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Hilbert space of probability density functions based on Aitchison geometry. (English) Zbl 1113.46016
Summary: 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, generalized perturbation and power transformation are used, 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.

MSC:
46C99 Inner product spaces and their generalizations, Hilbert spaces
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