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A backdoor adjustment set contains no descendant of the exposure and blocks every backdoor path from the exposure to the outcome. Averaging the conditional outcome distribution over such a set identifies the total causal effect.

A backdoor path from $A$ to $Y$ is a path whose first edge has an arrowhead at $A$. The set $X_I$ satisfies the backdoor criterion when it contains no descendant of $A$ and blocks every backdoor path from $A$ to $Y$. Under this criterion, consistency, and positivity,so adjustment for $X_I$ identifies the intervention mean.

WriteSince $X_1$ has no parents, its total effect has no backdoor path, and

For $X_2$, $X_3$ intercepts every directed path to $X_4$. Conditional on $X_1$, there is no unblocked backdoor path from $X_2$ to $X_3$, and $(X_1,X_2)$ blocks every backdoor path from $X_3$ to $X_4$. The conditional front-door adjustment therefore givesandPart ii shows that the inner sum, after also averaging $X_4$, does not actually depend on $x_1$.

For $X_3$, $(X_1,X_2)$ is a sufficient backdoor adjustment set. HenceThese three formulas identify the requested average treatment effects from the observed joint distribution.

The $X_1$ coefficient is the total causal effect of $X_1$ on $Y$. The node $X_1$ has no parents, so there is no backdoor path into it, and the unadjusted regression sums all directed paths from $X_1$ to $Y$.