{ "dc.contributor.author": "Boogaart, Karl Gerald van den" , "dc.contributor.author": "Egozcue, Juan José" , "dc.contributor.author": "Pawlowsky-Glahn, Vera" , "dc.date.accessioned": "2015-09-16T12:50:49Z" , "dc.date.available": "2015-09-16T12:50:49Z" , "dc.date.issued": "2014-06" , "dc.identifier.issn": "1369-1473 (versió paper)" , "dc.identifier.issn": "1467-842X (versió electrònica)" , "dc.identifier.uri": "http://hdl.handle.net/10256/10929" , "dc.description.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" , "dc.description.sponsorship": "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" , "dc.format.mimetype": "application/pdf" , "dc.language.iso": "eng" , "dc.publisher": "Wiley" , "dc.relation": "MINECO/PN 2013-2015/MTM2012-33236" , "dc.relation": "AGAUR/2009-2014/2009 SGR-424" , "dc.relation": "MICINN/PN 2010-2012/MTM2009-13272" , "dc.relation": "MEC/2006-2011/CSD2006-00032" , "dc.relation.isformatof": "Reproducció digital del document publicat a: http://dx.doi.org/10.1111/anzs.12074" , "dc.relation.ispartof": "© Australian and New Zealand Journal of Statistics, 2014, vol. 56, núm. 2, p. 171-194" , "dc.relation.ispartofseries": "Articles publicats (D-IMA)" , "dc.rights": "Tots els drets reservats" , "dc.subject": "Hilbert, Espais de" , "dc.subject": "Hilbert space" , "dc.subject": "Anàlisi multivariable" , "dc.subject": "Multivariate analysis" , "dc.subject": "Estadística bayesiana" , "dc.subject": "Bayesian statistical decision theory" , "dc.subject": "Funcional de densitat, Teoria del" , "dc.subject": "Density functionals" , "dc.subject": "Probabilitats" , "dc.subject": "Probabilities" , "dc.title": "Bayes hilbert spaces" , "dc.type": "info:eu-repo/semantics/article" , "dc.rights.accessRights": "info:eu-repo/semantics/embargoedAccess" , "dc.embargo.terms": "Cap" , "dc.type.version": "info:eu-repo/semantics/publishedVersion" , "dc.identifier.doi": "http://dx.doi.org/10.1111/anzs.12074" , "dc.identifier.idgrec": "022388" , "dc.contributor.funder": "Ministerio de Economía y Competitividad (Espanya)" , "dc.contributor.funder": "Generalitat de Catalunya. Agència de Gestió d'Ajuts Universitaris i de Recerca" , "dc.contributor.funder": "Ministerio de Ciencia e Innovación (Espanya)" , "dc.contributor.funder": "Ministerio de Educación y Ciencia (Espanya)" }