Structural Causal Model (SCM)

The deterministic or probabilistic functions that define how each endogenous variable is generated from its direct causes. An equation X = f(PA, U) maps parents and exogenous noise to an effect.

• Form: X := f_X(PA_X, U_X)
• Asymmetry: The := assignment operator enforces the direction of causality, distinguishing it from standard algebraic equality.
• Mechanism: Represents the invariant physical or behavioral process generating the data, which remains stable under intervention.

Latent background factors that capture all unmodeled or random influences on the endogenous variables. They represent the inherent stochasticity of the real world.

• Independence: Often assumed to be jointly independent, though this can be relaxed.
• Role: They are the source of uncertainty that distinguishes a statistical model from a purely deterministic simulation.
• Counterfactual Key: The specific values of U for an individual unit are the key to computing counterfactuals at the unit level.

A Directed Acyclic Graph where nodes represent variables and directed edges represent direct causal relationships. It encodes the qualitative causal assumptions of the model.

• Parents: The direct causes of a node X are denoted as PA_X.
• d-Separation: A graphical criterion for reading off conditional independencies implied by the model.
• Acyclicity: The absence of directed cycles ensures no variable can cause itself, enforcing logical consistency.

The mathematical operator do(X=x) that represents an external intervention forcing a variable to a specific value, surgically removing its incoming edges in the graph.

• Truncated Factorization: The joint distribution under an intervention is computed by deleting the factor for the intervened variable: P(v | do(x)) = ∏_{V_i ≠ X} P(v_i | pa_i) when X=x.
• Distinction: P(Y|X=x) (seeing) is fundamentally different from P(Y|do(X=x)) (doing).
• Purpose: This is the core mechanism for answering policy and action-based queries.

The three-step process for answering 'what if' questions about a specific observed unit, combining the SCM with observed evidence.

• Step 1 (Abduction): Infer the posterior distribution of the exogenous noise U given the observed factual evidence.
• Step 2 (Action): Modify the SCM by applying the do-operator to set a variable to its counterfactual value.
• Step 3 (Prediction): Compute the resulting value of the target variable using the modified model and the inferred U.

A complete set of three inference rules developed by Judea Pearl that allows an interventional distribution P(y|do(x)) to be transformed into an estimable expression from observational data alone, whenever possible.

• Rule 1 (Insertion/Deletion of Observations): Allows adding or removing a conditioning variable if the target is d-separated from it after the intervention.
• Rule 2 (Action/Observation Exchange): Allows replacing an intervention with a conditioning observation if they have the same effect.
• Rule 3 (Insertion/Deletion of Actions): Allows adding or removing an intervention on a variable when it has no causal effect on the outcome.