# Two Modeling Strategies for Empirical Bayes Estimation

arXiv: https://arxiv.org/abs/1409.2677 · HTML: https://mlsanity.com/paper/1409.2677/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Sep 2014.

## Built on by (most cited first, 16 of 16)

- [Optimal score estimation via empirical Bayes smoothing](https://mlsanity.com/paper/2402.07747/) (2024)
- [Learning Diffusion Priors from Observations by Expectation Maximization](https://mlsanity.com/paper/2405.13712/) (2024)
- [Sharp regret bounds for empirical Bayes and compound decision problems](https://mlsanity.com/paper/2109.03943/) (2021)
- [A Mean Field Approach to Empirical Bayes Estimation in High-dimensional Linear Regression](https://mlsanity.com/paper/2309.16843/) (2023)
- [A New Perspective On Denoising Based On Optimal Transport](https://mlsanity.com/paper/2312.08135/) (2023)
- [Distribution-dependent Generalization Bounds for Tuning Linear Regression Across Tasks](https://mlsanity.com/paper/2507.05084/) (2025)
- [Variational Learning of Disentangled Representations](https://mlsanity.com/paper/2506.17182/) (2025)
- [Sharp Inequalities between Total Variation and Hellinger Distances for Gaussian Mixtures](https://mlsanity.com/paper/2602.03202/) (2026)
- [Multi-Domain Empirical Bayes for Linearly-Mixed Causal Representations](https://mlsanity.com/paper/2603.18404/) (2026)
- [Simulation-Based Empirical Bayes](https://mlsanity.com/paper/2607.21843/) (2026)
- [Nonparametric Empirical Bayes Estimation and Testing for Sparse and Heteroscedastic Signals](https://mlsanity.com/paper/2106.08881/) (2021)
- [Neural-g: A Deep Learning Framework for Mixing Density Estimation](https://mlsanity.com/paper/2406.05986/) (2024)
- [Quasi-Bayes empirical Bayes: a sequential approach to the Poisson compound decision problem](https://mlsanity.com/paper/2411.07651/) (2024)
- [Distributional Shrinkage II: Higher-Order Scores Encode Brenier Map](https://mlsanity.com/paper/2512.09295/) (2025)
- [A Flexible Empirical Bayes Approach to Generalized Linear Models, with Applications to Sparse Logistic Regression](https://mlsanity.com/paper/2601.21217/) (2026)
- [Empirical Bayes for correlated Gaussian sequence model](https://mlsanity.com/paper/2607.03596/) (2026)

## Abstract (arXiv)

Empirical Bayes methods use the data from parallel experiments, for instance, observations Xk~N(Thetak,1) for k=1,2,ldots,N, to estimate the conditional distributions Thetak|Xk. There are two main estimation strategies: modeling on the θ space, called "g-modeling" here, and modeling on the x space, called "f-modeling." The two approaches are described and compared. A series of computational formulas are developed to assess their frequentist accuracy. Several examples, both contrived and genuine, show the strengths and limitations of the two strategies.