Bayes space is a function space defined as an equivalence class of measures with the same null-sets. Two measures are defined to be equivalent if they are proportional. The basic ideas of Bayes spaces have their roots in Compositional Data Analysis and the Aitchison geometry. Applications are mainly in statistics, specifically functional data analysis of density functions, aka density data analysis.
The basic structure of the Bayes space is that of a vector space, with addition and multiplication being defined by perturbation and powering. The space is formed over a formula_1-finite reference/base measure, denoted formula_2 or formula_3 depending on whether it is infinite or finite. Densities are considered as Radon-Nikodym derivatives of the measures with same null-sets as the base measure, and are equivalent if they are proportional. In case of finite base measures, Hilbert space structure can be achieved by defining a centered log-ratio transformation on the measures, mapping them to a subset of formula_4 consisting of functions integrating to 0.
Definitions and main results.
Consider a finite base measure formula_3 (not necessarily a probability measure) on a domain formula_6. This may be a uniform distribution on a bounded interval, or it can be a Radon-Nikodym derivative of the Lebesgue measure (the Gaussian distribution, for example). If we take two densities formula_7 with respect to formula_3, they are said to be B-equivalent if there exists a formula_9 s.t formula_10, denoted formula_11 (the convention formula_12 is used in cases where a measure is infinite). It can be shown that formula_13 is an equivalence relation. The Bayes space formula_14 is defined as the quotient space of all measures with the same null-sets in formula_6 as formula_3 under the equivalence relation formula_13.
The first challenge to analysing density functions is that formula_14 is not linear space under ordinary addition and multiplication since the ordinary difference between two densities would not be non-negative everywhere. Like in the Aitchison geometry for finite dimensional data, perturbation and powering is defined for densities:
Perturbation
formula_19
Powering
formula_20
where formula_21 are densities in formula_14 and formula_23 is some real number. It can be shown using the properties of multiplication and powering of real numbers that formula_24 forms a vector space over the real numbers.
The definition of Bayes space does not strictly require a finite reference measure formula_3. If Bayes space is defined over an infinite reference measure formula_2, it must be formula_1-finite (like the Lebesgue measure). The finite reference measure is, however, necessary for adding Hilbert space structure to a subset of formula_14. Consider the subspace
formula_29. For formula_30, this is a linear subspace and isometrically isomorphic to the Hilbert space formula_31 via the centered log-ratio (clr) transformation formula_32. The subspace of log-square integrable functions is termed the Bayes Hilbert space. It can be shown that the clr transformation is a linear isomorphism between the two spaces. Defining an inner product on formula_33 as the inner product of the clr transformations will provide the Hilbert space structure for formula_33, obtaining the centered log-ratio transformation as a linear isometry.
Multivariate densities.
The measure formula_3 does not have to be univariate (one-dimensional), but can also be defined as a product measure on Cartesian products, characterising bivariate (two-dimensional) or multivariate densities. The geometric structure of Hilbert spaces can be used to decompose multivariate densities orthogonally into independent and interaction parts using the concept of "geometric marginals". This decomposition has relations to copula theory. The geometry in formula_33 defines norms on densities that can be used to quantify "relative simplicial deviance", which is measure of how much of a bivariate distribution can be explained by the interaction part; in the multivariate case the relative simplicial deviance can be generalised to the "information composition".