# Causal Reasoning with Bipartite Graphical Causal Models

arXiv: https://arxiv.org/abs/2608.19831 · HTML: https://mlsanity.com/paper/2608.19831/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Causal reasoning in LLMs, one of 17 directions in LLM reasoning.
- Builds on 9 of its 27 references that are in the map.
- First posted on arXiv 20 Aug 2026.

## Placed in

- LLM reasoning / [Causal reasoning in LLMs](https://mlsanity.com/lr/causal-reasoning-in-llms/)

## Builds on (most cited first, 9 of 9)

- [Foundations of Structural Causal Models with Cycles and Latent Variables](https://mlsanity.com/paper/1611.06221/) (2016)
- [Causal Networks: Semantics and Expressiveness](https://mlsanity.com/paper/1304.2379/) (2013)
- [Markov Properties for Graphical Models with Cycles and Latent Variables](https://mlsanity.com/paper/1710.08775/) (2017)
- [Directed Cyclic Graphical Representations of Feedback Models](https://mlsanity.com/paper/1302.4982/) (2013)
- [Bipartite Causal Inference with Interference](https://mlsanity.com/paper/1807.08660/) (2018)
- [Conditional independences and causal relations implied by sets of equations](https://mlsanity.com/paper/2007.07183/) (2020)
- [Causality and independence in perfectly adapted dynamical systems](https://mlsanity.com/paper/2101.11885/) (2021)
- [Transitional Conditional Independence](https://mlsanity.com/paper/2104.11547/) (2021)
- [Near-Optimal Experiment Design in Linear non-Gaussian Cyclic Models](https://mlsanity.com/paper/2509.21423/) (2025)

## Abstract (arXiv)

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(fj : Xv = xiv) 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.