  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.
