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In statistical inference, problem 9.6b, a "Highest Density Region (HDR)" is mentioned. However, I didn't find the definition of this term in the book.

One similar term is the Highest Posterior Density (HPD). But it doesn't fit in this context, since 9.6b doesn't mention anything about a prior. And in the suggested solution, it only says that "obviously $c(y)$ is a HDR".

Or is the HDR a region containing the mode(s) of a pdf?

What is a Highest Density Region (HDR)?

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  • $\begingroup$ Yes. The amazon page is the book (purchase page). The pdf is the solution to the problems in the book. $\endgroup$ – user3813057 Apr 27 '15 at 14:17
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I recommend Rob Hyndman's 1996 article "Computing and Graphing Highest Density Regions" in The American Statistician. Here is the definition of the HDR, taken from that article:

Let $f(x)$ be the density function of a random variable $X$. Then the $100(1-\alpha)\%$ HDR is the subset $R(f_\alpha)$ of the sample space of $X$ such that $$R(f_\alpha) = \{x\colon f(x)\geq f_\alpha\},$$ where $f_\alpha$ is the largest constant such that $$P\big(X\in R(f_\alpha)\big)\geq 1-\alpha.$$

Figure 1 from that article illustrates the difference between the 75% HDR (so $\alpha=0.25$) and various other 75% Probability Regions for a mixture of two normals ($c_q$ is the $q$-th quantile, $\mu$ the mean and $\sigma$ the standard deviation of the density):

HDR

The idea in one dimension is to take a horizontal line and shift it up (to $y=f_\alpha$) until the area above it and under the density is $1-\alpha$. Then the HDR $R_\alpha$ is the projection to the $x$ axis of this area.

Of course, all this works with any density, whether Bayesian posterior or other.

Here is a link to R code, which is the hdrcdepackage (and to the article on JSTOR).

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A highest posterior density [interval] is basically the shortest interval on a posterior density for some given confidence level. A highest density region is probably the same idea applied to any arbitrary density, so not necessarily a posterior distribution.

If $1-\alpha$ is your confidence level, you can always find two quantiles $q_{1-\alpha/2 + c}$, $q_{\alpha/2 -c}$ that will give you a working interval. There are a bunch though, and they all have different lengths. You want the shortest.

If your density $f(\cdot)$ is unimodal, then the shortest interval will happen at the two quantiles $a$ and $b$ such that $f(a) = f(b)$.

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