When triangulating a belief network we aim to obtain a junction tree of minimum state space. According to (Rose, 1970), searching for the optimal triangulation can be cast as a search over all the permutations of the graph's vertices. Our approach is to embed the discrete set of permutations in a convex continuous domain D. By suitably extending the cost function over D and solving the continous nonlinear optimization task we hope to obtain a good triangulation with respect to the aformentioned cost. This paper presents two ways of embedding the triangulation problem into continuous domain and shows that they perform well compared to the best known heuristic.

The full Bayesian method for applying neural networks to a prediction problem is to set up the prior/hyperprior structure for the net and then perform the necessary integrals. However, these integrals are not tractable analytically, and Markov Chain Monte Carlo (MCMC) methods are slow, especially if the parameter space is high-dimensional. Using Gaussian processes we can approximate the weight space integral analytically, so that only a small number of hyperparameters need be integrated over by MCMC methods. We have applied this idea to classification problems, obtaining excellent results on the real-world problems investigated so far. 1 INTRODUCTION To make predictions based on a set of training data, fundamentally we need to combine our prior beliefs about possible predictive functions with the data at hand. In the Bayesian approach to neural networks a prior on the weights in the net induces a prior distribution over functions.

These include Boltzmann machines (Hinton and Sejnowski 1986), binary sigmoidal belief networks (Neal 1992) and Helmholtz machines (Hinton et al. 1995; Dayan et al. 1995). However, some hidden variables, such as translation or scaling in images of shapes, are best represented using continuous values. Continuous-valued Boltzmann machines have been developed (Movellan and McClelland 1993), but these suffer from long simulation settling times and the requirement of a "negative phase" during learning. Tibshirani (1992) and Bishop et al. (1996) consider learning mappings from a continuous latent variable space to a higher-dimensional input space. MacKay (1995) has developed "density networks" that can model both continuous and categorical latent spaces using stochasticity at the topmost network layer. In this paper I consider a new hierarchical top-down connectionist model that has stochastic hidden variables at all layers; moreover, these variables can adapt to be continuous or categorical. The proposed top-down model can be viewed as a continuous-valued belief network, which can be simulated by performing a quick top-down pass (Pearl 1988).

The classes in classification tasks often have a natural ordering, and the training and testing examples are often incomplete. We propose a nonlinear ordinal model for classification into ordered classes. Predictive, simulation-based approaches are used to learn from past and classify future incomplete examples. These techniques are illustrated by making prognoses for patients who have suffered severe head injuries.

We develop a recursive node-elimination formalism for efficiently approximating large probabilistic networks. No constraints are set on the network topologies. Yet the formalism can be straightforwardly integrated with exact methods whenever they are/become applicable. The approximations we use are controlled: they maintain consistently upper and lower bounds on the desired quantities at all times. We show that Boltzmann machines, sigmoid belief networks, or any combination (i.e., chain graphs) can be handled within the same framework.

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.

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.

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.

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.

For neural networks with a wide class of weight-priors, it can be shown that in the limit of an infinite number of hidden units the prior over functions tends to a Gaussian process. In this paper analytic forms are derived for the covariance function of the Gaussian processes corresponding to networks with sigmoidal and Gaussian hidden units. This allows predictions to be made efficiently using networks with an infinite number of hidden units, and shows that, somewhat paradoxically, it may be easier to compute with infinite networks than finite ones. 1 Introduction To someone training a neural network by maximizing the likelihood of a finite amount of data it makes no sense to use a network with an infinite number of hidden units; the network will "overfit" the data and so will be expected to generalize poorly. However, the idea of selecting the network size depending on the amount of training data makes little sense to a Bayesian; a model should be chosen that reflects the understanding of the problem, and then application of Bayes' theorem allows inference to be carried out (at least in theory) after the data is observed. In the Bayesian treatment of neural networks, a question immediately arises as to how many hidden units are believed to be appropriate for a task. Neal (1996) has argued compellingly that for real-world problems, there is no reason to believe that neural network models should be limited to nets containing only a "small" number of hidden units. He has shown that it is sensible to consider a limit where the number of hidden units in a net tends to infinity, and that good predictions can be obtained from such models using the Bayesian machinery. He has also shown that for fixed hyperparameters, a large class of neural network models will converge to a Gaussian process prior over functions in the limit of an infinite number of hidden units.