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Bayes hilbert spaces

A Bayes linear space is a linear space of equivalence classes of proportional σ-finite measures, including probability measures. Measures are identified with their density functions. Addition is given by Bayes’ rule and substraction by Radon-Nikodym derivatives. The present contribution shows the subspace of square-log-integrable densities to be a Hilbert space, which can include probability and infinite measures, measures on the whole real line or discrete measures. It extends the ideas from the Hilbert space of densities on a finite support towards Hilbert spaces on general measure spaces. It is also a generalisation of the Euclidean structure of the simplex, the sample space of random compositions. In this framework, basic notions of mathematical statistics get a simple algebraic interpretation. A key tool is the centred-log-ratio transformation, a generalization of that used in compositional data analysis, which maps the Hilbert space of measures into a subspace of square-integrable functions. As a consequence of this structure, distances between densities, orthonormal bases, and Fourier series representing measures become available. As an application, Fourier series of normal distributions and distances between them are derived, and an example related to grain size distributions is presented. The geometry of the sample space of random compositions, known as Aitchison geometry of the simplex, is obtained as a particular case of the Hilbert space when the measures have discrete and finite support

This research was supported by the Spanish Ministries of Education and Science and of Economy and Competitiveness under three projects: ’Ingenio Mathematica (i-MATH)’ Ref. No. CSD2006-00032; ’CODA-RSS’ Ref. MTM2009-13272; and ’METRICS’, Ref. MTM2012-33236. It was also supported by the Agencia de Gestio d’Ajuts Universitaris i de Recerca of the Generalitat de Catalunya under the project Ref: 2009SGR424

© Australian and New Zealand Journal of Statistics, 2014, vol. 56, núm. 2, p. 171-194

Wiley

Autor: Van den Boogaart, Karl Gerald
Egozcue, Juan José
Pawlowsky-Glahn, Vera
Data: juny 2014
Resum: A Bayes linear space is a linear space of equivalence classes of proportional σ-finite measures, including probability measures. Measures are identified with their density functions. Addition is given by Bayes’ rule and substraction by Radon-Nikodym derivatives. The present contribution shows the subspace of square-log-integrable densities to be a Hilbert space, which can include probability and infinite measures, measures on the whole real line or discrete measures. It extends the ideas from the Hilbert space of densities on a finite support towards Hilbert spaces on general measure spaces. It is also a generalisation of the Euclidean structure of the simplex, the sample space of random compositions. In this framework, basic notions of mathematical statistics get a simple algebraic interpretation. A key tool is the centred-log-ratio transformation, a generalization of that used in compositional data analysis, which maps the Hilbert space of measures into a subspace of square-integrable functions. As a consequence of this structure, distances between densities, orthonormal bases, and Fourier series representing measures become available. As an application, Fourier series of normal distributions and distances between them are derived, and an example related to grain size distributions is presented. The geometry of the sample space of random compositions, known as Aitchison geometry of the simplex, is obtained as a particular case of the Hilbert space when the measures have discrete and finite support
This research was supported by the Spanish Ministries of Education and Science and of Economy and Competitiveness under three projects: ’Ingenio Mathematica (i-MATH)’ Ref. No. CSD2006-00032; ’CODA-RSS’ Ref. MTM2009-13272; and ’METRICS’, Ref. MTM2012-33236. It was also supported by the Agencia de Gestio d’Ajuts Universitaris i de Recerca of the Generalitat de Catalunya under the project Ref: 2009SGR424
Format: application/pdf
ISSN: 1369-1473 (versió paper)
1467-842X (versió electrònica)
Accés al document: http://hdl.handle.net/10256/10929
Llenguatge: eng
Editor: Wiley
Col·lecció: MINECO/PN 2013-2015/MTM2012-33236
AGAUR/2009-2014/2009 SGR-424
MICINN/PN 2010-2012/MTM2009-13272
MEC/2006-2011/CSD2006-00032
Reproducció digital del document publicat a: http://dx.doi.org/10.1111/anzs.12074
Articles publicats (D-IMA)
És part de: © Australian and New Zealand Journal of Statistics, 2014, vol. 56, núm. 2, p. 171-194
Drets: Tots els drets reservats
Matèria: Hilbert, Espais de
Hilbert space
Anàlisi multivariable
Multivariate analysis
Estadística bayesiana
Bayesian statistical decision theory
Funcional de densitat, Teoria del
Density functionals
Probabilitats
Probabilities
Títol: Bayes hilbert spaces
Tipus: info:eu-repo/semantics/article
Repositori: DUGiDocs

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