Causal Inference

Origin. Pearl (1995, 2000). From association to causation. Structural causal models. Do-calculus. Counterfactual reasoning. Foundation for causal AI.

Models. DAG of causal relationships. Interventions vs observations. Do-operator: set variable. Identify causal effects from observational data.

Formalism.

Structural Causal Model (SCM): M = (U, V, F, P(U)) U: exogenous (background) variables V: endogenous (model) variables F: structural equations V = f(Pa(V), U_V) P(U): distribution over exogenous

Causal graph: DAG with V as nodes. Edge X → Y: X direct cause of Y. Pa(Y): parents of Y.

Do-operator: P(Y | do(X = x)): probability of Y when X set to x. Intervention, not observation. Mutilated graph: remove edges into X.

Adjustment formula: If Z satisfies back-door criterion: P(Y | do(X)) = Σ_z P(Y | X, Z=z) P(Z=z)

Back-door criterion: Z blocks all back-door paths X ← ... → Y. Z contains no descendants of X.

Front-door criterion: Alternative identification via mediators.

Do-calculus rules: Rule 1: Insertion/deletion of observations. Rule 2: Action/observation exchange. Rule 3: Insertion/deletion of actions. Complete for identifiability.

Counterfactuals: Y_x: value of Y had X been x. P(Y_x = y | X = x', Y = y') Requires full SCM.

Symbols.

Symbol	Unicode	Name	Meaning
do(X=x)	—	Do	Intervention
→	U+2192	Causes	Direct cause
Y_x	—	Counterfactual	Y under X=x
Pa(X)	—	Parents	Direct causes

Metatheory. Do-calculus complete. Identifiability decidable. Counterfactual consistency. Structural invariance.

Applies to. Epidemiology. Economics. AI fairness. Policy evaluation. Scientific discovery.

Limitations. Graph knowledge required. Unobserved confounding. Faithfulness assumptions. Counterfactuals controversial.