1 Introduction↩︎

Classical causal inference is built around mean-based targets such as the average treatment effect \[\mathrm{ATE} = \mathbb{E}\big[Y(1)-Y(0)\big]\] and the conditional average treatment effect \[\mathrm{CATE}(z) = \mathbb{E}\big[Y(1)-Y(0)\mid Z=z\big].\] These estimands are indispensable when the scientific question concerns shifts in mean response. However, many modern problems involve outcomes whose scientifically important changes are not mean shifts but shape changes in the full outcome law: cluster splitting, multimodality, loop formation, filamentary structure, or changes in connectedness. In such settings, ATE and CATE may be exactly zero while the treatment effect is scientifically substantial.

A minimal example illustrates the problem. Fix a covariate value \(z\) and let \[Y(0)\mid Z=z \sim \mathcal{N}(m(z),\sigma^2), \qquad Y(1)\mid Z=z \sim \tfrac12\mathcal{N}(m(z)-\Delta,\sigma^2)+\tfrac12\mathcal{N}(m(z)+\Delta,\sigma^2). \label{eq:motivating-example}\tag{1}\] Then \[\mathbb{E}[Y(0)\mid Z=z]=\mathbb{E}[Y(1)\mid Z=z]=m(z),\] so \[\mathrm{CATE}(z)=0 \qquad\text{and hence}\qquad \mathrm{ATE}=0.\] Yet for sufficiently large \(\Delta/\sigma\), the treated law in 1 is bimodal while the control law is unimodal. If topology is extracted from density superlevel sets, a distance-to-a-measure filtration, or a smoothed point-cloud filtration, the treated law develops two long-lived \(0\)-dimensional components whereas the control law has only one dominant component. Mean-based estimands therefore miss a scientifically meaningful effect.

The paper develops a causal framework that targets precisely this kind of structural change. Our starting point is a persistent-homology summary map \[G_k:\mathcal{M}(\mathcal{Y})\to \mathcal{D}_k,\] which sends an outcome law to a \(k\)-dimensional persistence diagram. This allows us to replace mean differences with distances or embedded differences between conditional persistence diagrams.

The key contributions are:

1. We formalize a persistent-homology conditional ignorability condition that aligns the causal assumption with the topological target.

2. We define topological analogues of CATE and ATE, as well as embedded analogues obtained by vectorizing persistence diagrams.

3. We prove approximate identification results with explicit error bounds. Under an \(\varepsilon\)-level topological ignorability condition, the observable topological CATE and ATE differ from their counterfactual targets by at most \(2\varepsilon\); for \(L_\phi\)-Lipschitz embeddings, the embedded effects are identified up to \(2L_\phi\varepsilon\).

4. We retain the original marginal topological effect as a motivating quantity, but show why it is not exactly identifiable from conditional topological ignorability alone without an additional mixture-stability assumption.

5. We provide a synthetic experiment showing that mean-based causal estimands remain near zero under a mean-preserving topology change, while the proposed topological effect grows with cluster separation and remains recoverable after covariate adjustment.

2 Related Work↩︎

Our formulation sits at the intersection of three literatures.

2.0.0.1 Causal inference with potential outcomes.

The potential-outcomes framework and ignorability-based identification are classical in statistics and econometrics. The foundations of treatment-effect estimation trace back to Rubin’s formulation of causal effects and to the role of strongly ignorable treatment assignment and balancing scores in observational studies. The propensity score provides a coarsest balancing score that can remove bias due to observed covariates under strong ignorability [1]–[3].

2.0.0.2 Topological data analysis and persistent homology.

Persistent homology provides multiscale summaries of shape and structure in data, and has become a central tool in topological data analysis (TDA) [4]–[8]. A major reason persistent homology is useful statistically is stability: persistence diagrams are Lipschitz-stable under perturbations of tame functions, and algebraic stability extends this viewpoint to persistence modules [9], [10]. Statistical inference for persistence diagrams and robust topological inference through distance-to-a-measure and kernel-distance constructions are now well developed [11], [12].

2.0.0.3 Vectorizations and statistical summaries of persistence diagrams.

Because persistence diagrams do not form a linear space, a substantial literature studies stable embeddings and summaries. Persistence landscapes place diagrams in a separable Banach space and support laws of large numbers, central limit theorems, and classical statistical tools [13]. Persistence images and kernel embeddings of diagrams make it possible to use standard machine-learning pipelines while preserving stability [14]–[16]. These representations are especially relevant for causal estimation because they allow us to construct embedded versions of topological treatment effects that can be averaged across covariates.

2.0.0.4 Causal inference for non-Euclidean and random-object outcomes.

Recent work has broadened causal inference beyond scalar outcomes to random objects and general metric-space-valued data. Fréchet regression provides a fundamental conditional-mean-type object for random objects [17]. Building on this perspective, recent work develops geodesic treatment effects and doubly robust estimation procedures for non-Euclidean outcomes [18], [19]. These papers show that causal inference can be meaningfully extended beyond vector spaces. Our contribution is complementary: instead of geodesic means, we target treatment-induced changes in topological structure.

2.0.0.5 Topology and causality.

2.0.0.6 Topology and causality.

There is already conceptual interest in the interaction between topology and causality. Ibeling and Icard introduce a topological learning-theoretic perspective on causal inference by equipping spaces of structural causal models with topologies [20]. More closely related to the present treatment-effect setting, Kim and Lee define topological causal effects for non-Euclidean outcomes using persistence-based summaries of potential outcomes [21]. Farzam et al.connect topology to treatment-effect estimation from a different angle, using topological summaries to regularize representation-balancing networks for causal effect estimation, but without defining a topological causal estimand or identification result of the kind studied here [22]. By contrast, Bouchattaoui uses persistent-homology features for bivariate causal discovery—that is, inferring causal direction rather than estimating treatment effects—so this line is only indirectly related to the present work [23]. Relative to these efforts, the present paper emphasizes the precise alignment of assumption, estimand, and identification argument: the ignorability condition is stated directly at the level of persistent-homology summaries, and the central estimands are the conditional topological analogues of CATE and ATE that this assumption actually identifies.

3 Setup and Topological Objects↩︎

Let \(T\in\{0,1\}\) denote treatment, let \(Z\in\mathcal{Z}\) denote observed covariates, and let \(Y(1),Y(0)\) denote potential outcomes taking values in a metric space \((\mathcal{Y},d_{\mathcal{Y}})\). The observed outcome is \[Y = T Y(1) + (1-T)Y(0)\] when \(\mathcal{Y}\subseteq\mathbb{R}^p\); more generally, we use the standard consistency relation \(Y=Y(T)\).

Let \(\mathcal{M}(\mathcal{Y})\) be a class of Borel probability measures on \(\mathcal{Y}\) and let \(\mathcal{D}_k\) denote the space of \(k\)-dimensional persistence diagrams equipped with a metric \(d_{\mathcal{D}}\), such as the bottleneck distance or a \(p\)-Wasserstein distance.

The choice of \(G_k\) matters scientifically. If one wants to detect multimodality, density-based or measure-based filtrations are more appropriate than raw support-based topology because the support of a smooth bimodal law may remain connected even when its mass concentrates in separated regions. This paper therefore treats \(G_k\) abstractly and allows density, DTM, or empirical point-cloud constructions depending on the application.

Define the counterfactual conditional diagrams \[D_k^{\mathrm{cf}}(t,z) := G_k\big(\mathcal{L}(Y(t)\mid Z=z)\big), \qquad t\in\{0,1\},\] and the observable conditional diagrams \[D_k^{\mathrm{obs}}(t,z) := G_k\big(\mathcal{L}(Y\mid T=t,Z=z)\big).\]

These objects are the topological counterparts of conditional potential-outcome laws and conditional observed-outcome laws.

4 Persistent-Homology Ignorability and Topological Causal Effects↩︎

This assumption is the persistent-homology analogue of conditional ignorability. It does not say that treatment assignment is conditionally independent of the full potential-outcome law. It only requires the topological summaries of these laws to match up to error \(\varepsilon\).

The converse is false in general because \(G_k\) is many-to-one: two distinct laws may have the same persistent-homology summary.

4.1 Topological analogues of CATE and ATE↩︎

These estimands are direct analogues of CATE and ATE, but with difference in conditional means replaced by distance between conditional persistence diagrams. The replacement is natural because diagrams do not admit a canonical signed subtraction.

The embedded estimands are the closest literal analogues of CATE and ATE because they are true differences and averages in a linear space. They become especially convenient when \(\phi\) is a stable embedding.

5 Approximate Identification Theorems↩︎

We now prove the main identification results. The core principle is simple: if the treated and control persistence diagrams observed within each covariate stratum are each within \(\varepsilon\) of their counterfactual targets, then the distance between the observed treated and control diagrams is within \(2\varepsilon\) of the counterfactual topological effect.

6 The Marginal Topological Causal Effect: Motivation and Technical Caution↩︎

The original motivating quantity in our discussion was the marginal topological causal effect \[\mathrm{TCE}_k^{\mathrm{marg}} := d_{\mathcal{D}}\!\left(G_k\big(\mathcal{L}(Y(1))\big),\, G_k\big(\mathcal{L}(Y(0))\big)\right). \label{eq:marginal-tce}\tag{3}\] This is a perfectly meaningful causal quantity: it compares the topological summaries of the two marginal counterfactual outcome laws.

A natural first proof sketch tries to mimic the classical ATE argument: \[\mathcal{L}(Y(t)) = \int \mathcal{L}(Y(t)\mid Z=z)\,\mathrm{d}P_Z(z),\] and, heuristically, \[G_k\big(\mathcal{L}(Y(t))\big) \approx \int G_k\big(\mathcal{L}(Y(t)\mid Z=z)\big)\,\mathrm{d}P_Z(z) \approx \int G_k\big(\mathcal{L}(Y\mid T=t,Z=z)\big)\,\mathrm{d}P_Z(z). \label{eq:heuristic-mixture}\tag{4}\] One would then hope to conclude that \[\mathrm{TCE}_k^{\mathrm{marg}} \approx d_{\mathcal{D}}\!\left(G_k(\mathcal{L}(Y\mid T=1)), G_k(\mathcal{L}(Y\mid T=0))\right).\]

6.0.0.1 Why this is not an exact proof.

The step in 4 is generally invalid. Persistent homology is nonlinear and does not in general commute with mixtures over covariates. More concretely, for a generic persistence map \(G_k\), \[G_k\!\left(\int \mu_z\,\mathrm{d}P_Z(z)\right) \neq \int G_k(\mu_z)\,\mathrm{d}P_Z(z)\] in any canonical sense because persistence diagrams do not form a linear space, and even stable embeddings of diagrams are applied after diagram construction.

6.0.0.2 What is scientifically correct.

The quantity in 3 should therefore be treated as a motivating marginal effect, not as the primary object identified by Assumption ?? . Exact or approximate identification of 3 requires additional structure, such as a mixture-stability condition tailored to the chosen summary map \(G_k\). The present paper does not assume such a property. Instead, it centers the conditional estimands in ?? –?? , which are precisely aligned with the ignorability assumption.

7 Synthetic Experiments↩︎

This section provides an illustrative experiment rather than a definitive benchmark. The goal is to visualize when and why mean-based causal estimands fail, and to show that the proposed topological estimand remains informative.

7.1 Synthetic design↩︎

We use a binary covariate \(Z\in\{0,1\}\) with \(\mathbb{P}(Z=1)=1/2\), and define two-dimensional outcomes. Let \(e_1=(1,0)^\top\), let \(m_0=(-1.5,0)^\top\), let \(m_1=(1.5,0)^\top\), and let \(\sigma=0.18\). The control potential outcome is unimodal, \[Y(0)\mid Z=z \sim \mathcal{N}_2(m_z,\sigma^2 I_2),\] while the treated potential outcome is a symmetric bimodal mixture with the same conditional mean, \[Y(1)\mid Z=z \sim \tfrac12\mathcal{N}_2(m_z-\Delta e_1,\sigma^2 I_2) + \tfrac12\mathcal{N}_2(m_z+\Delta e_1,\sigma^2 I_2). \label{eq:synthetic-laws}\tag{5}\] Therefore, \[\mathbb{E}[Y(1)\mid Z=z]=\mathbb{E}[Y(0)\mid Z=z]=m_z,\] so the adjusted mean effect is zero for every \(z\), and hence the true ATE is zero as well.

To create confounding, treatment assignment depends on \(Z\): \[\mathbb{P}(T=1\mid Z=0)=0.2, \qquad \mathbb{P}(T=1\mid Z=1)=0.8. \label{eq:propensity-design}\tag{6}\] Thus unadjusted treated and control groups differ in their covariate composition.

7.1.0.1 Topological summary used in the experiment.

For illustration we compute the empirical \(0\)-dimensional Vietoris–Rips persistence diagram of the outcome cloud within each stratum \((T=t,Z=z)\). Because births are zero in \(0\)-dimensional persistence, this diagram is equivalent to the multiset of edge lengths in the Euclidean minimum spanning tree. We embed each diagram using the first three persistence-landscape layers evaluated on a fixed grid, and compute the \(L^2\) distance between the resulting vectors. This yields an empirical approximation to the embedded topological effect. Although our theorems are agnostic to the exact filtration, this finite-sample construction is simple, reproducible, and visually transparent.

7.2 Mean-preserving topology change as separation increases↩︎

We first vary \(\Delta\in\{0,0.2,0.4,0.6,0.8,1.0,1.2\}\) in 5 without confounding and estimate the mean effect and the topological effect from 50 Monte Carlo replications with 200 observations per arm. Figure 2 shows the result.

The visual pattern is the key message. For small \(\Delta\), the treated mixture is nearly unimodal and its topology is close to that of the control law. As \(\Delta\) grows, the treated cloud splits into two separated regions and the topological effect increases. Meanwhile the mean effect remains near zero because the treated law is symmetric around the same conditional center. This demonstrates the power of the proposed framework to detect structural changes that are invisible to ATE and CATE.

7.3 Confounded observational sample↩︎

We next generate observational data under 5 and 6 with \(\Delta=1.0\), sample size \(n=600\) per replication, and 50 replications. We compare four quantities:

1. the unadjusted mean contrast, computed by pooling treated and control units;

2. the adjusted mean effect, computed by stratifying on \(Z\) and averaging with respect to the empirical law of \(Z\);

3. the unadjusted topological contrast, computed from the pooled treated and control persistence summaries;

4. the adjusted topological effect, computed by stratifying on \(Z\) and averaging the within-stratum topological contrasts.

Table 1 summarizes the results.

Two points are worth emphasizing. First, the adjusted mean effect is essentially zero, which is correct but scientifically incomplete: the treatment changes the topology of the outcome cloud without changing its conditional mean. Second, the unadjusted topological contrast is strongly biased because the treatment groups have different \(Z\)-composition, while the adjusted topological effect moves close to the counterfactual benchmark. This is exactly what the identification theorems predict: once the assumption and estimand are aligned at the conditional topological level, covariate adjustment restores the relevant causal quantity.

8 Discussion↩︎

The framework developed here is deliberately modest in one sense and ambitious in another. It is modest because it does not claim that persistent homology replaces all mean-based causal inference. It does not. Rather, it complements classical estimands by targeting structure that averages cannot see. It is ambitious because it argues that causal questions should sometimes be asked directly at the level of shape, topology, and multiscale organization.

Several extensions are natural. First, one can replace diagram distances with specific stable embeddings—landscapes, images, silhouettes, or kernel means—and develop semiparametric estimators for the corresponding embedded effects. Second, one can study doubly robust and orthogonalized estimators in the spirit of modern causal machine learning, but with persistence summaries replacing Euclidean outcomes. Third, one can adapt the framework to continuous treatments, longitudinal interventions, or synthetic-control settings for metric-space-valued outcomes.

The most important theoretical caution is the one emphasized in Section 6: persistence diagrams are nonlinear objects, so conditional topological ignorability does not by itself identify a marginal persistence-diagram effect. This is not a weakness of the framework; it is a sign that topological causal inference must respect the mathematical geometry of its own summaries.

9 Conclusion↩︎

ATE and CATE can miss treatment effects that operate through changes in the geometry and topology of outcome distributions. Persistent homology provides a principled way to summarize those changes. By defining a persistent-homology ignorability condition and matching it with conditional topological analogues of CATE and ATE, we obtain causal estimands that are identifiable up to an explicit error bound and that respond to structural changes invisible to mean-based summaries. The synthetic experiment demonstrates the intended use case: when treatment preserves the mean but changes the shape of the outcome distribution, the proposed topological effect is informative, stable, and recoverable after adjustment for confounding.

9.0.0.1 Reproducibility note.

All numerical values reported in Figures 2 and Table 1 come from a simple synthetic simulation with 0D Vietoris–Rips persistence, minimum spanning trees, and three-layer persistence landscapes. The experiment is meant to be illustrative of the theory rather than a definitive empirical benchmark.

10 Appendix A: Heuristic Marginal Argument Preserved from the Original Discussion↩︎

Because the original development of this work centered on a marginal persistence-diagram effect, we preserve that heuristic argument here without deleting its formulas. This appendix is intentionally faithful to the original idea, but it is followed immediately by the correction already stated in Section 6.

The motivating marginal causal metric is \[\mathrm{TCE}_k^{\mathrm{marg}} = d_{\mathcal{D}}\!\left(\mathrm{PD}_k(\mathcal{L}(Y(1))),\,\mathrm{PD}_k(\mathcal{L}(Y(0)))\right).\] The original topological ignorability condition was written as \[\sup_{t,z} W\!\left( \mathrm{PD}_k\big(\mathcal{L}(Y(t)\mid T=t,Z=z)\big), \mathrm{PD}_k\big(\mathcal{L}(Y(t)\mid Z=z)\big) \right) \le \varepsilon. \label{eq:original-ignorability}\tag{7}\] Using consistency, \[\mathcal{L}(Y(t)\mid T=t,Z=z)=\mathcal{L}(Y\mid T=t,Z=z),\] so 7 becomes \[W\!\left( \mathrm{PD}_k\big(\mathcal{L}(Y\mid T=t,Z=z)\big), \mathrm{PD}_k\big(\mathcal{L}(Y(t)\mid Z=z)\big) \right) \le \varepsilon.\] The original sketch then tried to pass from conditional to marginal topology via \[\mathcal{L}(Y(t)) = \int \mathcal{L}(Y(t)\mid Z=z)\,\mathrm{d}P(z)\] and heuristically wrote \[\mathrm{PD}_k\big(\mathcal{L}(Y(t))\big) \approx \int \mathrm{PD}_k\big(\mathcal{L}(Y\mid T=t,Z=z)\big)\,\mathrm{d}P(z).\] This would suggest an observable proxy \[\widehat{\mathrm{TCE}}_k^{\mathrm{marg}} = W\!\left( \mathrm{PD}_k\big(\mathcal{L}(Y\mid T=1)\big),\, \mathrm{PD}_k\big(\mathcal{L}(Y\mid T=0)\big) \right)\] and a bound of the form \[\big|\widehat{\mathrm{TCE}}_k^{\mathrm{marg}} - \mathrm{TCE}_k^{\mathrm{marg}}\big| \le \delta, \qquad \delta = 2C\varepsilon.\]

The purpose of preserving this appendix is to keep the original conceptual path visible. The correction, however, is exactly the one given in Section 6: the step from conditional diagrams to the marginal diagram is not justified without additional mixture-stability structure. Accordingly, the main results of the paper are stated only for the conditional topological estimands that the assumption actually identifies.

11 Appendix B: Why the motivating example must use a density- or measure-based filtration↩︎

There is a subtle point in the opening example. If \(Y(0)\) and \(Y(1)\) are absolutely continuous on \(\mathbb{R}\), then both supports are connected intervals (indeed all of \(\mathbb{R}\)), so the support topology alone does not distinguish unimodal from bimodal distributions. For that reason the motivating example should be interpreted through one of the following choices of \(G_k\):

1. the persistence of density superlevel sets;

2. the persistence of a kernel-distance or distance-to-a-measure function;

3. a finite-sample point-cloud filtration on empirical draws when the practically relevant object is the sampled cloud rather than the ideal support.

The theory in the main text is compatible with all three viewpoints because \(G_k\) was left abstract from the start.

12 Appendix C: A compact summary of the scientific logic↩︎

The framework can be summarized in one table.

The important point is not that topology replaces all other summaries, but that it targets a different and often scientifically essential aspect of the effect.

References↩︎