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.

Data augmentation and Gibbs sampling are two closely related, sampling-based approaches to the calculation of posterior moments. The fact that each produces a sample whose constituents are neither independent nor identically distributed complicates the assessment of convergence and numerical accuracy of the approximations to the expected value of functions of interest under the posterior. In this paper methods from spectral analysis are used to evaluate numerical accuracy formally and construct diagnostics for convergence. These methods are illustrated in the normal linear model with informative priors, and in the Tobit-censored regression model.

Evolutionary programming (EP) has been applied with success to many numerical and combinatorial optimization problems in recent years. EP has rather slow convergence rates, however, on some function optimization problems. In this paper, a "fast EP" (FEP) is proposed which uses a Cauchy instead of Gaussian mutation as the primary search operator. The relationship between FEP and classical EP (CEP) is similar to that between fast simulated annealing and the classical version. Both analytical and empirical studies have been carried out to evaluate the performance of FEP and CEP for different function optimization problems. This paper shows that FEP is very good at search in a large neighborhood while CEP is better at search in a small local neighborhood. For a suite of 23 benchmark problems, FEP performs much better than CEP for multimodal functions with many local minima while being comparable to CEP in performance for unimodal and multimodal functions with only a few local minima. This paper also shows the relationship between the search step size and the probability of finding a global optimum and thus explains why FEP performs better than CEP on some functions but not on others. In addition, the importance of the neighborhood size and its relationship to the probability of finding a near-optimum is investigated. Based on these analyses, an improved FEP (IFEP) is proposed and tested empirically. This technique mixes different search operators (mutations). The experimental results show that IFEP performs better than or as well as the better of FEP and CEP for most benchmark problems tested.

Modern risk management calls for an understanding of stochastic dependence going beyond simple linear correlation. This paper deals with the static (non-time-dependent) case and emphasizes the copula representation of dependence for a random vector. Linear correlation is a natural dependence measure for multivariate normally and, more generally, elliptically distributed risks but other dependence concepts like comonotonicity and rank correlation should also be understood by the risk management practitioner. Using counterexamples the falsity of some commonly held views on correlation is demonstrated; in general, these fallacies arise from the naive assumption that dependence properties of the elliptical world also hold in the non-elliptical world. In particular, the problem of finding multivariate models which are consistent with prespecified marginal distributions and correlations is addressed. Pitfalls are highlighted and simulation algorithms avoiding these problems are constructed. ...

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We present a statistical view of the texture retrieval problem by combining the two related tasks, namely feature extraction (FE) and similarity measurement (SM), into a joint modeling and classification scheme. We show that using a consistent estimator of texture model parameters for the FE step followed by computing the Kullback--Leibler distance (KLD) between estimated models for the SM step is asymptotically optimal in term of retrieval error probability. The statistical scheme leads to a new wavelet-based texture retrieval method that is based on the accurate modeling of the marginal distribution of wavelet coefficients using generalized Gaussian density (GGD) and on the existence a closed form for the KLD between GGDs. The proposed method provides greater accuracy and flexibility in capturing texture information, while its simplified form has a close resemblance with the existing methods which uses energy distribution in the frequency domain to identify textures. Experimental results on a database of 640 texture images indicate that the new method significantly improves retrieval rates, e.g., from 65% to 77%, compared with traditional approaches, while it retains comparable levels of computational complexity.

The construction and implementation of a Gibbs sampler for efficient simulation from the truncated multivariate normal and Student-t distributions is described. It is shown how the accuracy and convergence of integrals based on the Gibbs sample may be constructed, and how an estimate of the probability of the constraint set under the unrestricted distribution may be produced. Keywords: Bayesian inference; Gibbs sampler; Monte Carlo; multiple integration; truncated normal This paper was prepared for a presentation at the meeting Computing Science and Statistics: the Twenty-Third Symposium on the Interface, Seattle, April 22-24, 1991. Research assistance from Zhenyu Wang and financial support from National Science Foundation Grant SES-8908365 are gratefully acknowledged. The software for the examples may be requested by electronic mail, and will be returned by that medium. 2 1. Introduction The generation of random samples from a truncated multivariate normal distribution, that is, a ...

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This paper provides a practical simulation-based Bayesian and non-Bayesian analysis of correlated binary data using the multivariate probit model. The posterior distribution is simulated by Markov chain Monte Carlo methods and maximum likelihood estimates are obtained by a Monte Carlo version of the EM algorithm. A practical approach for the computation of Bayes factors from the simulation output is also developed. The methods are applied to a dataset with a bivariate binary response, to a four-year longitudinal dataset from the Six Cities study of the health effects of air pollution and to a sevenvariate binary response dataset on the labour supply of married women from the Panel Survey of Income Dynamics.

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