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A highest posterior density interval is a credible interval whose included points have at least as high posterior density as excluded points, apart from boundary and null-set choices. It is a highest density region of prescribed posterior probability that happens to be an interval. For a continuous unimodal posterior with two interior endpoints, the endpoint densities agree. A monotone posterior on a half-line instead starts at a support boundary, as in a Pareto highest density interval. For a multimodal posterior distribution, the highest-density region can be disconnected and cannot always be represented by a single interval. Density-level regions use a specified parameter coordinate and reference measure; they are not generally preserved by nonlinear reparametrization.

Let $y_{(n)}={\max }_i{y}_i$ and $c=\max (\beta ,y_{(n)})$. For the observed positive data, the uniform distribution likelihood function isMultiplying by the Pareto distribution prior density gives the posterior density kernel ${\theta }^{-(\alpha +n+1)}{1}_{\{\theta >c\}}$. Sincethe normalized posterior distribution isThis is uniform-Pareto conjugacy. The posterior density is strictly decreasing over its support, so the highest posterior density interval starts at $c$, where the admissible density is largest in the limiting sense. Its upper endpoint $u$ satisfiesHence a 95% highest posterior density interval isInclusion of endpoints does not change its posterior probability. Every point inside has higher posterior density than every admissible point to its right, and points below $c$ have density zero. This is the Pareto highest density interval; equal endpoint densities are unnecessary because one endpoint is the support boundary.

Let $M=\max \{b,y_1,\dots ,y_n\}$. The continuous uniform distribution likelihood contributes ${\varphi }^{-n}{1}_{\{\varphi \ge {\max }_i{y}_i\}}$, while the Pareto distribution prior contributes ${\varphi }^{-a-1}{1}_{\{\varphi \ge b\}}$. Their product proves Uniform-Pareto conjugacy:Thus the posterior distribution is Pareto with shape $a+n$ and lower bound $M$. Its density is strictly decreasing on its support, so the highest posterior density interval begins at $M$. Since the upper-tail probability is $(M/u{)}^{a+n}$, the required upper endpoint satisfies $(M/u{)}^{a+n}=0.05$. The Pareto highest density interval is therefore