One of the open questions that remains is how to set the'tunable' parameters of an SVM algorithm: While methods for choosing the width of the kernel function and the noise parameter C (which controls how closely the training data are fitted) have been proposed [4, 5] (see also, very recently, [6]), the effect of the overall shape of the kernel function remains imperfectly understood [1]. Error bars (class probabilities) for SVM predictions - important for safety-critical applications, for example - are also difficult to obtain. In this paper I suggest that a probabilistic interpretation of SVMs could be used to tackle these problems. It shows that the SVM kernel defines a prior over functions on the input space, avoiding the need to think in terms of high-dimensional feature spaces. It also allows one to define quantities such as the evidence (likelihood) for a set of hyperparameters (C, kernel amplitude Ko etc). I give a simple approximation to the evidence which can then be maximized to set such hyperparameters. The evidence is sensitive to the values of C and Ko individually, in contrast to properties (such as cross-validation error) of the deterministic solution, which only depends on the product CKo. It can thfrefore be used to assign an unambiguous value to C, from which error bars can be derived.

In this paper we will treat input selection for a radial basis function (RBF) like classifier within a Bayesian framework. We approximate the a-posteriori distribution over both model coefficients and input subsets by samples drawn with Gibbs updates and reversible jump moves. Using some public datasets, we compare the classification accuracy of the method with a conventional ARD scheme. These datasets are also used to infer the a-posteriori probabilities of different input subsets. 1 Introduction Methods that aim to determine relevance of inputs have always interested researchers in various communities. Classical feature subset selection techniques, as reviewed in [1], use search algorithms and evaluation criteria to determine one optimal subset.

A new method for multivariate density estimation is developed based on the Support Vector Method (SVM) solution of inverse ill-posed problems. The solution has the form of a mixture of densities. This method with Gaussian kernels compared favorably to both Parzen's method and the Gaussian Mixture Model method. For synthetic data we achieve more accurate estimates for densities of 2, 6, 12, and 40 dimensions. 1 Introduction The problem of multivariate density estimation is important for many applications, in particular, for speech recognition [1] [7]. When the unknown density belongs to a parametric set satisfying certain conditions one can estimate it using the maximum likelihood (ML) method. Often these conditions are too restrictive. Therefore, nonparametric methods were proposed. The most popular of these, Parzen's method [5], uses the following estimate given data

Fishers linear discriminant analysis (LDA) is a classical multivariate technique both for dimension reduction and classification. The data vectors are transformed into a low dimensional subspace such that the class centroids are spread out as much as possible. In this subspace LDA works as a simple prototype classifier with linear decision boundaries. However, in many applications the linear boundaries do not adequately separate the classes. We present a nonlinear generalization of discriminant analysis that uses the kernel trick of representing dot products by kernel functions.

The project pursued in this paper is to develop from first information-geometric principles a general method for learning the similarity between text documents. Each individual document is modeled as a memoryless information source. Based on a latent class decomposition of the term-document matrix, a lowdimensional (curved) multinomial subfamily is learned. From this model a canonical similarity function - known as the Fisher kernel - is derived. Our approach can be applied for unsupervised and supervised learning problems alike.

Dual estimation refers to the problem of simultaneously estimating the state of a dynamic system and the model which gives rise to the dynamics. Algorithms include expectation-maximization (EM), dual Kalman filtering, and joint Kalman methods. These methods have recently been explored in the context of nonlinear modeling, where a neural network is used as the functional form of the unknown model. Typically, an extended Kalman filter (EKF) or smoother is used for the part of the algorithm that estimates the clean state given the current estimated model. An EKF may also be used to estimate the weights of the network. This paper points out the flaws in using the EKF, and proposes an improvement based on a new approach called the unscented transformation (UT) [3]. A substantial performance gain is achieved with the same order of computational complexity as that of the standard EKF. The approach is illustrated on several dual estimation methods.

In particular, Mackay showed that by approximating the distributions of the weights with Gaussians and adopting smoothing priors, it is possible to obtain estimates of the weights and output variances and to automatically set the regularisation coefficients. Neal (1996) cast the net much further by introducing advanced Bayesian simulation methods, specifically the hybrid Monte Carlo method, into the analysis of neural networks [3]. Bayesian sequential Monte Carlo methods have also been shown to provide good training results, especially in time-varying scenarios [4]. More recently, Rios Insua and Muller (1998) and Holmes and Mallick (1998) have addressed the issue of selecting the number of hidden neurons with growing and pruning algorithms from a Bayesian perspective [5,6]. In particular, they apply the reversible jump Markov Chain Monte Carlo (MCMC) algorithm of Green [7] to feed-forward sigmoidal networks and radial basis function (RBF) networks to obtain joint estimates of the number of neurons and weights. We also apply the reversible jump MCMC simulation algorithm to RBF networks so as to compute the joint posterior distribution of the radial basis parameters and the number of basis functions. However, we advance this area of research in two important directions. Firstly, we propose a full hierarchical prior for RBF networks.

This paper presents a novel practical framework for Bayesian model averaging and model selection in probabilistic graphical models. Our approach approximates full posterior distributions over model parameters and structures, as well as latent variables, in an analytical manner. These posteriors fall out of a free-form optimization procedure, which naturally incorporates conjugate priors. Unlike in large sample approximations, the posteriors are generally non Gaussian and no Hessian needs to be computed. Predictive quantities are obtained analytically. The resulting algorithm generalizes the standard Expectation Maximization algorithm, and its convergence is guaranteed. We demonstrate that this approach can be applied to a large class of models in several domains, including mixture models and source separation. 1 Introduction

We give necessary and sufficient conditions for uniqueness of the support vector solution for the problems of pattern recognition and regression estimation, for a general class of cost functions. We show that if the solution is not unique, all support vectors are necessarily at bound, and we give some simple examples of non-unique solutions. We note that uniqueness of the primal (dual) solution does not necessarily imply uniqueness of the dual (primal) solution. We show how to compute the threshold b when the solution is unique, but when all support vectors are at bound, in which case the usual method for determining b does not work. 1 Introduction Support vector machines (SVMs) have attracted wide interest as a means to implement structural risk minimization for the problems of classification and regression estimation. The fact that training an SVM amounts to solving a convex quadratic programming problem means that the solution found is global, and that if it is not unique, then the set of global solutions is itself convex; furthermore, if the objective function is strictly convex, the solution is guaranteed to be unique [1]1.

In this paper we discuss the semi parametric statistical model for blind deconvolution. First we introduce a Lie Group to the manifold of noncausal FIR filters. Then blind deconvolution problem is formulated in the framework of a semiparametric model, and a family of estimating functions is derived for blind deconvolution. A natural gradient learning algorithm is developed for training noncausal filters. Stability of the natural gradient algorithm is also analyzed in this framework.