= Solution
The joint <prior predictive distribution> is the <Bayesian model evidence>, obtained by integrating over the shared parameter:
$$
\begin{aligned}
p(\mathbf y\mid\alpha,\beta)
&=\alpha\beta^\alpha\int_{\max(\beta,M)}^\infty
\theta^{-(\alpha+n+1)}\,d\theta\\
&=\boxed{\frac{\alpha\beta^\alpha}
{(\alpha+n)\max(\beta,M)^{\alpha+n}}},
\qquad y_j>0.
\end{aligned}
$$
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.
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