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16
Estimating the integrated likelihood via posterior simulation using the harmonic mean identity
 Bayesian Statistics
, 2007
"... The integrated likelihood (also called the marginal likelihood or the normalizing constant) is a central quantity in Bayesian model selection and model averaging. It is defined as the integral over the parameter space of the likelihood times the prior density. The Bayes factor for model comparison a ..."
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The integrated likelihood (also called the marginal likelihood or the normalizing constant) is a central quantity in Bayesian model selection and model averaging. It is defined as the integral over the parameter space of the likelihood times the prior density. The Bayes factor for model comparison and Bayesian testing is a ratio of integrated likelihoods, and the model weights in Bayesian model averaging are proportional to the integrated likelihoods. We consider the estimation of the integrated likelihood from posterior simulation output, aiming at a generic method that uses only the likelihoods from the posterior simulation iterations. The key is the harmonic mean identity, which says that the reciprocal of the integrated likelihood is equal to the posterior harmonic mean of the likelihood. The simplest estimator based on the identity is thus the harmonic mean of the likelihoods. While this is an unbiased and simulationconsistent estimator, its reciprocal can have infinite variance and so it is unstable in general. We describe two methods for stabilizing the harmonic mean estimator. In the first one, the parameter space is reduced in such a way that the modified estimator involves a harmonic mean of heaviertailed densities, thus resulting in a finite variance estimator. The resulting
On predictive probability matching priors
, 805
"... Abstract: We revisit the question of priors that achieve approximate matching of Bayesian and frequentist predictive probabilities. Such priors may be thought of as providing frequentist calibration of Bayesian prediction or simply ..."
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Abstract: We revisit the question of priors that achieve approximate matching of Bayesian and frequentist predictive probabilities. Such priors may be thought of as providing frequentist calibration of Bayesian prediction or simply
Recent Developments in Bootstrap Methodology
"... Abstract. Ever since its introduction, the bootstrap has provided both a powerful set of solutions for practical statisticians, and a rich source of theoretical and methodological problems for statistics. In this article, some recent developments in bootstrap methodology are reviewed and discussed. ..."
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Abstract. Ever since its introduction, the bootstrap has provided both a powerful set of solutions for practical statisticians, and a rich source of theoretical and methodological problems for statistics. In this article, some recent developments in bootstrap methodology are reviewed and discussed. After a brief introduction to the bootstrap, we consider the following topics at varying levels of detail: the use of bootstrapping for highly accurate parametric inference; theoretical properties of nonparametric bootstrapping with unequal probabilities; subsampling and the m out of n bootstrap; bootstrap failures and remedies for superefficient estimators; recent topics in significance testing; bootstrap improvements of unstable classifiers and resampling for dependent data. The treatment is telegraphic rather than exhaustive. Key words and phrases: Bagging, bootstrap, conditional inference, empirical strength probability, parametric bootstrap, subsampling, superefficient
On A Property Of Probability Matching Priors: Matching The Alternative Coverage Probabilities
"... this paper how far a prior satisfying (1.1) also matches P ` f` 1 + ..."
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this paper how far a prior satisfying (1.1) also matches P ` f` 1 +
Simple and Accurate OneSided Inference From Signed Roots of Likelihood Ratios
, 2000
"... The authors propose two methods based on the signed root of the likelihood ratio statistic for onesided testing of a simple null hypothesis about a scalar parameter in the presence of nuisance parameters. Both methods are thirdorder accurate and utilise simulation to avoid the need for onerous ana ..."
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The authors propose two methods based on the signed root of the likelihood ratio statistic for onesided testing of a simple null hypothesis about a scalar parameter in the presence of nuisance parameters. Both methods are thirdorder accurate and utilise simulation to avoid the need for onerous analytical calculations characteristic of competing saddlepoint procedures. Moreover, the new methods do not require specification of ancillary statistics. The methods respect the conditioning associated with similar tests up to an error of third order, and conditioning on ancillary statistics to an error of second order. R ESUM E Les auteurs proposent deux methodes permettant, a partir de la racine signee du rapport des vraisemblances, d'e#ectuer un test unilateral d'une hypothese nulle simple sur un parametre d'echelle, en presence de parametres nuisibles. Par le biais de simulations, ces methodes permettent d'obtenir une precision du troisieme ordre tout en evitant les calculs analytique...
Comparison Of Test Statistics Via Expected Lengths Of Associated Confidence Intervals
"... With reference to a large class of test statistics, higher order asymptotics on expected lengths of associated confidence intervals are investigated in a possibly noniid setting. The connection with Bartlett adjustability is also indicated. ..."
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With reference to a large class of test statistics, higher order asymptotics on expected lengths of associated confidence intervals are investigated in a possibly noniid setting. The connection with Bartlett adjustability is also indicated.
Asymptotic Expansions
, 1996
"... (1) The sequence fb kn g = f1; b 1n ; b 2n ; . . .g determines the asymptotic behavior of the expansion: in particular how the reexpression approximates the original function. Usual choices of fb kn g are f1; n \Gamma1=2 ; n \Gamma1 ; . . .g or f1; n \Gamma1 ; n \Gamma2 ; . . .g; in any cas ..."
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(1) The sequence fb kn g = f1; b 1n ; b 2n ; . . .g determines the asymptotic behavior of the expansion: in particular how the reexpression approximates the original function. Usual choices of fb kn g are f1; n \Gamma1=2 ; n \Gamma1 ; . . .g or f1; n \Gamma1 ; n \Gamma2 ; . . .g; in any case it is required that b kn = o(b k\Gamma1;n ) as n !1. For sequences of constants fa n g, fb n g, we write a n =<F13.5
Asymptotic Expansion of Null Distribution of Likelihood Ratio Statistic in Multiparameter Exponential Family to an Arbitrary Order
"... Consider likelihood ratio test of a simple null hypothesis in a multiparameter exponential family.We study the asymptotic expansion of the null distribution of log likelihood ratio statistic to an arbitrary order. Bartlett correctability of the O(n 01 )termiswell known. We show that higher order te ..."
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Consider likelihood ratio test of a simple null hypothesis in a multiparameter exponential family.We study the asymptotic expansion of the null distribution of log likelihood ratio statistic to an arbitrary order. Bartlett correctability of the O(n 01 )termiswell known. We show that higher order terms exhibit a similar simplicity. Moreover we give a combinatorially explicit expression for all terms of the asymptotic expansion of the characteristic function of log likelihood ratio statistic.