Cycles in Causal Learning

TL;DR

This paper explores the unique properties of cyclic causal structures, showing they can exist even with observational independence, and discusses implications for causal learning methods.

Contribution

It provides three theoretical results about distributions with cycles, highlighting challenges for current causal learning approaches and encouraging future research on cyclic causal structures.

Findings

Self-referential distributions in two variables are independent.

Distributions in N variables have zero mutual information.

Cyclic factorization is equivalent to reversed cycle factorization.

Abstract

In the causal learning setting, we wish to learn cause-and-effect relationships between variables such that we can correctly infer the effect of an intervention. While the difference between a cyclic structure and an acyclic structure may be just a single edge, cyclic causal structures have qualitatively different behavior under intervention: cycles cause feedback loops when the downstream effect of an intervention propagates back to the source variable. We present three theoretical observations about probability distributions with self-referential factorizations, i.e. distributions that could be graphically represented with a cycle. First, we prove that self-referential distributions in two variables are, in fact, independent. Second, we prove that self-referential distributions in N variables have zero mutual information. Lastly, we prove that self-referential distributions that factorize in a cycle, also factorize as though the cycle were reversed. These results suggest that cyclic causal dependence may exist even where observational data suggest independence among variables. Methods based on estimating mutual information, or heuristics based on independent causal mechanisms, are likely to fail to learn cyclic casual structures. We encourage future work in causal learning that carefully considers cycles.