# A Topological Perspective on Causal Inference

arXiv: https://arxiv.org/abs/2107.08558 · HTML: https://mlsanity.com/paper/2107.08558/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 51 references that are in the map.
- First posted on arXiv 18 Jul 2021.

## Builds on (most cited first, 8 of 8)

- [Foundations of Structural Causal Models with Cycles and Latent Variables](https://mlsanity.com/paper/1611.06221/) (2016)
- [Probabilities of Causation: Bounds and Identification](https://mlsanity.com/paper/1301.3898/) (2013)
- [Strong Completeness and Faithfulness in Bayesian Networks](https://mlsanity.com/paper/1302.4973/) (2013)
- [Probabilistic Reasoning across the Causal Hierarchy](https://mlsanity.com/paper/2001.02889/) (2020)
- [On Learning Causal Structures from Non-Experimental Data without Any Faithfulness Assumption](https://mlsanity.com/paper/1802.07051/) (2018)
- [On Open-Universe Causal Reasoning](https://mlsanity.com/paper/1907.02170/) (2019)
- [Absolutely No Free Lunches!](https://mlsanity.com/paper/2005.04791/) (2020)
- [The Topology of Statistical Verifiability](https://mlsanity.com/paper/1707.09378/) (2017)

## Built on by (most cited first, 10 of 10)

- [Robust agents learn causal world models](https://mlsanity.com/paper/2402.10877/) (2024)
- [Comparing Causal Frameworks: Potential Outcomes, Structural Models, Graphs, and Abstractions](https://mlsanity.com/paper/2306.14351/) (2023)
- [Are Bayesian networks typically faithful?](https://mlsanity.com/paper/2410.16004/) (2024)
- [Causality from Bottom to Top: A Survey](https://mlsanity.com/paper/2403.11219/) (2024)
- [On Probabilistic and Causal Reasoning with Summation Operators](https://mlsanity.com/paper/2405.03069/) (2024)
- [Beyond Means: Topological Causal Effects under Persistent-Homology Ignorability](https://mlsanity.com/paper/2603.14169/) (2026)
- [A Mathematical Framework for Topological Causal Data Analysis](https://mlsanity.com/paper/2607.28161/) (2026)
- [Causal Feature Learning in the Social Sciences](https://mlsanity.com/paper/2503.12784/) (2025)
- [C2MIL: Synchronizing Semantic and Topological Causalities in Multiple Instance Learning for Robust and Interpretable Survival Analysis](https://mlsanity.com/paper/2509.20152/) (2025)
- [Topological Ignorability for Structural Causal Effects Beyond Means](https://mlsanity.com/paper/2606.01184/) (2026)

## Abstract (arXiv)

This paper presents a topological learning-theoretic perspective on causal inference by introducing a series of topologies defined on general spaces of structural causal models (SCMs). As an illustration of the framework we prove a topological causal hierarchy theorem, showing that substantive assumption-free causal inference is possible only in a meager set of SCMs. Thanks to a known correspondence between open sets in the weak topology and statistically verifiable hypotheses, our results show that inductive assumptions sufficient to license valid causal inferences are statistically unverifiable in principle. Similar to no-free-lunch theorems for statistical inference, the present results clarify the inevitability of substantial assumptions for causal inference. An additional benefit of our topological approach is that it easily accommodates SCMs with infinitely many variables. We finally suggest that the framework may be helpful for the positive project of exploring and assessing alternative causal-inductive assumptions.