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24
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 1769 (74 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.
Predictive Model Selection
 Journal of the Royal Statistical Society, Ser. B
, 1995
"... this article we propose three criteria that can be used to address model selection. These emphasize observables rather than parameters and are based on a certain Bayesian predictive density. They have a unifying basis that is simple and interpretable,are free of asymptotic de#nitions,and allow the i ..."
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Cited by 97 (5 self)
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this article we propose three criteria that can be used to address model selection. These emphasize observables rather than parameters and are based on a certain Bayesian predictive density. They have a unifying basis that is simple and interpretable,are free of asymptotic de#nitions,and allow the incorporation of prior information. Moreover,two of these criteria are readily calibrated.
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 93 (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
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 83 (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.
Bayes factors and marginal distributions in invariant situations
, 1996
"... SUMMARY. In Bayesian analysis with a “minimal ” data set and common noninformative priors, the (formal) marginal density of the data is surprisingly often independent of the error distribution. This results in great simplifications in certain model selection methodologies; for instance, the Intrinsi ..."
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Cited by 48 (13 self)
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SUMMARY. In Bayesian analysis with a “minimal ” data set and common noninformative priors, the (formal) marginal density of the data is surprisingly often independent of the error distribution. This results in great simplifications in certain model selection methodologies; for instance, the Intrinsic Bayes Factor for models with this property reduces simply to the Bayes factor with respect to the noninformative priors. The basic result holds for comparison of models which are invariant with respect to the same group structure. Indeed the condition reduces to a condition on the distributions of the common maximal invariant. In these situations, the marginal density of a “minimal ” data set is typically available in closed form, regardless of the error distribution. This provides very useful expressions for computation of Intrinsic Bayes Factors in more general settings. The conditions for the results to hold are explored in some detail for nonnormal linear models and various transformations thereof. 1.
Sequential Ordinal Modeling with Applications to Survival Data
 Biometrics
, 2001
"... This paper considers the class of sequential probit models in relation to other models for ordinal data. Hierarchical and other extensions of the model are proposed for applications involving discrete time (grouped) survival data. Computationally practical Markov chain Monte Carlo algorithms are dev ..."
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Cited by 33 (2 self)
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This paper considers the class of sequential probit models in relation to other models for ordinal data. Hierarchical and other extensions of the model are proposed for applications involving discrete time (grouped) survival data. Computationally practical Markov chain Monte Carlo algorithms are developed for the fitting of these models. The ideas and methods are illustrated in detail with a real data example on the length of hospital stay for patients undergoing heart surgery. A notable aspect of this analysis is the comparison, based on marginal likelihoods and training sample priors, of several nonnested models, such as the sequential model, the cumulative ordinal model and Weibull and loglogistic models. Keywords: Bayes factor; Discrete hazard function; Gibbs sampling; Marginal likelihood; MetropolisHastings algorithm; Nonnested models; Sequential probit; Training sample prior; Model comparison. 1 Introduction Ordinal response data is generally analyzed using the cumulative o...
The Asymmetric Business Cycle *
, 2010
"... ABSTRACT: The “business cycle ” is a fundamental, yet elusive concept in macroeconomics. In this paper, we consider the problem of measuring the business cycle. First, we argue for the ‘outputgap ’ view that the business cycle corresponds to transitory deviations in economic activity away from a pe ..."
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Cited by 18 (1 self)
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ABSTRACT: The “business cycle ” is a fundamental, yet elusive concept in macroeconomics. In this paper, we consider the problem of measuring the business cycle. First, we argue for the ‘outputgap ’ view that the business cycle corresponds to transitory deviations in economic activity away from a permanent or “trend ” level. Then, we investigate the extent to which a general modelbased approach to estimating trend and cycle for the U.S. economy leads to measures of the business cycle that reflect models versus the data. We find empirical support for a nonlinear time series model that produces a business cycle measure with an asymmetric shape across NBER expansion and recession phases. Specifically, this business cycle measure suggests that recessions are periods of relatively large and negative transitory fluctuations in output. However, several close competitors to the nonlinear model produce business cycle measures of widely differing shapes and magnitudes. Given this modelbased uncertainty, we construct a modelaveraged measure of the business cycle. The modelaveraged measure also displays an asymmetric shape and is closely related to other measures of economic
Bayesian Methods for Cumulative, Sequential and Twostep Ordinal Data Regression Models
, 1997
"... This paper considers the fitting, criticism and comparison of three ordinal regression models  the cumulative, sequential and twostep models. Efficient algorithms based on Markov chain Monte Carlo methods are developed for each model. In the case of the cumulative model, a new MetropolisHastings ..."
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Cited by 8 (1 self)
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This paper considers the fitting, criticism and comparison of three ordinal regression models  the cumulative, sequential and twostep models. Efficient algorithms based on Markov chain Monte Carlo methods are developed for each model. In the case of the cumulative model, a new MetropolisHastings procedure to sample the cut points is proposed. This procedure relies on a simple transformation of the cutpoints that leaves the transformed cutpoints unordered. For comparing these models, we develop a coherent approach based on marginal likelihoods and Bayes factors. To help in the assignment of prior distributions to regression parameters and the cutpoints, different methods for forming and representing prior beliefs are provided. One set of methods is based on the idea of a training sample and a prior imaginary sample. Another method is based on the direct assessment of distributions on the multinomial response, followed by change of variable to a distribution on the parameters of t...
Predictive specification of prior model probabilities in variable selection
 Biometrika
, 1996
"... ..."
Sparse Linear Identifiable Multivariate Modeling
"... In this paper we consider sparse and identifiable linear latent variable (factor) and linear Bayesian network models for parsimonious analysis of multivariate data. We propose a computationally efficient method for joint parameter and model inference, and model comparison. It consists of a fully Bay ..."
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Cited by 4 (1 self)
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In this paper we consider sparse and identifiable linear latent variable (factor) and linear Bayesian network models for parsimonious analysis of multivariate data. We propose a computationally efficient method for joint parameter and model inference, and model comparison. It consists of a fully Bayesian hierarchy for sparse models using slab and spike priors (twocomponent δfunction and continuous mixtures), nonGaussian latent factors and a stochastic search over the ordering of the variables. The framework, which we call SLIM (Sparse Linear Identifiable Multivariate modeling), is validated and benchmarked on artificial and real biological data sets. SLIM is closest in spirit to LiNGAM (Shimizu et al., 2006), but differs substantially in inference, Bayesian network structure learning and model comparison. Experimentally, SLIM performs equally well or better than LiNGAM with comparable computational complexity. We attribute this mainly to the stochastic search strategy used, and to parsimony (sparsity and identifiability), which is an explicit part of the model. We propose two extensions to the basic i.i.d. linear framework: nonlinear dependence on observed variables, called SNIM (Sparse Nonlinear Identifiable Multivariate modeling) and allowing for correlations between latent variables, called CSLIM (Correlated SLIM), for the temporal and/or spatial data. The source code and scripts are available from