U_{\text{cs}}^{(k)}\left(S^{-1} w\right).$$ Expanding the above equation, we get the expression in (\[eq:cvcs-U-study\]).

In general, $\mathcal D$ is not necessarily study-specific and it might contains $D_t$’s that overlap. The utility function for ${\text{CV}_{\text{cs}}}$ in the general case can be constructed in the similar manner as when $\mathcal D$ is study-specific. In the first place, we modify $\hat U^{(t)}_{\text{cs}}(w)$, which estimates the expected utility of the PF combining the library of SPFs with weight $w^{(t)}$, into $$\hat U^{(t)}_{\text{cs}}(w) = \frac{1}{|D_t|} \sum_{(i,k) \in D_t} u\left( \sum_{\ell, t'}\mathbb I(s_t\cap s_{t'}=\emptyset)w_{\ell,t'}\hat Y_{t'}^\ell(x_{i,k}), y_{i,k}\right),$$ where $s_t = \{k:(i,k)\in D_t\text{ for some }i=1,\ldots,n_k\}$ is the list of studies with at least one sample in $D_t$. This modified $\hat U^{(t)}_{\text{cs}}$ guarantees no-date-reuse even if $D_t$’s are overlapped, since the set of studies that are involved in evaluating the utility of the stacked PF is mutually exclusive to the set of studies that are used in training SPFs in the considered stacked PF.

With the no-data-reuse property, it follows that with exchangeable distributions $P_1,P_2,\ldots$ $$\mathbb E\left[\hat U_{\text{cs}}^{(t)}(w)\right] = \mathbb E\left[U_g(w^{(t)})\right],$$ where $w^{(t)}$ is equal to $w$ except for all elements $w_{\ell,t'}$, such that $s_{t'}\cap s_t\neq \emptyset$ and $\ell=1,\ldots,L$, which are equal to zero.

In the study-specific $\mathcal D$ scenario, 