A CDP can always be discretized in state space and time and thus reduced to a Markov Decision Problem. Algorithms like Q-Iearning and RTDP as described in [1] can then be applied to produce controls or optimal value functions for a fixed discretization. Problems arise when the discretization needs to be refined, or when multi-grid information needs to be extracted to accelerate the algorithm. The relation of time to state space discretization parameters is crucial in both cases. Therefore 1034 S. Pareigis a mathematical model of the discretized process is introduced, which reflects the properties of the converged empirical process.

In this paper we apply the method of complexity regularization to derive estimation bounds for nonlinear function estimation using a single hidden layer radial basis function network.

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.

The perturbation method which we have presented overcomes the limitations of standard approaches, which are only appropriate for models with a single layer of adjustable weights, albeit at considerable computational expense. It has the added bonus of automatically taking into account the effect of regularisation techniques such as weight decay. The experimental results illustrate the application of the technique to two simple problems. As expected the number of degrees of freedom in the models is found to be related to the amount of weight decay used during training. The equivalent kernels are found to vary significantly in different regions of input space and the functions reconstructed from the estimated smoother matrices closely match the origna!

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.

Standard recurrent nets cannot deal with long minimal time lags between relevant signals. Several recent NIPS papers propose alternative methods. We first show: problems used to promote various previous algorithms can be solved more quickly by random weight guessing than by the proposed algorithms. We then use LSTM, our own recent algorithm, to solve a hard problem that can neither be quickly solved by random search nor by any other recurrent net algorithm we are aware of.

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.

The Self-Organizing Map (SOM) algorithm has been extensively studied and has been applied with considerable success to a wide variety of problems. However, the algorithm is derived from heuristic ideas and this leads to a number of significant limitations. In this paper, we consider the problem of modelling the probability density of data in a space of several dimensions in terms of a smaller number of latent, or hidden, variables. We introduce a novel form of latent variable model, which we call the GTM algorithm (for Generative Topographic Mapping), which allows general nonlinear transformations from latent space to data space, and which is trained using the EM (expectation-maximization) algorithm. Our approach overcomes the limitations of the SOM, while introducing no significant disadvantages. We demonstrate the performance of the GTM algorithm on simulated data from flow diagnostics for a multiphase oil pipeline.

The convergence properties of the gradient descent algorithm in the case of the linear perceptron may be obtained from the response function. We derive a general expression for the response function and apply it to the case of data with simple input correlations. It is found that correlations severely may slow down learning. This explains the success of PCA as a method for reducing training time. Motivated by this finding we furthermore propose to transform the input data by removing the mean across input variables as well as examples to decrease correlations. Numerical findings for a medical classification problem are in fine agreement with the theoretical results.