 Beta highest posterior density endpoint characterization

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.