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。预测推断方法已在多种假设下发展起来，通常——例如在标准共形预测中——依赖于数据分布在特定变换群（如置换群）下的不变性。此外，许多现有的预测推断方法旨在预测特征-观测序列中未观测到的结果。与此同时，人们对于更一般观测模型（例如部分观测特征）下以及满足更一般分布对称性（例如物理学中的旋转不变性或坐标无关观测）的数据进行预测推断存在兴趣。本文提出SymmPI，一种适用于任意观测模型中数据分布具有一般群对称性的预测推断方法。我们的方法利用了分布等变变换这一新概念，该变换在保持数据分布不变性的同时处理数据。我们证明SymmPI在分布不变性下具有有效覆盖性，并刻画了其在分布偏移下的性能，将近期相关结果作为特例予以涵盖。我们将SymmPI应用于预测网络中与顶点相关的未观测值，其中分布在保持网络结构不变的重新标记下保持不变。在双层层次模型的若干模拟实验以及一个实证数据分析案例中，SymmPI相较于现有方法表现出更优的性能。