# Gaussian BROJA Partial Information Decomposition in Closed Form

arXiv: https://arxiv.org/abs/2610.13464 · HTML: https://mlsanity.com/paper/2610.13464/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 28 references that are in the map.
- First posted on arXiv 9 Oct 2026.

## Builds on (most cited first, 10 of 10)

- [Quantifying & Modeling Multimodal Interactions: An Information Decomposition Framework](https://mlsanity.com/paper/2302.12247/) (2023)
- [The Algonauts Project 2023 Challenge: How the Human Brain Makes Sense of Natural Scenes](https://mlsanity.com/paper/2301.03198/) (2023)
- [A Rigorous Information-Theoretic Definition of Redundancy and Relevancy in Feature Selection Based on (Partial) Information Decomposition](https://mlsanity.com/paper/2105.04187/) (2021)
- [A Novel Approach to the Partial Information Decomposition](https://mlsanity.com/paper/1908.08642/) (2019)
- [Gaussian Partial Information Decomposition: Bias Correction and Application to High-dimensional Data](https://mlsanity.com/paper/2307.10515/) (2023)
- [Exploration of synergistic and redundant information sharing in static and dynamical Gaussian systems](https://mlsanity.com/paper/1411.2832/) (2014)
- [A Measure of the Complexity of Neural Representations based on Partial Information Decomposition](https://mlsanity.com/paper/2209.10438/) (2022)
- [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)
- [Partial Information Decomposition for Data Interpretability and Feature Selection](https://mlsanity.com/paper/2405.19212/) (2024)

## Abstract (arXiv)

The BROJA partial information decomposition provides an operationally motivated way to separate the information that two sources carry about a target into unique, redundant, and synergistic components. For continuous and high-dimensional variables, however, computing the BROJA decomposition requires solving a difficult optimization problem. Previous work showed that, when each source-target pair is jointly Gaussian, the problem can be reduced to finite-dimensional numerical optimization. Here, we derive a closed-form solution to this optimization problem, enabling direct computation of the BROJA decomposition in the general multivariate Gaussian setting. The resulting closed-form expression enables speedups exceeding two orders of magnitude for high-dimensional Gaussian BROJA PID. Code is available at https://github.com/brainsandmachines/Eigen_PID.