A method to construct exponential families by representation theory
This provides a foundational approach for statisticians and machine learning researchers to derive invariant exponential families, though it appears incremental as it builds on existing representation theory concepts.
The paper tackles the problem of systematically constructing exponential families by using representation theory on homogeneous spaces, resulting in a method that generates widely used distributions such as normal, gamma, and von Mises.
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 an 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, Fisher-Bingham and hyperboloid distributions.