- \nu_1\overline{y_1^2} - \nu_2\overline{y_2^2},\end{aligned}$$ where $\overline{y_k^2} = n^{-1}\sum_{i}y_{i,k}^2$. The first two terms on the right-hand side exist because of data reuse, that is, we evaluate the utility of $w_1\hat Y_1 + w_2\hat Y_2$ using the training data of $\hat Y_1$ and $\hat Y_2$.

It follows that $$\mathbb E\left(\hat U(w;K^{-1}{\mathbf{1}}_K) - U_g(w)\right)
 = \frac{2(n+1)}{n}(\nu_1w_1 + \nu_2w_2).$$ We can see data-reuse introduces a non-zero bias to $\hat U(w;K^{-1}{\mathbf{1}}_K)$. This bias term is not always maximized at $w_g$. In fact, if $\nu_1\geq \nu_2$, $\nu_1w_1 + \nu_2w_2$ is maximized at $w = (1,0)$. In this case, the bias term would shift $\hat w_1$ towards $1$, which would make $\hat w_1$ larger than $w_{g,1}$ if $w_{g,1}\neq 1$. The strength of this shift increases as $\nu_1$ increases, which explains the reason that $\hat w_1$ increases as $\nu_1$ increases.

The effect of the bias term on the discrepancy of $\hat w$ to $w_g$ is particularly pronounced when training specialist PFs for study $k$ with $\hat U(w;e_k)$. In our example, $\hat w = e_k$ regardless of the values of cMSE of $\hat Y_k$, which in our setting also captures the prediction accuracy of $\hat Y$ on future samples in study $k$. This result also generalizes to $K>2$ and 