Past exam of the advanced topics in statistical machine learning course of the University of Oxford / 2025 / 1 / b / iii

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 $V=q(\theta )/r(\theta )$,Consequently,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 $\mathbb{E}{\hat{p}}_2\ge Z$.