Item
Mateu i Figueras, GlÃ²ria
BarcelÃ³ i Vidal, Carles 

Universitat de Girona. Departament dâ€™InformÃ tica i MatemÃ tica Aplicada  
Boogaart, K. Gerald van den  
2005 October  
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 Mestimation 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 Geologische Vereinigung; Institut dâ€™EstadÃstica de Catalunya; International Association for Mathematical Geology; Patronat de lâ€™Escola PolitÃ¨cnica Superior de la Universitat de Girona; FundaciÃ³ privada: Girona, Universitat i Futur; CÃ tedra LluÃs SantalÃ³ dâ€™Aplicacions de la MatemÃ tica; Consell Social de la Universitat de Girona; Ministerio de Ciencia i TecnologÃa. 

application/pdf  
Boogaart, K.G. â€™Aitchison Geometry for Probability and Likelihood as a new approach to mathematical statisticsâ€™ a CODAWORKâ€™05. Girona: La Universitat, 2005 [consulta: 2 maig 2008]. Necessita Adobe Acrobat. Disponible a Internet a: http://hdl.handle.net/10256/653  
8484582221  
http://hdl.handle.net/10256/653  
eng  
Universitat de Girona. Departament dâ€™InformÃ tica i MatemÃ tica Aplicada  
Tots els drets reservats  
EstadÃstica matemÃ tica  
Aitchison Geometry for Probability and Likelihood as a new approach to mathematical statistics  
info:eurepo/semantics/conferenceObject  
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