In statistical theory, exponential families defined on a finite sample space \(\Omega \) are determined by tuples of functions \((C,F_{1},\ldots ,F_{n})\) defined on \(\Omega \) . However, this representation in terms of functions is not unique, leading to the problem of classifying equivalent tuples of functions \((C,F_{1},\ldots ,F_{n})\) . This paper 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 \(\Omega \) and the affine Grassmannian of a related function space.
