Abstract
The paper generalizes and extends the notions of dual connections and of statistical manifold, with and without torsion. Links with the deformation algebras and with the Riemannian Rinehart algebras are established. The semi-Riemannian manifolds admitting flat dual connections with torsion are characterized, thus solving a problem suggested in 2000 by S. Amari and H. Nagaoka. New examples of statistical manifolds are constructed, within and beyond the classical setting. The invariant statistical structures on Lie groups are characterized and the dimension of their set is determined. Examples for the new defined geometrical objects are found in the theory of Information Geometry.
Highlights
A triple (M, g, ∇) is called statistical manifold if (M, g) is a semi-Riemannian manifold and ∇ is a torsion-free affine connection on M such thatAcademic Editor: Vlad Stefan BarbuReceived: 27 June 2021Accepted: 12 July 2021Published: 14 July 2021Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.(∇ Z g)( X, Y ) = (∇Y g)( X, Z ).This notion was defined by S
Amari in [1], as a geometrical model for some facts in Statistics: M is a parameters space of distributions of probability, g is the Rao–Fisher metric deduced from the Bolzmann–Gibbs–Shannon entropy function, and ∇ is a tool for asymptotic estimations
We review in a creative manner the fundamentals of dual connections and of statistical manifolds and we give new examples (Sections 2 and 4)
Summary
A triple (M, g, ∇) is called statistical manifold if (M, g) is a semi-Riemannian manifold and ∇ is a torsion-free affine connection on M such that. The notion traces back to Norden and was adapted on statistical manifolds, where remarkable families of dual connections contain information about the dualistic properties of exponential families of probability distributions [7]. This initial (and already classical) setting was generalized in several ways. We review in a creative manner the fundamentals of dual connections and of statistical manifolds and we give new examples (Sections 2 and 4).
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