Bayes spaces: use of improper distributions and exponential families

  • J. J. Egozcue
  • V. Pawlowsky-Glahn
  • R. Tolosana-Delgado
  • M. I. Ortego
  • K. G. van den  Boogaart
Original Paper

Abstract

Bayes spaces are vector spaces of sigma-additive positive measures. Proportional measures are considered equivalent and can be represented by densities with respect to a fixed dominating measure. The addition in these spaces is perturbation. It corresponds to Bayes theorem, which appears as a linear operation. Bayes spaces, with continuous dominating measures, contain finite and infinite measures. Finite measures are equivalent to probability measures. Infinite measures include what in Bayesian statistics are called improper priors and non-integrable likelihood functions, justifying the use of such improper densities in Bayes theorem. Many concepts of probability theory can be handled in a natural way in the context of Bayes spaces. Particularly, an exponential family of probability densities appears as a cone contained in an affine subspace of the Bayes space. The framework of Bayes spaces allows an easy handling of exponential families and their extensions to improper distributions. Furthermore, the vector space structure of Bayes spaces allows the definition of derivatives of densities. In Bayesian statistics, these derivatives are a new tool to examine sensitivity of posterior distributions with respect to both observed data and prior changes.

Keywords

Simplex Aitchison geometry Derivative Sensitivity 

Mathematics Subject Classification

62F15 28E99 62E10 60A99 

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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • J. J. Egozcue
    • 1
    • 2
  • V. Pawlowsky-Glahn
    • 3
  • R. Tolosana-Delgado
    • 4
  • M. I. Ortego
    • 1
  • K. G. van den  Boogaart
    • 5
  1. 1.Dept. Matemática Aplicada IIIUniversidad Politécnica de CatalunyaBarcelonaSpain
  2. 2.Mod. C2 Campus Nord UPCBarcelonaSpain
  3. 3.Dept. Informática y Matemática AplicadaUniversidad de GironaGironaSpain
  4. 4.Centre Internacional de Recerca en Recursos Costaners (CIIRC)BarcelonaSpain
  5. 5.Institut für Stochastik, Fakultät für Mathematik und InformatikTU Bergakademie FreibergFreibergGermany

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