Exponential families

In probability and statistics, an exponential family is a set of probability distributions of a certain form, specified below. This special form is chosen for mathematical convenience, on account of some useful algebraic properties, as well as for generality, as exponential families are in a sense very natural sets of distributions to consider. The concept of exponential families is credited toE. J. G. Pitman,G. Darmois, and B. O. Koopman in 1935–36. The term exponential class is sometimes used in place of "exponential family".

The exponential family of distributions provides a general framework for selecting a possible alternative parameterisation of the distribution, in terms of natural parameters, and for defining useful sample statistics, called the natural sufficient statistics of the family. For more information, see below.

Most of the commonly used distributions are in the exponential family, listed in the subsection below. The subsections following it are a sequence of increasingly more general mathematical definitions of an exponential family. A casual reader may wish to restrict attention to the first and simplest definition, which corresponds to a single-parameter family of discrete or continuous probability distributions.

The exponential families include many of the most common distributions. Among many others, the family includes the following:

A number of common distributions are exponential families, but only when certain parameters are fixed and known. For example:

Notice that in each case, the parameters which must be fixed determine a limit on the size of observation values.

Examples of common distributions that are not exponential families are Student's t, most mixture distributions, and even the family of uniform distributions when the bounds are not fixed. See the section below on examples for more discussion.

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