Is it true $E(E(X|Y)|Z)=E(E(X|Y,Z)|Z)$?

Question

Intuitively I think this is true since given more information about $Z$ in the inner expectation seems to have no effect. But with Adam's law, we see that this would imply $E(E(X|Y)|Z)=E(E(X|Y,Z)|Z)=E(X|Z)$, which I'm not quite sure. Can someone confirm this is indeed true and provide a proof or a source of proof? Thank you!

Answer 1

No, it is not true.

First, recall the law of iterated expectations: $$E(E(X|Y))=E(X)$$ Consider the expression $E(E(X|Y,Z)|Z)$. Given $Y=y,Z=z$ the inner expectation is evaluated in the conditional universe where $Y=y,Z=z$; the outer expectation averages over all $Y=y$ given $Z=z$, and we end up with the law of iterated expectations in the conditional universe $Z=z$ $$E(E(X|Y,Z=z)|Z=z)=E(X|Z=z)$$ or in r.v. notation $$E(E(X|Y,Z)|Z)=E(X|Z)$$ On the other hand, the expression $E(E(X|Y)|Z)$ is something different. Given $Y=y,Z=z$ the inner expectation is evaluated in the conditional universe where $Y=y$, and there is no reason for $$E(E(X|Y)|Z=z)\overset{?}{=}E(X|Z=z)$$ to be true. For example, if $X$ and $Y$ are independent, then $E(X|Y)=E(X)=constant$, and we end up with $$E(X)\overset{?}{=}E(X|Z=z)$$ which cannot be generally true.