Adaptive exactness is unbiased with an independent pilot: if the exact-or-sampled decision uses samples independent of the estimate, E[output] = θ whenever every branch is unbiased

For all $\htmlData{mx=1}{\Omega_{1}, \Omega_{2}, D \in \htmlData{mx=0}{\operatorname{Type}}}$, $\htmlData{mx=3}{w_{1} : \htmlData{mx=0}{\Omega_{1}} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}$, $\htmlData{mx=3}{w_{2} : \htmlData{mx=0}{\Omega_{2}} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}$, $\htmlData{mx=3}{\mathit{dec} : \htmlData{mx=0}{\Omega_{1}} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{D}}$, $\htmlData{mx=5}{X : \htmlData{mx=0}{D} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=1}{\Omega_{2}} \mathrel{\htmlData{mx=3}{\to}} \htmlData{mx=2}{\mathbb{R}}}$ and $\htmlData{mx=1}{\theta \in \htmlData{mx=0}{\mathbb{R}}}$, if $\htmlData{mx=3}{\sum_{x} \htmlData{mx=2}{\htmlData{mx=0}{w_{1}}(\htmlData{mx=1}{x})}} \mathrel{\htmlData{mx=4}{=}} 1$ and $\htmlData{mx=11}{\forall d,\ \htmlData{mx=8}{\sum_{y} \htmlData{mx=7}{\htmlData{mx=2}{\htmlData{mx=0}{w_{2}}(\htmlData{mx=1}{y})} \cdot \htmlData{mx=6}{\htmlData{mx=3}{X}(\htmlData{mx=4}{d}, \htmlData{mx=5}{y})}}} \mathrel{\htmlData{mx=10}{=}} \htmlData{mx=9}{\theta}}$ (given $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{\Omega_{1}}}$ and $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{\Omega_{2}}}$), then