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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

Wiley

Manager: Ministerio de Economía y Competitividad (Espanya)
Generalitat de Catalunya. Agència de Gestió d’Ajuts Universitaris i de Recerca
Ministerio de Ciencia e Innovación (Espanya)
Ministerio de Educación y Ciencia (Espanya)
Author: Van den Boogaart, Karl Gerald
Egozcue, Juan José
Pawlowsky-Glahn, Vera
Abstract: 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
Document access: http://hdl.handle.net/2072/296149
Language: eng
Publisher: Wiley
Rights: Tots els drets reservats
Subject: Hilbert, Espais de
Hilbert space
Anàlisi multivariable
Multivariate analysis
Estadística bayesiana
Bayesian statistical decision theory
Funcional de densitat, Teoria del
Density functionals
Probabilitats
Probabilities
Title: Bayes hilbert spaces
Type: info:eu-repo/semantics/article
Repository: Recercat

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