In a 1935 paper, and in his book Theory of Probability, Jeffreys developed a methodology for quantifying the evidence in favor of a scientific theory. The centerpiece was a number, now called the Bayes factor, which is the posterior odds of the null hypothesis when the prior probability on the null is one-half. Although there has been much discussion of Bayesian hypothesis testing in the context of criticism of P -values, less attention has been given to the Bayes factor as a practical tool of applied statistics. In this paper we review and discuss the uses of Bayes factors in the context of five scientific applications in genetics, sports, ecology, sociology and psychology.

With ideal spatial adaptation, an oracle furnishes information about how best to adapt a spatially variable estimator, whether piecewise constant, piecewise polynomial, variable knot spline, or variable bandwidth kernel, to the unknown function. Estimation with the aid of an oracle o ers dramatic advantages over traditional linear estimation by nonadaptive kernels � however, it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone. We describe a new principle for spatially-adaptive estimation: selective wavelet reconstruction. Weshowthatvariableknot spline ts and piecewise-polynomial ts, when equipped with an oracle to select the knots, are not dramatically more powerful than selective wavelet reconstruction with an oracle. We develop a practical spatially adaptive method, RiskShrink, which works by shrinkage of empirical wavelet coe cients. RiskShrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so. A new inequality inmultivariate normal decision theory which wecallthe oracle inequality shows that attained performance di ers from ideal performance by at most a factor 2logn, where n is the sample size. Moreover no estimator can give a better guarantee than this. Within the class of spatially adaptive procedures, RiskShrink is essentially optimal. Relying only on the data, it comes within a factor log 2 n of the performance of piecewise polynomial and variable-knot spline methods equipped with an oracle. In contrast, it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone.

We address the problem of finding a subset of features that allows a supervised induction algorithm to induce small high-accuracy concepts. We examine notions of relevance and irrelevance, and show that the definitions used in the machine learning literature do not adequately partition the features into useful categories of relevance. We present definitions for irrelevance and for two degrees of relevance. These definitions improve our understanding of the behavior of previous subset selection algorithms, and help define the subset of features that should be sought. The features selected should depend not only on the features and the target concept, but also on the induction algorithm. We describe a method for feature subset selection using cross-validation that is applicable to any induction algorithm, and discuss experiments conducted with ID3 and C4.5 on artificial and real datasets.

This paper surveys locally weighted learning, a form of lazy learning and memorybased learning, and focuses on locally weighted linear regression. The survey discusses distance functions, smoothing parameters, weighting functions, local model structures, regularization of the estimates and bias, assessing predictions, handling noisy data and outliers, improving the quality of predictions by tuning t parameters, interference between old and new data, implementing locally weighted learning e ciently, and applications of locally weighted learning. A companion paper surveys how locally weighted learning can be used in robot learning and control.

Abstract. Compressive sampling offers a new paradigm for acquiring signals that are compressible with respect to an orthonormal basis. The major algorithmic challenge in compressive sampling is to approximate a compressible signal from noisy samples. This paper describes a new iterative recovery algorithm called CoSaMP that delivers the same guarantees as the best optimization-based approaches. Moreover, this algorithm offers rigorous bounds on computational cost and storage. It is likely to be extremely efficient for practical problems because it requires only matrix–vector multiplies with the sampling matrix. For compressible signals, the running time is just O(N log 2 N), where N is the length of the signal. 1.

Abstract—Many problems in signal processing and statistical inference involve finding sparse solutions to under-determined, or ill-conditioned, linear systems of equations. A standard approach consists in minimizing an objective function which includes a quadratic (squared ℓ2) error term combined with a sparseness-inducing (ℓ1) regularization term.Basis pursuit, the least absolute shrinkage and selection operator (LASSO), waveletbased deconvolution, and compressed sensing are a few wellknown examples of this approach. This paper proposes gradient projection (GP) algorithms for the bound-constrained quadratic programming (BCQP) formulation of these problems. We test variants of this approach that select the line search parameters in different ways, including techniques based on the Barzilai-Borwein method. Computational experiments show that these GP approaches perform well in a wide range of applications, often being significantly faster (in terms of computation time) than competing methods. Although the performance of GP methods tends to degrade as the regularization term is de-emphasized, we show how they can be embedded in a continuation scheme to recover their efficient practical performance. A. Background I.

Abstract. This technical report demonstrates theoretically and empirically that a greedy algorithm called Orthogonal Matching Pursuit (OMP) can reliably recover a signal with m nonzero entries in dimension d given O(m ln d) random linear measurements of that signal. This is a massive improvement over previous results for OMP, which require O(m 2) measurements. The new results for OMP are comparable with recent results for another algorithm called Basis Pursuit (BP). The OMP algorithm is faster and easier to implement, which makes it an attractive alternative to BP for signal recovery problems. 1.

We consider the problem of accounting for model uncertainty in linear regression models. Conditioning on a single selected model ignores model uncertainty, and thus leads to the underestimation of uncertainty when making inferences about quantities of interest. A Bayesian solution to this problem involves averaging over all possible models (i.e., combinations of predictors) when making inferences about quantities of

Decision trees have proved to be valuable tools for the description, classification and generalization of data. Work on constructing decision trees from data exists in multiple disciplines such as statistics, pattern recognition, decision theory, signal processing, machine learning and artificial neural networks. Researchers in these disciplines, sometimes working on quite different problems, identified similar issues and heuristics for decision tree construction. This paper surveys existing work on decision tree construction, attempting to identify the important issues involved, directions the work has taken and the current state of the art. Keywords: classification, tree-structured classifiers, data compaction 1. Introduction Advances in data collection methods, storage and processing technology are providing a unique challenge and opportunity for automated data exploration techniques. Enormous amounts of data are being collected daily from major scientific projects e.g., Human Genome...

Model selection is important in many areas of supervised learning. Given a dataset and a set of models for predicting with that dataset, we must choose the model which is expected to best predict future data. In some situations, such as online learning for control of robots or factories, data is cheap and human expertise costly. Cross validation can then be a highly effective method for automatic model selection. Large scale cross validation search can, however, be computationally expensive. This paper introduces new algorithms to reduce the computational burden of such searches. We show how experimental design methods can achieve this, using a technique similar to a Bayesian version of Kaelbling’s Interval Estimation. Several improvements are then given, including (1) the use of blocking to quickly spot near-identical models, and (2) schemata search: a new method for quickly finding families of relevant features. Experiments are presented for robot data and noisy synthetic datasets. The new algorithms speed up computation without sacrificing reliability, and in some cases are more reliable than conventional techniques. 1