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Ministerio de EconomÃa y Competitividad (Espanya)
Generalitat de Catalunya. AgÃ¨ncia de GestiÃ³ dâ€™Ajuts Universitaris i de Recerca Ministerio de Ciencia e InnovaciÃ³n (Espanya) Ministerio de EducaciÃ³n y Ciencia (Espanya) 

Van den Boogaart, Karl Gerald
Egozcue, Juan JosÃ© PawlowskyGlahn, Vera 

A Bayes linear space is a linear space of equivalence classes of proportional Ïƒfinite measures, including probability measures. Measures are identified with their density functions. Addition is given by Bayesâ€™ rule and substraction by RadonNikodym derivatives. The present contribution shows the subspace of squarelogintegrable 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 centredlogratio transformation, a generalization of that used in compositional data analysis, which maps the Hilbert space of measures into a subspace of squareintegrable 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: â€™Ingenio Mathematica (iMATH)â€™ Ref. No. CSD200600032; â€™CODARSSâ€™ Ref. MTM200913272; and â€™METRICSâ€™, Ref. MTM201233236. It was also supported by the Agencia de Gestio dâ€™Ajuts Universitaris i de Recerca of the Generalitat de Catalunya under the project Ref: 2009SGR424 

http://hdl.handle.net/2072/296149  
eng  
Wiley  
Tots els drets reservats  
Hilbert, Espais de
Hilbert space AnÃ lisi multivariable Multivariate analysis EstadÃstica bayesiana Bayesian statistical decision theory Funcional de densitat, Teoria del Density functionals Probabilitats Probabilities 

Bayes hilbert spaces  
info:eurepo/semantics/article  
Recercat 