# Invariant components of synergy, redundancy, and unique information among three variables

## TL;DR

This paper extends the Partial Information Decomposition framework to fully characterize three-variable dependencies without relying on source-target classifications, unveiling an invariant structure that captures all interdependencies.

## Contribution

It introduces a general, target/source classification-independent PID structure with seven nonnegative information atoms that fully describe three-variable dependencies.

## Key findings

- Identifies an invariant PID structure applicable to any three-variable system.
- Splits redundancy into source and non-source components based on correlations.
- Provides a complete characterization of interdependencies through the invariant structure.

## Abstract

In a system of three stochastic variables, the Partial Information Decomposition (PID) of Williams and Beer dissects the information that two variables (sources) carry about a third variable (target) into nonnegative information atoms that describe redundant, unique, and synergistic modes of dependencies among the variables. However, the classification of the three variables into two sources and one target limits the dependency modes that can be quantitatively resolved, and does not naturally suit all systems. Here, we extend the PID to describe trivariate modes of dependencies in full generality, without introducing additional decomposition axioms or making assumptions about the target/source nature of the variables. By comparing different PID lattices of the same system, we unveil a finer PID structure made of seven nonnegative information subatoms that are invariant to different target/source classifications and that are sufficient to construct any PID lattice. This finer structure naturally splits redundant information into two nonnegative components: the source redundancy, which arises from the pairwise correlations between the source variables, and the non-source redundancy, which does not, and relates to the synergistic information the sources carry about the target. The invariant structure is also sufficient to construct the system's entropy, hence it characterizes completely all the interdependencies in the system.