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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
Bayesian Calibration of Computer Models
 Journal of the Royal Statistical Society, Series B, Methodological
, 2000
"... this paper a Bayesian approach to the calibration of computer models. We represent the unknown inputs as a parameter vector `. Using the observed data we derive the posterior distribution of `, which in particular quantifies the `residual uncertainty' about ..."
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Cited by 62 (1 self)
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this paper a Bayesian approach to the calibration of computer models. We represent the unknown inputs as a parameter vector `. Using the observed data we derive the posterior distribution of `, which in particular quantifies the `residual uncertainty' about
Inference from a Deterministic Population Dynamics Model for Bowhead Whales
 Journal of the American Statistical Association
, 1995
"... We consider the problem of inference about a quantity of interest given different sources of information linked by a deterministic population dynamics model. Our approach consists of translating all the available information into a joint premodel distribution on all the model inputs and outputs, an ..."
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Cited by 41 (21 self)
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We consider the problem of inference about a quantity of interest given different sources of information linked by a deterministic population dynamics model. Our approach consists of translating all the available information into a joint premodel distribution on all the model inputs and outputs, and then restricting this to the submanifold defined by the model to obtain the joint postmodel distribution. Marginalizing this yields inference, conditional on the model, about quantities of interest which can be functions of model inputs, model outputs, or both. Samples from the postmodel distribution are obtained by importance sampling and Rubin's SIR algorithm. The framework includes as a special case the situation where the premodel information about the outputs consists of measurements with error; this reduces to standard Bayesian inference. The results are in the form of a sample from the postmodel distribution and so can be examined using the full range of exploratory data analysis...
Estimating Bayes Factors via Posterior Simulation with the LaplaceMetropolis Estimator
 Journal of the American Statistical Association
, 1994
"... The key quantity needed for Bayesian hypothesis testing and model selection is the marginal likelihood for a model, also known as the integrated likelihood, or the marginal probability of the data. In this paper we describe a way to use posterior simulation output to estimate marginal likelihoods. W ..."
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Cited by 33 (11 self)
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The key quantity needed for Bayesian hypothesis testing and model selection is the marginal likelihood for a model, also known as the integrated likelihood, or the marginal probability of the data. In this paper we describe a way to use posterior simulation output to estimate marginal likelihoods. We describe the basic LaplaceMetropolis estimator for models without random effects. For models with random effects the compound LaplaceMetropolis estimator is introduced. This estimator is applied to data from the World Fertility Survey and shown to give accurate results. Batching of simulation output is used to assess the uncertainty involved in using the compound LaplaceMetropolis estimator. The method allows us to test for the effects of independent variables in a random effects model, and also to test for the presence of the random effects. KEY WORDS: LaplaceMetropolis estimator; Random effects models; Marginal likelihoods; Posterior simulation; World Fertility Survey. 1 Introduction...
Methods for Approximating Integrals in Statistics with Special Emphasis on Bayesian Integration Problems
 Statistical Science
"... This paper is a survey of the major techniques and approaches available for the numerical approximation of integrals in statistics. We classify these into five broad categories; namely, asymptotic methods, importance sampling, adaptive importance sampling, multiple quadrature and Markov chain method ..."
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Cited by 32 (4 self)
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This paper is a survey of the major techniques and approaches available for the numerical approximation of integrals in statistics. We classify these into five broad categories; namely, asymptotic methods, importance sampling, adaptive importance sampling, multiple quadrature and Markov chain methods. Each method is discussed giving an outline of the basic supporting theory and particular features of the technique. Conclusions are drawn concerning the relative merits of the methods based on the discussion and their application to three examples. The following broad recommendations are made. Asymptotic methods should only be considered in contexts where the integrand has a dominant peak with approximate ellipsoidal symmetry. Importance sampling, and preferably adaptive importance sampling, based on a multivariate Student should be used instead of asymptotics methods in such a context. Multiple quadrature, and in particular subregion adaptive integration, are the algorithms of choice for...
Computing Nonparametric Hierarchical Models
, 1998
"... Bayesian models involving Dirichlet process mixtures are at the heart of the modern nonparametric Bayesian movement. Much of the rapid development of these models in the last decade has been a direct result of advances in simulationbased computational methods. Some of the very early work in thi ..."
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Cited by 22 (2 self)
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Bayesian models involving Dirichlet process mixtures are at the heart of the modern nonparametric Bayesian movement. Much of the rapid development of these models in the last decade has been a direct result of advances in simulationbased computational methods. Some of the very early work in this area, circa 19881991, focused on the use of such nonparametric ideas and models in applications of otherwise standard hierarchical models. This chapter provides some historical review and perspective on these developments, with a prime focus on the use and integration of such nonparametric ideas in hierarchical models. We illustrate the ease with which the strict parametric assumptions common to most standard Bayesian hierarchical models can be relaxed to incorporate uncertainties about functional forms using Dirichlet process components, partly enabled by the approach to computation using MCMC methods. The resulting methology is illustrated with two examples taken from an unpub...
Inference and Hierarchical Modeling in the Social Sciences
, 1995
"... this paper I (1) examine three levels of inferential strength supported by typical social science datagathering methods, and call for a greater degree of explicitness, when HMs and other models are applied, in identifying which level is appropriate; (2) reconsider the use of HMs in school effective ..."
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Cited by 21 (6 self)
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this paper I (1) examine three levels of inferential strength supported by typical social science datagathering methods, and call for a greater degree of explicitness, when HMs and other models are applied, in identifying which level is appropriate; (2) reconsider the use of HMs in school effectiveness studies and metaanalysis from the perspective of causal inference; and (3) recommend the increased use of Gibbs sampling and other Markovchain Monte Carlo (MCMC) methods in the application of HMs in the social sciences, so that comparisons between MCMC and betterestablished fitting methodsincluding full or restricted maximum likelihood estimation based on the EM algorithm, Fisher scoring or iterative generalized least squaresmay be more fully informed by empirical practice.
An Iterative Monte Carlo Method for Nonconjugate Bayesian Analysis
 Statistics and Computing
, 1991
"... The Gibbs sampler has been proposed as a general method for Bayesian calculation in Gelfand and Smith (1990). However, the predominance of experience to date resides in applications assuming conjugacy where implementation is reasonably straightforward. This paper describes a tailored approximate rej ..."
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Cited by 17 (0 self)
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The Gibbs sampler has been proposed as a general method for Bayesian calculation in Gelfand and Smith (1990). However, the predominance of experience to date resides in applications assuming conjugacy where implementation is reasonably straightforward. This paper describes a tailored approximate rejection method approach for implementation of the Gibbs sampler when nonconjugate structure is present. Several challenging applications are presented for illustration.
Bayesian Selection of LogLinear Models
 Canadian Journal of Statistics
, 1995
"... A general methodology is presented for finding suitable Poisson loglinear models with applications to multiway contingency tables. Mixtures of multivariate normal distributions are used to model prior opinion when a subset of the regression vector is believed to be nonzero. This prior distribution ..."
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Cited by 7 (2 self)
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A general methodology is presented for finding suitable Poisson loglinear models with applications to multiway contingency tables. Mixtures of multivariate normal distributions are used to model prior opinion when a subset of the regression vector is believed to be nonzero. This prior distribution is studied for two and threeway contingency tables, in which the regression coefficients are interpretable in terms of oddsratios in the table. Efficient and accurate schemes are proposed for calculating the posterior model probabilities. The methods are illustrated for a large number of twoway simulated tables and for two threeway tables. These methods appear to be useful in selecting the best loglinear model and in estimating parameters of interest that reflect uncertainty in the true model. Key words and phrases: Bayes factors, Laplace method, Gibbs sampling, Model selection, Odds ratios. AMS subject classifications: Primary 62H17, 62F15, 62J12. 1 Introduction 1.1 Bayesian testing...
Benefits of a Bayesian Approach for Synthesizing Multiple Sources of Evidence and Uncertainty Linked by a Deterministic Model
, 1993
"... A Bayesian synthesis approach has been proposed by Raftery, Givens, and Zeh (1992) for making inferences from a deterministic model with many inputs and outputs. The approach was applied to population dynamics models for bowhead whales. The approach consists of establishing a joint prior, or pre ..."
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Cited by 7 (6 self)
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A Bayesian synthesis approach has been proposed by Raftery, Givens, and Zeh (1992) for making inferences from a deterministic model with many inputs and outputs. The approach was applied to population dynamics models for bowhead whales. The approach consists of establishing a joint prior, or premodel distribution, on the model inputs and outputs for which there exists evidence independent of the model. The restriction of this distribution to a subspace defined by the model mapping then constitutes a postmodel distribution, from which inferences are drawn. We briefly review a methodology for implementing the Bayesian synthesis approach, and then consider in detail the potential uses of the results and the strengths and weaknesses of the approach compared to past methodologies. 1 Introduction Complex deterministic models reflect scientists' simplified conceptions of natural mechanisms about which they have incomplete understanding. Such a model depends on a set of input para...