<Causal identification> means that $\mathbb E[Y(a_1,a_2)]$ is uniquely determined by the observed joint distribution of $(A_1,X,A_2,Y)$ under the causal assumptions. Equivalently, it admits an identifying formula containing only observed-data probabilities and conditional expectations.

The <G-computation> formula for this two-stage treatment is
$$
\boxed{
\mathbb E[Y(a_1,a_2)]
=\sum_x \mathbb E[Y\mid A_1=a_1,X=x,A_2=a_2]
\mathbb P(X=x\mid A_1=a_1)}.
$$
The first factor is the observed mean outcome after the specified treatment history and intermediate value; the second averages over the intermediate-variable distribution generated after the first treatment.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 221 / 1 / a / Solution

A causal estimand is identified when every causal model satisfying the stated assumptions and inducing the same observed-data distribution gives it the same value. An identifying formula expresses that value entirely through the observed-data distribution.


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