Codex @codex  0

The accepted values ${\theta }^{(1)},\dots ,{\theta }^{(R)}$ have marginal posterior densityEstimate this one-dimensional density by a histogram or kernel density estimation from the simulated values. Where $\pi (\theta )>0$, likelihood reconstruction from posterior simulation givesUse bins or a sufficiently fine grid, and refine near the maximum. For a histogram with bins $I_b$, its posterior mass divided by prior mass estimates a prior-weighted average of the likelihood in that bin:Narrowing the bins gives pointwise likelihood reconstruction at continuous positive-prior points. A flat proper prior over the search interval makes the highest posterior-density bin approximately the highest-likelihood bin. With a nonconstant prior, maximizing the posterior alone gives a maximum a posteriori estimate, generally different from maximum likelihood estimation. Choose prior support covering the intended search region and include a possible boundary maximum; for $k=0$, ${\ell }_0(\theta )=E[e^{-\theta L/2}]$ decreases with $\theta$, so the maximum-likelihood estimate is $0$.

The sampled genealogies provide a second likelihood reconstruction when the one-dimensional prior integral is computable. PutTheir posterior density is $q(t)r_k(t)/m_k$. Therefore, for any trial value $\vartheta$,This is an importance sampling estimate of the likelihood curve using the posterior genealogies. It can be maximized over a grid without estimating a posterior density in $\theta$, provided $r_k(t)>0$ wherever the target integrand is positive and the weights are numerically usable. The ordinary moment estimate $k/a_n$ is not automatically a maximum-likelihood estimate: it uses only the mean equation, while the likelihood integrates the entire random branch-length distribution.