Transductive Conformal Inference With Adaptive Scores: Exact Formulas for Pn,m

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28 Feb 2024

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.

A Exact formulas for Pn,m

Theorem A.1 (ii) provides the exact dependency structure between the p-values: for instance, M(j)! = 1 when the coordinates of j = (j1, . . . , jm) are all distinct, while M(j)! = m! when the coordinates of j = (j1, . . . , jm) are the same. This means that the distribution slightly favors the j with repeated entries. This shows that the conformal p-values are not i.i.d. but have a positive structure of dependency. This is in accordance with the specific positive dependence property (called PRDS) already shown by Bates et al. (2023); Marandon et al. (2022).