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Beyond Means: Topological Causal Effects under Persistent-Homology Ignorability

Amir Saki, Usef Faghihi

Abstract

Average treatment effects (ATE) and conditional average treatment effects (CATE) are foundational causal estimands, but they target changes in expected outcomes and can miss treatment-induced changes in the shape of outcome distributions. A canonical failure mode occurs when control outcomes are unimodal, treated outcomes become bimodal, and both distributions have the same mean. In such cases mean-based causal estimands are zero even though the geometry and topology of the outcome law change substantially. This paper develops a topological causal framework based on persistent homology. We formalize a persistent-homology ignorability condition, define topological analogues of CATE and ATE, and prove that these estimands are identifiable up to an explicit error bound under approximate topological ignorability. We also clarify a subtle but important point: a marginal persistence-diagram effect is not identified from conditional topological ignorability alone because persistent homology does not in general commute with mixtures over covariates. To preserve the original intuition while ensuring scientific correctness, we retain the marginal effect as a motivating quantity, but place the mathematically sound conditional estimands at the center of the theory. A synthetic experiment with mean-preserving topology change shows that mean-based causal estimands remain near zero while the proposed topological effect increases sharply and remains recoverable after adjustment for confounding.

Beyond Means: Topological Causal Effects under Persistent-Homology Ignorability

Abstract

Average treatment effects (ATE) and conditional average treatment effects (CATE) are foundational causal estimands, but they target changes in expected outcomes and can miss treatment-induced changes in the shape of outcome distributions. A canonical failure mode occurs when control outcomes are unimodal, treated outcomes become bimodal, and both distributions have the same mean. In such cases mean-based causal estimands are zero even though the geometry and topology of the outcome law change substantially. This paper develops a topological causal framework based on persistent homology. We formalize a persistent-homology ignorability condition, define topological analogues of CATE and ATE, and prove that these estimands are identifiable up to an explicit error bound under approximate topological ignorability. We also clarify a subtle but important point: a marginal persistence-diagram effect is not identified from conditional topological ignorability alone because persistent homology does not in general commute with mixtures over covariates. To preserve the original intuition while ensuring scientific correctness, we retain the marginal effect as a motivating quantity, but place the mathematically sound conditional estimands at the center of the theory. A synthetic experiment with mean-preserving topology change shows that mean-based causal estimands remain near zero while the proposed topological effect increases sharply and remains recoverable after adjustment for confounding.
Paper Structure (26 sections, 3 theorems, 61 equations, 2 figures, 1 table)

This paper contains 26 sections, 3 theorems, 61 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Causal inference with potential outcomes.
  4. Topological data analysis and persistent homology.
  5. Vectorizations and statistical summaries of persistence diagrams.
  6. Causal inference for non-Euclidean and random-object outcomes.
  7. Topology and causality.
  8. Topology and causality.
  9. Setup and Topological Objects
  10. Persistent-Homology Ignorability and Topological Causal Effects
  11. Topological analogues of CATE and ATE
  12. Approximate Identification Theorems
  13. The Marginal Topological Causal Effect: Motivation and Technical Caution
  14. Why this is not an exact proof.
  15. What is scientifically correct.
  16. ...and 11 more sections

Key Result

Proposition 1

Suppose standard conditional ignorability holds: Then, under consistency, for each $t\in\{0,1\}$ and $P_Z$-almost every $z$. In particular, eq:ph-ignorability holds with $\varepsilon=0$.

Figures (2)

  • Figure 1: A motivating mean-preserving topology change. Mean-based estimands vanish, but a density-based or measure-based persistent-homology summary detects the split from one dominant connected component to two.
  • Figure 2: Mean-preserving topology change under the synthetic design. The right panel shows that the mean effect remains statistically negligible across all values of $\Delta$. The left panel shows that the proposed topological effect remains close to zero when the treated distribution is nearly unimodal and then rises sharply as the two clusters separate. This is exactly the failure mode we want to capture: ATE misses the effect, while topology detects it.

Theorems & Definitions (12)

  • Definition 1: Persistent-homology summary map
  • Proposition 1: Classical ignorability implies exact PH-ignorability
  • proof
  • Definition 2: Distance-based topological CATE and ATE
  • Definition 3: Embedded topological CATE and ATE
  • Remark 1: Do topological effects replace ATE and CATE?
  • Theorem 1: Approximate identification of distance-based topological effects
  • proof
  • Theorem 2: Approximate identification of embedded topological effects
  • proof
  • ...and 2 more