📂Mathematical Statistics

Highest Posterior Density Credible Interval

Definition ¹

For the parameter space $\Theta$, a subset $C \subset \Theta$ satisfying $C : = \left\{ \theta \in \Theta \ | \ p ( \theta | y ) \ge k (\alpha) \right\}$ with respect to the significance level $\alpha$ is called the $100(1 - \alpha) % $ highest posterior density credible interval for $\theta$ given the data $y$.

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• Here, $k(\alpha)$ is the largest constant satisfying $p(\theta \in C | y ) \ge 1 - \alpha$.

Explanation

It is far easier to understand through a picture than through formulas and words.

In actual computation as well, a numerical method is used that keeps changing $k$ as above so that the integral value approximates $1 - \alpha$.

• The first one is taken too narrow, so it fails to become a credible interval at all.
• The second one conservatively takes a wide interval, so it does become a credible interval, but the range is too wide to be useful.
• If the area of the third, the green region, equals $1 - \alpha$, then the interval obtained in this case is called the HPD (highest posterior density) credible interval.

The explanation that such intervals become credible intervals is [much more intuitive than the confidence interval, which used to rely on the sample mean and the standard error](../752).

Equal-Tailed Credible Interval

Meanwhile, since such an HPD credible interval is very difficult to compute in practice, something called the equal-tailed credible interval is also used. As its name suggests, the equal-tailed credible interval refers to $[a,b]$ such that $$ \int_{-\infty}^{ a } p(\theta | y) d \theta = \int_{b}^{ \infty } p(\theta | y) d \theta = {{ \alpha } \over {2}} $$ holds.