Self-normalized importance sampling

Let $(\mathsf{X},\mathcal{X},\mu )$ be a measure space. The unnormalized target density is a measurable $\tilde{p}\ge 0$ with $0<Z_p=\int \tilde{p}d\mu <\mathrm{\infty }$ and target $\pi (dx)=\tilde{p}(x)\mu (dx)\mathrm{/}{Z}_p$. Let $q\ge 0$ be a measurable proposal probability density against $\mu$, let $Q(dx)=q(x)\mu (dx)$, and let $f$ be a measurable real-valued integrand. The input $X_1,\dots ,X_n$ is an IID sample from $Q$ in the sense of Identical distribution and IID families, with $n\ge 1$. For target interpretation require $q>0$ $\pi$-almost everywhere. Write $W_i:=\{\begin{array}{ll}\tilde{p}(X_i)\mathrm{/}q(X_i),& q(X_i)>0,\\ 0,& q(X_i)=0.\end{array}$ This is a nonnegative unnormalized importance weight; the value on $q=0$ is immaterial under $Q$ and under the stated target support condition.

The reciprocal map that is $1\mathrm{/}s$ for $s>0$ and $0$ for $s\le 0$ is Borel. Its composition with $q$, multiplied by $\tilde{p}$, makes $W_i$ measurable by Composition with a Borel measurable outer map preserves measurability and Arithmetic and lattice operations preserve measurability whenever they are defined.

Procedure and output: Compute $S_n={\sum }_{i=1}^n{W}_i$. If $S_n=0$, return the status no estimate: zero observed target weight. If $S_n>0$, return ${\widehat{\theta }}_{\mathrm{S}\mathrm{N}}:=\frac{{\sum }_{i=1}^n{W}_{i}f(X_i)}{S_n}\mathrm{.}$ For consistency require ${\mathbb{E}}_Q{W}_1=Z_p\in (0,\mathrm{\infty })$ and ${\mathbb{E}}_Q[W_1\mathrm{\mid }f(X_1)\mathrm{\mid }]<\mathrm{\infty }$. A ratio CLT and its Monte Carlo standard error require finite second moments of $W_1$ and $W_{1}f(X_1)$.