Transductive Conformal Inference With Adaptive Scores: the Simes Inequality

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

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