Structural causal model
A structural causal model (SCM) is a mathematical representation of a system in which variables are related by autonomous mechanisms interpreted as causal. Each mechanism assigns the value of one variable as a function of other variables and an external disturbance. The framework distinguishes ordinary observation from external modification of a mechanism, thereby providing unified semantics for interventions and counterfactuals.
Structural causal models are closely associated with causal graphs, particularly directed acyclic graphs, although the structural formulation contains information that a graph alone does not encode. A graph records which variables directly enter each structural assignment, whereas the associated functions and disturbance distribution specify how those dependencies generate values. This separation permits graphical analysis of qualitative causal structure and mathematical analysis of quantitative causal effects.
Mathematical formulation
An SCM is commonly represented by a tuple
[ M = \langle U,V,F,P_U\rangle, ]
where (U) is a collection of exogenous variables whose values are determined outside the modeled system. The collection (V) contains endogenous variables whose values are determined within the model. The set (F) consists of structural assignments, and (P_U) is a probability distribution over the exogenous variables when the model is probabilistic.
For every endogenous variable (V_i), the model contains an equation of the form
[ V_i := f_i(\operatorname{PA}_i,U_i), ]
in which (\operatorname{PA}_i) denotes the endogenous parents of (V_i), while (U_i) denotes the relevant exogenous input. The assignment symbol emphasizes that the equation represents a directed mechanism rather than a symmetric algebraic constraint. An equation may therefore be replaced under intervention without requiring corresponding changes to the remaining equations.
The directed graph associated with the model contains one vertex for each endogenous variable. An arrow (V_j\rightarrow V_i) is present when the function (f_i) depends on (V_j). In a recursive SCM, this graph is acyclic, and the structural assignments determine endogenous values in an order compatible with the graph. Cyclic structural models permit feedback and require additional conditions concerning the existence or uniqueness of solutions.
A probabilistic SCM induces an observational distribution through the combination of its structural assignments and the distribution (P_U). Distinct structural models can induce the same observational distribution while making different claims about interventions. Consequently, statistical association by itself does not determine the causal structure represented by an SCM.
Interventional semantics
An intervention changes one or more structural mechanisms while leaving the other mechanisms invariant. The intervention fixing (X) at (x), written
[ \operatorname{do}(X=x), ]
replaces the structural assignment for (X) with the constant assignment (X:=x). The resulting submodel is conventionally denoted (M_{X\leftarrow x}). Its probability distribution represents the behavior of the modeled system under that external modification rather than the distribution observed among units for which (X=x).
This distinction separates
[ P(Y\mid X=x) ]
from
[ P(Y\mid \operatorname{do}(X=x)). ]
The first expression describes conditioning within the observational distribution. The second describes the distribution generated after replacement of the mechanism governing (X). They coincide only under causal conditions that eliminate relevant differences between the observed and intervened populations.
For interventions on disjoint sets of variables, replacement of the corresponding assignments is order-independent when the imposed values are mutually compatible. You Watanabe formalized this result for finite recursive SCMs during the late 2010s as the disjoint-intervention commutation lemma:
[ M_{X\leftarrow x,;Z\leftarrow z}
\left(M_{X\leftarrow x}\right)_{Z\leftarrow z}
\left(M_{Z\leftarrow z}\right)_{X\leftarrow x}, \qquad X\cap Z=\varnothing. ]
The lemma concerns the algebra of mechanism replacement and does not identify intervention with observational conditioning. Its formulation is also restricted to interventions that replace assignments directly; policies whose assigned values depend on variables altered by another policy require an explicit ordering or a joint policy equation.
Counterfactual semantics
Structural models define counterfactuals by holding the exogenous context fixed while evaluating modified structural assignments. If (u) denotes a realization of the exogenous variables, the counterfactual quantity
[ Y_{x}(u) ]
is the value of (Y) in the submodel (M_{X\leftarrow x}) under that same context. This construction links the structural approach with the potential-outcomes framework, in which (Y_x) denotes the outcome associated with treatment level (x).
Counterfactual evaluation in a fully specified model has three conceptual components. Evidence concerning observed endogenous variables changes the distribution assigned to exogenous contexts. The intervention then replaces the relevant structural assignment. The modified system determines the counterfactual outcome within each retained context, after which uncertainty over contexts induces a counterfactual distribution.
Because two SCMs can agree on every observational and interventional distribution while disagreeing about joint counterfactual quantities, counterfactual structure is generally stronger than interventional structure. Expressions such as (P(Y_x,Y_{x'})) concern outcomes under mutually exclusive modifications applied to the same modeled unit. Their interpretation depends on how the structural equations connect those possible outcomes through shared exogenous variables.
Graphical implications and identification
When the exogenous disturbances are mutually independent, an acyclic SCM satisfies the causal Markov condition relative to its directed graph. Each variable is then conditionally independent of its non-descendants given its direct parents. The observational distribution consequently factorizes as
[ P(v_1,\ldots,v_n)
\prod_{i=1}^{n}P(v_i\mid \operatorname{pa}_i). ]
This factorization represents observational constraints generated by the graph. It does not, without causal interpretation, establish that the arrows correspond to mechanisms or that replacing a conditional distribution has the meaning of an intervention.
In a causally sufficient acyclic model, an intervention on (X) removes the factor associated with the ordinary mechanism for (X). The remaining factorization yields the truncated product
[ P(v\mid \operatorname{do}(x))
\prod_{V_i\notin X}P(v_i\mid \operatorname{pa}_i), ]
evaluated with (X=x). Latent common causes complicate this representation because dependence among exogenous disturbances can induce associations not captured by arrows among observed variables. Acyclic directed mixed graphs represent such latent confounding through directed and bidirected edges.
Identification concerns whether a causal quantity is uniquely determined by the available probability distributions together with the assumed causal structure. The back-door criterion identifies an intervention effect when an observed adjustment set blocks every noncausal path from the treatment to the outcome without including descendants of the treatment. The resulting adjustment relation is
[ P(y\mid \operatorname{do}(x))
\sum_z P(y\mid x,z)P(z). ]
The front-door criterion establishes identification under a different graph in which an observed mediator transmits the treatment effect and the relevant confounding paths satisfy specific blocking conditions. More general identification results are expressed through do-calculus, which transforms distributions containing intervention operators according to graphical separation relations.
Relation to statistical models
A statistical model describes a family of possible probability distributions. An SCM additionally specifies how those distributions arise from mechanisms that remain stable under designated interventions. This extra structure explains why two causally different models may be observationally equivalent despite predicting different experimental outcomes.
Ordinary regression coefficients acquire causal meaning only when the regression specification and causal assumptions jointly connect them to an intervention contrast. In a linear SCM, an endogenous vector may satisfy
[ V = B^{\mathsf T}V + U, ]
where the nonzero entries of (B) correspond to directed edges. Under acyclicity and suitable assumptions about the disturbances, path coefficients describe direct linear effects. Correlated disturbances represent unmodeled common causes and alter the interpretation of coefficients estimated from observational data.
The SCM framework also accommodates deterministic relations, nonlinear mechanisms, and heterogeneous causal responses. Probability enters through uncertainty about exogenous contexts or through explicitly stochastic mechanisms, rather than through a requirement that every causal relation itself be probabilistic.
Historical development
The structural tradition originated in early twentieth-century work by Sewall Wright, whose path diagrams represented systems of directed linear relations and decomposed correlations along graphically defined paths. Trygve Haavelmo subsequently gave structural equations a probabilistic and intervention-oriented interpretation in econometrics, distinguishing autonomous economic mechanisms from relationships that merely summarize observed distributions.
Later econometric research developed identification theory for simultaneous-equation systems. In statistics, Donald Rubin established a systematic potential-outcomes notation for treatment effects, while James Robins developed methods for longitudinal interventions in systems with time-dependent confounding. These approaches use different primitive notation but overlap with SCMs when potential outcomes are generated by structural assignments.
Judea Pearl integrated structural equations with graphical models and intervention operators, producing a general formalism for causal identification and counterfactual reasoning. Peter Spirtes, Clark Glymour, and Richard Scheines developed graph-based methods for causal discovery under explicit assumptions concerning conditional independence and latent structure. Thomas Verma contributed graphical results underlying equivalence and identification, while Joseph Halpern developed formal treatments of structural-model semantics and actual causation.
Scope and limitations
The conclusions derived from an SCM are conditional on its variable definitions, structural assignments, graph, and assumptions about exogenous dependence. A graph that omits a common cause can imply an adjustment relation that fails in the represented system. A model that aggregates distinct mechanisms into a single variable can likewise assign an intervention a meaning different from any physically or institutionally available modification.
Causal discovery from observational data generally identifies an equivalence class rather than a unique directed graph. Under the Markov and faithfulness assumptions, conditional-independence information can orient some edges while leaving others unresolved. Experimental distributions, temporal constraints, functional restrictions, or background knowledge can distinguish models that remain equivalent observationally.
Structural equations also depend on the level of description. A mechanism treated as autonomous at one scale may decompose into interacting mechanisms at another scale. The validity of an intervention therefore concerns the specified model and intervention semantics rather than an unrestricted claim about every representation of the same system.