specialist prediction function for study $k$, we can optimize $\hat U(w; e_k)$ to select $w_s^{(k)}$, where $e_k$ is a $K-$dimensional vector with the $k$-th component to be one and all others zero. This approach reuses data. Training an SPF $\hat Y^{\ell}_t$ uses part of the data $D_t\subset \mathcal S$ that are then reused to compute $\hat U(w;\nu)$. Data reuse makes $\hat U(w;\nu)$ a biased estimator of $U_g(w)$ and $U_s(w;k)$. In the next paragraph we illustrate a simple example where the bias of $\hat U(w;\nu)$ due to data reuse makes the selection of $w$, denoted as $\hat w$, erroneously favors those $\hat Y_t^\ell$ generated from studies with large $\nu_k$.

Consider a scenario where $\mathcal D$ is study-specific and $K=2$. Let $u(y,y') = -(y-y')^2$. We only observe $y_{i,k}$ without any covariates and we assume $y_{i,k}\sim N(\mu_k,1)$ for $k=1,2$ where $\mu_k\sim N(0,1)$. Let $n_1=n_2 = n$. In this simple example, we generate a library of SPFs with two constant functions $\hat Y_1(\cdot) = \bar y_1$ and $\hat Y_2(\cdot) = \bar y_2$, where $\bar y_k = n^{-1}\sum_i y_{i,k}$. Under the constraint that $W = \Delta_1$, where $\Delta_1$ is the standard 1-simplex, the weights that optimize $\hat U(w;\nu)$ is $\hat w = (\hat w_1, \hat 