# Empirical Bayes for correlated Gaussian sequence model

arXiv: https://arxiv.org/abs/2607.03596 · HTML: https://mlsanity.com/paper/2607.03596/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 65 references that are in the map.
- First posted on arXiv 3 Jul 2026.

## Builds on (most cited first, 18 of 18)

- [Gradient Descent with Random Initialization: Fast Global Convergence for Nonconvex Phase Retrieval](https://mlsanity.com/paper/1803.07726/) (2018)
- [Moving Beyond Sub-Gaussianity in High-Dimensional Statistics: Applications in Covariance Estimation and Linear Regression](https://mlsanity.com/paper/1804.02605/) (2018)
- [On the nonparametric maximum likelihood estimator for Gaussian location mixture densities with application to Gaussian denoising](https://mlsanity.com/paper/1712.02009/) (2017)
- [Self-regularizing Property of Nonparametric Maximum Likelihood Estimator in Mixture Models](https://mlsanity.com/paper/2008.08244/) (2020)
- [Two Modeling Strategies for Empirical Bayes Estimation](https://mlsanity.com/paper/1409.2677/) (2014)
- [A flexible empirical Bayes approach to multiple linear regression and connections with penalized regression](https://mlsanity.com/paper/2208.10910/) (2022)
- [Stein's unbiased risk estimate and Hyv"arinen's score matching](https://mlsanity.com/paper/2502.20123/) (2025)
- [Two-component Mixture Model in the Presence of Covariates](https://mlsanity.com/paper/1810.07897/) (2018)
- [ebnm: An R Package for Solving the Empirical Bayes Normal Means Problem Using a Variety of Prior Families](https://mlsanity.com/paper/2110.00152/) (2021)
- [A Mean Field Approach to Empirical Bayes Estimation in High-dimensional Linear Regression](https://mlsanity.com/paper/2309.16843/) (2023)
- [Gradient flows for empirical Bayes in high-dimensional linear models](https://mlsanity.com/paper/2312.12708/) (2023)
- [Long-time dynamics and universality of nonconvex gradient descent](https://mlsanity.com/paper/2509.11426/) (2025)
- [Empirical Bayes When Estimation Precision Predicts Parameters](https://mlsanity.com/paper/2212.14444/) (2022)
- [Empirical Bayes inference in sparse high-dimensional generalized linear models](https://mlsanity.com/paper/2303.07854/) (2023)
- [Gradient descent inference in empirical risk minimization](https://mlsanity.com/paper/2412.09498/) (2024)
- [Compound decisions and empirical Bayes via Bayesian nonparametrics](https://mlsanity.com/paper/2602.20115/) (2026)
- [Parametric Mean-Field empirical Bayes in high-dimensional linear regression](https://mlsanity.com/paper/2601.16842/) (2026)
- [Empirical Bayes Estimation and Inference via Smooth Nonparametric Maximum Likelihood](https://mlsanity.com/paper/2603.27843/) (2026)

## Abstract (arXiv)

Empirical Bayes methods are among the most widely used statistical methods for large-scale inference. A central paradigm is the NPMLE, whose theoretical guarantees are by now well understood for the independent Gaussian sequence model. In this paper, we study empirical Bayes estimation from dependent observations in the Gaussian sequence model. We show that the maximum Composite Marginal Likelihood (CML) estimator, which ignores all correlations in the likelihood, converges in weighted Hellinger distance at the rate n_*-1/2, where n_*=n/kappa0 is the `effective sample size' determined solely by the number of observations n and the spectral radius kappa0 of the correlation matrix of the Gaussian observations. A complementary minimax lower bound shows that n_* indeed serves as the right complexity measure, and that the CML estimator is nearly rate optimal under general dependence. We consider two concrete applications. In the first, we consider Bayesian linear regression, where the signal prior is estimated via CML applied to the least squares estimator. In the second, we consider the more challenging Bayesian nonlinear single-index model, where prior is estimated by CML applied to a one-step debiased gradient descent. In both applications, although the full likelihood landscape can be arbitrarily complicated and intractable, our CML method is facilitated by exploiting the high-dimensional distribution of the auxiliary statistics through a correlated Gaussian sequence model. The key ingredient in the proof of our results is a sharp local maximal inequality for the log composite marginal likelihood process under dependent Gaussian observations. In contrast to standard empirical process methods, we prove this inequality by leveraging a recent geometric Brascamp-Lieb inequality for Gaussian measures.