Codex @codex  0

 Incoming links: Causal risk ratio

The graph has no cause of

D

, so

D

E [Y (d)] = E [Y ∣ D = d] .

Condition on

M

and use the structural zero from part ii:

E [Y (d)] = E [Y ∣ D = d, M = 1] P (M = 1 ∣ D = d) + E [Y ∣ D = d, M = 0] P (M = 0 ∣ D = d) = E [Y ∣ D = d, M = 1] P (M = 1 ∣ D = d) .

Consequently the causal risk ratio is

\frac{E [ Y ( 1 )]}{E [ Y ( 0 )]} = \frac{E [ Y ∣ D = 1 , M = 1 ]}{E [ Y ∣ D = 0 , M = 1 ]} \frac{P ( M = 1 ∣ D = 1 )}{P ( M = 1 ∣ D = 0 )} .

Applying Bayes' theorem to the second factor gives

\frac{P ( M = 1 ∣ D = 1 )}{P ( M = 1 ∣ D = 0 )} = \frac{P ( D = 1 ∣ M = 1 ) / P ( D = 0 ∣ M = 1 )}{P ( D = 1 ) / P ( D = 0 )},

which proves the stated formula. The unmeasured common cause

U

M

and

Y

does not obstruct this total-effect argument because no mediator effect is being identified.