The graph has no cause of $D$, so $D$ is independent of its potential outcomes. By <consistency of potential outcomes>, the <g-formula> therefore gives
$$
\mathbb E[Y(d)]=\mathbb E[Y\mid D=d].
$$
Condition on $M$ and use the structural zero from part ii:
$$
\begin{aligned}
\mathbb E[Y(d)]
&=\mathbb E[Y\mid D=d,M=1]\mathbb P(M=1\mid D=d)\\
&\quad+\mathbb E[Y\mid D=d,M=0]\mathbb P(M=0\mid D=d)\\
&=\mathbb E[Y\mid D=d,M=1]\mathbb P(M=1\mid D=d).
\end{aligned}
$$
Consequently the <causal risk ratio> is
$$
\frac{\mathbb E[Y(1)]}{\mathbb E[Y(0)]}
=\frac{\mathbb E[Y\mid D=1,M=1]}{\mathbb E[Y\mid D=0,M=1]}
\frac{\mathbb P(M=1\mid D=1)}{\mathbb P(M=1\mid D=0)}.
$$
Applying <Bayes theorem> to the second factor gives
$$
\frac{\mathbb P(M=1\mid D=1)}{\mathbb P(M=1\mid D=0)}
=\frac{\mathbb P(D=1\mid M=1)/\mathbb P(D=0\mid M=1)}
{\mathbb P(D=1)/\mathbb P(D=0)},
$$
which proves the stated formula. The unmeasured common cause $U$ of $M$ and $Y$ does not obstruct this total-effect argument because no mediator effect is being identified.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 221 / 4 / iii / Solution

{title2=CRR}

For binary potential outcomes, the causal risk ratio compares intervention risks:
$$
\operatorname{CRR}=\frac{\mathbb E[Y(1)]}{\mathbb E[Y(0)]}.
$$


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