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On warped product on information geometry: statistical manifolds and statistical models

Nicolás Martínez Alba, Olga Garatejo Escobar

Abstract

We study a differential geometric construction, the warped product, on the background geometry for information theory. Divergences, dual structures and symmetric 3-tensor are studied under this construction, and we show that warped product of manifolds endow with such structure also is endowed with the same geometric notion. However, warped product does not preserve canonical divergences, which in particular shows that warped product lacks of meaning in the information theory setting.

On warped product on information geometry: statistical manifolds and statistical models

Abstract

We study a differential geometric construction, the warped product, on the background geometry for information theory. Divergences, dual structures and symmetric 3-tensor are studied under this construction, and we show that warped product of manifolds endow with such structure also is endowed with the same geometric notion. However, warped product does not preserve canonical divergences, which in particular shows that warped product lacks of meaning in the information theory setting.

Paper Structure

This paper contains 14 sections, 20 theorems, 101 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Geometrical setting
  3. Basics on Riemann geometry
  4. Warped product of Riemannian manifolds
  5. Statistical model and its structure
  6. Finite sample space
  7. Fisher metric and statistical model
  8. Exponential and mixture families in geometry
  9. Geometric concepts for information geometry
  10. Statistical models and dual structure
  11. Tensor Amari-Chentsov
  12. Isostatistical immersion
  13. Canonical divergence
  14. The warped product for statistical structure

Key Result

Theorem 2.1

Jo Let $M$ be a Riemannian manifold, $x\in M$ and $v\in T_xM$. Then there exist $\epsilon>0$ and precisely one geodesic $c : [0, \epsilon] \to M$ with $c(0) = x, c'(0)= v$. In addition, $c$ depends smoothly on $x$ and $v$.

Figures (2)

  • Figure 1: Balls in Fisher metric centered at $\nu$ and $\mu$ with same radii
  • Figure 2: Shortest path between the normal distributions $p=\mathcal{N}(-0.75,0.5)$ and $q=\mathcal{N}(1.25,0.5)$ in the Fisher metric Rit .

Theorems & Definitions (52)

  • Theorem 2.1
  • Definition 2.2
  • Lemma 2.3
  • Proposition 2.4
  • Remark 2.5
  • Proposition 2.6
  • Example 3.1
  • Definition 3.2: jost
  • Definition 3.3
  • Example 3.4
  • ...and 42 more