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The joint prior predictive distribution is the Bayesian model evidence, obtained by integrating over the shared parameter:It is zero otherwise. This uniform-Pareto model evidence is a joint density, not the product of separately marginalized observation densities: mixing over the common parameter induces dependence.

Use the intended hierarchical Bayesian model: the breed parameters are conditionally independent given $\alpha ,\beta$, and observations are independent given their breed parameters. Put $m_i=\max (\beta ,M_i)$. The uniform-Pareto model evidence factorizes over breeds:Identical marginal prior distributions alone would not determine this product; conditional independence is the additional assumption.