Highest Posterior Density (HPD) region of the marginals vs. of the joint distribution

Question

In a Bayesian context, to analyse the posterior distribution, one can define the Highest Posterior Density (HPD) region or interval as $$\{\theta; \pi(\theta \mid x) \geq k\} $$ in both unidimensional and multidimensional case (Robert - The Bayesian Choice, p 25).

In the unidimensional case, the HPD region is an interval or an union of intervals. In multidimensional case, the HPD region is more complicated and in general it is not possible to summarise it as simply as a union of intervals.

To keep things simple, a solution would be to compute HPD regions of the marginals. I suppose that this is what is most often done in the literature, for summary tables for example, is that right?

Second question: more generally, how do HPD regions of the marginals relate to the HPD region of the joint distribution?

Answer 1

Both your questions apply equally to calculating or summarizing highest-density-regions that come from other sources than Bayesian posteriors.

1. I can't say what is done most frequently in the literature, but yes, it's quite frequent to summarize a multidimensional distribution by its marginals.

2. Again, this is not specific to Bayesian posteriors. Unless you specify some additional structure, marginal distributions can be consistent with a wide range of joint distributions (see here or here). The same of course also holds for marginal vs. joint HDRs.

The tag wiki contains some additional information. You might be especially interested in the literature that is implemented in the hdrcde package for R.