# The Partial Entropy Decomposition: Decomposing multivariate entropy and mutual information via pointwise common surprisal

arXiv: https://arxiv.org/abs/1702.01591 · HTML: https://mlsanity.com/paper/1702.01591/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Feb 2017.

## Built on by (most cited first, 5 of 5)

- [Interaction Measures, Partition Lattices and Kernel Tests for High-Order Interactions](https://mlsanity.com/paper/2306.00904/) (2023)
- [Information-Theoretic Measures on Lattices for High-Order Interactions](https://mlsanity.com/paper/2408.07533/) (2024)
- [Covered Information Disentanglement: Model Transparency via Unbiased Permutation Importance](https://mlsanity.com/paper/2111.09744/) (2021)
- [Surveying the space of descriptions of a composite system with machine learning](https://mlsanity.com/paper/2411.18579/) (2024)
- [Interpretable Information-Decomposed Brain Graph Learning for fMRI-based Disease Diagnosis](https://mlsanity.com/paper/2608.20380/) (2026)

## Abstract (arXiv)

Obtaining meaningful quantitative descriptions of the statistical dependence within multivariate systems is a difficult open problem. Recently, the Partial Information Decomposition (PID) was proposed to decompose mutual information (MI) about a target variable into components which are redundant, unique and synergistic within different subsets of predictor variables. Here, we propose to apply the elegant formalism of the PID to multivariate entropy, resulting in a Partial Entropy Decomposition (PED). We implement the PED with an entropy redundancy measure based on pointwise common surprisal; a natural definition which is closely related to the definition of MI. We show how this approach can reveal the dyadic vs triadic generative structure of multivariate systems that are indistinguishable with classical Shannon measures. The entropy perspective also shows that misinformation is synergistic entropy and hence that MI itself includes both redundant and synergistic effects. We show the relationships between the PED and MI in two predictors, and derive two alternative information decompositions which we illustrate on several example systems. This reveals that in entropy terms, univariate predictor MI is not a proper subset of the joint MI, and we suggest this previously unrecognised fact explains in part why obtaining a consistent PID has proven difficult. The PED also allows separate quantification of mechanistic redundancy (related to the function of the system) versus source redundancy (arising from dependencies between inputs); an important distinction which no existing methods can address. The new perspective provided by the PED helps to clarify some of the difficulties encountered with the PID approach and the resulting decompositions provide useful tools for practical data analysis across a wide range of application areas.