If $X,Y$ are independent, does $E(X\mid Y,Z)= E(X\mid Z)$ hold?

Question

I wonder if $X,Y$ are independent random variables and $Z$ is another random variable, will we have $E(X\mid Y,Z)= E(X\mid Z)$?

Answer 1

The answer is NO.

let $X$ and $Y$ are i.i.d random variables from $Bernoulli(p=\frac{1}{2})$.

Define $$Z=X+Y$$

$$E(X|Y,Z)=E(Z-Y|Y,Z)=Z-Y$$

but

$$E(X|Z)=\frac{Z}{2}$$.since

$$Z=E(Z|Z)=E(X+Y|Z)=E(X|Z)+E(Y|Z)=2E(X|Z)$$

(because $E(X|Z)=E(Y|Z)$).

so

$$E(X|Z)=\frac{Z}{2}$$

but

$$E(X|Y,Z)=Z-Y$$