Cyclic functional causal models beyond unique solvability with a graph separation theorem
Carla Ferradini, Victor Gitton, V. Vilasini
TL;DR
The paper tackles the challenge of cyclic functional causal models by introducing a complete framework for finite-cardinality fCMs that ensures a unique probability distribution even when models are not uniquely solvable. It constructs a mapping from cyclic to acyclic models using classical post-selected teleportation graphs (G_tp), establishing a probability rule that is independent of the particular teleportation construction and yields a post-selected distribution consistent with acyclic causal structure. A central contribution is p-separation, a graph-separation property that is sound and complete for all consistent cyclic fCMs and reduces to d-separation in DAGs, thereby generalizing causal-independence reasoning beyond acyclic graphs. The work links classical cyclic causality with quantum-inspired methods, introduces averagely uniquely solvable models as the largest class with Markov factorization, and opens avenues for causal discovery, compatibility problems, and cross-fertilization with quantum causality and higher-order processes.
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.