New Geometry of Parametric Statistical Models

Abstract

We provide an alternative differential geometric framework of the manifold \\(\\mathbb M\\) of parametric statistical models. While adopting the Fisher-Rao metric as the Riemannian metric g on \\(\\mathbb M\\), we treat the original parameterization of the statistical model as affine coordinate chart on the manifold endowed with a flat connection, instead of using a pair of torsion-free affine connections with generally non-vanishing curvature. We then construct its g-conjugate connection which, while necessarily curvature-free, carries torsion in general. So instead of associating a statistical structure to \\(\\mathbb M\\), we construct a statistical manifold admitting torsion (SMAT). We show that \\(\\mathbb M\\) is dually flat if and only if torsion of the conjugate connection vanishes.