We present a technique for constructing random fields from a set of training samples. The learning paradigm builds increasingly complex fields by allowing potential functions, or features, that are supported by increasingly large subgraphs. Each feature has a weight that is trained by minimizing the Kullback-Leibler divergence between the model and the empirical distribution of the training data. A greedy algorithm determines how features are incrementally added to the field and an iterative scaling algorithm is used to estimate the optimal values of the weights. The random field models and techniques introduced in this paper differ from those common to much of the computer vision literature in that the underlying random fields are non-Markovian and have a large number of parameters that must be estimated. Relations to other learning approaches, including decision trees, are given. As a demonstration of the method, we describe its application to the problem of automatic word classifica...

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Markov chains having the same stationary distribution ß can be partially ordered by performance in the central limit theorem. We say that one chain is at least as good as another in the efficiency partial ordering if the variance in the central limit theorem is at least as small for every L 2 (ß) functional of the chain. Peskun partial ordering implies efficiency partial ordering [25, 30]. Here we show that Peskun partial ordering implies, for finite state spaces, ordering of all the eigenvalues of the transition matrices, and, for general state spaces, ordering of the suprema of the spectra of the transition operators. We also define a covariance partial ordering based on lag one autocovariances and show that it is equivalent to the efficiency partial ordering when restricted to reversible Markov chains. Similar but weaker results are provided for non-reversible Markov chains. Keywords: Peskun ordering, Eigenvalues, Spectral decomposition, Non-reversible kernels. 1 Introduction I...

Markov chain Monte Carlo is a method of approximating the integral of a function f with respect to a distribution ß. A Markov chain that has ß as its stationary distribution is simulated producing samples X 1 ; X 2 ; : : : . The integral is approximated by taking the average of f(X n ) over the sample path. The standard way to construct such Markov chains is the Metropolis-Hastings algorithm. The class P of all Markov chains having ß as their unique stationary distribution is very large, so it is important to have criteria telling when one chain performs better than another. The Peskun ordering is a partial ordering on P. If two Markov chains are Peskun ordered, then the better chain has smaller variance in the central limit theorem for every function f that has a variance. Peskun ordering is sufficient for this but not necessary. We study the implications of the Peskun ordering both in finite and general state spaces. Unfortunately there are many Metropolis-Hastings samplers that are...

Random field models in image analysis and spatial statistics usually have local interactions. They can be simulated by Markov chains which update a single site at a time. The updating rules typically condition on only a few neighboring sites. If we want to approximate the expectation of a bounded function, can we make better use of the simulations than through the empirical estimator? We describe symmetrizations of the empirical estimator which are computationally feasible and can lead to considerable variance reduction. The method is reminiscent of the idea behind generalized von Mises statistics. To simplify the exposition, we consider mainly nearest neighbor random fields and the Gibbs sampler. Key words and Phrases. Asymptotic relative efficiency, Gibbs sampler, Ising model, Markov chain Monte Carlo, Metropolis algorithm, parallel updating, variance reduction. Research partially supported by NSF Grant ATM-9417528. 1 All three authors were partially supported by NSERC, Canada. ...

If we wish to efficiently estimate the expectation of an arbitrary function on the basis of the output of a Gibbs sampler, which is better: deterministic or random sweep? In each case we calculate the asymptotic variance of the empirical estimator, the average of the function over the output, and determine the minimal asymptotic variance for estimators that use no information about the underlying distribution. The empirical estimator has noticeably smaller variance for deterministic sweep. The variance bound for random sweep is in general smaller than for deterministic sweep, but the two are equal if the target distribution is continuous. If the components of the target distribution are not strongly dependent, the empirical estimator is close to efficient under deterministic sweep, and its asymptotic variance approximately doubles under random sweep. 1 Introduction The Gibbs sampler is a widely used Markov chain Monte Carlo (MCMC) method for estimating analytically intractable feature...

The class of nite state space Markov chains, stationary with respect to a common pre-specied distribution, is considered. An easy to check partial ordering is dened on this class. The ordering provides a sucient condition for the dominating Markov chain to be more ecient. Eciency is measured by the asymptotic variance of the estimator of the integral of a specic function with respect to the stationary distribution of the chains. A class of transformations that, when applied to a transition matrix, preserves its stationary distribution and improves its eciency is dened and studied. Keywords: Eciency ordering, Markov chain Monte Carlo, Metropolis-Hastings algorithm, Stationarity preserving and eciency increasing transfers. 1 Introduction Consider a distribution of interest (x), x 2 X , possibly known up to a normalizing constant. Assume that X is an N dimensional nite state space. In order to gather information about we construct a Harris recurrent Markov chain with tr...

We propose a general synchronous model of lattice random fields which could be used similarly to Gibbs distributions in a bayesian framework for image analysis, leading to algorithms ideally designed for an implementation on massively parallel hardware. After a theoretical description of the model, we give an experimental illustration in the context of image restoration. Keywords: Random fields, Image processing, Parallelism, Monte-Carlo sampling 1 Introduction. The use of random fields in practical problems involving complex interactions in high dimensional systems is by now a widespreaded technique. The range of applications includes statistical mechanics, spatial statistics, image analysis, neural network modeling, etc. . . Models which are employed generally have recourse to the Gibbs representation of the field, by means of a potential. This representation, which is simple and natural, has found applications in many situations, since the evidence of its feasibility in a bayesian ...

This communication presents new results about convergence of stochastic gradient algorithms for maximum likelihood estimation of Markov random fields. We first present theoretical results dealing with the convergence of a generalized Robbins-Monro procedure. These results provide rigorous justifications for simple numerical strategies which can be employed in practice; they are illustrated by numerical experiments. 1. INTRODUCTION Markov random field models have become a standard, efficient tool, for low-level Bayesian analysis of images. In many cases, a good estimation of the prior distribution brings significant improvement to subsequent analyses based on the posterior. We describe here some strategies which evaluate with accuracy the true parameters of a statistical model of random fields, on the basis of an observed training data. The statistical procedure which will be used is maximum likelihood estimation. For Markov random fields, this is a hard numerical problem which can be...

We outline the theory of efficient estimation for semiparametric Markov chain models, and illustrate in a number of simple cases how the theory can be used to determine lower bounds for the asymptotic variance of estimators and to construct efficient estimators. In particular, we consider estimation of stationary distributions of Markov chains, of autoregression parameters and innovation distributions in AR- and ARCH-models and more general time series, and of parameters in quasi-likelihood models. AMS 1991 subject classifications. Primary 62M05; secondary 62F12, 62F35, 62G20, 62M10. Key words and Phrases. Variance bound, empirical estimator, martingale approximation, maximum likelihood estimator, Kullback-Leibler information, estimating equation, misspecified model, weighted least squares, conditional heteroscedasticity, quasi-likelihood, Markov chain Monte Carlo, Gibbs sampler. 1 Introduction The basic example of a time series is the autoregressive process X i = ffX i\Gamma1 + " ...