Exponential Family

In probability and statistics, an exponential family is an important class of probability distributions sharing 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 distributions to consider. The concept of exponential families is credited to E. J. G. Pitman, G. Darmois, and B. O. Koopman in 1935–6. The term exponential class is sometimes used in place of "exponential family".

The exponential families include many of the most common distributions, including the normal, exponential, gamma, chi-squared, beta, Dirichlet, Bernoulli, categorical, Poisson, Wishart, Inverse Wishart and many others. A number of common distributions are exponential families only when certain parameters are considered fixed and known, e.g. binomial (with fixed number of trials), multinomial (with fixed number of trials), and negative binomial (with fixed number of failures). Examples of common distributions that are not exponential families are Student's t, most mixture distributions, and even the family of uniform distributions with unknown bounds. See the section below on examples for more discussion.

Consideration of exponential-family 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 statistics of the family. See below for more information.

Read more about Exponential Family:  Definition, The Meaning of "exponential Family", Interpretation, Properties, Examples, Table of Distributions, Maximum Entropy Derivation

Other articles related to "exponential family, exponential":

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... The normal-gamma distribution is a four-parameter exponential family with natural parameters and natural statistics ...
Exponential Family - Role in Statistics - Generalized Linear Models
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Natural Exponential Family - Definition - Probability Distribution Function (PDF) of The Univariate Case (scalar Domain, Scalar Parameter)
... The natural exponential family (NEF) is a subset of the exponential family ... NEF is an exponential family in which the natural parameter η and the natural statistic T(x) are both the identity ... A distribution in the exponential family with parameter θ can be written with probability density function (PDF) where and are known functions ...
Natural Exponential Family
... In probability and statistics, a natural exponential family (NEF) is a class of probability distributions that is a special case of an exponential family (EF) ... function is a member of a natural exponential family, and the use of such distributions simplifies the theory and computation of generalized linear models ...
Natural Exponential Family - Examples
... Distributions such as the exponential, chi-squared, Rayleigh, Weibull, Bernoulli, and geometric distributions are special cases of the above five distributions ... The exponential distribution is a gamma distribution with shape parameter α = 1 (or k = 1 ) ... The Rayleigh and Weibull distributions can each be written in terms of an exponential distribution ...

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