Posterior Bayes Factors

Aitkin, Murray

doi:10.1111/j.2517-6161.1991.tb01812.x

Cited by 265 publications

(258 citation statements)

References 23 publications

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“…Difficulties with this approach are well known and are not repeated here, except for the insoluble difficulty from using diffuse priors on unbounded parameter spaces, when the value of B is undefined. Aitkin (1991) has an extensive discussion of this issue.…”

Section: Lff = I Lj(oj)trj(oj)dojmentioning

confidence: 99%

“…The new general approach described in Aitkin (1991) is to replace integration with respect to the prior by integration with respect to the posterior, giving the 'posterior mean' of the likelihood with L~ = s Lj(Oj)~j(Oj l y')dOj / The 'posterior Bayes factor' is then A Li4 and is interpreted in the same way as B: values of 10 ~ l0 -l and l0 -2 represent no, mild, and strong evidence against MI in favour of M2.…”

Section: Lff = I Lj(oj)trj(oj)dojmentioning

confidence: 99%

“…For the posterior density in p, we integrate L with respect to the prior in 7 and a; for subsequent use we give a more general result (Aitkin 1991) Ye and Berger (1991).…”

Section: Bayes Analysismentioning

confidence: 99%

“…In a recent paper (Aitkin 1991), I developed a general likelihood inferential framework for arbitrary model comparisons problems, including problems of inference about a single parameter in the presence of nuisance parameters. This may be applied directly to the exponential regression model, and leads to a very simple integrated likelihood for p which is effectively the profile likelihood.…”

Section: Introductionmentioning

confidence: 99%

“…Computations are particularly simple, requiring only standard regression model fitting. Section 6 briefly considers hypothesis testing and interval construction for p using the general theory of Aitkin (1991). Section 7 gives brief comments on the Bayes and likelihood approaches.…”

Section: Introductionmentioning

confidence: 99%

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Posterior Bayes factor analysis for an exponential regression model

Aitkin

1993

Stat Comput

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In the exponential regression model, Bayesian inference concerning the non-linear regression parameter p has proved extremely difficult. In particular, standard improper diffuse priors for the usual parameters lead to an improper posterior for the non-linear regression parameter. In a recent paper Ye and Berger (1991) applied the reference prior approach of Bernardo (1979) and Berger and Bernardo (1989) yielding a proper informative prior for p. This prior depends on the values of the explanatory variable, goes to 0 as p goes to 1, and depends on the specification of a hierarchical ordering of importance of the parameters.This paper explains the failure of the uniform prior to give a proper posterior: the reason is the appearance of the determinant of the information matrix in the posterior density for p. We apply the posterior Bayes factor approach of Aitkin ( 1991) to this problem; in this approach we integrate out nuisance parameters with respect to their conditional posterior density given the parameter of interest. The resulting integrated likelihood for p requires only the standard diffuse prior for all the parameters, and is unaffected by orderings of importance of the parameters. Computation of the likelihood for p is extremely simple. The approach is applied to the three examples discussed by Berger and Ye and the likelihoods compared with their posterior densities.

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Section: Lff = I Lj(oj)trj(oj)dojmentioning

confidence: 99%

Section: Lff = I Lj(oj)trj(oj)dojmentioning

confidence: 99%

“…For the posterior density in p, we integrate L with respect to the prior in 7 and a; for subsequent use we give a more general result (Aitkin 1991) Ye and Berger (1991).…”

Section: Bayes Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Posterior Bayes factor analysis for an exponential regression model

Aitkin

1993

Stat Comput

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Bayesian Hypothesis Testing

Press¹

2002

Subjective and Objective Bayesian Statistics

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In this chapter we discuss Bayesian hypothesis testing. We begin with some historical background regarding how hypothesis testing has been treated in science in the past, and show how the Bayesian approach to the subject has really provided the statistical basis for its development. We then discuss some of the problems that have plagued fiequentist methods of hypothesis testing during the twentieth century. We will treat two Bayesian approaches to the subject:1. The vague prior approach of Lindley (which is somewhat limited but easy to implement); and 2. The very general approach of Jefkys, which is the current, commonly accepted Bayesian niethod of hypothesis testing, although it is somewhat more complicated to carry out. A BRIEF HISTORY OF SCIENTIFIC HYPOTHESIS TESTINGThere is considerable evidence of ad hoc tests of hypotheses that were developed to serve particular applications (especially in astronomy), as science has developed. But there had been no underlying theory that could serve as the basis for generating appropriate tests in general until Bayes' theorem was expounded. Moreover, researchers had difficulty applying the theorem even when they wanted to use it. Karl Pearson (1 892), initiated the development of a formal theory of hypothesis testing with his development of chi-squared testing for multinomial proportions. Me liked the idea of applying Bayes' theorem to test hypotheses, but he could not quite figure out how to generate prior distributions to support the Bayesian approach. Moreover, he did not recognize that corwideration of one or more! alternative hypotheses might be relevant for testing a basic scientific hypothesis.

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Bayesian Model Selection

George¹

2014

Wiley StatsRef: Statistics Reference Online

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Posterior Bayes Factors

Cited by 265 publications

References 23 publications

Posterior Bayes factor analysis for an exponential regression model

Posterior Bayes factor analysis for an exponential regression model

Bayesian Hypothesis Testing

Bayesian Model Selection

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