The highest density interval (HDI) is a specific type of interval used in Bayesian statistics that represents the range within which a parameter is most likely to fall, based on a probability distribution.
Understanding Highest Density Intervals
The key distinction of an HDI is that it includes the values with the highest probability density. This means:
- All points inside the HDI have a higher probability density than any point outside the interval. As stated in the reference, "the interval which contains the required mass such that all points within the interval have a higher probability density than points outside the interval".
- It's not necessarily centered around the mean or median.
- It can be asymmetric, especially if the probability distribution is skewed.
How HDIs Differ from Symmetric Intervals
Traditional symmetric intervals, like those defined by quantiles (e.g., the 10% and 90% percentiles), might include values with lower probability density while excluding those with higher.
For example:
| Feature | Highest Density Interval (HDI) | Symmetric Interval (e.g., Quantiles) |
|---|---|---|
| Definition | Interval containing the required probability mass where all included points have a higher density. | Interval between two specified quantiles (e.g., 90% and 10%). |
| Probability Density | All points inside have higher density than those outside. | May include lower probability areas and exclude higher probability areas. |
| Shape | Can be asymmetrical and not necessarily centered on the median or mean. | Typically symmetrical around a center point, based on quantiles. |
| Use Case | For representing the most credible range of a parameter. | For indicating a spread around the central tendency. |
Practical Insights and Use Cases:
- Bayesian Inference: HDIs are commonly used in Bayesian statistics to report credible intervals, providing the range of parameter values that are most plausible given the observed data and prior knowledge.
- Credibility, not Probability: Unlike confidence intervals used in frequentist statistics, HDIs represent the probability that the true parameter value lies within the interval, based on a Bayesian posterior distribution.
- Asymmetric Distributions: They are especially valuable when working with skewed or complex probability distributions.
Key Takeaway:
In summary, the highest density interval is a powerful tool that precisely identifies the most credible region within a probability distribution, ensuring that no excluded point has a higher probability density than the included ones.
