4 and Corollary 3.3 to make certain expectations zero and to apply moment bounds for the error terms.
  • standard math Standard Gaussian tools: Poincaré inequality, second-order Poincaré inequality, Stein's lemma, Lindeberg-Feller CLT.
    Used as background for the variance and CLT computations in the appendix.

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Cite this review

Pith. "Pith review of CLT in high-dimensional Bayesian linear regression with low SNR." pith.science (2026). https://pith.science/paper/NURHQVHN

@misc{pith2026250723285,
  author       = {Pith},
  title        = {Pith review of: CLT in high-dimensional Bayesian linear regression with low SNR},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NURHQVHN}},
  note         = {Machine review of arXiv:2507.23285}
}
read the original abstract

We study central limit theorems for linear statistics in high-dimensional Bayesian linear regression with product priors. Unlike the existing literature where the focus is on posterior contraction, we work under a non-contracting regime where neither the likelihood nor the prior dominates the other. This is motivated by modern high-dimensional datasets characterized by a bounded signal-to-noise ratio. This work takes a first step towards understanding limit distributions for one-dimensional projections of the posterior, as well as the posterior mean, in such regimes. Analogous to contractive settings, the resulting limiting distributions are Gaussian, but they heavily depend on the chosen prior and center around the Mean-Field approximation of the posterior. We study two concrete models of interest to illustrate this phenomenon -- the white noise design, and the (misspecified) Bayesian model. As an application, we construct credible intervals and compute their coverage probability under any misspecified prior. Our proofs rely on a combination of recent developments in Berry-Esseen type bounds for Random Field Ising models and both first and second order Poincar\'{e} inequalities. Notably, our results do not require any sparsity assumptions on the prior.

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Reference graph

Works this paper leans on

82 extracted references · 60 canonical work pages

  1. [1]

    , Montiel Olea , Jos\'e Luis J

    barticle [author] Avella Medina , Marco M. , Montiel Olea , Jos\'e Luis J. L. , Rush , Cynthia C. Velez , Amilcar A. ( 2022 ). On the robustness to misspecification of -posteriors and their variational approximations . J. Mach. Learn. Res. 23 Paper No. [147], 51 . 4577099 barticle

    2022

  2. [2]

    Schwartzman , Armin A

    barticle [author] Azriel , David D. Schwartzman , Armin A. ( 2020 ). Estimation of linear projections of non-sparse coefficients in high-dimensional regression . Electronic Journal of Statistics 14 174 -- 206 . 10.1214/19-EJS1656 barticle

    doi:10.1214/19-ejs1656 2020

  3. [3]

    , Macris , Nicolas N

    barticle [author] Barbier , Jean J. , Macris , Nicolas N. , Dia , Mohamad M. Krzakala , Florent F. ( 2020 ). Mutual information and optimality of approximate message-passing in random linear estimation . IEEE Transactions on Information Theory 66 4270--4303 . barticle

    2020

  4. [4]

    Mukherjee , Sumit S

    barticle [author] Basak , Anirban A. Mukherjee , Sumit S. ( 2017 ). Universality of the mean-field for the P otts model . Probability Theory and Related Fields 168 557--600 . barticle

    work page 2017

  5. [5]

    barticle [author] Bellec , Pierre C. P. C. Zhang , Cun-Hui C.-H. ( 2023 ). Debiasing convex regularized estimators and interval estimation in linear models . The Annals of Statistics 51 391 -- 436 . 10.1214/22-AOS2243 barticle

    work page doi:10.1214/22-aos2243 2023

  6. [6]

    , Deb , Nabarun N

    barticle [author] Bhattacharya , Sohom S. , Deb , Nabarun N. Mukherjee , Sumit S. ( 2023 ). Gibbs Measures with Multilinear Forms . arXiv preprint arXiv:2307.14600 . barticle

    work page arXiv 2023

  7. [7]

    , Deb , Nabarun N

    barticle [author] Bhattacharya , Sohom S. , Deb , Nabarun N. Mukherjee , Sumit S. ( 2024 ). LDP for inhomogeneous U-statistics . The Annals of Applied Probability 34 5769--5808 . barticle

    work page 2024

  8. [8]

    , Pati , Debdeep D

    barticle [author] Bhattacharya , Anirban A. , Pati , Debdeep D. Yang , Yun Y. ( 2019 ). Bayesian fractional posteriors . Ann. Statist. 47 39--66 . 10.1214/18-AOS1712 3909926 barticle

    doi:10.1214/18-aos1712 2019

Show all 82 references
  1. [9]

    barticle [author] Bickel , Peter J P. J. Yahav , Joseph A J. A. ( 1969 ). Some contributions to the asymptotic theory of B ayes solutions . Zeitschrift f \"u r Wahrscheinlichkeitstheorie und verwandte Gebiete 11 257--276 . barticle

    1969

  2. [10]

    ( 2006 )

    barticle [author] Bishop , Christopher C. ( 2006 ). Pattern recognition and machine learning . Springer google scholar 2 5--43 . barticle

    2006

  3. [11]

    barticle [author] Blei , David M D. M. , Kucukelbir , Alp A. McAuliffe , Jon D J. D. ( 2017 ). Variational inference: A review for statisticians . Journal of the American statistical Association 112 859--877 . barticle

    2017

  4. [12]

    bbook [author] Bogachev , V. I. V. I. ( 2007 ). Measure theory. V ol. I , II . Springer-Verlag, Berlin . 10.1007/978-3-540-34514-5 2267655 bbook

    2007 doi

  5. [13]

    , Bousquet , Olivier O

    barticle [author] Boucheron , St\'ephane S. , Bousquet , Olivier O. , Lugosi , G\'abor G. Massart , Pascal P. ( 2005 ). Moment inequalities for functions of independent random variables . Ann. Probab. 33 514--560 . 10.1214/009117904000000856 2123200 barticle

    2005 doi

  6. [14]

    , Schmidt-Hieber , Johannes J

    barticle [author] Castillo , Isma \"e l I. , Schmidt-Hieber , Johannes J. van der Vaart , Aad A. ( 2015 ). Bayesian linear regression with sparse priors . The Annals of Statistics 43 1986 -- 2018 . 10.1214/15-AOS1334 barticle

    2015 doi

  7. [15]

    van der Vaart , Aad A

    barticle [author] Castillo , Isma\"el I. van der Vaart , Aad A. ( 2012 ). Needles and straw in a haystack: posterior concentration for possibly sparse sequences . Ann. Statist. 40 2069--2101 . 10.1214/12-AOS1029 3059077 barticle

    2012 doi

  8. [16]

    , Fan , Zhou Z

    barticle [author] Celentano , Michael M. , Fan , Zhou Z. , Lin , Licong L. Mei , Song S. ( 2023 ). Mean-field variational inference with the TAP free energy: Geometric and statistical properties in linear models . arXiv preprint arXiv:2311.08442 . barticle

    2023 arXiv

  9. [17]

    , Kuchibhotla , Arun Kumar A

    barticle [author] Chang , Woonyoung W. , Kuchibhotla , Arun Kumar A. K. Rinaldo , Alessandro A. ( 2023 ). Inference for Projection Parameters in Linear Regression: beyond d=o(n^ 1/2 ) . arXiv preprint arXiv:2307.00795 . barticle

    2023 arXiv

  10. [18]

    ( 2009 )

    barticle [author] Chatterjee , Sourav S. ( 2009 ). Fluctuations of eigenvalues and second order P oincar \'e inequalities . Probability Theory and Related Fields 143 1--40 . barticle

    2009

  11. [19]

    ( 2024 )

    barticle [author] Chatterjee , Sourav S. ( 2024 ). Features of a spin glass in the random field I sing model . Communications in Mathematical Physics 405 93 . barticle

    2024

  12. [20]

    Shao , Qi-Man Q.-M

    barticle [author] Chatterjee , Sourav S. Shao , Qi-Man Q.-M. ( 2011 ). Nonnormal approximation by Stein’s method of exchangeable pairs with application to the Curie--Weiss model . The Annals of Applied Probability 21 464--483 . barticle

    2011

  13. [21]

    ( 2024 )

    bbook [author] Chewi , Sinho S. ( 2024 ). Log-Concave Sampling . Preprint . bbook

    2024

  14. [22]

    barticle [author] de Matos , JMG Amaro J. A. Perez , J Fernando J. F. ( 1991 ). Fluctuations in the C urie- W eiss version of the random field I sing model . Journal of Statistical Physics 62 587--608 . barticle

    1991

  15. [23]

    Mukherjee , Sumit S

    barticle [author] Deb , Nabarun N. Mukherjee , Sumit S. ( 2023 ). Fluctuations in mean-field I sing models . The Annals of Applied Probability 33 1961--2003 . barticle

    2023

  16. [24]

    , Mukherjee , Rajarshi R

    barticle [author] Deb , Nabarun N. , Mukherjee , Rajarshi R. , Mukherjee , Sumit S. Yuan , Ming M. ( 2024 ). Detecting structured signals in I sing models . Ann. Appl. Probab. 34 1--45 . 10.1214/23-aap1929 4696272 barticle

    2024 doi

  17. [25]

    , Guan , Leying L

    barticle [author] Fan , Zhou Z. , Guan , Leying L. , Shen , Yandi Y. Wu , Yihong Y. ( 2023 ). Gradient flows for empirical Bayes in high-dimensional linear models . arXiv preprint arXiv:2312.12708 . barticle

    2023 arXiv

  18. [26]

    , Ko , Justin J

    barticle [author] Fan , Zhou Z. , Ko , Justin J. , Loureiro , Bruno B. , Lu , Yue M Y. M. Shen , Yandi Y. ( 2025 ). Dynamical mean-field analysis of adaptive Langevin diffusions: Replica-symmetric fixed point and empirical Bayes . arXiv preprint arXiv:2504.15558 . barticle

    2025

  19. [27]

    barticle [author] George , Edward I E. I. McCulloch , Robert E R. E. ( 1993 ). Variable selection via Gibbs sampling . Journal of the American Statistical Association 88 881--889 . barticle

    1993

  20. [28]

    barticle [author] George , Edward I E. I. McCulloch , Robert E R. E. ( 1997 ). Approaches for Bayesian variable selection . Statistica sinica 339--373 . barticle

    1997

  21. [29]

    , Lubetzky , Eyal E

    barticle [author] Gheissari , Reza R. , Lubetzky , Eyal E. Peres , Yuval Y. ( 2018 ). Concentration inequalities for polynomials of contracting I sing models . Electron. Commun. Probab. 23 Paper No. 76, 12 . 10.1214/18-ECP173 3873783 barticle

    2018 doi

  22. [30]

    ( 1999 )

    barticle [author] Ghosal , Subhashis S. ( 1999 ). Asymptotic normality of posterior distributions in high-dimensional linear models . Bernoulli 315--331 . barticle

    1999

  23. [31]

    ( 2000 )

    barticle [author] Ghosal , Subhashis S. ( 2000 ). Asymptotic normality of posterior distributions for exponential families when the number of parameters tends to infinity . J. Multivariate Anal. 74 49--68 . 10.1006/jmva.1999.1874 1790613 barticle

    2000

  24. [32]

    , Broderick , Tamara T

    barticle [author] Giordano , Ryan R. , Broderick , Tamara T. et al. ( 2025 ). How good is your Laplace approximation of the Bayesian posterior? Finite-sample computable error bounds for a variety of useful divergences . Journal of Machine Learning Research 26 1--81 . barticle

    2025

  25. [33]

    ( 2012 )

    bincollection [author] Gr\"unwald , Peter P. ( 2012 ). The safe B ayesian: learning the learning rate via the mixability gap . In Algorithmic learning theory . Lecture Notes in Comput. Sci. 7568 169--183 . Springer, Heidelberg . 10.1007/978-3-642-34106-9\_16 3042889 bincollection

    2012 doi

  26. [34]

    Stephens , Matthew M

    barticle [author] Guan , Yongtao Y. Stephens , Matthew M. ( 2011 ). Bayesian variable selection regression for G enome-wide association studies and other large-scale problems . Ann. Appl. Stat. 5 1780--1815 . 10.1214/11-AOAS455 2884922 barticle

    2011 doi

  27. [35]

    barticle [author] Hastings , W. K. W. K. ( 1970 ). Monte C arlo sampling methods using M arkov chains and their applications . Biometrika 57 97--109 . 10.1093/biomet/57.1.97 3363437 barticle

    1970 doi

  28. [36]

    barticle [author] Ishwaran , Hemant H. Rao , J. Sunil J. S. ( 2005 ). Spike and slab variable selection: frequentist and B ayesian strategies . Ann. Statist. 33 730--773 . 10.1214/009053604000001147 2163158 barticle

    2005 doi

  29. [37]

    , Koehler , Frederic F

    binproceedings [author] Jain , Vishesh V. , Koehler , Frederic F. Mossel , Elchanan E. ( 2018 ). The mean-field approximation: Information inequalities, algorithms, and complexity . In Conference On Learning Theory 1326--1347 . PMLR . binproceedings

    2018

  30. [38]

    , L\"owe , Matthias M

    barticle [author] Kabluchko , Zakhar Z. , L\"owe , Matthias M. Schubert , Kristina K. ( 2022 ). Fluctuations of the magnetization for I sing models on E rd o s- R \'e nyi random graphs---the regimes of low temperature and external magnetic field . ALEA Lat. Am. J. Probab. Math...

    2022 doi

  31. [39]

    L \"o we , Matthias M

    barticle [author] Kabluchko , Zakhar Z. L \"o we , Matthias M. ( 2024 ). Propagation of chaos in the random field C urie-- W eiss model . Journal of Statistical Mechanics: Theory and Experiment 2024 113206 . barticle

    2024

  32. [40]

    ( 2024 )

    barticle [author] Katsevich , Anya A. ( 2024 ). The L aplace asymptotic expansion in high dimensions: a nonasymptotic analysis . arXiv preprint arXiv:2406.12706 . barticle

    2024 arXiv

  33. [41]

    ( 2025 )

    barticle [author] Katsevich , Anya A. ( 2025 ). Improved dimension dependence in the Bernstein--von Mises theorem via a new Laplace approximation bound . Information and Inference: A Journal of the IMA 14 iaaf020 . barticle

    2025

  34. [42]

    barticle [author] Kleijn , B. J. K. B. J. K. van der Vaart , A. W. A. W. ( 2012 ). The B ernstein- V on- M ises theorem under misspecification . Electron. J. Stat. 6 354--381 . 10.1214/12-EJS675 2988412 barticle

    2012 doi

  35. [43]

    , Mukherjee , Sumit S

    barticle [author] Lacker , Daniel D. , Mukherjee , Sumit S. Yeung , Lane Chun L. C. ( 2024 ). Mean field approximations via log-concavity . International Mathematics Research Notices 2024 6008--6042 . barticle

    2024

  36. [44]

    ( 1953 )

    barticle [author] Le Cam , Lucien L. ( 1953 ). On some asymptotic properties of maximum likelihood estimates and related B ayes' estimates . Univ. Calif. Publ. in Statist. 1 277--330 . barticle

    1953

  37. [45]

    , Deb , Nabarun N

    barticle [author] Lee , Seunghyun S. , Deb , Nabarun N. Mukherjee , Sumit S. ( 2025 ). Fluctuations in random field Ising models . arXiv preprint . barticle

    2025

  38. [46]

    bbook [author] Lehmann , Erich L E. L. Casella , George G. ( 2006 ). Theory of point estimation . Springer Science & Business Media . bbook

    2006

  39. [47]

    barticle [author] Lenk , Peter J P. J. Rao , Ambar G A. G. ( 1990 ). New models from old: Forecasting product adoption by hierarchical Bayes procedures . Marketing Science 9 42--53 . barticle

    1990

  40. [48]

    ( 2017 )

    barticle [author] Lu , Yulong Y. ( 2017 ). On the B ernstein-von M ises theorem for high dimensional nonlinear B ayesian inverse problems . arXiv preprint arXiv:1706.00289 . barticle

    2017 arXiv

  41. [49]

    , Foini , Laura L

    barticle [author] Maillard , Antoine A. , Foini , Laura L. , Castellanos , Alejandro Lage A. L. , Krzakala , Florent F. , M \'e zard , Marc M. Zdeborov \'a , Lenka L. ( 2019 ). High-temperature expansions and message passing algorithms . Journal of Statistical Mechanics: Theor...

    2019

  42. [50]

    binproceedings [author] Margossian , Charles C C. C. Saul , Lawrence K L. K. ( 2023 ). The shrinkage-delinkage trade-off: An analysis of factorized gaussian approximations for variational inference . In Uncertainty in Artificial Intelligence 1358--1367 . PMLR . binproceedings

    2023

  43. [51]

    , Rosenbluth , Arianna W A

    barticle [author] Metropolis , Nicholas N. , Rosenbluth , Arianna W A. W. , Rosenbluth , Marshall N M. N. , Teller , Augusta H A. H. Teller , Edward E. ( 1953 ). Equation of state calculations by fast computing machines . The Journal of Chemical Physics 21 1087--1092 . barticle

    1953

  44. [52]

    barticle [author] Miller , Jeffrey W. J. W. Dunson , David B. D. B. ( 2019 ). Robust B ayesian inference via coarsening . J. Amer. Statist. Assoc. 114 1113--1125 . 10.1080/01621459.2018.1469995 4011766 barticle

    2019

  45. [53]

    barticle [author] Minka , Tom T. et al. ( 2005 ). Divergence measures and message passing . barticle

    2005

  46. [54]

    Wu , Yuchen Y

    barticle [author] Montanari , Andrea A. Wu , Yuchen Y. ( 2024 ). Provably efficient posterior sampling for sparse linear regression via measure decomposition . arXiv preprint arXiv:2406.19550 . barticle

    2024 arXiv

  47. [55]

    barticle [author] Moon , Hyungsik Roger H. R. Schorfheide , Frank F. ( 2012 ). Bayesian and frequentist inference in partially identified models . Econometrica 80 755--782 . barticle

    2012

  48. [56]

    , Carbonetto , Peter P

    barticle [author] Morgante , Fabio F. , Carbonetto , Peter P. , Wang , Gao G. , Zou , Yuxin Y. , Sarkar , Abhishek A. Stephens , Matthew M. ( 2023 ). A flexible empirical Bayes approach to multivariate multiple regression, and its improved accuracy in predicting multi-tissue g...

    2023

  49. [57]

    , Qiu , Jiaze J

    barticle [author] Mukherjee , Sumit S. , Qiu , Jiaze J. Sen , Subhabrata S. ( 2024 ). On Naive Mean-Field Approximation for high-dimensional canonical GLM s . arXiv preprint arXiv:2406.15247 . barticle

    2024 arXiv

  50. [58]

    Sen , Subhabrata S

    barticle [author] Mukherjee , Sumit S. Sen , Subhabrata S. ( 2022 ). Variational Inference in high-dimensional linear regression . The Journal of Machine Learning Research 23 13703--13758 . barticle

    2022

  51. [59]

    , Sen , Bodhisattva B

    barticle [author] Mukherjee , Sumit S. , Sen , Bodhisattva B. Sen , Subhabrata S. ( 2023 ). A mean field approach to empirical B ayes estimation in high-dimensional linear regression . arXiv preprint arXiv:2309.16843 . barticle

    2023 arXiv

  52. [60]

    ( 2018 )

    bbook [author] Nesterov , Yurii Y. ( 2018 ). Lectures on convex optimization , second ed. Springer Optimization and Its Applications 137 . Springer, Cham . 10.1007/978-3-319-91578-4 3839649 bbook

    2018 doi

  53. [61]

    Wright , Stephen J

    bbook [author] Nocedal , Jorge J. Wright , Stephen J. S. J. ( 1999 ). Numerical optimization . Springer Series in Operations Research . Springer-Verlag, New York . 10.1007/b98874 1713114 bbook

    1999 doi

  54. [62]

    West , Mike M

    bbook [author] O'Hagan , Anthony A. West , Mike M. , eds. ( 2013 ). The O xford handbook of applied B ayesian analysis . Oxford University Press, Oxford . 3157454 bbook

    2013

  55. [63]

    ( 2024 )

    barticle [author] Qiu , Jiaze J. ( 2024 ). Sub-optimality of the Naive Mean Field approximation for proportional high-dimensional Linear Regression . Advances in Neural Information Processing Systems 36 . barticle

    2024

  56. [64]

    bbook [author] Rachev , Svetlozar T S. T. , Hsu , John SJ J. S. , Bagasheva , Biliana S B. S. Fabozzi , Frank J F. J. ( 2008 ). Bayesian methods in finance . John Wiley & Sons . bbook

    2008

  57. [65]

    , Medina , Marco Avella M

    binproceedings [author] Ray , Ruchira R. , Medina , Marco Avella M. A. Rush , Cynthia C. ( 2023 ). Asymptotics for power posterior mean estimation . In 2023 59th Annual Allerton Conference on Communication, Control, and Computing (Allerton) 1--8 . IEEE . binproceedings

    2023

  58. [66]

    ( 2018 )

    barticle [author] Ro c kov\'a , Veronika V. ( 2018 ). Bayesian estimation of sparse signals with a continuous spike-and-slab prior . Ann. Statist. 46 401--437 . 10.1214/17-AOS1554 3766957 barticle

    2018 doi

  59. [67]

    ( 2013 )

    barticle [author] Spokoiny , Vladimir V. ( 2013 ). Bernstein-von M ises theorem for growing parameter dimension . arXiv preprint arXiv:1302.3430 . barticle

    2013 arXiv

  60. [68]

    ( 2023 )

    barticle [author] Spokoiny , Vladimir V. ( 2023 ). Dimension free nonasymptotic bounds on the accuracy of high-dimensional L aplace approximation . SIAM/ASA J. Uncertain. Quantif. 11 1044--1068 . 10.1137/22M1495688 4646387 barticle

    2023 doi

  61. [69]

    barticle [author] Stein , Charles M C. M. ( 1981 ). Estimation of the mean of a multivariate normal distribution . The A nnals of S tatistics 1135--1151 . barticle

    1981

  62. [70]

    ( 2010 )

    barticle [author] Tamer , Elie E. ( 2010 ). Partial identification in econometrics . Annu. Rev. Econ. 2 167--195 . barticle

    2010

  63. [71]

    bbook [author] Van der Vaart , Aad W A. W. ( 2000 ). Asymptotic statistics 3 . Cambridge university press . bbook

    2000

  64. [72]

    ( 2014 )

    barticle [author] Van Handel , Ramon R. ( 2014 ). Probability in high dimension . Lecture Notes (Princeton University) . barticle

    2014

  65. [73]

    ( 2018 )

    bbook [author] Vershynin , Roman R. ( 2018 ). High-dimensional probability: An introduction with applications in data science 47 . Cambridge university press . bbook

    2018

  66. [74]

    barticle [author] Wainwright , Martin J M. J. , Jordan , Michael I M. I. et al. ( 2008 ). Graphical models, exponential families, and variational inference . Foundations and Trends in Machine Learning 1 1--305 . barticle

    2008

  67. [75]

    Blei , David M D

    barticle [author] Wang , Yixin Y. Blei , David M D. M. ( 2018 ). Frequentist consistency of variational Bayes . Journal of the American Statistical Association . barticle

    2018

  68. [76]

    , Pati , Debdeep D

    barticle [author] Yang , Yun Y. , Pati , Debdeep D. Bhattacharya , Anirban A. ( 2020 ). -variational inference with statistical guarantees . Ann. Statist. 48 886--905 . 10.1214/19-AOS1827 4102680 barticle

    2020 doi

  69. [77]

    Gao , Chao C

    barticle [author] Zhang , Fengshuo F. Gao , Chao C. ( 2020 ). Convergence rates of variational posterior distributions . The Annals of Statistics 48 2180--2207 . barticle

    2020

  70. [78]

    , Qi , Guanghao G

    barticle [author] Zhang , Yan Y. , Qi , Guanghao G. , Park , Ju-Hyun J.-H. Chatterjee , Nilanjan N. ( 2018 a). Estimation of complex effect-size distributions using summary-level statistics from genome-wide association studies across 32 complex traits . Nature genetics 50 1318...

    2018

  71. [79]

    u tepage , Judith J. , Kjellstr \

    barticle [author] Zhang , Cheng C. , B \"u tepage , Judith J. , Kjellstr \"o m , Hedvig H. Mandt , Stephan S. ( 2018 b). Advances in variational inference . IEEE transactions on pattern analysis and machine intelligence 41 2008--2026 . barticle

    2018

  72. [80]

    , Zhou , Yang Y

    barticle [author] Zhao , Junlong J. , Zhou , Yang Y. Liu , Yufeng Y. ( 2023 ). Estimation of Linear Functionals in High-Dimensional Linear Models: From Sparsity to Nonsparsity . Journal of the American Statistical Association 1--13 . barticle

    2023

  73. [81]

    Zhao , Hongyu H

    barticle [author] Zhou , Geyu G. Zhao , Hongyu H. ( 2021 ). A fast and robust Bayesian nonparametric method for prediction of complex traits using summary statistics . PLoS genetics 17 e1009697 . barticle

    2021

  74. [82]

    Bradic , Jelena J

    barticle [author] Zhu , Yinchu Y. Bradic , Jelena J. ( 2018 ). Linear hypothesis testing in dense high-dimensional linear models . Journal of the American Statistical Association 113 1583--1600 . barticle

    2018

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