= Solution
For <reciprocal importance sampling>, use the same $r$ and $Z$ but draw from the <posterior density> $\pi=r/Z$. Here the support requirement is reversed: the normalized density $q$ must vanish almost everywhere that $r=0$. Thus $q$ may be confined to a subset of the <posterior distribution>'s support. For $V=q(\theta)/r(\theta)$,
$$
\mathbb E_\pi V=\int_{\{r>0\}}\frac{q(\theta)}{r(\theta)}\frac{r(\theta)}Z\,d\theta=\frac1Z\int q(\theta)\,d\theta=\frac1Z.
$$
Consequently,
$$
\boxed{\mathbb E\bigl[\widehat p_2(\mathcal D)^{-1}\bigr]=p(\mathcal D)^{-1}}.
$$
It is the estimate of the reciprocal that is an <unbiased estimator>. Taking its reciprocal does not generally produce an <unbiased estimator> of $Z$: inversion is nonlinear. In fact, when the average is strictly positive and the relevant <expectation> exists, <Jensen's inequality> gives $\mathbb E\widehat p_2\geq Z$.
Back to article page
