AI 中文总结
该研究提出拓扑因果数据分析（TCDA）框架，区分结果级与分布级TCDA，构建对应识别与表示方法，明确观测拓扑在因果发现中的作用，为结构化对象的因果分析提供数学基础。
AI 中文摘要
许多现代结果，包括图像、点云、网络和空间场，都是结构化对象，对于这些对象，\nY^1-Y^0\n可能没有定义，或者在科学上不够充分。我们引入了\textit{拓扑因果数据分析}（Topological Causal Data Analysis，TCDA），这是一种将观测空间、因果模型类、拓扑表示和因果查询分离开的框架。拓扑并不定义干预；它在指定因果假设后提供稳定的、对形状敏感的摘要。我们区分了结果级TCDA（转换个体潜在结果）和分布级TCDA（转换干预结果分布），并刻画了结果级和分布级对比一致的情况。基于最近的结果级理论，我们为巴拿赫空间值摘要构建了识别和双重稳健表示。在分布级，我们通过标准因果\textit{g}-公式识别目标，并推导了稳定性转移界和插件一致性。我们还在该框架中放置了特定目标的拓扑不可知性，阐明了在不识别完整干预分布的情况下，何时可以识别协变量标准化的粗效果。最后，我们界定了观测拓扑在因果发现中的作用：它可以辅助对受限模型类的诊断，但本身无法识别因果结构。
英文摘要
Many modern outcomes, including images, point clouds, networks, and spatial fields, are structured objects for which \(Y^1-Y^0\) may be undefined or scientifically inadequate. We introduce \emph{Topological Causal Data Analysis} (TCDA), a framework separating the observation space, causal-model class, topological representation, and causal query. Topology does not define interventions; it supplies stable, shape-sensitive summaries after causal assumptions have been specified. We distinguish outcome-level TCDA, which transforms individual potential outcomes, from distribution-level TCDA, which transforms interventional outcome laws, and characterize when outcome and distribution level contrasts agree. Building on recent outcome-level theory, we formulate identification and doubly robust representations for Banach-space-valued summaries. At the distribution level, we identify targets through the standard causal \(g\)-formula and derive stability-transfer bounds and plug-in consistency. We also place target-specific topological ignorability within the framework, clarifying when a covariate-standardized coarse effect can be identified without identifying the full interventional laws. Finally, we delimit the role of observational topology in causal discovery: it can assist diagnosis on restricted model classes but cannot by itself identify causal structure.
