Advertisement

Parametrized Measure Models

  • Nihat Ay
  • Jürgen Jost
  • Hông Vân Lê
  • Lorenz Schwachhöfer
Chapter
Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics book series (MATHE3, volume 64)

Abstract

This chapter represents the most important technical achievement of this book, a combination of functional analysis and geometry as the natural framework for families of probability measures on general sample spaces. In order to work on such a sample space, one needs a base or reference measure. Other measures, like those in a parametric family, are then described by densities w.r.t. this base measure. Such a base measure, however, is not canonical, and it can be changed by multiplication with an \(L^{1}\)-function. But then, also the description of a parametric family by densities changes. Keeping track of the resulting functorial behavior and pulling it back to the parameter spaces of a parametric family is the key that unlocks the natural functional analytical properties of parametric families. We develop the appropriate differentiability and integrability concepts. In particular, we shall need roots (half-densities) and other fractional powers of densities. For instance, when the sample space is a differentiable manifold, its diffeomorphism group operates isometrically on the space of half-densities with their \(L^{2}\)-product. The latter again yields the Fisher metric. At the end of this chapter, we compare our framework with that of Pistone–Sempi which depends on an analysis of integrability properties under exponentiation.

References

  1. 5.
    Aliprantis, C., Border, K.: Infinite Dimensional Analysis. Springer, Berlin (2007) zbMATHGoogle Scholar
  2. 8.
    Amari, S.: Differential-Geometric Methods in Statistics. Lecture Notes in Statistics, vol. 28. Springer, Heidelberg (1985) CrossRefzbMATHGoogle Scholar
  3. 9.
    Amari, S.: Differential geometrical theory of statistics. In: Differential Geometry in Statistical Inference, Institute of Mathematical Statistics, California. Lecture Notes–Monograph Series, vol. 10 (1987) Google Scholar
  4. 16.
    Amari, S., Nagaoka, H.: Methods of Information Geometry. Translations of Mathematical Monographs, vol. 191. Am. Math. Soc./Oxford University Press, Providence/London (2000) zbMATHGoogle Scholar
  5. 25.
    Ay, N., Jost, J., Lê, H.V., Schwachhöfer, L.: Information geometry and sufficient statistics. Probab. Theory Relat. Fields 162, 327–364 (2015) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 26.
    Ay, N., Jost, J., Lê, H.V., Schwachhöfer, L.: Parametrized measure models. Bernoulli (2015). To appear, arXiv:1510.07305
  7. 99.
    Friedrich, Th.: Die Fisher-Information und symplektische Strukturen. Math. Nachr. 152, 273–296 (1991) MathSciNetCrossRefzbMATHGoogle Scholar
  8. 101.
    Fukumizu, K.: Exponential manifold by reproducing kernel Hilbert spaces. In: Gibilisco, P., Riccomagno, E., Rogantin, M.-P., Winn, H. (eds.) Algebraic and Geometric Methods in Statistics, pp. 291–306. Cambridge University Press, Cambridge (2009) CrossRefGoogle Scholar
  9. 106.
    Gibilisco, P., Pistone, G.: Connections on non-parametric statistical models by Orlicz space geometry. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1(2), 325–347 (1998) MathSciNetCrossRefzbMATHGoogle Scholar
  10. 153.
    Krasnosel’skii, M.A., Rutickii, Ya.B.: Convex functions and Orlicz spaces. Fizmatgiz, Moscow (1958) (In Russian); English translation: P. Noordfoff Ltd., Groningen (1961) Google Scholar
  11. 160.
    Lauritzen, S.: Statistical manifolds. In: Differential Geometry in Statistical Inference, Institute of Mathematical Statistics, California. Lecture Note-Monograph Series, vol. 10 (1987) Google Scholar
  12. 169.
    Lovrić, M., Min-Oo, M., Ruh, E.: Multivariate normal distributions parametrized as a Riemannian symmetric space. J. Multivar. Anal. 74, 36–48 (2000) MathSciNetCrossRefzbMATHGoogle Scholar
  13. 190.
    Moser, J.: On the volume elements on a manifold. Trans. Am. Math. Soc. 120, 286–294 (1965) MathSciNetCrossRefzbMATHGoogle Scholar
  14. 192.
    Murray, M., Rice, J.: Differential Geometry and Statistics. Chapman & Hall, London (1993) CrossRefzbMATHGoogle Scholar
  15. 200.
    Neveu, J.: Bases Mathématiques du Calcul de Probabilités, deuxième édition. Masson, Paris (1970) zbMATHGoogle Scholar
  16. 201.
    Newton, N.: An infinite-dimensional statistical manifold modelled on Hilbert space. J. Funct. Anal. 263, 1661–1681 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  17. 216.
    Pistone, G., Sempi, C.: An infinite-dimensional structure on the space of all the probability measures equivalent to a given one. Ann. Stat. 23(5), 1543–1561 (1995) MathSciNetCrossRefzbMATHGoogle Scholar
  18. 219.
    Rao, C.R.: Information and the accuracy attainable in the estimation of statistical parameters. Bull. Calcutta Math. Soc. 37, 81–89 (1945) MathSciNetzbMATHGoogle Scholar
  19. 230.
    Santacroce, M., Siri, P., Trivellato, B.: New results on mixture and exponential models by Orlicz spaces. Bernoulli 22(3), 1431–1447 (2016) MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Nihat Ay
    • 1
    • 5
  • Jürgen Jost
    • 2
    • 5
  • Hông Vân Lê
    • 3
  • Lorenz Schwachhöfer
    • 4
  1. 1.Information Theory of Cognitive SystemsMPI for Mathematics in the SciencesLeipzigGermany
  2. 2.Geometric Methods and Complex SystemsMPI for Mathematics in the SciencesLeipzigGermany
  3. 3.Mathematical Institute of ASCRCzech Academy of SciencesPraha 1Czech Republic
  4. 4.Department of MathematicsTU Dortmund UniversityDortmundGermany
  5. 5.Santa Fe InstituteSanta FeUSA

Personalised recommendations