Independent fixed-level splitting

Input the nested levels and exact conditional samplers for ${\pi }_{\mathrm{\ell }-1}$ from Nested rare-event levels and conditional laws, a particle count $N\ge 1$, and a declared per-draw work cost $c_{\mathrm{\ell }}>0$ for each level. If any conditional sampler is unavailable, return conditional-sampler-unavailable.

For each level $\mathrm{\ell }=1,\dots ,L$, independently of every other level, draw $X_{\mathrm{\ell },1},\dots ,X_{\mathrm{\ell },N}$ IID from ${\pi }_{\mathrm{\ell }-1}$. Set ${\widehat{p}}_{\mathrm{\ell }}=N^{-1}{\sum }_{i=1}^N{\mathbf{1}}_{A_{\mathrm{\ell }}}(X_{\mathrm{\ell },i}),\widehat{p}={\prod }_{\mathrm{\ell }=1}^L{\widehat{p}}_{\mathrm{\ell }}\mathrm{.}$ Return $\widehat{p}$, the level counts, total draw work $W=N{\sum }_{\mathrm{\ell }=1}^L{c}_{\mathrm{\ell }}$, and status success if all level counts are positive or zero-level-estimate otherwise. The zero estimate is not a proof that $P(A_L)=0$.

The procedure is an ideal independent-level baseline. It does not assert that a branching or Markov mutation method produces independent exact conditional draws.