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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 1769 (74 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 121 (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 33 (2 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 13 (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...
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
and
, 2001
"... SUMMARY. This paper considers the class of sequential ordinal models in relation to other models for ordinal response data. Markov chain Monte Carlo (MCMC) algorithms, based on the approach of Albert and Chib (1993, Journal of the American Statistical Association 88, 669479), are developed for the f ..."
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SUMMARY. This paper considers the class of sequential ordinal models in relation to other models for ordinal response data. Markov chain Monte Carlo (MCMC) algorithms, based on the approach of Albert and Chib (1993, Journal of the American Statistical Association 88, 669479), 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.
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.
ESTIMATION OF THE NUMBER OF SPECIES FROM A RANDOM
"... Dedicated to the memory of Thyagaraju Chelluri, a wonderful human being who would have become a fine mathematician had his life not been cut tragically short. Abstract. We consider the classical problem of estimating T, the total number of species in a population, from repeated counts in a simple ra ..."
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Dedicated to the memory of Thyagaraju Chelluri, a wonderful human being who would have become a fine mathematician had his life not been cut tragically short. Abstract. We consider the classical problem of estimating T, the total number of species in a population, from repeated counts in a simple random sample and propose a new algorithm for treating it. In order to produce an estimator T ̂ we actually start from the estimation of a related quantity, the unobserved probability U. In fact, we first show that an estimation of T can be obtained by requiring compatibility between the Laplace addone (or addλ) estimator and the TuringGood estimator ÛTG of U; the estimators obtained in this way concide with those of ChaoLee and of HorvitzThompson, depending on λ. On the other hand, since the Laplace formula can be derived as the mean of a Bayesian posterior with a uniform (or Dirichlet) prior, we later modify the structure of the likelihood and, by requiring the compatibility of the new posterior with ÛTG, determine a modified Bayesian estimator T ̂ ′. The form of T ̂ ′ can be again related to that of ChaoLee, but provides a better justified term for their estimated variance. T ̂ ′ appears to be extremely effective in estimating T, for instance improving upon all existing estimators for the standard fully explicit Carothers data. In addition, we can derive estimations of the population distribution, confidence intervals for U and confidence intervals for T; these last appear to be the first in the literature not based on resampling. 1 2Keywords and phrases: simple random sample, unobserved species, unobserved probability, point estimator, confidence interval, Dirichlet prior, Bayesian posterior. 1.