📂Mathematical Statistics

Highest Posterior Density Credible Interval

Definition ¹

A subset $C \subset \Theta$ of the parameter space $\Theta$ is called the Highest Posterior Density (HPD) Credible Interval for $100(1 - \alpha) % $ given the data $y$ at the significance level $\alpha$ if it satisfies $C : = \left\{ \theta \in \Theta \ | \ p ( \theta | y ) \ge k (\alpha) \right\}$.

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

Description

It is much easier to understand through a diagram than through formulas and words.

In actual calculations, a numerical method is used to find the value of $k$ by continuously adjusting it so that the integral value approximately equals $1 - \alpha$.

• The first is too narrow to become a credible interval.
• The second chooses a conservatively wide range, making it a credible interval but the range is too broad to be useful.
• If the area of the third, the green part, equals $1 - \alpha$, then the interval obtained at this time is called the HPD (Highest Posterior Density) Credible Interval.

This explanation that such intervals become credible intervals is much more intuitive than confidence intervals, which depended on sample means and standard error.

Equal Tail Credible Interval

Meanwhile, because calculating the HPD Credible Interval can be quite difficult, Equal Tail Credible Interval is also used. As the name suggests, it refers to $[a,b]$ that makes $$ \int_{-\infty}^{ a } p(\theta | y) d \theta = \int_{b}^{ \infty } p(\theta | y) d \theta = {{ \alpha } \over {2}} $$ equal.