# Back-door Criterion

The backdoor criterion provides us with conditions under which we can estimate causal effects given a causal Bayesian Network.

Let \(\mathcal{G}=(\mathbf{V},\mathbf{E})\) be the corresponding DAG, with \(T,Y \in \mathbf{V}\) being the treatment and outcome variables respectively. A set of variables \(X \subset \mathbf{V}\) satisfies the backdoor criterion relative to \((T,V)\) if:

- No node in \(X\) is a descendant of \(T\).
- \(X\) blocks every path between \(T\) and \(Y\) that contains an arrow into \(T\).

If \(X\) satisfies the backdoor criterion then the causal effect of \(T\) on \(Y\) assuming the BN has graph structure \(\mathcal{G}\) is: \[ \mathbb{P}(Y=y|do(T=t)) = \sum_x \mathbb{P}(Y=y|T=t,X=x)\mathbb{P}(X=x) \]

Here, \(X\) is a set of nodes which blocks all spurious paths between \(T\) and \(Y\), leaves all the direct paths from \(T\) to \(Y\) unchanged and does not create any new spurious paths i.e. does not unblock any new paths.