Skip to main content
SpringerLink
Log in
Book cover

International Conference on Geometric Science of Information

GSI 2015: Geometric Science of Information pp 213–221Cite as

Hessian Structures and Non-invariant (F, G)-Geometry on a Deformed Exponential Family

  • Conference paper
  • First Online:

  • 2026 Accesses

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 9389))

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.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Amari, S.I., Nagaoka, H.: Methods of Information Geometry. Translations of Mathematical Monographs. Oxford University Press, Oxford (2000)

    MATH  Google Scholar 

  2. Harsha, K.V., Subrahamanian Moosath, K.S.: \(F\)-geometry and Amari’s \(\alpha \)-geometry on a statistical manifold. Entropy 16(5), 2472–2487 (2014)

    Article  MathSciNet  Google Scholar 

  3. Naudts, J.: Estimators, escort probabilities, and phi-exponential families in statistical physics. J. Ineq. Pure Appl. Math., 5(4) (2004). Article no 102

    Google Scholar 

  4. Eguchi, S., Komori, O., Ohara, A.: Duality of maximum entropy and minimum divergence. Entropy 16, 3552–3572 (2014)

    Article  MathSciNet  Google Scholar 

  5. 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)

    Article  MathSciNet  Google Scholar 

  6. 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)

    Chapter  Google Scholar 

  7. Amari, S.I., Ohara, A.: Geometry of q-exponential family of probability distributions. Entropy 13, 1170–1185 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. 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)

    Google Scholar 

  9. 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)

    Chapter  Google Scholar 

  10. 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)

    Google Scholar 

  11. Harsha, K.V., Subrahamanian Moosath, K.S.: Dually flat geometries of the deformed exponential family. Phys. A 433, 136–147 (2015)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Space Science and Technology, Valiamala P.O, 695547, Thiruvananthapuram, Kerala, India

    K. V. Harsha & K. S. Subrahamanian Moosath

Authors
  1. K. V. Harsha
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. S. Subrahamanian Moosath
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to K. S. Subrahamanian Moosath .

Editor information

Editors and Affiliations

  1. Bâtiment Alan Turing, CS35003, École Polytechnique, Palaiseau, France

    Frank Nielsen

  2. Thales Land\& Air Systems, Limours, France

    Frédéric Barbaresco

Rights and permissions

Reprints 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)

Publish with us

Policies and ethics

Search

Navigation