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.

SaM can be said to do clustering/vector quantization (VQ) and at the same time to preserve the spatial ordering of the input data reflected by an ordering of the code book vectors (cluster centroids) in a one or two dimensional output space, where the latter property is closely related to multidimensional scaling (MDS) in statistics. Although the level of activity and research around the SaM algorithm is quite large (a recent overview by [Kohonen 95] contains more than 1000 citations), only little comparison among the numerous existing variants of the basic approach and also to more traditional statistical techniques of the larger frameworks of VQ and MDS is available. Additionally, thereis only little advice in the literature about how to properly use 446 A.Flexer SOM in order to get optimal results in terms of either vector quantization (VQ) or multidimensional scaling or maybe even both of them. To make the notion of SOM being a tool for "data visualization" more precise, the following question has to be answered: Should SOM be used for doing VQ, MDS, both at the same time or none of them? Two recent comprehensive studies comparing SOM either to traditional VQ or MDS techniques separately seem to indicate that SOM is not competitive when used for either VQ or MDS: [Balakrishnan et al. 94J compare SOM to K-means clustering on 108 multivariate normal clustering problems with known clustering solutions and show that SOM performs significantly worse in terms of data points misclassified

Closed-loop control relies on sensory feedback that is usually assumed tobe free . But if sensing incurs a cost, it may be costeffective totake sequences of actions in open-loop mode. We describe a reinforcement learning algorithm that learns to combine open-loop and closed-loop control when sensing incurs a cost. Although weassume reliable sensors, use of open-loop control means that actions must sometimes be taken when the current state of the controlled system is uncertain. This is a special case of the hidden-state problem in reinforcement learning, and to cope, our algorithm relies on short-term memory.

In most treatments of the regression problem it is assumed that the distribution of target data can be described by a deterministic function of the inputs, together with additive Gaussian noise having constantvariance. The use of maximum likelihood to train such models then corresponds to the minimization of a sum-of-squares error function. In many applications a more realistic model would allow the noise variance itself to depend on the input variables. However, the use of maximum likelihood to train such models would give highly biased results. In this paper we show how a Bayesian treatment can allow for an input-dependent variance while overcoming thebias of maximum likelihood. 1 Introduction In regression problems it is important not only to predict the output variables but also to have some estimate of the error bars associated with those predictions.

This paper shows how the prices of option contracts traded in financial marketscan be tracked sequentially by means of the Extended Kalman Filter algorithm. I consider call and put option pairs with identical strike price and time of maturity as a two output nonlinear system.The Black-Scholes approach popular in Finance literature andthe Radial Basis Functions neural network are used in modelling the nonlinear system generating these observations. I show how both these systems may be identified recursively using the EKF algorithm. I present results of simulations on some FTSE 100 Index options data and discuss the implications of viewing the pricing problem in this sequential manner. 1 INTRODUCTION Data from the financial markets has recently been of much interest to the neural computing community. The complexity of the underlying macroeconomic system and how traders react to the flow of information leads to highly nonlinear relationships betweenobservations.

Furthermore it is shown that networks of noisy spiking neurons with temporal coding have a strictly larger computational power than sigmoidal neural nets with the same number of units. 1 Introduction and Definitions We consider a formal model SNN for a §piking neuron network that is basically a reformulation of the spike response model (and of the leaky integrate and fire model) without using 6-functions (see [Maass, 1996a] or [Maass, 1996b] for further backgrou nd).

A hint is any piece of side information about the target function to be learned. We consider the monotonicity hint, which states that the function to be learned is monotonic in some or all of the input variables. The application of monotonicity hints is demonstrated on two real-world problems-a credit card application task, and a problem in medical diagnosis. A measure of the monotonicity error of a candidate function is defined and an objective function for the enforcement of monotonicity is derived from Bayesian principles. We report experimental results which show that using monotonicity hints leads to a statistically significant improvement in performance on both problems.

Recently, there has been a vigorous debate concerning the nature of neural coding (Rieke et al. 1996; Stevens and Zador 1995; Shadlen and Newsome 1994). The prevailing viewhas been that the mean firing rate conveys all information about the sensory stimulus in a spike train and the precise timing of the individual spikes is noise. This belief is, in part, based on a lack of correlation between the precise timing ofthe spikes and the sensory qualities of the stimulus under study, particularly, on a lack of spike timing repeatability when identical stimulation is delivered. This view has been challenged by a number of recent studies, in which highly repeatable temporal patterns of spikes can be observed both in vivo (Bair and Koch 1996; Abeles et al. 1993) and in vitro (Mainen and Sejnowski 1994). Furthermore, application ofinformation theory to the coding problem in the frog and house fly (Bialek et al. 1991; Bialek and Rieke 1992) suggested that additional information could be extracted from spike timing. In the absence of direct evidence for a timing code in the cerebral cortex, the role of spike timing in neural coding remains controversial.

We propose and analyze an algorithm that approximates solutions to the problem of optimal stopping in a discounted irreducible aperiodic Markovchain. The scheme involves the use of linear combinations offixed basis functions to approximate a Q-function. The weights of the linear combination are incrementally updated through an iterative process similar to Q-Iearning, involving simulation ofthe underlying Markov chain. Due to space limitations, we only provide an overview of a proof of convergence (with probability 1)and bounds on the approximation error. This is the first theoretical result that establishes the soundness of a Q-Iearninglike algorithmwhen combined with arbitrary linear function approximators tosolve a sequential decision problem.

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 ideasand this leads to a number of significant limitations. In this paper, we consider the problem of modelling the probability densityof 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 (forGenerative Topographic Mapping), which allows general nonlinear transformations from latent space to data space, and which is trained using the EM (expectation-maximization) algorithm. Ourapproach overcomes the limitations of the SOM, while introducing no significant disadvantages. We demonstrate the performance ofthe GTM algorithm on simulated data from flow diagnostics for a multiphase oil pipeline.