Estimating the Unique Information of Continuous Variables

TL;DR

This paper introduces a novel method to estimate the unique information in continuous variables using copula decompositions and variational autoencoders, applicable to neural systems and complex networks.

Contribution

It presents the first approach for estimating unique information in continuous distributions with one versus two variables, extending PID analysis beyond Gaussian and discrete cases.

Findings

Accurately estimates unique information in Gaussian distributions.

Recovers effective connectivity in neural network models.

Reveals complex information trade-offs in recurrent networks.

Abstract

The integration and transfer of information from multiple sources to multiple targets is a core motive of neural systems. The emerging field of partial information decomposition (PID) provides a novel information-theoretic lens into these mechanisms by identifying synergistic, redundant, and unique contributions to the mutual information between one and several variables. While many works have studied aspects of PID for Gaussian and discrete distributions, the case of general continuous distributions is still uncharted territory. In this work we present a method for estimating the unique information in continuous distributions, for the case of one versus two variables. Our method solves the associated optimization problem over the space of distributions with fixed bivariate marginals by combining copula decompositions and techniques developed to optimize variational autoencoders. We obtain excellent agreement with known analytic results for Gaussians, and illustrate the power of our new approach in several brain-inspired neural models. Our method is capable of recovering the effective connectivity of a chaotic network of rate neurons, and uncovers a complex trade-off between redundancy, synergy and unique information in recurrent networks trained to solve a generalized XOR task.