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157
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 ..."
Abstract

Cited by 973 (70 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.
Marginal likelihood from the Gibbs output
 J. Am. Stat. Assoc
, 1995
"... Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at ..."
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Cited by 321 (19 self)
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Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at
Using simulation methods for Bayesian econometric models: Inference, development and communication
 Econometric Review
, 1999
"... This paper surveys the fundamental principles of subjective Bayesian inference in econometrics and the implementation of those principles using posterior simulation methods. The emphasis is on the combination of models and the development of predictive distributions. Moving beyond conditioning on a ..."
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Cited by 199 (15 self)
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This paper surveys the fundamental principles of subjective Bayesian inference in econometrics and the implementation of those principles using posterior simulation methods. The emphasis is on the combination of models and the development of predictive distributions. Moving beyond conditioning on a fixed number of completely specified models, the paper introduces subjective Bayesian tools for formal comparison of these models with as yet incompletely specified models. The paper then shows how posterior simulators can facilitate communication between investigators (for example, econometricians) on the one hand and remote clients (for example, decision makers) on the other, enabling clients to vary the prior distributions and functions of interest employed by investigators. A theme of the paper is the practicality of subjective Bayesian methods. To this end, the paper describes publicly available software for Bayesian inference, model development, and communication and provides illustrations using two simple econometric models. *This paper was originally prepared for the Australasian meetings of the Econometric Society in Melbourne, Australia,
Simulating Normalized Constants: From Importance Sampling to Bridge Sampling to Path Sampling
 Statistical Science, 13, 163–185. COMPARISON OF METHODS FOR COMPUTING BAYES FACTORS 435
, 1998
"... Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at ..."
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Cited by 145 (4 self)
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Bayesian measures of model complexity and fit
 Journal of the Royal Statistical Society, Series B
, 2002
"... [Read before The Royal Statistical Society at a meeting organized by the Research ..."
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Cited by 132 (2 self)
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[Read before The Royal Statistical Society at a meeting organized by the Research
Simulating ratios of normalizing constants via a simple identity: A theoretical exploration
 Statistica Sinica
, 1996
"... Abstract: Let pi(w),i =1, 2, be two densities with common support where each density is known up to a normalizing constant: pi(w) =qi(w)/ci. We have draws from each density (e.g., via Markov chain Monte Carlo), and we want to use these draws to simulate the ratio of the normalizing constants, c1/c2. ..."
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Cited by 109 (4 self)
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Abstract: Let pi(w),i =1, 2, be two densities with common support where each density is known up to a normalizing constant: pi(w) =qi(w)/ci. We have draws from each density (e.g., via Markov chain Monte Carlo), and we want to use these draws to simulate the ratio of the normalizing constants, c1/c2. Such a computational problem is often encountered in likelihood and Bayesian inference, and arises in fields such as physics and genetics. Many methods proposed in statistical and other literature (e.g., computational physics) for dealing with this problem are based on various special cases of the following simple identity: c1 c2 = E2[q1(w)α(w)] E1[q2(w)α(w)]. Here Ei denotes the expectation with respect to pi (i =1, 2), and α is an arbitrary function such that the denominator is nonzero. A main purpose of this paper is to provide a theoretical study of the usefulness of this identity, with focus on (asymptotically) optimal and practical choices of α. Using a simple but informative example, we demonstrate that with sensible (not necessarily optimal) choices of α, we can reduce the simulation error by orders of magnitude when compared to the conventional importance sampling method, which corresponds to α =1/q2. We also introduce several generalizations of this identity for handling more complicated settings (e.g., estimating several ratios simultaneously) and pose several open problems that appear to have practical as well as theoretical value. Furthermore, we discuss related theoretical and empirical work.
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.
Bayes factors and model uncertainty
 DEPARTMENT OF STATISTICS, UNIVERSITY OFWASHINGTON
, 1993
"... 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 ..."
Abstract

Cited by 89 (6 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 Pvalues, 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. The points we emphasize are: from Jeffreys's Bayesian point of view, the purpose of hypothesis testing is to evaluate the evidence in favor of a scientific theory; Bayes factors offer a way of evaluating evidence in favor ofa null hypothesis; Bayes factors provide a way of incorporating external information into the evaluation of evidence about a hypothesis; Bayes factors are very general, and do not require alternative models to be nested; several techniques are available for computing Bayes factors, including asymptotic approximations which are easy to compute using the output from standard packages that maximize likelihoods; in "nonstandard " statistical models that do not satisfy common regularity conditions, it can be technically simpler to calculate Bayes factors than to derive nonBayesian significance
The TimeVarying Volatility of Macroeconomic Fluctuations
, 2006
"... In this paper we investigate the sources of the important shifts in the volatility of U.S. macroeconomic variables in the postwar period. To this end, we propose the estimation of DSGE models allowing for time variation in the volatility of the structural innovations. We apply our estimation strate ..."
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Cited by 89 (1 self)
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In this paper we investigate the sources of the important shifts in the volatility of U.S. macroeconomic variables in the postwar period. To this end, we propose the estimation of DSGE models allowing for time variation in the volatility of the structural innovations. We apply our estimation strategy to a largescale model of the business cycle and find that investment specific technology shocks account for most of the sharp decline in volatility of the last two decades.
BUGS  Bayesian inference Using Gibbs Sampling Version 0.50
, 1995
"... e wrong, which is even worse. Please let us know of any successes or failures. Beware  Gibbs sampling can be dangerous!. BUGS c flcopyright MRC Biostatistics Unit 1995. ALL RIGHTS RESERVED. The support of the Economic and Social Research Council (UK) is gratefully acknowledged. The work was funde ..."
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Cited by 64 (0 self)
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e wrong, which is even worse. Please let us know of any successes or failures. Beware  Gibbs sampling can be dangerous!. BUGS c flcopyright MRC Biostatistics Unit 1995. ALL RIGHTS RESERVED. The support of the Economic and Social Research Council (UK) is gratefully acknowledged. The work was funded in part by ESRC (UK) Award Number H519 25 5023. 1 2 Contents 1 Introduction 5 1.1 What is BUGS? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 For what kind of problems is BUGS best suited? . . . . . . . . . . . . . . . . . . . . . 5 1.3 Markov Chain Monte Carlo (MCMC) techniques . . . . . . . . . . . . . . . . . . . . 5 1.4 A simple example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 Hardware platforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.6 Software . . .