# Stacking weights to get posterior of quantity of interest

Let’s assume that I have a type of model that produces a quantity of interest \Delta, such as a marginal effect or a ratio of two parameters. My goal is to estimate many models M_k with different interactions, assumptions and explanatory variables to get the posterior of \Delta.

If I use model averaging to obtain “weights” p(M_k|y), I can get a posterior distribution of \Delta by calculating

p(\Delta | y) = \sum_{k=1}^K p(\Delta | M_k, y) p(M_k | y).

If I use stacking, then I know I can use weights w_k to get the predictive density of new observations \hat{p}(\tilde{y} | y) (eq. 5 of Yao, Vehtari, Simpson and Gelman, 2018). Can I use the same weights w to calculate \hat{p}(\Delta | y)? Or would I have to solve \max_\omega S(\sum_k \omega_k p(\Delta | y, M_k), p_t(\Delta | y)) (eq. 3)?

Thanks!