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A Mean Field Approach to Empirical Bayes Estimation in High-dimensional Linear Regression

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arxiv 2309.16843 v3 pith:F7L7XH7V submitted 2023-09-28 math.ST stat.MEstat.MLstat.TH

Sumit Mukherjee , Bodhisattva Sen , Subhabrata Sen This is my paper · ORCID
classification math.STstat.MEstat.MLstat.TH
keywords bayesestimationempiricalregressioncomputationallyfieldhigh-dimensionalmean
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We study empirical Bayes estimation in high-dimensional linear regression. To facilitate computationally efficient estimation of the underlying prior, we adopt a variational empirical Bayes approach, introduced originally in Carbonetto and Stephens (2012) and Kim et al. (2022). We establish asymptotic consistency of the nonparametric maximum likelihood estimator (NPMLE) and its (computable) naive mean field variational surrogate under mild assumptions on the design and the prior. Assuming, in addition, that the naive mean field approximation has a dominant optimizer, we develop a computationally efficient approximation to the oracle posterior distribution, and establish its accuracy under the 1-Wasserstein metric. This enables computationally feasible Bayesian inference; e.g., construction of posterior credible intervals with an average coverage guarantee, Bayes optimal estimation for the regression coefficients, estimation of the proportion of non-nulls, etc. Our analysis covers both deterministic and random designs, and accommodates correlations among the features. To the best of our knowledge, this provides the first rigorous nonparametric empirical Bayes method in a high-dimensional regression setting without sparsity.

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Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Variational Inference for Latent Variable Models in High Dimensions

    math.ST 2025-06 accept novelty 8.0 of 10

    The paper characterizes the exact regimes where mean-field variational inference is accurate for LDA and MMSB, with tight finite-sample KL bounds.

  2. Empirical Bayes for correlated Gaussian sequence model

    math.ST 2026-07 accept novelty 7.0 of 10

    CML for the correlated Gaussian sequence model converges at rate n_*^{-1/2} in weighted Hellinger distance, with matching minimax lower bound, and applies to linear GLS and one-step debiased nonlinear regression.

  3. CLT in high-dimensional Bayesian linear regression with low SNR

    math.ST 2025-07 conditional novelty 7.0 of 10

    In low-SNR high-dimensional Bayesian linear regression with product priors, one-dimensional posterior projections and the posterior mean are asymptotically Gaussian, centered at the mean-field approximation, with vari...

  4. Stability of Mean-Field Variational Inference

    math.PR 2025-06 conditional novelty 7.0 of 10

    The mean-field variational inference optimizer is Lipschitz stable and differentiable in the target potential under strong log-concavity, with an explicit PDE for the derivative.

  5. Solving Empirical Bayes via Transformers

    cs.LG 2025-02 conditional novelty 6.0 of 10

    A transformer pre-trained on synthetic Poisson data can beat the classical NPMLE estimator on several empirical Bayes tasks and run about 100x faster.

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