FDR_Bayes_figures

This repository provides the codes used to generate figures for the paper:

Near-optimal multiple testing in Bayesian linear models with finite-sample FDR control.
Taejoo Ahn, Licong Lin, Song Mei.
Paper: https://arxiv.org/abs/2211.02778.

Figure 1 in the paper. Figure 2 in the paper.

Testing procedures

Our primary goal of the paper is to provide multiple-testing procedures that control FDR from finite samples and achieve near-optimal power under well-specified Bayesian linear models.

We introduce 2 base procedures TPoP and CPoP, which are procedures that simply truncate the local false discovery rate (local fdr).

Based on these two procedures, we introduce PoPCe and PoEdCe. These procedures control frequentist FDR from finite samples and achieve near-optimal power under well-specified Bayesian linear models. They use local fdrs as base statistics, wrap them using CRT and dCRT to generate valid p-values and e-values, then apply the eBH procedure to provide finite-sample FDR control. EPoEdCe is an empirical Bayes variant of PoEdCe.

Bayesian linear models

In Bayesian linear models, we have ${\bf Y}={\bf X \bf \beta_0}+{\bf \varepsilon}$ where ${\bf Y},{\bf \varepsilon}\in\mathbb{R}^n$, ${\bf X}\in\mathbb{R}^{n\times d}$ and ${\bf \beta_0}\in\mathbb{R}^d$. Here, $\varepsilon_i\sim N(0,\sigma^2)$, $X_{ij} \sim N(0,1/n)$ and $\beta_{0i}\sim\Pi$ independently.

Figure 1 and Figure 2

Figure 1 compares TPoP and CPoP with the thresholded LASSO procedure, and reports the theoretical asymptotic curves of FDP and TPP, as well as the realization of 10 simulated instances.

Figure 2 reports the FDP and TPP for PoPCe, PoEdCE and EPoEdCe on 10 instances of the Bayesian linear model.

How to use the code to generate the figures

To generate Figure 1 in the paper, simply run fig1_data.py to generate the data file fig1data.npy first. Then run fig1_plot.py to generate the plot fig1.pdf. Other figures are generated similarly.

Environments

- python==3.7.2
 - numpy==1.16.1
 - cvxpy==1.1.5
 - scipy==1.2.1
 - sklearn==0.24.1

Miscellanous

More information about the procedures can be found in the original paper. If you use this code in your research, please cite our paper

@misc{ahn2023nearoptimal,
      title={Near-optimal multiple testing in Bayesian linear models with finite-sample FDR control}, 
      author={Taejoo Ahn and Licong Lin and Song Mei},
      year={2023},
      eprint={2211.02778},
      archivePrefix={arXiv},
      primaryClass={math.ST}
}