PICPIs：预测区间条件预测区间
PICPIs: Prediction-Interval-Conditional Prediction Intervals
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中文总结 AI 辅助
针对保形预测中边际有效性与完全条件性之间的差距，提出预测区间条件预测区间（PICPIs）框架，通过自一致性条件实现数据自适应分层，并给出构建算法、理论保证及实证比较。
中文摘要 AI 辅助
统计学中的一个经典问题是，在推断不可观测目标时，应基于哪些可观测量进行条件化。在非参数不确定性量化中的保形预测领域，标准边际有效性在决策所依据的预测值处提供的分辨率有限，而相对于协变量的完全条件性保证已被证明是无法实现的。我们通过引入一种基于预测的条件化框架来解决这一差距，该框架被称为预测区间条件预测区间（PICPIs）。形式上，PICPI是一个满足自一致性条件的区间$I$：$$\mathbb{E} [Y \mid p(X) \in I] \in I,$$其中$p$为预测模型，$X$为上下文协变量，$Y$为结果变量。因此，一个区间同时定义了预测值的一个分层，并证明该分层中的平均结果位于同一区间内。这种自一致性条件在不改变原始预测的情况下产生了数据自适应的分层。此类区间可以通过实用算法构建。在预测分布的正则性条件下，所构建的区间覆盖了除任意小比例之外的所有预测值，并且其宽度以$n^{-1/3}$的速率减小（忽略对数因子和预测误差）。此外，识别这些局部校准的区间反过来可以为下游决策提供信息。我们推导了在概率预测和多类分类中PICPIs的推断程序，并附有理论保证。我们提供了将PICPIs与现有基于区间的基线进行比较的实证结果。
英文摘要
A classical question in statistics is which observable quantities to condition on when drawing inferences about unobservable targets. For conformal prediction in nonparametric uncertainty quantification, standard marginal validity offers limited resolution at the prediction values on which decisions are based, and fully conditional guarantees with respect to the covariates are provably unattainable. We address this gap by introducing a prediction-based conditioning framework that we refer to as Prediction-Interval-Conditional Prediction Intervals (PICPIs). Formally, a PICPI is an interval $I$ satisfying a self-consistency condition: $$\mathbb{E} [Y \mid p(X) \in I] \in I,$$ for predictive model $p$, contextual covariate $X$, and outcome $Y$. Thus, an interval simultaneously defines a stratum of prediction values and certifies that the mean outcome in that stratum lies in the same interval. This self-consistency condition yields data-adaptive strata without altering the original prediction. Such intervals can be constructed using practical algorithms. Under regularity of the prediction distribution, the constructed intervals cover all but an arbitrarily small fraction of prediction values and have widths that decrease at rate $n^{-1/3}$, up to logarithmic factors and the prediction error. Moreover, identifying these locally calibrated intervals can, in turn, inform downstream decision-making. We derive inference procedures for PICPIs in probabilistic prediction and multi-class classification, accompanied by theoretical guarantees. Empirical results are provided that compare PICPIs with existing interval-based baselines.
发表机构
- University of California, Berkeley（加州大学伯克利分校）
- Stanford University（斯坦福大学）
- Inria Paris（法国国家信息与自动化研究所巴黎分院）
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