The highest density region (HDR) refers to the smallest possible area within a sample space that contains a specified probability, often expressed as 1-α.
Understanding Highest Density Regions
The concept of the highest density region is crucial in statistical analysis, especially when dealing with probability distributions. It's not just about finding areas of high density but also about defining the smallest region that captures a predetermined amount of probability.
Definition and Key Characteristics
- Probability Coverage: The HDR contains a specific probability mass (e.g., 95% or 1-α).
- Smallest Region: It is defined as the smallest possible region in the sample space that encloses this probability.
- Convex Contours: For easier computation and interpretation, HDRs are often represented by convex contours, particularly in two-dimensional spaces.
- This means the shape bulges outward, without any indentations.
- Univariate Density: When dealing with one variable (univariate), finding the HDR is easier. We locate the range containing the required probability.
- Multivariate Density: In higher dimensions (like a two-dimensional space), the HDR becomes more complex, involving contours.
Practical Insights and Examples
Imagine a heat map representing the probability of finding a particular value in a dataset.
- 95% HDR: The 95% HDR is the area containing 95% of the total probability mass of the data, taking the smallest area that accomplishes this.
- Visual Representation: In a two-dimensional space, the HDR can often be visualized as a contour line, which encompasses the densest parts of the probability distribution.
Heuristic for Determining HDR (Univariate)
The reference suggests this heuristic for determining the HDR in a univariate density estimate:
- Sort the density values (e.g. values from a density function).
- Sum the probability mass associated with the highest densities until the desired probability (1-α) is reached.
- The region corresponding to those densities is the HDR.
Why HDR is Important
- Estimation: It's helpful for estimating the uncertainty associated with model parameters.
- Bayesian Statistics: It helps in visualizing uncertainty in Bayesian inference
- Data Visualization: It allows for concisely summarizing regions of most importance within complex datasets.
| Feature | Description |
|---|---|
| Core Concept | Smallest area containing a set probability |
| Probability | Target probability is typically 1-α |
| Region Size | The region must be the smallest possible |
| Common Shape | Often represented by convex contours, particularly in 2D spaces |
| Multivariate Spaces | Finding HDR is more complex in multivariate density compared to univariate density estimates. |
