The Exponential Family And Statistical Applications
PROBABILITY FOR STATISTICS AND MACHINE LEARNING: FUNDAMENTALS AND ADVANCED TOPICS(2010)
摘要
The exponential family is a practically convenient and widely used unified family of distributions on finite-dimensional Euclidean
spaces parametrized by a finite-dimensional parameter vector. Specialized to the case of the real line, the exponential family
contains as special cases most of the standard discrete and continuous distributions that we use for practical modeling, such
as the normal, Poisson, binomial, exponential, Gamma, multivariate normal, and so on. The reason for the special status of
the exponential family is that a number of important and useful calculations in statistics can be done all at one stroke within
the framework of the exponential family. This generality contributes to both convenience and larger-scale understanding. The
exponential family is the usual testing ground for the large spectrum of results in parametric statistical theory that require
notions of regularity or Cramér–Rao regularity. In addition, the unified calculations in the exponential family have an element of mathematical neatness. Distributions
in the exponential family have been used in classical statistics for decades. However, it has recently obtained additional
importance due to its use and appeal to the machine learning community. A fundamental treatment of the general exponential
family is provided in this chapter. Classic expositions are available in Barndorff-Nielsen (1978), Brown (1986), and Lehmann
and Casella (1998). An excellent recent treatment is available in Bickel and Doksum (2006).
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