Gibbs Probability Research Articles

Level-2 IFS thermodynamic formalism: Gibbs probabilities in the space of probabilities and the push-forward map

We will denote by M the space of Borel probabilities on the symbolic space Ω = { 1 , 2 ⋯ , m } N . M is equipped Monge–Kantorovich metric. We consider here the push-forward map T : M → M as a dynamical system. The space of Borel probabilities on M is denoted by M . Given a continuous function A : M → R , an a priori probability Π 0 on M , and a certain convolution operation acting on pairs of probabilities on M , we define an associated Level-2 IFS Ruelle operator. We show the existence of an eigenfunction and an eigenprobability Π ^ ∈ M for such an operator. Under a normalization condition for A, we show the existence of some T -invariant probabilities Π ^ ∈ M . We are able to define the variational entropy of such Π ^ and a related maximization pressure problem associated to A. In some particular examples, we show how to get eigenprobabilities solutions on M for the Level-2 Thermodynamic Formalism problem from eigenprobabilities on M for the classical (Level-1) Thermodynamic Formalism; this shows that our approach is a natural generalization of the classic case.

Asymptotic Analysis for the Generalized Langevin Equation with Singular Potentials

We consider a system of interacting particles governed by the generalized Langevin equation (GLE) in the presence of external confining potentials, singular repulsive forces, as well as memory kernels. Using a Mori–Zwanzig approach, we represent the system by a class of Markovian dynamics. Under a general set of conditions on the nonlinearities, we study the large-time asymptotics of the multi-particle Markovian GLEs. We show that the system is always exponentially attractive toward the unique invariant Gibbs probability measure. The proof relies on a novel construction of Lyapunov functions. We then establish the validity of the small-mass approximation for the solutions by an appropriate equation on any finite-time window. Important examples of singular potentials in our results include the Lennard–Jones and Coulomb functions.

Gibbs probability entropy and its implication to combinatorial entropy models

We show that the class of combinatorial entropy models, such as the Guggenheim–Staverman model, in which the many conformations of a molecule are taken into account, does not fulfill the Gibbs probability normalization condition. The root cause for this deviation lies in the definition of the pure and mixture state. In the athermal limit, mandatory to define the combinatorial entropy, the number of molecules in a particular conformation does not change upon mixing. Therefore each set of molecules with a particular conformation in the pure state can be regarded as a distinguishable subclass of rigid molecules. When this subdivision is applied to the ‘shape’ models, they fulfill the Gibbs probability normalization condition. The resulting equations simplify to the Flory–Huggins entropy model. Implications of this finding to the existing activity coefficient models are discussed.

Generalization Analysis of Machine Learning Algorithms via the Worst-Case Data-Generating Probability Measure

In this paper, the worst-case probability measure over the data is introduced as a tool for characterizing the generalization capabilities of machine learning algorithms. More specifically, the worst-case probability measure is a Gibbs probability measure and the unique solution to the maximization of the expected loss under a relative entropy constraint with respect to a reference probability measure. Fundamental generalization metrics, such as the sensitivity of the expected loss, the sensitivity of the empirical risk, and the generalization gap are shown to have closed-form expressions involving the worst-case data-generating probability measure. Existing results for the Gibbs algorithm, such as characterizing the generalization gap as a sum of mutual information and lautum information, up to a constant factor, are recovered. A novel parallel is established between the worst-case data-generating probability measure and the Gibbs algorithm. Specifically, the Gibbs probability measure is identified as a fundamental commonality of the model space and the data space for machine learning algorithms.

On the asymptotic behavior of solutions to a class of grand canonical master equations

In this article, we investigate the long-time behavior of solutions to a class of infinitely many master equations defined from transition rates that are suitable for the description of a quantum system approaching thermodynamical equilibrium with a heat bath at fixed temperature and a reservoir consisting of one species of particles characterized by a fixed chemical potential.We do so by proving a result which pertains to the spectral resolution of the semigroup generated by the equations, whose infinitesimal generator is realized as a trace-class self-adjoint operator defined in a suitably weighted sequence space. This allows us to prove the existence of global solutions which all stabilize toward the grand canonical equilibrium probability distribution as the time variable becomes large, some of them doing so exponentially rapidly but not all. When we set the chemical potential equal to zero, the stability statements continue to hold in the sense that all solutions converge toward the Gibbs probability distribution of the canonical ensemble which characterizes the equilibrium of the given system with a heat bath at fixed temperature.

Dynamical hypothesis tests and decision theory for Gibbs distributions

We consider the problem of testing for two Gibbs probabilities $ \mu_0 $ and $ \mu_1 $ defined for a dynamical system $ (\Omega, T) $. Due to the fact that in general full orbits are not observable or computable, one needs to restrict to subclasses of tests defined by a finite time series $ h(x_0), h(x_1) = h(T(x_0)), ..., h(x_n) = h(T^n(x_0)) $, $ x_0\in \Omega $, $ n\ge 0 $, where $ h:\Omega\to\mathbb R $ denotes a suitable measurable function. We determine in each class the Neyman-Pearson tests, the minimax tests, and the Bayes solutions and show the asymptotic decay of their risk functions as $ n\to\infty $. In the case of $ \Omega $ being a symbolic space, for each $ n\in \mathbb{N} $, these optimal tests rely on the information of the measures for cylinder sets of size $ n $.

Decision Theory and large deviations for dynamical hypotheses tests: The Neyman-Pearson Lemma, Min-Max and Bayesian tests

<p style='text-indent:20px;'>We analyze hypotheses tests using classical results on large deviations to compare two models, each one described by a different Hölder Gibbs probability measure. One main difference to the classical hypothesis tests in Decision Theory is that here the two measures are singular with respect to each other. Among other objectives, we are interested in the decay rate of the wrong decisions probability, when the sample size <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula> goes to infinity. We show a dynamical version of the Neyman-Pearson Lemma displaying the ideal test within a certain class of similar tests. This test becomes exponentially better, compared to other alternative tests, when the sample size goes to infinity. We are able to present the explicit exponential decay rate. We also consider both, the Min-Max and a certain type of Bayesian hypotheses tests. We shall consider these tests in the log likelihood framework by using several tools of Thermodynamic Formalism. Versions of the Stein's Lemma and Chernoff's information are also presented.</p>

Invariant probabilities for discrete time linear dynamics via thermodynamic formalism * *AOL partially supported by CNPq. AM partially supported by CNPq, project 311018/2018-1, and FAPESP, project 2013/24541-0. MS partially supported by CNPq, project 312632/2018-5. VV supported by INCTMat.

We show the existence of invariant ergodic σ-additive probability measures with full support on X for a class of linear operators L : X → X, where L is a weighted shift operator and X either is the Banach space or for 1 ⩽ p < ∞. In order to do so, we adapt ideas from thermodynamic formalism as follows. For a given bounded Hölder continuous potential , we define a transfer operator which acts on continuous functions on X and prove that this operator satisfies a Ruelle–Perron–Frobenius theorem. That is, we show the existence of an eigenfunction for which provides us with a normalised potential and an action of the dual operator on the one-Wasserstein space of probabilities on X with a unique fixed point, to which we refer to as Gibbs probability. It is worth noting that the definition of requires an a priori probability on the kernel of L. These results are extended to a wide class of operators with a non-trivial kernel defined on separable Banach spaces.

Lie Group Cohomology and (Multi)Symplectic Integrators: New Geometric Tools for Lie Group Machine Learning Based on Souriau Geometric Statistical Mechanics.

In this paper, we describe and exploit a geometric framework for Gibbs probability densities and the associated concepts in statistical mechanics, which unifies several earlier works on the subject, including Souriau’s symplectic model of statistical mechanics, its polysymplectic extension, Koszul model, and approaches developed in quantum information geometry. We emphasize the role of equivariance with respect to Lie group actions and the role of several concepts from geometric mechanics, such as momentum maps, Casimir functions, coadjoint orbits, and Lie-Poisson brackets with cocycles, as unifying structures appearing in various applications of this framework to information geometry and machine learning. For instance, we discuss the expression of the Fisher metric in presence of equivariance and we exploit the property of the entropy of the Souriau model as a Casimir function to apply a geometric model for energy preserving entropy production. We illustrate this framework with several examples including multivariate Gaussian probability densities, and the Bogoliubov-Kubo-Mori metric as a quantum version of the Fisher metric for quantum information on coadjoint orbits. We exploit this geometric setting and Lie group equivariance to present symplectic and multisymplectic variational Lie group integration schemes for some of the equations associated with Souriau symplectic and polysymplectic models, such as the Lie-Poisson equation with cocycle.

Weak Gibbs measures and equilibrium states

ABSTRACT We present a simple proof of the fairly general fact that an invariant right-weak Gibbs probability measure is an equilibrium state, using a generalization of the formula of Brin and Katok for the local metric entropy. This new formula also allows us to clarify the role of the topological pressure on the standard definition of the Gibbs property.

AN ENERGY SEGMENTATION METHOD OF HIGH-RESOLUTION SAR IMAGE BASED ON MULTIPLE FEATURES

Abstract. To achieve the optimal image segmentation, an energy segmentation method based on multiple features and blocks for high resolution Synthetic Aperture Radar (SAR) image is proposed in this paper. First of all, a feature vector of pixel is formed with the texture feature extracted by curvelet transform and means function, the boundary feature extracted by curvelet transform and Canny Operator, and the original spectral feature; a feature set is formed by all feature vectors of pixels in the image. The feature vector is considered as segmentation basis, and its domain is partitioned by regular tessellation. On the partitioned image domain, a label variable is assigned to a regular block; each homogeneous region is fitted by one or more regular blocks; Obviously, a label field is formed by all the label variables of regular blocks. The model of label field is built by using energy function of neighborhood relationship. The feature set is considered as a realization of a random filed of multiple features (multiple features field for short). A heterogeneous energy function is used to establish the model of multiple features field. Then the established models of the label field and multiple features field are combined to define global energy function of image segmentation, and non-constrained Gibbs probability distribution is used to describe the global energy function to build the energy segmentation model based on multiple features. Further, a RJMCMC algorithm is designed to simulate from the model to segment SAR image. To verify the feasibility and superiority of the proposed approach, real SAR images are tested.

МЕТОД КОНТРОЛЮ ПОСЛІДОВНОСТІ РЕАЛІЗАЦІЇ АТАКУЮЧИХ ДІЙ ПІД ЧАС АКТИВНОГО АНАЛІЗУ ЗАХИЩЕНОСТІ КОРПОРАТИВНИХ МЕРЕЖ

The article proposes an approach to increase the efficiency of vulnerability validation during automatic active analysis of security of corporate networks based on control of the sequence of implementation of offensive actions (exploits) according to softmax action selection strategy using Gibbs probability distribution. At the same time, based on a practical analysis of the process of validation of vulnerabilities, the coefficient of erroneous decisions on the implementation of the exploit was introduced, which allows to dynamically change the key parameter of the Gibbs distribution - temperature, which in turn balances the probability of choosing the next attack. when implementing the validation of the identified vulnerabilities of a specific target system.

Gutenberg–Richter’s b Value and Earthquake Asperity Models

The Gutenberg–Richter (G–R) relationship can be derived as the Gibbs distribution. For a given earthquake set (all earthquakes in a given region, time period, magnitude range, tectonic settings) the Gibbs probability density function for magnitudes, with a given b value in its exponent, is the most uniform distribution under the constraints of the magnitude range and mean value. Therefore, it represents our limited knowledge about the system output: the only pieces of information are the mean value and the magnitude range. Honest earthquake forecasts can be based on such a distribution, since it represents all and only available information about the seismic system. The b value can change among different earthquake sets (in time, space, magnitude ranges, or tectonic settings), since it is related to earthquake rupture dynamics, or seismic source characteristics. The relationship between the b value and the exponent $$\beta$$ in the rupture area vs. maximum slip scaling, $$A\propto D^{\beta }$$, results from viewing earthquake recurrence time in connection with the slip budget. This makes a link between earthquake statistics (the G–R law) and physics (fault characteristics). Specifically, the relationship enables us to explain different ranges of b values at megathrust faults, in dependence on interplate asperity and coupling distributions, as well as on amounts of sediments and fluids in subduction channels. The approach differs from common interpretations of the G–R law in that the b value becomes a field variable, not a constant. It is always the Gibbs distribution for a given magnitude range that we use due to our ignorance about the system outcome, and it is the b value that variates, depending on our knowledge about the system physics. This is important for seismic forecasts, which are mostly based on the G–R relationship. First, because the physical processes leading to the largest earthquakes can be revealed by observing the b value variations. Second, because earthquake generation process can be thought of as sampling with constraints, where the b value and the magnitude range are the constraints.

Gibbs states and Gibbsian specifications on the space ℝℕ

We are interested in the study of Gibbs and equilibrium probabilities on the space state . We consider the unilateral full-shift σ defined on the non-compact set , that is , and a Hölder continuous potential . From a suitable class of a priori probability measures ν we define the Ruelle operator associated to A and we show the existence of eigenfunctions, conformal probability measures and equilibrium states associated to A. Moreover, we prove the existence of an involution kernel for A, we build a Gibbsian specification for the Borelian sets on and we show that this family of probabilities satisfies a FKG-inequality.

Quantum spin probabilities at positive temperature are Hölder Gibbs probabilities

We consider the KMS state associated to the Hamiltonian [Formula: see text] over the quantum spin lattice [Formula: see text] For a fixed observable of the form [Formula: see text] where [Formula: see text] is self-adjoint, and for positive temperature [Formula: see text] one can get a naturally defined stationary probability [Formula: see text] on the Bernoulli space [Formula: see text]. The Jacobian of [Formula: see text] can be expressed via a certain continued fraction expansion. We will show that this probability is a Gibbs probability for a Hölder potential. Therefore, this probability is mixing for the shift map. For such probability [Formula: see text] we will show the explicit deviation function for a certain class of functions. When decreasing temperature we will be able to exhibit the explicit transition value [Formula: see text] where the set of values of the Jacobian of the Gibbs probability [Formula: see text] changes from being a Cantor set to being an interval. We also present some properties for quantum spin probabilities at zero temperature (for instance, the explicit value of the entropy).

An Energy-Based SAR Image Segmentation Method with Weighted Feature

To extract more structural features, which can contribute to segment a synthetic aperture radar (SAR) image accurately, and explore their roles in the segmentation procedure, this paper presents an energy-based SAR image segmentation method with weighted features. To precisely segment a SAR image, multiple structural features are incorporated into a block- and energy-based segmentation model in weighted way. In this paper, the multiple features of a pixel, involving spectral feature obtained from original SAR image, texture and boundary features extracted by a curvelet transform, form a feature vector. All the pixels’ feature vectors form a feature set of a SAR image. To automatically determine the roles of the multiple features in the segmentation procedure, weight variables are assigned to them. All the weight variables form a weight set. Then the image domain is partitioned into a set of blocks by regular tessellation. Afterwards, an energy function and a non-constrained Gibbs probability distribution are used to combine the feature and weight sets to build a block-based energy segmentation model with feature weighted on the partitioned image domain. Further, a reversible jump Markov Chain Monte Carlo (RJMCMC) algorithm is designed to simulate from the segmentation model. In the RJMCMC algorithm, three move types were designed according to the segmentation model. Finally, the proposed method was tested on the SAR images, and the quantitative and qualitative results demonstrated its effectiveness.

On the critical behavior of a homopolymer model

We begin with the reference measure P0 induced by simple, symmetric nearest neighbor continuous time random walk on Zd starting at 0 with jump rate 2d and then define, for β ⩾ 0, t > 0, the Gibbs probability measure Pβ,t by specifying its density with respect to P0 as $${{d{P_{\beta, t}}} \over {d{P^0}}} = {Z_{\beta, t}}{(0)^{-1}}{{\rm{e}}^{\beta \int_0^t {{\delta _0}({x_s})ds}}},$$ (0.1) where $${Z_{\beta, t}}(0) \equiv {E^0}{\rm{[}}{{\rm{e}}^{\beta \;\int_0^t {{\delta _0}({x_s})ds}}}].$$. This Gibbs probability measure provides a simple model for a homopolymer with an attractive potential at the origin. In a previous paper (Cranston and Molchanov, 2007), we showed that for dimensions d ⩾ 3 there is a phase transition in the behavior of these paths from the diffusive behavior for β below a critical parameter to the positive recurrent behavior for β above this critical value. The critical value was determined by means of the spectral properties of the operator Δ + βδ0, where Δ is the discrete Laplacian on Zd. This corresponds to a transition from a diffusive or stretched-out phase to a globular phase for the polymer. In this paper we give a description of the polymer at the critical value where the phase transition takes place. The behavior at the critical parameter is dimension-dependent.

The KMS condition for the homoclinic equivalence relation and Gibbs probabilities

D. Ruelle considered a general setting where he is able to characterize equilibrium states for Holder potentials based on properties of conjugating homeomorphism in the so called Smale spaces. On this setting he also shows a relation of KMS states of $$C^*$$ -algebras with equilibrium probabilities of Thermodynamic Formalism. A later paper by N. Haydn and D. Ruelle presents a shorter proof of this equivalence. Here we consider similar problems but now on the symbolic space $$\Omega = \{1,2,\ldots ,d\}^{{\mathbb {Z}} - \{ 0 \} }$$ and the dynamics will be given by the shift $$\tau $$ . In the case of potentials depending on a finite coordinates we will present a simplified proof of the equivalence mentioned above which is the main issue of the papers by D. Ruelle and N. Haydn. The class of conjugating homeomorphism is explicit and reduced to a minimal set of conditions. We also present with details (following D. Ruelle) the relation of these probabilities with the KMS dynamical $$C^*$$ -state on the $$C^*$$ -algebra associated to the groupoid defined by the homoclinic equivalence relation. The topics presented here are not new but we believe the main ideas of the proof of the results by Ruelle and Haydn will be quite transparent in our exposition.

Active image restoration

We study active restoration of noise-corrupted images generated via the Gibbs probability of an Ising ferromagnet in external magnetic field. Ferromagnetism accounts for the prior expectation of data smoothness, i.e. a positive correlation between neighbouring pixels (Ising spins), while the magnetic field refers to the bias. The restoration is actively supervised by requesting the true values of certain pixels after a noisy observation. This additional information improves restoration of other pixels. The optimal strategy of active inference is not known for realistic (two-dimensional) images. We determine this strategy for the mean-field version of the model and show that it amounts to supervising the values of spins (pixels) that do not agree with the sign of the average magnetization. The strategy leads to a transparent analytical expression for the minimal Bayesian risk, and shows that there is a maximal number of pixels beyond of which the supervision is useless. We show numerically that this strategy applies for two-dimensional images away from the critical regime. Within this regime the strategy is outperformed by its local (adaptive) version, which supervises pixels that do not agree with their Bayesian estimate. We show on transparent examples how active supervising can be essential in recovering noise-corrupted images and advocate for a wider usage of active methods in image restoration.

RETRACTED ARTICLE: Mobile client data security storage protocol based on multifactor node evaluation

To improve calculation and storage capacity of mobile cloud data storage algorithm and solve the problem of low reliability and low energy utilization efficiency in remote server mode, the resampling mobile cloud data storage algorithm based on Gibbs probability allocation estimate is proposed. This algorithm firstly uses vote data allocation and vote data processing model to construct the calculation model for expected propagation time of resampling model in the condition of node failure probability and establishes vote dynamic network integrating energy efficiency and fault tolerance. Secondly, for the constructed dynamic network model, the storage path is optimized via sample probability allocation estimate. In addition, to improve process performance of allocation estimate, the Gibbs sampling process is used to realize high-dimensional coupling and non-supervised training for sample data in the process of allocation estimate. Lastly the effectiveness of proposed model algorithm was verified by experiment.