Let \(X=(X_1,\ldots , X_p)\) be the vector of covariates in a regression problem and let \({\widetilde{X}}\) be a knockoff copy of X (in the sense of Candes et al. 2018). In a number of applications, mainly in genetics, there is a finite set F such that \(X_i\in F\) for each \(i=1,\ldots ,p\) . Despite the latter fact, to make variable selection with the knockoff procedure, X is usually modeled as an absolutely continuous random vector. While comprehensible from the point of view of applications, this approximate procedure does not make sense theoretically, since X is supported by the finite set \(F^p\) . In this paper, explicit formulae for the joint distribution of \((X,{\widetilde{X}})\) are provided when \(P(X\in F^p)=1\) and X is partially exchangeable. In fact, when \(X_i\in F\) for all i, assuming X partially exchangeable is often a good strategy. In a few situations, even if extreme, it may be also reasonable to assume X exchangeable. Hence, some attention is paid to the exchangeable special case. The robustness of \({\widetilde{X}}\) , with respect to the de Finetti’s measure \(\pi \) of X, is investigated as well. Let \({\mathcal {L}}_\pi ({\widetilde{X}}\mid X=x)\) be the conditional distribution of \({\widetilde{X}}\) , given \(X=x\) , when X is exchangeable and the de Finetti’s measure of X is \(\pi \) . It is shown that \(\Vert {\mathcal {L}}_{\pi _1}({\widetilde{X}}\mid X=x)-{\mathcal {L}}_{\pi _2}({\widetilde{X}}\mid X=x)\Vert \le c(x)\,\Vert \pi _1-\pi _2\Vert \) where \(\Vert \cdot \Vert \) is total variation distance and c(x) a suitable constant. Finally, a numerical experiment is performed. Overall, the knockoffs of this paper outperform the alternatives (i.e., the knockoffs obtained by giving X an absolutely continuous distribution) as regards the false discovery rate but are slightly weaker in terms of power.
