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Generalized entropies and logarithms and their duality relations.

ABSTRACT: For statistical systems that violate one of the four Shannon-Khinchin axioms, entropy takes a more general form than the Boltzmann-Gibbs entropy. The framework of superstatistics allows one to formulate a maximum entropy principle with these generalized entropies, making them useful for understanding distribution functions of non-Markovian or nonergodic complex systems. For such systems where the composability axiom is violated there exist only two ways to implement the maximum entropy principle, one using escort probabilities, the other not. The two ways are connected through a duality. Here we show that this duality fixes a unique escort probability, which allows us to derive a complete theory of the generalized logarithms that naturally arise from the violation of this axiom. We then show how the functional forms of these generalized logarithms are related to the asymptotic scaling behavior of the entropy.

SUBMITTER: Hanel R

PROVIDER: S-EPMC3511158 | biostudies-other | 2012 Nov

REPOSITORIES: biostudies-other

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Generalized entropies and logarithms and their duality relations.

Hanel Rudolf R Thurner Stefan S Gell-Mann Murray M

Proceedings of the National Academy of Sciences of the United States of America 20121105 47

For statistical systems that violate one of the four Shannon-Khinchin axioms, entropy takes a more general form than the Boltzmann-Gibbs entropy. The framework of superstatistics allows one to formulate a maximum entropy principle with these generalized entropies, making them useful for understanding distribution functions of non-Markovian or nonergodic complex systems. For such systems where the composability axiom is violated there exist only two ways to implement the maximum entropy principle ...[more]

PMID: 23129618

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Project description:BackgroundIntolerance of uncertainty (IU) is considered as a specific risk factor in the development and maintenance of generalized anxiety disorder (GAD). Yet, researches have investigated the relations between IU and GAD (or worry) using total scores on self-report measures. This ignores that there are different components exist in IU and the heterogeneity of GAD symptoms. In the present study, we explored the relations among different components of IU and symptoms of GAD.MethodsA dimensional approach which take individual differences into consideration in different components of IU along a full range of normal to abnormal symptom severity levels of GAD were used in this study. Components of IU were measured by 12-item Intolerance of Uncertainty Scale and symptoms of GAD were measured by Generalized Anxiety Disorder 7-Item Questionnaire. Regularized partial-correlation network was estimated using cross-sectional data from 624 university students.ResultsFour strongest edges are between components of IU, like "Unforeseen events upset me greatly" and "It frustrates me not having all the information I need". Two strongest edges are between symptoms of GAD, like "Being so restless that it is hard to sit still" and "Feeling afraid as if something awful might happen". Symptom "Worrying too much about different things" and component "It frustrates me not having all the information I need" have the highest expected influences in the present network. In the community of IU, component "It frustrates me not having all the information I need" has the highest bridge expected influence. And in the community of GAD, symptoms "Worrying too much about different things" and "Not being able to stop or control worrying" have the highest bridge expected influence.ConclusionsThis study reveals potential pathways between different components of IU and various symptoms of GAD. Understanding how putative risk factors such as different components of IU are related to symptoms of GAD may provide some references for related preventions and interventions, such as targeting component "It frustrates me not having all the information I need" may be more effective at reducing symptoms of GAD than targeting other components of IU.

Dataset Information

Generalized entropies and logarithms and their duality relations.

Publications

Generalized entropies and logarithms and their duality relations.

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