 Solution

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.