Backward Conformal Prediction

We introduce textitBackward Conformal Prediction, a method that guarantees conformal coverage while providing flexible control over the size of prediction sets. Unlike standard conformal prediction, which fixes the coverage level and allows the conformal set size to vary, our approach defines a rule that constrains how prediction set sizes behave based on the observed data, and adapts the coverage level accordingly. Our method builds on two key foundations: (i) recent results by Gauthier et al. [2025] on post-hoc validity using e-values, which ensure marginal coverage of the form mathbbP(Yrm test ∈ hat Cn^tildeα(Xrm test)) ≥ 1 - mathbbE[tildeα] up to a first-order Taylor approximation for any data-dependent miscoverage tildeα, and (ii) a novel leave-one-out estimator hatαrm LOO of the marginal miscoverage mathbbE[tildeα] based on the calibration set, ensuring that the theoretical guarantees remain computable in practice. This approach is particularly useful in applications where large prediction sets are impractical such as medical diagnosis. We provide theoretical results and empirical evidence supporting the validity of our method, demonstrating that it maintains computable coverage guarantees while ensuring interpretable, well-controlled prediction set sizes.