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285
Regression Shrinkage and Selection Via the Lasso
 Journal of the Royal Statistical Society, Series B
, 1994
"... We propose a new method for estimation in linear models. The "lasso" minimizes the residual sum of squares subject to the sum of the absolute value of the coefficients being less than a constant. Because of the nature of this constraint it tends to produce some coefficients that are exactly zero and ..."
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Cited by 1828 (36 self)
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We propose a new method for estimation in linear models. The "lasso" minimizes the residual sum of squares subject to the sum of the absolute value of the coefficients being less than a constant. Because of the nature of this constraint it tends to produce some coefficients that are exactly zero and hence gives interpretable models. Our simulation studies suggest that the lasso enjoys some of the favourable properties of both subset selection and ridge regression. It produces interpretable models like subset selection and exhibits the stability of ridge regression. There is also an interesting relationship with recent work in adaptive function estimation by Donoho and Johnstone. The lasso idea is quite general and can be applied in a variety of statistical models: extensions to generalized regression models and treebased models are briefly described. Keywords: regression, subset selection, shrinkage, quadratic programming. 1 Introduction Consider the usual regression situation: we h...
Bayes Factors
, 1995
"... 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 ..."
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Cited by 981 (70 self)
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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 onehalf. 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.
Model selection and estimation in regression with grouped variables
 Journal of the Royal Statistical Society, Series B
, 2006
"... We consider the problem of selecting grouped variables (factors) for accurate prediction in regression. Such a problem arises naturally in many practical situations with the multifactor ANOVA problem as the most important and well known example. Instead of selecting factors by stepwise backward el ..."
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Cited by 509 (7 self)
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We consider the problem of selecting grouped variables (factors) for accurate prediction in regression. Such a problem arises naturally in many practical situations with the multifactor ANOVA problem as the most important and well known example. Instead of selecting factors by stepwise backward elimination, we focus on estimation accuracy and consider extensions of the LASSO, the LARS, and the nonnegative garrote for factor selection. The LASSO, the LARS, and the nonnegative garrote are recently proposed regression methods that can be used to select individual variables. We study and propose efficient algorithms for the extensions of these methods for factor selection, and show that these extensions give superior performance to the traditional stepwise backward elimination method in factor selection problems. We study the similarities and the differences among these methods. Simulations and real examples are used to illustrate the methods.
Has the U.S. Economy Become More Stable? A Bayesian Approach Based on a MarkovSwitching Model of Business Cycle
, 1999
"... We hope to be able to provide answers to the following questions: 1) Has there been a structural break in postwar U.S. real GDP growth toward more stabilization? 2) If so, when would it have been? 3) What's the nature of the structural break? For this purpose, we employ a Bayesian approach to dealin ..."
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Cited by 255 (13 self)
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We hope to be able to provide answers to the following questions: 1) Has there been a structural break in postwar U.S. real GDP growth toward more stabilization? 2) If so, when would it have been? 3) What's the nature of the structural break? For this purpose, we employ a Bayesian approach to dealing with structural break at an unknown changepoint in a Markovswitching model of business cycle. Empirical results suggest that there has been a structural break in U.S. real GDP growth toward more stabilization, with the posterior mode of the break date around 1984:1. Furthermore, we #nd a narrowing gap between growth rates during recessions and booms is at least as important as a decline in the volatility of shocks. Key Words: Bayes Factor, Gibbs sampling, Marginal Likelihood, MarkovSwitching, Stabilization, Structural Break. JEL Classi#cations: C11, C12, C22, E32. 1. Introduction In the literature, the issue of postwar stabilization of the U.S. economy relative to the prewar period has...
Bayesian Model Averaging for Linear Regression Models
 Journal of the American Statistical Association
, 1997
"... 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 in ..."
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Cited by 184 (13 self)
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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
Model Selection and the Principle of Minimum Description Length
 Journal of the American Statistical Association
, 1998
"... This paper reviews the principle of Minimum Description Length (MDL) for problems of model selection. By viewing statistical modeling as a means of generating descriptions of observed data, the MDL framework discriminates between competing models based on the complexity of each description. This ..."
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Cited by 145 (5 self)
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This paper reviews the principle of Minimum Description Length (MDL) for problems of model selection. By viewing statistical modeling as a means of generating descriptions of observed data, the MDL framework discriminates between competing models based on the complexity of each description. This approach began with Kolmogorov's theory of algorithmic complexity, matured in the literature on information theory, and has recently received renewed interest within the statistics community. In the pages that follow, we review both the practical as well as the theoretical aspects of MDL as a tool for model selection, emphasizing the rich connections between information theory and statistics. At the boundary between these two disciplines, we find many interesting interpretations of popular frequentist and Bayesian procedures. As we will see, MDL provides an objective umbrella under which rather disparate approaches to statistical modeling can coexist and be compared. We illustrate th...
Nonparametric regression using Bayesian variable selection
 Journal of Econometrics
, 1996
"... This paper estimates an additive model semiparametrically, while automatically selecting the significant independent variables and the app~opriatc power transformation of the dependent variable. The nonlinear variables arc modeled as regression splincs, with significant knots selected fiom a large ..."
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Cited by 136 (10 self)
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This paper estimates an additive model semiparametrically, while automatically selecting the significant independent variables and the app~opriatc power transformation of the dependent variable. The nonlinear variables arc modeled as regression splincs, with significant knots selected fiom a large number of candidate knots. The estimation is made robust by modeling the errors as a mixture of normals. A Bayesian approach is used to select the significant knots, the power transformation, and to identify oatliers using the Gibbs sampler to curry out the computation. Empirical evidence is given that the sampler works well on both simulated and real examples and that in the univariate case it compares faw)rably with a kernelweighted local linear smoother, The variable selection algorithm in the paper is substantially fasler than previous Bayesian variable sclcclion algorithms. K('I ' word~': Additive nlodel, Pov¢¢r Iransformalio:l: Robust cslinlalion
Approaches for Bayesian variable selection
 Statistica Sinica
, 1997
"... Abstract: This paper describes and compares various hierarchical mixture prior formulations of variable selection uncertainty in normal linear regression models. These include the nonconjugate SSVS formulation of George and McCulloch (1993), as well as conjugate formulations which allow for analytic ..."
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Cited by 124 (5 self)
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Abstract: This paper describes and compares various hierarchical mixture prior formulations of variable selection uncertainty in normal linear regression models. These include the nonconjugate SSVS formulation of George and McCulloch (1993), as well as conjugate formulations which allow for analytical simplification. Hyperparameter settings which base selection on practical significance, and the implications of using mixtures with point priors are discussed. Computational methods for posterior evaluation and exploration are considered. Rapid updating methods are seen to provide feasible methods for exhaustive evaluation using Gray Code sequencing in moderately sized problems, and fast Markov Chain Monte Carlo exploration in large problems. Estimation of normalization constants is seen to provide improved posterior estimates of individual model probabilities and the total visited probability. Various procedures are illustrated on simulated sample problems and on a real problem concerning the construction of financial index tracking portfolios.
Multiple Shrinkage and Subset Selection in Wavelets
, 1997
"... This paper discusses Bayesian methods for multiple shrinkage estimation in wavelets. Wavelets are used in applications for data denoising, via shrinkage of the coefficients towards zero, and for data compression, by shrinkage and setting small coefficients to zero. We approach wavelet shrinkage by u ..."
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Cited by 118 (16 self)
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This paper discusses Bayesian methods for multiple shrinkage estimation in wavelets. Wavelets are used in applications for data denoising, via shrinkage of the coefficients towards zero, and for data compression, by shrinkage and setting small coefficients to zero. We approach wavelet shrinkage by using Bayesian hierarchical models, assigning a positive prior probability to the wavelet coefficients being zero. The resulting estimator for the wavelet coefficients is a multiple shrinkage estimator that exhibits a wide variety of nonlinear shrinkage patterns. We discuss fast computational implementations, with a focus on easytocompute analytic approximations as well as importance sampling and Markov chain Monte Carlo methods. Multiple shrinkage estimators prove to have excellent mean squared error performance in reconstructing standard test functions. We demonstrate this in simulated test examples, comparing various implementations of multiple shrinkage to commonly used shrinkage rules. Finally, we illustrate our approach with an application to the socalled "glint" data.
Calibration and Empirical Bayes Variable Selection
 Biometrika
, 1997
"... this paper, is that with F =2logp. This choice was proposed by Foster &G eorge (1994) where it was called the Risk Inflation Criterion (RIC) because it asymptotically minimises the maximum predictive risk inflation due to selection when X is orthogonal. This choice and its minimax property were also ..."
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Cited by 114 (19 self)
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this paper, is that with F =2logp. This choice was proposed by Foster &G eorge (1994) where it was called the Risk Inflation Criterion (RIC) because it asymptotically minimises the maximum predictive risk inflation due to selection when X is orthogonal. This choice and its minimax property were also discovered independently by Donoho & Johnstone (1994) in the wavelet regression context, where they refer to it as the universal hard thresholding rule