I have a problem about intuition: substracting the mean of iid RVs seems to increase the mutual information.
Say $X,Y$ are real iid RVs, then $\frac{X-Y}{2}$ and $\frac{Y-X}{2}$ are not independent because one is just the negative of the other?
If this is right, it seems contraintuitive to me, as subtracting the mean is such a usual preprocessing step for iid samples. What am I missing or where did I go completely wrong? It's probably rather stupid but my intuition fails or tricks me here. Guess I am mixing up things. Sorry.