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53
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 981 (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.
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 ..."
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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
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 62 (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.
Markov Chain Monte Carlo methods and the label switching problem in Bayesian mixture modelling
 Statistical Science
"... Abstract. In the past ten years there has been a dramatic increase of interest in the Bayesian analysis of finite mixture models. This is primarily because of the emergence of Markov chain Monte Carlo (MCMC) methods. While MCMC provides a convenient way to draw inference from complicated statistical ..."
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Cited by 51 (4 self)
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Abstract. In the past ten years there has been a dramatic increase of interest in the Bayesian analysis of finite mixture models. This is primarily because of the emergence of Markov chain Monte Carlo (MCMC) methods. While MCMC provides a convenient way to draw inference from complicated statistical models, there are many, perhaps underappreciated, problems associated with the MCMC analysis of mixtures. The problems are mainly caused by the nonidentifiability of the components under symmetric priors, which leads to socalled label switching in the MCMC output. This means that ergodic averages of component specific quantities will be identical and thus useless for inference. We review the solutions to the label switching problem, such as artificial identifiability constraints, relabelling algorithms and label invariant loss functions. We also review various MCMC sampling schemes that have been suggested for mixture models and discuss posterior sensitivity to prior specification.
Expected Returns, Realized Return, and Asset Pricing Tests
 Journal of Finance
, 1999
"... Richardson were especially helpful on this manuscript. Thanks to Deepak Agrawal for computational assistance and thoughtful comments. I would also like to thank Yakov Amihud, Anthony Lynch, Jennifer Carpenter, Paul Wachtel and Cliff Green for their comments and help. As always, none of the aforement ..."
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Cited by 44 (2 self)
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Richardson were especially helpful on this manuscript. Thanks to Deepak Agrawal for computational assistance and thoughtful comments. I would also like to thank Yakov Amihud, Anthony Lynch, Jennifer Carpenter, Paul Wachtel and Cliff Green for their comments and help. As always, none of the aforementioned are responsible for any opinions expressed or any errors. 1 One of the fundamental issues in finance is what are the factors that affect expected return on assets, the sensitivity of expected return to these factors, and the reward for bearing this sensitivity. There is a long history of testing in this area, and it is clearly one of the most investigated areas in finance. Almost all of the testing I am aware of involves using realized returns as a proxy for expected returns. The use of average realized returns as a proxy for expected returns relies on a belief that information surprises tend to cancel out over the period of the study and realized returns are therefore an unbiased estimate of expected returns. However, I believe that there is ample evidence that this belief is misplaced. There are periods longer than ten years where stock market realized returns are on average less than the riskfree rate (1973 to 1984). There are periods longer than fifty years in which risky longterm bonds on average underperformed the risk free
Bayesian variable selection in multinomial models with application to spectral data and DNA microarrays
, 2002
"... Summary. Here we focus on discrimination problems where the number of predictors substantially exceeds the sample size and we propose a Bayesian variable selection approach to multinomial probit models. Our method makes use of mixture priors and Markov chain Monte Carlo techniques to select sets of ..."
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Cited by 26 (10 self)
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Summary. Here we focus on discrimination problems where the number of predictors substantially exceeds the sample size and we propose a Bayesian variable selection approach to multinomial probit models. Our method makes use of mixture priors and Markov chain Monte Carlo techniques to select sets of variables that differ among the classes. We apply our methodology to a problem in functional genomics using gene expression profiling data. The aim of the analysis is to identify molecular signatures that characterize two different stages of rheumatoid arthritis.
Bayesian inference procedures derived via the concept of relative surprise
 Communications in Statistics
, 1997
"... of least relative surprise; model checking; change of variable problem; crossvalidation. We consider the problem of deriving Bayesian inference procedures via the concept of relative surprise. The mathematical concept of surprise has been developed by I.J. Good in a long sequence of papers. We make ..."
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Cited by 17 (6 self)
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of least relative surprise; model checking; change of variable problem; crossvalidation. We consider the problem of deriving Bayesian inference procedures via the concept of relative surprise. The mathematical concept of surprise has been developed by I.J. Good in a long sequence of papers. We make a modiÞcation to this development that permits the avoidance of a serious defect; namely, the change of variable problem. We apply relative surprise to the development of estimation, hypothesis testing and model checking procedures. Important advantages of the relative surprise approach to inference include the lack of dependence on a particular loss function and complete freedom to the statistician in the choice of prior for hypothesis testing problems. Links are established with common Bayesian inference procedures such as highest posterior density regions, modal estimates and Bayes factors. From a practical perspective new inference
Bayesian hypothesis testing: A reference approach
 Internat. Statist. Rev
, 2002
"... For any probability model M ≡{p(x  θ, ω), θ ∈ Θ, ω ∈ Ω} assumed to describe the probabilistic behaviour of data x ∈ X, it is argued that testing whether or not the available data are compatible with the hypothesis H0 ≡{θ = θ0} is best considered as a formal decision problem on whether to use (a0), ..."
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Cited by 17 (5 self)
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For any probability model M ≡{p(x  θ, ω), θ ∈ Θ, ω ∈ Ω} assumed to describe the probabilistic behaviour of data x ∈ X, it is argued that testing whether or not the available data are compatible with the hypothesis H0 ≡{θ = θ0} is best considered as a formal decision problem on whether to use (a0), or not to use (a1), the simpler probability model (or null model) M0 ≡{p(x  θ0, ω), ω ∈ Ω}, where the loss difference L(a0, θ, ω) − L(a1, θ, ω) is proportional to the amount of information δ(θ0, θ, ω) which would be lost if the simplified model M0 were used as a proxy for the assumed model M. For any prior distribution π(θ, ω), the appropriate normative solution is obtained by rejecting the null model M0 whenever the corresponding posterior expectation ∫ ∫ δ(θ0, θ, ω) π(θ, ω  x) dθ dω is sufficiently large. Specification of a subjective prior is always difficult, and often polemical, in scientific communication. Information theory may be used to specify a prior, the reference prior, which only depends on the assumed model M, and mathematically describes a situation where no prior information is available about the quantity of interest. The reference posterior expectation, d(θ0, x) = ∫ δπ(δ  x) dδ, of the amount of information δ(θ0, θ, ω) which could be lost if the null model were used, provides an attractive nonnegative test function, the intrinsic statistic, which is
Bayesian information criterion for censored survival models
 Biometrics
"... We investigate the Bayesian Information Criterion (BIC) for variable selection in models for censored survival data. Kass and Wasserman (1995) showed that BIC provides a close approximation to the Bayes factor when a unitinformation prior on the parameter space is used. We propose a revision of the ..."
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Cited by 15 (2 self)
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We investigate the Bayesian Information Criterion (BIC) for variable selection in models for censored survival data. Kass and Wasserman (1995) showed that BIC provides a close approximation to the Bayes factor when a unitinformation prior on the parameter space is used. We propose a revision of the penalty term in BIC so that it is de ned in terms of the number of uncensored events instead of the number of observations. For the simplest censored data model, that of exponential distributions of survival times (i.e. a constant hazard rate), this revision results in a better approximation to the exact Bayes factor based on a conjugate unitinformation prior. In the Cox proportional hazards regression model, we propose de ning BIC in terms of the maximized partial likelihood. Using the number of deaths rather than the number of individuals in the BIC penalty term corresponds to a more realistic prior on the parameter space, and is shown to improve predictive performance for assessing stroke risk in the Cardiovascular Health Study.