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.

翻译：在贝叶斯统计学中，最高后验密度（HPD）区间常被用于描述后验分布的特性。作为一种置信区间（CI）估计方法，HPD区间具备两个主要理想性质：其一，它是达到指定覆盖概率的最短区间；其二，区间内任意点的密度值均高于区间外任意点。然而，该方法因缺乏变换不变性而时有争议。本文论证HPD区间实质上是频率学派轮廓似然比置信区间（LRCI）的自然类比。首先，我们系统阐述HPD区间的基本原理，以及似然比检验统计量及其通过逆推生成渐近正确置信区间的方法。核心研究结果表明：HPD区间具有与轮廓LRCI相似的理想性质，例如在单调函数变换下关于众数具有不变性。继而通过典型案例应用该结论，比较二项分布概率参数p的轮廓LRCI与贝塔分布密度函数的贝叶斯HPD区间——两者均为估计总体比例的常用方法。