We introduce $\textit{Backward Conformal Prediction}$, 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 $\mathbb{P}(Y_{\rm test} \in \hat C_n^{\tildeα}(X_{\rm test})) \ge 1 - \mathbb{E}[\tildeα]$ for any data-dependent miscoverage $\tildeα$, and (ii) a novel leave-one-out estimator $\hatα^{\rm LOO}$ of the marginal miscoverage $\mathbb{E}[\tildeα]$ 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.

翻译：本文提出$\textit{后向共形预测}$方法，该方法在保证共形覆盖的同时，为预测集大小的控制提供了灵活的手段。与标准共形预测固定覆盖水平而允许共形集大小自由变化不同，我们的方法定义了一种基于观测数据约束预测集大小行为的规则，并据此自适应调整覆盖水平。本方法建立在两个关键基础之上：(i) Gauthier等人[2025]最近利用e值实现事后有效性的研究成果，该成果保证了对于任意数据相关的误覆盖率$\tildeα$，其边际覆盖满足$\mathbb{P}(Y_{\rm test} \in \hat C_n^{\tildeα}(X_{\rm test})) \ge 1 - \mathbb{E}[\tildeα]$；(ii) 基于校准集构建的边际误覆盖率$\mathbb{E}[\tildeα]$的新型留一估计量$\hatα^{\rm LOO}$，确保理论保证在实际中可计算。该方法在大型预测集不切实际的应用场景（如医学诊断）中具有特殊价值。我们提供了支持本方法有效性的理论结果与实证证据，证明其在确保可解释且受控的预测集大小的同时，仍能保持可计算的覆盖保证。