Abstract
Heavily tailed probability distributions are important objects in anomalous statistical physics. For such probability distributions, expectations do not exist in general. Therefore, an escort distribution and an escort expectation have been introduced. In this paper, by generalizing such escort distributions, a sequence of escort distributions is introduced. For a deformed exponential family, we study the fundamental properties of statistical manifold structures derived from the sequence of escort expectations.
H. Matsuzoe—This research was partially supported by JSPS (Japan Society for the Promotion of Science), KAKENHI (Grants-in-Aid for Scientific Research) Grant Numbers JP26108003, JP15K04842 and JP16KT0132.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Amari, S., Nagaoka, H.: Method of Information Geometry. Amer. Math. Soc., Providence, Oxford University Press, Oxford (2000)
Amari, S.: Information Geometry and Its Applications. AMS, vol. 194. Springer, Tokyo (2016). doi:10.1007/978-4-431-55978-8
Amari, S., Ohara, A., Matsuzoe, H.: Geometry of deformed exponential families: invariant, dually-flat and conformal geometry. Phys. A 391, 4308–4319 (2012)
Beck, C., Schlögl, F.: Thermodynamics of Chaotic Systems: An Introduction. Cambridge University Press, Cambridge (1993)
Kurose, T.: On the divergences of \(1\)-conformally flat statistical manifolds. Tôhoku Math. J. 46, 427–433 (1994)
Lauritzen, S.L.: Statistical manifolds. In: Differential Geometry in Statistical Inferences. IMS Lecture Notes Monograph Series, vol. 10, pp. 96–163. Hayward California, Institute of Mathematical Statistics (1987)
Matsuzoe, H.: A sequence of escort distributions and generalizations of expectations on \(q\)-exponential family. Entropy 19(1), 7 (2017)
Matsuzoe, H., Henmi, M.: Hessian structures and divergence functions on deformed exponential families. In: Nielsen, F. (ed.) Geometric Theory of Information. SCT, pp. 57–80. Springer, Cham (2014). doi:10.1007/978-3-319-05317-2_3
Matsuzoe, H., Wada, T.: Deformed algebras and generalizations of independence on deformed exponential families. Entropy 17(8), 5729–5751 (2015)
Murata, N., Takenouchi, T., Kanamori, T., Eguchi, S.: Information geometry of U-boost and Bregman divergence. Neural Comput. 16, 1437–1481 (2004)
Naudts, J.: Generalised Thermostatistics. Springer, London (2011). doi:10.1007/978-0-85729-355-8
Sakamoto, M., Matsuzoe, H.: A generalization of independence and multivariate student’s t-distributions. In: Nielsen, F., Barbaresco, F. (eds.) GSI 2015. LNCS, vol. 9389, pp. 740–749. Springer, Cham (2015). doi:10.1007/978-3-319-25040-3_79
Shima, H.: The Geometry of Hessian Structures. World Scientific, Singapore (2007)
Tanaka, M.: Meaning of an escort distribution and \(\tau \)-transformation. J. Phys: Conf. Ser. 201, 012007 (2010)
Tsallis, C.: Introduction to Nonextensive Statistical Mechanics: Approaching a Complex World. Springer, New York (2009). doi:10.1007/978-0-387-85359-8
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Matsuzoe, H., Scarfone, A.M., Wada, T. (2017). A Sequential Structure of Statistical Manifolds on Deformed Exponential Family. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2017. Lecture Notes in Computer Science(), vol 10589. Springer, Cham. https://doi.org/10.1007/978-3-319-68445-1_26
Download citation
DOI: https://doi.org/10.1007/978-3-319-68445-1_26
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-68444-4
Online ISBN: 978-3-319-68445-1
eBook Packages: Computer ScienceComputer Science (R0)