Segregating-site likelihood under the neutral coalescent

For $n\ge 2$, the probability generating function of the segregating-site count conditional on mutation parameter $\theta$ isThis follows by integrating the Poisson generating function over the independent exponential epoch lengths. With $W=L/2$, its density is $(n-1)e^{-w}(1-e^{-w}{)}^{n-2}$. Expanding the binomial factor and integrating $w^k{e}^{-(\theta +r+1)w}$ givesFor $k=0$ use ${\theta }^0=1$, including at $\theta =0$. A posterior density divided by a positive prior density is proportional to this likelihood, allowing posterior simulation output to estimate its maximizer.