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

TL;DR

This paper introduces a geometric framework in Bayes spaces for decomposing multivariate densities into independent and interaction components, linking it with copula representations and extending previous bivariate results.

Contribution

It reformulates orthogonal decomposition of distributions using Hilbert space theory and develops a multivariate extension with a Hoeffding-Sobol analog, connecting to copulas.

Findings

Decomposition into independent and interaction parts using Bayes spaces.

Extension of bivariate results to multivariate densities.

Connection established between decomposition and copula representations.

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 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.