 Bayesian rejection sampling for a segregating-site count

With proper prior $\pi (\theta )$ independent of the neutral Kingman's coalescent times, propose $(\theta ,T)$ from their joint prior and accept with probability $e^{-\theta L/2}(\theta L/2{)}^k/k!$. The accepted density is proportional to the prior times the observed-count likelihood, hence is the posterior given $S=k$. The acceptance rate is the prior predictive probability $P(S=k)$. For $k>0$, divide the acceptance function by its maximum $M_k=e^{-k}{k}^k/k!$, achieved at $\theta L/2=k$, to improve the rate to $P(S=k)/M_k$ without altering the accepted distribution. For $k=0$, $M_0=1$.