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A Lecture About the Use of Orlicz Spaces in Information Geometry

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Geometric Structures of Statistical Physics, Information Geometry, and Learning (SPIGL 2020)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 361))

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Abstract

This chapter is a revised version of a tutorial lecture that I presented at the École de Physique des Houches on July 26–31 2020. Topics include: Non-parametric Information Geometry, the Statistical bundle, exponential Orlicz spaces, and Gaussian Orlicz-Sobolev spaces.

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References

  1. Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, Pure and Applied Mathematics (Amsterdam), 2nd edn., vol. 140. Elsevier/Academic Press, Amsterdam (2003)

    Google Scholar 

  2. Amari, S.I.: Information geometry and its applications. Appl. Math. Sci. 194 (2016). https://doi.org/10.1007/978-4-431-55978-8

  3. Amari, S., Nagaoka, H.: Methods of information geometry. Am. Math. Soc. (2000). Translated from the 1993 Japanese original by Daishi Harada

    Google Scholar 

  4. Ay, N., Jost, J., Lê, H.V., Schwachhöfer, L.: Information geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 64. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-56478-4

  5. Brezis, H.: Functional analysis. In: Sobolev Spaces and Partial Differential Equations. Universitext. Springer, New York (2011)

    Google Scholar 

  6. Brown, L.D.: Fundamentals of statistical exponential families with applications in statistical decision theory, No. 9 in IMS Lecture Notes. Monograph Series, Institute of Mathematical Statistics (1986)

    Google Scholar 

  7. Buldygin, V.V., Kozachenko, Y.V.: Metric characterization of random variables and random processes, Translations of Mathematical Monographs, vol. 188. American Mathematical Society, Providence, RI (2000). Translated from the 1998 Russian original by V. Zaiats

    Google Scholar 

  8. Cena, A.: Geometric structures on the non-parametric statistical manifold. Ph.D. thesis, Università degli Studi di Milano (2002)

    Google Scholar 

  9. Chirco, G., Malagò, L., Pistone, G.: Lagrangian and Hamiltonian mechanics for probabilities on the statistical manifold (2020). arXiv:2009.09431

  10. Efron, B., Hastie, T.: Computer age statistical inference, Institute of Mathematical Statistics (IMS) Monographs, vol. 5. Cambridge University Press, New York (2016). https://doi.org/10.1017/CBO9781316576533. Algorithms, evidence, and data science

  11. Gibilisco, P., Pistone, G.: Connections on non-parametric statistical manifolds by Orlicz space geometry. IDAQP 1(2), 325–347 (1998)

    MathSciNet  MATH  Google Scholar 

  12. Hyvärinen, A.: Estimation of non-normalized statistical models by score matching. J. Mach. Learn. Res. 6, 695–709 (2005)

    MathSciNet  MATH  Google Scholar 

  13. Kass, R.E., Vos, P.W.: Geometrical foundations of asymptotic inference. Wiley Series in Probability and Statistics: Probability and Statistics. Wiley, New York (1997). https://doi.org/10.1002/9781118165980. A Wiley-Interscience Publication

  14. Kriegl, A., Michor, P.W.: The convenient setting of global analysis. Math. Surv. Monographs, vol. 53. American Mathematical Society, Providence, RI (1997). https://doi.org/10.1090/surv/053.

  15. Lang, S.: Differential and Riemannian Manifolds. In: Graduate Texts in Mathematics, 3rd edn., vol. 160. Springer, New York (1995)

    Google Scholar 

  16. Ledoux, M.: The concentration of measure phenomenon. Math. Surv. Monogr. 89 (2001). https://doi.org/10.1090/surv/089

  17. Lods, B., Pistone, G.: Information geometry formalism for the spatially homogeneous Boltzmann equation. Entropy 17(6), 4323–4363 (2015)

    Article  MathSciNet  Google Scholar 

  18. Lott, J.: Some geometric calculations on Wasserstein space. Comm. Math. Phys.277(2), 423–437 (2008). https://doi.org/10.1007/s00220-007-0367-3

  19. Malliavin, P.: Integration and Probability. Graduate Texts in Mathematics, vol. 157. Springer, New York (1995). With the collaboration of Héléne Airault, Leslie Kay and Gérard Letac, Edited and translated from the French by Kay, With a foreword by Mark Pinsky

    Google Scholar 

  20. Malliavin, P.: Stochastic Analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 313. Springer, Heidelberg (1997)

    Google Scholar 

  21. Musielak, J.: Orlicz Spaces and Modular Spaces. Lecture Notes in Mathematics, vol. 1034. Springer, Heidelberg (1983)

    Google Scholar 

  22. Newton, N.J.: Sobolev statistical manifolds and exponential models. In: Geometric science of information, Lecture Notes in Computer Science, vol. 11712, pp. 443–452. Springer, Cham (2019)

    Google Scholar 

  23. Nielsen, F.: An elementary introduction to information geometry. Entropy 22(10) (2020). https://doi.org/10.3390/e22101100

  24. Nourdin, I., Peccati, G.: Normal approximations with Malliavin calculus. InL Cambridge Tracts in Mathematics, vol. 192. Cambridge University Press, Cambridge (2012). https://doi.org/10.1017/CBO9781139084659. from Stein’s method to universality

  25. Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Differ. Equ. 26(1-2), 101–174 (2001)

    Google Scholar 

  26. Pistone, G.: Examples of the application of nonparametric information geometryto statistical physics. Entropy 15(10), 4042–4065 (2013). https://doi.org/10.3390/e15104042

  27. Pistone, G.: Nonparametric information geometry. In: Nielsen, F., Barbaresco, F. (eds.) Geometric science of information, Lecture Notes in Computer Science, First International Conference, GSI 2013 Paris, France, 28–30 August 2013, Proceedings, vol. 8085, pp. 5–36. Springer, Heidelberg (2013)

    Google Scholar 

  28. Pistone, G.: Translations in the exponential Orlicz space with Gaussian weight. In: Nielsen, F., Barbaresco, F. (eds.) Geometric Science of Information. Third International Conference, GSI 2017, Paris, France, 7–9 November 2017, Proceedings, LNCS, No. 10589, pp. 569–576. Springer (2017)

    Google Scholar 

  29. Pistone, G.: Information geometry of the Gaussian space. In: Information Geometry and Its Applications. Springer Proceedings in Mathematics and Statistics, vol. 252, pp. 119–155. Springer, Cham (2018)

    Google Scholar 

  30. Pistone, G.: Lagrangian function on the finite state space statistical bundle. Entropy 20(2),  139 (2018). https://doi.org/10.3390/e20020139

  31. Pistone, G.: Information geometry of the probability simplex: a short course. Nonlinear Pheno. Complex Syst. 23(2), 221–242 (2020). arXiv:1911.01876

  32. Pistone, G., Rogantin, M.: The exponential statistical manifold: mean parameters, orthogonality and space transformations. Bernoulli 5(4), 721–760 (1999)

    Article  MathSciNet  Google Scholar 

  33. Pistone, G., Sempi, C.: An infinite-dimensional geometric structure on the space of all the probability measures equivalent to a given one. Ann. Statist. 23(5), 1543–1561 (1995)

    Article  MathSciNet  Google Scholar 

  34. Santacroce, M., Siri, P., Trivellato, B.: New results on mixture andexponential models by Orlicz spaces. Bernoulli 22(3), 1431–1447(2016). https://doi.org/10.3150/15-BEJ698

  35. Santacroce, M., Siri, P., Trivellato, B.: Exponential models by Orlicz spacesand applications. J. Appl. Probab. 55(3), 682–700 (2018).https://doi.org/10.1017/jpr.2018.45

  36. Siri, P., Trivellato, B.: Robust concentration inequalities in maximalexponential models. Statist. Probab. Lett. 170, 109001 (2021). https://doi.org/10.1016/j.spl.2020.109001

  37. Vershynin, R.: High-Dimensional Probability: An Introduction with Applications in Data Science. Cambridge Series in Statistical and Probabilistic Mathematics, vol. 47. Cambridge University Press, Cambridge (2018). https://doi.org/10.1017/9781108231596. With a foreword by Sara van de Geer

  38. Wainwright, M.J.: High-dimensional statistics: a non-asymptotic viewpoint. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge (2019). https://doi.org/10.1017/9781108627771

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Acknowledgements

The author acknowledges the support by de Castro Statistics, Collegio Carlo Alberto, Turin, Italy. He is a member of GNAMPA-INDAM. The author thanks an anonymous reviewer and L. Malagò for their helpful comments.

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Authors and Affiliations

  1. de Castro Statistics, Collegio Carlo Alberto, Turin, Italy

    Giovanni Pistone

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  1. Giovanni Pistone
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Correspondence to Giovanni Pistone .

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Editors and Affiliations

  1. Thales Land & Air Systems, Technical Directorate, Thales, Limours, France

    Frédéric Barbaresco

  2. Sony Computer Science Laboratories Inc., Tokyo, Japan

    Frank Nielsen

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Pistone, G. (2021). A Lecture About the Use of Orlicz Spaces in Information Geometry. In: Barbaresco, F., Nielsen, F. (eds) Geometric Structures of Statistical Physics, Information Geometry, and Learning. SPIGL 2020. Springer Proceedings in Mathematics & Statistics, vol 361. Springer, Cham. https://doi.org/10.1007/978-3-030-77957-3_10

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