A characterization of the alpha-connections on the statistical manifold of normal distributions
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A characterization of the alpha-connections on the statistical manifold of normal distributions Hitoshi Furuhata1 · Jun-ichi Inoguchi2 · Shimpei Kobayashi1 Received: 3 September 2019 / Revised: 2 September 2020 / Accepted: 12 October 2020 © Springer Nature Singapore Pte Ltd. 2020
Abstract We show that the statistical manifold of normal distributions is homogeneous. In particular, it admits a 2-dimensional solvable Lie group structure. In addition, we give a geometric characterization of the Amari–Chentsov α-connections on the Lie group. Keywords Statistical manifolds · The Amari–Chentsov α-connection · Lie groups Mathematics Subject Classification Primary 53A15; Secondary 22E25
Introduction The set of normal distributions is parametrized by R × R+ as 1 (t − μ)2 , t ∈ R, R × R+ θ = (μ, σ ) → p(t, θ ) = √ exp − 2σ 2 2π σ 2 where μ is the mean, and σ 2 is the variance. For tangent vectors X , Y , Z of an manifold R × R+ at θ , we define
Jun-ichi Inoguchi is partially supported by Kakenhi 19K03461. Shimpei Kobayashi is partially supported by Kakenhi 18K03265.
B
Shimpei Kobayashi [email protected] Hitoshi Furuhata [email protected] Jun-ichi Inoguchi [email protected]
1
Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan
2
Institute of Mathematics, University of Tsukuba, Tsukuba 305-8571, Japan
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Information Geometry
g F (X , Y ) = E θ [(X log p)(Y log p)], C(X , Y , Z ) = E θ [(X log p)(Y log p)(Z log p)], where E θ [ f ] =
R
f (t) p(t, θ ) dt for an integrable function f on R, see [1]. For a
constant α, we define ∇ (α) by (α)
gF
g F (∇ X Y , Z ) = g F (∇ X Y , Z ) −
α C(X , Y , Z ), 2
F
where ∇ g is the Levi–Civita connection of the Riemannian metric g F . In information geometry, g F and ∇ (α) are well known as the Fisher metric and the Amari–Chentsov α-connection for the space of normal distributions, respectively. In the same fashion, we have the Fisher metric and the Amari–Chentsov α-connection for spaces of certain probability densities. Abstracting an essence from these, we reach the notion of statistical manifolds. The Fisher metric and the Amari–Chentsov α-connection for the space of all the positive probability densities on a finite set, that is, the space of multinomial distributions, are characterized from a viewpoint of statistics, which is known as the Chentsov theorem and has been generalized for other spaces, see [4] for example and references therein. The Fisher metric and the Amari–Chentsov α-connection for the space of normal distributions are expressed as d x 2 + 2dy 2 , y2 1−α 1+α 1 + 2α (α) (α) (α) (α) ∂ y , ∇∂x ∂ y = ∇∂ y ∂x = − ∂ x , ∇∂ y ∂ y = − ∂y , ∇∂ x ∂ x = 2y y y gF =
where ∂x = ∂/∂ x, ∂ y = ∂/∂ y and (x, y) = (μ, σ ) ∈ R × R+ . The goal of this article is to characterize affine connections on the statistical manifold of normal distributions with the Fisher metric from a purely differential geometric viewpoint, especially from a viewpoint of homogeneity. For this purpose, in Sect. 3, we discuss Lie gro
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