Topological data analysis has emerged as a powerful tool for extracting the metric, geometric and topological features underlying the data as a multi-resolution summary statistic, and has found applications in several areas where data arises from complex sources. In this paper, we examine the use of topological summary statistics through the lens of statistical inference. We investigate necessary and sufficient conditions under which \textit{valid statistical inference} is possible using {topological summary statistics}. Additionally, we provide examples of models that demonstrate invariance with respect to topological summaries.

翻译：拓扑数据分析作为一种多分辨率汇总统计量，已成为提取数据背后度量、几何和拓扑特征的有力工具，并在数据源自复杂源的多个领域得到应用。本文从统计推断的视角审视拓扑汇总统计量的使用，探究利用拓扑汇总统计量实现有效统计推断的充要条件，并给出模型示例以证明其对拓扑汇总统计量的不变性。