Abstract
A deformed exponential family has two kinds of dual Hessian structures, the U-geometry and the \(\chi \)-geometry. In this paper, we discuss the relation between the non-invariant (F, G)-geometry and the two Hessian structures on a deformed exponential family. A generalized likelihood function called F-likelihood function is defined and proved that the Maximum F-likelihood estimator is a Maximum a posteriori estimator.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Amari, S.I., Nagaoka, H.: Methods of Information Geometry. Translations of Mathematical Monographs. Oxford University Press, Oxford (2000)
Harsha, K.V., Subrahamanian Moosath, K.S.: \(F\)-geometry and Amari’s \(\alpha \)-geometry on a statistical manifold. Entropy 16(5), 2472–2487 (2014)
Naudts, J.: Estimators, escort probabilities, and phi-exponential families in statistical physics. J. Ineq. Pure Appl. Math., 5(4) (2004). Article no 102
Eguchi, S., Komori, O., Ohara, A.: Duality of maximum entropy and minimum divergence. Entropy 16, 3552–3572 (2014)
Amari, S.I., Ohara, A., Matsuzoe, H.: Geometry of deformed exponential families: invariant, dually flat and conformal geometries. Phys. A Stat. Mech. Appl. 391, 4308–4319 (2012)
Matsuzoe, H., Henmi, M.: Hessian structures on deformed exponential families. In: Nielsen, F., Barbaresco, F. (eds.) GSI 2013. LNCS, vol. 8085, pp. 275–282. Springer, Heidelberg (2013)
Amari, S.I., Ohara, A.: Geometry of q-exponential family of probability distributions. Entropy 13, 1170–1185 (2011)
Matsuzoe, H. and Ohara, A. : Geometry for \(q-\)exponential families. In: Proceedings of the 2nd International Colloquium on Differential Geometry and its Related Fields, Veliko Tarnovo, 6–10 September (2010)
Fujimoto, Y., Murata, N.: A generalization of independence in naive bayes model. In: Fyfe, C., Tino, P., Charles, D., Garcia-Osorio, C., Yin, H. (eds.) IDEAL 2010. LNCS, vol. 6283, pp. 153–161. Springer, Heidelberg (2010)
Harsha, K.V., Subrahamanian Moosath, K.S.: Geometry of \(F\)-likelihood Estimators and \(F\)-Max-Ent Theorem. In: AIP Conference Proceedings Bayesian Inference and Maximum Entropy Methods in Science and Engineering (MaxEnt 2014), Amboise, France, 21–26 September 2014, vol. 1641, pp. 263–270 (2015)
Harsha, K.V., Subrahamanian Moosath, K.S.: Dually flat geometries of the deformed exponential family. Phys. A 433, 136–147 (2015)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Harsha, K.V., Moosath, K.S.S. (2015). Hessian Structures and Non-invariant (F, G)-Geometry on a Deformed Exponential Family. 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_24
Download citation
DOI: https://doi.org/10.1007/978-3-319-25040-3_24
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)