Editing Synergy (section)

==Information theory==
{{see also| Partial Information Decomposition}}

Mathematical formalizations of synergy have been proposed using [[information theory]] to rigorously define the relationships between "wholes" and "parts".<ref>{{cite journal | vauthors = Gutknecht AJ, Wibral M, Makkeh A | title = Bits and pieces: understanding information decomposition from part-whole relationships and formal logic | journal = Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences| volume = 477 | issue = 2251 | pages = 20210110 | date = July 2021 | pmid = 35197799 | pmc = 8261229 | doi = 10.1098/rspa.2021.0110 | arxiv = 2008.09535 | bibcode = 2021RSPSA.47710110G }}</ref> In this context, synergy is said to occur when there is information present in the joint state of multiple variables that cannot be extracted from the individual parts considered individually. For example, consider the logical [[XOR gate]]. If <math>Y=XOR(X_1,X_2)</math> for three binary variables, the [[mutual information]] between any individual source and the target is 0 bit. However, the joint mutual information <math>I(X_1,X_2;Y)=1</math> bit. There is information about the target that can only be extracted from the joint state of the inputs considered jointly, and not any others. 

There is, thus far, no universal agreement on how synergy can best be quantified, with different approaches that decompose information into redundant, unique, and synergistic components appearing in the literature.<ref>{{Cite arXiv | vauthors = Williams PL, Beer RD |date=2010-04-14 |title=Nonnegative Decomposition of Multivariate Information |class=cs.IT |eprint=1004.2515 }}</ref><ref>{{Cite journal | vauthors = Quax R, Har-Shemesh O, Sloot PM |date=February 2017 |title=Quantifying Synergistic Information Using Intermediate Stochastic Variables |journal=Entropy |language=en |volume=19 |issue=2 |pages=85 |doi=10.3390/e19020085 |issn=1099-4300|doi-access=free |arxiv=1602.01265 }}</ref><ref>{{Cite journal | vauthors = Rosas FE, Mediano PA, Rassouli B, Barrett AB |date=2020-12-04 |title=An operational information decomposition via synergistic disclosure |journal=Journal of Physics A: Mathematical and Theoretical |volume=53 |issue=48 |pages=485001 |doi=10.1088/1751-8121/abb723 |arxiv=2001.10387 |bibcode=2020JPhA...53V5001R |s2cid=210932609 |issn=1751-8113}}</ref><ref>{{cite journal | vauthors = Kolchinsky A | title = A Novel Approach to the Partial Information Decomposition | journal = Entropy | volume = 24 | issue = 3 | pages = 403 | date = March 2022 | pmid = 35327914 | doi = 10.3390/e24030403 | pmc = 8947370 | arxiv = 1908.08642 | bibcode = 2022Entrp..24..403K | doi-access = free }}</ref> Despite the lack of universal agreement, information-theoretic approaches to statistical synergy have been applied to diverse fields, including climatology,<ref>{{Cite journal | vauthors = Goodwell AE, Jiang P, Ruddell BL, Kumar P |date= February 2020 |title=Debates—Does Information Theory Provide a New Paradigm for Earth Science? Causality, Interaction, and Feedback |journal=Water Resources Research |language=en |volume=56 |issue=2 |doi=10.1029/2019WR024940 |bibcode= 2020WRR....5624940G |s2cid= 216201598 |issn=0043-1397|doi-access=free }}</ref> neuroscience<ref>{{cite journal | vauthors = Newman EL, Varley TF, Parakkattu VK, Sherrill SP, Beggs JM | title = Revealing the Dynamics of Neural Information Processing with Multivariate Information Decomposition | journal = Entropy | volume = 24 | issue = 7 | pages = 930 | date = July 2022 | pmid = 35885153 | doi = 10.3390/e24070930 | pmc = 9319160 | bibcode = 2022Entrp..24..930N | doi-access = free }}</ref><ref>{{cite journal | vauthors = Luppi AI, Mediano PA, Rosas FE, Holland N, Fryer TD, O'Brien JT, Rowe JB, Menon DK, Bor D, Stamatakis EA | display-authors = 6 | title = A synergistic core for human brain evolution and cognition | journal = Nature Neuroscience | volume = 25 | issue = 6 | pages = 771–782 | date = June 2022 | pmid = 35618951 | doi = 10.1038/s41593-022-01070-0 | s2cid = 249096746 | pmc = 7614771 }}</ref><ref>{{cite journal | vauthors = Wibral M, Priesemann V, Kay JW, Lizier JT, Phillips WA | title = Partial information decomposition as a unified approach to the specification of neural goal functions | journal = Brain and Cognition | volume = 112 | pages = 25–38 | date = March 2017 | pmid = 26475739 | doi = 10.1016/j.bandc.2015.09.004 | series = Perspectives on Human Probabilistic Inferences and the 'Bayesian Brain' | s2cid = 4394452 | doi-access = free | arxiv = 1510.00831 }}</ref> sociology,<ref>{{Cite journal | vauthors = Varley TF, Kaminski P |date= October 2022 |title=Untangling Synergistic Effects of Intersecting Social Identities with Partial Information Decomposition  |journal=Entropy |language=en |volume=24 |issue=10 |pages=1387 |doi=10.3390/e24101387 |pmid= 37420406 |pmc= 9611752 |bibcode= 2022Entrp..24.1387V |issn=1099-4300|doi-access= free }}</ref> and machine learning<ref>{{Cite journal | vauthors = Tax TM, Mediano PA, Shanahan M |date= September 2017 |title=The Partial Information Decomposition of Generative Neural Network Models |journal=Entropy |language=en |volume=19 |issue=9 |pages=474 |doi=10.3390/e19090474 |bibcode= 2017Entrp..19..474T |issn=1099-4300|doi-access= free |hdl=10044/1/50586 |hdl-access=free }}</ref> Synergy has also been proposed as a possible foundation on which to build a mathematically robust definition of [[Emergence|emergence in complex systems]]<ref>{{cite journal | vauthors = Varley TF, Hoel E | title = Emergence as the conversion of information: a unifying theory | journal = Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences | volume = 380 | issue = 2227 | pages = 20210150 | date = July 2022 | pmid = 35599561 | pmc = 9131462 | doi = 10.1098/rsta.2021.0150 }}</ref><ref>{{cite journal | vauthors = Mediano PA, Rosas FE, Luppi AI, Jensen HJ, Seth AK, Barrett AB, Carhart-Harris RL, Bor D | display-authors = 6 | title = Greater than the parts: a review of the information decomposition approach to causal emergence | journal = Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences | volume = 380 | issue = 2227 | pages = 20210246 | date = July 2022 | pmid = 35599558 | pmc = 9125226 | doi = 10.1098/rsta.2021.0246 }}</ref> and may be relevant to formal theories of consciousness.<ref>{{cite journal | vauthors = Luppi AI, Mediano PA, Rosas FE, Harrison DJ, Carhart-Harris RL, Bor D, Stamatakis EA | title = What it is like to be a bit: an integrated information decomposition account of emergent mental phenomena | journal = Neuroscience of Consciousness | volume = 2021 | issue = 2 | pages = niab027 | date = 2021 | pmid = 34804593 | pmc = 8600547 | doi = 10.1093/nc/niab027 }}</ref>