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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
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...
Belief Ratios
, 1105
"... Abstract: We discuss the definition of a Bayes factor, the SavageDickey result, and develop some inequalities relevant to Bayesian inferences. We consider the implications of these inequalities for the Bayes factor approach to hypothesis assessment. An approach to hypothesis assessment based on the ..."
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Abstract: We discuss the definition of a Bayes factor, the SavageDickey result, and develop some inequalities relevant to Bayesian inferences. We consider the implications of these inequalities for the Bayes factor approach to hypothesis assessment. An approach to hypothesis assessment based on the computation of a Bayes factor, a measure of reliability of the Bayes factor, and the point where the Bayes factor is maximized is recommended. This can be seen to deal with many of the issues and controversies associated with hypothesis assessment. It is noted that an inconsistency in prior assignments can arise when priors are placed on hypotheses that do not arise from a parameter of interest. It is recommended that this inconsistency be avoided by choosing a distance measure from the hypothesis as the parameter of interest. An application is made to assessing the goodness of fit for a logistic regression model and it is shown that this leads to resolving some diffi culties associated with assigning priors for this model. Key words and phrases: Bayes factors, relative belief ratios, inequalities, concentration. 1
unknown title
"... Scientists who study cognition infer underlying processes either by observing behavior (e.g., response times, percentage correct) or by observing neural activity (e.g., the BOLD response). These two types of observations have traditionally supported two separate lines of study. The first is led by c ..."
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Scientists who study cognition infer underlying processes either by observing behavior (e.g., response times, percentage correct) or by observing neural activity (e.g., the BOLD response). These two types of observations have traditionally supported two separate lines of study. The first is led by cognitive modelers, who rely on behavior alone to support their computational theories. The second is led by cognitive neuroimagers, who rely on statistical models to link patterns of neural activity to experimental manipulations, often without any attempt to make a direct connection to an explicit computational theory. Here we present a flexible Bayesian framework for combining neural and cognitive models. Joining neuroimaging and computational modeling in a single hierarchical framework allows the neural data to influence the parameters of the cognitive model and allows behavioral data, even in the absence of neural data, to constrain the neural model. Critically, our Bayesian approach can reveal interactions between behavioral and neural parameters, and hence between neural activity and cognitive mechanisms. We demonstrate the utility of our approach with applications to simulated fMRI data with a recognition model and to diffusionweighted imaging data
Unimodal contaminations in testing point null hypothesis
"... The problem of testing a point null hypothesis from the Bayesian perspective is considered. The uncertainties are modelled through use of ε–contamination class with the class of contaminations including: i) All unimodal distributions and ii) All unimodal and symmetric distributions. Over these class ..."
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The problem of testing a point null hypothesis from the Bayesian perspective is considered. The uncertainties are modelled through use of ε–contamination class with the class of contaminations including: i) All unimodal distributions and ii) All unimodal and symmetric distributions. Over these classes, the infimum of the posterior probability of the point null hypothesis is computed and compared with the p–value and a better approach than the one known is obtained. Key Words: p–values. ε–contaminated class, point null hypothesis, posterior probability, AMS subject classification: 62F15, 62A15 Resumen Contaminaciones unimodales en el contraste de hipótesis nula puntual Se considera el problema del contraste de hipótesis nula puntual desde el punto de vista Bayesiano. La incertidumbre se modeliza mediante el uso de la clase de las distribuciones ε–contaminadas, cuando la clase de las contaminaciones incluye: i) todas las distribuciones unimodales y ii) todas las distribuciones unimodales y simétricas. Se calcula el ínfimo de las probabilidades a posteriori de la hipótesis nula
NeuroImage 72 (2013) 193–206 Contents lists available at SciVerse ScienceDirect
"... journal homepage: www.elsevier.com/locate/ynimg ..."
1 HYPOTHESIS ASSESSMENT USING THE BAYES FACTOR AND RELATIVE BELIEF RATIO
"... www.utstat.toronto.edu/mikevans/ The Bayes factor is commonly used for assessing the evidence for or against a given hypothesis H0: θ ∈ Θ0, where Θ0 is a subset of the parameter space. In this paper we discuss the Bayes factor and various issues associated with its use. A Bayes factor is seen to be ..."
Abstract
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www.utstat.toronto.edu/mikevans/ The Bayes factor is commonly used for assessing the evidence for or against a given hypothesis H0: θ ∈ Θ0, where Θ0 is a subset of the parameter space. In this paper we discuss the Bayes factor and various issues associated with its use. A Bayes factor is seen to be intimately connected with a relative belief ratio which provides a somewhat simpler approach to assessing the evidence in favor of H0. It is noted that, when there is a parameter of interest generating H0, then a Bayes factor for H0 can be defined as a limit and there is no need to introduce a discrete prior mass for Θ0 or a prior within Θ0. It is further noted that when a prior on Θ0 does not correspond to a conditional prior induced by a parameter of interest generating H0, then there is an inconsistency in prior assignments. This inconsistency can be avoided by choosing a parameter of interest that generates the hypothesis. A natural choice of a parameter of interest is given by a measure of distance of the model parameter from Θ0. This leads to a Bayes factor for H0 that is comparing the concentration of the posterior about Θ0 with the concentration of the prior about Θ0. The issue of calibrating a Bayes factor is also discussed and is seen to be equivalent to computing a posterior probability that measures the reliability of the evidence provided by the Bayes factor.