A Topological Perspective on Causal Inference

As an approach to the workshop theme of causal assumptions, we offer a topological learning-theoretic perspective on causal inference by introducing a series of topologies defined on general spaces of structural causal models (SCMs). To illustrate the power 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 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. We furthermore suggest that the framework may be helpful for the positive project of exploring and assessing alternative causal-inductive assumptions.