Quantifying the uncertainty of predictions is a core problem in modern statistics. Methods for predictive inference have been developed under a variety of assumptions, often -- for instance, in standard conformal prediction -- relying on the invariance of the distribution of the data under special groups of transformations such as permutation groups. Moreover, many existing methods for predictive inference aim to predict unobserved outcomes in sequences of feature-outcome observations. Meanwhile, there is interest in predictive inference under more general observation models (e.g., for partially observed features) and for data satisfying more general distributional symmetries (e.g., rotationally invariant or coordinate-independent observations in physics). Here we propose SymmPI, a methodology for predictive inference when data distributions have general group symmetries in arbitrary observation models. Our methods leverage the novel notion of distributional equivariant transformations, which process the data while preserving their distributional invariances. We show that SymmPI has valid coverage under distributional invariance and characterize its performance under distribution shift, recovering recent results as special cases. We apply SymmPI to predict unobserved values associated to vertices in a network, where the distribution is unchanged under relabelings that keep the network structure unchanged. In several simulations in a two-layer hierarchical model, and in an empirical data analysis example, SymmPI performs favorably compared to existing methods.

翻译：量化预测的不确定性是现代统计学的核心问题。在多种假设下已发展了预测推断方法——例如在标准共形预测中——通常依赖于数据分布在特殊变换群（如置换群）下的不变性。此外，许多现有预测推断方法旨在预测特征-结果观测序列中未观测到的结果。与此同时，在更一般的观测模型（如部分特征可观测）下，以及满足更一般分布对称性（如物理学中旋转不变性或坐标无关观测）的数据上进行预测推断也备受关注。本文提出SymmPI方法，适用于数据分布具有任意观测模型下一般群对称性的预测推断。该方法利用分布等变变换的新概念，在处理数据时保持其分布不变性。我们证明SymmPI在分布不变性下具有有效的覆盖特性，并刻画了其在分布偏移下的性能表现，恢复近期多项研究作为特例。我们将SymmPI应用于预测网络中与顶点相关的未观测值，其中网络结构在保持不变的重新标记下分布保持不变。在两层次分层模型的多次模拟及实证数据分析示例中，SymmPI的性能优于现有方法。