Exponential family refers to a broad class of probability distributions with certain structural characteristics. This family of distributions is known for its ease of use and flexibility across various statistical applications. A distribution belongs to the exponential family if its pdf can be expressed in a specific form involving exponentials.
The exponential family is characterized by its simplicity and power, as many common distributions such as Normal, Binomial, and Poisson all belong to this family. These distributions can be represented through a canonical form involving certain functions of the data (often called the ‘natural parameter’, ‘sufficient statistic’, and ‘normalizer’).
- **Natural Parameters**: These are the parameters used in the formulation of the distribution's pdf.
- **Sufficient Statistics**: Functions of the data used in the pdf that capture essential information about the natural parameters.
- **Normalizer**: A function that ensures the entire expression is a valid probability distribution.
Across a wide range of statistical models, the exponential family's form simplifies many calculations, particularly those involving parameter estimation.