Over-relaxation method for the Monte Carlo evaluation of the partition function for multiquadratic actions
TL;DR: In this paper, a successive over-relaxation (SOR) procedure for the Monte Carlo evaluation of the Euclidean partition function for multiquadratic actions (such as the Yang-Mills action with canonical gauge fixing).
Abstract: I formulate a successive over-relaxation (SOR) procedure for the Monte Carlo evaluation of the Euclidean partition function for multiquadratic actions (such as the Yang-Mills action with canonical gauge fixing). A convergence analysis for the quadratic-action (Abelian) case shows that as thermalization proceeds the mean nodal fields relax according to the difference equation arising from the standard SOR analysis of the associated classical Euclidean field equation. Hence, SOR should accelerate the thermalization process, just as it accelerates convergence in the numerical solution of second-order elliptic differential equations.
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01 Jan 2006
TL;DR: Probability distributions of linear models for regression and classification are given in this article, along with a discussion of combining models and combining models in the context of machine learning and classification.
Abstract: Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.
10,141 citations
Book•
06 Oct 2003
TL;DR: A fun and exciting textbook on the mathematics underpinning the most dynamic areas of modern science and engineering.
Abstract: Fun and exciting textbook on the mathematics underpinning the most dynamic areas of modern science and engineering.
8,091 citations
Cites methods from "Over-relaxation method for the Mont..."
...In Adler’s (1981) overrelaxation method, one instead samples x (t+1) i from a Gaussian that is biased to the opposite side of the conditional distribution....
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Dissertation•
01 Jan 2003
TL;DR: A unified variational Bayesian (VB) framework which approximates computations in models with latent variables using a lower bound on the marginal likelihood and is compared to other methods including sampling, Cheeseman-Stutz, and asymptotic approximations such as BIC.
Abstract: The Bayesian framework for machine learning allows for the incorporation of prior knowledge in a coherent way, avoids overfitting problems, and provides a principled basis for selecting between alternative models. Unfortunately the computations required are usually intractable. This thesis presents a unified variational Bayesian (VB) framework which approximates these computations in models with latent variables using a lower bound on the marginal likelihood.
Chapter 1 presents background material on Bayesian inference, graphical models, and propagation algorithms. Chapter 2 forms the theoretical core of the thesis, generalising the expectation- maximisation (EM) algorithm for learning maximum likelihood parameters to the VB EM algorithm which integrates over model parameters. The algorithm is then specialised to the large family of conjugate-exponential (CE) graphical models, and several theorems are presented to pave the road for automated VB derivation procedures in both directed and undirected graphs (Bayesian and Markov networks, respectively).
Chapters 3–5 derive and apply the VB EM algorithm to three commonly-used and important models: mixtures of factor analysers, linear dynamical systems, and hidden Markov models. It is shown how model selection tasks such as determining the dimensionality, cardinality, or number of variables are possible using VB approximations. Also explored are methods for combining sampling procedures with variational approximations, to estimate the tightness of VB bounds and to obtain more effective sampling algorithms. Chapter 6 applies VB learning to a long-standing problem of scoring discrete-variable directed acyclic graphs, and compares the performance to annealed importance sampling amongst other methods. Throughout, the VB approximation is compared to other methods including sampling, Cheeseman-Stutz, and asymptotic approximations such as BIC. The thesis concludes with a discussion of evolving directions for model selection including infinite models and alternative approximations to the marginal likelihood.
1,930 citations
01 Jan 2011
TL;DR: The role of probabilistic inference in artificial intelligence is outlined, the theory of Markov chains is presented, and various Markov chain Monte Carlo algorithms are described, along with a number of supporting techniques.
Abstract: Probabilistic inference is an attractive approach to uncertain reasoning and empirical learning in artificial intelligence. Computational difficulties arise, however, because probabilistic models with the necessary realism and flexibility lead to complex distributions over high-dimensional spaces. Related problems in other fields have been tackled using Monte Carlo methods based on sampling using Markov chains, providing a rich array of techniques that can be applied to problems in artificial intelligence. The “Metropolis algorithm” has been used to solve difficult problems in statistical physics for over forty years, and, in the last few years, the related method of “Gibbs sampling” has been applied to problems of statistical inference. Concurrently, an alternative method for solving problems in statistical physics by means of dynamical simulation has been developed as well, and has recently been unified with the Metropolis algorithm to produce the “hybrid Monte Carlo” method. In computer science, Markov chain sampling is the basis of the heuristic optimization technique of “simulated annealing”, and has recently been used in randomized algorithms for approximate counting of large sets. In this review, I outline the role of probabilistic inference in artificial intelligence, present the theory of Markov chains, and describe various Markov chain Monte Carlo algorithms, along with a number of supporting techniques. I try to present a comprehensive picture of the range of methods that have been developed, including techniques from the varied literature that have not yet seen wide application in artificial intelligence, but which appear relevant. As illustrative examples, I use the problems of probabilistic inference in expert systems, discovery of latent classes from data, and Bayesian learning for neural networks.
1,185 citations
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TL;DR: The basic principles and the most common Monte Carlo algorithms are reviewed, among which rejection sampling, importance sampling and Monte Carlo Markov chain (MCMC) methods are reviewed.
Abstract: Bayesian inference often requires integrating some function with respect to a posterior distribution. Monte Carlo methods are sampling algorithms that allow to compute these integrals numerically when they are not analytically tractable. We review here the basic principles and the most common Monte Carlo algorithms, among which rejection sampling, importance sampling and Monte Carlo Markov chain (MCMC) methods. We give intuition on the theoretical justification of the algorithms as well as practical advice, trying to relate both. We discuss the application of Monte Carlo in experimental physics, and point to landmarks in the literature for the curious reader.
1,067 citations