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Bayes Hilbert spaces. (English) Zbl 1335.62025
Summary: A Bayes linear space is a linear space of equivalence classes of proportional $$\sigma$$-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.

##### MSC:
 62C10 Bayesian problems; characterization of Bayes procedures 60B11 Probability theory on linear topological spaces
sm
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##### References:
 [1] Aitchison, The Statistical Analysis of Compositional Data pp 416– (1986) · Zbl 0688.62004 · doi:10.1007/978-94-009-4109-0 [2] Ash, Real Analysis and Probability pp 476– (1972) [3] Barceló-Vidal, Proceedings of IAMG’01 - The Sixth Annual Conference of the International Association for Mathematical Geology pp 1– (2001) [4] Bauer, Ma$$\beta$$- und Integrationstheorie, 2 überarb. Auflage pp 260– (1992) [5] Bernardo, Bayesian Theory pp 608– (1984) [6] Bickel, On some global measures of the deviations of density function estimates, Ann. Statist. 1 pp 1071– (1973) · Zbl 0275.62033 · doi:10.1214/aos/1176342558 [7] Billheimer, Statistical interpretation of species composition, J. Amer. Statist. Assoc. 96 pp 1205– (2001) · Zbl 1073.62573 · doi:10.1198/016214501753381850 [8] Birnbaum, On the foundations of statistical inference (with discussion), J. Amer. Statist. Assoc. 57 pp 269– (1962) · Zbl 0107.36505 · doi:10.1080/01621459.1962.10480660 [9] Boogaart, Bayes linear spaces, SORT 34 pp 201– (2010) [10] Bowman, Applied Smoothing Techniques for Data Analysis: The Kernel Approach with S-Plus Ilustrations pp 193– (1997) [11] Durbin, Distribution Theory for Tests Based on the Sample Distribution Function pp 64– (1973) · Zbl 0267.62002 · doi:10.1137/1.9781611970586 [12] Egozcue, Groups of parts and their balances in compositional data analysis, Math. Geol. 37 pp 795– (2005) · Zbl 1177.86018 · doi:10.1007/s11004-005-7381-9 [13] Egozcue, Isometric logratio transformations for compositional data analysis, Math. Geol. 35 pp 279– (2003) · Zbl 1302.86024 · doi:10.1023/A:1023818214614 [14] Egozcue, Hilbert space of probability density functions based on Aitchison geometry, Acta Math. Sin. (Engl. Ser.) 22 pp 1175– (2006) · Zbl 1113.46016 · doi:10.1007/s10114-005-0678-2 [15] Egozcue, Compositional Data Analysis: Theory and Applications pp 141– (2011) [16] Egozcue, Bayes spaces: Use of improper distributions and exponential families, Revista de la Real Academia de Ciencias Exactas, Fı’sicas y Naturales. Serie A, Matemáticas, RACSAM 107 pp 475– (2013) · Zbl 1280.62030 · doi:10.1007/s13398-012-0082-6 [17] Eynatten, Sediment generation in modern glacial settings: Grain-size and source-rock control on sediment composition, Sed. Geol. 280 pp 80– (2012) · doi:10.1016/j.sedgeo.2012.03.008 [18] Fréchet, Les éléments Aléatoires de Nature Quelconque dans une Espace Distancié, Ann. Inst. Henri Poincaré 10 pp 215– (1948) [19] Kullback, On information and sufficiency, Ann. Math. Stat. 22 pp 79– (1951) · Zbl 0042.38403 · doi:10.1214/aoms/1177729694 [20] Parzen, On estimation of a probability density function and mode, Ann. Math. Stat 33 pp 1065– (1962) · Zbl 0116.11302 · doi:10.1214/aoms/1177704472 [21] Pawlowsky-Glahn, Geometric approach to statistical analysis on the simplex, SERRA 15 pp 384– (2001) · Zbl 0987.62001 [22] Quintana, A non-parametric method for the measurement of size diversity, with emphasis on data standardisation, Limnology and Oceanography: Methods 6 pp 75– (2008) · doi:10.4319/lom.2008.6.75 [23] Robert, The Bayesian Choice. A Decision-Theoretic Motivation pp 436– (1994) · Zbl 0808.62005 [24] Rosenblatt, Remarks on some nonparametric estimates of a density function, Ann. Math. Stat 27 pp 832– (1956) · Zbl 0073.14602 · doi:10.1214/aoms/1177728190 [25] Silverman, Density Estimation for Statistics and Data Analysis pp 175– (1986) · doi:10.1007/978-1-4899-3324-9
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