Uniform-Pareto model evidence ($p(y\mid \alpha ,\beta )=\alpha {\beta }^{\alpha }/[(\alpha +n)m^{\alpha +n}]$)

For positive observations and $m=\max (\beta ,{\max }_j{y}_j)$, the Bayesian model evidence follows by integrating the uniform distribution likelihood against a Pareto distribution prior. For independent groups with shared fixed hyperparameters, these evidences multiply. The resulting hyperparameter log likelihood is $I\log \alpha -{\sum }_i\log (\alpha +n_i)-\alpha {\sum }_i\log (m_i/\beta )$ up to a constant.