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For a beta distribution with shape parameters $A,B>1$, the probability density function vanishes at zero and one and is strictly increasing up to $m=(A-1)/(A+B-2)$ and strictly decreasing afterwards. This follows because the derivative of its logarithm is $(A-1)/t-(B-1)/(1-t)$, whose sign changes at $m$.

For each $0<\alpha <1$, the highest posterior density interval when this beta distribution is a posterior distribution has unique endpoints $0<l<m<u<1$ determined byThe second equation gives equal endpoint posterior densities. For every density threshold strictly between zero and the modal height there is precisely one crossing on each side. The enclosed probability is continuous and strictly decreasing from one to zero as the threshold increases, proving existence, uniqueness and the highest posterior density interval property. The restrictions $A,B>1$ are essential to this two-interior-endpoint characterization; monotone cases use a support boundary instead.

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.

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 $\max (b,{\max }_i{y}_i)$.

For a specified posterior probability $1-\alpha$, an interval $I=({\theta }_1,{\theta }_2)$ is a highest posterior density interval ifand every included point has posterior density at least as large as every excluded point, apart from boundary and null-set choices. Equivalently, $I$ is a level set $\{\theta :\pi (\theta \mid x)>k\}$, with possible choices on its boundary, whose posterior probability is $1-\alpha$.

This selects the most densely supported values of the statistical parameter. When the relevant level set is one interval and has no plateau of positive probability, it is the shortest credible interval of that posterior probability. With several separated peaks, the corresponding highest density region can instead be disconnected, so an interval need not exist. The definition is in the stated parameter coordinate: posterior density ordering can change under a nonlinear change of parameter.

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.

Put $M_0=\alpha +x_0\beta$. As a linear combination of independent normal random variables under the posterior distribution,The shortest 95% credible interval isHere $\Phi$ is the standard normal distribution cumulative distribution function. To justify shortest length, consider all intervals of a fixed length $d$. If $f$ is this normal distribution posterior density, the interval probability ${\int }_a^{a+d}f(t)dt$ has derivative $f(a+d)-f(a)$ with respect to $a$. The normal distribution is symmetric and its density strictly decreases with distance from its mean, so this probability is uniquely maximized when the midpoint is $m_0$. The centered interval can therefore attain 95% posterior probability with the least length. It is both an equal-tailed credible interval and a highest posterior density interval.

The target is the experiment's mean at $x_0$. A posterior predictive distribution for a new noisy observation would instead have variance $v_0+{\sigma }^2$; that extra observation variance does not belong in the requested interval.

An equal-tailed credible interval for a continuous posterior distribution has probability $\alpha /2$ below its lower endpoint and $\alpha /2$ above its upper endpoint. If $F$ is the posterior cumulative distribution function, the endpoints are $F^{-1}(\alpha /2)$ and $F^{-1}(1-\alpha /2)$.

A highest posterior density interval is a credible interval consisting of points of highest posterior density: every included point has density at least as high as every excluded point, apart from null-set and boundary choices. It is a posterior highest density region when that region is an interval. For a continuous unimodal density with two interior endpoints, their densities agree. A mode at a support boundary can instead force one endpoint to that boundary; a multimodal posterior can have a disconnected highest-density region rather than one interval. Equal tails and highest density need not give the same interval.

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