A Topological Perspective on Causal Inference

TL;DR

This paper introduces a topological framework for causal inference, revealing that assumption-free causal inference is only possible in a negligible set of models and highlighting the necessity of unverifiable assumptions.

Contribution

It develops a novel topological perspective on causal models, demonstrating the limitations of assumption-free inference and accommodating models with infinitely many variables.

Findings

Assumption-free causal inference is only possible in a meager set of models.

Statistically verifiable hypotheses correspond to open sets in the weak topology.

Substantial assumptions are necessary for valid causal inferences.

Abstract

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.