Bayes Hilbert Spaces

Abstract

Summary 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.

A kriging approach based on Aitchison geometry for the characterization of particle-size curves in heterogeneous aquifers

“…Furthermore, we show in Sect. 3.3 that an isometric isomorphism exists between A 2 ðT Þ and L 2 ðT Þ: Additional properties and generalizations are reported in (Egozcue et al 2006;van den Boogaart et al 2010).…”

“…The problem of kriging PDFs has been considered also by Salazar Buelvas (2011), who exploits a logarithmic transformation to deal with the data constraint. Our approach is however different and move from the geostatistical methodology proposed in (Menafoglio et al 2013) and the mathematical construction developed by Egozcue et al (2006) and further investigated in (van den Boogaart et al 2010). Our approach shares with FDA and CoDa the foundational role of geometry.…”

Profile Monitoring of Probability Density Functions via Simplicial Functional PCA With Application to Image Data

“…This space was designed by and Van den Boogaart et al (2014) precisely to represent the salient features of density functions when interpreted in the light of compositional data analysis. For instance, the information conveyed by compositional data is well-known to be relative (Aitchison, 1986;Pawlowsky-Glahn and Egozcue, 2001), that is, the relevant information is provided by ratios between the parts (i.e., the point evaluation of the functions) rather that by the absolute value of the functions themselves.…”

“…A number of authors (e.g., , Van den Boogaart et al, 2010, Delicado (2011, Van den Boogaart et al (2014), Menafoglio et al (2014Menafoglio et al ( , 2016aMenafoglio et al ( , 2016b, Hron et al (2016)) pointed out that PDFs can be interpreted as functional compositional data, i.e., functional observations carrying only relative information, which are usually collected in the form of constrained data integrating to a constant. Traditional FDA techniques operate in the space of square-integrable real measurable functions L 2 , whereas compositional data entails the use of a different space, known as Bayes space, B 2 , Van den Boogaart et al, 2010, Egozcue et al (2013, Van den Boogaart et al (2014)), that generalizes to the functional setting the well-known Aitchison geometry for compositional data (Aitchison, 1986;Pawlowsky-Glahn and Egozcue, 2001;Egozcue, 2009;Pawlowsky-Glahn and Buccianti, 2011). Therefore, the theory of Bayes spaces can be used to extend the applicative domain of FDA techniques to probability density curves.…”

Changing the Reference Measure in the Simplex and its Weighting Effects

“…Bayes Hilbert spaces are spaces of measures and densities, and their algebraicgeometric structure is an extension of the Aitchison geometry of the simplex. In fact, in (Boogaart et al 2014), it is shown that the simplex, endowed with the Aitchison geometry, is a particular case of a Bayes Hilbert space. In the development of Bayes Hilbert spaces, a reference probability measure is introduced as a parameter regulating the geometry of the measures and densities in the space.…”