$(Y_i - \hat{Y}_i)(\hat{Y}_i - \bar{Y}) = 0$ in the image below (third and fourth line of the proof!). Why?
2026-03-30 10:17:04.1774865824
$(Y_i - \hat{Y}_i)(\hat{Y}_i - \bar{Y}_i) = 0$
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By $\hat{Y}_i$ you mean the MMSE estimator (or the posterior mean) ?
If so, remember a very important property of this estimator is that the MMSE estimator of $Y$ is that the error $\hat Y - Y$ is orthogonal to any function of $Y$. Thus $E[(Y_i - \hat{Y}_i)(\hat{Y}_i - \bar{Y})] = 0$. But in your case there are no expectations. Please verify that or give the full context of this derivation