To complement: the HPD has the nice property that any point **within** the interval has a higher density than any other point outside. Thus, HPD interval is **the** collection of most likely values of the parameters.

To answer your more general question, I believe there is no **universal** requirement for _good intervals_. It's up to the specific case, and your objectives. If the parameter of interest is greater than zero, you might be interested in the leftmost interval as well.

In the end, intervals are only summaries of your posterior.

It is perhaps also worth noting that HPD does not always result in an interval estimate, When the posterior density is multimodal, then HPD can yield non-interval set estimators, in contrast to quantile limits. As such, the HPD is a truer summary of your posterior. Gelman et al's Bayesian Data Analysis, 2nd Edition, has a nice short discussion at Section 2.3.

Can I add another question? Are HPD and HDI (high density interval) both synonyms that are just used in different disciplines? I  have learned about HDI and never heard about HPD.

To the best of my knowledge, HPD and HDI are synonyms.

HPD seems more specific, as it means a HDI for a posterior density.

While HDI can be for a prior, a predictive or in general any density function.

In any dimension if one draws the level curve in densities one obtains the shortest set with a specific volume p. The level curve defines the credible set (HDS) with credibility p.

It is perhaps worth remembering that the posterior and its summaries are also conditional on the model as well as the data.  My epsilon's worth.

HPD is utmost useful for investigating of faint sources with low count rates (with significant background signal)