Talk by Koichi Tojo (U. Tokyo) A method to construct exponential families by representation theory

Exponential families are important subjects of study in the field
of information geometry and are often used for Bayes inference
because they have conjugate priors. By definition, there are
infinitely many exponential families, however, only a small part
of them are widely used. Most of the useful families have
the same symmetry as the sample space. More precisely,
the sample space can be regarded as a homogeneous space
$G/H$ and the family of distributions are $G$-invariant.
So, we suggest a method to construct exponential families
invariant under the transformation groups systematically
by using representation theory. Though the families obtained
by this method are limited, in fact, the method generates
many practical families of distributions such as normal, gamma,
Bernoulli, categorical, Wishart, von Mises and Fisher-Bingham
distributions. We can also construct a good family of distributions
on the upper half plane which is compatible with the Poincaré metric.