# Exploration of synergistic and redundant information sharing in static and dynamical Gaussian systems

arXiv: https://arxiv.org/abs/1411.2832 · HTML: https://mlsanity.com/paper/1411.2832/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Nov 2014.

## Built on by (most cited first, 16 of 16)

- [Disentanglement Analysis with Partial Information Decomposition](https://mlsanity.com/paper/2108.13753/) (2021)
- [Synergistic information supports modality integration and flexible learning in neural networks solving multiple tasks](https://mlsanity.com/paper/2210.02996/) (2022)
- [Partial Information Decomposition via Normalizing Flows in Latent Gaussian Distributions](https://mlsanity.com/paper/2510.04417/) (2025)
- [Many Perception Tasks are Highly Redundant Functions of their Input Data](https://mlsanity.com/paper/2407.13841/) (2024)
- [Disentangling Interactions and Dependencies in Feature Attribution](https://mlsanity.com/paper/2410.23772/) (2024)
- [SOmegaI: Score-based O-INFORMATION Estimation](https://mlsanity.com/paper/2402.05667/) (2024)
- [Decomposing Interventional Causality into Synergistic, Redundant, and Unique Components](https://mlsanity.com/paper/2501.11447/) (2025)
- [Broadcast Channel Cooperative Gain: An Operational Interpretation of Partial Information Decomposition](https://mlsanity.com/paper/2502.10878/) (2025)
- [Cooperative effects in feature importance of individual patterns: application to air pollutants and Alzheimer disease](https://mlsanity.com/paper/2508.00930/) (2025)
- [Information-theoretic signatures of causality in Bayesian networks and hypergraphs](https://mlsanity.com/paper/2512.20552/) (2025)
- [Task Relevance Is Not Local Replaceability: A Two-Axis View of Channel Information](https://mlsanity.com/paper/2605.07086/) (2026)
- [SPICE: Synergy and Partial Information Based Curriculum Evolution](https://mlsanity.com/paper/2606.16639/) (2026)
- [DIPHINE: Diffusion-based Phi-ID Neural Estimator](https://mlsanity.com/paper/2606.18997/) (2026)
- [Information Dynamics of Language Communication](https://mlsanity.com/paper/2606.30096/) (2026)
- [Partial Information Decomposition as a Multi-Contrast 3D MRI Selection Strategy for Resource-Constrained Deep Neural Network Training in Brain Tumor Segmentation](https://mlsanity.com/paper/2607.15396/) (2026)
- [Gaussian BROJA Partial Information Decomposition in Closed Form](https://mlsanity.com/paper/2610.13464/) (2026)

## Abstract (arXiv)

To fully characterize the information that two `source' variables carry about a third `target' variable, one must decompose the total information into redundant, unique and synergistic components, i.e. obtain a partial information decomposition (PID). However Shannon's theory of information does not provide formulae to fully determine these quantities. Several recent studies have begun addressing this. Some possible definitions for PID quantities have been proposed, and some analyses have been carried out on systems composed of discrete variables. Here we present the first in-depth analysis of PIDs on Gaussian systems, both static and dynamical. We show that, for a broad class of Gaussian systems, previously proposed PID formulae imply that: (i) redundancy reduces to the minimum information provided by either source variable, and hence is independent of correlation between sources; (ii) synergy is the extra information contributed by the weaker source when the stronger source is known, and can either increase or decrease with correlation between sources. We find that Gaussian systems frequently exhibit net synergy, i.e. the information carried jointly by both sources is greater than the sum of informations carried by each source individually. Drawing from several explicit examples, we discuss the implications of these findings for measures of information transfer and information-based measures of complexity, both generally and within a neuroscience setting. Importantly, by providing independent formulae for synergy and redundancy applicable to continuous time-series data, we open up a new approach to characterizing and quantifying information sharing amongst complex system variables.