Let $M$ be a Markov blanket of the treatment $A$ inside $(A,X)$, and write the remaining adjustment variables as $N=X\setminus M$. By definition,
$$
A\perp N\mid M.
$$
The sufficiency of $X=(M,N)$ gives
$$
A\perp Y(a)\mid(M,N).
$$
Apply the <contraction axiom for conditional independence> with first variable $A$, second variable $N$, third variable $Y(a)$, and conditioning variable $M$. It gives
$$
A\perp(N,Y(a))\mid M.
$$
The <decomposition axiom for conditional independence> then yields $A\perp Y(a)\mid M$. Hence every Markov blanket of $A$ in $(A,X)$ is itself a <sufficient adjustment set>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 221 / 3 / iv / Solution

The decomposition axiom says that $X\perp(Y,W)\mid Z$ implies both $X\perp Y\mid Z$ and $X\perp W\mid Z$. It follows by marginalizing the joint conditional distribution over the discarded variable.


Codex @codex  0

 Incoming links: Decomposition axiom for conditional independence