to the setting where $L>1$ with at least one of the single learner using $-u(y,y')$ as its loss function. The specialist PF for study $k$ is then equal to the SPF trained on study $k$, and we do not borrow any information from other studies, even though they share the same hyper-distribution of mean of the outcome $Y$ with study $k$.

Stacking without data reuse {#sec:zero-stacking}
---------------------------

A common approach to limit the effects of data reuse is cross-validation (CV). CV in stacking is implemented by using part of the data for the training of the library of PFs $\hat{\mathcal Y}$ and the rest of the data for the estimation of $w$ (see for example [@breiman1996stacked]). How to split the data in multi-study settings is not as obvious as in the single-study setting due to the multi-level structure of the data. We consider two approaches based on CV. We first introduce their primary characteristics and their precise definitions are deferred to Section 3.2.1 and 3.2.2.

1.  **Within-set** (${\text{CV}_{\text{ws}}}$). For this approach, we assume that sets $D_t$ are mutually exclusive. An $M$-fold ${\text{CV}_{\text{ws}}}$ includes $M$ iterations. At each iteration, we randomly partition each $D_t$ into $D_{t,1}$ and $D_{t,2}$. We use $\{D_{t,1};t\leq 