= Solution
\b[No.] A permitted counterexample is $a=1$ with all observed counts zero. Then $A=1$ and the <posterior distribution> is an <exponential distribution> with rate $B=b+n$, whose density decreases strictly from the left boundary. Its <equal-tailed credible interval> is
$$
[L,U]=\left[-\frac{\log(1-\alpha/2)}B,\ -\frac{\log(\alpha/2)}B\right].
$$
Here $L>0$. This excludes values just above zero whose posterior density is larger than that of retained values near $U$, violating the highest-density criterion. The actual <highest posterior density interval> starts at the support boundary and solves $1-e^{-Bh}=1-\alpha$:
$$
\boxed{[0,h],\qquad h=-\frac{\log\alpha}{B}.}
$$
Writing zero as the endpoint causes no problem because a single point has zero posterior <probability>. Thus even within the gamma family the two intervals need not agree. More generally, for an interior unimodal <gamma distribution>, highest-density endpoints have equal densities, whereas equal-tailed endpoints need not.
\Image[/past-exam-of-the-statistics-course-of-the-university-of-oxford/2014-credible-intervals.png]
{title=Equal-tailed and highest-density intervals for an exponential posterior}
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