# A Partial Information Decomposition for Multivariate Gaussian Systems Based on Information Geometry

## TL;DR

This paper extends a method for analyzing information in multivariate Gaussian systems using information geometry, showing its effectiveness and limitations.

## Contribution

The paper extends a partial information decomposition algorithm to multivariate Gaussian systems and proves its theoretical properties.

## Key findings

- Explicit expressions for partial information decomposition components are derived for multivariate Gaussian systems.
- The Iig algorithm satisfies non-negativity, self-redundancy, symmetry, and monotonicity properties.
- Iig sometimes overestimates synergy and shared information while underestimating unique information.

## Abstract

There is much interest in the topic of partial information decomposition, both in developing new algorithms and in developing applications. An algorithm, based on standard results from information geometry, was recently proposed by Niu and Quinn (2019). They considered the case of three scalar random variables from an exponential family, including both discrete distributions and a trivariate Gaussian distribution. The purpose of this article is to extend their work to the general case of multivariate Gaussian systems having vector inputs and a vector output. By making use of standard results from information geometry, explicit expressions are derived for the components of the partial information decomposition for this system. These expressions depend on a real-valued parameter which is determined by performing a simple constrained convex optimisation. Furthermore, it is proved that the theoretical properties of non-negativity, self-redundancy, symmetry and monotonicity, which were proposed by Williams and Beer (2010), are valid for the decomposition Iig derived herein. Application of these results to real and simulated data show that the Iig algorithm does produce the results expected when clear expectations are available, although in some scenarios, it can overestimate the level of the synergy and shared information components of the decomposition, and correspondingly underestimate the levels of unique information. Comparisons of the Iig and Idep (Kay and Ince, 2018) methods show that they can both produce very similar results, but interesting differences are provided. The same may be said about comparisons between the Iig and Immi (Barrett, 2015) methods.