Suppose $X$ is distributed with a unimodal pdf $f(x)$ and let $Y = g(X)$ for some strictly monotone function $g$. Hence $g$ is invertible. Is there an analytically tractable relationship between the highest posterior interval of $X$ and the highest posterior interval of $Y$? For example, let $X$ be $\chi^2$ with $d$ degrees of freedom, and let $Y = \sqrt(X)$.
Comparison to transformations on equal-tailed intervals
Clearly, the answer would be yes if we were considering equal-tailed intervals, since if (as above) $P(X<q) \leq 1−α/2$, say, then $P(Y^2<q) = P(Y < \sqrt q) \leq 1−α/2$.
Therefore, if both $X$ and $Y$ are symmetric and unimodal, then the HPI case is equivalent to the equal-tailed case, and inverting $g$ provides a mapping between intervals without ever needing to evaluate anything about the distribution of $Y$. I would be interested in any less trivial relationship between the HPI intervals.