Higher-Order Interactions and Their Duals Reveal Synergy and Logical Dependence beyond Shannon-Information
Information-theoretic quantities reveal dependencies among variables in the structure of joint, marginal, and conditional entropies while leaving certain fundamentally different systems indistinguishable. Furthermore, there is no consensus on the correct higher-order generalisation of mutual informa...
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| Published in | Entropy (Basel, Switzerland) Vol. 25; no. 4; p. 648 |
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| Format | Journal Article |
| Language | English |
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| Abstract | Information-theoretic quantities reveal dependencies among variables in the structure of joint, marginal, and conditional entropies while leaving certain fundamentally different systems indistinguishable. Furthermore, there is no consensus on the correct higher-order generalisation of mutual information (MI). In this manuscript, we show that a recently proposed model-free definition of higher-order interactions among binary variables (MFIs), such as mutual information, is a Möbius inversion on a Boolean algebra, except of surprisal instead of entropy. This provides an information-theoretic interpretation to the MFIs, and by extension to Ising interactions. We study the objects dual to mutual information and the MFIs on the order-reversed lattices. We find that dual MI is related to the previously studied differential mutual information, while dual interactions are interactions with respect to a different background state. Unlike (dual) mutual information, interactions and their duals uniquely identify all six 2-input logic gates, the dy- and triadic distributions, and different causal dynamics that are identical in terms of their Shannon information content. |
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| AbstractList | Information-theoretic quantities reveal dependencies among variables in the structure of joint, marginal, and conditional entropies while leaving certain fundamentally different systems indistinguishable. Furthermore, there is no consensus on the correct higher-order generalisation of mutual information (MI). In this manuscript, we show that a recently proposed model-free definition of higher-order interactions among binary variables (MFIs), such as mutual information, is a Möbius inversion on a Boolean algebra, except of surprisal instead of entropy. This provides an information-theoretic interpretation to the MFIs, and by extension to Ising interactions. We study the objects dual to mutual information and the MFIs on the order-reversed lattices. We find that dual MI is related to the previously studied differential mutual information, while dual interactions are interactions with respect to a different background state. Unlike (dual) mutual information, interactions and their duals uniquely identify all six 2-input logic gates, the dy- and triadic distributions, and different causal dynamics that are identical in terms of their Shannon information content. Information-theoretic quantities reveal dependencies among variables in the structure of joint, marginal, and conditional entropies while leaving certain fundamentally different systems indistinguishable. Furthermore, there is no consensus on the correct higher-order generalisation of mutual information (MI). In this manuscript, we show that a recently proposed model-free definition of higher-order interactions among binary variables (MFIs), such as mutual information, is a Möbius inversion on a Boolean algebra, except of surprisal instead of entropy. This provides an information-theoretic interpretation to the MFIs, and by extension to Ising interactions. We study the objects dual to mutual information and the MFIs on the order-reversed lattices. We find that dual MI is related to the previously studied differential mutual information, while dual interactions are interactions with respect to a different background state. Unlike (dual) mutual information, interactions and their duals uniquely identify all six 2-input logic gates, the dy- and triadic distributions, and different causal dynamics that are identical in terms of their Shannon information content.Information-theoretic quantities reveal dependencies among variables in the structure of joint, marginal, and conditional entropies while leaving certain fundamentally different systems indistinguishable. Furthermore, there is no consensus on the correct higher-order generalisation of mutual information (MI). In this manuscript, we show that a recently proposed model-free definition of higher-order interactions among binary variables (MFIs), such as mutual information, is a Möbius inversion on a Boolean algebra, except of surprisal instead of entropy. This provides an information-theoretic interpretation to the MFIs, and by extension to Ising interactions. We study the objects dual to mutual information and the MFIs on the order-reversed lattices. We find that dual MI is related to the previously studied differential mutual information, while dual interactions are interactions with respect to a different background state. Unlike (dual) mutual information, interactions and their duals uniquely identify all six 2-input logic gates, the dy- and triadic distributions, and different causal dynamics that are identical in terms of their Shannon information content. |
| Audience | Academic |
| Author | Jansma, Abel |
| AuthorAffiliation | 3 Biomedical AI Lab, School of Informatics, University of Edinburgh, Edinburgh EH8 9YL, UK 2 Higgs Centre for Theoretical Physics, School of Physics & Astronomy, University of Edinburgh, Edinburgh EH8 9YL, UK 1 MRC Human Genetics Unit, Institute of Genetics & Cancer, University of Edinburgh, Edinburgh EH8 9YL, UK; a.a.a.jansma@sms.ed.ac.uk |
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| Author_xml | – sequence: 1 givenname: Abel orcidid: 0000-0002-1734-637X surname: Jansma fullname: Jansma, Abel organization: Biomedical AI Lab, School of Informatics, University of Edinburgh, Edinburgh EH8 9YL, UK |
| BackLink | https://www.ncbi.nlm.nih.gov/pubmed/37190436$$D View this record in MEDLINE/PubMed |
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| SubjectTerms | Algebra Analysis Boolean algebra Decomposition Entropy higher-order information Information theory Ising model Lattices Logic circuits Mathematical functions Möbius inversions Probability Probability distribution Random variables synergy triadic |
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| Title | Higher-Order Interactions and Their Duals Reveal Synergy and Logical Dependence beyond Shannon-Information |
| URI | https://www.ncbi.nlm.nih.gov/pubmed/37190436 https://www.proquest.com/docview/2806516581 https://www.proquest.com/docview/2814526024/abstract/ https://pubmed.ncbi.nlm.nih.gov/PMC10137660 https://doaj.org/article/9f69f14fdd5c46a2a52cee78b9cc197e |
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