Many real world patterns have a hierarchical underlying structure in which simple features have a higher order structure among themselves. Because these relationships are often statistical in nature, it is natural to view the process of discovering such structures as a statistical inference problem in which a hierarchical model is fit to data. Hierarchical statistical structure can be conveniently represented with Bayesian belief networks (Pearl, 1988; Lauritzen and Spiegelhalter, 1988; Neal, 1992). These 530 M. S. Lewicki and T. 1. Sejnowski models are powerful, because they can capture complex statistical relationships among the data variables, and also mathematically convenient, because they allow efficient computation of the joint probability for any given set of model parameters.

This paper describes how the early visual process of contour organisation can be realised using the EM algorithm. The underlying computational representation is based on fine spline coverings. According to our EM approach the adjustment of spline parameters draws on an iterative weighted least-squares fitting process. The expectation step of our EM procedure computes the likelihood of the data using a mixture model defined over the set of spline coverings. These splines are limited in their spatial extent using Gaussian windowing functions.

The essential property of BBNs is summarized by the Markov condition, which asserts that each variable is independent of its non-descendants given its parents.

In the Bayes approach to statistical inference [Berger, 1985] one assumes that the prior uncertainty about parameters of an unknown data generating mechanism can be encoded in a probability distribution, the so called prior. Using the prior and the likelihood of the data given the parameters, the posterior distribution of the parameters can be derived from Bayes rule. From this posterior, various estimates for functions ofthe parameter, like predictions about unseen data, can be calculated. However, in general, those predictions cannot be realised by specific parameter values, but only by an ensemble average over parameters according to the posterior probability. Hence, exact implementations of Bayes method for neural networks require averages over network parameters which in general can be performed by time consuming 226 M. Opper and O. Winther Monte Carlo procedures.

Coarse codes are widely used throughout the brain to encode sensory and motor variables. Methods designed to interpret these codes, such as population vector analysis, are either inefficient, i.e., the variance of the estimate is much larger than the smallest possible variance, or biologically implausible, like maximum likelihood. Moreover, these methods attempt to compute a scalar or vector estimate of the encoded variable. Neurons are faced with a similar estimation problem. They must read out the responses of the presynaptic neurons, but, by contrast, they typically encode the variable with a further population code rather than as a scalar. We show how a nonlinear recurrent network can be used to perform these estimation in an optimal way while keeping the estimate in a coarse code format. This work suggests that lateral connections in the cortex may be involved in cleaning up uncorrelated noise among neurons representing similar variables.

Neural Computing Research Group Aston University, Birmingham, B4 7ET, U.K. http://www.ncrg.aston.ac.uk/ Abstract The techniques of Bayesian inference have been applied with great success to many problems in neural computing including evaluation of regression functions, determination of error bars on predictions, and the treatment of hyper-parameters. However, the problem of model comparison is a much more challenging one for which current techniques have significant limitations. In this paper we show how an extended form of Markov chain Monte Carlo, called chaining, is able to provide effective estimates of the relative probabilities of different models. We present results from the robot arm problem and compare them with the corresponding results obtained using the standard Gaussian approximation framework. Initially this is chosen to be some prior distribution p(wIM), which can be combined with a likelihood function p( Dlw, M) using Bayes' theorem to give a posterior distribution p(wID, M) in the form (ID M) p(Dlw,M)p(wIM) (1) p w, p(DIM) where D is the data set. Predictions of the model are obtained by performing integrations weighted by the posterior distribution.

This paper develops arguments for a family of temporal log-linear models to represent spatiotemporal correlations among the spiking events in a group of neurons. The models can represent not just pairwise correlations but also correlations of higher order. Methods are discussed for inferring the existence or absence of correlations and estimating their strength. A frequentist and a Bayesian approach to correlation detection are compared.

Images are ambiguous at each of many levels of a contextual hierarchy. Nevertheless, the high-level interpretation of most scenes is unambiguous, as evidenced by the superior performance of humans. This observation argues for global vision models, such as deformable templates. Unfortunately, such models are computationally intractable for unconstrained problems. We propose a compositional model in which primitives are recursively composed, subject to syntactic restrictions, to form tree-structured objects and object groupings. Ambiguity is propagated up the hierarchy in the form of multiple interpretations, which are later resolved by a Bayesian, equivalently minimum-description-Iength, cost functional.

Coarse codes are widely used throughout the brain to encode sensory andmotor variables. Methods designed to interpret these codes, such as population vector analysis, are either inefficient, i.e., the variance of the estimate is much larger than the smallest possible variance,or biologically implausible, like maximum likelihood. Moreover, these methods attempt to compute a scalar or vector estimate of the encoded variable. Neurons are faced with a similar estimationproblem. They must read out the responses of the presynaptic neurons, but, by contrast, they typically encode the variable with a further population code rather than as a scalar. We show how a nonlinear recurrent network can be used to perform theseestimation in an optimal way while keeping the estimate in a coarse code format. This work suggests that lateral connections inthe cortex may be involved in cleaning up uncorrelated noise among neurons representing similar variables.

In the Bayes approach to statistical inference [Berger, 1985] one assumes that the prior uncertainty about parameters of an unknown data generating mechanism can be encoded in a probability distribution, the so called prior. Using the prior and the likelihood of the data given the parameters, the posterior distribution of the parameters can be derived from Bayes rule. From this posterior, various estimates for functions ofthe parameter, like predictions about unseen data, can be calculated. However, in general, those predictions cannot be realised by specific parameter values, but only by an ensemble average over parameters according to the posterior probability. Hence,exact implementations of Bayes method for neural networks require averages over network parameters which in general can be performed by time consuming 226 M.Opper and O. Winther Monte Carlo procedures.