Independent-pilot cross-entropy proposal fitting and production

Let $(\mathsf{X},\mathcal{X},\mu )$ be a measure space, let $A\in \mathcal{X}$, and let the target probability measure $P$ have a pointwise finite measurable density $f:\mathsf{X}\to [0,\mathrm{\infty })$ with respect to $\mu$, so $P(B)={\int }_{B}fd\mu$. Let $\{q_{\theta }:\theta \in \mathrm{\Theta }\}$ be pointwise finite probability densities with respect to $\mu$, and write $Q_{\theta }(dx)=q_{\theta }(x)\mu (dx)$, where $\mathrm{\Theta }\subset {\mathbb{R}}^d$ is a finite nonempty set equipped with lexicographic order. For each $\theta \in \mathrm{\Theta }$, supply an exact admissibility checker $\mathcal{C}(A,f,q_{\theta })$ that returns admissible if and only if $P{\mathrm{\mid }}_A\ll Q_{\theta }$ and the contribution

$$w_{\theta }(x)=\{\begin{array}{ll}{\mathbf{1}}_A(x)f(x)\mathrm{/}{q}_{\theta }(x),& q_{\theta }(x)>0,\\ 0,& q_{\theta }(x)=0\end{array}$$

is integrable under $Q_{\theta }$, where $P{\mathrm{\mid }}_A(B):=P(B\cap A)$ for measurable $B$. Supply an initial parameter ${\theta }_0\in \mathrm{\Theta }$, a pilot size $m\ge 1$, and a production size $n\ge 1$.

1. Draw an IID pilot $Y_1,\dots ,Y_m\sim q_{{\theta }_0}$.
2. If no $Y_j$ lies in $A$, return pilot-no-hit.
3. For every $\theta \in \mathrm{\Theta }$, compute $s_{\theta }(y)=\{\begin{array}{ll}\log q_{\theta }(y),& y\in A,\\ 0,& y\mathrm{\notin }A,\end{array}S(\theta )={\sum }_{j=1}^m{s}_{\theta }(Y_j),$ with $\log 0=-\mathrm{\infty }$. Pointwise finiteness rules out $+\mathrm{\infty }$, and the piecewise score avoids the undefined product $0\cdot (-\mathrm{\infty })$. If all scores equal $-\mathrm{\infty }$, return no-finite-maximizer. Otherwise choose the lexicographically least maximizer $\widehat{\theta }$ among the finite scores and freeze it.
4. Evaluate $\mathcal{C}(A,f,q_{\widehat{\theta }})$. If it returns inadmissible, return inadmissible-production-proposal. Otherwise draw an IID production sample $X_1,\dots ,X_n\sim q_{\widehat{\theta }}$, independent of the pilot conditional on $\widehat{\theta }$, and return the ordinary importance-sampling estimate ${\widehat{p}}_{\mathrm{C}\mathrm{E}}=\frac{1}{n}{\sum }_{i=1}^n{w}_{\widehat{\theta }}(X_i)$ using the piecewise-defined contribution $w_{\widehat{\theta }}$ above.

The finite candidate set makes the score comparisons and tie rule deterministic. The supplied exact checker is an input oracle; the procedure does not infer support or integrability from a pilot fit. If its check fails, no production estimate or unbiasedness or finite-variance claim is returned.