A Topological Perspective on Causal Inference
This work addresses foundational limitations in causal inference for researchers in statistics and machine learning, clarifying the inevitability of assumptions, but it is incremental as it builds on existing topological and no-free-lunch concepts.
The paper tackles the problem of causal inference by introducing a topological framework for structural causal models, proving that assumption-free causal inference is only possible in a meager set of models, which shows that necessary inductive assumptions are statistically unverifiable.
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.