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

Put $r(\theta )=p(\mathcal{D}\mid \theta )p(\theta )$ and $Z=\int r(\theta )d\theta$. The support condition for importance sampling is essential: $q(\theta )>0$ almost everywhere that $r(\theta )>0$. For one draw $\theta \sim q$, the importance weight $W=r(\theta )/q(\theta )$ satisfiesBy linearity of expectation,This proves that it is an unbiased estimator. If the proposal omits a region of positive target mass, the expected value instead misses that region's contribution; the unconditional unbiasedness claim needs the stated support condition for importance sampling.