Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 34 / 1 / f / Solution

The Type II maximum likelihood estimator maximizes the Bayesian model evidence after integrating out the breed parameters:For positive sample sizes define $S={\sum }_i\log (m_i/\beta )\ge 0$. Up to an additive constant, the log-likelihood isDifferentiation gives the interior score equationAlso ${\ell }^{\prime \prime }(\alpha )=-I/{\alpha }^2+{\sum }_i(\alpha +n_i{)}^{-2}<0$. If $S>0$, the derivative decreases from infinity to $-S$, so a unique finite maximum exists. When all sample sizes equal $n$, the equation becomes $In/[\alpha (\alpha +n)]=S$, givingFor $S=0$ no finite maximum exists. Plugging this hyperparameter estimate into the breed posterior distributions is an Empirical Bayes method.