Causal Reasoning with Bipartite Graphical Causal Models

Causal {Bayesian} networks (CBNs) and structural causal models (SCMs) are the dominant frameworks for graphical causal reasoning, but they cannot adequately represent all real-world causal systems. In particular, systems at equilibrium—where feedback mechanisms create cyclic causal dependencies—can exhibit causal semantics that are fundamentally incompatible with these frameworks: different interventions that enforce the same variable value may have different effects, rendering the standard “perfect intervention” $\mathrm{do}(X=x)$ ambiguous. We propose _bipartite graphical causal models_ (BGCMs), in which the structure of a system of equations is encoded by a bipartite graph with variable and equation nodes. In this framework, a hard intervention $\mathrm{do}(f_j : X_v=\xi_v)$ specifies which equation is replaced, which variable is targeted, and at what value—resolving the ambiguity of the standard notion. We demonstrate, through a detailed case study of a physical system, that this representation naturally corresponds to distinct real-world interventions. We formulate a {Markov} property in terms of a new graphical separation criterion ($B$-separation) that exploits the functional determinism inherent in the equations, and we extend it to settings with non-random inputs. We show how this gives rise to a do-calculus for reasoning about domain invariances. BGCMs strictly generalize CBNs and SCMs while retaining the ability to perform graphical causal reasoning.