Solution

Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 45 / 3 / v / Solution

Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 45 / 4 / g / Solution

Watterson estimator

For $n\ge2$, the <probability generating function> of the segregating-site count conditional on mutation parameter $\theta$ is
$$
E[z^S\mid\theta]=\prod_{i=1}^{n-1}\frac{i}{i+\theta(1-z)}.
$$
This 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 $w^k e^{-(\theta+r+1)w}$ gives
$$
P(S=k\mid\theta)=(n-1)\theta^k\sum_{r=0}^{n-2}\frac{(-1)^r\binom{n-2}{r}}{(\theta+r+1)^{k+1}}.
$$
For $k=0$ 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.

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 Incoming links: Segregating-site likelihood under the neutral coalescent