Rare-event efficiency and work criteria

Let $(P_{\lambda },A_{\lambda }{)}_{\lambda \in \mathrm{\Lambda }}$ be a family of probability measures and measurable events with $p_{\lambda }=P_{\lambda }(A_{\lambda })>0$ and $p_{\lambda }\to 0$ along the stated rare-event limit. An estimator ${\widehat{p}}_{\lambda }$ is unbiased with finite variance when $\mathbb{E}[{\widehat{p}}_{\lambda }]=p_{\lambda }$ and $\mathrm{Var}({\widehat{p}}_{\lambda })<\mathrm{\infty }$. Its relative error is $\mathrm{RE}({\widehat{p}}_{\lambda })=\frac{\sqrt{\mathrm{Var}({\widehat{p}}_{\lambda })}}{p_{\lambda }}\mathrm{.}$ It has bounded relative error if these relative errors remain bounded for all sufficiently rare indices. For such an unbiased estimator family, it is logarithmically efficient if ${\widehat{p}}_{\lambda }\ge 0$, its second moment is finite and positive, and ${\lim }_{\lambda }\frac{\log \mathbb{E}[{\widehat{p}}_{\lambda }^2]}{\log (p_{\lambda }^2)}=1.$ Among nonnegative unbiased estimator families, logarithmic efficiency is a weaker asymptotic property than bounded relative error; the definitions impose no implication between an algorithm's efficiency and its proposal-fitting success.

For a specified cost model, let $W_{\lambda }$ be the expected work of one estimator replication. Record both its variance and its variance-work product $W_{\lambda }\mathrm{Var}({\widehat{p}}_{\lambda })$; for relative precision, record $W_{\lambda }\mathrm{Var}({\widehat{p}}_{\lambda })\mathrm{/}{p}_{\lambda }^2$. These quantities depend on the cost model. Comparing algorithms requires the same declared work convention.

For a single event $A$ under a probability measure $P$, put $p=P(A)>0$. If $P$ has density $f$ with respect to $\mu$, the zero-variance event proposal is the probability density $q^{\ast }(x)=\frac{{\mathbf{1}}_A(x)f(x)}{p}\mathrm{.}$ It is a formal benchmark because its normalizing constant is the unknown event probability $p$. Ordinary importance-sampling estimators and their weighted-second-moment condition are defined in Ordinary importance sampling and Ordinary importance sampling has finite variance and a CLT under a weighted second moment.