In a system of three stochastic variables, the Partial InformationDecomposition (PID) of Williams and Beer dissects the information that twovariables (sources) carry about a third variable (target) into nonnegativeinformation atoms that describe redundant, unique, and synergistic modes ofdependencies among the variables. However, the classification of the threevariables into two sources and one target limits the dependency modes that canbe quantitatively resolved, and does not naturally suit all systems. Here, weextend the PID to describe trivariate modes of dependencies in full generality,without introducing additional decomposition axioms or making assumptions aboutthe target/source nature of the variables. By comparing different PID latticesof the same system, we unveil a finer PID structure made of seven nonnegativeinformation subatoms that are invariant to different target/sourceclassifications and that are sufficient to construct any PID lattice. Thisfiner structure naturally splits redundant information into two nonnegativecomponents: the source redundancy, which arises from the pairwise correlationsbetween 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.
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