= Solution
{c}
For a specified <posterior probability> $1-\alpha$, an interval $I=(\theta_1,\theta_2)$ is a <highest posterior density interval> if
$$
\int_{\theta_1}^{\theta_2}\pi(\theta\mid x)\,d\theta=1-\alpha
$$
and 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.
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