Pareto highest density interval ($[b,b{\alpha }^{-1/a}]$)

For the Pareto distribution of shape $a>0$ and lower bound $b>0$, the density $a{b}^a{x}^{-a-1}$ is strictly decreasing on $[b,\infty )$. Its highest-density interval of probability $1-\alpha$, $0<\alpha <1$, must begin at $b$. The tail probability $\mathbb{P}(X>u)=(b/u{)}^a$ makes the upper endpoint $u=b{\alpha }^{-1/a}$. If this is a posterior distribution, the result is its highest posterior density interval. In particular, Uniform-Pareto conjugacy supplies posterior shape $a+n$ and lower bound $\max (b,{\max }_i{y}_i)$.