Abstract:
This work addresses two classification problems. First, we classify 1-dimensional connected dually
flat manifolds M that are toric in the sense of (1), and show that the corresponding torifications
are complex space forms. Special emphasis is put on the case where M is an exponential family
defined over a finite set.
The second problem addresses a classification question in statistical theory. Exponential families
defined on a finite sample space Ω are determined by (n + 1)-uples of functions (C , F 1 , ..., F n )
defined on Ω. However, this representation in terms of functions is not unique, leading to the
problem of classifying equivalent tuples of functions (C , F 1 , ..., F n ). This work presents a systematic
Lie group theoretical approach to this classification problem. We explicitly describe the underlying
symmetry group and, using a reduction by stages method, establish a one-to-one correspondence
between the set of n-dimensional exponential families on Ω and the affine Grassmannian of a
related function space.
