Orthogonal decomposition of multivariate densities in Bayes spaces and its connection with copulas

Bayes spaces were initially designed to provide a geometric framework for the modeling and analysis of distributional data. It has recently come to light that this methodology can be exploited to provide an orthogonal decomposition of bivariate probability distributions into an independent and an interaction part. In this paper, new insights into these results are provided by reformulating them using Hilbert space theory and a multivariate extension is developed using a distributional analog of the Hoeffding-Sobol identity. A connection between the resulting decomposition of a multivariate density and its copula-based representation is also provided.