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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 1176 (71 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 accounting for model uncertainty in graphical models using Occam's window
, 1993
"... We consider the problem of model selection and accounting for model uncertainty in highdimensional contingency tables, motivated by expert system applications. The approach most used currently is a stepwise strategy guided by tests based on approximate asymptotic Pvalues leading to the selection o ..."
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Cited by 293 (46 self)
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We consider the problem of model selection and accounting for model uncertainty in highdimensional contingency tables, motivated by expert system applications. The approach most used currently is a stepwise strategy guided by tests based on approximate asymptotic Pvalues leading to the selection of a single model; inference is then conditional on the selected model. The sampling properties of such a strategy are complex, and the failure to take account of model uncertainty leads to underestimation of uncertainty about quantities of interest. In principle, a panacea is provided by the standard Bayesian formalism which averages the posterior distributions of the quantity of interest under each of the models, weighted by their posterior model probabilities. Furthermore, this approach is optimal in the sense of maximising predictive ability. However, this has not been used in practice because computing the posterior model probabilities is hard and the number of models is very large (often greater than 1011). We argue that the standard Bayesian formalism is unsatisfactory and we propose an alternative Bayesian approach that, we contend, takes full account of the true model uncertainty byaveraging overamuch smaller set of models. An efficient search algorithm is developed for nding these models. We consider two classes of graphical models that arise in expert systems: the recursive causal models and the decomposable
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 95 (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
Comparing Dynamic Equilibrium Models to Data: A Bayesian Approach
, 2002
"... This paper studies the properties of the Bayesian approach to estimation and comparison of dynamic equilibrium economies. Both tasks can be performed even if the models are nonnested, misspecified, and nonlinear. First, we show that Bayesian methods have a classical interpretation: asymptotically ..."
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Cited by 69 (12 self)
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This paper studies the properties of the Bayesian approach to estimation and comparison of dynamic equilibrium economies. Both tasks can be performed even if the models are nonnested, misspecified, and nonlinear. First, we show that Bayesian methods have a classical interpretation: asymptotically, the parameter point estimates converge to their pseudotrue values, and the best model under the KullbackLeibler distance will have the highest posterior probability. Second, we illustrate the strong small sample behavior of the approach using a wellknown application: the U.S. cattle cycle. Bayesian estimates outperform maximum likelihood results, and the proposed model is easily compared with a set of BVARs.
Bayesian model selection in structural equation models
, 1993
"... A Bayesian approach to model selection for structural equation models is outlined. This enables us to compare individual models, nested or nonnested, and also to search through the (perhaps vast) set of possible models for the best ones. The approach selects several models rather than just one, whe ..."
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Cited by 39 (10 self)
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A Bayesian approach to model selection for structural equation models is outlined. This enables us to compare individual models, nested or nonnested, and also to search through the (perhaps vast) set of possible models for the best ones. The approach selects several models rather than just one, when appropriate, and so enables us to take account, both informally and formally, of uncertainty about model structure when making inferences about quantities of interest. The approach tends to select simpler models than strategies based on multiple Pvaluebased tests. It may thus help to overcome the criticism of structural
Estimating dynamic equilibrium economies: linear versus nonlinear likelihood
 Journal of Applied Econometrics
, 2005
"... This paper compares twomethods for undertaking likelihoodbased inference in dynamic equilibrium economies: a Sequential Monte Carlo filter and the Kalman filter. The Sequential Monte Carlo filter exploits the nonlinear structure of the economy and evaluates the likelihood function of the model by s ..."
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Cited by 31 (13 self)
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This paper compares twomethods for undertaking likelihoodbased inference in dynamic equilibrium economies: a Sequential Monte Carlo filter and the Kalman filter. The Sequential Monte Carlo filter exploits the nonlinear structure of the economy and evaluates the likelihood function of the model by simulation methods. The Kalman filter estimates a linearization of the economy around the steady state. We report two main results. First, both for simulated and for real data, the Sequential Monte Carlo filter delivers a substantially better fit of the model to the data as measured by the marginal likelihood. This is true even for a nearly linear case. Second, the differences in terms of point estimates, although relatively small in absolute values, have important effects on the moments of the model. We conclude that the nonlinear filter is a superior procedure for taking models to the data.
A Bayesian Approach to Detection of Small Low Emission Sources, preprint 2010
"... Abstract. The article addresses the problem of detecting presence and location of a small low emission source inside of an object, when the background noise dominates. This problem arises, for instance, in some homeland security applications. The goal is to reach the signaltonoise ratio (SNR) leve ..."
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Cited by 1 (1 self)
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Abstract. The article addresses the problem of detecting presence and location of a small low emission source inside of an object, when the background noise dominates. This problem arises, for instance, in some homeland security applications. The goal is to reach the signaltonoise ratio (SNR) levels on the order of 10−3. A Bayesian approach to this problem is implemented in 2D. The method allows inference not only about the existence of the source, but also about its location. We derive Bayes factors for model selection and estimation of location based on Markov Chain Monte Carlo simulation. A simulation study shows that with sufficiently high total emission level, our method can effectively locate the source or indicate its absence. 1.