Bayes space

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. ^[1] Theoretical applications are mainly in statistics, specifically functional data analysis of density functions, aka density data analysis. ^{[2] [3] [4] [5]} Practical applications are in geochemistry ^[6] , COVID-19 modelling ^[7] , sediment analysis ^[8] and developmental research ^[9] . Alternative approaches to density analysis are based on the Wasserstein metric, often termed Wasserstein regression, have also been applied to medicine ^[10] .

The basic structure of the Bayes space is that of a vector space, with addition and multiplication being defined by perturbation and powering. ^[11] The space is formed over a ${\displaystyle \sigma }$-finite reference/base measure, denoted ${\displaystyle \lambda }$ or ${\displaystyle P}$ 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 ${\displaystyle L^{2}(P)}$ consisting of functions integrating to 0. ^[12]

Definitions and main results

Consider a finite base measure ${\displaystyle P}$ (not necessarily a probability measure) on a domain ${\displaystyle \Omega }$. 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 ${\displaystyle f,g}$ with respect to ${\displaystyle P}$, they are said to be B-equivalent if there exists a ${\displaystyle c>0}$ s.t ${\displaystyle f(x)=c\cdot g(x)}$, denoted ${\displaystyle f=_{B}g}$ (the convention ${\displaystyle c\cdot \infty =\infty }$ is used in cases where a measure is infinite). It can be shown that ${\displaystyle (=_{B})}$ is an equivalence relation. The Bayes space ${\displaystyle B(P)}$ is defined as the quotient space of all measures with the same null-sets in ${\displaystyle \Omega }$ as ${\displaystyle P}$ under the equivalence relation ${\displaystyle (=_{B})}$.

The first challenge to analysing density functions is that ${\displaystyle B(P)}$ 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

${\textstyle (f\oplus g)(x)=_{B}f(x)\cdot g(x)}$

Powering

${\textstyle \alpha \odot f(x)=_{B}f(x)^{\alpha }}$

where ${\displaystyle f(x),{\text{ }}g(x)}$ are densities in ${\displaystyle B(P)}$ and ${\displaystyle \alpha }$ is some real number. It can be shown using the properties of multiplication and powering of real numbers that ${\displaystyle (B(P),\oplus ,\odot )}$ forms a vector space over the real numbers.

The definition of Bayes space does not strictly require a finite reference measure ${\displaystyle P}$. If Bayes space is defined over an infinite reference measure ${\displaystyle \lambda }$, it must be ${\displaystyle \sigma }$-finite (like the Lebesgue measure). The finite reference measure is, however, necessary for adding Hilbert space structure to a subset of ${\displaystyle B(P)}$. Consider the subspace

${\textstyle B^{p}(P)=\{f\in B(P)|\int _{\Omega }|\log(f(x))|^{p}dP(x)<\infty \}}$. For ${\displaystyle p=2}$, this is a linear subspace and isometrically isomorphic to the Hilbert space ${\textstyle L_{0}^{2}(P)=\{g\in L^{2}(P):\int _{\Omega }g(x)dP(x)=0\}}$ via the centered log-ratio (clr) transformation ${\displaystyle {\text{clr}}(f(x))=\log(f(x))-{\frac {1}{P(\Omega )}}\int \log(f(x))dP(x)\in L_{0}^{2}(P)}$. 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 ${\displaystyle B^{2}(P)}$ as the inner product of the clr transformations will provide the Hilbert space structure for ${\displaystyle B^{2}(P)}$, obtaining the centered log-ratio transformation as a linear isometry.

Multivariate densities

The measure ${\displaystyle P}$ 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". ^{[13] [14] [9]} This decomposition has relations to copula theory. ^[14] The geometry in ${\displaystyle B^{2}(P)}$ 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; ^[13] in the multivariate case the relative simplicial deviance can be generalised to the "information composition". ^[14]

See also

• Compositional data analysis
• Functional data analysis
• Copulas
• Information geometry

References

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9. Škorňa, S.; Machalová, J.; Burkotová, J.; Hron, K.; Greven, S. (2024). "Approximation of bivariate densities with compositional splines". arXiv: 2405.11615 [stat.ME].
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11. van den Boogart, K. G.; Egozcue, J. J.; Pawlowsky-Glahn, V. (2010). "Bayes linear spaces". SORT: Statistics and Operations Research Transactions. 34: 201–222.
12. van den Boogart, K. G.; Egozcue, J. J.; Pawlowsky-Glahn, V. (2014). "Bayes Hilbert spaces" . Australian & New Zealand Journal of Statistics. 56 (2): 171–194. doi:10.1111/anzs.12074.
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14. Genest, C.; Hron, K.; Nešlehová, J. G. (2023). "Orthogonal decomposition of multivariate densities in Bayes spaces and relation with their copula-based representation". Journal of Multivariate Analysis. 198 (5) 105228. doi: 10.1016/j.jmva.2023.105228 .