Abstract not found

The output of a classifier should be a calibrated posterior probability to enable post-processing. Standard SVMs do not provide such probabilities. One method to create probabilities is to directly train a kernel classifier with a logit link function and a regularized maximum likelihood score. However, training with a maximum likelihood score will produce non-sparse kernel machines. Instead, we train an SVM, then train the parameters of an additional sigmoid function to map the SVM outputs into probabilities. This chapter compares classification error rate and likelihood scores for an SVM plus sigmoid versus a kernel method trained with a regularized likelihood error function. These methods are tested on three data-mining-style data sets. The SVM+sigmoid yields probabilities of comparable quality to the regularized maximum likelihood kernel method, while still retaining the sparseness of the SVM.

This paper introduces a general Bayesian framework for obtaining sparse solutions to regression and classification tasks utilising models linear in the parameters. Although this framework is fully general, we illustrate our approach with a particular specialisation that we denote the `relevance vector machine’ (RVM), a model of identical functional form to the popular and state-of-the-art `support vector machine ’ (SVM). We demonstrate that by exploiting a probabilistic Bayesian learning framework, we can derive accurate prediction models which typically utilise dramatically fewer basis functions than a comparable SVM while offering a number of additional advantages. These include the benefits of probabilistic predictions, automatic estimation of `nuisance’ parameters, and the facility to utilise arbitrary basis functions (e.g. non-`Mercer’ kernels). We detail the Bayesian framework and associated learning algorithm for the RVM, and give some illustrative examples of its application along with some comparative benchmarks. We offer some explanation for the exceptional degree of sparsity obtained, and discuss and demonstrate some of the advantageous features, and potential extensions, of Bayesian relevance learning.

Motivation: DNA microarrays are now capable of providing genome-wide patterns of gene expression across many different conditions. The first level of analysis of these patterns requires determining whether observed differences in expression are significant or not. Current methods are unsatisfactory due to the lack of a systematic framework that can accommodate noise, variability, and low replication often typical of microarray data. Results: We develop a Bayesian probabilistic framework for microarray data analysis. At the simplest level, we model log-expression values by independent normal distributions, parameterized by corresponding means and variances with hierarchical prior distributions. We derive point estimates for both parameters and hyperparameters, and regularized expressions for the variance of each gene by combining the empirical variance with a local background variance associated with neighboring genes. An additional hyperparameter, inversely related to the number of empirical observations, determines the strength of the background variance. Simulations show that these point estimates, combined with a t-test, provide a systematic inference approach that compares favorably with simple t-test or fold methods, and partly compensate for the lack of replication. Availability: The approach is implemented in a software called Cyber-T accessible through a Web interface at www.genomics.uci.edu/software.html. The code is available as Open Source and is written in the freely available statistical language R. and Department of Biological Chemistry, College of Medicine, University of California, Irvine. To whom all correspondence should be addressed. Contact: pfbaldi@ics.uci.edu, tdlong@uci.edu. 1

[Read before The Royal Statistical Society at a meeting organized by the Research

The data of interest are assumed to be represented as N-dimensional real vectors, and these vectors are compressible in some linear basis B, implying that the signal can be reconstructed accurately using only a small number M ≪ N of basis-function coefficients associated with B. Compressive sensing is a framework whereby one does not measure one of the aforementioned N-dimensional signals directly, but rather a set of related measurements, with the new measurements a linear combination of the original underlying N-dimensional signal. The number of required compressive-sensing measurements is typically much smaller than N, offering the potential to simplify the sensing system. Let f denote the unknown underlying N-dimensional signal, and g a vector of compressive-sensing measurements, then one may approximate f accurately by utilizing knowledge of the (under-determined) linear relationship between f and g, in addition to knowledge of the fact that f is compressible in B. In this paper we employ a Bayesian formalism for estimating the underlying signal f based on compressive-sensing measurements g. The proposed framework has the following properties: (i) in addition to estimating the underlying signal f, “error bars ” are also estimated, these giving a measure of confidence in the inverted signal; (ii) using knowledge of the error bars, a principled means is provided for determining when a sufficient

The support vector machine (SVM) is a state-of-the-art technique for regression and classification, combining excellent generalisation properties with a sparse kernel representation. However, it does suffer from a number of disadvantages, notably the absence of probabilistic outputs, the requirement to estimate a trade-off parameter and the need to utilise `Mercer' kernel functions. In this paper we introduce the Relevance Vector Machine (RVM), a Bayesian treatment of a generalised linear model of identical functional form to the SVM. The RVM suffers from none of the above disadvantages, and examples demonstrate that for comparable generalisation performance, the RVM requires dramatically fewer kernel functions.

We introduce the independent factor analysis (IFA) method for recovering independent hidden sources from their observed mixtures. IFA generalizes and unifies ordinary factor analysis (FA), principal component analysis (PCA), and independent component analysis (ICA), and can handle not only square noiseless mixing, but also the general case where the number of mixtures differs from the number of sources and the data are noisy. IFA is a two-step procedure. In the first step, the source densities, mixing matrix and noise covariance are estimated from the observed data by maximum likelihood. For this purpose we present an expectation-maximization (EM) algorithm, which performs unsupervised learning of an associated probabilistic model of the mixing situation. Each source in our model is described by a mixture of Gaussians, thus all the probabilistic calculations can be performed analytically. In the second step, the sources are reconstructed from the observed data by an optimal non-linear ...

The human visual system appears to be capable of temporally integrating information in a video sequence in such a way that the perceived spatial resolution of a sequence appears much higher than the spatial resolution of an individual frame. While the mechanisms in the human visual system which do this are unknown, the effect is not too surprising given that temporally adjacent frames in a video sequence contain slightly different, but unique, information. This paper addresses how to utilize both the spatial and temporal information present in a short image sequence to create a single high-resolution video frame. A novel observation model based on motion compensated subsampling is proposed for a video sequence. Since the reconstruction problem is ill-posed, Bayesian restoration with a discontinuity-preserving prior image model is used to extract a high-resolution video still given a short low-resolution sequence. Estimates computed from a low-resolution image sequence containing a subp...

this paper I discuss a Bayesian approach to solving this problem that has long been available in principle but is only now becoming routinely feasible, by virtue of recent computational advances, and examine its implementation in examples that involve forecasting the price of oil and estimating the chance of catastrophic failure of the U.S. Space Shuttle.