= Solution
Here $m_i=\beta$ for every breed, so $S=0$ and
$$
\ell'(\alpha)=\sum_i\frac{n_i}{\alpha(\alpha+n_i)}>0.
$$
The <Bayesian model evidence> increases towards its supremum as $\alpha\to\infty$. Thus
$$
\boxed{\widehat\alpha=+\infty\quad\text{(no finite maximizer).}}
$$
For a <Pareto distribution>, $\mathbb P(\theta>\beta(1+\varepsilon))=(1+\varepsilon)^{-\alpha}\to0$. The limiting empirical prior, and each corresponding <posterior distribution>, collapses onto $\beta$.
The boundary result is coherent within the assumed model: the observations favour the smallest allowed upper limits. Nevertheless, reporting exact concentration on a prespecified bound is overconfident for finite data and ignores <hyperparameter> uncertainty. A proper hyperprior on $\alpha$, sensitivity analysis for $\beta$, or a scientifically justified restriction on concentration avoids treating this limit as certain biological knowledge.
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