KV Cohomology in Information Geometry

  • Michel Nguiffo Boyom
  • Paul Mirabeau Byande


Statistical structures and some information geometry invariants are discussed from the cohomology point of view. Some comparison criteria for statistical models are studied. The KV anomaly of an algebra structure as well as the Maurer-Cartan polynomial functions of KV complexes are used to discuss the linear convexity problems for various kind of linear connections. Deformation of statistical structures is discussed as well. Through the paper the differential geometry of Hessian manifolds is involved in its KV cohomology versus.


Curvature Tensor Fisher Information Linear Connection Information Geometry Cochain Complex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Amari, S.I., Nagaoka, H.: Methods of Information Geometry. Translation of Mathetmatical Monogragphs. AMS, OxfordGoogle Scholar
  2. 2.
    Barbaresco, F.: Modèles autorégressifs: du coefficient de reflexion à la géométrie Riemannienne de l’information, Manuscrit 2 avril 1998, disponible sur le web (1998)Google Scholar
  3. 3.
    Byande, P.M.: des structures affines à la géométrie de l’information, thèse de Doctorat 2010, Univ Montpellier 2 (2010)Google Scholar
  4. 4.
    Carrire, Yves: Autour de la conjecture de L Markus sur les variétés affines. Inv. Math. 95, 615–628 (1989)CrossRefGoogle Scholar
  5. 5.
    Chentsov, N.N.: Statistical decision rules and optimal inference. Trans. Math. Mono 53, 12 (1972)Google Scholar
  6. 6.
    Eilenberg, C.: Cohomology theory of lie groups and lie algebras. Trans. Am. Math. Soc. 63, 85–124 (1948)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Dorfmeister, J.: Homogeneous Kahler manifolds admitting a transitive solvable group of automorphisms. Ann. Sci. Ec. Norm. Sup. 4(18), 143–180 (1985)MathSciNetGoogle Scholar
  8. 8.
    Dzhumadil’daev, A.: Cohomologies and deformations of right-symmetric algebras. J. Math. Sci. 93(6), 836–876 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Fried, D., Goldman, W., Hirsh, M.: Affine manifolds with nilpotent holonomy. Comment. Helvetici Math. 56, 487–523 (1983)CrossRefGoogle Scholar
  10. 10.
    Gerstenhaber, M.: On deformations of rings and algebras. Ann. Math. 79(1), 59–103 (1964)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Gindikin, S.G., Pyateckii-Sapiro, I.I., Vinberg, E.B.: Homogeneous Kahler manifolds, in Geometry of homogeneous bounded domains, CIME III Circ Urbino 3, 3rd edn. Cremonese, Rome (1968)Google Scholar
  12. 12.
    Goldman, W., Baues, O.: Is the deformation space of affine structures on the \(2\)-torus smooth. Contemp. Math. 389, 69–89 (2005)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vols. 1 and 2. Wiley-Interscience, New york (1969)Google Scholar
  14. 14.
    Lauritzen, S.L.: Statistical manifolmds. In: Amari, S., Barndorff-Nielsen, O., Kass, R., Lauritzen, S., Rao, C.R. (eds.) Differential Geometry in Statistical Inference, IMS Lecture Notes, vol. 10, pp. 163–216 (1987)Google Scholar
  15. 15.
    Naudts, J.: Genralized Thermostatistics. Springer, London (2011)CrossRefGoogle Scholar
  16. 16.
    Koszul, J.-L.: Déformations des variétés localement plates. Ann. Inst. Fourier 18(1), 103–114 (1968)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Nguiffo Boyom, M.: The cohomology of Koszul-Vinberg algebras. Pac. J. Math. 225(1), 119–153 (2006)zbMATHCrossRefGoogle Scholar
  18. 18.
    Nguiffo Boyom, M.: Some Lagrangian Invariants of Symplectic Manifolds. Geometry and Topology of Manifolds, Banach Center Publications, vol. 76, pp. 515–525. (2007)Google Scholar
  19. 19.
    Nguiffo Boyom, M.: Réductions Kahlériennes dans les groupes de Lie Résolubles et applications. Osaka J. Math. 47, 237–283 (2010)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Nguiffo Boyom, M., Ngakeu, F.: Cohomology and Homology of Abelian Groups Graded Koszul-Vinberg Algebras, Manuscript 2008Google Scholar
  21. 21.
    Nguiffo Boyom, M., Wolak, R.A.: Affine structures and KV-cohomology. J. Geom. Phys. 42(4), 307–317 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Nguiffo Boyom, M., Wolak, R.: Local structure of Koszul-Vinberg and of Lie algebroids. Bull. Sci. Math. 128(6), 467–479 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Nijenhuis, A.: Sur une classe de propriétés communes à quelques types différents d’algèbres. Enseignement Math. 2(14), 225–277 (1968)MathSciNetGoogle Scholar
  24. 24.
    Picard, D.: Statistical morphisms and related invariance properties. Ann Inst. Stat. Math. 44(1), 45–61 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Rao, C.R.: Information and accuracy attainable in the estimation of statistical parameters. Bull. Calcutta Math. Soc. 37, 81–91 (1945)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Shima, H.: Homogeneous Hessian manifolds. Ann Inst Fourier 30, 91–128 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Shima, H.: The Differential Geometry of Hessian Manifolds. World scientific Publishing Co, Pte. Ltd, Hackensack (2007)Google Scholar
  28. 28.
    Wolf, J.: Spaces of Constant Curvature. McGraw-Hill, New York (1967)zbMATHGoogle Scholar
  29. 29.
    Zhan, J.: A note on curvature of connection of a statistical model. Ann. Inst. Stat. Math. 59, 161–170 (2007)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Université Montpellier 2Montpellier Cedex 5France

Personalised recommendations