Estimating causal effects is particularly challenging when outcomes arise in complex, non-Euclidean spaces, where conventional methods often fail to capture meaningful structural variation. We develop a framework for topological causal inference that defines treatment effects through differences in the topological structure of potential outcomes, summarized by power-weighted silhouette functions of persistence diagrams. We develop an efficient, doubly robust estimator in a fully nonparametric model, establish functional weak convergence, and construct a formal test of the null hypothesis of no topological effect. Empirical studies illustrate that the proposed method reliably quantifies topological treatment effects across diverse complex outcome types.
Figures & tables
Figure 1: Left: Example of untreated vs. treated macromolecule structures. Right: Corresponding persistence diagrams, highlighting treatment-induced changes in the 1st-order homology features.
Figure 2: Silhouette functions revealing differences in 1 -dimensional features in the ORBIT dataset ( Adams et al., 2017 ) . (a) Point clouds A,B,C,D ; (b) their power-weighted silhouettes; (c) silhouette contrasts ϕA−ϕB (top) and ϕC−ϕD (bottom). Strong signals in ϕA−ϕB indicate new 1 -dimensional features in A relative to B , whereas near-zero ϕC−ϕD shows little change between C and D .
Figure 3: (a) CT-scan images of non-infected (top) and infected (bottom) patients; (b) corresponding 0-dimensional persistence diagrams; (c) average silhouettes of non-infected and infected patients given r=0.1 (top) and difference in average silhouettes between non-infected and infected patients.
Figure 4: The true silhouette function and its PI, IPW, AIPW estimates. (a) 0-dimensional silhouettes computed from the SARS-CoV-2 dataset, (b) 0-dimensional silhouettes computed from the GEOM-Drugs dataset, and (c) 1-dimensional silhouettes computed from the GEOM-Drugs dataset.
Appendix figures & tables11 assets
Supplementary material from the paper’s appendix.
Appendix
SARS-CoV-2
GEOM-Drugs
ORBIT
H0
H0
H1
H0
H1
L1 dist.
Std.
L1 dist.
Std.
L1 dist.
Std.
L1 dist.
Std.
L1 dist.
Std.
PI
4.544
2.712
3.215
0.740
7.041
1.283
0.571
0.530
36.063
37.704
IPW
2.936
9.035
1.670
5.756
18.886
8.337
0.073
3.722
18.210
93.978
AIPW
1.246
3.099
3.296
1.470
2.136
1.553
0.121
0.802
6.920
47.127
Appendix
Table 1: L1 distance between the average estimate and the true effect, together with the standard deviation of each estimator over 20 repetitions. Hd denotes homology dimension d , and the scale is in 1e-05 for SARS-CoV-2 and ORBIT and 1e-02 for GEOM-Drugs.
Data (Sample Size)
SARS-CoV-2 (2481)
GEOM-Drugs (30433)
ORBIT (2000)
Runtime
244 seconds
21 second
25 seconds
Appendix
Table 2: The runtime of computing weighted silhouettes for each dataset in seconds.
Figure 5: Visualization of the point-wise mean (dotted line) and the point-wise 1-standard deviation error bands (shaded area) of PI, IPW, and AIPW estimators on the SARS-CoV-2 dataset. The true topological causal effect is shown as a blue line.
Figure 6: (a) Propensity score distribution by subgroup; (b–c) representative molecular graphs.
Figure 7: Visualization of the point-wise mean (dotted line) and the point-wise 1-standard deviation error bands (shaded area) of 0-dimensional (top) and 1-dimensional (bottom) PI, IPW, and AIPW estimators on the GEOM-Drugs dataset. The true topological causal effect is shown as a blue line.
Figure 8: 0-dimensional (left) and 1-dimensional (right) true silhouette functions and its PI, IPW, AIPW estimates for the ORBIT dataset.
Figure 9: Visualization of the pointwise mean (dotted line) and the pointwise 1-standard deviation error bands (shaded area) of 1-dimensional PI, IPW, and AIPW estimators on the ORBIT dataset. The true topological causal effect is shown as a blue line.
Figure 10: True silhouette and its PI, IPW, AIPW estimates for 0-dimensional homological features on the SARS-CoV-2 dataset when one of the nuisance functions is misspecified.
Figure 11: True silhouette and its PI, IPW, AIPW estimates for 0- and 1-dimensional homolgical features on the GEOM-Drugs dataset when one of the nuisance functions is misspecified.
Figure 12: True silhouette and its PI, IPW, AIPW estimates for 0- and 1-dimensional homolgical features on the ORBIT dataset when one of the nuisance functions is misspecified.
SARS-CoV-2 ( H0 )
π is misspecified, μ is correct
μ is misspecified, π is correct
L1 dist.
Std.
L1 dist.
Std.
PI
1.390
3.132
7.985
3.132
IPW
9.674
4.421
2.243
4.328
AIPW
0.786
2.761
1.529
3.395
Appendix
Table 3: L1 distance between the average estimate and the true effect, together with the standard deviation of each estimator over 20 repetitions, when either of the nuisance functions is misspecified. Hd denotes homology dimension d , and the scale is in 1e-05 for SARS-CoV-2 and ORBIT and 1e-02 for GEOM-Drugs.
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.
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.
A fundamental challenge in causal inference with observational data is correct specification of a causal model. When there is model uncertainty, analysts may seek to use estimates from multiple candidate models that rely on distinct, and possibly partially overlapping, sets of identifying assumptions to infer the causal effect, a process known as triangulation. Principled methods for triangulation, however, remain underdeveloped. Here, we develop a framework for causal effect triangulation that combines model testability methods from causal discovery with statistical inference methods from semiparametric theory, while avoiding explicit model selection and post-selection inference problems. We propose a triangulation functional that combines identified functionals from each model with data-driven measures of model validity. We provide a bound on the distance of the functional from the true causal effect along with conditions under which this distance can be taken to zero. Finally, we derive valid statistical inference for this functional. Our framework formalizes robustness under causal pluralism without requiring agreement across models or commitment to a single specification. We demonstrate its performance through simulations and an empirical application.
Rohit Bhattacharya, Ina Ocelli, Ted Westling
Dept. of Computer Science, Williams College · Dept. of Mathematics and Statistics, University of Massachusetts Amherst