Abstract
Many interventions alter the structure of an outcome distribution rather than its mean: they can split a population into disconnected regimes, create loops or holes, generate branches, or reorganize an outcome cloud while leaving the average response nearly unchanged. In such settings, mean-based causal estimands such as the average treatment effect may miss important structural effects. We introduce topological-geometrical causal metrics based on summaries of interventional outcome laws, including density-superlevel Betti summaries, Euler signatures, and persistent-homology summaries. These metrics quantify structural differences between treated and untreated outcome laws beyond averages. We also study the assumptions needed for causal interpretation. We introduce topological ignorability, a topological analogue of conditional ignorability that requires invariance of the chosen structural feature rather than the full counterfactual distribution. When the chosen summary is injective, this condition coincides with weak ignorability; for noninjective summaries, it can identify the structural feature of interest without identifying the full interventional law. We define a covariate-standardized topological-geometrical causal effect and develop practical estimators. We validate the framework in two hidden-confounding benchmarks: a fully synthetic exact benchmark and a real-covariate semi-synthetic benchmark using Wisconsin breast-cancer covariates. In both, weak ignorability fails and balancing observed covariates nearly eliminates standardized mean differences, yet the coordinate-mean average treatment effect remains biased. By contrast, selected finite density-superlevel Betti and Euler contrasts remain stable across oracle, observational, and weighted analyses.
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