Information-theoretic signatures of causality in Bayesian networks and hypergraphs

Analyzing causality in multivariate systems involves establishing how information is generated, distributed and combined, and thus requires tools that capture interactions beyond pairwise relations. Higher-order information theory provides such tools. In particular, Partial Information Decomposition (PID) allows the decomposition of the information that a set of sources provides about a target into redundant, unique, and synergistic components. Yet the mathematical connection between such higher-order information-theoretic measures and causal structure remains undeveloped. Here we establish the first theoretical correspondence between PID components and causal structure in both Bayesian networks and hypergraphs. We first show that in Bayesian networks unique information precisely characterizes direct causal neighbors, while synergy identifies collider relationships. This establishes a localist causal discovery paradigm in which the structure surrounding each variable can be recovered from its immediate informational footprint, eliminating the need for global search over graph space. Extending these results to higher-order systems, we prove that PID signatures in Bayesian hypergraphs differentiate parents, children, co-heads, and co-tails, revealing a higher-order collider effect unique to multi-tail hyperedges. We also present procedures by which our results can be used to characterize systematically the causal structure of Bayesian networks and hypergraphs. Our results position PID as a rigorous, model-agnostic foundation for inferring both pairwise and higher-order causal structure, and introduce a fundamentally local information-theoretic viewpoint on causal discovery.