Exact partial information decompositions for Gaussian systems based on dependency constraints

TL;DR

This paper extends the Idep partial information decomposition method to Gaussian systems, providing closed-form solutions and comparing it with existing methods to better understand information sharing structures.

Contribution

It derives closed-form solutions for Idep PID in Gaussian systems and compares its estimates with the minimum mutual information PID, highlighting differences in redundancy and synergy.

Findings

Closed-form solutions for Gaussian systems are derived.

Idep generally estimates lower redundancy and synergy than Immi.

Numerical and graphical illustrations demonstrate the method's effectiveness.

Abstract

The Partial Information Decomposition (PID) [arXiv:1004.2515] provides a theoretical framework to characterize and quantify the structure of multivariate information sharing. A new method (Idep) has recently been proposed for computing a two-predictor PID over discrete spaces. [arXiv:1709.06653] A lattice of maximum entropy probability models is constructed based on marginal dependency constraints, and the unique information that a particular predictor has about the target is defined as the minimum increase in joint predictor-target mutual information when that particular predictor-target marginal dependency is constrained. Here, we apply the Idep approach to Gaussian systems, for which the marginally constrained maximum entropy models are Gaussian graphical models. Closed form solutions for the Idep PID are derived for both univariate and multivariate Gaussian systems. Numerical and graphical illustrations are provided, together with practical and theoretical comparisons of the Idep PID with the minimum mutual information PID (Immi). [arXiv:1411.2832] In particular, it is proved that the Immi method generally produces larger estimates of redundancy and synergy than does the Idep method. In discussion of the practical examples, the PIDs are complemented by the use of deviance tests for the comparison of Gaussian graphical models.