cs.AIAug 20, 2026

Causal Reasoning with Bipartite Graphical Causal Models

Authors: Joris M. Mooij

Organizations: Korteweg-De Vries Institute for Mathematics, University of Amsterdam, Amsterdam, the Netherlands

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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=xX = 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(fj:Xv=ξvf_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.

Figures & tables

Figure 1: The bathtub system (rendered by Google Gemini).
Figure 2: Three different interventions on the bathtub that all set the water depth to a fixed value (Example 20 ).
do(fj:Xv=ξv)\mathrm{do}(f_{j}:X_{v}=\xi_{v})f1f_{1}f2f_{2}f3f_{3}
XO=ξOX_{O}=\xi_{O}✓\checkmark∅\emptyset∅\emptyset
XP=ξPX_{P}=\xi_{P}✓\checkmark✓\checkmark∅\emptyset
XD=ξDX_{D}=\xi_{D}✓\checkmark✓\checkmark✓\checkmark
Table 1: Feasibility of hard interventions do(fj:Xv=ξv)\mathrm{do}(f_{j}:X_{v}=\xi_{v}) for the bathtub. Checkmarks indicate uniquely solvable systems; ∅\emptyset indicates the system is not uniquely solvable.
 XOX_{O}XPX_{P}XDX_{D}
observationalXIX_{I}XI2XK2\frac{X_{I}^{2}}{X_{K}^{2}}XI2XK2Xg\frac{X_{I}^{2}}{X_{K}^{2}X_{g}}
do(XI=ξI)\mathrm{do}(X_{I}=\xi_{I})ξI\xi_{I}ξI2XK2\frac{\xi_{I}^{2}}{X_{K}^{2}}ξI2XK2Xg\frac{\xi_{I}^{2}}{X_{K}^{2}X_{g}}
do(XK=ξK)\mathrm{do}(X_{K}=\xi_{K})XIX_{I}XI2ξK2\frac{X_{I}^{2}}{\xi_{K}^{2}}XI2ξK2Xg\frac{X_{I}^{2}}{\xi_{K}^{2}X_{g}}
do(Xg=ξg)\mathrm{do}(X_{g}=\xi_{g})XIX_{I}XI2XK2\frac{X_{I}^{2}}{X_{K}^{2}}XI2XK2ξg\frac{X_{I}^{2}}{X_{K}^{2}\xi_{g}}
do(f1:XO=ξO)\mathrm{do}(f_{1}:X_{O}=\xi_{O})ξO\xi_{O}ξO2XK2\frac{\xi_{O}^{2}}{X_{K}^{2}}ξO2XK2Xg\frac{\xi_{O}^{2}}{X_{K}^{2}X_{g}}
do(f1:XP=ξP)\mathrm{do}(f_{1}:X_{P}=\xi_{P})ξP XK\sqrt{\xi_{P}}\,X_{K}ξP\xi_{P}ξPXg\frac{\xi_{P}}{X_{g}}
Table 2: Solution functions for all hard interventions on the bathtub model.
Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 3: Partially oriented bipartite graphs for the bathtub model under all hard interventions (cf. Table 1 ). The observational graph is shown top-right for reference. Intervened equation nodes are shown in red; edges whose orientation changes relative to the observational case are drawn in red. Note how replacing different equations can lead to fundamentally different causal orderings: e.g., do(f1:XD=ξD)\mathrm{do}(f_{1}:X_{D}=\xi_{D}) reverses the causal flow entirely. For three hard interventions, the intervened bipartite graph cannot be oriented (no perfect matching exists), so the undirected intervened bipartite graph is displayed instead.
Figure 4: Partially oriented bipartite graphs of the joint models for the four domain invariance examples (Section 6 ). In each graph, the exogenous domain indicator RR is connected to the equation(s) that differ between domains. The BB -separation statement below each graph summarizes the graphical criterion from which the corresponding domain invariance is derived via the Markov property. In Example 23 , all endogenous nodes and equation nodes form a single cluster, so no non-trivial BB -separation involving RR exists.
Figure 5: The supply–demand system and its acyclification. (a) The partial orientation \mkern 2.0mu\textstyle\mathrel{\mathchoice{{}\hbox{\displaystyle{\montraitd}}}{{}\hbox{\textstyle{\montraitd}}}{{}\hbox{\scriptstyle{\montraitd}}}{{}\hbox{\scriptscriptstyle{\montraitd}}}}\mkern-1.5mu\leaders{\hbox{\textstyle\mkern 0.0mu\mathrel{\mathchoice{{}\hbox{\montraita\displaystyle{\montraita}}}{{}\hbox{\montraita\textstyle{\montraita}}}{{}\hbox{\montraita\scriptstyle{\montraita}}}{{}\hbox{\montraita\scriptscriptstyle{\montraita}}}}\mkern 0.0mu}}{\hfill}\mkern-1.5mu\fldr \hfilG\hfil\hfil\textstyle G\hfil : the three endogenous variables and three equations form a single feedback cluster (dashed box) with the exogenous shifts XUS,XUDX_{U_{S}},X_{U_{D}} as its parents; all intra-cluster edges are double-undirected. (b) The acyclification \mathchoice{\vbox{\halign{#\cr\kern-0.7pt\cr\mkern 2.0mu\displaystyle\mathrel{\mathchoice{{}\hbox{\montraitd\displaystyle{\montraitd}}}{{}\hbox{\montraitd\textstyle{\montraitd}}}{{}\hbox{\montraitd\scriptstyle{\montraitd}}}{{}\hbox{\montraitd\scriptscriptstyle{\montraitd}}}}\mkern-1.5mu\leaders{\hbox{\displaystyle\mkern 0.0mu\mathrel{\mathchoice{{}\hbox{\displaystyle{\montraita}}}{{}\hbox{\textstyle{\montraita}}}{{}\hbox{\scriptstyle{\montraita}}}{{}\hbox{\scriptscriptstyle{\montraita}}}}\mkern 0.0mu}}{\hfill}\mkern-1.5mu\fldr\crcr\cr\crcr\kern-0.3pt\cr\hfil\displaystyle G\hfil\crcr}}}{\vbox{\halign{#\cr\kern-0.7pt\cr\mkern 2.0mu\textstyle\mathrel{\mathchoice{{}\hbox{\montraitd\displaystyle{\montraitd}}}{{}\hbox{\montraitd\textstyle{\montraitd}}}{{}\hbox{\montraitd\scriptstyle{\montraitd}}}{{}\hbox{\montraitd\scriptscriptstyle{\montraitd}}}}\mkern-1.5mu\leaders{\hbox{\textstyle\mkern 0.0mu\mathrel{\mathchoice{{}\hbox{\displaystyle{\montraita}}}{{}\hbox{\textstyle{\montraita}}}{{}\hbox{\scriptstyle{\montraita}}}{{}\hbox{\scriptscriptstyle{\montraita}}}}\mkern 0.0mu}}{\hfill}\mkern-1.5mu\fldr\crcr\cr\crcr\kern-0.3pt\cr\hfil\textstyle G\hfil\crcr}}}{\vbox{\halign{#\cr\kern-0.7pt\cr\mkern 2.0mu\scriptstyle\mathrel{\mathchoice{{}\hbox{\montraitd\displaystyle{\montraitd}}}{{}\hbox{\montraitd\textstyle{\montraitd}}}{{}\hbox{\montraitd\scriptstyle{\montraitd}}}{{}\hbox{\montraitd\scriptscriptstyle{\montraitd}}}}\mkern-1.5mu\leaders{\hbox{\scriptstyle\mkern 0.0mu\mathrel{\mathchoice{{}\hbox{\displaystyle{\montraita}}}{{}\hbox{\textstyle{\montraita}}}{{}\hbox{\scriptstyle{\montraita}}}{{}\hbox{\scriptscriptstyle{\montraita}}}}\mkern 0.0mu}}{\hfill}\mkern-1.5mu\fldr\crcr\cr\crcr\kern-0.3pt\cr\hfil\scriptstyle G\hfil\crcr}}}{\vbox{\halign{#\cr\kern-0.7pt\cr\mkern 2.0mu\scriptscriptstyle\mathrel{\mathchoice{{}\hbox{\montraitd\displaystyle{\montraitd}}}{{}\hbox{\montraitd\textstyle{\montraitd}}}{{}\hbox{\montraitd\scriptstyle{\montraitd}}}{{}\hbox{\montraitd\scriptscriptstyle{\montraitd}}}}\mkern-1.5mu\leaders{\hbox{\scriptscriptstyle\mkern 0.0mu\mathrel{\mathchoice{{}\hbox{\displaystyle{\montraita}}}{{}\hbox{\textstyle{\montraita}}}{{}\hbox{\scriptstyle{\montraita}}}{{}\hbox{\scriptscriptstyle{\montraita}}}}\mkern 0.0mu}}{\hfill}\mkern-1.5mu\fldr\crcr\cr\crcr\kern-0.3pt\cr\hfil\scriptscriptstyle G\hfil\crcr}}}^{\mathrm{acy}} , the DAG on the variable nodes with an edge v→v′v\to v^{\prime} whenever v\in\mathrm{pa}_{\mathchoice{\vbox{\halign{#\cr\kern-0.7pt\cr\mkern 2.0mu\displaystyle\mathrel{\mathchoice{{}\hbox{\montraitd\displaystyle{\montraitd}}}{{}\hbox{\montraitd\textstyle{\montraitd}}}{{}\hbox{\montraitd\scriptstyle{\montraitd}}}{{}\hbox{\montraitd\scriptscriptstyle{\montraitd}}}}\mkern-1.5mu\leaders{\hbox{\displaystyle\mkern 0.0mu\mathrel{\mathchoice{{}\hbox{\displaystyle{\montraita}}}{{}\hbox{\textstyle{\montraita}}}{{}\hbox{\scriptstyle{\montraita}}}{{}\hbox{\scriptscriptstyle{\montraita}}}}\mkern 0.0mu}}{\hfill}\mkern-1.5mu\fldr\crcr\cr\crcr\kern-0.3pt\cr\hfil\displaystyle G\hfil\crcr}}}{\vbox{\halign{#\cr\kern-0.7pt\cr\mkern 2.0mu\textstyle\mathrel{\mathchoice{{}\hbox{\montraitd\displaystyle{\montraitd}}}{{}\hbox{\montraitd\textstyle{\montraitd}}}{{}\hbox{\montraitd\scriptstyle{\montraitd}}}{{}\hbox{\montraitd\scriptscriptstyle{\montraitd}}}}\mkern-1.5mu\leaders{\hbox{\textstyle\mkern 0.0mu\mathrel{\mathchoice{{}\hbox{\displaystyle{\montraita}}}{{}\hbox{\textstyle{\montraita}}}{{}\hbox{\scriptstyle{\montraita}}}{{}\hbox{\scriptscriptstyle{\montraita}}}}\mkern 0.0mu}}{\hfill}\mkern-1.5mu\fldr\crcr\cr\crcr\kern-0.3pt\cr\hfil\textstyle G\hfil\crcr}}}{\vbox{\halign{#\cr\kern-0.7pt\cr\mkern 2.0mu\scriptstyle\mathrel{\mathchoice{{}\hbox{\montraitd\displaystyle{\montraitd}}}{{}\hbox{\montraitd\textstyle{\montraitd}}}{{}\hbox{\montraitd\scriptstyle{\montraitd}}}{{}\hbox{\montraitd\scriptscriptstyle{\montraitd}}}}\mkern-1.5mu\leaders{\hbox{\scriptstyle\mkern 0.0mu\mathrel{\mathchoice{{}\hbox{\displaystyle{\montraita}}}{{}\hbox{\textstyle{\montraita}}}{{}\hbox{\scriptstyle{\montraita}}}{{}\hbox{\scriptscriptstyle{\montraita}}}}\mkern 0.0mu}}{\hfill}\mkern-1.5mu\fldr\crcr\cr\crcr\kern-0.3pt\cr\hfil\scriptstyle G\hfil\crcr}}}{\vbox{\halign{#\cr\kern-0.7pt\cr\mkern 2.0mu\scriptscriptstyle\mathrel{\mathchoice{{}\hbox{\montraitd\displaystyle{\montraitd}}}{{}\hbox{\montraitd\textstyle{\montraitd}}}{{}\hbox{\montraitd\scriptstyle{\montraitd}}}{{}\hbox{\montraitd\scriptscriptstyle{\montraitd}}}}\mkern-1.5mu\leaders{\hbox{\scriptscriptstyle\mkern 0.0mu\mathrel{\mathchoice{{}\hbox{\displaystyle{\montraita}}}{{}\hbox{\textstyle{\montraita}}}{{}\hbox{\scriptstyle{\montraita}}}{{}\hbox{\scriptscriptstyle{\montraita}}}}\mkern 0.0mu}}{\hfill}\mkern-1.5mu\fldr\crcr\cr\crcr\kern-0.3pt\cr\hfil\scriptscriptstyle G\hfil\crcr}}}}([v^{\prime}]) ; the two shifts become common parents of all three endogenous variables, which are mutually nonadjacent. BB -separation in \mkern 2.0mu\textstyle\mathrel{\mathchoice{{}\hbox{\displaystyle{\montraitd}}}{{}\hbox{\textstyle{\montraitd}}}{{}\hbox{\scriptstyle{\montraitd}}}{{}\hbox{\scriptscriptstyle{\montraitd}}}}\mkern-1.5mu\leaders{\hbox{\textstyle\mkern 0.0mu\mathrel{\mathchoice{{}\hbox{\montraita\displaystyle{\montraita}}}{{}\hbox{\montraita\textstyle{\montraita}}}{{}\hbox{\montraita\scriptstyle{\montraita}}}{{}\hbox{\montraita\scriptscriptstyle{\montraita}}}}\mkern 0.0mu}}{\hfill}\mkern-1.5mu\fldr \hfilG\hfil\hfil\textstyle G\hfil coincides with DD -separation in \mathchoice{\vbox{\halign{#\cr\kern-0.7pt\cr\mkern 2.0mu\displaystyle\mathrel{\mathchoice{{}\hbox{\montraitd\displaystyle{\montraitd}}}{{}\hbox{\montraitd\textstyle{\montraitd}}}{{}\hbox{\montraitd\scriptstyle{\montraitd}}}{{}\hbox{\montraitd\scriptscriptstyle{\montraitd}}}}\mkern-1.5mu\leaders{\hbox{\displaystyle\mkern 0.0mu\mathrel{\mathchoice{{}\hbox{\displaystyle{\montraita}}}{{}\hbox{\textstyle{\montraita}}}{{}\hbox{\scriptstyle{\montraita}}}{{}\hbox{\scriptscriptstyle{\montraita}}}}\mkern 0.0mu}}{\hfill}\mkern-1.5mu\fldr\crcr\cr\crcr\kern-0.3pt\cr\hfil\displaystyle G\hfil\crcr}}}{\vbox{\halign{#\cr\kern-0.7pt\cr\mkern 2.0mu\textstyle\mathrel{\mathchoice{{}\hbox{\montraitd\displaystyle{\montraitd}}}{{}\hbox{\montraitd\textstyle{\montraitd}}}{{}\hbox{\montraitd\scriptstyle{\montraitd}}}{{}\hbox{\montraitd\scriptscriptstyle{\montraitd}}}}\mkern-1.5mu\leaders{\hbox{\textstyle\mkern 0.0mu\mathrel{\mathchoice{{}\hbox{\displaystyle{\montraita}}}{{}\hbox{\textstyle{\montraita}}}{{}\hbox{\scriptstyle{\montraita}}}{{}\hbox{\scriptscriptstyle{\montraita}}}}\mkern 0.0mu}}{\hfill}\mkern-1.5mu\fldr\crcr\cr\crcr\kern-0.3pt\cr\hfil\textstyle G\hfil\crcr}}}{\vbox{\halign{#\cr\kern-0.7pt\cr\mkern 2.0mu\scriptstyle\mathrel{\mathchoice{{}\hbox{\montraitd\displaystyle{\montraitd}}}{{}\hbox{\montraitd\textstyle{\montraitd}}}{{}\hbox{\montraitd\scriptstyle{\montraitd}}}{{}\hbox{\montraitd\scriptscriptstyle{\montraitd}}}}\mkern-1.5mu\leaders{\hbox{\scriptstyle\mkern 0.0mu\mathrel{\mathchoice{{}\hbox{\displaystyle{\montraita}}}{{}\hbox{\textstyle{\montraita}}}{{}\hbox{\scriptstyle{\montraita}}}{{}\hbox{\scriptscriptstyle{\montraita}}}}\mkern 0.0mu}}{\hfill}\mkern-1.5mu\fldr\crcr\cr\crcr\kern-0.3pt\cr\hfil\scriptstyle G\hfil\crcr}}}{\vbox{\halign{#\cr\kern-0.7pt\cr\mkern 2.0mu\scriptscriptstyle\mathrel{\mathchoice{{}\hbox{\montraitd\displaystyle{\montraitd}}}{{}\hbox{\montraitd\textstyle{\montraitd}}}{{}\hbox{\montraitd\scriptstyle{\montraitd}}}{{}\hbox{\montraitd\scriptscriptstyle{\montraitd}}}}\mkern-1.5mu\leaders{\hbox{\scriptscriptstyle\mkern 0.0mu\mathrel{\mathchoice{{}\hbox{\displaystyle{\montraita}}}{{}\hbox{\textstyle{\montraita}}}{{}\hbox{\scriptstyle{\montraita}}}{{}\hbox{\scriptscriptstyle{\montraita}}}}\mkern 0.0mu}}{\hfill}\mkern-1.5mu\fldr\crcr\cr\crcr\kern-0.3pt\cr\hfil\scriptscriptstyle G\hfil\crcr}}}^{\mathrm{acy}} (Lemma 42 ), so the two encode the same conditional independences; but only \mkern 2.0mu\textstyle\mathrel{\mathchoice{{}\hbox{\displaystyle{\montraitd}}}{{}\hbox{\textstyle{\montraitd}}}{{}\hbox{\scriptstyle{\montraitd}}}{{}\hbox{\scriptscriptstyle{\montraitd}}}}\mkern-1.5mu\leaders{\hbox{\textstyle\mkern 0.0mu\mathrel{\mathchoice{{}\hbox{\montraita\displaystyle{\montraita}}}{{}\hbox{\montraita\textstyle{\montraita}}}{{}\hbox{\montraita\scriptstyle{\montraita}}}{{}\hbox{\montraita\scriptscriptstyle{\montraita}}}}\mkern 0.0mu}}{\hfill}\mkern-1.5mu\fldr \hfilG\hfil\hfil\textstyle G\hfil , which retains the equation nodes, can model interventions.
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