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Transductive Conformal Inference With Adaptive Scores: the Simes Inequalityby@transduction
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Transductive Conformal Inference With Adaptive Scores: the Simes Inequality

by Transduction University PapersFebruary 28th, 2024
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Conformal inference is a fundamental and versatile tool that provides distribution-free guarantees for many machine learning tasks.
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This paper is available on arxiv under CC 4.0 license.

Authors:

(1) Ulysse Gazin, Universit´e Paris Cit´e and Sorbonne Universit´e, CNRS, Laboratoire de Probabilit´es, Statistique et Mod´elisation,

(2) Gilles Blanchard, Universit´e Paris Saclay, Institut Math´ematique d’Orsay,

(3) Etienne Roquain, Sorbonne Universit´e and Universit´e Paris Cit´e, CNRS, Laboratoire de Probabilit´es, Statistique et Mod´elisation.

F The Simes inequality

As proved in Marandon et al. (2022), and since the joint distribution of the conformal p-values does not change from one context to another (Proposition 2.2), the conformal p-values are positively regressively dependent on each one of a subset (PRDS) under (Exch) and (NoTies), see Benjamini and Yekutieli (2001) for a formal definition of the latter.



Hence, by Benjamini and Yekutieli (2001), the Simes inequality (Simes, 1986) is valid, that is, for all λ > 0, we have



This envelope can be applied in the two applications of the paper as follows:


(PI) Under the condition of Corollary 3.1, the bound



is valid for (17).


(ND) Under the condition of Corollary 4.1 the following control is valid



for



for any estimator