Fisher information and \(\alpha\)-connections for a class of transformational models (Q1573629)

The Fisher information metric and the \(\alpha\)-connections are geometric objects attached to statistical parametric models. By definition they are given by some integrals over the sample space. On the other hand a transformational model is a statistical model generated by the action of a group on the sample space. In such models the parameter space turns out to be a coset space of the acting group. This paper develops a method for computing the Fisher information and the \(\alpha\)-connections for transformational models parametrized by a symmetric space. Exploiting the invariance of the geometric objects, the explicit computation of the integrals appearing in their definition is avoided. A basic tool is the knowledge of the invariant polynomials for the representation of the Weyl group associated with the symmetric space. There are some classical models whose parameter space is symmetric, like the von Mises-Fisher model, the hyperboloid model, the multivariable zero mean normal model with determinant one covariant matrix and the Wishard model.