Bernoulli

  • Bernoulli
  • Volume 24, Number 3 (2018), 1692-1725.

Parametrized measure models

Nihat Ay, Jürgen Jost, Hông Vân Lê, and Lorenz Schwachhöfer

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Abstract

We develop a new and general notion of parametric measure models and statistical models on an arbitrary sample space $\Omega$ which does not assume that all measures of the model have the same null sets. This is given by a differentiable map from the parameter manifold $M$ into the set of finite measures or probability measures on $\Omega$, respectively, which is differentiable when regarded as a map into the Banach space of all signed measures on $\Omega$. Furthermore, we also give a rigorous definition of roots of measures and give a natural characterization of the Fisher metric and the Amari–Chentsov tensor as the pullback of tensors defined on the space of roots of measures. We show that many features such as the preservation of this tensor under sufficient statistics and the monotonicity formula hold even in this very general set-up.

Article information

Source
Bernoulli, Volume 24, Number 3 (2018), 1692-1725.

Dates
Received: December 2015
Revised: September 2016
First available in Project Euclid: 2 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.bj/1517540458

Digital Object Identifier
doi:10.3150/16-BEJ910

Mathematical Reviews number (MathSciNet)
MR3757513

Zentralblatt MATH identifier
06839250

Keywords
Amari–Chentsov tensor Fisher quadratic form monotonicity sufficient statistic

Citation

Ay, Nihat; Jost, Jürgen; Lê, Hông Vân; Schwachhöfer, Lorenz. Parametrized measure models. Bernoulli 24 (2018), no. 3, 1692--1725. doi:10.3150/16-BEJ910. https://projecteuclid.org/euclid.bj/1517540458


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References

  • [1] Amari, S. (1980). Theory of information space: A differential-geometrical foundation of statistics. RAAG Res. Notes 106 1–54.
  • [2] Amari, S. (1982). Differential geometry of curved exponential families – curvatures and information loss. Ann. Statist. 10 357–385.
  • [3] Amari, S. and Nagaoka, H. (2000). Methods of Information Geometry. Translations of Mathematical Monographs 191. Providence, RI: Amer. Math. Soc.; Oxford: Oxford Univ. Press. Translated from the 1993 Japanese original by Daishi Harada.
  • [4] Amari, S.-I., Barndorff-Nielsen, O.E., Kass, R.E., Lauritzen, S.L. and Rao, C.R. (1987). Differential Geometry in Statistical Inference. Institute of Mathematical Statistics Lecture Notes – Monograph Series 10. Hayward, CA: IMS.
  • [5] Ay, N., Jost, J., Lê, H.V. and Schwachhöfer, L. (2015). Information geometry and sufficient statistics. Probab. Theory Related Fields 162 327–364.
  • [6] Bauer, H. (2001). Measure and Integration Theory. De Gruyter Studies in Mathematics 26. Berlin: de Gruyter. Translated from the German by Robert B. Burckel.
  • [7] Bauer, M., Bruveris, M. and Michor, P.W. (2016). Uniqueness of the Fisher–Rao metric on the space of smooth densities. Bull. Lond. Math. Soc. 48 499–506.
  • [8] Borovkov, A.A. (1998). Mathematical Statistics. Amsterdam: Gordon and Breach Science Publishers. Translated from the Russian by A. Moullagaliev and revised by the author.
  • [9] Cena, A. and Pistone, G. (2007). Exponential statistical model. Ann. Inst. Statist. Math. 59 27–56.
  • [10] Chentsov, N. (1965). Category of mathematical statistics. Dokl. Acad. Nauk USSR 164 511–514.
  • [11] Chentsov, N. (1972). Statistical Decision Rules and Optimal Inference. Moscow: Nauka (in Russian). English translation in: Translation of Math. Monograph 53, AMS, Providence, 1982.
  • [12] Chentsov, N. (1978). Algebraic foundation of mathematical statistics. Math. Oper.forsch. Stat., Ser. Stat. 9 267–276.
  • [13] Efron, B. (1975). Defining the curvature of a statistical problem (with applications to second order efficiency). Ann. Statist. 3 1189–1242. With a discussion by by C.R. Rao, Don A. Pierce, D.R. Cox, D.V. Lindley, Lucien LeCam, J.K. Ghosh, J. Pfanzagl, Niels Keiding, A.P. Dawid, Jim Reeds and with a reply by the author.
  • [14] Fisher, R.A. (1922). On the mathematical foundations of theoretical statistics. Philos. Trans. R. Soc. Lond. Ser. A 222 309–368.
  • [15] Guillemin, V. and Sternberg, S. (1977). Geometric Asymptotics. Mathematical Surveys 14. Providence, RI: Amer. Math. Soc.
  • [16] Jeffreys, H. (1946). An invariant form for the prior probability in estimation problems. Proc. R. Soc. Lond. Ser. A 186 453–461.
  • [17] Lang, S. (2002). Introduction to Differentiable Manifolds, 2nd ed. Universitext. New York: Springer.
  • [18] Lauritzen, S. (1987). Statistical manifolds. In Differential Geometry in Statistical Inference. Institute of Mathematical Statistics, Lecture Note – Monograph Series 10. Hayward, CA: IMS.
  • [19] Lê, H.V. (2017). The uniqueness of the Fisher metric as information metric. Ann. Inst. Statist. Math. 69 879–896.
  • [20] Morozova, E. and Chentsov, N. (1991). Natural geometry on families of probability laws. In Probability theory, 8. Itogi Nauki i Techniki 83 133–265. Moscow: VINITI.
  • [21] Morse, N. and Sacksteder, R. (1966). Statistical isomorphism. Ann. Math. Stat. 37 203–214.
  • [22] Murray, M.K. and Rice, J.W. (1993). Differential Geometry and Statistics. Monographs on Statistics and Applied Probability 48. London: Chapman & Hall.
  • [23] Neveu, J. (1965). Mathematical Foundations of the Calculus of Probability. Translated by Amiel Feinstein. San Francisco, CA–London–Amsterdam: Holden–Day, Inc.
  • [24] Neyman, J. (1935). Sur un teorema concernente le cosidette statistiche sufficienti. G. Ist. Ital. Attuari 6 320–334.
  • [25] Pistone, G. (2013). Nonparametric information geometry. In Geometric Science of Information. Lecture Notes in Computer Science 8085 5–36. Heidelberg: Springer.
  • [26] Pistone, G. and Sempi, C. (1995). An infinite-dimensional geometric structure on the space of all the probability measures equivalent to a given one. Ann. Statist. 23 1543–1561.
  • [27] Rao, C.R. (1945). Information and the accuracy attainable in the estimation of statistical parameters. Bull. Calcutta Math. Soc. 37 81–91.
  • [28] Schwachhöfer, L., Ay, N., Jost, J. and Lê, H.V. (2015). Invariant geometric structures on statistical models. In Geometric Science of Information. Lecture Notes in Computer Science 9389 150–158. Cham: Springer.