Abstract
The authors suggested a family of Poincaré distributions on the upper half plane, which is essentially the same as a family of hyperboloid distributions on the two dimensional hyperbolic space. This family has an explicit form of normalizing constant and is \(SL(2,\mathbb {R})\)-invariant. In this paper, as a q-analogue of Poincaré distributions, we propose a q-exponential family on the upper half plane with an explicit form of normalizing constant and show that it is also \(SL(2,\mathbb {R})\)-invariant.
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References
Amari, S., Ohara, A.: Geometry of q-exponential family of probability distributions. Entropy 13, 1170–1185 (2011)
Barbaresco, F., Gay-Balmaz, F.: Lie group cohomology and (multi)symplectic integrators: new geometric tools for Lie group machine learning based on Souriau Geometric statistical mechanics. Entropy 22(5), 498 (2022)
Barbaresco, F.: Symplectic theory of heat and information geometry. In: Handbook of Statistics, chap. 4, vol. 46, pp. 107–143. Elsevier (2022)
Barndorff-Nielsen, O.E.: Information and exponential families in statistical theory. Wiley Series in Probability and Statistics, John Wiley & Sons Ltd., Chichester (1978)
Barndorff-Nielsen, O.E.: Hyperbolic distributions and distribution on hyperbolae. Scand. J. Stat. 8, 151–157 (1978)
Barndorff-Nielsen, O.E., Blæsild, P., Eriksen, P.S.: Decomposition and invariance of measures, and statistical transformation models. Springer-Verlag Lecture Note in Statistics (1989)
Barndorff-Nielsen, O.E., Blæsild, P., Jensen, J.L., Jørgensen, B.: Exponential transformation models. Proc. Roy. Soc. Lond. Ser. A 379(1776), 41–65 (1982)
Bourbaki, N.: Integration (translated by Sterlin K. Beberian). Springer, Heidelberg (2004)
Folland, G.B.: A Course in Abstract Harmonic Analysis. CRC Press, Boca Raton (1995)
Jensen, J.L.: On the hyperboloid distribution. Scand. J. Statist. 8, 193–206 (1981)
Kiral, E.M., Möllenhoff, T., Khan, M.E.: The lie-group bayesian learning rule. In: Proceedings of 26th International Conference on Artificial Intelligence and Statistics (AISTATS), vol. 206. PMLR (2023)
Marle, C.-M.: On Gibbs states of mechanical systems with symmetries. J. Geom. Symm. Phys. 57, 45–85 (2020)
Matsuzoe, H., Ohara, A.: Geometry of q-exponential families. Recent Progress in Differential Geometry and its Related Fields (2011)
Tojo, K., Yoshino, T.: Harmonic exponential families on homogeneous spaces. Inf. Geo. 4, 215–243 (2021)
Tojo, K., Yoshino, T.: An exponential family on the upper half plane and its conjugate prior. In: Barbaresco, F., Nielsen, F. (eds.) SPIGL 2020. SPMS, vol. 361, pp. 84–95. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-77957-3_4
Tojo, K., Yoshino, T.: A method to construct exponential families by representation theory. Inf. Geo. 5, 493–510 (2022)
Tsallis, C.: What are the numbers that experiments provide? Quimica Nova 17, 468–471 (1994)
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The first author is supported by Special Postdoctoral Researcher Program at RIKEN.
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Tojo, K., Yoshino, T. (2023). A q-Analogue of the Family of Poincaré Distributions on the Upper Half Plane. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2023. Lecture Notes in Computer Science, vol 14071. Springer, Cham. https://doi.org/10.1007/978-3-031-38271-0_17
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DOI: https://doi.org/10.1007/978-3-031-38271-0_17
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