In many modern statistical problems, the limited available data must be used both to develop the hypotheses to test, and to test these hypotheses-that is, both for exploratory and confirmatory data analysis. Reusing the same dataset for both exploration and testing can lead to massive selection bias, leading to many false discoveries. Selective inference is a framework that allows for performing valid inference even when the same data is reused for exploration and testing. In this work, we are interested in the problem of selective inference for data clustering, where a clustering procedure is used to hypothesize a separation of the data points into a collection of subgroups, and we then wish to test whether these data-dependent clusters in fact represent meaningful differences within the data. Recent work by Gao et al. [2022] provides a framework for doing selective inference for this setting, where the hierarchical clustering algorithm is used for producing the cluster assignments, which was then extended to k-means clustering by Chen and Witten [2022]. Both these works rely on assuming a known covariance structure for the data, but in practice, the noise level needs to be estimated-and this is particularly challenging when the true cluster structure is unknown. In our work, we extend to the setting of noise with unknown variance, and provide a selective inference method for this more general setting. Empirical results show that our new method is better able to maintain high power while controlling Type I error when the true noise level is unknown.

翻译：在许多现代统计问题中，有限的可利用数据必须既用于构建待检验的假设，又用于检验这些假设——即同时用于探索性数据分析和验证性数据分析。对同一数据集重复用于探索和检验可能导致严重的选择偏差，进而产生大量虚假发现。选择性推断是一个框架，允许在相同数据被重复用于探索和检验时进行有效的统计推断。本文关注数据聚类的选择性推断问题：通过聚类过程假设数据点被划分为若干子组，随后检验这些依赖于数据的聚类是否确实代表数据中的有意义差异。Gao等人[2022]近期的工作为这一场景提供了选择性推断框架，其中使用层次聚类算法生成聚类分配，随后由Chen和Witten[2022]将其扩展至k-means聚类。这两项工作均依赖于假设数据具有已知的协方差结构，但在实践中，噪声水平需要被估计——当真实聚类结构未知时，这一估计尤为困难。本文将其推广至噪声方差未知的场景，并为这一更一般化的场景提供了选择性推断方法。实验结果表明，当真实噪声水平未知时，我们的新方法能够在控制第一类错误的同时更好地维持统计功效。