Front-door Criterion

In cases where we cannot use the back-door criterion, for e.g. because the variables we need to condition on are unobserved, we might be able to make use of the front-door criterion.

Let \(\mathcal{G}=(\mathbf{V},\mathbf{E})\) be the DAG for a Bayesian Network. Let \(T,Y \in \mathbf{V}\) be the intervention and outcome variables respectively. A set of variables \(Z\subset \mathbf{V}\) satisfies the front-door criterion relative to \((X,Y)\) if:

• \(Z\) blocks all directed paths from \(T\) to \(Y\).
• There is no unblocked path from \(T\) to \(Z\).
• All back-door paths from \(Z\) to \(Y\) are blocked by \(T\).

If \(Z\) satisfies the front-door criterion relative to \((T,Y)\) and if \(\mathbb{P}(T,Z)>0\), then the causal effect of \(T\) on \(Y\) is identifiable and can be computed via the front-door adjustment formula: \[ \mathbb{P}(Y=y|do(T=t)) = \sum_z \mathbf{P}(Z=z|T=t)\sum_{t'}\mathbb{P}(Y=y|T=t',Z=z)\mathbb{P}(T=t')\]