= Reciprocal importance sampling
{title2=$\widehat{Z^{-1}}$}
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 $T^{-1}\sum_tq(\theta_t)/r(\theta_t)$. Its <expectation> is $Z^{-1}\int q=Z^{-1}$ and its <variance> is $(TZ^2)^{-1}(\int q^2/\pi-1)$. 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 $Z$.
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