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14
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 137 (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.
The practical implementation of Bayesian model selection
 Institute of Mathematical Statistics
, 2001
"... In principle, the Bayesian approach to model selection is straightforward. Prior probability distributions are used to describe the uncertainty surrounding all unknowns. After observing the data, the posterior distribution provides a coherent post data summary of the remaining uncertainty which is r ..."
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Cited by 94 (3 self)
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In principle, the Bayesian approach to model selection is straightforward. Prior probability distributions are used to describe the uncertainty surrounding all unknowns. After observing the data, the posterior distribution provides a coherent post data summary of the remaining uncertainty which is relevant for model selection. However, the practical implementation of this approach often requires carefully tailored priors and novel posterior calculation methods. In this article, we illustrate some of the fundamental practical issues that arise for two different model selection problems: the variable selection problem for the linear model and the CART model selection problem.
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 54 (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 Variable Selection for Proportional Hazards Models
, 1996
"... The authors consider the problem of Bayesian variable selection for proportional hazards regression models with right censored data. They propose a semiparametric approach in which a nonparametric prior is specified for the baseline hazard rate and a fully parametric prior is specified for the regr ..."
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Cited by 17 (1 self)
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The authors consider the problem of Bayesian variable selection for proportional hazards regression models with right censored data. They propose a semiparametric approach in which a nonparametric prior is specified for the baseline hazard rate and a fully parametric prior is specified for the regression coe#cients. For the baseline hazard, they use a discrete gamma process prior, and for the regression coe#cients and the model space, they propose a semiautomatic parametric informative prior specification that focuses on the observables rather than the parameters. To implement the methodology, they propose a Markov chain Monte Carlo method to compute the posterior model probabilities. Examples using simulated and real data are given to demonstrate the methodology. R ESUM E Les auteurs abordent d'un point de vue bayesien le problemedelaselection de variables dans les modeles de regression des risques proportionnels en presence de censure a droite. Ils proposent une approche semip...
Bayesian Variable Selection Using the Gibbs Sampler
, 2000
"... Specification of the linear predictor for a generalised linear model requires determining which variables to include. We consider Bayesian strategies for performing this variable selection. In particular we focus on approaches based on the Gibbs sampler. Such approaches may be implemented using the ..."
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Cited by 13 (2 self)
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Specification of the linear predictor for a generalised linear model requires determining which variables to include. We consider Bayesian strategies for performing this variable selection. In particular we focus on approaches based on the Gibbs sampler. Such approaches may be implemented using the publically available software BUGS. We illustrate the methods using a simple example. BUGS code is provided in an appendix. 1 Introduction In a Bayesian analysis of a generalised linear model, model uncertainty may be incorporated coherently by specifying prior probabilities for plausible models and calculating posterior probabilities using f(mjy) = f(m)f(yjm) P m2M f(m)f(y jm) ; m 2 M (1.1) where m denotes the model, M is the set of all models under consideration, f (m) is the prior probability of model m and f (yjm; fi m ) the likelihood of the data y under model m. The observed data y contribute to the posterior model probabilities through f(yjm), the marginal likelihood calculated...
Criterionbased methods for Bayesian model assessment
 Statist. Sin
, 2001
"... Abstract: We propose a general Bayesian criterion for model assessment. The criterion is constructed from the posterior predictive distribution of the data, and can be written as a sum of two components, one involving the means of the posterior predictive distribution and the other involving the va ..."
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Cited by 5 (2 self)
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Abstract: We propose a general Bayesian criterion for model assessment. The criterion is constructed from the posterior predictive distribution of the data, and can be written as a sum of two components, one involving the means of the posterior predictive distribution and the other involving the variances. It can be viewed as a Bayesian goodnessoffit statistic which measures the performance of a model by a combination of how close its predictions are to the observed data and the variability of the predictions. We call this proposed predictive criterion the L measure, it is motivated by earlier work of Ibrahim and Laud (1994) and related to a criterion of Gelfand and Ghosh (1998). We examine the L measure in detail for the class of generalized linear models and survival models with right censored or interval censored data. We also propose a calibration of the L measure, defined as the prior predictive distribution of the difference between the L measures of the candidate model and the criterion minimizing model, and call it the calibration distribution. The calibration distribution will allow us to formally compare two competing models based on their L measure values. We discuss theoretical properties of the calibration distribution in detail, and provide Monte Carlo methods for computing it. For the linear model, we derive an analytic closed form expression for the L measure and the calibration distribution, and also derive a closed form expression for the mean of the calibration distribution. These novel developments will enable us to fully characterize the properties of the L measure for each model under consideration and will facilitate a direct formal comparison between several models, including nonnested models. Informative priors based on historical data and computational techniques are discussed. Several simulated and real datasets are used to demonstrate the proposed methodology. Key words and phrases: Calibration, model selection, predictive criterion, predictive distribution, variable selection.
On Bayesian Calculations for Mixture Likelihoods and Priors
, 1997
"... this paper and in a few others ..."
A Hierarchical Bayes Approach to Variable Selection for Generalized Linear Models
, 2004
"... For the problem of variable selection in generalized linear models, we develop various adaptive Bayesian criteria. Using a hierarchical mixture setup for model uncertainty, combined with an integrated Laplace approximation, we derive Empirical Bayes and Fully Bayes criteria that can be computed easi ..."
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Cited by 2 (0 self)
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For the problem of variable selection in generalized linear models, we develop various adaptive Bayesian criteria. Using a hierarchical mixture setup for model uncertainty, combined with an integrated Laplace approximation, we derive Empirical Bayes and Fully Bayes criteria that can be computed easily and quickly. The performance of these criteria is assessed via simulation and compared to other criteria such as AIC and BIC on normal, logistic and Poisson regression model classes. A Fully Bayes criterion based on a restricted region hyperprior seems to be the most promising.
ADAPTIVE BAYESIAN CRITERIA IN VARIABLE SELECTION FOR GENERALIZED LINEAR MODELS
"... Abstract: For the problem of variable selection in generalized linear models, we develop various adaptive Bayesian criteria. Using a hierarchical mixture setup for model uncertainty, combined with an integrated Laplace approximation, we derive Empirical Bayes and Fully Bayes criteria that can be com ..."
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Cited by 1 (0 self)
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Abstract: For the problem of variable selection in generalized linear models, we develop various adaptive Bayesian criteria. Using a hierarchical mixture setup for model uncertainty, combined with an integrated Laplace approximation, we derive Empirical Bayes and Fully Bayes criteria that can be computed easily and quickly. The performance of these criteria is assessed via simulation and compared to other criteria such as AIC and BIC on normal, logistic and Poisson regression model classes. A Fully Bayes criterion based on a restricted region hyperprior seems to be the most promising. Finally, our criteria are illustrated and compared with competitors on a data example.