# Asymptotically Optimal Knockoff Statistics via the Masked Likelihood Ratio

**Authors:** Asher Spector, William Fithian  
**arXiv:** [2212.08766](https://arxiv.org/abs/2212.08766)  
**Date:** 2022-12-17  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/2212.08766v1/

## Key points

- Asher Spector and William Fithian introduce a new class of knockoff statistics that are asymptotically most powerful
- The masked likelihood ratio (MLR) statistic is introduced, which estimates the oracle MLR
- MLR statistics are asymptotically average-case optimal, meaning they maximize the expected number of discoveries made by knockoffs when averaging over a user-specified prior on unknown parameters
- This optimality result places no explicit restrictions on problem dimensions or the unknown relationship between response and covariates; instead, it assumes a "local dependence" condition dependent only on simple quantities calculated from the data
- In simulations and three real data applications, the authors show that MLR statistics outperform state-of-the-art feature statistics, even in settings where the prior is highly misspecified
- The authors implement MLR statistics in an open-source Python package called knockpy; their implementation is often faster than computing a cross-validated lasso.

## Summary

In their paper titled "Asymptotically Optimal Knockoff Statistics via the Masked Likelihood Ratio," Asher Spector and William Fithian introduce a new class of knockoff statistics that are asymptotically most powerful. These ratios are Neyman-Pearson statistics used to discriminate between features and knockoffs using partially observed (masked) data. The authors introduce the masked likelihood ratio (MLR) statistic, a knockoff statistic that estimates the oracle MLR. They demonstrate that MLR statistics are asymptotically average-case optimal, meaning they maximize the expected number of discoveries made by knockoffs when averaging over a user-specified prior on unknown parameters. This optimality result places no explicit restrictions on problem dimensions or the unknown relationship between response and covariates; instead, it assumes a "local dependence" condition dependent only on simple quantities calculated from the data. In simulations and three real data applications, the authors show that MLR statistics outperform state-of-the-art feature statistics, even in settings where the prior is highly misspecified. The authors implement MLR statistics in an open-source Python package called knockpy; their implementation is often faster than computing a cross-validated lasso. Overall, this paper presents a significant contribution to statistical inference by introducing a novel approach for selecting variables in high-dimensional regression models with strong theoretical guarantees and good performance in practical applications.