Bayesian Statistics: An Introduction, 4th Edition

2.6 Highest density regions

2.6.1 Need for summaries of posterior information

In the case of our example on Ennerdale granophyre, all the information available after the experiment is contained in the posterior distribution. One of the best ways of conveying this information would be to sketch the posterior density (though this procedure is more difficult in cases where we have several parameters to estimate, so that θ is multi-dimensional). It is less trouble to the statistician to say simply that

although those without experience may need tables to appreciate what this assertion means.

Sometimes the probability that the parameter lies in a particular interval may be of interest. Thus, there might be geological reasons why, in the above example, we wanted to know the chance that the rocks were less than 400 million years old. If this is the case, the probability required is easily found by use of tables of the normal distribution. More commonly, there are no limits of any special interest, but it seems reasonable to specify an interval in which ‘most of the distribution’ lies. It would appear sensible to look for an interval which is such that the density at any point inside it is greater than the density at any point outside it, and it would also appear sensible to seek (for a given probability level) an interval that is as short as possible (in several dimensions, this means ...