Special Issue "Distance in Information and Statistical Physics Volume 2"

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.

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.

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.

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.

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.

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.

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.