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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.
Model Selection and the Principle of Minimum Description Length
 Journal of the American Statistical Association
, 1998
"... This paper reviews the principle of Minimum Description Length (MDL) for problems of model selection. By viewing statistical modeling as a means of generating descriptions of observed data, the MDL framework discriminates between competing models based on the complexity of each description. This ..."
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Cited by 145 (5 self)
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This paper reviews the principle of Minimum Description Length (MDL) for problems of model selection. By viewing statistical modeling as a means of generating descriptions of observed data, the MDL framework discriminates between competing models based on the complexity of each description. This approach began with Kolmogorov's theory of algorithmic complexity, matured in the literature on information theory, and has recently received renewed interest within the statistics community. In the pages that follow, we review both the practical as well as the theoretical aspects of MDL as a tool for model selection, emphasizing the rich connections between information theory and statistics. At the boundary between these two disciplines, we find many interesting interpretations of popular frequentist and Bayesian procedures. As we will see, MDL provides an objective umbrella under which rather disparate approaches to statistical modeling can coexist and be compared. We illustrate th...
Prior distributions for variance parameters in hierarchical models
 Bayesian Analysis
, 2006
"... Various noninformative prior distributions have been suggested for scale parameters in hierarchical models. We construct a new foldednoncentralt family of conditionally conjugate priors for hierarchical standard deviation parameters, and then consider noninformative and weakly informative priors i ..."
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Cited by 140 (13 self)
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Various noninformative prior distributions have been suggested for scale parameters in hierarchical models. We construct a new foldednoncentralt family of conditionally conjugate priors for hierarchical standard deviation parameters, and then consider noninformative and weakly informative priors in this family. We use an example to illustrate serious problems with the inversegamma family of “noninformative ” prior distributions. We suggest instead to use a uniform prior on the hierarchical standard deviation, using the halft family when the number of groups is small and in other settings where a weakly informative prior is desired.
A Reference Bayesian Test for Nested Hypotheses And its Relationship to the Schwarz Criterion
 Journal of the American Statistical Association
, 1994
"... To compute a Bayes factor for testing H 0 : / = / 0 in the presence of a nuisance parameter fi, priors under the null and alternative hypotheses must be chosen. As in Bayesian estimation, an important problem has been to define automatic or "reference" methods for determining priors based only on t ..."
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Cited by 125 (4 self)
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To compute a Bayes factor for testing H 0 : / = / 0 in the presence of a nuisance parameter fi, priors under the null and alternative hypotheses must be chosen. As in Bayesian estimation, an important problem has been to define automatic or "reference" methods for determining priors based only on the structure of the model. In this paper we apply the heuristic device of taking the amount of information in the prior on / equal to the amount of information in a single observation. Then, after transforming fi to be "null orthogonal" to /, we take the marginal priors on fi to be equal under the null and alternative hypotheses. Doing so, and taking the prior on / to be Normal, we find that the log of the Bayes factor may be approximated by the Schwarz criterion with an error of order O(n \Gamma1=2 ), rather than the usual error of order O(1). This result suggests the Schwarz criterion should provide sensible approximate solutions to Bayesian testing problems, at least when the hypothese...
Practical Bayesian Density Estimation Using Mixtures Of Normals
 Journal of the American Statistical Association
, 1995
"... this paper, we propose some solutions to these problems. Our goal is to come up with a simple, practical method for estimating the density. This is an interesting problem in its own right, as well as a first step towards solving other inference problems, such as providing more flexible distributions ..."
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Cited by 116 (2 self)
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this paper, we propose some solutions to these problems. Our goal is to come up with a simple, practical method for estimating the density. This is an interesting problem in its own right, as well as a first step towards solving other inference problems, such as providing more flexible distributions in hierarchical models. To see why the posterior is improper under the usual reference prior, we write the model in the following way. Let Z = (Z 1 ; : : : ; Z n ) and X = (X 1 ; : : : ; X n ). The Z
Calibration and Empirical Bayes Variable Selection
 Biometrika
, 1997
"... this paper, is that with F =2logp. This choice was proposed by Foster &G eorge (1994) where it was called the Risk Inflation Criterion (RIC) because it asymptotically minimises the maximum predictive risk inflation due to selection when X is orthogonal. This choice and its minimax property were also ..."
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Cited by 114 (19 self)
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this paper, is that with F =2logp. This choice was proposed by Foster &G eorge (1994) where it was called the Risk Inflation Criterion (RIC) because it asymptotically minimises the maximum predictive risk inflation due to selection when X is orthogonal. This choice and its minimax property were also discovered independently by Donoho & Johnstone (1994) in the wavelet regression context, where they refer to it as the universal hard thresholding rule
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 Model Selection and Model Averaging
, 1999
"... This paper reviews the Bayesian approach to model selection and model averaging. In this review, I emphasize objective Bayesian methods based on noninformative priors. I will also discuss implementation details, approximations and relationships to other methods. KEY WORDS AND PHRASES: AIC, Bayes Fac ..."
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Cited by 58 (0 self)
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This paper reviews the Bayesian approach to model selection and model averaging. In this review, I emphasize objective Bayesian methods based on noninformative priors. I will also discuss implementation details, approximations and relationships to other methods. KEY WORDS AND PHRASES: AIC, Bayes Factors, BIC, Consistency, Default Bayes Methods, Markov Chain Monte Carlo.
Markov Chain Monte Carlo Model Determination for Hierarchical and Graphical Loglinear Models
 Biometrika
, 1996
"... this paper, we will only consider undirected graphical models. For details of Bayesian model selection for directed graphical models see Madigan et al (1995). An (undirected) graphical model is determined by a set of conditional independence constraints of the form `fl 1 is independent of fl 2 condi ..."
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Cited by 55 (8 self)
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this paper, we will only consider undirected graphical models. For details of Bayesian model selection for directed graphical models see Madigan et al (1995). An (undirected) graphical model is determined by a set of conditional independence constraints of the form `fl 1 is independent of fl 2 conditional on all other fl i 2 C'. Graphical models are so called because they can each be represented as a graph with vertex set C and an edge between each pair fl 1 and fl 2 unless fl 1 and fl 2 are conditionally independent as described above. Darroch, Lauritzen and Speed (1980) show that each graphical loglinear model is hierarchical, with generators given by the cliques (complete subgraphs) of the graph. The total number of possible graphical models is clearly given by 2 (