Maximum Entropy distributions of correlated variables with prespecified marginals

TL;DR

This paper investigates the maximum entropy approach to derive joint distributions of correlated variables with fixed marginals, providing exact solutions for Gaussian cases and perturbative methods for others.

Contribution

It introduces methods to compute maximum entropy distributions for correlated variables with given marginals, including exact solutions for Gaussian marginals and perturbation techniques for general cases.

Findings

Exact maximum entropy solutions for Gaussian marginals.

Perturbation methods for non-Gaussian marginals.

Framework for constructing joint distributions with specified marginals.

Abstract

The problem of determining the joint probability distributions for correlated random variables with pre-specified marginals is considered. When the joint distribution satisfying all the required conditions is not unique, the "most unbiased" choice corresponds to the distribution of maximum entropy. The calculation of the maximum entropy distribution requires the solution of rather complicated nonlinear coupled integral equations, exact solutions to which are obtained for the case of Gaussian marginals; otherwise, the solution can be expressed as a perturbation around the product of the marginals if the marginal moments exist.