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. Table of Links Abstract & Introduction Main results Application to prediction intervals Application to novelty detection Conclusion, Acknowledgements and References Appendix A: Exact formulas for Pn,m Appendix B: Numerical bounds and templates Appendix C: Proof Appendix D: Explicit control of (16) for α=0 Appendix E: Proof of Corollary 4.1 Appendix F: The Simes inequality Appendix G: Uniform FDP bound for AdaDetect Appendix H:  Additional experiments 5 Conclusion The main takeaway from this work is the characterization of a “universal” joint distribution Pn,m for conformal p-values based on n calibration points and m test points. We derived as a consequence a non-asymptotic concentration inequality for the p-value empirical distribution function; numerical procedures can also be of use for calibration in practice. This entails uniform error bounds on the false coverage/false discovery proportion that hold with high probability, while standard results are only marginal or in expectation and not uniform in the decision. Since the results hold under the score exchangeability assumption only, they are applicable to adaptive score procedures using the calibration and test sets for training. Acknowledgements We would like to thank Anna Ben-Hamou and Claire Boyer for constructive discussions and Ariane Marandon for her support with the code. The authors acknowledge the grants ANR-21-CE23-0035 (ASCAI) and ANR-19-CHIA-0021-01 (BISCOTTE) of the French National Research Agency ANR and the Emergence project MARS. References Balasubramanian, V., Ho, S.-S., and Vovk, V. (2014). Conformal prediction for reliable machine learning: theory, adaptations and applications. Morgan Kaufmann books. Bashari, M., Epstein, A., Romano, Y., and Sesia, M. (2023). 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Transductive Conformal Inference With Adaptive Scores: Conclusion, Acknowledgements and References

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