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[Significance] The maximum entropy principle (MEP) states that for many statistical systems the entropy that is associated with an observed distribution function is a maximum, given that prior information is taken into account appropriately. Usually systems where the MEP applies are simple systems, such as gases and independent processes. The MEP has found thousands of practical applications. Whether a MEP holds for complex systems, where elements interact strongly and have memory and path dependence, remained unclear over the past half century. Here we prove that a MEP indeed exists for complex systems and derive the generalized entropy. We find that it belongs to the class of the recently proposed (c,d)-entropies. The practical use of the formalism is shown for a path-dependent random walk. [Abstract] The maximum entropy principle (MEP) is a method for obtaining the most likely distribution functions of observables from statistical systems by maximizing entropy under constraints. The MEP has found hundreds of applications in ergodic and Markovian systems in statistical mechanics, information theory, and statistics. For several decades there has been an ongoing controversy over whether the notion of the maximum entropy principle can be extended in a meaningful way to nonextensive, nonergodic, and complex statistical systems and processes. In this paper we start by reviewing how Boltzmann-Gibbs-Shannon entropy is related to multiplicities of independent random processes. We then show how the relaxation of independence naturally leads to the most general entropies that are compatible with the first three Shannon-Khinchin axioms, the Graphic-entropies. We demonstrate that the MEP is a perfectly consistent concept for nonergodic and complex statistical systems if their relative entropy can be factored into a generalized multiplicity and a constraint term. The problem of finding such a factorization reduces to finding an appropriate representation of relative entropy in a linear basis. In a particular example we show that path-dependent random processes with memory naturally require specific generalized entropies. The example is to our knowledge the first exact derivation of a generalized entropy from the microscopic properties of a path-dependent random process.

@article{hanelHowMultiplicityDetermines2014, title = {How Multiplicity Determines Entropy and the Derivation of the Maximum Entropy Principle for Complex Systems}, author = {Hanel, Rudolf and Thurner, Stefan and {Gell-Mann}, Murray}, year = {2014}, month = may, volume = {111}, pages = {6905--6910}, issn = {1091-6490}, doi = {10.1073/pnas.1406071111}, abstract = {[Significance] The maximum entropy principle (MEP) states that for many statistical systems the entropy that is associated with an observed distribution function is a maximum, given that prior information is taken into account appropriately. Usually systems where the MEP applies are simple systems, such as gases and independent processes. The MEP has found thousands of practical applications. Whether a MEP holds for complex systems, where elements interact strongly and have memory and path dependence, remained unclear over the past half century. Here we prove that a MEP indeed exists for complex systems and derive the generalized entropy. We find that it belongs to the class of the recently proposed (c,d)-entropies. The practical use of the formalism is shown for a path-dependent random walk. [Abstract] The maximum entropy principle (MEP) is a method for obtaining the most likely distribution functions of observables from statistical systems by maximizing entropy under constraints. The MEP has found hundreds of applications in ergodic and Markovian systems in statistical mechanics, information theory, and statistics. For several decades there has been an ongoing controversy over whether the notion of the maximum entropy principle can be extended in a meaningful way to nonextensive, nonergodic, and complex statistical systems and processes. In this paper we start by reviewing how Boltzmann-Gibbs-Shannon entropy is related to multiplicities of independent random processes. We then show how the relaxation of independence naturally leads to the most general entropies that are compatible with the first three Shannon-Khinchin axioms, the Graphic-entropies. We demonstrate that the MEP is a perfectly consistent concept for nonergodic and complex statistical systems if their relative entropy can be factored into a generalized multiplicity and a constraint term. The problem of finding such a factorization reduces to finding an appropriate representation of relative entropy in a linear basis. In a particular example we show that path-dependent random processes with memory naturally require specific generalized entropies. The example is to our knowledge the first exact derivation of a generalized entropy from the microscopic properties of a path-dependent random process.}, journal = {Proceedings of the National Academy of Sciences}, keywords = {*imported-from-citeulike-INRMM,~INRMM-MiD:c-13168201,complexity,entropy,mathematical-reasoning,multiplicity}, lccn = {INRMM-MiD:c-13168201}, number = {19} }

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