Harmonic mean estimator of Bayesian model evidence

Let $\pi$ be a proper prior density, let $L>0$ for prior-almost every parameter, and let $Z=\int L\pi \in (0,\infty )$. For posterior draws ${\theta }_i$, the harmonic mean estimator isThe identity ${\mathbb{E}}_{\pi (\cdot \mid y)}[1/L]=1/Z$ follows by cancelling $L$ in the posterior density. The variance of $1/L$ is often infinite, making the estimate unstable even when an ergodic average is consistent.^[ref]

The support qualification matters. If $a={\int }_{\{L>0\}}\pi <1$, cancellation instead gives $a/Z$. For a uniform prior on $[0,1]$ and $L(\theta )=1_{[0,1/2]}(\theta )$, the evidence is $1/2$, but every posterior likelihood value is one and the displayed estimator is always one. An improper prior also leaves the absolute evidence normalization undefined.