Structural Causal Model (SCM)

The deterministic functional backbone of an SCM. Each endogenous variable $X_i$ is defined by a function $f_i$ of its direct causes (parents $PA_i$) and an exogenous noise term $U_i$: $X_i := f_i(PA_i, U_i)$. Unlike standard regression, these equations represent asymmetric, directional causal mechanisms that remain invariant under interventions on other variables. This modularity allows the model to predict the effect of surgically altering one part of the system without changing the others.

The qualitative structure of an SCM, typically represented as a Directed Acyclic Graph (DAG). Nodes represent variables, and directed edges represent direct causal relationships. The acyclic constraint ensures no variable can cause itself, directly or indirectly. This graph encodes the Causal Markov Condition: a variable is independent of its non-descendants given its parents. The DAG is the primary tool for visually communicating causal assumptions and systematically identifying confounding paths.

The mathematical formalization of an action. The do-operator, $do(X=x)$, represents an external intervention that surgically sets a variable to a specific value, severing all incoming edges to that node in the graph. This creates a mutilated graph. The distribution $P(Y | do(X=x))$ is fundamentally different from the conditional distribution $P(Y | X=x)$ because it simulates a controlled experiment, eliminating confounding bias. This is the core of Pearl's do-calculus.

The source of non-determinism in an otherwise deterministic system. Each endogenous variable $X_i$ is influenced by an unobserved, background variable $U_i$. These $U$ terms encapsulate all unmodeled causes, latent confounding, and inherent randomness. The joint distribution over all exogenous variables, $P(U)$, fully determines the joint distribution over all endogenous variables. Counterfactual reasoning is only possible by reasoning about the specific, fixed values of these $U$ variables for a given observed individual.

The highest rung of the Ladder of Causation. A counterfactual answers the question: 'What would have happened for this specific unit, had I acted differently?' This requires a three-step process: 1) Abduction: Use observed evidence to update the distribution of the exogenous noise variables $P(U | evidence)$. 2) Action: Perform the $do$-operation to modify the structural equations. 3) Prediction: Compute the outcome in the modified model using the updated $U$ distribution. This enables individual-level 'what-if' analysis.

A three-level taxonomy of causal reasoning abilities that an SCM can support. Level 1 (Association) involves seeing and answering statistical questions like $P(Y|X)$. Level 2 (Intervention) involves doing and answering $P(Y|do(X))$. Level 3 (Counterfactuals) involves imagining and answering questions about a specific past event. A key theorem states that a model capable of Level 3 reasoning is strictly more powerful and can answer queries from the lower levels, but not vice-versa.