Past exam of the simulation and statistical programming course of the University of Oxford / 2023 / 2 / b / i

Fix a proposal $y$ with $\tilde{q}(y)>0$, and abbreviate $B=M\tilde{q}(y)$, $H=h(y)$ and $P=\tilde{p}(y)$. The envelope says $0\le H\le P\le B$. Assuming fresh independent uniform distribution draws at both stages, the law of total probability givesThe second term includes both reaching the residual test and passing it. If $H=B$, the envelope forces $P=B$ and the first test accepts with probability one; the second-stage $0/0$ ratio is never evaluated. This is squeeze rejection sampling.