Search

This chapter describes a sequence of Monte Carlo methods: importance sampling, rejection sampling, the Metropolis method, and Gibbs sampling. For each method, we discuss whether the method is expected to be useful for high—dimensional problems such as arise in inference with graphical models. After the methods have been described, the terminology of Markov chain Monte Carlo methods is presented. The chapter concludes with a discussion of advanced methods, including methods for reducing random walk behaviour.

Probabilistic models for images are analysed quantitatively using Bayesian hypothesis comparison on a set of image data sets. One motivation for this study is to produce models which can be used as better priors in image reconstruction problems.

The 1993 energy prediction competition involved the prediction of a series of building energy loads from a series of environmental input variables. Non-linear regression using ‘neural networks’ is a popular technique for such modeling tasks. Since it is not obvious how large a time-window of inputs is appropriate, or what preprocessing of inputs is best, this can be viewed as a regression problem in which there are many possible input variables, some of which may actually be irrelevant to the prediction of the output variable. Because a finite data set will show random correlations between the irrelevant inputs and the output, any conventional neural network (even with regularisation or ‘weight decay’) will not set the coefficients for these junk inputs to zero. Thus the irrelevant variables will hurt the model’s performance.

I define a latent variable model in the form of a neural network for which only target outputs are specified; the inputs are unspecified. Although the inputs are missing, it is still possible to train this model by placing a simple probability distribution on the unknown inputs and maximizing the probability of the data given the parameters. The model can then discover for itself a description of the data in terms of an underlying latent variable space of lower dimensionality. I present preliminary results of the application of these models to protein data.

I examine two approximate methods for computational implementation of Bayesian hierarchical models, that is, models which include unknown hyperparameters such as regularization constants. In the ‘evidence framework’ the model parameters are integrated over, and the resulting evidence is maximized over the hyperparameters. The optimized hyperparameters are used to define a Gaussian approximation to the posterior distribution. In the alternative ‘MAP’ method, the true posterior probability is found by integrating over the hyperparameters. The true posterior is then maximized over the model parameters, and a Gaussian approximation is made. The similarities of the two approaches, and their relative merits, are discussed, and comparisons are made with the ideal hierarchical Bayesian solution.

We examine the problem of deconvolving blurred text. This is a task in which there is strong prior knowledge (e.g., font characteristics) that is hard to express computationally. These priors are implicit, however, in mock data for which the true image is known. When trained on such mock data, a neural network is able to learn a solution to the image deconvolution problem which takes advantage of this implicit prior knowledge. Prior knowledge of image positivity can be hard-wired into the functional architecture of the network, but we leave it to the network to learn most of the parameters of the task from the data. We do not need to tell the network about the point spread function, the intrinsic correlation function, or the noise process.

A number of methods have been developed to analyze the response of the linear phased array radar. These perform remarkably well when the number of sources is known, but in cases where a determination of this number is required, problems are often encountered. These problems can be resolved by a Bayesian approach.

A quantitative and practical Bayesian framework is described for learning of mappings in feedforward networks. The framework makes possible: (1) objective comparisons between solutions using alternative network architectures; (2) objective stopping rules for network pruning or growing procedures; (3) objective choice of magnitude and type of weight decay terms or additive regularisers (for penalising large weights, etc.); (4) a measure of the effective number of well-determined parameters in a model; (5) quantified estimates of the error bars on network parameters and on network output; (6) objective comparisons with alternative learning and interpolation models such as splines and radial basis functions. The Bayesian ‘evidence’ automatically embodies ‘Occam’s razor,’ penalising over-flexible and over-complex models. The Bayesian approach helps detect poor underlying assumptions in learning models. For learning models well matched to a problem, a good correlation between generalisation ability and the Bayesian evidence is obtained.