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Generating knockoffs via conditional independence

19 December 2022
E. Dreassi
Fabrizio Leisen
L. Pratelli
P. Rigo
    UQCV
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Abstract

Let XXX be a ppp-variate random vector and X~\widetilde{X}X a knockoff copy of XXX (in the sense of \cite{CFJL18}). A new approach for constructing X~\widetilde{X}X (henceforth, NA) has been introduced in \cite{JSPI}. NA has essentially three advantages: (i) To build X~\widetilde{X}X is straightforward; (ii) The joint distribution of (X,X~)(X,\widetilde{X})(X,X) can be written in closed form; (iii) X~\widetilde{X}X is often optimal under various criteria. However, for NA to apply, X1,…,XpX_1,\ldots, X_pX1​,…,Xp​ should be conditionally independent given some random element ZZZ. Our first result is that any probability measure μ\muμ on Rp\mathbb{R}^pRp can be approximated by a probability measure μ0\mu_0μ0​ of the form \mu_0\bigl(A_1\times\ldots\times A_p\bigr)=E\Bigl\{\prod_{i=1}^p P(X_i\in A_i\mid Z)\Bigr\}. The approximation is in total variation distance when μ\muμ is absolutely continuous, and an explicit formula for μ0\mu_0μ0​ is provided. If X∼μ0X\sim\mu_0X∼μ0​, then X1,…,XpX_1,\ldots,X_pX1​,…,Xp​ are conditionally independent. Hence, with a negligible error, one can assume X∼μ0X\sim\mu_0X∼μ0​ and build X~\widetilde{X}X through NA. Our second result is a characterization of the knockoffs X~\widetilde{X}X obtained via NA. It is shown that X~\widetilde{X}X is of this type if and only if the pair (X,X~)(X,\widetilde{X})(X,X) can be extended to an infinite sequence so as to satisfy certain invariance conditions. The basic tool for proving this fact is de Finetti's theorem for partially exchangeable sequences. In addition to the quoted results, an explicit formula for the conditional distribution of X~\widetilde{X}X given XXX is obtained in a few cases. In one of such cases, it is assumed Xi∈{0,1}X_i\in\{0,1\}Xi​∈{0,1} for all iii.

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