Cyclic functional causal models beyond unique solvability with a graph separation theorem

TL;DR

This paper extends causal modeling to cyclic graphs with finite variables, introducing a new probability rule and graph-separation criterion that generalize existing theories for acyclic models, and connects quantum protocols to causal inference.

Contribution

It develops a comprehensive framework for cyclic functional causal models, including a new probability rule and p-separation criterion that generalize d-separation for DAGs.

Findings

Introduces a probability rule for cyclic fCMs that assigns unique distributions.

Defines p-separation, a new graph-separation property complete for cyclic models.

Identifies averagely uniquely solvable fCMs as the largest class with Markov factorization.

Abstract

Functional causal models (fCMs) specify functional dependencies between random variables associated to the vertices of a graph. In directed acyclic graphs (DAGs), fCMs are well-understood: a unique probability distribution on the random variables can be easily specified, and a crucial graph-separation result called the d-separation theorem allows one to characterize conditional independences between the variables. However, fCMs on cyclic graphs pose challenges due to the absence of a systematic way to assign a unique probability distribution to the fCM's variables, the failure of the d-separation theorem, and lack of a generalization of this theorem that is applicable to all consistent cyclic fCMs. In this work, we develop a causal modeling framework applicable to all cyclic fCMs involving finite-cardinality variables, except inconsistent ones admitting no solutions. Our probability rule assigns a unique distribution even to non-uniquely solvable cyclic fCMs and reduces to the known rule for uniquely solvable fCMs. We identify a class of fCMs, called averagely uniquely solvable, that we show to be the largest class where the probabilities admit a Markov factorization. Furthermore, we introduce a new graph-separation property, p-separation, and prove this to be sound and complete for all consistent finite-cardinality cyclic fCMs while recovering the d-separation theorem for DAGs. These results are obtained by considering classical post-selected teleportation protocols inspired by analogous protocols in quantum information theory. We discuss further avenues for exploration, linking in particular problems in cyclic fCMs and in quantum causality.