分层数据下的无分布推断
Distribution-free inference with hierarchical data
AI总结：
针对分层数据下标准可交换性不成立的问题，提出分层可交换性以拓展无分布方法，还在独立重复测量场景下构建了具备二阶矩覆盖率的无分布预测集，仿真验证其条件未覆盖率低且模型准确时代价轻微。
摘要：
本文研究数据集具有分层结构（例如成组观测值或重复测量）场景下的无分布推断。在此类场景中，标准的可交换性概念可能不成立。为应对这一挑战，本文推导了一种分层形式的可交换性，为共形预测（conformal prediction）、jackknife+等无分布方法的拓展提供了基础。\n尽管共形预测框架的标准理论保证是边际预测覆盖率保证，但在独立重复测量的特殊情况下，可以实现更强的覆盖率形式——“二阶矩覆盖率”特性，以更好地控制条件未覆盖率，本文还构建了满足该特性的无分布预测集。\n仿真实验表明，该保证确实能带来一致较小的条件未覆盖率。从经验上看，在拟合模型校准不佳的场景中，这种更强的保证会以预测集宽度更大为代价，但在拟合模型准确的情况下，该代价非常轻微。
英文摘要：
This paper studies distribution-free inference in settings where the data set has a hierarchical structure -- for example, groups of observations, or repeated measurements. In such settings, standard notions of exchangeability may not hold. To address this challenge, a hierarchical form of exchangeability is derived, facilitating extensions of distribution-free methods, including conformal prediction and jackknife+. While the standard theoretical guarantee obtained by the conformal prediction framework is a marginal predictive coverage guarantee, in the special case of independent repeated measurements, it is possible to achieve a stronger form of coverage -- the "second-moment coverage" property -- to provide better control of conditional miscoverage rates, and distribution-free prediction sets that achieve this property are constructed. Simulations illustrate that this guarantee indeed leads to uniformly small conditional miscoverage rates. Empirically, this stronger guarantee comes at the cost of a larger width of the prediction set in scenarios where the fitted model is poorly calibrated, but this cost is very mild in cases where the fitted model is accurate.
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