What a Structural Causal Model is.

Structure

An SCM has three components. First, a set of endogenous variables X — the variables whose values are determined within the model. Second, a set of exogenous variables U — the background factors that make each individual case unique; these are unobserved but their distributions are specified. Third, a set of structural equations, one per endogenous variable, of the form X⃰ = f⃰(pa(X⃰), U⃰) where pa(X⃰) is the set of causal parents of X⃰.

The structural equations are not regression equations. A regression equation describes the conditional expectation of a variable given others. A structural equation encodes the mechanism by which causes produce effects — it specifies what would happen to X⃰ if its parents were forcibly changed, holding everything else constant. This is the do() operator: intervening on X⃰ means replacing its structural equation with the constant X⃰ = x, while leaving all other equations intact.

What It Adds Over a Bayesian Network

A Bayesian network encodes the joint probability distribution over a set of variables and supports forward and backward inference. An SCM adds the U variables — the unobserved individual-level factors — which is exactly what is required to answer Rung 3 counterfactual questions.

The counterfactual question requires reasoning about the same individual under two different conditions: what actually happened, and what would have happened under the alternative. The U variables carry the individual-specific information that makes the individual the same individual across both conditions. Without them, you can compute population-level counterfactuals but not individual-level ones. Pearl’s Ladder covers the formal proof of why this matters.

The Three Layers

The SCM supports all three rungs from a single model. The observational distribution P(X) is recovered by marginalising over U. The interventional distribution P(Y | do(X=x)) is computed by replacing the structural equation for X. The counterfactual distribution P(Yₓ | X=x, Y=y) — what Y would have been for an individual who was observed at X=x — is computed by first inferring the posterior over U given the observed (X=x, Y=y), then running the model forward under the alternative condition. These three operations correspond exactly to the three rungs. They are not separate models — they are three query modes on the same object.