= Joint posterior of mutation rate and coalescent times
For a proper <prior distribution> $\pi(\theta)$ independent of coalescent times $T$, let $q(T)=\prod_{j=2}^n\lambda_j e^{-\lambda_jT_j}$ with $\lambda_j=\binom j2$ and $L=\sum jT_j$. Observing $S=k$ under an <infinite sites mutation model> gives <posterior density> $\pi(\theta)q(T)e^{-\theta L/2}(\theta L/2)^k/(k!m_k)$, where $m_k$ is the prior predictive <probability> of $S=k$. Multiplication by the conditional Poisson mass and normalization is an application of <Bayes' theorem>.
// Target: statistical-inference.bigb
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