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12
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
Benchmark Priors for Bayesian Model Averaging
 FORTHCOMING IN THE JOURNAL OF ECONOMETRICS
, 2001
"... In contrast to a posterior analysis given a particular sampling model, posterior model probabilities in the context of model uncertainty are typically rather sensitive to the specification of the prior. In particular, “diffuse” priors on modelspecific parameters can lead to quite unexpected consequ ..."
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Cited by 94 (5 self)
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In contrast to a posterior analysis given a particular sampling model, posterior model probabilities in the context of model uncertainty are typically rather sensitive to the specification of the prior. In particular, “diffuse” priors on modelspecific parameters can lead to quite unexpected consequences. Here we focus on the practically relevant situation where we need to entertain a (large) number of sampling models and we have (or wish to use) little or no subjective prior information. We aim at providing an “automatic” or “benchmark” prior structure that can be used in such cases. We focus on the Normal linear regression model with uncertainty in the choice of regressors. We propose a partly noninformative prior structure related to a Natural Conjugate gprior specification, where the amount of subjective information requested from the user is limited to the choice of a single scalar hyperparameter g0j. The consequences of different choices for g0j are examined. We investigate theoretical properties, such as consistency of the implied Bayesian procedure. Links with classical information criteria are provided. More importantly, we examine the finite sample implications of several choices of g0j in a simulation study. The use of the MC3 algorithm of Madigan and York (1995), combined with efficient coding in Fortran, makes it feasible to conduct large simulations. In addition to posterior criteria, we shall also compare the predictive performance of different priors. A classic example concerning the economics of crime will also be provided and contrasted with results in the literature. The main findings of the paper will lead us to propose a “benchmark” prior specification in a linear regression context with model uncertainty.
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...
Model Selection and Accounting for Model Uncertainty in Linear Regression Models
, 1993
"... We consider the problems of variable selection and 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. The complete B ..."
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Cited by 47 (6 self)
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We consider the problems of variable selection and 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. The complete Bayesian solution to this problem involves averaging over all possible models when making inferences about quantities of interest. This approach is often not practical. In this paper we offer two alternative approaches. First we describe a Bayesian model selection algorithm called "Occam's "Window" which involves averaging over a reduced set of models. Second, we describe a Markov chain Monte Carlo approach which directly approximates the exact solution. Both these model averaging procedures provide better predictive performance than any single model which might reasonably have been selected. In the extreme case where there are many candidate predictors but there is no relationship between any of them and the response, standard variable selection procedures often choose some subset of variables that yields a high R² and a highly significant overall F value. We refer to this unfortunate phenomenon as "Freedman's Paradox" (Freedman, 1983). In this situation, Occam's vVindow usually indicates the null model as the only one to be considered, or else a small number of models including the null model, thus largely resolving the paradox.
Mixtures of gpriors for Bayesian variable selection
 Journal of the American Statistical Association
, 2008
"... Zellner’s gprior remains a popular conventional prior for use in Bayesian variable selection, despite several undesirable consistency issues. In this paper, we study mixtures of gpriors as an alternative to default gpriors that resolve many of the problems with the original formulation, while mai ..."
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Cited by 36 (4 self)
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Zellner’s gprior remains a popular conventional prior for use in Bayesian variable selection, despite several undesirable consistency issues. In this paper, we study mixtures of gpriors as an alternative to default gpriors that resolve many of the problems with the original formulation, while maintaining the computational tractability that has made the gprior so popular. We present theoretical properties of the mixture gpriors and provide real and simulated examples to compare the mixture formulation with fixed gpriors, Empirical Bayes approaches and other default procedures.
Bayesian Adaptive Sampling for Variable Selection and Model Averaging
"... For the problem of model choice in linear regression, we introduce a Bayesian adaptive sampling algorithm (BAS), that samples models without replacement from the space of models. For problems that permit enumeration of all models BAS is guaranteed to enumerate the model space in 2 p iterations where ..."
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Cited by 9 (4 self)
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For the problem of model choice in linear regression, we introduce a Bayesian adaptive sampling algorithm (BAS), that samples models without replacement from the space of models. For problems that permit enumeration of all models BAS is guaranteed to enumerate the model space in 2 p iterations where p is the number of potential variables under consideration. For larger problems where sampling is required, we provide conditions under which BAS provides perfect samples without replacement. When the sampling probabilities in the algorithm are the marginal variable inclusion probabilities, BAS may be viewed as sampling models “near ” the median probability model of Barbieri and Berger. As marginal inclusion probabilities are not known in advance we discuss several strategies to estimate adaptively the marginal inclusion probabilities within BAS. We illustrate the performance of the algorithm using simulated and real data and show that BAS can outperform Markov chain Monte Carlo methods. The algorithm is implemented in the R package BAS available at CRAN.
Effective IS Security
, 1990
"... this paper to refer either to the set of measures that are, taken togther, effects or causes, or individual measures that stand for an effect or a cause. 8. Results Statistical techniques chosen for the study included LInear Structural RELations modelling (LISREL) and multivariate and univariate c ..."
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Cited by 3 (0 self)
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this paper to refer either to the set of measures that are, taken togther, effects or causes, or individual measures that stand for an effect or a cause. 8. Results Statistical techniques chosen for the study included LInear Structural RELations modelling (LISREL) and multivariate and univariate correlational tests. LISREL was used to investigate the simultaneous effects of deterrents and rival explanations. The additional corroborative tests offered strengths such as nonstructural tests of covariance equality (canonical correlation), distributionfree tests (KruskalWallis), and zeroorder or directeffect tests (ChiSquare Contingency Tables). 8.1 LISREL Modelling The Security Impact Model (Figure 1) was tested using the LISREL statistical package. Observed values for measures were entered as were parameters (constraints) specifying causal relationships between constructs. The LISREL package uses confirmatory factor analysis to generate loadings that best describe the specified relationships between measures and constructs. Through structural equations, LISREL then generates causal coefficients that best model the fit between constructs
Supplement to “Computational approaches for empirical Bayes methods and Bayesian sensitivity analysis
, 2011
"... We consider situations in Bayesian analysis where we have a family of priors νh on the parameter θ, where h varies continuously over a space H, and we deal with two related problems. The first involves sensitivity analysis and is stated as follows. Suppose we fix a function f of θ. How do we efficie ..."
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Cited by 2 (2 self)
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We consider situations in Bayesian analysis where we have a family of priors νh on the parameter θ, where h varies continuously over a space H, and we deal with two related problems. The first involves sensitivity analysis and is stated as follows. Suppose we fix a function f of θ. How do we efficiently estimate the posterior expectation of f(θ) simultaneously for all h in H? The second problem is how do we identify subsets of H which give rise to reasonable choices of νh? We assume that we are able to generate Markov chain samples from the posterior for a finite number of the priors, and we develop a methodology, based on a combination of importance sampling and the use of control variates, for dealing with these two problems. The methodology applies very generally, and we show how it applies in particular to a commonly used model for variable selection in Bayesian linear regression, and give an illustration on the US crime data of Vandaele. 1. Introduction. In
Perfect Simulation for orthogonal model mixing
, 1998
"... In this article we demonstrate how to generate independent and identically distributed samples from the model space of the Bayes linear model with orthogonal predictors. We use the method of coupled Markov chains from the past as introduced by Propp and Wilson (1996). This procedure alleviates any c ..."
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Cited by 1 (1 self)
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In this article we demonstrate how to generate independent and identically distributed samples from the model space of the Bayes linear model with orthogonal predictors. We use the method of coupled Markov chains from the past as introduced by Propp and Wilson (1996). This procedure alleviates any concerns over convergence and sample mixing. We present a number of examples including a perfect simulation of Bayesian wavelet selection in a 1024 dimensional model space, a knot selection problem for spline smoothers and, a standard linear regression variable selection problem. Keywords: Exact sampling, perfect simulation, wavelets, variable selection, Markov chain Monte Carlo, knot selection, radial basis functions. 1 Introduction Accounting for model uncertainty is an important issue in statistical data modelling. Failure to do so can lead to poorer performance and over confident predictions (Draper, 1995). An important component of model uncertainty is determining which predictors to ...
Exercises from book c ○ SpringerVerlag, 1997 Further exercises and answers
"... Selectable links are in this colour. 1 ..."