Abstract
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 \(L^{1}\)-function. But then, also the description of a parametric family by densities changes. Kee** 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 \(L^{2}\)-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.
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Notes
- 1.
\({\varOmega }\) will take over the role of the finite sample space \(I\) in Sect. 2.1.
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- 3.
Again, we are employing a fundamental mathematical principle here: Instead of considering objects in isolation, we rather focus on the transformations between them. Thus, instead of an individual basis, we consider the transformation that generates it from some (arbitrarily chosen) standard basis. This automatically gives a very powerful structure, that of a group (of transformations).
- 4.
More precisely, \({\mathcal {M}}_{+}({\varOmega }, \mu_{0})\) is not open unless \({\varOmega }\) is the disjoint union of finitely many \(\mu_{0}\)-atoms, where \(A {\subseteq }{\varOmega }\) is a \(\mu _{0}\)-atom if for each \(B {\subseteq }A\) either \(B\) or \(A\backslash B\) is a \(\mu_{0}\)-null set.
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Ay, N., Jost, J., Lê, H.V., Schwachhöfer, L. (2017). Parametrized Measure Models. In: Information Geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol 64. Springer, Cham. https://doi.org/10.1007/978-3-319-56478-4_3
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