Reciprocal importance sampling ($\widehat{Z^{-1}}$)

Let $r$ be an unnormalized density, $Z=\int r$ with $0<Z<\infty$, and $\pi =r/Z$. Given independent draws ${\theta }_t\sim \pi$ and an evaluable normalized density $q$ supported where $r>0$, reciprocal importance sampling estimates $Z^{-1}$ by $T^{-1}{\sum }_{t}q({\theta }_t)/r({\theta }_t)$. Its expectation is $Z^{-1}\int q=Z^{-1}$ and its variance is $(T{Z}^2{)}^{-1}(\int q^2/\pi -1)$. Broad tails of $q$ relative to $\pi$ can make this infinite; reversing the importance sampling ratio reverses the dangerous tail mismatch. Taking the reciprocal of the estimate is generally biased for $Z$.