Interventionist Causation

Origin. Judea Pearl's Causality (2000, 2009) developed the interventionist/structural approach, building on his work on Bayesian networks and the do-calculus. James Woodward's Making Things Happen (2003) provided philosophical foundations. Extends work by Sewall Wright (path analysis), Haavelmo (econometric interventions), and Rubin (potential outcomes).

Models. Causation as invariance under intervention. X causes Y if intervening to change X would change Y, holding the causal mechanism fixed. Distinguishes "seeing" (observation) from "doing" (intervention). Core insight: P(Y|X=x) ≠ P(Y|do(X=x)). Observing that the barometer is low predicts rain; setting the barometer low doesn't cause rain.

Formalism.

Structural Causal Models (SCM): A tuple M = (U, V, F) where:

U: exogenous (background) variables
V: endogenous variables
F: structural equations V_i = f_i(PA_i, U_i) expressing how each variable is determined by its parents and noise

Causal graphs: Directed acyclic graph (DAG) where edges represent direct causal relationships.

Intervention do(X=x): Replace the structural equation for X with X := x, removing all incoming arrows to X. Compute the resulting distribution over other variables. This differs from conditioning: intervention breaks the causal mechanism, conditioning doesn't.

do-calculus: Three rules for transforming interventional distributions:

Rule 1: Insertion/deletion of observations
Rule 2: Action/observation exchange
Rule 3: Insertion/deletion of actions When applicable, these rules reduce interventional queries to observational quantities.

Counterfactuals in SCMs: Y_{X=x}(u) — the value Y would take if X were set to x, given background U=u. Computed by: (1) Abduction: infer U from evidence, (2) Action: set X=x, (3) Prediction: compute Y.

Identifiability: An interventional distribution P(Y|do(X)) is identifiable from observational data if it can be computed without experiments, using the graph structure.

Symbols.

Symbol	Unicode	Name	Meaning
do(X=x)	—	Intervention	Set X to value x
P(Y\|do(X))	—	Interventional distribution	Distribution of Y under intervention
Y_x	—	Potential outcome	Y if X were set to x
→	U+2192	Causes	Direct causal relationship
⊥	U+22A5	Independence	Probabilistic independence
PA(X)	—	Parents	Direct causes of X
⊥⊥	—	d-separation	Conditional independence given graph
X ⇝ Y	U+21DD	Causal path	Path of causal influence

Metatheory. do-calculus is complete: if an interventional query is identifiable from a DAG, the rules can derive it. Causal discovery: under assumptions (causal sufficiency, faithfulness, Markov condition), the causal graph is identifiable up to Markov equivalence class from observational data. Interventions and counterfactuals form a hierarchy: associational (seeing) < interventional (doing) < counterfactual (imagining). Each level requires more information.

Applies to. Epidemiology (causal effects of treatments). Economics (policy evaluation). Machine learning (causal inference, fairness). Medicine (clinical trials, observational studies). AI (reasoning about actions). Social science (causal identification).

Limitations. Requires knowing or discovering the causal graph — often the hard part. Causal sufficiency assumption (no hidden confounders) is strong and often violated. Faithfulness can fail in designed systems. Linear SCMs are tractable; nonlinear SCMs are harder. DAGs don't capture feedback loops (cyclic causation requires different machinery). Counterfactual queries often not identifiable without experiments or strong assumptions. The do-calculus tells you when identification is possible but doesn't always provide practical estimators.