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Quantifying Unique Information
Abstract
We propose new measures of shared information, unique information and synergistic information that can be used to decompose the mutual information of a pair of random variables (Y, Z) with a third random variable X. Our measures are motivated by an operational idea of unique information, which suggests that shared information and unique information should depend only on the marginal distributions of the pairs (X, Y) and (X,Z). Although this invariance property has not been studied before, it is satisfied by other proposed measures of shared information. The invariance property does not uniquely determine our new measures, but it implies that the functions that we define are bounds to any other measures satisfying the same invariance property. We study properties of our measures and compare them to other candidate measures.
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![Figure 1 The shared information measures ˜ SI and I red in the dice example depending on the correlation parameter λ . The figure on the right is reproduced from [6] (copyright 2013 by the American Physical Society). In both figures, the summation parameter α varies from one (uppermost line) to six (lowest line).](/paper-figures/b8b828f15160429186b8c3bac103f7a3/quantifying-unique-information-figure-1.png)
![Figure A.1. The function CoI Q in Example 31 (figure created with Mathematica [12]). Darker colors indicate larger values of CoI Q . In this example, ∆ P is a square. The uniform distribution lies at the center of this square and is the maximum of CoI Q . In the two dark corners, X is independent of Y and Z , and either Y = Z or Y = ¬ Z . In the two light corners, Y and Z are independent, and either X = Y XOR Z or X = ¬ ( Y XOR Z ) .](/paper-figures/56c514b9af1847eb8ea883b48a94b1b8/quantifying-unique-information-figure-2.png)
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“It satisfies many desirable properties including additivity of all PID components for independent systems of sources and targets (proven for discrete33,46 and empirically validated for Gaussian PID30) and has been widely applied to neural data.13,36,37,47,48”
From Sampling bias corrections for discrete and Gaussian partial information decompositions (2026) · cites this paper as [33]
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“Among the advantages of the BROJA PID, we note that this definition ensures that all PID terms satisfy the additivity property”
From Sampling bias corrections for discrete and Gaussian partial information decompositions (2026) · cites this paper as [33]
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“Although neither the discrete BROJA PID33 nor its Gaussian counterpart gPID30 have been defined for more than two sources as yet”
From Sampling bias corrections for discrete and Gaussian partial information decompositions (2026) · cites this paper as [33]
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Research works printed but not linked: 11
- McGill, 1954, Multivariate information transmission, Psychometrika, 19, 97, 10.1007/BF02289159 · DOI
- Schneidman, 2003, Synergy, redundancy, and independence in population codes, J. Neurosci, 23, 11539, 10.1523/JNEUROSCI.23-37-11539.2003 · DOI
- Latham, 2005, Synergy, redundancy, and independence in population codes, revisited, J. Neurosci, 25, 5195, 10.1523/JNEUROSCI.5319-04.2005 · DOI
- Williams, P., and Beer, R. (2010). Nonnegative decomposition of multivariate information. arXiv:1004.2515v1. · arXiv
- Griffith, V., and Koch, C. (2013). Quantifying synergistic mutual information. arXiv:1205.4265. · DOI
- Harder, 2013, Bivariate measure of redundant information, Phys. Rev. E, 87, 012130, 10.1103/PhysRevE.87.012130 · DOI
- Bertschinger, N., Rauh, J., Olbrich, E., and Jost, J. (2012, January September). Shared information–new insights and problems in decomposing Information in Complex Systems. Brussels. · DOI · Paper page
- Bertschinger, N., and Rauh, J. (2014). The blackwell relation defines no lattice. arXiv:1401.3146. · DOI
- Blackwell, 1953, Equivalent comparisons of experiments, Ann. Math. Stat, 24, 265, 10.1214/aoms/1177729032 · DOI
- Rauh, J., Bertschinger, N., Olbrich, E., and Jost, J. (2014). Reconsidering unique information: Towards a multivariate information Decomposition. Accepted for ISIT 2014. · DOI
- Sullivant, 2002, Gröbner bases and polyhedral geometry of reducible and cyclic Models, J. Combin. Theor. A, 100, 277, 10.1006/jcta.2002.3301 · DOI
Cited by
Sorted by citations
Measuring integrated information: comparison of candidate measures in theory and simulation
Mediano, P., Seth, A., & Barrett, A. (2018) · Entropy · doi:10.3390/e21010017 · Cited by 145
Phillips, W. A., Bachmann, T., & Storm, J. F. (2018) · Frontiers in Neural Circuits · doi:10.3389/fncir.2018.00050 · Cited by 62
Causal composition: Structural differences among dynamically equivalent systems
Albantakis, L., & Tononi, G. (2019) · Entropy · doi:10.3390/e21100989 · Cited by 30
Sampling bias corrections for discrete and Gaussian partial information decompositions
Lorenz, G. M., Engel, N. M., Koçillari, L., Celotto, M., Orsenigo, D., Curreli, S., Malerba, S. B., Engel, A. K., Kayser, C., Fellin, T., et al. (2026) · Patterns · doi:10.1016/j.patter.2026.101619 · Cited by 9
Metadata
- Authors
- Bertschinger, N., Rauh, J., Olbrich, E., Jost, J., & Ay, N.
- Affiliations
- Max Planck Institute for Mathematics in the Sciences, Inselstraße 23, 04109 Leipzig, Germany
- ORCID
- Rauh, Johannes
- Venue
- Entropy
- Publisher
- MDPI AG
- Published
- 2014-04-15
- Volume
- 16
- Issue
- 4
- Pages
- 2161-2183
- ISSN
- 1099-4300
- Cited by
- 253
- References
- 0
- Language
- en
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