 Solution

The accepted mutation parameters have marginal probability density functionEstimate this probability density function with a suitably smoothed histogram or a boundary-aware kernel density estimator from the accepted draws. On a parameter region where $\pi (\theta )>0$, divide the estimate by the known prior density and maximize:The unknown $Z_k$ is constant in $\theta$ and cancels. Maximizing the posterior density itself gives a posterior mode, not generally a maximum-likelihood estimate; the two coincide for a flat prior on a region containing the maximum. A prior excluding candidate maximizers cannot recover them through its posterior output. Boundary behavior also matters: for $k=0$, the likelihood is decreasing in $\theta$, so the unrestricted maximum is at $\theta =0$.

The exact likelihood provides a useful independent check. Put $W=L/2$. Its Laplace transform is ${\prod }_{i=1}^{n-1}i/(i+s)$, since $j{T}_j/2$ is exponential of rate $j-1$. The probability density functionhas that same transform: substitute $u=e^{-w}$ to obtain $(n-1){\int }_0^1{u}^s(1-u{)}^{n-2}du={\prod }_{i=1}^{n-1}i/(i+s)$. Thus, expanding its binomial factor and integrating each term,This is the segregating-site likelihood under the neutral coalescent. It can be maximized numerically to check the posterior-density estimate; the generating-function representation avoids cancellation in large alternating sums. For $n=2$, it reduces to ${\theta }^k/(1+\theta {)}^{k+1}$, whose maximum is at $\theta =k$ for $k>0$.

Alternatively, differentiating the integrated likelihood givesAn interior maximum therefore satisfies $\theta =k/E[W\mid S=k,\theta ]$. Conditional expectations of $W$ can be estimated by smoothing the accepted pairs $(\theta ,W)$ near each candidate parameter value, yielding another simulation-based score equation. The simple estimator $k/a_{n-1}$ is the unbiased Watterson estimator from part (b), and is not generally the likelihood maximizer.