= Solution
The <Type II maximum likelihood estimator> maximizes the <Bayesian model evidence> after integrating out the breed parameters:
$$
\widehat\alpha=\arg\max_{\alpha>0}
p(\mathbf y_1,\ldots,\mathbf y_I\mid\alpha,\beta).
$$
For positive sample sizes define $S=\sum_i\log(m_i/\beta)\ge0$. Up to an additive constant, the <log-likelihood> is
$$
\ell(\alpha)=I\log\alpha-\sum_i\log(\alpha+n_i)-\alpha S.
$$
Differentiation gives the interior score equation
$$
\boxed{\frac I{\widehat\alpha}
-\sum_i\frac1{\widehat\alpha+n_i}
=\sum_i\log\max(1,M_i/\beta).}
$$
Also $\ell''(\alpha)=-I/\alpha^2+\sum_i(\alpha+n_i)^{-2}<0$. 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 $In/[\alpha(\alpha+n)]=S$, giving
$$
\boxed{S\widehat\alpha^2+Sn\widehat\alpha-In=0,\qquad
\widehat\alpha=\frac{\sqrt{n^2+4In/S}-n}{2}\quad(S>0).}
$$
For $S=0$ no finite maximum exists. Plugging this <hyperparameter> estimate into the breed <posterior distributions> is an <Empirical Bayes method>.
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