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MiS Preprint
69/2015

Parametrized measure models

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

Abstract

We develope a new and general notion of parametric measure models and statistical models on an arbitrary sample space $\Omega$. This is given by a diffferentiable 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 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.

Received:
Oct 26, 2015
Published:
Oct 26, 2015
MSC Codes:
53C99, 62B05
Keywords:
Fisher quadratic form, Amari-Chentsov tensor, sufficient statistic, monotonicity

Related publications

inBook
2017 Repository Open Access
Nihat Ay, Jürgen Jost, Hông Vân Lê and Lorenz J. Schwachhöfer

Parametrized measure models

In: Information geometry
Cham : Springer, 2017. - pp. 121-184
(Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A series of modern surveys in mathematics ; 64)
inJournal
2018 Repository Open Access
Nihat Ay, Jürgen Jost, Hông Vân Lê and Lorenz J. Schwachhöfer

Parametrized measure models

In: Bernoulli, 24 (2018) 3, pp. 1692-1725