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Let $t=(t_2,\dots ,t_n)$, with all $t_j>0$, and put $L(t)={\sum }_{j=2}^{n}j{t}_j$. The independent exponential epoch times have joint densityTake a proper prior distribution density $\pi (\theta )$ on $\theta \ge 0$, independent of the neutral genealogy. Independence is appropriate because the specified neutral Kingman's coalescent does not depend on the mutation parameter. The conditional observed-count likelihood function isIts prior predictive probability, or Bayesian model evidence, isAssuming $m_k>0$, Bayes' theorem gives the joint posterior of mutation rate and coalescent times:It is zero outside the prior support and positive-time region. For $k=0$, use the usual convention $0^0=1$ in the Poisson mass. An improper prior cannot be sampled as the proposal in the following algorithm, even in cases where its formal posterior can be normalized.