Abstract
This paper develops Sobolev variants of the non-parametric statistical manifolds appearing in [10] and [11]. The manifolds are modelled on a particular class of weighted, mixed-norm Sobolev spaces, including a Hilbert-Sobolev space. Densities are expressed in terms of a deformed exponential function having linear growth, which lifts to a continuous nonlinear superposition (Nemytskii) operator. This property is used in the construction of finite-dimensional mixture and exponential submanifolds, on which approximations can be based. The manifolds of probability measures are developed in their natural setting, as embedded submanifolds of those of finite measures.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Amari, S.-I., Nagaoka, H.: Methods of Information Geometry. American Mathematical Society, Providence (2000)
Cena, A., Pistone, G.: Exponential statistical manifold. Ann. Inst. Stat. Math. 59, 27–56 (2007)
Chentsov, N.N.: Optimal Decision Rules and Optimal Inference. American Mathematical Society, Providence (1982)
Eguchi, S.: Second order efficiency of minimum contrast estimators in a curved exponential family. Ann. Stat. 11, 793–803 (1983)
Gibilisco, P., Pistone, G.: Connections on non-parametric statistical manifolds by Orlicz space geometry. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1, 325–347 (1998)
Lang, S.: Fundamentals of Differential Geometry. Springer, New York (2001)
Lods, B., Pistone, G.: Information geometry formalism for the spatially homogeneous Boltzmann equation. Entropy 17, 4323–4363 (2015)
Montrucchio, L., Pistone, G.: A class of non-parametric deformed exponential statistical models, arXiv:1709.01430 (2018)
Naudts, J.: Generalised Thermostatistics. Springer, London (2011). https://doi.org/10.1007/978-0-85729-355-8
Newton, N.J.: An infinite-dimensional statistical manifold modelled on Hilbert space. J. Funct. Anal. 263, 1661–1681 (2012)
Newton, N.J.: Infinite-dimensional statistical manifolds based on a balanced chart. Bernoulli 22, 711–731 (2016)
Newton, N.J.: A class of non-parametric statistical manifolds modelled on Sobolev space, arXiv:1808.06451 (2018)
Pistone, G., Rogantin, M.P.: The exponential statistical manifold: mean parameters, orthogonality and space transformaions. Bernoulli 5, 721–760 (1999)
Pistone, G., Sempi, C.: An infinite-dimensional geometric structure on the space of all the probability measures equivalent to a given one. Ann. Stat. 23, 1543–1561 (1995)
Runst, T., Sickel, W.: Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations. de Gruyter, Berlin (2011)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Newton, N.J. (2019). Sobolev Statistical Manifolds and Exponential Models. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2019. Lecture Notes in Computer Science(), vol 11712. Springer, Cham. https://doi.org/10.1007/978-3-030-26980-7_46
Download citation
DOI: https://doi.org/10.1007/978-3-030-26980-7_46
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-26979-1
Online ISBN: 978-3-030-26980-7
eBook Packages: Computer ScienceComputer Science (R0)