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

Assume $n\ge 2$, $k\ge 0$, a proper prior density $\pi$ supported on nonnegative mutation rates, and independence of the mutation-rate prior from the neutral genealogy. PutBy Bayes' theorem, the joint posterior isThe normalizing constant is the prior predictive probability $P(S=k)$ and must be positive. For $k=0$, interpret the factor $(\theta L/2{)}^0$ as 1 even at zero. If $n=1$, there are no segregating sites: $S=0$ surely, so that case supplies no information about $\theta$.