Hilbert Space of Probability Density Functions Based on Aitchison Geometry  

作　　者：J.J.EGOZCUE J.L.DIAZ-BARRERO V.PAWLOWSKY-GLAHN 

机构地区：[1]Applied Mathematics Ⅲ,Universitat Politàcnica de Catalunya,Jordi Girona 1-3,C2,08034,Barcelona,Spain [2]Applied Mathematics Ⅲ,Universitat Politàcnica de Catalunya,Jordi Girona 13,C2,08034,Barcelona,Spain [3]Informatics and Applied Mathematics,Universitat de Girona,Campus Montilivi,P4,17071,Girona,Spain

出　　处：《Acta Mathematica Sinica,English Series》2006年第4期1175-1182,共8页数学学报（英文版）

基　　金：the Dirección General de Investigación of the Spanish Ministry for Science;Technology through the project BFM2003-05640/MATE and from the Departament d'Universitats,Recerca i Societat de la Informac

摘　　要：The set of probability functions is a convex subset of L1 and it does not have a linear space structure when using ordinary sum and multiplication by real constants. Moreover, difficulties arise when dealing with distances between densities. The crucial point is that usual distances are not invariant under relevant transformations of densities. To overcome these limitations, Aitchison＇s ideas on compositional data analysis are used, generalizing perturbation and power transformation, as well as the Aitchison inner product, to operations on probability density functions with support on a finite interval. With these operations at hand, it is shown that the set of bounded probability density functions on finite intervals is a pre-Hilbert space. A Hilbert space of densities, whose logarithm is square-integrable, is obtained as the natural completion of the pre-Hilbert space.The set of probability functions is a convex subset of L1 and it does not have a linear space structure when using ordinary sum and multiplication by real constants. Moreover, difficulties arise when dealing with distances between densities. The crucial point is that usual distances are not invariant under relevant transformations of densities. To overcome these limitations, Aitchison＇s ideas on compositional data analysis are used, generalizing perturbation and power transformation, as well as the Aitchison inner product, to operations on probability density functions with support on a finite interval. With these operations at hand, it is shown that the set of bounded probability density functions on finite intervals is a pre-Hilbert space. A Hilbert space of densities, whose logarithm is square-integrable, is obtained as the natural completion of the pre-Hilbert space.

关 键 词：Bayes＇ theorem Fourier coefficients Haar basis Aitchison distance SIMPLEX Least squares approximation 

分 类 号：O177.1[理学—数学]