Bucketed tilt proposal along a direction u: bucket b w.p. ∝ W_b e^{a z_b}, then j ∝ π₀ inside ⇒ P(j) ∝ π₀(j) e^{a z_{b(j)}}

For all $\htmlData{mx=1}{n, B \in \htmlData{mx=0}{\mathbb{N}}}$, $\htmlData{mx=7}{\mathit{bk} : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}} \mathrel{\htmlData{mx=6}{\to}} \htmlData{mx=5}{\htmlData{mx=3}{\operatorname{Fin}}\,\htmlData{mx=4}{B}}}$, $\htmlData{mx=5}{\pi_{0} : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$, $\htmlData{mx=5}{\mathit{zc} : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{B}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$, $\htmlData{mx=1}{a \in \htmlData{mx=0}{\mathbb{R}}}$ and $\htmlData{mx=3}{j \in \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}}}$, if $\htmlData{mx=4}{\forall i,\ 0 \mathrel{\htmlData{mx=3}{<}} \htmlData{mx=2}{\htmlData{mx=0}{\pi_{0}}(\htmlData{mx=1}{i})}}$, then