I have been wondering how to calculate $E[X\mid E[X\mid Y]]$. The lecturer gave this question after explaining how the error $X-\widehat{X}$ will always be perpendicular to any vector $Z-\widehat{X}$, which is in the linear space $S$ built from any linear combination of functions $Y_{1},Y_{2},\cdots,Y_{n}$.
I know that $E[X\mid Y]$ will be a function of $Y$, so we are basically calculating the mean of $X$ given the optimal estimator of $X$. I'm just stuck here with getting a final answer, and I can't draw this in a graphical way. I hope if someone can shed some light on this.
The answer is $E(X|Y)$. To show that we have to show that $$\int_E XdP= \int_E E(X|Y) dP$$ for all $E \in \sigma (E(X|Y))$. But note that $\sigma (E(X|Y)) \subseteq \sigma (Y)$. Hence the equation holds by deinition of $E(X|Y)$.