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

Average the conditional probability in (i) over $Y\sim q=\tilde{q}/Z_q$:This is exactly the acceptance mass for an unnormalized rejection envelope. In normalized form the envelope is $p\le (M{Z}_q/Z_p)q$, giving the same acceptance probability. Here $M$ bounds the ratio of the unnormalized functions, so its value need not equal the envelope constant for the normalized probability density functions.

The squeeze changes the number of expensive target evaluations, while preserving the per-proposal acceptance probability. The expected number of proposals per accepted draw is $M{Z}_q/Z_p$.