= Beta highest posterior density endpoint characterization
{title2=$F_{A,B}(u)-F_{A,B}(l)=1-\alpha,\quad f_{A,B}(l)=f_{A,B}(u)$}
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 by
$$
F_{A,B}(u)-F_{A,B}(l)=1-\alpha,\qquad
l^{A-1}(1-l)^{B-1}=u^{A-1}(1-u)^{B-1}.
$$
The 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.
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