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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.
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,
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.
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
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.
Hypothesis Testing and Model Selection Via Posterior Simulation
 In Practical Markov Chain
, 1995
"... Introduction To motivate the methods described in this chapter, consider the following inference problem in astronomy (Soubiran, 1993). Until fairly recently, it has been believed that the Galaxy consists of two stellar populations, the disk and the halo. More recently, it has been hypothesized tha ..."
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Cited by 24 (1 self)
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Introduction To motivate the methods described in this chapter, consider the following inference problem in astronomy (Soubiran, 1993). Until fairly recently, it has been believed that the Galaxy consists of two stellar populations, the disk and the halo. More recently, it has been hypothesized that there are in fact three stellar populations, the old (or thin) disk, the thick disk, and the halo, distinguished by their spatial distributions, their velocities, and their metallicities. These hypotheses have different implications for theories of the formation of the Galaxy. Some of the evidence for deciding whether there are two or three populations is shown in Figure 1, which shows radial and rotational velocities for n = 2; 370 stars. A natural model for this situation is a mixture model with J components, namely y i = J X j=1 ae j
A Discussion of Parameter and Model Uncertainty in Insurance
 in Insurance,” Insurance: Mathematics and Economics
, 2000
"... In this paper we consider the process of modelling uncertainty. In particular we are concerned with making inferences about some quantity of interest which, at present, has been unobserved. Examples of such a quantity include the probability of ruin of a surplus process, the accumulation of an inves ..."
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Cited by 22 (7 self)
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In this paper we consider the process of modelling uncertainty. In particular we are concerned with making inferences about some quantity of interest which, at present, has been unobserved. Examples of such a quantity include the probability of ruin of a surplus process, the accumulation of an investment, the level or surplus or deficit in a pension fund and the future volume of new business in an insurance company. Uncertainty in this quantity of interest, y, arises from three sources: . uncertainty due to the stochastic nature of a given model; . uncertainty in the values of the parameters in a given model; . uncertainty in the model underlying what we are able to observe and determining the quantity of interest. It is common in actuarial science to find that the first source of uncertainty is the only one which receives rigorous attention. A limited amount of research in recent years has considered the effect of parameter uncertainty, while there is still considerable scope ...
Statistical Ideas for Selecting Network Architectures
 Invited Presentation, Neural Information Processing Systems 8
, 1995
"... Choosing the architecture of a neural network is one of the most important problems in making neural networks practically useful, but accounts of applications usually sweep these details under the carpet. How many hidden units are needed? Should weight decay be used, and if so how much? What type of ..."
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Cited by 18 (3 self)
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Choosing the architecture of a neural network is one of the most important problems in making neural networks practically useful, but accounts of applications usually sweep these details under the carpet. How many hidden units are needed? Should weight decay be used, and if so how much? What type of output units should be chosen? And so on. We address these issues within the framework of statistical theory for model choice, which provides a number of workable approximate answers. This paper is principally concerned with architecture selection issues for feedforward neural networks (also known as multilayer perceptrons). Many of the same issues arise in selecting radial basis function networks, recurrent networks and more widely. These problems occur in a much wider context within statistics, and applied statisticians have been selecting and combining models for decades. Two recent discussions are [4, 5]. References [3, 20, 21, 22] discuss neural networks from a statistical perspecti...
Estimating Ratios of Normalizing Constants for Densities with Different Dimensions
 STATISTICA SINICA
, 1997
"... In Bayesian inference, a Bayes factor is defined as the ratio of posterior odds versus prior odds where posterior odds is simply a ratio of the normalizing constants of two posterior densities. In many practical problems, the two posteriors have different dimensions. For such cases, the current Mont ..."
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Cited by 17 (2 self)
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In Bayesian inference, a Bayes factor is defined as the ratio of posterior odds versus prior odds where posterior odds is simply a ratio of the normalizing constants of two posterior densities. In many practical problems, the two posteriors have different dimensions. For such cases, the current Monte Carlo methods such as the bridge sampling method (Meng and Wong 1996), the path sampling method (Gelman and Meng 1994), and the ratio importance sampling method (Chen and Shao 1994) cannot directly be applied. In this article, we extend importance sampling, bridge sampling, and ratio importance sampling to problems of different dimensions. Then we find global optimal importance sampling, bridge sampling, and ratio importance sampling in the sense of minimizing asymptotic relative meansquare errors of estimators. Implementation algorithms, which can asymptotically achieve the optimal simulation errors, are developed and two illustrative examples are also provided.
The use of Bayes factors for model selection in structural reliability
 In: Proc. of 8th Int. Conf. on Structural Safety and Reliability (ICOSSAR). June 2001
, 2001
"... ABSTRACT: Probabilistic design of structures is usually based on estimates of a design load with a high average return period. Design loads are often estimated using classical statistical methods. A shortcoming of this approach is that statistical uncertainties are not taken into account. In this pa ..."
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Cited by 5 (3 self)
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ABSTRACT: Probabilistic design of structures is usually based on estimates of a design load with a high average return period. Design loads are often estimated using classical statistical methods. A shortcoming of this approach is that statistical uncertainties are not taken into account. In this paper, a method based on Bayesian statistics is presented. Using Bayes ’ theorem, the prior distribution representing information about the uncertainty of the statistical parameters can be updated to the posterior distribution as soon as data becomes available. Seven predictive probability distributions are considered for determining extreme quantiles of loads: the exponential, Rayleigh, normal, lognormal, gamma, Weibull and Gumbel. The Bayesian method has been successfully applied to estimate the design discharge of the river Rhine while taking account of the statistical uncertainties involved. As a prior the noninformative Jeffreys prior was chosen. The Bayes estimates are compared to the classical maximumlikelihood estimates. Furthermore, socalled Bayes factors are used to determine weights corresponding to how well a probability distribution fits the observed data; that is, the better the fit, the higher the weighting. 1