Shortest posterior intervals

[andrewgelman]

You can take the entire posterior distribution, or you can use the posterior distribution to make decisions, integrating a utility function over the posterior. But there’s no decision question for which the posterior 90% interval (of any sort) is the answer. This is not to say that posterior intervals are useless. It’s just to say that they are summaries to aid understanding, in the same way that graphs are summaries to aid understanding. Some graphs can be better than others, but it will depend on context. Similarly, there is no right or wrong way to define a posterior interval; what to do will depend on your communication goals.