72,127 research outputs found
Generalized entropies and logarithms and their duality relations
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 non-ergodic 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.Comment: 4 pages, 1 page supporting informatio
Generalized maximum entropy (GME) estimator: formulation and a monte carlo study
The origin of entropy dates back to 19th century. In 1948, the entropy concept as a measure of uncertainty was developed by Shannon. A decade after in 1957, Jaynes formulated Shannonâs entropy as a method for estimation and inference particularly for ill-posed problems by proposing the so called Maximum Entropy (ME) principle. More recently, Golan et al. (1996) developed the Generalized Maximum Entropy (GME) estimator and started a new discussion in econometrics. This paper is divided into two parts. The first part considers the formulation of this new technique (GME). Second, by Monte Carlo simulations the estimation results of GME will be discussed in the context of non-normal disturbances.Entropy, Maximum Entropy, ME, Generalized Maximum Entropy, GME, Monte Carlo Experiment, Shannonâs Entropy, Non-normal disturbances
How multiplicity determines entropy and the derivation of the maximum entropy principle for complex systems
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 exists an ongoing controversy whether the
notion of the maximum entropy principle can be extended in a meaningful way to
non-extensive, non-ergodic, 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
(c,d)-entropies. We demonstrate that the MEP is a perfectly consistent concept
for non-ergodic 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 the first exact derivation of a
generalized entropy from the microscopic properties of a path-dependent random
process.Comment: 6 pages, 1 figure. To appear in PNA
Algorithm of arithmetical operations with fuzzy numerical data
In this article the theoretical generalization for representation of arithmetic operations with fuzzy numbers is considered. Fuzzy numbers are generalized by means of fuzzy measures. On the basis of this generalization the new algorithm of fuzzy arithmetic which uses a principle of entropy maximum is created. As example, the summation of two fuzzy numbers is considered. The algorithm is realized in the software "Fuzzy for Microsoft Excel".fuzzy measure (Sugeno), fuzzy integral (Sugeno), fuzzy numbers; arithmetical operations; principle of entropy maximum
Generalized Maximum Entropy for Supervised Classification
The maximum entropy principle advocates to
evaluate eventsâ probabilities using a distribution that maximizes
entropy among those that satisfy certain expectationsâ constraints. Such principle can be generalized for arbitrary decision
problems where it corresponds to minimax approaches. This
paper establishes a framework for supervised classification based
on the generalized maximum entropy principle that leads to
minimax risk classifiers (MRCs). We develop learning techniques
that determine MRCs for general entropy functions and provide
performance guarantees by means of convex optimization. In
addition, we describe the relationship of the presented techniques
with existing classification methods, and quantify MRCs performance in comparison with the proposed bounds and conventional
methods.RYC-2016-1938
Implications of Form Invariance to the Structure of Nonextensive Entropies
The form invariance of the statement of the maximum entropy principle and the
metric structure in quantum density matrix theory, when generalized to
nonextensive situations, is shown here to determine the structure of the
nonextensive entropies. This limits the range of the nonextensivity parameter
to so as to preserve the concavity of the entropies. The Tsallis entropy is
thereby found to be appropriately renormalized.Comment: 8 page
Abolishing the maximum tension principle
We find the series of example theories for which the relativistic limit of
maximum tension represented by the entropic force can be
abolished. Among them the varying constants theories, some generalized entropy
models applied both for cosmological and black hole horizons as well as some
generalized uncertainty principle models.Comment: 5 pages, no figures, REVTEX4-1, a typo in abstract correcte
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