What is the interval that relates to the mean as the equal tailed interval relates to the median and the highest density interval relates to the mode? When summarizing a one dimensional continuous distribution (e.g. a posterior distribution) it is common to use either an equal tailed interval (aka quantile-based) or a highest density interval. The 95% equal tailed interval conceptually corresponds to the median in that when the coverage of the interval $\rightarrow 0\%$ the interval converges to the median. In the same way the highest density interval corresponds to the mode as the mode is the point of the highest density. But another popular point summary of a continuous distribution is the mean and my question is:
What is the interval that corresponds to the mean?
That is:


*

*What is that interval called?

*How is it defined/calculated?


If someone has a comment on the conundrum why it is the case that the mean is a really popular way of summarizing a distribution while the corresponding interval is not as popular (as is my impression).
 A: I don't know if the interval centered on the mean has a special name, but I can think of more than one way that one could define an interval centered on the mean in the sense that the interval will converge to the mean as the width of the inverval goes to zero. The most simple would perhaps be the interval
$$C = (\mu-k, \mu+k),$$ 
where $\mu$ is the mean and $k$ is the smallest value for which
$P(x\in C)\ge 1-\alpha$ holds.
More general, one could define the mean centered interval as
$$C = (\mu-a, \mu+b),$$ 
where $a$ and $b$ are positive and chosen by some procedure such that again the interval has the desired credibility.
However, I think there are good reasons for mainly considering the highest density region and the equal tailed interval. To quote Christian Robert (The Bayesian Choice):

To consider only HPD [highest posterior density] regions is motivated by the fact that they minimize the volume among $\alpha$-credible regions and, therefore, can be envisioned as optimal solutions in a decision setting.

Highest density regions are not necessarily connected intervals, which can happen for example when the distribution is multimodal. On this, Robert writes:

When (...) the confidence region is not connected (...), the usual
  solution is to replace the HPD $\alpha$-credible region by an interval with equal tails.

