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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...
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.
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.
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 85 (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.
Bayesian Compressed Sensing via Belief Propagation
, 2010
"... Compressive sensing (CS) is an emerging field based on the revelation that a small collection of linear projections of a sparse signal contains enough information for stable, subNyquist signal acquisition. When a statistical characterization of the signal is available, Bayesian inference can comple ..."
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Cited by 51 (12 self)
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Compressive sensing (CS) is an emerging field based on the revelation that a small collection of linear projections of a sparse signal contains enough information for stable, subNyquist signal acquisition. When a statistical characterization of the signal is available, Bayesian inference can complement conventional CS methods based on linear programming or greedy algorithms. We perform asymptotically optimal Bayesian inference using belief propagation (BP) decoding, which represents the CS encoding matrix as a graphical model. Fast computation is obtained by reducing the size of the graphical model with sparse encoding matrices. To decode a length signal containing large coefficients, our CSBP decoding algorithm uses ( log ()) measurements and ( log 2 ()) computation. Finally, although we focus on a twostate mixture Gaussian model, CSBP is easily adapted to other signal models.
Spike and slab variable selection: frequentist and bayesian strategies
 The Annals of Statistics
"... Variable selection in the linear regression model takes many apparent faces from both frequentist and Bayesian standpoints. In this paper we introduce a variable selection method referred to as a rescaled spike and slab model. We study the importance of prior hierarchical specifications and draw con ..."
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Cited by 40 (7 self)
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Variable selection in the linear regression model takes many apparent faces from both frequentist and Bayesian standpoints. In this paper we introduce a variable selection method referred to as a rescaled spike and slab model. We study the importance of prior hierarchical specifications and draw connections to frequentist generalized ridge regression estimation. Specifically, we study the usefulness of continuous bimodal priors to model hypervariance parameters, and the effect scaling has on the posterior mean through its relationship to penalization. Several model selection strategies, some frequentist and some Bayesian in nature, are developed and studied theoretically. We demonstrate the importance of selective shrinkage for effective variable selection in terms of risk misclassification, and show this is achieved using the posterior from a rescaled spike and slab model. We also show how to verify a procedure’s ability to reduce model uncertainty in finite samples using a specialized forward selection strategy. Using this tool, we illustrate the effectiveness of rescaled spike and slab models in reducing model uncertainty. 1. Introduction. We
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.
On the Relationship Between Markov Chain Monte Carlo Methods for Model Uncertainty
 JOURNAL OF COMPUTATIONAL AND GRAPHICAL STATISTICS
, 2001
"... This article considers Markov chain computational methods for incorporating uncertainty about the dimension of a parameter when performing inference within a Bayesian setting. A general class of methods is proposed for performing such computations, based upon a product space representation of the ..."
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Cited by 31 (3 self)
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This article considers Markov chain computational methods for incorporating uncertainty about the dimension of a parameter when performing inference within a Bayesian setting. A general class of methods is proposed for performing such computations, based upon a product space representation of the problem which is similar to that of Carlin and Chib. It is shown that all of the existing algorithms for incorporation of model uncertainty into Markov chain Monte Carlo (MCMC) can be derived as special cases of this general class of methods. In particular, we show that the popular reversible jump method is obtained when a special form of MetropolisHastings (MH) algorithm is applied to the product space. Furthermore, the Gibbs sampling method and the variable selection method are shown to derive straightforwardly from the general framework. We believe that these new relationships between methods, which were until now seen as diverse procedures, are an important aid to the understanding of MCMC model selection procedures and may assist in the future development of improved procedures. Our discussion also sheds some light upon the important issues of "pseudoprior" selection in the case of the Carlin and Chib sampler and choice of proposal distribution in the case of reversible jump. Finally, we propose efficient reversible jump proposal schemes that take advantage of any analytic structure that may be present in the model. These proposal schemes are compared with a standard reversible jump scheme for the problem of model order uncertainty in autoregressive time series, demonstrating the improvements which can be achieved through careful choice of proposals
Bayesian wavelet regression on curves with application to a spectroscopic calibration problem
 Journal of the American Statistical Association
, 2001
"... Motivated by calibration problems in nearinfrared (N IR) spectroscopy, we consider the linear regression setting in which the many predictor variables arise from sampling an essentially continuous curve at equally spaced points and there may be multiple predictands. We tackle this regression proble ..."
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Cited by 28 (4 self)
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Motivated by calibration problems in nearinfrared (N IR) spectroscopy, we consider the linear regression setting in which the many predictor variables arise from sampling an essentially continuous curve at equally spaced points and there may be multiple predictands. We tackle this regression problem by calculating the wavelet transforms of the discretized curves, then applying a Bayesian variable selection method using mixture priors to the multivariate regression of predictands on wavelet coef � cients. For prediction purposes, we average over a set of likely models. Applied to a particular problem in N IR spectroscopy, this approach was able to � nd subsets of the wavelet coef � cients with overall better predictive performance than the more usual approaches. In the application, the available predictors are measurements of the N IR re � ectance spectrum of biscuit dough pieces at 256 equally spaced wavelengths. The aim is to predict the composition (i.e., the fat, � our, sugar, and water content) of the dough pieces using the spectral variables. Thus we have a multivariate regression of four predictands on 256 predictors with quite high intercorrelation among the predictors. A training set of 39 samples is available to � t this regression. Applying a wavelet transform replaces the 256 measurements on each spectrum with 256 wavelet coef � cients that carry the same information. The variable selection method could use subsets of these coef � cients that gave good predictions for all four compositional variables on a separate test set of samples. Selecting in the wavelet domain rather than from the original spectral variables is appealing in this application, because a single wavelet coef � cient can carry information from a band of wavelengths in the original spectrum. This band can be narrow or wide, depending on the scale of the wavelet selected.