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	Rightwards arrow / Causes	Direct causal relationship
⊥	U+22A5	Up tack / Independence	Probabilistic independence
PA(X)	—	Parents	Direct causes of X
⊥⊥	—	Doubled up tack / d-separation	Conditional independence given graph
⇝	U+21DD	Rightwards squiggle arrow	X ⇝ Y: a directed path of causal influence
:=	—	Assignment	Replacement of a structural equation under do
M	—	Structural causal model	The tuple (U, V, F)
U	—	Exogenous variables	Background variables and noise
V	—	Endogenous variables	The modelled variables
F	—	Structural equations	One per endogenous variable
f_i	—	Mechanism	The function determining V_i
u	—	Exogenous assignment	A specific background context

Metatheory. Pearl (1995) proved the three rules of do-calculus sound in any semi-Markovian model with the given graph; Shpitser and Pearl (2006) and, independently, Huang and Valtorta (2006) proved them complete, so if P(Y | do(X)) is identifiable from the observational distribution and the graph, the rules derive it, and identifiability is decidable. Non-identifiability is certified structurally by a hedge, the simplest instance being the bow arc — a single edge with an unobserved common cause — whose effect no function of observational data recovers. Causal discovery is bounded from the other side: under the causal Markov condition, faithfulness and causal sufficiency, the graph is recoverable only up to its Markov equivalence class, which by Verma and Pearl (1990) is the class of DAGs with the same skeleton and the same v-structures, so observation fixes the graph exactly to that resolution and no further. The three levels are strictly separated. Association does not determine intervention, since distinct graphs share an observational distribution, and intervention does not determine counterfactuals, since the joint distribution of potential outcomes is not fixed by any set of experiments.

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.