Ordinary importance sampling

Let $(\mathsf{X},\mathcal{X},\mu )$ be a measure space. An unnormalized target density is a measurable function $\tilde{p}:\mathsf{X}\to [0,\mathrm{\infty })$ with $Z_p:={\int }_{\mathsf{X}}\tilde{p}d\mu \in (0,\mathrm{\infty })\mathrm{.}$ The target probability measure is $\pi (dx)=\tilde{p}(x)\mu (dx)\mathrm{/}{Z}_p$. A proposal density is a measurable $q:\mathsf{X}\to [0,\mathrm{\infty })$ with $\int qd\mu =1$; write $Q(dx)=q(x)\mu (dx)$. The input integrand $f:\mathsf{X}\to \mathbb{R}$ is measurable, and the input sample $X_1,\dots ,X_n$ is IID with law $Q$, in the sense of Identical distribution and IID families. The normalizing constant $Z_p$ is known for this algorithm.

Admissible conditions for an unbiased estimate: $n\ge 1$, $q>0$ $\mu$-almost everywhere on $\{x:f(x)\tilde{p}(x)\mathrm{\ne }0\}$, and the random variable $Y:=\{\begin{array}{ll}f(X_1)\tilde{p}(X_1)\mathrm{/}(Z_{p}q(X_1)),& q(X_1)>0,\\ 0,& q(X_1)=0\end{array}$ is integrable under $Q$. The support condition makes the value on $q=0$ immaterial to the target integral.

Well-defined weights: The reciprocal map $r(s)=1\mathrm{/}s$ for $s>0$ and $r(s)=0$ for $s\le 0$ is Borel. Its composition with $q$, multiplied by the measurable inputs, makes the piecewise weight and weighted integrand measurable (Composition with a Borel measurable outer map preserves measurability, Arithmetic and lattice operations preserve measurability whenever they are defined).

Procedure and output: For each supplied draw $X_i$, set $Y_i=f(X_i)\tilde{p}(X_i)\mathrm{/}(Z_{p}q(X_i))$ when $q(X_i)>0$ and set $Y_i=0$ otherwise. Return ${\widehat{\mu }}_{\mathrm{I}\mathrm{S}}:=\frac{1}{n}{\sum }_{i=1}^n{Y}_i\mathrm{.}$ Separate variance and central limit claims require the corresponding weighted second-moment hypothesis.