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.

翻译：本文研究数据集具有分层结构（例如观测组或重复测量）时的分布自由推断问题。在此类场景中，标准交换性概念可能不成立。为解决这一挑战，本文推导出一种分层交换性形式，从而扩展了包括共形预测和jackknife+在内的分布自由方法。尽管共形预测框架的标准理论保证是边际预测覆盖保证，但在独立重复测量的特殊情况下，可以实现更强形式的覆盖——"二阶矩覆盖"性质——以更好地控制条件错误覆盖概率，并构建了满足该性质的分布自由预测集。仿真结果表明，该保证确实能产生均匀较小的条件错误覆盖概率。实证分析显示，当拟合模型校准不佳时，这种更强保证会以增大预测集宽度为代价，但在拟合模型准确的情况下，这一代价非常轻微。