# Near-optimal multiple testing in Bayesian linear models with finite-sample FDR control

arXiv: https://arxiv.org/abs/2211.02778 · HTML: https://mlsanity.com/paper/2211.02778/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 63 references that are in the map.
- First posted on arXiv 4 Nov 2022.

## Builds on (most cited first, 20 of 20)

- [Statistical physics of inference: Thresholds and algorithms](https://mlsanity.com/paper/1511.02476/) (2015)
- [Optimal Errors and Phase Transitions in High-Dimensional Generalized Linear Models](https://mlsanity.com/paper/1708.03395/) (2017)
- [Controlling the false discovery rate via knockoffs](https://mlsanity.com/paper/1404.5609/) (2014)
- [The Lasso with general Gaussian designs with applications to hypothesis testing](https://mlsanity.com/paper/2007.13716/) (2020)
- [The estimation error of general first order methods](https://mlsanity.com/paper/2002.12903/) (2020)
- [False Discoveries Occur Early on the Lasso Path](https://mlsanity.com/paper/1511.01957/) (2015)
- [Fast and Powerful Conditional Randomization Testing via Distillation](https://mlsanity.com/paper/2006.03980/) (2020)
- [False Discovery Rate Control via Data Splitting](https://mlsanity.com/paper/2002.08542/) (2020)
- [Conditional calibration for false discovery rate control under dependence](https://mlsanity.com/paper/2007.10438/) (2020)
- [Empirical Bayes PCA in high dimensions](https://mlsanity.com/paper/2012.11676/) (2020)
- [Global and Simultaneous Hypothesis Testing for High-Dimensional Logistic Regression Models](https://mlsanity.com/paper/1805.06970/) (2018)
- [Controlling False Discovery Rate Using Gaussian Mirrors](https://mlsanity.com/paper/1911.09761/) (2019)
- [A Scale-free Approach for False Discovery Rate Control in Generalized Linear Models](https://mlsanity.com/paper/2007.01237/) (2020)
- [Improving the Stability of the Knockoff Procedure: Multiple Simultaneous Knockoffs and Entropy Maximization](https://mlsanity.com/paper/1810.11378/) (2018)
- [Powerful Knockoffs via Minimizing Reconstructability](https://mlsanity.com/paper/2011.14625/) (2020)
- [Performance of Bayesian linear regression in a model with mismatch](https://mlsanity.com/paper/2107.06936/) (2021)
- [A Power and Prediction Analysis for Knockoffs with Lasso Statistics](https://mlsanity.com/paper/1712.06465/) (2017)
- [Estimation, Confidence Intervals, and Large-Scale Hypotheses Testing for High-Dimensional Mixed Linear Regression](https://mlsanity.com/paper/2011.03598/) (2020)
- [Characterizing the SLOPE Trade-off: A Variational Perspective and the Donoho-Tanner Limit](https://mlsanity.com/paper/2105.13302/) (2021)
- [Power analysis of knockoff filters for correlated designs](https://mlsanity.com/paper/1910.12428/) (2019)

## Built on by (most cited first, 3 of 3)

- [Online multiple testing with e-values](https://mlsanity.com/paper/2311.06412/) (2023)
- [CatNet: Controlling the False Discovery Rate in LSTM with SHAP Feature Importance and Gaussian Mirrors](https://mlsanity.com/paper/2411.16666/) (2024)
- [Model-Agnostic FDR Control via Group Gaussian Mirror and Permutation SHAP](https://mlsanity.com/paper/2608.00989/) (2026)

## Abstract (arXiv)

In high dimensional variable selection problems, statisticians often seek to design multiple testing procedures that control the False Discovery Rate (FDR), while concurrently identifying a greater number of relevant variables. Model-X methods, such as Knockoffs and conditional randomization tests, achieve the primary goal of finite-sample FDR control, assuming a known distribution of covariates. However, whether these methods can also achieve the secondary goal of maximizing discoveries remains uncertain. In fact, designing procedures to discover more relevant variables with finite-sample FDR control is a largely open question, even within the arguably simplest linear models. In this paper, we develop near-optimal multiple testing procedures for high dimensional Bayesian linear models with isotropic covariates. We introduce Model-X procedures that provably control the frequentist FDR from finite samples, even when the model is misspecified, and conjecturally achieve near-optimal power when the data follow the Bayesian linear model. Our proposed procedure, PoEdCe, incorporates three key ingredients: Posterior Expectation, distilled Conditional randomization test (dCRT), and the Benjamini-Hochberg procedure with e-values (eBH). The optimality conjecture of PoEdCe is based on a heuristic calculation of its asymptotic true positive proportion (TPP) and false discovery proportion (FDP), which is supported by methods from statistical physics as well as extensive numerical simulations. Our result establishes the Bayesian linear model as a benchmark for comparing the power of various multiple testing procedures.