Structural Causal Models

The deterministic functional backbone of an SCM. Each endogenous variable X is defined as a function of its direct causal parents PA(X) and an exogenous noise variable U: X = f(PA(X), U). These equations represent the data-generating mechanism, not mere correlations. Unlike standard regression, these functions remain invariant under interventions on other variables, capturing the asymmetry of cause and effect.

A Directed Acyclic Graph (DAG) where nodes represent variables and directed edges represent direct causal relationships. The graph encodes conditional independence assumptions and the factorization of the joint distribution. Key structures include:

• Chains: Mediation paths (X → Y → Z)
• Forks: Common causes (X ← Y → Z)
• Colliders: Common effects (X → Y ← Z) The topology is critical for identifying valid adjustment sets to block confounding.

Unobserved background factors U that account for all unmodeled causes of an endogenous variable. Each U is assumed to be independently distributed, encapsulating the inherent stochasticity or ignorance in the model. The independence of these noise terms is a core assumption that enables the factorization of the joint distribution and the calculation of counterfactuals. Violations occur when unobserved confounders create dependence between noise terms.

The mathematical formalization of an action that forces a variable to a specific value, severing its incoming causal edges. The do(X=x) operator mutilates the graph by removing arrows into X and replacing its structural equation with a constant. This distinguishes P(Y | X=x) (seeing) from P(Y | do(X=x)) (doing). This distinction is the foundation for computing the causal effect of a treatment or policy change.

A three-step process for answering 'what if' questions about a specific past event:

1. Abduction: Infer the posterior distribution of the exogenous noise U given the observed factual evidence.
2. Action: Apply the do-operator to modify the structural equations according to the hypothetical scenario.
3. Prediction: Compute the resulting value of the outcome variable using the updated model and inferred noise. This enables queries like 'Would this patient have survived with a different treatment?'

A classification based on the independence of exogenous variables:

• Markovian: All noise terms U are mutually independent. The causal DAG is a complete representation, and the causal effect is identifiable via the back-door criterion or truncated factorization formula.
• Semi-Markovian: Dependence exists between some noise terms, represented by bidirected dashed arcs indicating unobserved confounding. Identification requires more complex do-calculus derivations.

SCMs enable the formal evaluation of interventions using the do() operator, which simulates setting a variable to a specific value while severing its incoming causal edges. This is distinct from conditioning on an observation.

• Mechanism: Replaces the structural equation for a variable with a constant, creating a mutilated graph.
• Application: In a molecular GNN, do(X=1) on a functional group predicts the effect of physically replacing that group, isolating its causal role from confounding substituents.
• Key Distinction: P(Y|do(X)) is not equivalent to P(Y|X) when confounders exist.

SCMs are the only framework capable of answering counterfactual queries—retrospective 'what if' questions about a specific instance. This requires an SCM to compute the exogenous noise U from the factual observation before altering the structural equations.

• Three-Step Process: 1) Abduction (infer U), 2) Action (apply do()), 3) Prediction (compute new outcome).
• GNN Use Case: For a node misclassified by a GNN, a counterfactual SCM identifies the minimal structural change to its ego-network that would have resulted in the correct classification, providing actionable recourse.
• Example: 'What would this molecule's solubility be if we had used an oxygen atom instead of a nitrogen atom at this position?'

Instead of relying on correlational heuristics, SCMs drive causal graph discovery to learn the true underlying Directed Acyclic Graph (DAG) from observational or interventional data.

• Constraint-Based Methods: Use conditional independence tests (e.g., PC algorithm) to prune edges.
• Score-Based Methods: Optimize a goodness-of-fit score (e.g., BIC) over the space of DAGs.
• GNN Integration: A discovered causal graph serves as the input structure for a GNN, ensuring the model's reasoning is grounded in cause-effect mechanisms rather than spurious correlations, dramatically improving out-of-distribution generalization.

SCMs decompose the total causal effect between a treatment T and outcome Y into distinct path-specific effects through intermediate mediators M.

• Natural Direct Effect (NDE): The effect of T on Y not mediated by M.
• Natural Indirect Effect (NIE): The effect of T on Y exclusively through the pathway T -> M -> Y.
• GNN Explainability: In a social network GNN predicting churn, path analysis can distinguish the direct influence of a user's age on churn from the indirect influence mediated by their peer group's activity, offering granular intervention points.

SCMs formalize the principle of invariant prediction across different environments. A predictor is causal if it relies on features that are direct causes of the outcome, as these relationships remain stable under intervention.

• Core Idea: Learn a representation Φ(X) such that the optimal classifier on Φ(X) is the same across all training environments.
• GNN Application: IRM forces a GNN to ignore unstable, environment-specific correlations (e.g., background color in image graphs) and focus on the invariant causal subgraph (e.g., the object's shape), leading to robust graph classification.

An SCM is a generative model defined by a set of structural equations X_i = f_i(PA_i, U_i), where PA_i are the direct causes (parents) of X_i and U_i are independent exogenous noise variables.

• Mechanism Independence: A core assumption is that causal mechanisms are modular and can be independently modified without affecting others.
• GNN Rationalization: A GNN rationalization module can be trained to explicitly model these structural equations, extracting a subgraph that serves as the PA_i for a target node's label, providing a causal rationale for the prediction.

The process of inferring cause-and-effect relationships from observational data to construct an SCM's directed acyclic graph (DAG). Unlike correlation-based graph learning, discovery algorithms like PC and GES use conditional independence tests to orient edges directionally.

• Distinguishes confounders from mediators
• Essential when the true causal structure is unknown
• Outputs a Markov equivalence class when multiple DAGs fit the data

The minimal structural perturbations to a graph that would alter a GNN's prediction to a different outcome. In the SCM framework, this corresponds to intervening on exogenous variables and computing the downstream effect through structural equations.

• Answers: 'Which edges must I remove to change the classification?'
• Provides actionable recourse for graph-structured decisions
• Evaluated using fidelity and sparsity metrics

A formal mathematical framework developed by Judea Pearl for reasoning about interventions in causal models. The do-operator do(X=x) represents setting a variable to a value by external force, severing its incoming edges.

• Three rules for transforming interventional distributions into observational ones
• Enables identifiability analysis — determining if a causal effect can be estimated from data
• Foundational to generating valid counterfactuals from SCMs

A principle for learning explainable GNNs by compressing the input graph into a minimal subgraph that retains maximal mutual information about the label. This connects to SCMs by isolating the causal substructure from spurious correlations.

• Objective: max I(Y; G_sub) - β I(G; G_sub)
• Discards non-causal shortcut features
• Produces inherently interpretable graph representations

A quantitative evaluation score measuring how accurately an explanation subgraph reflects the true causal reasoning of the GNN. For SCM-based explanations, faithfulness is assessed by the drop in prediction accuracy when the identified causal variables are removed.

• High faithfulness: removing the explanation breaks the prediction
• Low faithfulness: the explanation missed critical causal edges
• Complements fidelity, which measures prediction agreement on the subgraph alone

Node embeddings where each latent dimension corresponds to an independent, interpretable generative factor of the graph data. In SCM terms, this maps to learning the exogenous independent variables that causally determine observed structure.

• Enables fine-grained intervention: modify one factor without affecting others
• Supports counterfactual generation in latent space
• Used in CausalVAE and similar causal representation learning models