frontdoorcriteria

The front-door criterion is a principle in causal inference for identifying the causal effect of a treatment variable X on an outcome Y when there may be unobserved confounding between X and Y. It relies on a mediator Z that transmits the effect of X to Y and on structural assumptions that allow the effect to be recovered from observational data.

First, Z must intercept all directed paths from X to Y, meaning every causal path from X to Y goes through Z and there is no direct X → Y link or alternative route that bypasses Z.

Second, there must be no unblocked back-door paths from X to Z. In other words, all confounding between X and Z must be captured by observed variables, so P(z|x) reflects a causal influence of X on Z.

Third, all back-door paths from Z to Y must be blocked by X. After conditioning on X, there should be no remaining unblocked confounding path from Z to Y.

Under these conditions, the causal effect of X on Y is identifiable from observational data via a rendering formula such as P(y|do(x)) = sum_z P(z|x) sum_{x'} P(y|x', z) P(x'). This allows estimation even when X and Y share unobserved confounders, provided a suitable mediator Z exists.

Notes: the front-door criterion complements the back-door criterion and is applicable only when the mediator satisfies the three conditions. It is a standard tool within structural causal models and observational analyses where direct unconfounded adjustment is not possible.