Invariant statistical connections on the multivariate centered Gaussian model and their moduli spaces

Abstract

We study invariant statistical connections on the space ${\mathcal{N}}_0^n$ of zero-mean multivariate normal distributions (the multivariate centered Gaussian model) equipped with the Fisher metric $g^F$. We introduce moduli spaces of invariant statistical connections on homogeneous Riemannian manifolds via two natural equivalence relations arising from a categorical viewpoint, and apply this framework to $({\mathcal{N}}_0^n,g^F)$. We explicitly determine the $GL(n,\mathbb{R})$-invariant and $\mathrm{Isom}({\mathcal{N}}_0^n,g^F)$-invariant statistical connections, with particular emphasis on the dually flat case, and describe the corresponding moduli spaces.

40 pages. Comments are welcome!