On the Relationship between Bayesian Networks and Probabilistic Structural Causal Models

Peter J. F. Lucas; Eleonora Zullo; Fabio Stella

cs.AIMar 28, 2026

On the Relationship between Bayesian Networks and Probabilistic Structural Causal Models

Authors: Peter J. F. Lucas, Eleonora Zullo, Fabio Stella

Organizations: ICIS, Radboud University, Nijmegen, the Netherlands · Dipartimento di Informatica, Sistemistica e comunicazion University of Milano-Bicocca, Milan, Italy

Abstract

In this paper, the relationship between probabilistic graphical models, in particular Bayesian networks, and causal diagrams, also called structural causal models, is studied. Structural causal models are deterministic models, based on structural equations or functions, that can be provided with uncertainty by adding independent, unobserved random variables to the models, equipped with probability distributions. One question that arises is whether a Bayesian network that has obtained from expert knowledge or learnt from data can be mapped to a probabilistic structural causal model, and whether or not this has consequences for the network structure and probability distribution. We show that linear algebra and linear programming offer key methods for the transformation, and examine properties for the existence and uniqueness of solutions based on dimensions of the probabilistic structural model. Finally, we examine in what way the semantics of the models is affected by this transformation. Keywords: Causality, probabilistic structural causal models, Bayesian networks, linear algebra, experimental software.

Figures & tables

Panel of Figure 1 — Figure 1: Two different Bayesian network representations of the same distribution; the top one a regular Bayesian network, whereas the one at the bottom is a Bayesian network representation of a probabilistic structured causal model; (a): priors; (b) posteriors.

Panel of Figure 2 — Figure 2: Two different Bayesian network representations with two conditional distributions and more values for $R$ , before and after entering an intervention $T$ ; (a): priors; (b): posteriors.

Panel of Figure 3 — Figure 3: Two different Bayesian network representations of the same distribution with one extra arc from $T$ to $R$ with (a): priors; (b) posteriors.

Structural Causal Models

Structural causal models describe how variables are generated by causal mechanisms, enabling reasoning about interventions and counterfactuals.

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Causal Atlases from Entropic Inference: Bayesian Networks beyond Optimal DAGs

Hazhir Aliahmadi, Irina Babayan, Greg van Anders

Department of Physics, Engineering Physics and Astronomy, Queen’s University, Kingston ON, Canada

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Data-driven causal relationship identification is pertinent to advancing understanding of complex systems both within and beyond science. Bayesian networks offer a probabilistic method for modelling generic causal relationships via directed acyclic graphs (DAGs). However, typical techniques for constructing Bayesian networks rely on optimization, which can be ill-suited for learning causal relationships because the underlying data may admit multiple chains of causation. More data-faithful representations of causal relationships would provide frameworks for constructing multiple causal maps that are consistent with the variability that is inherent in underlying data. Here, we show that entropy-based inference generates atlases of plausible causal relationships that are consistent with underlying data. On simulated noisy data of 2- and 20-node linear structural equation models, we sample a maximum-entropy ensemble of graphs that allow us to quantify the inherent structural ambiguity in underlying causal relationships. Our method shows that "optimized" DAGs can contain causal artifacts are not consistent across equivalently accurate topologies.

Abstract

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