= 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 $k=0$, $M_0=1$.
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