$f$-statistical connections and Miao-Tam statistical manifolds

Abstract

We introduce $f$-statistical connections as a family of statistical connections and study some geometric objects associated to these connections such as divergence, curvature and Ricci tensors, Hessian and Laplacian operators. We construct examples of $f$-statistical connections and study the introducing concepts on them. Finally we introduce Miao-Tam statistical manifolds and study properties of them.

Keywords

• f-statistical connction
• Hessian operator
• Laplacian operator
• Miao-Tam equation
• statistical manifold

References

1. [1] S. Amari, Information geometry of the EM and em algorithms for neural networks, Neural Networks 8 (9), 1379–1408, 1995.
2. [2] K. Arwini and C.T.J. Dodson, Neighbourhoods of independence and associated geometry in manifolds of bivariate Gaussian and Freund distributions, Open Math. 5 (1), 50–83, 2007.
3. [3] B. Balcerzak, Linear connection and secondary characteristic classes of Lie algebroids, Monographs of Lodz University of Technology, 2021. Doi:10.34658/9788366741287.
4. [4] H. Baltazar and A. Da Silva, On static manifolds and related critical spaces with cyclic parallel Ricci tensor, Advances in Geometry 21 (3), 443–450, 2021.
5. [5] A. Barros, R. Diógenes and E. Ribeiro Jr, Bach-Flat Critical Metrics of the Volume Functional on 4-Dimensional Manifolds with Boundary, J. Geom. Anal. 25, 2698– 2715, 2015.
6. [6] M. Belkin, P. Niyogi and V. Sindhwani, Manifold regularization: a geometric framework for learning from labeled and unlabeled examples, Journal of Machine Learning Research 7, 2399–2434, 2006.
7. [7] O. Calin and C. Udrişte, Geometric Modeling in Probability and Statistics, Springer, Cham, Switzerland, 2014.
8. [8] A. Caticha, Geometry from information geometry, arxiv.org/abs/1512.09076v1.

9. [9] A. Caticha, The information geometry of space and time, Proceedings 33 (1), 15, 2019.
10. [10] R. A. Fisher, On the mathematical foundations of theoretical statistics, Phil. Trans. Roy. Soc. London. 222, 309–368, 1922.
11. [11] S. Kobayashi and Y. Ohno, On a constant curvature statistical manifold, Inf. Geom. 5 (1), 31-46, 2022.
12. [12] P. Miao and L.F. Tam, On the volume functional of compact manifolds with boundary with constant scalar curvature, Calc. Var. PDE. 36, 141–171, 2009.
13. [13] P. Miao and L.F. Tam, Einstein and conformally flat critical metrics of the volume functional, Trans. Amer. Math. Soc. 363, 2907–2937, 2011.
14. [14] B. Opozda, Bochners technique for statistical structures, Ann. Global Anal. Geom. 48, 357–395, 2015.
15. [15] 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.