Abstract
The set of probability functions is a convex subset of L 1 and it does not have a linear space structure when using ordinary sum and multiplication by real constants. Moreover, difficulties arise when dealing with distances between densities. The crucial point is that usual distances are not invariant under relevant transformations of densities. To overcome these limitations, Aitchison's ideas on compositional data analysis are used, generalizing perturbation and power transformation, as well as the Aitchison inner product, to operations on probability density functions with support on a finite interval. With these operations at hand, it is shown that the set of bounded probability density functions on finite intervals is a pre–Hilbert space. A Hilbert space of densities, whose logarithm is square–integrable, is obtained as the natural completion of the pre–Hilbert space.
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This research has received financial support from the Dirección General de Investigación of the Spanish Ministry for Science and Technology through the project BFM2003–05640/MATE and from the Departament d'Universitats, Recerca i Societat de la Informació of the Generalitat de Catalunya through the project 2003XT 00079
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Egozcue, J.J., Díaz–Barrero, J.L. & Pawlowsky–Glahn, V. Hilbert Space of Probability Density Functions Based on Aitchison Geometry. Acta Math Sinica 22, 1175–1182 (2006). https://doi.org/10.1007/s10114-005-0678-2
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DOI: https://doi.org/10.1007/s10114-005-0678-2