Title: Topological Ignorability for Structural Causal Effects Beyond Means

Abstract:

Interventions often modify the underlying structure of an outcome distribution without significantly shifting its mean. Such interventions may divide a population into separate regimes, introduce loops or voids, produce branching structures, or rearrange the distribution’s geometry while keeping the average response largely intact. In these scenarios, traditional mean-based causal measures, such as the average treatment effect, may overlook critical structural changes. To address this, we propose topological-geometrical causal metrics derived from summaries of interventional outcome laws, including density-superlevel Betti summaries, Euler signatures, and persistent-homology summaries. These metrics capture structural disparities between treated and untreated outcome laws that go beyond simple averages.

We also examine the assumptions necessary for valid causal interpretation. We define topological ignorability, a topological counterpart to conditional ignorability, which demands the invariance of the selected structural feature rather than the entire counterfactual distribution. If the chosen summary is injective, this condition aligns with weak ignorability; however, for noninjective summaries, it allows for the identification of the specific structural feature of interest without requiring the full interventional law to be identified. We establish a covariate-standardized topological-geometrical causal effect and develop practical estimators for it.

The framework is validated using two benchmarks involving hidden confounding: a fully synthetic exact benchmark and a semi-synthetic benchmark utilizing Wisconsin breast-cancer covariates. In both cases, weak ignorability is violated, and balancing observed covariates nearly removes standardized mean differences, yet the coordinate-mean average treatment effect remains biased. Conversely, selected finite density-superlevel Betti and Euler contrasts remain stable across oracle, observational, and weighted analyses.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC