Bayes-optimal snapshot proposal for normalised attention: the least expected cost factor under a drift prior is (Σ_j e^{s_j}√E[e^{2δ_j}/Z_δ²])², attained by r_j ∝ e^{s_j}√E[e^{2δ_j}/Z_δ²]

For all $\htmlData{mx=1}{\Omega, J \in \htmlData{mx=0}{\operatorname{Type}}}$, $\htmlData{mx=3}{\mathit{lam} : \htmlData{mx=0}{\Omega} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}$, $\htmlData{mx=3}{s : \htmlData{mx=0}{J} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}$, $\htmlData{mx=5}{\delta : \htmlData{mx=0}{\Omega} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=1}{J} \mathrel{\htmlData{mx=3}{\to}} \htmlData{mx=2}{\mathbb{R}}}$, $\htmlData{mx=3}{Z : \htmlData{mx=0}{\Omega} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}$ and $\htmlData{mx=3}{r : \htmlData{mx=0}{J} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}$, if $\htmlData{mx=4}{\forall \omega,\ 0 \mathrel{\htmlData{mx=3}{\leq}} \htmlData{mx=2}{\htmlData{mx=0}{\mathit{lam}}(\htmlData{mx=1}{\omega})}}$, $\htmlData{mx=3}{\sum_{\omega} \htmlData{mx=2}{\htmlData{mx=0}{\mathit{lam}}(\htmlData{mx=1}{\omega})}} \mathrel{\htmlData{mx=4}{=}} 1$, $\htmlData{mx=4}{\forall \omega,\ 0 \mathrel{\htmlData{mx=3}{<}} \htmlData{mx=2}{\htmlData{mx=0}{Z}(\htmlData{mx=1}{\omega})}}$, $\htmlData{mx=4}{\forall j,\ 0 \mathrel{\htmlData{mx=3}{<}} \htmlData{mx=2}{\htmlData{mx=0}{r}(\htmlData{mx=1}{j})}}$ and $\htmlData{mx=3}{\sum_{j} \htmlData{mx=2}{\htmlData{mx=0}{r}(\htmlData{mx=1}{j})}} \mathrel{\htmlData{mx=4}{=}} 1$ (given $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{\Omega}}$, $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{J}}$ and $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Nonempty}}\,\htmlData{mx=1}{J}}$), then