different $D_t$ is held out. We first introduce this CV scheme when $\mathcal D$ is study-specific hence $T = K$:

1.  Generate the library of SPFs $\hat{\mathcal Y}$ using every set in $\mathcal D$. Note that this library remains the same across $T$ iterations.

2.  At iteration $t$, evaluate the utility of $w$ using $D_t$ with SPFs that are not trained on $D_t$: $$\hat U_{\text{cs}}^{(t)}(w;\nu) = \frac{1}{n_t}\sum_{i = 1}^{n_t} u\left(\sum_{\ell,t}\mathbb I(t'\neq t) w_{\ell,t'} \hat Y_{t'}^\ell(x_{i,t}), y_{i,t}\right).
    \label{eq:loo-ind}$$

3.  Combine all $\hat U_{\text{cs}}^{(t)}$ across $T$ iterations and evaluated at a scaled $w$, yielding the utility function $\hat U_{\text{cs}}(w;\nu)$ for the selection of $w$ in ${\text{CV}_{\text{cs}}}$: $$\hat U_{\text{cs}}(w;\nu) = \sum_t \nu_t\hat U_{\text{cs}}^{(t)}(w;\nu) = \sum_{t=1}^K \frac{\nu_t}{n_t}\sum_i u\left(\sum_{\ell,t'}\frac{\mathbb I(t'\neq t)}{1-\nu_t}w_{\ell,t'}\hat Y_{t'}^\ell(x_{i,t}), y_{i,t}\right).
    \label{eq:cvcs-U-study}$$ The scaling factor $(1-\nu_t)^{-1}$ is used to extrapolate the predicted value given by the full ensemble using the prediction from the partial ensemble. For example, in generalist predictions for exchangeable studies with $\nu_t = K^{-1}$, we expect that $\hat Y_k^\ell(x)\approx \hat Y_{k'}^\ell(x)$ for $k'\neq k$ and hence $(\sum_{k,\ell}\hat Y_k^\ell(x))/(\sum_{k\neq k',\ell}\hat Y_k^\ell(x))\approx K/(K-1)$.

4.  Let $\hat w^{\text{cs}} = \operatorname*{arg\,max}_{w\in W} \hat U_{\text{cs}}(w;\nu)$, ${\text{CV}_{\text{cs}}}$ stacked PF is $$\hat Y_{w}^{\text{cs}} = \sum_{\ell, t} \hat w_{\ell, t}^{\text{cs}}\hat Y_{t}^\ell.$$

To understand the rationale for (\[eq:cvcs-U-study\]), we 