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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.
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
Sequential Ordinal Modeling with Applications to Survival Data
 Biometrics
, 2001
"... This paper considers the class of sequential probit models in relation to other models for ordinal data. Hierarchical and other extensions of the model are proposed for applications involving discrete time (grouped) survival data. Computationally practical Markov chain Monte Carlo algorithms are dev ..."
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Cited by 14 (0 self)
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This paper considers the class of sequential probit models in relation to other models for ordinal data. Hierarchical and other extensions of the model are proposed for applications involving discrete time (grouped) survival data. Computationally practical Markov chain Monte Carlo algorithms are developed for the fitting of these models. The ideas and methods are illustrated in detail with a real data example on the length of hospital stay for patients undergoing heart surgery. A notable aspect of this analysis is the comparison, based on marginal likelihoods and training sample priors, of several nonnested models, such as the sequential model, the cumulative ordinal model and Weibull and loglogistic models. Keywords: Bayes factor; Discrete hazard function; Gibbs sampling; Marginal likelihood; MetropolisHastings algorithm; Nonnested models; Sequential probit; Training sample prior; Model comparison. 1 Introduction Ordinal response data is generally analyzed using the cumulative o...
Statistical Techniques for Language Recognition: An Introduction and Guide for Cryptanalysts
 Cryptologia
, 1993
"... We explain how to apply statistical techniques to solve several languagerecognition problems that arise in cryptanalysis and other domains. Language recognition is important in cryptanalysis because, among other applications, an exhaustive key search of any cryptosystem from ciphertext alone requir ..."
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Cited by 11 (2 self)
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We explain how to apply statistical techniques to solve several languagerecognition problems that arise in cryptanalysis and other domains. Language recognition is important in cryptanalysis because, among other applications, an exhaustive key search of any cryptosystem from ciphertext alone requires a test that recognizes valid plaintext. Written for cryptanalysts, this guide should also be helpful to others as an introduction to statistical inference on Markov chains. Modeling language as a finite stationary Markov process, we adapt a statistical model of pattern recognition to language recognition. Within this framework we consider four welldefined languagerecognition problems: 1) recognizing a known language, 2) distinguishing a known language from uniform noise, 3) distinguishing unknown 0thorder noise from unknown 1storder language, and 4) detecting nonuniform unknown language. For the second problem we give a most powerful test based on the NeymanPearson Lemma. For the oth...
Bayesian Assessment of GoodnessofFit against Nonparametric Alternatives
, 2000
"... The classical chisquare test of goodnessoffit compares the hypothesis that data arise from some parametric family of distributions, against the nonparametric alternative that they arise from some other distribution. However, the chisquare test requires continuous data to be grouped into arbitrar ..."
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The classical chisquare test of goodnessoffit compares the hypothesis that data arise from some parametric family of distributions, against the nonparametric alternative that they arise from some other distribution. However, the chisquare test requires continuous data to be grouped into arbitrary categories. Furthermore, as the test is based upon an approximation, it can only be used if there is su#cient data. In practice, these requirements are often wasteful of information and overly restrictive. The authors explore the use of the fractional Bayes factor to obtain a Bayesian alternative to the chisquare test when no specific prior information is available. They consider the extent to which their methodology can handle small data sets and continuous data without arbitrary grouping. R ESUM E Le test classique d'ajustement du khideux confronte l'hypothese que les observations proviennent d'une famille parametrique de lois a l'hypothese non parametrique qu'elles sont issues d'un...
Bayes Factors for Goodness of Fit Testing
, 2003
"... We propose the use of the generalized fractional Bayes factor for testing fit in multinomial models. This is a nonasymptotic method that can be used to quantify the evidence for or against a submodel. We give expressions for the generalized fractional Bayes factor and we study its properties. In p ..."
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We propose the use of the generalized fractional Bayes factor for testing fit in multinomial models. This is a nonasymptotic method that can be used to quantify the evidence for or against a submodel. We give expressions for the generalized fractional Bayes factor and we study its properties. In particular, we show that the generalized fractional Bayes factor has better properties than the fractional Bayes factor. Keywords: generalized fractional Bayes factor, Dirichlet process, BetaStacy process. 1. Introduction. In this paper we propose a Bayesian method for testing fit in multinomial models. Specifically, we will use the Bayes factor for evaluating the evidence for or against a null submodel of the multinomial. The advantages of using a Bayesian approach for this problem are that it does not rely
Summer Term
"... The decision tree is a wellknown methodology for classification and regression. In this dissertation, we focus on the minimization of the misclassification rate for decision tree classifiers. We derive the necessary equations that provide the optimal tree prediction, the estimated risk of the tree’ ..."
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The decision tree is a wellknown methodology for classification and regression. In this dissertation, we focus on the minimization of the misclassification rate for decision tree classifiers. We derive the necessary equations that provide the optimal tree prediction, the estimated risk of the tree’s prediction, and the reliability of the tree’s risk estimation. We carry out an extensive analysis of the application of Lidstone’s law of succession for the estimation of the class probabilities. In contrast to existing research, we not only compute the expected values of the risks but also calculate the corresponding reliability of the risk (measured by standard deviations). We also provide an explicit expression of the knorm estimation for the tree’s misclassification rate that combines both the expected value and the reliability. Furthermore, our proposed and proven theorem on knorm estimation suggests an efficient pruning algorithm that has a clear theoretical interpretation, is easily implemented, and does not require a validation set. Our experiments show that our proposed pruning algorithm produces accurate trees quickly that compares very favorably with two other wellknown pruning algorithms, CCP of CART and EBP of C4.5. Finally, our work provides a