scholarly journals Mean Field Methods for a Special Class of Belief Networks

2001 ◽  
Vol 15 ◽  
pp. 91-114 ◽  
Author(s):  
C. Bhattacharyya ◽  
S. S. Keerthi
Keyword(s):  
Broad Class ◽  
Order Approximation ◽  
Mean Field Theory ◽  
Mean Field ◽  
Small Scale ◽  
Belief Networks ◽  
Field Methods ◽  
Special Cases ◽  
First Order Approximation ◽  
Mean Field Approximations

The chief aim of this paper is to propose mean-field approximations for a broad class of Belief networks, of which sigmoid and noisy-or networks can be seen as special cases. The approximations are based on a powerful mean-field theory suggested by Plefka. We show that Saul, Jaakkola and Jordan' s approach is the first order approximation in Plefka's approach, via a variational derivation. The application of Plefka's theory to belief networks is not computationally tractable. To tackle this problem we propose new approximations based on Taylor series. Small scale experiments show that the proposed schemes are attractive.

scholarly journals A one-dimensional modified TASEP model on a track of variable length: analytical and computational results

2021 ◽  
Vol 2090 (1) ◽  
pp. 012025
Author(s):  
B. Reed ◽  
E. Aldrich ◽  
L. Stoleriu ◽  
D.A. Mazilu ◽  
I. Mazilu
Keyword(s):  
Monte Carlo ◽  
Molecular Motors ◽  
Mean Field Theory ◽  
Mean Field ◽  
Limited Range ◽  
Field Methods ◽  
One Dimensional ◽  
Emergent Features ◽  
Simulation Results ◽  
Mean Field Methods

Abstract We present analytical solutions and Monte Carlo simulation results for a one-dimensional modified TASEP model inspired by the interplay between molecular motors and their cellular tracks of variable lengths, known as microtubules. Our TASEP model incorporates rules for changes in the length of the track based on the occupation of the first two sites. Using mean-field theory, we derive analytical results for the particle densities and particle currents and compare them with Monte Carlo simulations. These results show the limited range of mean-field methods for models with localized high correlation between particles. The variability in length adds to the complexity of the model, leading to emergent features for the evolution of particle densities and particle currents compared to the traditional TASEP model.

Mean-Field Approximations for Coupled Populations of Generalized Linear Model Spiking Neurons with Markov Refractoriness

2009 ◽  
Vol 21 (5) ◽  
pp. 1203-1243 ◽  
Author(s):  
Taro Toyoizumi ◽  
Kamiar Rahnama Rad ◽  
Liam Paninski
Keyword(s):  
Linear Model ◽  
Generalized Linear Model ◽  
Mean Field ◽  
Process Models ◽  
Network Activity ◽  
Direct Monte Carlo Simulation ◽  
Field Methods ◽  
Firing Rates ◽  
Cross Correlations ◽  
Mean Field Approximations

There has recently been a great deal of interest in inferring network connectivity from the spike trains in populations of neurons. One class of useful models that can be fit easily to spiking data is based on generalized linear point process models from statistics. Once the parameters for these models are fit, the analyst is left with a nonlinear spiking network model with delays, which in general may be very difficult to understand analytically. Here we develop mean-field methods for approximating the stimulus-driven firing rates (in both the time-varying and steady-state cases), auto- and cross-correlations, and stimulus-dependent filtering properties of these networks. These approximations are valid when the contributions of individual network coupling terms are small and, hence, the total input to a neuron is approximately gaussian. These approximations lead to deterministic ordinary differential equations that are much easier to solve and analyze than direct Monte Carlo simulation of the network activity. These approximations also provide an analytical way to evaluate the linear input-output filter of neurons and how the filters are modulated by network interactions and some stimulus feature. Finally, in the case of strong refractory effects, the mean-field approximations in the generalized linear model become inaccurate; therefore, we introduce a model that captures strong refractoriness, retains all of the easy fitting properties of the standard generalized linear model, and leads to much more accurate approximations of mean firing rates and cross-correlations that retain fine temporal behaviors.

scholarly journals Mean Field Theory for Sigmoid Belief Networks

1996 ◽  
Vol 4 ◽  
pp. 61-76 ◽  
Author(s):  
L. K. Saul ◽  
T. Jaakkola ◽  
M. I. Jordan
Keyword(s):  
Field Theory ◽  
Pattern Recognition ◽  
Mean Field Theory ◽  
Mean Field ◽  
Statistical Pattern Recognition ◽  
Belief Networks ◽  
Benchmark Problem ◽  
True Probability ◽  
Statistical Pattern

We develop a mean field theory for sigmoid belief networks based on ideas from statistical mechanics. Our mean field theory provides a tractable approximation to the true probability distribution in these networks; it also yields a lower bound on the likelihood of evidence. We demonstrate the utility of this framework on a benchmark problem in statistical pattern recognition---the classification of handwritten digits.

scholarly journals Dynamics of fingering convection. Part 1 Small-scale fluxes and large-scale instabilities

2011 ◽  
Vol 677 ◽  
pp. 530-553 ◽  
Author(s):  
A. TRAXLER ◽  
S. STELLMACH ◽  
P. GARAUD ◽  
T. RADKO ◽  
N. BRUMMELL
Keyword(s):  
Large Scale ◽  
Molecular Diffusion ◽  
Turbulent Transport ◽  
Mean Field Theory ◽  
Density Ratio ◽  
Mean Field ◽  
Field Measurements ◽  
Small Scale ◽  
Diapycnal Mixing ◽  
Density Ratios

Double-diffusive instabilities are often invoked to explain enhanced transport in stably stratified fluids. The most-studied natural manifestation of this process, fingering convection, commonly occurs in the ocean's thermocline and typically increases diapycnal mixing by 2 orders of magnitude over molecular diffusion. Fingering convection is also often associated with structures on much larger scales, such as thermohaline intrusions, gravity waves and thermohaline staircases. In this paper, we present an exhaustive study of the phenomenon from small to large scales. We perform the first three-dimensional simulations of the process at realistic values of the heat and salt diffusivities and provide accurate estimates of the induced turbulent transport. Our results are consistent with oceanic field measurements of diapycnal mixing in fingering regions. We then develop a generalized mean-field theory to study the stability of fingering systems to large-scale perturbations using our calculated turbulent fluxes to parameterize small-scale transport. The theory recovers the intrusive instability, the collective instability and the γ-instability as limiting cases. We find that the fastest growing large-scale mode depends sensitively on the ratio of the background gradients of temperature and salinity (the density ratio). While only intrusive modes exist at high density ratios, the collective and γ instabilities dominate the system at the low density ratios where staircases are typically observed. We conclude by discussing our findings in the context of staircase-formation theory.

scholarly journals Mean-Field Approaches to Independent Component Analysis

2002 ◽  
Vol 14 (4) ◽  
pp. 889-918 ◽  
Author(s):  
Pedro A.d.F.R. Højen-Sørensen ◽  
Ole Winther ◽  
Lars Kai Hansen
Keyword(s):  
Independent Component Analysis ◽  
Mean Field Theory ◽  
Mean Field ◽  
Component Analysis ◽  
Independent Component ◽  
Field Methods ◽  
Field Approach ◽  
Mixing Matrix ◽  
The Mean ◽  
Field Approaches

We develop mean-field approaches for probabilistic independent component analysis (ICA). The sources are estimated from the mean of their posterior distribution and the mixing matrix (and noise level) is estimated by maximum a posteriori (MAP). The latter requires the computation of (a good approximation to) the correlations between sources. For this purpose, we investigate three increasingly advanced mean-field methods: the variational (also known as naive mean field) approach, linear response corrections, and an adaptive version of the Thouless, Anderson and Palmer (1977) (TAP) mean-field approach, which is due to Opper and Winther (2001). The resulting algorithms are tested on a number of problems. On synthetic data, the advanced mean-field approaches are able to recover the correct mixing matrix in cases where the variational mean-field theory fails. For handwritten digits, sparse encoding is achieved using nonnegative source and mixing priors. For speech, the mean-field method is able to separate in the underdetermined (overcomplete) case of two sensors and three sources. One major advantage of the proposed method is its generality and algorithmic simplicity. Finally, we point out several possible extensions of the approaches developed here.

scholarly journals On the helicity characteristics and induced emf of magnetic-Coriolis wave packets

2020 ◽  
Vol 223 (2) ◽  
pp. 1398-1411
Author(s):  
B R McDermott ◽  
P A Davidson
Keyword(s):  
Magnetic Field ◽  
Large Scale ◽  
Rotation Axis ◽  
Wave Packets ◽  
Mean Field Theory ◽  
Mean Field ◽  
Low Frequency ◽  
Dynamo Model ◽  
Small Scale ◽  
Inertial Wave

SUMMARY In a rapidly rotating Boussinesq fluid, buoyant anomalies radiate low-frequency inertial wave packets that disperse along the rotation axis. The wave packets lead to axially elongated vortices, which propagate negative (positive) kinetic helicity upwards (downwards) with respect to the rotation vector. The kinetic helicity carried by the inertial wave packets is near-maximal relative to the velocity and vorticity fields. In classical mean-field theory, kinetic helicity is often associated with the α-effect, which is thought to be an important ingredient for planetary dynamos. The modification of inertial wave packets in the presence of a transverse uniform magnetic field is investigated here, motivated by small-scale dynamics in planetary cores, where a large-scale magnetic field affects fluid motions. We study numerically the dispersion of wave packets from an isolated buoyant source and from a random layer of buoyant anomalies, while varying the Lehnert number Le—the ratio of the frequencies of Alfvén and inertial waves. We find that for Le < 0.1, the vortices are columnar and continue to segregate kinetic helicity so that it is negative (positive) above (below) the buoyant source. Importantly, the wave packets induce an α-effect, which remains strong and coherent for Earth-like values of the Lehnert number (Le < 0.1). The interaction of wave packets emitted by multiple neighbouring buoyant sources results in an α-effect that is stronger than the α-effect induced by wave packets launched from an isolated buoyant source, and we provide an analytical explanation for this. The coherence of the α-effect induced by the wave packets, for Earth-like values of the Lehnert number, lends support to the α2 dynamo model driven by helical waves.

scholarly journals Energy transfer in Hall-MHD turbulence: cascades, backscatter, and dynamo action

2007 ◽  
Vol 73 (3) ◽  
pp. 377-401 ◽  
Author(s):  
PABLO D. MININNI ◽  
ALEXANDROS ALEXAKIS ◽  
ANNICK POUQUET
Keyword(s):  
Large Scale ◽  
Magnetic Energy ◽  
Transport Coefficients ◽  
Mean Field Theory ◽  
Mean Field ◽  
Small Scale ◽  
Mhd Turbulence ◽  
Dynamo Action ◽  
Large Scale Magnetic Field ◽  
Hall Mhd

AbstractScale interactions in Hall magnetohydrodynamics (MHDs) are studied using both the mean field theory derivation of transport coefficients, and direct numerical simulations in three space dimensions. In the magnetically dominated regime, the eddy resistivity is found to be negative definite, leading to large-scale instabilities. A direct cascade of the total energy is observed, although as the amplitude of the Hall effect is increased, backscatter of magnetic energy to large scales is found, a feature not present in MHD flows. The coupling between the magnetic and velocity fields is different than in the MHD case, and backscatter of energy from small-scale magnetic fields to large-scale flows is also observed. For the magnetic helicity, a strong quenching of its transfer is found. We also discuss non-helical magnetically forced Hall-MHD simulations where growth of a large-scale magnetic field is observed.

scholarly journals Slow-roll inflation with exponential potential in scalar-tensor models

2019 ◽  
Vol 79 (9) ◽  
Author(s):  
L. N. Granda ◽  
D. F. Jimenez
Keyword(s):  
Scalar Field ◽  
Order Approximation ◽  
Energy Scale ◽  
Model Parameters ◽  
Exponential Potential ◽  
Minimal Coupling ◽  
Slow Roll ◽  
Special Cases ◽  
First Order Approximation ◽  
Slow Roll Inflation

Abstract A study of the slow-roll inflation for an exponential potential in the frame of the scalar-tensor theory is performed, where non-minimal kinetic coupling to curvature and non-minimal coupling of the scalar field to the Gauss-Bonnet invariant are considered. Different models were considered with couplings given by exponential functions of the scalar field, that lead to graceful exit from inflation and give values of the scalar spectral index and the tensor-to-scalar ratio in the region bounded by the current observational data. Special cases were found, where the coupling functions are inverse of the potential, that lead to inflation with constant slow-roll parameters, and it was possible to reconstruct the model parameters for given ns and r. In first-order approximation the standard consistency relation maintains its validity in the model with non-minimal coupling, but it modifies in presence of Gauss–Bonnet coupling. The obtained Hubble parameter during inflation, $$H\sim 10^{-5} M_p$$H∼10-5Mp and the energy scale of inflation $$V^{1/4}\sim 10^{-3} M_p$$V1/4∼10-3Mp, are consistent with the upper bounds set by latest observations.

scholarly journals A mean-field toolbox for spiking neuronal network model analysis

2021 ◽  
Author(s):  
Moritz Layer ◽  
Johanna Senk ◽  
Simon Essink ◽  
Alexander van Meegen ◽  
Hannah Bos ◽  
...  
Keyword(s):  
Network Model ◽  
Neuronal Network ◽  
High Performance ◽  
Neuronal Networks ◽  
Mean Field Theory ◽  
Mean Field ◽  
Power Spectra ◽  
Model Analysis ◽  
Field Methods ◽  
Neuronal Network Model

Mean-field theory of spiking neuronal networks has led to numerous advances in our analytical and intuitive understanding of the dynamics of neuronal network models during the past decades. But, the elaborate nature of many of the developed methods, as well as the difficulty of implementing them, may limit the wider neuroscientific community from taking maximal advantage of these tools. In order to make them more accessible, we implemented an extensible, easy-to-use open-source Python toolbox that collects a variety of mean-field methods for the widely used leaky integrate-and-fire neuron model. The Neuronal Network Mean-field Toolbox (NNMT) in its current state allows for estimating properties of large neuronal networks, such as firing rates, power spectra, and dynamical stability in mean-field and linear response approximation, without running simulations on high performance systems. In this article we describe how the toolbox is implemented, show how it is used to calculate neuronal network properties, and discuss different use-cases, such as extraction of network mechanisms, parameter space exploration, or hybrid modeling approaches. Although the initial version of the toolbox focuses on methods that are close to our own past and present research, its structure is designed to be open and extensible. It aims to provide a platform for collecting analytical methods for neuronal network model analysis and we discuss how interested scientists can share their own methods via this platform.

Sign in / Sign up

Export Citation Format

Share Document