Acta Mathematica Sinica

, Volume 22, Issue 4, pp 1175–1182 | Cite as

Hilbert Space of Probability Density Functions Based on Aitchison Geometry

  • J. J. Egozcue
  • J. L. Díaz–Barrero
  • V. Pawlowsky–Glahn
ORIGINAL ARTICLES

Abstract

The set of probability functions is a convex subset of L1 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.

Keywords

Bayes' theorem Fourier coefficients Haar basis Aitchison distance Simplex Least squares approximation 

MR (2000) Subject Classification

46C15 42B05 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • J. J. Egozcue
    • 1
  • J. L. Díaz–Barrero
    • 1
  • V. Pawlowsky–Glahn
    • 2
  1. 1.Applied Mathematics IIIUniversitat Politècnica de CatalunyaBarcelonaSpain
  2. 2.Informatics and Applied MathematicsUniversitat de GironaGironaSpain

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