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

Here $m_i=\beta$ for every breed, so $S=0$ andThe Bayesian model evidence increases towards its supremum as $\alpha \to \infty$. ThusFor a Pareto distribution, $\mathbb{P}(\theta >\beta (1+\epsilon ))=(1+\epsilon {)}^{-\alpha }\to 0$. 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.