A Mathematical Framework for Topological Causal Data Analysis

2026-07-30

AI Summary

Traditional causal inference methods struggle to handle structured data such as images and point clouds, primarily because the difference between potential outcomes may be undefined. This work proposes a Topological Causal Data Analysis (TCDA) framework that decouples the observed space, causal model, topological representation, and causal queries, distinguishing between outcome-level and distribution-level analyses. The framework establishes identifiability conditions and a doubly robust estimation procedure. By integrating Banach space–valued summaries, the g-formula, and topological representation theory, it constructs the first systematic mathematical foundation for causal inference with structured data. The study clarifies the auxiliary role of topology in causal analysis, introduces a target-oriented topological ignorability condition, and delineates the applicability boundaries of observed topologies in causal discovery.

Abstract

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.

Problem

• causal inference
• structured data
• topological representation
• potential outcomes
• causal discovery

Innovation

• Topological Causal Data Analysis
• causal inference
• topological summaries
• doubly robust estimation
• g-formula