denote the only index $m$ such that $(i,k)\in D_{t,m}$ by $m(i,k)$. The estimated utility function for $w$ is $$\hat U_{\text{ws}}(w;\nu) = \sum_{k=1}^K\frac{\nu_k}{n_k}\sum_{i=1}^{n_k}u\left(\sum_{\ell,t}w_{\ell,t}\hat Y_{t,m(i,k)}^\ell(x_{i,k}), y_{i,k}\right).
        \label{eq:cvws-utility}$$ and $$\hat w^{\text{ws}} = \operatorname*{arg\,max}_{w\in W} \hat U_{\text{ws}}(w;\nu).$$

4.  The ${\text{CV}_{\text{ws}}}$ stacked PF is $$\hat Y_{w}^{\text{ws}} = \sum_{\ell, t} \hat w^{\text{ws}}_{\ell,t}\hat Y_t^\ell.$$

For specialist predictions, ${\text{CV}_{\text{ws}}}$ can solve the data-reuse related problem in Section 3.1. In particular, we consider the example in Section 3.1. Assume $D_t$ is study-specific. Denote $\hat w_{s}^{\text{ws}} = \operatorname*{arg\,max}_{w} \hat U_{\text{ws}}(w;e_1)$ the ${\text{CV}_{\text{ws}}}$ selected weights for the specialist PF of study 1 and $\hat w_s = \operatorname*{arg\,max}_w \hat U(w;e_1)$. We measure the prediction accuracy of a PF $\hat Y_w$ with the expected MSE on study 1 is $\text{MSE}_1(w) = \int_{\mu_1}(\mu_1 - \hat Y_w)^2dP(\mu_1) + 1$, where $P(\mu_1)$ is the distribution of $\mu_1$.

Since there is no analytic expression for $\hat w^{\text{ws}}_s$, we use Monte Carlo simulation (1000 replications) to compare $\text{MSE}_1(\hat w_s^{\text{ws}})$ and $\text{MSE}_1(\hat w_1)$. We set $n = 90$ and $\mu_1\sim N(0,0.1)$, where borrowing information from study 2 is beneficial for the estimation of $\mu_1$. When varying $M$ from 3 to 15, we observe that $\mathbb E\left[\text{MSE}_1(\hat w_1) - \text{MSE}_1(\hat w_s^{\text{ws}})\right]$ first increases from 0.0014 to 0.002 then decrease when 