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-operator	Intervention
→	U+2192	Rightwards arrow / Causes	Direct cause
←	U+2190	Leftwards arrow	Edge into X on a back-door path
Y_x	—	Counterfactual	Y under X=x
Pa(X)	—	Parents	Direct causes
M	—	Structural causal model	The tuple (U, V, F, P(U))
U	—	Exogenous variables	Background, unmodelled causes
V	—	Endogenous variables	The modelled variables
F	—	Structural equations	One assignment per endogenous variable
Σ	U+03A3	Greek capital sigma / Summation	Sum over adjustment-set values in the back-door formula
P	—	Probability	Distribution over exogenous or observed variables
∣	U+2223	Divides / Given	Separator in a conditional or interventional query

Metatheory. Pearl (1995) gave the three rules of do-calculus and proved them sound: each is a valid transformation of interventional expressions in any semi-Markovian model with the given graph. Completeness was open for a decade and settled independently by Shpitser and Pearl (2006) and by Huang and Valtorta (2006): if P(y | do(x)) is identifiable from the observational distribution together with the graph, then some sequence of the three rules derives it, so identifiability is decidable and the ID algorithm is a decision procedure. Non-identifiability has a structural witness — the algorithm returns a hedge, a pair of C-components that no adjustment can separate — and the simplest case is the bow arc, a single confounded edge X → Y, whose effect is unidentifiable however much observational data is available. Shpitser and Pearl (2006) extended completeness to conditional interventional queries. The back-door and front-door criteria are the two named sufficient conditions the calculus subsumes; both are sound but neither is necessary.

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

Limitations. The calculus identifies effects relative to a graph it does not supply; the graph is an input, and structure learning from observational data recovers it only up to Markov equivalence and only under faithfulness, which fails when causal paths cancel. Identifiability genuinely fails in some structures rather than merely resisting proof: no function of the observational distribution equals P(y | do(x)) when a hedge is present, so the answer in those cases is that experiment is required. Counterfactual quantities sit strictly above interventional ones — the probability of necessity P(Y_{x'} = y' | X = x, Y = y) is not identified even from the full set of experimental distributions, and can only be bounded, because it depends on the joint distribution of potential outcomes that no experiment observes. Acyclicity is built into the semantics, so equilibrium and feedback systems have no structural causal model of this form, and the framework presupposes well-defined interventions on well-individuated variables, which excludes attributes that cannot coherently be set.