The region covering the sample space for a given probability 1-α, should have the smallest possible volume.
Every point inside the region should have probability density at least as large as every point outside the region.
such regions are called highest density regions (HDR’s)
One of the most distinctive property of HDR’s is that of all possible regions of probability coverage, the HDR has the smallest region possible in the sample space. “Smallest” mean with respect to some simple measure such as the usual Lebesgue measure; in the one-dimensional continuous case that would be the shortest interval, and in the two-dimensional case that would be the smallest area of the surface. In Bayesian analysis a similar approach is called the highest posterior density region (HPD) and the posterior density is used as a measure.
HPD is one of the methods for defining a credible interval in Bayesian statistics.
A credible interval is an interval within which an unobserved parameter value falls with a particular probability. It is an interval in the domain of a posterior probability distribution or a predictive distribution. The generalisation to multivariate problems is the credible region.
Credible intervals are not unique on a posterior distribution. Methods for defining a suitable credible interval include:
- Choosing the narrowest interval, which for a unimodal distribution will involve choosing those values of highest probability density including the mode (the maximum a posteriori). This is sometimes called the highest posterior density interval (HPDI).
- Choosing the interval where the probability of being below the interval is as likely as being above it. This interval will include the median. This is sometimes called the equal-tailed interval.
- Assuming that the mean exists, choosing the interval for which the mean is the central point.