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In probability and statistics, an exponential family is a parametric set of probability distributions of a certain form, specified below. This special form is chosen for mathematical convenience, including the enabling of the user to calculate expectations, covariances using differentiation based on some useful algebraic properties, as well as for generality, as exponential families are in a ...
The natural exponential families (NEF) are a subset of the exponential families.A NEF is an exponential family in which the natural parameter η and the natural statistic T(x) are both the identity.
The natural exponential family of a distribution may be realized by shifting or translating K(t), and adjusting it vertically so that it always passes through the origin: if f is the pdf with cumulant generating function () = (), and | is its natural exponential family, then () = (), and () = (+) ().
For a parametric family of distributions one compares a code with the best code based on one of the distributions in the parameterized family. The main result is that in exponential families , asymptotically for large sample size, the code based on the distribution that is a mixture of the elements in the exponential family with the Jeffreys ...
When a member of the exponential family has been specified, the variance function can easily be derived. [4]: 29 The general form of the variance function is presented under the exponential family context, as well as specific forms for Normal, Bernoulli, Poisson, and Gamma. In addition, we describe the applications and use of variance functions ...
Since the functions are the sufficient statistics of the Dirichlet distribution, the exponential family differential identities can be used to get an analytic expression for the expectation of (see equation (2.62) in [12]) and its associated covariance matrix:
Pages in category "Exponential family distributions" The following 42 pages are in this category, out of 42 total. This list may not reflect recent changes. A.
Exponential functions with bases 2 and 1/2. In mathematics, the exponential function is the unique real function which maps zero to one and has a derivative equal to its value. . The exponential of a variable is denoted or , with the two notations used interchangeab