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Use the joint prior density $g(\theta ,t)=\pi (\theta )q(t)$ as the proposal for rejection sampling. Since the Poisson probability $a_k(\theta ,t)$ lies in $[0,1]$, the following algorithm is valid:

The joint density of a proposed pair together with its acceptance event is $g(\theta ,t)a_k(\theta ,t)$. Its integral is $m_k$, so conditioning on acceptance gives precisely $g{a}_k/m_k$, the desired posterior density. Repeating independent proposal trials until each acceptance produces independent posterior observations. This is Bayesian rejection sampling for a segregating-site count.

An equivalent implementation simulates $S^{\ast }\sim \mathrm{Poisson}(\theta L/2)$ and accepts exactly when $S^{\ast }=k$. Its acceptance probability is the same Poisson mass. Propose from the joint prior and accept according to the exact observed-count likelihood. No approximation to the posterior distribution is involved in the accepted draws.

Use Bayesian rejection sampling for a segregating-site count. Independently propose $\theta \sim \pi$ and $T_j\sim \mathrm{Exp}({\lambda }_j)$ for $j=2,\dots ,n$, and calculate $L={\sum }_{j}j{T}_j$. Draw an independent uniform $U$ on $(0,1)$ and accept the whole proposed vector exactly whenRepeat after a rejection. The right side is a probability mass function value of a Poisson distribution and is therefore between zero and one. On acceptance, the probability density function of a proposed vector is its prior density multiplied by this acceptance probability, divided by the probability of acceptance. Part (c) shows that this is precisely $f(\theta ,T\mid S=k)$. Independent proposals and uniforms produce independent posterior draws. Equivalently, one could simulate a count from a Poisson distribution with mean $\theta L/2$ and retain the proposal when that count equals $k$; the uniform version avoids simulating that extra count.