Beyond Means: Topological Causal Effects under Persistent-Homology Ignorability

Amir Saki; Usef Faghihi

stat.MEMar 15, 2026

Beyond Means: Topological Causal Effects under Persistent-Homology Ignorability

Authors: Amir Saki, Usef Faghihi

Organizations: Université du Québec à Trois-Rivières (UQTR).

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.

Figures & tables

Panel of Figure 1 — Figure 1: Filtration induced by the density $f_{\mu}$ . Nested shaded regions depict superlevel sets $\{x:f_{\mu}(x)\geq\lambda\}$ for decreasing thresholds. Darker regions represent higher density. At high levels, two disconnected components are present; as the threshold decreases, these regions expand and eventually merge into a single connected set, revealing the multiscale topological structure captured by summaries such as persistent homology, Betti curves, and Euler characteristic.

Figure 2: Motivating structural effects beyond means. The control law (left) is connected. Intervention 1 deforms the distribution into an elongated or filament-like structure, while Intervention 2 produces a differently shaped but still connected cloud. Mean-based targets can be near zero even when the structural contrast is substantial.

Assumptions	Properties of $\Psi$	Identified topological target
Weak ignorability $+$ positivity	Arbitrary measurable $\Psi$	The interventional laws $P_{t}$ , the marginal contrast $\Theta_{\Psi}$ , the conditional contrast $\theta_{\Psi}(z)$ , and the standardized effect $\tau_{\Psi}$
Marginal topological ignorability $+$ positivity	Injective $\Psi$	The marginal interventional laws $P_{t}$ and the marginal contrast $\Theta_{\Psi}$
Marginal topological ignorability $+$ positivity	Noninjective $\Psi$	The marginal contrast $\Theta_{\Psi}$ only
Conditional topological ignorability $+$ positivity	Injective $\Psi$	The interventional laws $P_{t}$ , the marginal contrast $\Theta_{\Psi}$ , the conditional contrast $\theta_{\Psi}(z)$ , and the standardized effect $\tau_{\Psi}$
Conditional topological ignorability $+$ positivity	Noninjective, nonlinear $\Psi$	The conditional contrast $\theta_{\Psi}(z)$ and the standardized effect $\tau_{\Psi}$ ; the marginal contrast $\Theta_{\Psi}$ is not identified in general

Table 1: Identification of topological causal targets under different assumptions. Injectivity of

\Psi

determines whether topological equality implies equality of laws.

Figure 3: Illustration of two filtrations lying in the same $\Lambda$ -filtered topological chamber. The top row corresponds to the interventional mixing law $\Pi$ , and the bottom row to the reweighted factual mixing law $\Pi^{\prime}$ . The maps $h_{c_{i}}$ denote homotopy equivalences compatible with the inclusion maps across levels, yielding isomorphic persistence modules over $\Lambda=\{C_{1}>C_{2}>C_{3}\}$ .

Figure 4: Hyperplane walls in the probability simplex. Changing between interventional and factual weights moves the latent mixture weights $w_{i}=\Pi(U=i)$ . The finite-cell topology is preserved as long as the path does not cross any wall $\sum_{i}w_{i}a_{\sigma}(i)=c$ .

Figure 5: Illustration of the common deformation-retract skeleton criterion. At each density level $c_{i}$ , the interventional and factual superlevel sets deformation retract onto the same skeleton $M_{c_{i}}$ . The inclusion maps between levels commute with the deformation retractions, yielding isomorphic persistent homology across the two filtrations.

Figure 6: Topological bifurcations of Gaussian-mixture superlevel sets: component disappearance, saddle merging, and loop collapse under mixture-weight reweighting.

Quantity	Density	Superlevel set at $c=0.20$	$\beta_{0}$
Interventional control $f_{0}$	$\frac{4}{15}\mathbf{1}_{[0,1]}+\frac{19}{60}\mathbf{1}_{[1,2]}+\frac{5}{12}\mathbf{1}_{[2,3]}$	$[0,3]$	$1$
Factual control $h_{0}$	$\frac{23}{75}\mathbf{1}_{[0,1]}+\frac{49}{150}\mathbf{1}_{[1,2]}+\frac{11}{30}\mathbf{1}_{[2,3]}$	$[0,3]$	$1$
Interventional treatment $f_{1}$	$\frac{7}{20}\mathbf{1}_{[0,1]}+\frac{13}{20}\mathbf{1}_{[2,3]}$	$[0,1]\cup[2,3]$	$2$
Factual treatment $h_{1}$	$\frac{13}{50}\mathbf{1}_{[0,1]}+\frac{37}{50}\mathbf{1}_{[2,3]}$	$[0,1]\cup[2,3]$	$2$

Table 2: Superlevel sets at

c=0.20

. Hidden confounding changes the density values but not the selected superlevel-set topology.

Panel of Figure 7 — Figure 7: Observed factual outcome clouds and oracle interventional outcome clouds for the two hidden-confounding benchmarks. The point clouds visualize the geometry underlying the finite-superlevel summaries: the control distribution is designed to have one connected high-density region, while the treated distribution is designed to have two. The numerical balance, mean-bias, and finite-superlevel topology summaries are reported in Table 3 .

Dataset	Raw SMD	Weighted SMD	True ATE x	Naive bias	IPW bias	$\Delta\beta_{0}$ ref.	$\Delta\chi$ ref.
Synthetic exact	0.728	0.001	-0.029	0.257	0.221	1.000	1.000
Breast-cancer semi-synthetic	0.430	0.006	0.053	0.275	0.236	1.000	1.000

Table 3: Finite superlevel-set hidden-confounding benchmarks. The mean target is the ATE on the first native outcome coordinate. IPW uses only observed covariates.

Panel of Figure 8 — Figure 8: JOBS II randomized application. Panel (a) shows the observed joint outcome cloud, with standardized job-search self-efficacy on the horizontal axis and standardized negative depression score on the vertical axis. Panel (b) reports the finite ECT contrast under raw, IPW, and overlap analyses. The ECT signal suggests a distributional shape/location difference between intervention and control groups, complementing the modest mean contrasts reported in the text.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 9: A four-cycle nerve preserves one $H_{1}$ loop under reweightings that keep the same active pieces and overlaps; suppressing one piece turns the nerve into a path and destroys the loop.

Arm $a$	Latent state $b$	Density $f_{a,b}(y)$	$\mathbb{E}[Y(a)\mid U=b]$
0	0	$\frac{1}{3}\mathbf{1}_{[0,3]}(y)$	$\frac{3}{2}$
0	1	$\frac{1}{5}\mathbf{1}_{[0,1]}(y)+\frac{3}{10}\mathbf{1}_{[1,2]}(y)+\frac{1}{2}\mathbf{1}_{[2,3]}(y)$	$\frac{9}{5}$
1	0	$\frac{1}{2}\mathbf{1}_{[0,1]}(y)+\frac{1}{2}\mathbf{1}_{[2,3]}(y)$	$\frac{3}{2}$
1	1	$\frac{1}{5}\mathbf{1}_{[0,1]}(y)+\frac{4}{5}\mathbf{1}_{[2,3]}(y)$	$\frac{21}{10}$

Appendix

Table 4: Latent-state-specific densities generated by the structural maps in the worked SEM.

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