Exponential Families and Mixture Families of Probability Distributions

  • Shun-ichi AmariEmail author
Part of the Applied Mathematical Sciences book series (AMS, volume 194)


The present chapter studies the geometry of the exponential family of probability distributions. It is not only a typical statistical model, including many well-known families of probability distributions such as discrete probability distributions \(S_n\), Gaussian distributions, multinomial distributions, gamma distributions, etc., but is associated with a convex function known as the cumulant generating function or free energy. The induced Bregman divergence is the KL-divergence. It defines a dually flat Riemannian structure. The derived Riemannian metric is the Fisher information matrix and the two affine coordinate systems are the natural (canonical) parameters and expectation parameters, well-known in statistics.


Probability Density Function Exponential Family Fisher Information Matrix Maximum Entropy Principle Negative Entropy 
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Copyright information

© Springer Japan 2016

Authors and Affiliations

  1. 1.Brain Science InstituteRIKENWakoJapan

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