Partial Information Decomposition via Deficiency for Multivariate Gaussians

TL;DR

This paper develops a convex optimization framework to efficiently approximate partial information decompositions for multivariate Gaussian variables, overcoming computational challenges in high dimensions.

Contribution

It extends the analysis of Gaussian PIDs beyond scalar messages, characterizes when closed-form solutions exist, and proposes an efficient approximation method for high-dimensional cases.

Findings

Closed-form solutions exist only for certain Gaussian distributions.

The proposed convex optimization method accurately approximates PIDs in high dimensions.

Empirical results validate the efficiency and accuracy of the approximation.

Abstract

Bivariate partial information decompositions (PIDs) characterize how the information in a "message" random variable is decomposed between two "constituent" random variables in terms of unique, redundant and synergistic information components. These components are a function of the joint distribution of the three variables, and are typically defined using an optimization over the space of all possible joint distributions. This makes it computationally challenging to compute PIDs in practice and restricts their use to low-dimensional random vectors. To ease this burden, we consider the case of jointly Gaussian random vectors in this paper. This case was previously examined by Barrett (2015), who showed that certain operationally well-motivated PIDs reduce to a closed form expression for scalar messages. Here, we show that Barrett's result does not extend to vector messages in general, and characterize the set of multivariate Gaussian distributions that reduce to closed-form. Then, for all other multivariate Gaussian distributions, we propose a convex optimization framework for approximately computing a specific PID definition based on the statistical concept of deficiency. Using simplifying assumptions specific to the Gaussian case, we provide an efficient algorithm to approximately compute the bivariate PID for multivariate Gaussian variables with tens or even hundreds of dimensions. We also theoretically and empirically justify the goodness of this approximation.