each $\hat U_{\text{cs}}^{(t)}(w)$ is combined into $\hat U_{\text{cs}}(w;\nu)$ according a relative importance $\nu_t$. This relative importance can be interpreted as the total probability mass assigned to data in $D_t$ in the empirical distribution of $\mathcal S$: $$\hat \pi(x,y) = \sum_k\frac{\nu_k}{n_k}\sum_i \mathbb I\left((x,y) = (x_{i,k},y_{i,k})\right).$$

With this definition, in the general case, the relative importance of $D_t$ is $
\gamma_t = \sum_{k\in s_t} \nu_k n_{k,t}/n_k,
$ where $n_{k,t}$ is the number of samples from study $k$ that are present in $D_t$. As in the case for study-specific $\mathcal D$, we can use these $\gamma_t$ to combine $\hat U_{\text{ws}}^{(t)}(w)$.

With Taylor expansion of $U_g(w^{(t)})$ around ${\mathbf{0}}$, we get $$\mathbb E\left(\sum_t\gamma_t \hat U_{\text{cs}}^{(t)}(w)\right) = \left(\sum_t \gamma_t\right)\mathbb EU_g({\mathbf{0}}) + \mathbb E\left[\frac{\partial U_g}{\partial w^\intercal}({\mathbf{0}})\right]\sum_t\gamma_t w^{(t)} + o(\|w\|).$$ Let $\Gamma$ be a $KL\times KL$ diagonal matrix with the element corresponds to $w_{\ell,t}$ equal to $
\sum_{t'} \gamma_{t'}\mathbb I(s_{t'}\cap s_t = \emptyset)w_{\ell,t}$. By the definition of $w^{(t)}$, $\sum_t \gamma_t w^{(t)} = \Gamma w$. Therefore we have $$\left(\sum_t\gamma_t\right)^{-1}\mathbb E\left(\sum_{t}\gamma_t \hat U_{cs}^{(t)}(w)\right) = \mathbb EU_g({\mathbf{0}}) + \mathbb E\left[\frac{\partial U_g}{\partial w^\intercal}({\mathbf{0}})\right] \frac{\Gamma w}{\sum_t \gamma_t} + o(\|w\|).$$

If the linear term dominates higher order terms in the above expansion, we have $$\left(\sum_t\gamma_t\right)^{-1}\mathbb E\left(\sum_{t}\gamma_t \hat U_{cs}^{(t)}(w)\right) = \mathbb E U_g\left(\frac{\Gamma w}{\sum_t \gamma_t}\right) 