= Solution
An <equal-tailed credible interval> allocates equal posterior <probability> to its two excluded tails:
$$
\Pr(\theta<L\mid x)=\Pr(\theta>U\mid x)=\alpha/2.
$$
For a continuous strictly increasing posterior <cumulative distribution function> $F$, the endpoints are $F^{-1}(\alpha/2)$ and $F^{-1}(1-\alpha/2)$.
A <highest posterior density interval> instead includes the parameter values of greatest posterior density: there is a cutoff $c$ such that the retained region is a superlevel set of $\pi(\theta\mid x)$, of posterior mass $1-\alpha$. Every interior retained value has density at least as large as every excluded value, apart from possible ties. For a unimodal continuous density this is an interval; when both endpoints lie inside the support they have equal posterior density. For a multimodal <posterior distribution>, a highest-density region may be a union of intervals, rather than one interval. \b[Equal tail probabilities and highest density are different criteria.]
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