#
A Foliation by Deformed Probability Simplexes for Transition of α-Parameters^{ †}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Dualistic Structures and Divergences on the Probability Simplex

## 3. Deformed Probability Simplexes and Escort Distributions

## 4. Divergences Generated by Affine Immersions as Level Surfaces

**Theorem**

**1**

**.**Let M be a simply connected n-dimensional level surface of φ on an $(n+1)$-dimensional Hessian domain $(\mathsf{\Omega},\tilde{D},\tilde{g}=\tilde{D}d\phi )$ with a Riemannian metric $\tilde{g}$ and suppose that $n\ge 2$. If we consider $(\mathsf{\Omega},\tilde{D},\tilde{g})$ a flat statistical manifold, $(M,D,g)$ is a 1-conformally flat statistical submanifold of $(\mathsf{\Omega},\tilde{D},\tilde{g})$, where D and g denote the connection and the Riemannian metric on M induced by $\tilde{D}$ and $\tilde{g}$, respectively.

**Definition**

**1**

**.**Let $(N,\nabla ,h)$ be a 1-conformally flat statistical manifold realized by a non-degenerate affine immersion $(v,\xi )$ into ${\mathbf{A}}^{n+1}$, and w the conormal immersion for v. Then, the divergence ${\rho}_{\mathrm{c}onf}$ of $(N,\nabla ,h)$ is defined by

**Theorem**

**2**

**.**For a 1-conformally flat statistical submanifold $(M,D,g)$ of $(\mathsf{\Omega},\tilde{D},\tilde{g})$, two divergences ${\rho}_{\mathrm{c}onf}$ and ${\rho}_{\mathrm{s}ub}$ coincide.

## 5. Extended Divergence on a Foliation by Deformed Probability Simplexes

**Proposition**

**1.**

**Proof.**

**Definition**

**2.**

**Proposition**

**2.**

**Proof.**

## 6. Decomposition of an Extended Divergence

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

## 7. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Tsallis, C. Introduction to Nonextensive Statistical Mechanics: Approaching a Complex World; Springer: New York, NY, USA, 2009. [Google Scholar]
- Naudts, J. Generalised Thermostatistics; Springer: London, UK, 2011. [Google Scholar]
- Ohara, A.; Wada, T. Information geometry of q-Gaussian densities and behaviors of solutions to related diffusion equations. J. Phys. A Math. Theor.
**2010**, 43, 035002. [Google Scholar] [CrossRef] [Green Version] - Matsuzoe, H.; Ohara, A. Geometry for q-exponential families. In Recent Progress in Differential Geometry and Its Related Fields; Adachi, T., Hashimoto, H., Hristov, M.J., Eds.; World Scientific Publishing: Hackensack, NJ, USA, 2011; pp. 55–71. [Google Scholar]
- Amari, S.; Ohara, A.; Matsuzoe, H. Geometry of deformed exponential families: Invariant, dually-flat and conformal geometry. Physica A
**2012**, 391, 4308–4319. [Google Scholar] [CrossRef] - Matsuzoe, H.; Henmi, M. Hessian structures and divergence functions on deformed exponential families. In Geometric Theory of Information, Signals and Communication Technology; Nielsen, F., Ed.; Springer: Basel, Switzerland, 2014; pp. 57–80. [Google Scholar]
- Matsuzoe, H.; Wada, T. Deformed algebras and generalizations of independence on deformed exponential families. Entropy
**2015**, 17, 5729–5751. [Google Scholar] [CrossRef] [Green Version] - Wada, T.; Matsuzoe, H.; Scarfone, A.M. Dualistic Hessian structures among the thermodynamic potentials in the κ-thermostatistics. Entropy
**2015**, 17, 7213–7229. [Google Scholar] [CrossRef] [Green Version] - Amari, S. Information Geometry and Its Applications; Springer: Tokyo, Japan, 2016. [Google Scholar]
- Scarfone, A.M.; Matsuzoe, H.; Wada, T. Information geometry of κ-exponential families: Dually-flat, Hessian and Legendre structures. Entropy
**2018**, 20, 436. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Naudts, J. Estimators, escort probabilities, and ϕ-exponential families in statistical physics. J. Inequal. Pure Appl. Math.
**2004**, 5, 102. [Google Scholar] - Ohara, A. Geometry of distributions associated with Tsallis statistics and properties of relative entropy minimization. Phys. Lett. A
**2007**, 370, 184–193. [Google Scholar] [CrossRef] - Ohara, A. Geometric study for the Legendre duality of generalized entropies and its application to the porous medium equation. Eur. Phys. J. B
**2009**, 70, 15–28. [Google Scholar] [CrossRef] [Green Version] - Matsuzoe, H. A sequence of escort distributions and generalizations of expectations on q-exponential family. Entropy
**2017**, 19, 7. [Google Scholar] [CrossRef] [Green Version] - Shima, H. The Geometry of Hessian Structures; World Scientific: Singapore, 2007. [Google Scholar]
- Uohashi, K.; Ohara, A.; Fujii, T. 1-conformally flat statistical submanifolds. Osaka J. Math.
**2000**, 37, 501–507. [Google Scholar] - Uohashi, K.; Ohara, A.; Fujii, T. Foliations and divergences of flat statistical manifolds. Hiroshima Math. J.
**2000**, 30, 403–414. [Google Scholar] [CrossRef] - Nomizu, K.; Sasaki, T. Affine Differential Geometry: Geometry of Affine Immersions; Cambridge University Press: Cambridge, UK, 1994. [Google Scholar]
- Kurose, T. On the divergences of 1-conformally flat statistical manifolds. Tohoku Math. J.
**1994**, 46, 427–433. [Google Scholar] [CrossRef] - Nomizu, K.; Pinkal, U. On the geometry and affine immersions. Math. Z.
**1987**, 195, 165–178. [Google Scholar] [CrossRef] - Azoury, K.S.; Warmuth, M.K. Relative loss bounds for on-line density estimation with the exponential family of distributions. Mach. Learn.
**2001**, 43, 211–246. [Google Scholar] [CrossRef] [Green Version] - Nielsen, F. Statistical divergences between densities of truncated exponential families with nested supports: Duo Bregman and duo Jensen divergences. Entropy
**2022**, 24, 421. [Google Scholar] [CrossRef] [PubMed] - Fujiwara, A.; Amari, S. Gradient systems in view of information geometry. Physica D
**1995**, 80, 317–327. [Google Scholar] [CrossRef]

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Uohashi, K.
A Foliation by Deformed Probability Simplexes for Transition of *α*-Parameters. *Phys. Sci. Forum* **2022**, *5*, 53.
https://doi.org/10.3390/psf2022005053

**AMA Style**

Uohashi K.
A Foliation by Deformed Probability Simplexes for Transition of *α*-Parameters. *Physical Sciences Forum*. 2022; 5(1):53.
https://doi.org/10.3390/psf2022005053

**Chicago/Turabian Style**

Uohashi, Keiko.
2022. "A Foliation by Deformed Probability Simplexes for Transition of *α*-Parameters" *Physical Sciences Forum* 5, no. 1: 53.
https://doi.org/10.3390/psf2022005053