Orthogonal decomposition of bivariate densities using the Bayes space methodology

Abstract

Bivariate probability densities capture relationships within and between two continuous random variables. As such, they carry essentially relative information and follow the scale invariance property which is widely recognized in Bayesian statistics (e.g., when normalizing constant are neglected from computations). Both these properties are captured by the so called Bayes spaces, which are spaces of positive measures equipped with a Hilbert space structure built as a generalization of the log-ratio methodology for compositional data. In fact, Bayes spaces form a natural sample space for “scale invariant” measures and their respective densities. It is possible to decompose the bivariate densities orthogonally into independent and interactive parts, the former being product of revised definitions of marginal densities and the latter capturing the relationships between the random variables. This has several important consequences in the probability context. For instance, this yields the marginal invariance, i.e., when the bivariate density is shifted (in the Bayes space sense) by marginal densities, the interaction density is not changed. Furthermore, the centred logratio transformation of bivariate densities enables to move them from the Bayes space to the standard L 2 space where popular methods of functional data analysis can be applied. The novel theoretical framework here proposed has thus clear potential on the application side, allowing to analyse samples of densities arising, for example, as a result of aggregation of massive data coming from large-scale studies or automated collection of data