I asked this question on stats.stackexchange and even elsewhere, but it never received an answer.

I just state the probabilistic problem here. It is about the optimality of the conditional expectation.

Consider a random vector $(X_1,X_2,X_3,Y)$ with $Y\in L^1$, and let $f$ be the Borelian function such that $E[Y\mid X_1, X_2, X_3]=f(X_1, X_2, X_3)$.

Define $$I = E\left[{\bigl(Y-f(X'_1,X_2,X_3)\bigr)}^2\right] - E\left[{\bigl(Y-f(X_1,X_2,X_3)\bigr)}^2\right]$$ where $X'_1$ is a random variable having the same distribution as $X_1$ but is independent of all other random variables $X_2,X_3,Y$.

*The question: "is it possible that $I < 0$?"*

In case you are interested in a simple example, the paper Correlation and variable importance in random forests (Gregorutti & al) investigates some particular cases such as the additive case $f(X_1,X_2,X_3)=f_1(X_1)+f_2(X_2)+f_3(X_3)$. It is not difficult to get that $I=2\textrm{Var}\bigl(f_1(X_1)\bigr)$ in this case.