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Affine statistical bundle modeled on a Gaussian Orlicz–Sobolev space

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Abstract

The dually flat structure of statistical manifolds can be derived in a non-parametric way from a particular case of affine space defined on a qualified set of probability measures. The statistically natural displacement mapping of the affine space depends on the notion of Fisher’s score. The model space must be carefully defined if the state space is not finite. Among various options, we discuss how to use Orlicz–Sobolev spaces with Gaussian weight. Such a fully non-parametric set-up provides tools to discuss intrinsically infinite-dimensional evolution problems

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  1. The difference in terminology between Mathematical Statistics and Physics is sometimes confusing. The tutorial [51] and the textbook [8] should be helpful.

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Acknowledgements

It is a pleasure to acknowledge the contribution of many people to my work in non-parametric IG. In particular, I like to mention professor Sun-ichi Amari’s constant encouragement and the critical assessment by Nihat Ay, Jürgen Jost, Hông Vân Lê, Lorenz Schwachöfer in [9, 3.3]. I also like to mention friends and coworkers in order of appearance in this paper: Carlo Sempi, Paolo Gibilisco, Alberto Cena, Maria Piera Rogantin, Barbara Trivellato, Paola Siri, Marina Santacroce, Luigi Malagò, Luigi Montrucchio, Goffredo Chirco, Bertrand Lods. The author was partially supported by de Castro Statistics, Collegio Carlo Alberto, and is a member of GNAMPA, Istituto di Alta Matematica, Rome.

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The author is supported by de Castro Statistics, Collegio Carlo Alberto, and INdAM-Gnafa.

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  1. de Castro Statistics, Collegio Carlo Alberto, Piazza Vincenzo Arbarello 8, 10122, Turin, Italy

    Giovanni Pistone

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Pistone, G. Affine statistical bundle modeled on a Gaussian Orlicz–Sobolev space. Info. Geo. 7 (Suppl 1), 109–130 (2024). https://doi.org/10.1007/s41884-022-00078-6

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