发表机构
University of California, Berkeley; University of Chicago; Stanford University（加州大学伯克利分校; 芝加哥大学; 斯坦福大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文从可交换性原假设的置换检验视角阐释共形预测，借助经典假设检验理论重现共形预测的普适性与不可能性结果，并推导得出最优共形预测器的形式。
AI 中文摘要
共形预测与置换检验之间的联系在文献中已广为人知，部分研究者通过以下方式阐释共形预测：其计算假设 $H_0: Y_{n+1} = y$ 的置换 p 值，随后将其反转以形成 $Y_{n+1}$ 的预测集（即接受所有 p 值较大的 y 进入预测集）。本文探讨一种较少为人熟知的替代视角：我们仍通过反转置换检验来构建共形预测，但原假设为 $n+1$ 个样本联合分布的可交换性。这一视角的转变虽简单，却更贴合假设检验的传统形式化，带来了诸多益处。首先，我们利用共形集与检验的对偶性，证明共形预测文献中的基础普适性和不可能性结果可直接借助经典假设检验理论（源于 Neyman、Lehmann、Scheffé、Kraft、Le Cam 等人的研究）重现。此外，我们表明共形预测的最优性结果可通过标准 Neyman-Pearson 理论推导得出：对于任意协变量与响应的联合分布 $X,Y$ 以及任意样本量，最优预测集方法（在所有对可交换分布具有有效覆盖率的方法中，能提供最有效集的方法）是一种得分函数为 $Y|X$ 的条件密度倒数的共形预测器。
英文摘要
The connections between conformal prediction and permutation tests are already widely-known in the literature. Some authors motivate conformal prediction by saying that it computes a permutation p-value for the hypothesis $H_0 : Y_{n+1} = y$, and then inverts this to form a prediction set for $Y_{n+1}$ (i.e., accepts all values $y$ into the prediction set for which the p-value is large). In this paper, we examine an alternative view, which is less well-known: we again cast conformal prediction via the inversion of a permutation test, but for the null of exchangeability of the joint distribution of the $n+1$ samples. This change in perspective, while simple, adheres more closely to traditional formalization in hypothesis testing, which offers several benefits. First, we use the duality between conformal sets and testing to show that foundational universality and impossibility results in the conformal prediction literature can be reproduced directly using classical hypothesis testing theory (due to Neyman, Lehmann, Scheff{é}, Kraft, Le Cam, and others). Furthermore, we show that an optimality result for conformal prediction can be derived using standard Neyman-Pearson theory: for any joint distribution of the covariates and response $X,Y$, and any sample size, the optimal method for prediction sets---delivering the most efficient set among all methods with valid coverage for exchangeable distributions---is a conformal predictor whose score is the inverse conditional density of $Y|X$.
