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Parametrized measure models
Publikationstyp
Book Part
Date Issued
2017-08-26
Sprache
German
Author(s)
First published in
Number in series
64
Start Page
121
End Page
184
Citation
Ergebnisse der Mathematik und ihrer Grenzgebiete Folge 3, 64: 121-184 (2017)
Publisher DOI
Scopus ID
Publisher
Springer
This chapter represents the most important technical achievement of this book, a combination of functional analysis and geometry as the natural framework for families of probability measures on general sample spaces. In order to work on such a sample space, one needs a base or reference measure. Other measures, like those in a parametric family, are then described by densities w.r.t. this base measure. Such a base measure, however, is not canonical, and it can be changed by multiplication with an L1 -function. But then, also the description of a parametric family by densities changes. Keeping track of the resulting functorial behavior and pulling it back to the parameter spaces of a parametric family is the key that unlocks the natural functional analytical properties of parametric families. We develop the appropriate differentiability and integrability concepts. In particular, we shall need roots (half-densities) and other fractional powers of densities. For instance, when the sample space is a differentiable manifold, its diffeomorphism group operates isometrically on the space of half-densities with their L2 -product. The latter again yields the Fisher metric. At the end of this chapter, we compare our framework with that of Pistone–Sempi which depends on an analysis of integrability properties under exponentiation.
DDC Class
510: Mathematik