Results 1  10
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38
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.
Variable Selection and Model Comparison in Regression
, 1994
"... In the specification of linear regression models it is common to indicate a list of candidate variables from which a subset enters the model with nonzero coefficients. In some cases any combination of variables may enter, but in others certain necessary conditions must be satisfied: e.g., in time se ..."
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Cited by 62 (2 self)
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In the specification of linear regression models it is common to indicate a list of candidate variables from which a subset enters the model with nonzero coefficients. In some cases any combination of variables may enter, but in others certain necessary conditions must be satisfied: e.g., in time series applications it is common to allow a lagged variable only if all shorter lags for the same variable also enter. This paper interprets this specification as a mixed continuousdiscrete prior distribution for coefficient values. It then utilizes a Gibbs sampler to construct posterior moments. It is shown how this method can incorporate sign constraints and provide posterior probabilities for all possible subsets of regressors. The methods are illustrated using some standard data sets.
Bayesian Variable Selection with Related Predictors
 Canadian Journal of Statistics
, 1996
"... In data sets with many predictors, algorithms for identifying a good subset of predictors are often used. Most such algorithms do not account for any relationships between predictors. For example, stepwise regression might select a model containing an interaction AB but neither main effect A or B ..."
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Cited by 57 (4 self)
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In data sets with many predictors, algorithms for identifying a good subset of predictors are often used. Most such algorithms do not account for any relationships between predictors. For example, stepwise regression might select a model containing an interaction AB but neither main effect A or B. This paper develops mathematical representations of this and other relations between predictors, which may then be incorporated in a model selection procedure. A Bayesian approach that goes beyond the standard independence prior for variable selection is adopted, and preference for certain models is interpreted as prior information. Priors relevant to arbitrary interactions and polynomials, dummy variables for categorical factors, competing predictors, and restrictions on the size of the models are developed. Since the relations developed are for priors, they may be incorporated in any Bayesian variable selection algorithm for any type of linear model. The application of the method...
2005), “The conquest of U.S. inflation: Learning and robustness to model uncertainty
 Review of Economic Dynamics
"... Previous studies have interpreted the rise and fall of U.S. inflation after World War II in terms of the Fed’s changing views about the natural rate hypothesis but have left an important question unanswered. Why was the Fed so slow to implement the lowinflation policy recommended by a natural rate ..."
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Cited by 55 (6 self)
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Previous studies have interpreted the rise and fall of U.S. inflation after World War II in terms of the Fed’s changing views about the natural rate hypothesis but have left an important question unanswered. Why was the Fed so slow to implement the lowinflation policy recommended by a natural rate model even after economists had developed statistical evidence strongly in its favor? Our answer features model uncertainty. Each period a central bank sets the systematic part of the inflation rate in light of updated probabilities that it assigns to three competing models of the Phillips curve. Cautious behavior induced by model uncertainty can explain why the central bank presided over the inflation of the 1970s even after the data had convinced it to place much the highest probability on the natural rate model. 1
Prediction via Orthogonalized Model Mixing
 JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
, 1994
"... In this paper we introduce an approach and algorithms for model mixing in large prediction problems with correlated predictors. We focus on the choice of predictors in linear models, and mix over possible subsets of candidate predictors. Our approach is based on expressing the space of models in ter ..."
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Cited by 50 (9 self)
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In this paper we introduce an approach and algorithms for model mixing in large prediction problems with correlated predictors. We focus on the choice of predictors in linear models, and mix over possible subsets of candidate predictors. Our approach is based on expressing the space of models in terms of an orthogonalization of the design matrix. Advantages are both statistical and computational. Statistically, orthogonalization often leads to a reduction in the number of competing models by eliminating correlations. Computationally, large model spaces cannot be enumerated; recent approaches are based on sampling models with high posterior probability via Markov chains. Based on orthogonalization of the space of candidate predictors, we can approximate the posterior probabilities of models by products of predictorspecific terms. This leads to an importance sampling function for sampling directly from the joint distribution over the model space, without resorting to Markov chains. Comp...
Bayesian Model Averaging in proportional hazard models: Assessing the risk of a stroke
 Applied Statistics
, 1997
"... Evaluating the risk of stroke is important in reducing the incidence of this devastating disease. Here, we apply Bayesian model averaging to variable selection in Cox proportional hazard models in the context of the Cardiovascular Health Study, a comprehensive investigation into the risk factors for ..."
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Cited by 28 (5 self)
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Evaluating the risk of stroke is important in reducing the incidence of this devastating disease. Here, we apply Bayesian model averaging to variable selection in Cox proportional hazard models in the context of the Cardiovascular Health Study, a comprehensive investigation into the risk factors for stroke. We introduce a technique based on the leaps and bounds algorithm which e ciently locates and ts the best models in the very large model space and thereby extends all subsets regression to Cox models. For each independent variable considered, the method provides the posterior probability that it belongs in the model. This is more directly interpretable than the corresponding Pvalues, and also more valid in that it takes account of model uncertainty. Pvalues from models preferred by stepwise methods tend to overstate the evidence for the predictive value of a variable. In our data Bayesian model averaging predictively outperforms standard model selection methods for assessing
Variable selection and Bayesian model averaging in casecontrol studies
, 1998
"... Covariate and confounder selection in casecontrol studies is most commonly carried out using either a twostep method or a stepwise variable selection method in logistic regression. Inference is then carried out conditionally on the selected model, but this ignores the model uncertainty implicit in ..."
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Cited by 19 (7 self)
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Covariate and confounder selection in casecontrol studies is most commonly carried out using either a twostep method or a stepwise variable selection method in logistic regression. Inference is then carried out conditionally on the selected model, but this ignores the model uncertainty implicit in the variable selection process, and so underestimates uncertainty about relative risks. We report on a simulation study designed to be similar to actual casecontrol studies. This shows that pvalues computed after variable selection can greatly overstate the strength of conclusions. For example, for our simulated casecontrol studies with 1,000 subjects, of variables declared to be "significant" with pvalues between.01 and.05, only 49 % actually were risk factors when stepwise variable selection was used. We propose Bayesian model averaging as a formal way of taking account of model uncertainty in casecontrol studies. This yields an easily interpreted summary, the posterior probability that a variable is a risk factor, and our simulation study indicates this to be reasonably well calibrated in the situations simulated. The methods are applied and compared
The Posterior Probability of Bayes Nets with Strong Dependences
 Soft Computing
, 1999
"... Stochastic independence is an idealized relationship located at one end of a continuum of values measuring degrees of dependence. Modeling real world systems, we are often not interested in the distinction between exact independence and any degree of dependence, but between weak ignorable and strong ..."
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Cited by 14 (1 self)
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Stochastic independence is an idealized relationship located at one end of a continuum of values measuring degrees of dependence. Modeling real world systems, we are often not interested in the distinction between exact independence and any degree of dependence, but between weak ignorable and strong substantial dependence. Good models map significant deviance from independence and neglect approximate independence or dependence weaker than a noise threshold. This intuition is applied to learning the structure of Bayes nets from data. We determine the conditional posterior probabilities of structures given that the degree of dependence at each of their nodes exceeds a critical noise level. Deviance from independence is measured by mutual information. Arc probabilities are determined by the amount of mutual information the neighbors contribute to a node, is greater than a critical minimum deviance from independence. A Ø 2 approximation for the probability density function of mutual info...