Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 45 / 4 / e / Solution

The acceptance probability is the prior average of the Poisson likelihood:Thus the expected number of proposals per accepted draw is $1/Z_k$. A numerical rate depends on the specified prior, sample size and observed count; it is not determined by $k$ alone.

For a more explicit one-dimensional expression, condition on $\theta$. The probability generating function, integrating each independent exponential epoch, isLet $p_k(\theta )$ be its coefficient of $z^k$. Then $Z_k=\int \pi (\theta )p_k(\theta )d\theta$. This identifies the acceptance rate with a fully specified marginal likelihood average.