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).
A NEF is an exponential family in which the natural parameter η and the natural statistic T(x) are both the identity.
A distribution in an exponential family with parameter θ can be written with probability density function (PDF)
A distribution in a natural exponential family with parameter θ can thus be written with PDF
[Note that slightly different notation is used by the originator of the NEF, Carl Morris.
[1] Morris uses ω instead of η and ψ instead of A.]
, then a natural exponential family of order p has density or mass function of the form:
A member of a natural exponential family has moment generating function (MGF) of the form
The cumulant generating function is by definition the logarithm of the MGF, so it is
The Kullback–Leibler divergence of two natural exponential families with parameters
is The five most important univariate cases are: These five examples – Poisson, binomial, negative binomial, normal, and gamma – are a special subset of NEF, called NEF with quadratic variance function (NEF-QVF) because the variance can be written as a quadratic function of the mean.
Some exponential family distributions are not NEF.
The gamma distribution with two parameters is an exponential family but not a NEF and the chi-squared distribution is a special case of the gamma distribution with fixed scale parameter, and thus is also an exponential family but not a NEF (note that only a gamma distribution with fixed shape parameter is a NEF).
The inverse Gaussian distribution is a NEF with a cubic variance function.
The parameterization of most of the above distributions has been written differently from the parameterization commonly used in textbooks and the above linked pages.
, where λ is the mean parameter, and so that the density may be written as
This alternative parameterization can greatly simplify calculations in mathematical statistics.
For example, in Bayesian inference, a posterior probability distribution is calculated as the product of two distributions.
Normally this calculation requires writing out the probability distribution functions (PDF) and integrating; with the above parameterization, however, that calculation can be avoided.
Instead, relationships between distributions can be abstracted due to the properties of the NEF described below.
An example of the multivariate case is the multinomial distribution with known number of trials.
The properties of the natural exponential family can be used to simplify calculations involving these distributions.
In the multivariate case, the mean vector and covariance matrix are[citation needed]
A special case of the natural exponential families are those with quadratic variance functions.
Six NEFs have quadratic variance functions (QVF) in which the variance of the distribution can be written as a quadratic function of the mean.
The properties of these distributions were first described by Carl Morris.
The six NEF-QVF are written here in increasing complexity of the relationship between variance and mean.
The properties of NEF-QVF can simplify calculations that use these distributions.
Given independent identically distributed (iid)
If the original NEF-QVF had variance function