by a single study $k$ or combining multiple studies. The ensemble weights will approximately maximize a utility function $U$ which we estimate using the entire collection of $K$ studies (generalist prediction) or only data in study $k$ (specialist prediction). Notably, stacking as currently implemented in multi-study learning can potentially suffer from over-fitting due to data reuse (DR): the same datasets generate SPFs and contribute (with others) to guiding the optimization of the stacking weights. With the aim of mitigating overfitting we introduce a no data reuse (NDR) procedure that still includes three key components of the staking methodology: the training of SPFs, the estimation of the utility function $U$, and the optimal choice of the ensemble weights.

In this manuscript we compare procedures to weight SPFs with and without data reuse. We use the mean squared error (MSE) as our primary metric to evaluate prediction accuracy. Our results prove that, when the number of studies $K$ and the sample sizes $n_k$ become large, both stacking with DR and NDR achieve a performance similar to that of an oracle benchmark. The oracle is defined as the linear combination of the SPFs’ limits ($\lim_{n_k} Y_ k$) that minimizes the MSE in future 