Aitchison Geometry for Probability and Likelihood as a new approach to mathematical statistics

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dc.contributor Universitat de Girona. Departament d'Informàtica i Matemàtica Aplicada
dc.contributor.author Boogaart, K. Gerald van den
dc.contributor.editor Mateu i Figueras, Glòria
dc.contributor.editor Barceló i Vidal, Carles
dc.date.issued 2005-10
dc.identifier.citation Boogaart, K.G. 'Aitchison Geometry for Probability and Likelihood as a new approach to mathematical statistics' a CODAWORK’05. Girona: La Universitat, 2005 [consulta: 2 maig 2008]. Necessita Adobe Acrobat. Disponible a Internet a: http://hdl.handle.net/10256/653
dc.identifier.isbn 84-8458-222-1
dc.identifier.uri http://hdl.handle.net/10256/653
dc.description.abstract The Aitchison vector space structure for the simplex is generalized to a Hilbert space structure A2(P) for distributions and likelihoods on arbitrary spaces. Central notations of statistics, such as Information or Likelihood, can be identified in the algebraical structure of A2(P) and their corresponding notions in compositional data analysis, such as Aitchison distance or centered log ratio transform. In this way very elaborated aspects of mathematical statistics can be understood easily in the light of a simple vector space structure and of compositional data analysis. E.g. combination of statistical information such as Bayesian updating, combination of likelihood and robust M-estimation functions are simple additions/ perturbations in A2(Pprior). Weighting observations corresponds to a weighted addition of the corresponding evidence. Likelihood based statistics for general exponential families turns out to have a particularly easy interpretation in terms of A2(P). Regular exponential families form finite dimensional linear subspaces of A2(P) and they correspond to finite dimensional subspaces formed by their posterior in the dual information space A2(Pprior). The Aitchison norm can identified with mean Fisher information. The closing constant itself is identified with a generalization of the cummulant function and shown to be Kullback Leiblers directed information. Fisher information is the local geometry of the manifold induced by the A2(P) derivative of the Kullback Leibler information and the space A2(P) can therefore be seen as the tangential geometry of statistical inference at the distribution P. The discussion of A2(P) valued random variables, such as estimation functions or likelihoods, give a further interpretation of Fisher information as the expected squared norm of evidence and a scale free understanding of unbiased reasoning
dc.description.sponsorship Geologische Vereinigung; Institut d’Estadística de Catalunya; International Association for Mathematical Geology; Patronat de l’Escola Politècnica Superior de la Universitat de Girona; Fundació privada: Girona, Universitat i Futur; Càtedra Lluís Santaló d’Aplicacions de la Matemàtica; Consell Social de la Universitat de Girona; Ministerio de Ciencia i Tecnología.
dc.format.mimetype application/pdf
dc.language.iso eng
dc.publisher Universitat de Girona. Departament d’Informàtica i Matemàtica Aplicada
dc.rights Tots els drets reservats
dc.subject Estadística matemàtica
dc.title Aitchison Geometry for Probability and Likelihood as a new approach to mathematical statistics
dc.type info:eu-repo/semantics/conferenceObject


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