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Bayes Factors and BIC  Comment on “A Critique of the Bayesian Information Criterion for Model Selection”
, 1999
"... I would like to thank David L. Weakliem (1999 [this issue]) for a thoughtprovoking discussion of the basis of the Bayesian information criterion (BIC). We may be in closer agreement than one might think from reading his article. When writing about Bayesian model selection for social researchers, I ..."
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I would like to thank David L. Weakliem (1999 [this issue]) for a thoughtprovoking discussion of the basis of the Bayesian information criterion (BIC). We may be in closer agreement than one might think from reading his article. When writing about Bayesian model selection for social researchers, I focused on the BIC approximation on the grounds that it is easily implemented and often reasonable, and simplifies the exposition of an already technical topic. As Weakliem says, BIC corresponds to one of many possible priors, although I will argue that this prior is such as to make BIC appropriate for baseline reference use and reporting, albeit not necessarily always appropriate for drawing final conclusions. When writing about the same subject for statistical journals, however, I have paid considerable attention to the choice of priors for Bayes factors. I thank Weakliem for bringing this subtle but important topic to the attention of sociologists. In 1986, I proposed replacing P values by Bayes factors as the basis for hypothesis testing and model selection in social research, and I suggested BIC as a simple and convenient, albeit crude, approximation. Since then, a great deal has been learned about Bayes factors in general, and about BIC in particular. Weakliem seems to agree that the Bayes factor framework is a useful one for hypothesis testing and model selection; his concern is with how the Bayes factors are to be evaluated. Weakliem makes two main points about the BIC approximation. The first is that BIC yields an approximation to Bayes factors that corresponds closely to a particular prior (the unit information prior) on
Fortune or Virtue: TimeVariant Volatilities Versus Parameter Drifting in U.S. Data ∗
, 2010
"... participants at several seminars for useful comments, and Béla Személy for invaluable research assistance. Beyond the usual disclaimer, we must note that any views expressed herein are those of the authors and not necessarily those of the Federal Reserve Bank of Atlanta, the Federal Reserve Bank of ..."
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participants at several seminars for useful comments, and Béla Személy for invaluable research assistance. Beyond the usual disclaimer, we must note that any views expressed herein are those of the authors and not necessarily those of the Federal Reserve Bank of Atlanta, the Federal Reserve Bank of Philadelphia, or the Federal Reserve System. Finally, we also thank the NSF for financial support.
Testing Nonnested Models of International Relations: Reevaluating Realism
, 2001
"... Unknown to most world politics scholars and political scientists in general, traditional methods of model discrimination such as likelihood ratio tests, Ftests, and artificial nesting fail when applied to nonnested models. That the vast majority of models used throughout international relations ..."
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Cited by 10 (4 self)
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Unknown to most world politics scholars and political scientists in general, traditional methods of model discrimination such as likelihood ratio tests, Ftests, and artificial nesting fail when applied to nonnested models. That the vast majority of models used throughout international relations research have nonlinear functional forms complicates the problem. The purpose of this research is to suggest methods of properly discriminating between nonnested models and then to demonstrate how these techniques can shed light on substantive debates in international relations. Reanalysis of two wellknown articles that compare structural realism to various alternatives suggests that the evidence against realism in both articles is overstated.
Bayesian Regression Analysis With Scale Mixtures of Normals
, 1999
"... This paper considers a Bayesian analysis of the linear regression model under independent sampling from general scale mixtures of Normals. Using a common reference prior, we investigate the validity of Bayesian inference and the existence of posterior moments of the regression and scale parameters. ..."
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This paper considers a Bayesian analysis of the linear regression model under independent sampling from general scale mixtures of Normals. Using a common reference prior, we investigate the validity of Bayesian inference and the existence of posterior moments of the regression and scale parameters. We find that whereas existence of the posterior distribution does not depend on the choice of the design matrix or the mixing distribution, both of them can crucially intervene in the existence of posterior moments. We identify some useful characteristics that allow for an easy verification of the existence of a wide range of moments. In addition, we provide full characterizations under sampling from finite mixtures of Normals, Pearson VII or certain Modulated Normal distributions. For empirical applications, a numerical implementation based on the Gibbs sampler is recommended.
Bayesian Selection of LogLinear Models
 Canadian Journal of Statistics
, 1995
"... A general methodology is presented for finding suitable Poisson loglinear models with applications to multiway contingency tables. Mixtures of multivariate normal distributions are used to model prior opinion when a subset of the regression vector is believed to be nonzero. This prior distribution ..."
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A general methodology is presented for finding suitable Poisson loglinear models with applications to multiway contingency tables. Mixtures of multivariate normal distributions are used to model prior opinion when a subset of the regression vector is believed to be nonzero. This prior distribution is studied for two and threeway contingency tables, in which the regression coefficients are interpretable in terms of oddsratios in the table. Efficient and accurate schemes are proposed for calculating the posterior model probabilities. The methods are illustrated for a large number of twoway simulated tables and for two threeway tables. These methods appear to be useful in selecting the best loglinear model and in estimating parameters of interest that reflect uncertainty in the true model. Key words and phrases: Bayes factors, Laplace method, Gibbs sampling, Model selection, Odds ratios. AMS subject classifications: Primary 62H17, 62F15, 62J12. 1 Introduction 1.1 Bayesian testing...
Approximate Bayesian Inference for Quantiles
"... Suppose data consist of a random sample from a distribution function FY, which is unknown, and that interest focuses on inferences on θ, a vector of quantiles of FY. When the likelihood function is not fully specified, a posterior density cannot be calculated and Bayesian inference is difficult. Th ..."
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Suppose data consist of a random sample from a distribution function FY, which is unknown, and that interest focuses on inferences on θ, a vector of quantiles of FY. When the likelihood function is not fully specified, a posterior density cannot be calculated and Bayesian inference is difficult. This article considers an approach which relies on a substitution likelihood characterized by a vector of quantiles. Properties of the substitution likelihood are investigated, strategies for prior elicitation are presented, and a general framework is proposed for quantile regression modeling. Posterior computation proceeds via a Metropolis algorithm that utilizes a normal approximation to the posterior. Results from a simulation study are presented, and the methods are illustrated through application to data from a genotoxicity experiment.
The formal definition of reference priors
 ANN. STATIST
, 2009
"... Reference analysis produces objective Bayesian inference, in the sense that inferential statements depend only on the assumed model and the available data, and the prior distribution used to make an inference is least informative in a certain informationtheoretic sense. Reference priors have been r ..."
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Reference analysis produces objective Bayesian inference, in the sense that inferential statements depend only on the assumed model and the available data, and the prior distribution used to make an inference is least informative in a certain informationtheoretic sense. Reference priors have been rigorously defined in specific contexts and heuristically defined in general, but a rigorous general definition has been lacking. We produce a rigorous general definition here and then show how an explicit expression for the reference prior can be obtained under very weak regularity conditions. The explicit expression can be used to derive new reference priors both analytically and numerically.
A Compendium of Conjugate Priors
, 1997
"... This report reviews conjugate priors and priors closed under sampling for a variety of data generating processes where the prior distributions are univariate, bivariate, and multivariate. The effects of transformations on conjugate prior relationships are considered and cases where conjugate prior r ..."
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This report reviews conjugate priors and priors closed under sampling for a variety of data generating processes where the prior distributions are univariate, bivariate, and multivariate. The effects of transformations on conjugate prior relationships are considered and cases where conjugate prior relationships can be applied under transformations are identified. Univariate and bivariate prior relationships are verified using Monte Carlo methods. Contents 1