Causal Reasoning with Bipartite Graphical Causal Models

• URL: http://arxiv.org/abs/2608.19831v1
• Date: Thu, 20 Aug 2026 09:33:20 GMT
• Title: Causal Reasoning with Bipartite Graphical Causal Models
• Abstract summary: Causal Bayesian networks (CBNs) and structural causal models (SCMs) are the dominant frameworks for graphical causal reasoning.<n>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.
• Score: 0.7106986689736826
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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'' 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 do($f_j : X_v = ξ_v$) 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.

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