Parametrized measure models
Nihat Ay, Jürgen Jost, Hông Vân Lê, and Lorenz J. Schwachhöfer
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Submission date: 26. Oct. 2015
published in: Information geometry / N.Ay ... (eds.)
Cham : Springer, 2017. - P. 121 - 184
(Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A series of modern surveys in mathematics ; 64)
DOI number (of the published article): 10.1007/978-3-319-56478-4_3
MSC-Numbers: 53C99, 62B05
Keywords and phrases: Fisher quadratic form, Amari-Chentsov tensor, sufficient statistic, monotonicity
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We develope a new and general notion of parametric measure models and statistical models on an arbitrary sample space Ω. This is given by a diffferentiable map from the parameter manifold M into the set of finite measures or probability measures on Ω, respectively, which is differentiable when regarded as a map into the Banach space of all signed measures on Ω. Furthermore, we also give a rigorous definition of roots of measures and give a natural definition 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.