New perspectives on knockoffs construction

TL;DR

This paper explores new theoretical perspectives on constructing knockoff variables, including a characterization of all possible distributions, a copula-based subclass, and an analysis of conditionally independent components.

Contribution

It introduces a new characterization of the distribution set for knockoffs, a copula-based subclass, and an in-depth study of conditionally independent components.

Findings

Provided a new characterization of the distribution set for knockoffs.

Introduced a copula-based subclass of knockoff distributions.

Analyzed the case where components of X are conditionally independent.

Abstract

Let $\Lambda$ be the collection of all probability distributions for $(X,\widetilde{X})$, where $X$ is a fixed random vector and $\widetilde{X}$ ranges over all possible knockoff copies of $X$ (in the sense of \cite{CFJL18}). Three topics are developed in this paper: (i) A new characterization of $\Lambda$ is proved; (ii) A certain subclass of $\Lambda$, defined in terms of copulas, is introduced; (iii) The (meaningful) special case where the components of $X$ are conditionally independent is treated in depth. In real problems, after observing $X=x$, each of points (i)-(ii)-(iii) may be useful to generate a value $\widetilde{x}$ for $\widetilde{X}$ conditionally on $X=x$.