Conformal e-prediction
Vladimir Vovk
TL;DR
This work analyzes conformal e-prediction as a counterpart to conformal prediction, highlighting its definitions, validity properties, conditional constructions, and efficiency considerations. It shows that conformal e-predictors yield space-wise validity ($\mathbb{E}(E)\le1$) and time-wise ergodicity in online settings, while cross-conformal e-predictors provide guaranteed validity, unlike their p-value-based counterparts. The paper introduces split and cross-conformal variants to improve computational and predictive efficiency, and develops objective criteria (observed and prior log criteria) to design optimal nonconformity e-measures, with explicit optimal forms in the idealised setting. It concludes that conformal e-prediction offers practical advantages in conditional validity and cross-conformal guarantees, even as some strengths of conformal prediction (e.g., simple error bounds on sets and well-calibrated predictive distributions) do not directly translate. The work identifies a roadmap for leveraging e-values to build flexible, efficiently computable conformal methodologies with robust validity guarantees.
Abstract
This paper discusses a counterpart of conformal prediction for e-values, conformal e-prediction. Conformal e-prediction is conceptually simpler and had been developed in the 1990s as a precursor of conformal prediction. When conformal prediction emerged as result of replacing e-values by p-values, it seemed to have important advantages over conformal e-prediction without obvious disadvantages. This paper re-examines relations between conformal prediction and conformal e-prediction systematically from a modern perspective. Conformal e-prediction has advantages of its own, such as the ease of designing conditional conformal e-predictors and the guaranteed validity of cross-conformal e-predictors (whereas for cross-conformal predictors validity is only an empirical fact and can be broken with excessive randomization). Even where conformal prediction has clear advantages, conformal e-prediction can often emulate those advantages, more or less successfully.