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In the shape–minimum parameterization, the density is $\alpha {\beta }^{\alpha }{x}^{-\alpha -1}{1}_{\{x>\beta \}}$, with $\alpha ,\beta >0$. The survival function is $(\beta /x{)}^{\alpha }$ for $x\ge \beta$. Large shape concentrates the distribution near its lower bound. This convention is useful for uniform-Pareto conjugacy.

Multiplying the likelihood function by the Pareto distribution prior givesThe integral of this kernel is $m^{-(\alpha +n)}/(\alpha +n)$, so the normalized posterior distribution isThus it is $\mathrm{Pareto}(\alpha +n,m)$. This proves uniform-Pareto conjugacy: applying Bayes theorem preserves the family of prior distributions.