Generalized mean field approximation for parallel dynamics of the Ising model

The dynamics of the non-equilibrium Ising model with parallel updates is investigated using a generalized mean field approximation that incorporates multiple two-site correlations at any two time steps, which can be obtained recursively. The proposed method shows significant improvement in predicting local system properties compared to other mean field approximation techniques, particularly in systems with symmetric interactions. Results are also evaluated against those obtained from Monte Carlo simulations. The method is also employed to obtain parameter values for the kinetic inverse Ising modeling problem, where couplings and local field values of a fully connected spin system are inferred from data. © 2014 IOP Publishing Ltd and SISSA Medialab srl.

Gaussian processes for classification: mean field algorithms:mean-field algorithms

We derive a mean field algorithm for binary classification with Gaussian processes which is based on the TAP approach originally proposed in Statistical Physics of disordered systems. The theory also yields an approximate leave-one-out estimator for the generalization error which is computed with no extra computational cost. We show that from the TAP approach, it is possible to derive both a simpler 'naive' mean field theory and support vector machines (SVM) as limiting cases. For both mean field algorithms and support vectors machines, simulation results for three small benchmark data sets are presented. They show 1. that one may get state of the art performance by using the leave-one-out estimator for model selection and 2. the built-in leave-one-out estimators are extremely precise when compared to the exact leave-one-out estimate. The latter result is a taken as a strong support for the internal consistency of the mean field approach.

Advanced mean field methods:theory and practice

A major problem in modern probabilistic modeling is the huge computational complexity involved in typical calculations with multivariate probability distributions when the number of random variables is large. Because exact computations are infeasible in such cases and Monte Carlo sampling techniques may reach their limits, there is a need for methods that allow for efficient approximate computations. One of the simplest approximations is based on the mean field method, which has a long history in statistical physics. The method is widely used, particularly in the growing field of graphical models. Researchers from disciplines such as statistical physics, computer science, and mathematical statistics are studying ways to improve this and related methods and are exploring novel application areas. Leading approaches include the variational approach, which goes beyond factorizable distributions to achieve systematic improvements; the TAP (Thouless-Anderson-Palmer) approach, which incorporates correlations by including effective reaction terms in the mean field theory; and the more general methods of graphical models. Bringing together ideas and techniques from these diverse disciplines, this book covers the theoretical foundations of advanced mean field methods, explores the relation between the different approaches, examines the quality of the approximation obtained, and demonstrates their application to various areas of probabilistic modeling.

Mean field methods for classification with Gaussian processes

We discuss the Application of TAP mean field methods known from Statistical Mechanics of disordered systems to Bayesian classification with Gaussian processes. In contrast to previous applications, no knowledge about the distribution of inputs is needed. Simulation results for the Sonar data set are given.

Gaussian processes and SVM: Mean field results and leave-one-out

In this chapter, we elaborate on the well-known relationship between Gaussian processes (GP) and Support Vector Machines (SVM). Secondly, we present approximate solutions for two computational problems arising in GP and SVM. The first one is the calculation of the posterior mean for GP classifiers using a `naive' mean field approach. The second one is a leave-one-out estimator for the generalization error of SVM based on a linear response method. Simulation results on a benchmark dataset show similar performances for the GP mean field algorithm and the SVM algorithm. The approximate leave-one-out estimator is found to be in very good agreement with the exact leave-one-out error.

A mean field theory of coded CDMA systems

We present a mean field theory of code-division multiple access (CDMA) systems with error-control coding. On the basis of the relation between the free energy and mutual information, we obtain an analytical expression of the maximum spectral efficiency of the coded CDMA system, from which a mean field description of the coded CDMA system is provided in terms of a bank of scalar Gaussian channels whose variances in general vary at different code symbol positions. Regular low-density parity-check (LDPC)-coded CDMA systems are also discussed as an example of the coded CDMA systems.

Mean field methods for Gaussian process classifiers

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Variational mean-field algorithm for efficient inference in large systems of stochastic differential equations

This work introduces a Gaussian variational mean-field approximation for inference in dynamical systems which can be modeled by ordinary stochastic differential equations. This new approach allows one to express the variational free energy as a functional of the marginal moments of the approximating Gaussian process. A restriction of the moment equations to piecewise polynomial functions, over time, dramatically reduces the complexity of approximate inference for stochastic differential equation models and makes it comparable to that of discrete time hidden Markov models. The algorithm is demonstrated on state and parameter estimation for nonlinear problems with up to 1000 dimensional state vectors and compares the results empirically with various well-known inference methodologies.

Evolution of non-uniformly seeded warm clouds in idealized turbulent conditions

We present a mean-field model of cloud evolution that describes droplet growth due to condensation and collisions and droplet loss due to fallout. The model accounts for the effects of cloud turbulence both in a large-scale turbulent mixing and in a microphysical enhancement of condensation and collisions. The model allows for an effective numerical simulation by a scheme that is conservative in water mass and keeps accurate count of the number of droplets. We first study the homogeneous situation and determine how the rain-initiation time depends on the concentration of cloud condensation nuclei (CCN) and turbulence level. We then consider clouds with an inhomogeneous concentration of CCN and evaluate how the rain initiation time and the effective optical depth vary in space and time. We argue that over-seeding even a part of a cloud by small hygroscopic nuclei, one can substantially delay the onset and increase the amount of precipitation.

Non-zero mean Gaussian process prior wind field models

This report outlines the derivation and application of a non-zero mean, polynomial-exponential covariance function based Gaussian process which forms the prior wind field model used in 'autonomous' disambiguation. It is principally used since the non-zero mean permits the computation of realistic local wind vector prior probabilities which are required when applying the scaled-likelihood trick, as the marginals of the full wind field prior. As the full prior is multi-variate normal, these marginals are very simple to compute.

Error-correcting codes on a Bethe-like lattice

We analyse Gallager codes by employing a simple mean-field approximation that distorts the model geometry and preserves important interactions between sites. The method naturally recovers the probability propagation decoding algorithm as a minimization of a proper free-energy. We find a thermodynamical phase transition that coincides with information theoretical upper-bounds and explain the practical code performance in terms of the free-energy landscape.

The TAP approach to intensive and extensive connectivity systems

The Thouless-Anderson-Palmer (TAP) approach was originally developed for analysing the Sherrington-Kirkpatrick model in the study of spin glass models and has been employed since then mainly in the context of extensively connected systems whereby each dynamical variable interacts weakly with the others. Recently, we extended this method for handling general intensively connected systems where each variable has only O(1) connections characterised by strong couplings. However, the new formulation looks quite different with respect to existing analyses and it is only natural to question whether it actually reproduces known results for systems of extensive connectivity. In this chapter, we apply our formulation of the TAP approach to an extensively connected system, the Hopfield associative memory model, showing that it produces identical results to those obtained by the conventional formulation.

Sequential, Bayesian geostatistics:a principled method for large data sets

The principled statistical application of Gaussian random field models used in geostatistics has historically been limited to data sets of a small size. This limitation is imposed by the requirement to store and invert the covariance matrix of all the samples to obtain a predictive distribution at unsampled locations, or to use likelihood-based covariance estimation. Various ad hoc approaches to solve this problem have been adopted, such as selecting a neighborhood region and/or a small number of observations to use in the kriging process, but these have no sound theoretical basis and it is unclear what information is being lost. In this article, we present a Bayesian method for estimating the posterior mean and covariance structures of a Gaussian random field using a sequential estimation algorithm. By imposing sparsity in a well-defined framework, the algorithm retains a subset of “basis vectors” that best represent the “true” posterior Gaussian random field model in the relative entropy sense. This allows a principled treatment of Gaussian random field models on very large data sets. The method is particularly appropriate when the Gaussian random field model is regarded as a latent variable model, which may be nonlinearly related to the observations. We show the application of the sequential, sparse Bayesian estimation in Gaussian random field models and discuss its merits and drawbacks.

The charge effect on the hindrance factors for diffusion and convection of a solute in pores II

The diffusion and convection of a solute suspended in a fluid across porous membranes are known to be reduced compared to those in a bulk solution, owing to the fluid mechanical interaction between the solute and the pore wall as well as steric restriction. If the solute and the pore wall are electrically charged, the electrostatic interaction between them could affect the hindrance to diffusion and convection. In this study, the transport of charged spherical solutes through charged circular cylindrical pores filled with an electrolyte solution containing small ions was studied numerically by using a fluid mechanical and electrostatic model. Based on a mean field theory, the electrostatic interaction energy between the solute and the pore wall was estimated from the Poisson-Boltzmann equation, and the charge effect on the solute transport was examined for the solute and pore wall of like charge. The results were compared with those obtained from the linearized form of the Poisson-Boltzmann equation, i.e.the Debye-Hückel equation. © 2012 The Japan Society of Fluid Mechanics and IOP Publishing Ltd.

Charge effects on the hindered transport of macromolecules across the endothelial surface glycocalyx layer

A fluid mechanical and electrostatic model for the transport of solute molecules across the vascular endothelial surface glycocalyx layer (EGL) was developed to study the charge effect on the diffusive and convective transport of the solutes. The solute was assumed to be a spherical particle with a constant surface charge density, and the EGL was represented as an array of periodically arranged circular cylinders of like charge, with a constant surface charge density. By combining the fluid mechanical analyses for the flow around a solute suspended in an electrolyte solution and the electrostatic analyses for the free energy of the interaction between the solute and cylinders based on a mean field theory, we estimated the transport coefficients of the solute across the EGL. Both of diffusive and convective transports are reduced compared to those for an uncharged system, due to the stronger exclusion of the solute that results from the repulsive electrostatic interaction. The model prediction for the reflection coefficient for serum albumin agreed well with experimental observations if the charge density in the EGL is ranged from approximately -10 to -30 mEq/l.