Optimal Multiple Testing Under a Gaussian Prior on the Effect Sizes

TL;DR

This paper introduces a new Bayesian-inspired method for multiple hypothesis testing that optimally incorporates uncertain Gaussian prior information, improving discovery power in genome-wide association studies.

Contribution

It develops the Bayes weights approach, an efficient algorithm for optimal p-value weighting under Gaussian priors, addressing non-convex optimization challenges.

Findings

Discoveries in genome-wide association studies improved.

Method outperforms existing multiple testing procedures.

Open source implementation available.

Abstract

We develop a new method for frequentist multiple testing with Bayesian prior information. Our procedure finds a new set of optimal p-value weights called the Bayes weights. Prior information is relevant to many multiple testing problems. Existing methods assume fixed, known effect sizes available from previous studies. However, the case of uncertain information is usually the norm. For a Gaussian prior on effect sizes, we show that finding the optimal weights is a non-convex problem. Despite the non-convexity, we give an efficient algorithm that solves this problem nearly exactly. We show that our method can discover new loci in genome-wide association studies. On several data sets it compares favorably to other methods. Open source code is available.