On the Relationship between Bayesian Networks and Probabilistic Structural Causal Models

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.

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.

An artificial intelligence must have a model of its environment that is causal, supporting reasoning about interventions and counterfactuals, and also combinatorial, supporting generalization to unseen combinations of objects. In this work, we formally study when and how such a model can be learned. We develop relational structural causal models, extending structural causal models (Pearl 2009) to settings where objects and their relations vary. First, we show how answers to not only causal but also observational queries about unseen combinations of objects can not be identified without further assumptions. To enable such identification--including in the presence of unobserved confounding--we define relational causal graphs and derive symbolic identification criteria. Finally, we propose relational neural causal models, a provably correct approach that outperforms non-relational baselines on simulated traffic scenes with varying cars, signals, and pedestrians.