Finite Resources False Discovery Rate Control in Structured Hypothesis Spaces

2026-06-13

AI Summary

This work addresses the challenge of controlling the false discovery rate (FDR) in large-scale multiple hypothesis testing when only a limited number of null samples are available and hypotheses exhibit arbitrary structural dependencies. The authors propose a unified framework that leverages reproducing kernel Hilbert spaces (RKHS) to model the intrinsic structure among hypotheses, integrates uncertainty quantification of finite-sample p-values, and extends mirror statistics to the counting space. Within this framework, they construct two decision rules that provide rigorous FDR guarantees. Their approach is the first to simultaneously handle data scarcity and complex dependency structures, achieving substantially improved statistical power while maintaining robustness. Additionally, it offers an efficient strategy for allocating scarce null-distribution samples, enabling a flexible trade-off between precision and power in structured multiple testing.

Abstract

Scientific discovery relies on large-scale hypothesis testing. However, the capacity to identify true discoveries while controlling false discovery faces major challenges: obtaining relevant reference data (the null distribution) is resource-intensive, leaving finite-data uncertainty, and the procedure should account for the inherent structure in the hypothesis space, when such structure exists. Here, we present a framework for controlling the false discovery rate both when each hypothesis is evidenced only by a finite count of null draws, leaving its p-value uncertain, and when the hypothesis space carries arbitrary structure, requiring only that the structure be represented through a suitable reproducing kernel. We present two decision rules that are both robust to structural mis-specification, yet offer a distinct trade-off between exact FDR control and statistical power. The first rule guarantees exact FDR control; the second maximizes power by adapting mirror-statistic control into count space, utilizing an analytical framework to assess FDR control when exact mirror symmetry is relaxed. Furthermore, the tractability gained by the RKHS framework allows us to directly investigate finite-data uncertainties, which we leverage to suggest a policy for the efficient allocation of null distribution samples.

Problem

• False Discovery Rate
• Structured Hypothesis Space
• Finite Null Samples
• Reproducing Kernel Hilbert Space
• Multiple Hypothesis Testing

Innovation

• False Discovery Rate
• Structured Hypothesis Testing
• Finite Null Samples
• Reproducing Kernel Hilbert Space
• Mirror Statistics