Abstract
We review the notion of parametrized measure models and tensor fields on them, which encompasses all statistical models considered by Chentsov [6], Amari [3] and Pistone-Sempi [10]. We give a complete description of n-tensor fields that are invariant under sufficient statistics. In the cases \(n= 2\) and \(n = 3\), the only such tensors are the Fisher metric and the Amari-Chentsov tensor. While this has been shown by Chentsov [7] and Campbell [5] in the case of finite measure spaces, our approach allows to generalize these results to the cases of infinite sample spaces and arbitrary n. Furthermore, we give a generalisation of the monotonicity theorem and discuss its consequences.
L. Schwachhöfer—speaker
J. Jost—partially supported by ERC Advanced Grant FP7-267087
H.V. Lê—partially supported by RVO: 6798584.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Ay, N., Jost, J., Lê, H.V., Schwachhöfer, L.: Information geometry and sufficient statistics. Probab. Theor. Relat. Fields 162(1–2), 327–364 (2015)
Ay, N., Jost, J., Lê, H.V., Schwachhöfer, L.: Information geometry, book in preparation
Amari, S.: Differential Geometrical Theory of Statistics. In: Amari, S.-I., Barndorff-Nielsen, O.E., Kass, R.E., Lauritzen, S.L., Rao, C.R. (eds.) Differential Geometry in Statistical Inference. Lecture Note-Monograph Series, vol. 10. Institute of Mathematical Statistics, California (1987)
Amari, S., Nagaoka, H.: Methods of Information Geometry. Translations of Mathematical Monographs, vol. 191. American Mathematical Society, Providence (2000)
Campbell, L.L.: An extended Chentsov characterization of a Riemannian metric. Proc. AMS 98, 135–141 (1986)
Chentsov, N.: Category of mathematical statistics. Dokl. Acad. Nauk USSR 164, 511–514 (1965)
Chentsov, N.: Statistical Decision Rules and Optimal Inference. Translation of Mathematical Monograph, vol. 53. AMS, Providence (1982)
Bauer, M., Bruveris, M., Michor, P.: Uniqueness of the Fisher-Rao metric on the space of smooth densities (2014). arXiv:1411.5577
Lang, S.: Fundamentals Of Differential Geometry. Springer, NewYork (1999)
Pistone, G., Sempi, C.: An infinite-dimensional structure on the space of all the probability measures equivalent to a given one. Ann. Stat. 5, 1543–1561 (1995)
Rao, C.R.: Information and the accuracy attainable in the estimation of statistical parameters. Bull. Calcutta Math. Soc. 37, 81–89 (1945)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Schwachhöfer, L., Ay, N., Jost, J., Vân Lê, H. (2015). Invariant Geometric Structures on Statistical Models. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2015. Lecture Notes in Computer Science(), vol 9389. Springer, Cham. https://doi.org/10.1007/978-3-319-25040-3_17
Download citation
DOI: https://doi.org/10.1007/978-3-319-25040-3_17
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-25039-7
Online ISBN: 978-3-319-25040-3
eBook Packages: Computer ScienceComputer Science (R0)