Orthogonal decomposition of multivariate densities in Bayes spaces and relation with their copula-based representation

Abstract

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 construct an orthogonal decomposition of a bivariate probability density into an independence and an interaction part. In this paper, new insights into these results are given by reformulating them using Hilbert space theory, and a multivariate extension is developed using a distributional analog of the Hoeffding–Sobol identity. A connection is also made between the resulting decomposition of a multivariate density and its copula-based representation.