Highest Posterior Density Intervals As Analogues to Profile Likelihood Ratio Confidence Intervals for Modes of Unimodal Distributions

In Bayesian statistics, the highest posterior density (HPD) interval is often used to describe properties of a posterior distribution. As a method for estimating confidence intervals (CIs), the HPD has two main desirable properties. Firstly, it is the shortest interval to have a specified coverage probability. Secondly, every point inside the HPD interval has a density greater than every point outside the interval. However, it is sometimes criticized for being transformation invariant. We make the case that the HPD interval is a natural analog to the frequentist profile likelihood ratio confidence interval (LRCI). First we provide background on the HPD interval as well as the Likelihood Ratio Test statistic and its inversion to generate asymptotically-correct CIs. Our main result is to show that the HPD interval has similar desirable properties as the profile LRCI, such as transformation invariance with respect to the mode for monotonic functions. We then discuss an application of the main result, an example case which compares the profile LRCI for the binomial probability parameter p with the Bayesian HPD interval for the beta distribution density function, both of which are used to estimate population proportions.