Uncertainty quantification for robust variable selection and multiple testing

TL;DR

This paper introduces a unified procedure for robust variable selection and multiple testing under non-normal, dependent data, incorporating uncertainty quantification and analyzing its optimality across various risk measures.

Contribution

It proposes a novel, versatile method based on risk hull minimization for simultaneous variable selection and multiple testing in robust, dependent data settings, including uncertainty quantification.

Findings

The procedure effectively controls FDR, FPR, FWER, and other risks.

Introduces the first uncertainty quantification approach in robust variable selection.

Demonstrates weak optimality of the proposed method.

Abstract

We study the problem of identifying the set of \emph{active} variables, termed in the literature as \emph{variable selection} or \emph{multiple hypothesis testing}, depending on the pursued criteria. For a general \emph{robust setting} of non-normal, possibly dependent observations and a generalized notion of \emph{active set}, we propose a procedure that is used simultaneously for the both tasks, variable selection and multiple testing. The procedure is based on the \emph{risk hull minimization} method, but can also be obtained as a result of an empirical Bayes approach or a penalization strategy. We address its quality via various criteria: the Hamming risk, FDR, FPR, FWER, NDR, FNR,and various \emph{multiple testing risks}, e.g., MTR=FDR+NDR; and discuss a weak optimality of our results. Finally, we introduce and study, for the first time, the \emph{uncertainty quantification problem} in the variable selection and multiple testing context in our robust setting.