Statistical structures on the tangent bundle of a statistical manifold with Sasaki metric

Year 2020, Volume: 49 Issue: 1, 120 - 135, 06.02.2020

Vladimir Balan , E. Peyghan , Esa Sharahi

https://doi.org/10.15672/HJMS.2019.667

Cited By: 4

Abstract

The first part of the paper is devoted to the classification of the statistical structures which live on the tangent bundle of a statistical manifold endowed with a Sasaki metric. Further, considering a Kähler structure on the base statistical manifold, we introduce a family of almost complex structures on the tangent bundle equipped with the Sasaki metric, and find equivalent conditions for which this family induces a Kähler structure. Finally, we derive equivalent conditions for existence of holomorphic structures on the tangent bundle equipped with the Sasaki metric in the presence of a statistical structure. Several illustrative examples are provided, as well.

Keywords

Holomorphic statistical manifold, statistical structure, Kähler structure, Sasaki metric

References

• [1] S. Amari, Information geometry of the EM and em algorithms for neural networks, Neural Networks, 8 (9), 1379–1408, 1995.
• [2] S. Amari and H. Nagaoka, Methods of information geometry, American Mathematical Society, 2000.
• [3] V. Balan, H.V. Grushevskaya,N.G. Krylova, G.G. Krylov, and I.V. Lipnevich, Two- dimensional first-order phase transition as signature change event in contact statisti- cal manifolds with Finsler metric, Appl. Sci. 21, 11–26, 2019.
• [4] M. Belkin, P. Niyogi and V.Sindhwani, Manifold regularization: a geometric frame- work for learning from labeled and unlabeled examples, J. Mach. Learn. Res. 7, 2399– 2434, 2006.
• [5] M. Burgin, Theory of information: fundamentality, diversity and unification, World Scientific Series in Information Studies, 2009.
• [6] A. Caticha, Geometry from information geometry, https://arxiv.org/abs/1512. 09076v1.
• [7] A. Caticha, The information geometry of space and time, https://arxiv.org/abs/ gr-qc/0508108.
• [8] N.N. Cencov, Statistical decision rules and optimal inference, Amer. Mathematical Society: Translations of Mathematical Monographs, 1982.
• [9] R.A. Fisher, On the mathematical foundations of theoretical statistics, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 222, 309–368, 1922.
• [10] H. Furuhata, Hypersurfaces in statistical manifolds, Diff. Geom. Appl. 27, 420–429, 2009.
• [11] H. Furuhata, I. Hasegawa, Y. Okuyama, K. Sato and M.H. Shahid, Sasakian statistical manifolds, J. Geom. Phys. 117, 179–186, 2017.
• [12] S. Ianuş, Statistical manifolds and tangent bundles, Sci. Bull. Univ. Politechnica of Bucharest Ser. D, 56, 29–34, 1994.
• [13] S. Lauritzen, Statistical manifolds, in: Differential Geometry in Statistical Inference, IMS Lecture Notes 10, 163–216, 1987.
• [14] H. Matsuzoe and J.I. Inoguchi, Statistical structures on tangent bundles, APPS. Appl. Sci. 5 (1), 55–57, 2003.
• [15] J.M. Oller and C.N. Caudras, Rao’s distance for negative multinomial distributions, Sankhya A, 47 (1), 75–83, 1985.
• [16] C.R. Rao, Information and accuracy attainable in estimation of statistical parameters, Bull. Cal. Math. Soc. 37, 81–91, 1945.
• [17] K. Sun and S. Marchand-Maillet, An information geometry of statistical manifold learning, Proceedings of the 31st International Conference on Machine Learning (ICML-14), 1–9, 2014.
• [18] K. Yano and S. Ishihara, Tangent and cotangent bundles, Marcel Dekker Inc., New York, 1973.