Estimating differential entropy using recursive copula splitting

TL;DR

This paper introduces a recursive copula splitting method for estimating the differential entropy of multidimensional distributions using samples, effectively handling unknown or mixed supports and outperforming existing methods in high dimensions.

Contribution

The paper presents a novel recursive approach to estimate the copula entropy, improving accuracy and efficiency for high-dimensional data with unknown or mixed supports.

Findings

Accurate entropy estimation for distributions with unknown or mixed supports.

Superior performance and efficiency in high-dimensional settings (over 20 dimensions).

Effective decomposition of joint distribution into marginals and copula for entropy calculation.

Abstract

A method for estimating the Shannon differential entropy of multidimensional random variables using independent samples is described. The method is based on decomposing the distribution into a product of the marginal distributions and the joint dependency, also known as the copula. The entropy of marginals is estimated using one-dimensional methods. The entropy of the copula, which always has a compact support, is estimated recursively by splitting the data along statistically dependent dimensions. Numerical examples demonstrate that the method is accurate for distributions with compact and non-compact supports, which is imperative when the support is not known or of mixed type (in different dimensions). At high dimensions (larger than 20), our method is not only more accurate, but also significantly more efficient than existing approaches.