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p. 4622-4633
Received: 8 September 2013 / Revised: 21 October 2013 / Accepted: 22 October 2013 / Published: 28 October 2013
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Abstract: In a previous paper (C. Cafaro et al. , 2012), we compared an uncorrelated 3D Gaussian statistical model to an uncorrelated 2D Gaussian statistical model obtained from the former model by introducing a constraint that resembles the quantum mechanical canonical minimum uncertainty relation. Analysis was completed by way of the information geometry and the entropic dynamics of each system. This analysis revealed that the chaoticity of the 2D Gaussian statistical model, quantified by means of the Information Geometric Entropy (IGE), is softened or weakened with respect to the chaoticity of the 3D Gaussian statistical model, due to the accessibility of more information. In this companion work, we further constrain the system in the context of a correlation constraint among the system’s micro-variables and show that the chaoticity is further weakened, but only locally . Finally, the physicality of the constraints is briefly discussed, particularly in the context of quantum entanglement.
p. 3698-3713
Received: 21 June 2013 / Revised: 21 August 2013 / Accepted: 3 September 2013 / Published: 6 September 2013
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Abstract: The correlation distance quantifies the statistical independence of two classical or quantum systems, via the distance from their joint state to the product of the marginal states. Tight lower bounds are given for the mutual information between pairs of two-valued classical variables and quantum qubits, in terms of the corresponding classical and quantum correlation distances. These bounds are stronger than the Pinsker inequality (and refinements thereof) for relative entropy. The classical lower bound may be used to quantify properties of statistical models that violate Bell inequalities. Partially entangled qubits can have lower mutual information than can any two-valued classical variables having the same correlation distance. The qubit correlation distance also provides a direct entanglement criterion, related to the spin covariance matrix. Connections of results with classically-correlated quantum states are briefly discussed.
p. 3361-3378
Received: 16 July 2013 / Revised: 12 August 2013 / Accepted: 16 August 2013 / Published: 23 August 2013
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Abstract: Information geometry provides a tool to systematically investigate the parameter sensitivity of the state of a system. If a physical system is described by a linear combination of eigenstates of a complex (that is, non-Hermitian) Hamiltonian, then there can be phase transitions where dynamical properties of the system change abruptly. In the vicinities of the transition points, the state of the system becomes highly sensitive to the changes of the parameters in the Hamiltonian. The parameter sensitivity can then be measured in terms of the Fisher-Rao metric and the associated curvature of the parameter-space manifold. A general scheme for the geometric study of parameter-space manifolds of eigenstates of complex Hamiltonians is outlined here, leading to generic expressions for the metric.
p. 2989-3006
Received: 26 June 2013 / Revised: 17 July 2013 / Accepted: 18 July 2013 / Published: 26 July 2013
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Abstract: The entropy production paradox for anomalous diffusion processes describes a phenomenon where one-parameter families of dynamical equations, falling between the diffusion and wave equations, have entropy production rates (Shannon, Tsallis or Renyi) that increase toward the wave equation limit unexpectedly. Moreover, also surprisingly, the entropy does not order the bridging regime between diffusion and waves at all. However, it has been found that relative entropies, with an appropriately chosen reference distribution, do. Relative entropies, thus, provide a physically sensible way of setting which process is “nearer” to pure diffusion than another, placing pure wave propagation, desirably, “furthest” from pure diffusion. We examine here the time behavior of the relative entropies under the evolution dynamics of the underlying one-parameter family of dynamical equations based on space-fractional derivatives.
p. 1202-1220
Received: 15 January 2013 / Revised: 21 March 2013 / Accepted: 25 March 2013 / Published: 8 April 2013
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Abstract: Studies of learning algorithms typically concentrate on situations where potentially ever growing training sample is available. Yet, there can be situations (e.g., detection of differentially expressed genes on unreplicated data or estimation of time delay in non-stationary gravitationally lensed photon streams) where only extremely small samples can be used in order to perform an inference. On unreplicated data, the inference has to be performed on the smallest sample possible—sample of size 1. We study whether anything useful can be learnt in such extreme situations by concentrating on a Bayesian approach that can account for possible prior information on expected counts. We perform a detailed information theoretic study of such Bayesian estimation and quantify the effect of Bayesian averaging on its first two moments. Finally, to analyze potential benefits of the Bayesian approach, we also consider Maximum Likelihood (ML) estimation as a baseline approach. We show both theoretically and empirically that the Bayesian model averaging can be potentially beneficial.
p. 1606-1626
Received: 16 July 2012 / Revised: 25 August 2012 / Accepted: 27 August 2012 / Published: 4 September 2012
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Abstract: The aim of this work is to provide the tools to compute the well-known Kullback–Leibler divergence measure for the flexible family of multivariate skew-normal distributions. In particular, we use the Jeffreys divergence measure to compare the multivariate normal distribution with the skew-multivariate normal distribution, showing that this is equivalent to comparing univariate versions of these distributions. Finally, we applied our results on a seismological catalogue data set related to the 2010 Maule earthquake. Specifically, we compare the distributions of the local magnitudes of the regions formed by the aftershocks.
p. 1170-1185
Received: 11 February 2011 / Revised: 1 June 2011 / Accepted: 2 June 2011 / Published: 14 June 2011
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Abstract: The Gibbs distribution of statistical physics is an exponential family of probability distributions, which has a mathematical basis of duality in the form of the Legendre transformation. Recent studies of complex systems have found lots of distributions obeying the power law rather than the standard Gibbs type distributions. The Tsallis q -entropy is a typical example capturing such phenomena. We treat the q -Gibbs distribution or the q -exponential family by generalizing the exponential function to the q -family of power functions, which is useful for studying various complex or non-standard physical phenomena. We give a new mathematical structure to the q -exponential family different from those previously given. It has a dually flat geometrical structure derived from the Legendre transformation and the conformal geometry is useful for understanding it. The q -version of the maximum entropy theorem is naturally induced from the q -Pythagorean theorem. We also show that the maximizer of the q -escort distribution is a Bayesian MAP (Maximum A posteriori Probability) estimator.
p. 1055-1075
Received: 11 April 2011 / Accepted: 27 May 2011 / Published: 3 June 2011
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Abstract: Statistical complexity measures (SCM) are the composition of two ingredients: (i) entropies and (ii) distances in probability-space. In consequence, SCMs provide a simultaneous quantification of the randomness and the correlational structures present in the system under study. We address in this review important topics underlying the SCM structure, viz. , (a) a good choice of probability metric space and (b) how to assess the best distance-choice, which in this context is called a “disequilibrium” and is denoted with the letter Q . Q , indeed the crucial SCM ingredient, is cast in terms of an associated distance D . Since out input data consists of time-series, we also discuss the best way of extracting from the time series a probability distribution P . As an illustration, we show just how these issues affect the description of the classical limit of quantum mechanics.
p. 818-843
Received: 16 November 2009 / Revised: 28 March 2010 / Accepted: 2 April 2010 / Published: 9 April 2010
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Abstract: Given iid samples drawn from a distribution with known parametric form, we propose the minimization of expected Bregman divergence to form Bayesian estimates of differential entropy and relative entropy, and derive such estimators for the uniform, Gaussian, Wishart, and inverse Wishart distributions. Additionally, formulas are given for a log gamma Bregman divergence and the differential entropy and relative entropy for the Wishart and inverse Wishart. The results, as always with Bayesian estimates, depend on the accuracy of the prior parameters, but example simulations show that the performance can be substantially improved compared to maximum likelihood or state-of-the-art nonparametric estimators.
p. 262-274
Received: 29 December 2009 / Revised: 20 February 2010 / Accepted: 23 February 2010 / Published: 25 February 2010
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Abstract: In statistical physics, Boltzmann-Shannon entropy provides good understanding for the equilibrium states of a number of phenomena. In statistics, the entropy corresponds to the maximum likelihood method, in which Kullback-Leibler divergence connects Boltzmann-Shannon entropy and the expected log-likelihood function. The maximum likelihood estimation has been supported for the optimal performance, which is known to be easily broken down in the presence of a small degree of model uncertainty. To deal with this problem, a new statistical method, closely related to Tsallis entropy, is proposed and shown to be robust for outliers, and we discuss a local learning property associated with the method.
p. 748-765
Received: 11 September 2009 / Accepted: 26 October 2009 / Published: 3 November 2009
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Abstract: Current research is probing transport on ever smaller scales. Modeling of the electromagnetic interaction with nanoparticles or small collections of dipoles and its associated energy transport and nonequilibrium characteristics requires a detailed understanding of transport properties. The goal of this paper is to use a nonequilibrium statistical-mechanical method to obtain exact time-correlation functions, fluctuation-dissipation theorems (FD), heat and charge transport, and associated transport expressions under electromagnetic driving. We extend the time-symmetric Robertson statistical-mechanical theory to study the exact time evolution of relevant variables and entropy rate in the electromagnetic interaction with materials. In this exact statistical-mechanical theory, a generalized canonical density is used to define an entropy in terms of a set of relevant variables and associated Lagrange multipliers. Then the entropy production rate are defined through the relevant variables. The influence of the nonrelevant variables enter the equations through the projection-like operator and thereby influences the entropy. We present applications to the response functions for the electrical and thermal conductivity, specific heat, generalized temperature, Boltzmann’s constant, and noise. The analysis can be performed either classically or quantum-mechanically, and there are only a few modifications in transferring between the approaches. As an application we study the energy, generalized temperature, and charge transport equations that are valid in nonequilibrium and relate it to heat flow and temperature relations in equilibrium states.
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