Partial information decomposition

Partial Information Decomposition is an extension of information theory, that aims to generalize the pairwise relations described by information theory to the interaction of multiple variables. ^[1]

It has been shown, however, that Partial Information Decomposition is mathematically not an extension of classical information theory as core properties/axioms of classical information theory can generally not be simultaneously fulfilled for Partial Information Decomposition. ^[2]

Motivation

Information theory can quantify the amount of information a single source variable ${\displaystyle X_{1}}$ has about a target variable ${\displaystyle Y}$ via the mutual information ${\displaystyle I(X_{1};Y)}$. If we now consider a second source variable ${\displaystyle X_{2}}$, classical information theory can only describe the mutual information of the joint variable ${\displaystyle \{X_{1},X_{2}\}}$ with ${\displaystyle Y}$, given by ${\displaystyle I(X_{1},X_{2};Y)}$. In general however, it would be interesting to know how exactly the individual variables ${\displaystyle X_{1}}$ and ${\displaystyle X_{2}}$ and their interactions relate to ${\displaystyle Y}$.

Consider that we are given two source variables ${\displaystyle X_{1},X_{2}\in \{0,1\}}$ and a target variable ${\displaystyle Y=XOR(X_{1},X_{2})}$. In this case the total mutual information ${\displaystyle I(X_{1},X_{2};Y)=1}$, while the individual mutual information ${\displaystyle I(X_{1};Y)=I(X_{2};Y)=0}$. That is, there is synergistic information arising from the interaction of ${\displaystyle X_{1},X_{2}}$ about ${\displaystyle Y}$, which cannot be easily captured with classical information theoretic quantities.

As mutual information is able to capture non-linear, non-monotone statistical relationships between variables, the PID framework would be able to quantify mulitvariate statistical dependencies in arbitrary complex systems in a much more general way than e.g. the correlation coefficient and is especially able to distinguish between different kinds of interaction between variables.

Definition

Partial information decomposition further decomposes the mutual information between the source variables ${\displaystyle \{X_{1},X_{2}\}}$ with the target variable ${\displaystyle Y}$ as

${\displaystyle I(X_{1},X_{2};Y)={\text{Unq}}(X_{1};Y\setminus X_{2})+{\text{Unq}}(X_{2};Y\setminus X_{1})+{\text{Syn}}(X_{1},X_{2};Y)+{\text{Red}}(X_{1},X_{2};Y)}$

Here the individual information atoms are defined as

• ${\displaystyle {\text{Unq}}(X_{1};Y\setminus X_{2})}$ is the unique information that ${\displaystyle X_{1}}$ has about ${\displaystyle Y}$, which is not in ${\displaystyle X_{2}}$
• ${\displaystyle {\text{Syn}}(X_{1},X_{2};Y)}$ is the synergistic information that is in the interaction of ${\displaystyle X_{1}}$ and ${\displaystyle X_{2}}$ about ${\displaystyle Y}$
• ${\displaystyle {\text{Red}}(X_{1},X_{2};Y)}$ is the redundant information that is in both ${\displaystyle X_{1}}$ or ${\displaystyle X_{2}}$ about ${\displaystyle Y}$

There is, thus far, no universal agreement on how these terms should be defined, with different approaches that decompose information into redundant, unique, and synergistic components appearing in the literature. ^{[1] [3] [4] [5]}

However, once an appropriate definition of redundant information has been chosen, the decomposition reduces to a Möbius inversion ^[6] and can be calculated using the Fast Möbius Transform. ^[7]

Applications

Despite the lack of universal agreement, partial information decomposition has been applied to diverse fields, including climatology, ^[8] neuroscience ^{[9] [10] [11]} sociology, ^[12] and machine learning ^[13] Partial information decomposition has also been proposed as a possible foundation on which to build a mathematically robust definition of emergence in complex systems ^[14] and may be relevant to formal theories of consciousness. ^[15]

See also

• Mutual information
• Total correlation
• Dual total correlation
• Interaction information

References

1. Williams PL, Beer RD (2010-04-14). "Nonnegative Decomposition of Multivariate Information". arXiv: 1004.2515 [cs.IT].
2. Matthias, Philip; Makkeh, Abdullah; Wibral, Michael; Gutknecht, Aaron J. "Novel Inconsistencies for Partial Information Decomposition" . Retrieved 19 December 2025.
3. Quax R, Har-Shemesh O, Sloot PM (February 2017). "Quantifying Synergistic Information Using Intermediate Stochastic Variables". Entropy. 19 (2): 85. arXiv: 1602.01265 . doi: 10.3390/e19020085 . ISSN   1099-4300.
4. Rosas FE, Mediano PA, Rassouli B, Barrett AB (2020-12-04). "An operational information decomposition via synergistic disclosure". Journal of Physics A: Mathematical and Theoretical. 53 (48): 485001. arXiv: 2001.10387 . Bibcode:2020JPhA...53V5001R. doi:10.1088/1751-8121/abb723. ISSN   1751-8113. S2CID   210932609.
5. Kolchinsky A (March 2022). "A Novel Approach to the Partial Information Decomposition". Entropy. 24 (3): 403. arXiv: 1908.08642 . Bibcode:2022Entrp..24..403K. doi: 10.3390/e24030403 . PMC   8947370 . PMID   35327914.
6. Jansma A (2025). "Mereological approach to higher-order structure in complex systems: From macro to micro with Möbius". Physical Review Research. 7 (2) 023016. American Physical Society. arXiv: 2404.14423 . Bibcode:2025PhRvR...7b3016J. doi:10.1103/PhysRevResearch.7.023016.
7. Jansma A, Mediano PA, Rosas FE (2025). "Fast Möbius transform: An algebraic approach to information decomposition". Physical Review Research. 7 (3) 033049. American Physical Society. doi:10.1103/PhysRevResearch.7.033049 (inactive 6 September 2025).
8. Goodwell AE, Jiang P, Ruddell BL, Kumar P (February 2020). "Debates—Does Information Theory Provide a New Paradigm for Earth Science? Causality, Interaction, and Feedback". Water Resources Research. 56 (2) e2019WR024940. Bibcode:2020WRR....5624940G. doi: 10.1029/2019WR024940 . ISSN   0043-1397. S2CID   216201598.
9. Newman EL, Varley TF, Parakkattu VK, Sherrill SP, Beggs JM (July 2022). "Revealing the Dynamics of Neural Information Processing with Multivariate Information Decomposition". Entropy. 24 (7): 930. Bibcode:2022Entrp..24..930N. doi: 10.3390/e24070930 . PMC   9319160 . PMID   35885153.
10. Luppi AI, Mediano PA, Rosas FE, Holland N, Fryer TD, O'Brien JT, et al. (June 2022). "A synergistic core for human brain evolution and cognition". Nature Neuroscience. 25 (6): 771–782. doi:10.1038/s41593-022-01070-0. PMC   7614771 . PMID   35618951. S2CID   249096746.
11. Wibral M, Priesemann V, Kay JW, Lizier JT, Phillips WA (March 2017). "Partial information decomposition as a unified approach to the specification of neural goal functions". Brain and Cognition. Perspectives on Human Probabilistic Inferences and the 'Bayesian Brain'. 112: 25–38. arXiv: 1510.00831 . doi: 10.1016/j.bandc.2015.09.004 . PMID   26475739. S2CID   4394452.
12. Varley TF, Kaminski P (October 2022). "Untangling Synergistic Effects of Intersecting Social Identities with Partial Information Decomposition". Entropy. 24 (10): 1387. Bibcode:2022Entrp..24.1387V. doi: 10.3390/e24101387 . ISSN   1099-4300. PMC   9611752 . PMID   37420406.
13. Tax TM, Mediano PA, Shanahan M (September 2017). "The Partial Information Decomposition of Generative Neural Network Models". Entropy. 19 (9): 474. Bibcode:2017Entrp..19..474T. doi: 10.3390/e19090474 . hdl: 10044/1/50586 . ISSN   1099-4300.
14. Mediano PA, Rosas FE, Luppi AI, Jensen HJ, Seth AK, Barrett AB, et al. (July 2022). "Greater than the parts: a review of the information decomposition approach to causal emergence". Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences. 380 (2227) 20210246. doi:10.1098/rsta.2021.0246. PMC   9125226 . PMID   35599558.
15. Luppi AI, Mediano PA, Rosas FE, Harrison DJ, Carhart-Harris RL, Bor D, Stamatakis EA (2021). "What it is like to be a bit: an integrated information decomposition account of emergent mental phenomena". Neuroscience of Consciousness. 2021 (2) niab027. doi:10.1093/nc/niab027. PMC   8600547 . PMID   34804593.