Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 221 / 3 / iv

Table of contents
- Solution iv

Solution

This is the covariate-conditional front-door adjustment. The paths from

A

to

M

have no unblocked backdoor path after conditioning on

X

, and all backdoor paths from

M

to

Y

are blocked by

(A, X)

. Therefore

P (Y (a) = y) = \sum_{x, m} P (x) P (m ∣ a, x) \sum_{a^{'}} P (y ∣ m, a^{'}, x) P (a^{'} ∣ x) .

The inner sum identifies the effect of setting

M = m

at covariate value

x

by adjusting for

A

; the outer sum transports this through the mediator distribution generated by setting

A = a

and then averages over

X

.

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