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, the HPD interval is sometimes criticized for being transformation invariant. We make the case that under certain conditions the HPD interval is a natural analog to the frequentist profile likelihood ratio confidence interval (LRCI). Our main result is to derive a proof showing that under specified conditions, the HPD interval with respect to the density mode is transformation invariant for monotonic functions in a manner which is similar to a profile LRCI.

翻译：在贝叶斯统计中，最高后验密度（HPD）区间常用于描述后验分布的性质。作为估计置信区间的一种方法，HPD具有两个主要优点：首先，它是具有指定覆盖概率的最短区间；其次，HPD区间内部的每个点的密度值均大于区间外所有点的密度值。然而，HPD区间有时因缺乏变换不变性而受到批评。我们论证表明，在特定条件下，HPD区间是频率学派剖面似然比置信区间（LRCI）的自然类比。本文的主要结论是推导出一个证明：在指定条件下，对于单调函数，基于密度众数的HPD区间具有与剖面LRCI相似的变换不变性。