# Front-Door Adjustment formula: confusing notation

Pearl et al. "Causal Inference in Statistics: A Primer" (2016) p. 69 Theorem 3.4.1 provides the Front-Door Adjustment formula: $$P(y | \text{do}(X = x)) = \sum_z P(z | x) \sum_{x'} P(y|x', z)P(x').$$ Does it mean $$P(y | \text{do}(X = x)) = \sum_z \left[ P(z | x) \sum_{x'} \left\{ P(y|x', z)P(x') \right\} \right]$$ or $$P(y | \text{do}(X = x)) = \left[ \sum_z P(z | x) \right]\cdot \left[ \sum_{x'} \left\{ P(y|x', z)P(x') \right\} \right]?$$

• The last expression is not allowed, because $P(y|x',z)$ has $z$ in it, but that term has fallen out of scope of the $z$ summation. Apr 2, 2020 at 14:40

It means $$P(y|\text{do}(X=x)) = \sum_z \left[ P(z|x) \sum_{x'} \left\{ P(y|x',z) P(x') \right\} \right].$$ This can be seen from equation (3.15) on p. 68 which is $$P(y|\text{do}(X=x)) = \sum_z \sum_{x'} \left\{ P(y|x',z) P(x') P(z|x) \right\}$$ and can be rearranged to be $$P(y|\text{do}(X=x)) = \sum_z \sum_{x'} \left\{ P(z|x) P(y|x',z) P(x') \right\},$$ the latter ordering of conditional probabilities corresponding to the ordering in the equation at the top of this answer.
• Right. $x'$ is just a dummy variable, and does not show up in the $P(z|x),$ so that that term can move in or out of the second sum with impunity. Apr 2, 2020 at 14:31