A method to construct exponential families by representation theory
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In this paper, we give a method to construct "good" exponential families systematically by representation theory. More precisely, we consider a homogeneous space G/H as a sample space and construct a exponential family invariant under the transformation group G by using a representation of G. The method generates widely used exponential families such as normal, gamma, Bernoulli, categorical, Wishart, von Mises and Fisher-Bingham distributions. Moreover, we obtain a new family of distributions on the upper half plane compatible with the Poincaré metric.
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