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Reference Priors (reference + prior)

Distribution by Scientific Domains
Mathematics and Statistics50%

Selected Abstracts

Reference and probability-matching priors in Bayesian analysis of mixed linear models

JOURNAL OF ANIMAL BREEDING AND GENETICS, Issue 5 2002
A. L. Pretorius
Summary Determination of reasonable non-informative priors in multiparameter problems is not easy; common non-informative priors, such as Jeffrey's prior, can have features that have an unexpectedly dramatic effect on the posterior. In recognition of this problem Berger and Bernardo (Bayesian Statistics IV. Oxford University Press, Oxford, UK, pp. 35,70, 1992), proposed the Reference Prior approach to the development of non-informative priors. In the present paper the reference priors of Berger and Bernardo (1992) are derived for the mixed linear model. In spite of these difficulties, there is growing evidence, mainly through examples that reference priors provide ,sensible' answers from a Bayesian point of view. We also examine whether the reference priors satisfy the probability-matching criterion. The theory and results are applied to a real problem consisting of 879 weaning weight records, from the progeny of 17 sires. These important aspects are explored via Monte Carlo simulations. Zusammenfassung Reference und Probability-Matching Priors in Bayesian Analysen von gemischten linearen Modellen Die Festlegung von vernünftigen Non-Informative Priors in Multi-Paramter Analysen ist nicht leicht; gewöhnliche Non-Informative Priors, wie beispielsweise Jeffrey's Priors, können unerwartete dramatische Effekte auf die Lösungen haben. Zur Lösung dieses Problems schlagen Berger and Bernardo (Bayesian Statistics IV. Oxford University Press, Oxford, UK, pp. 35,70, 1992) den Reference Prior Ansatz zur Entwicklung von Non-Informative Priors vor. In der vorliegenden Untersuchung werden die von Berger and Bernardo (1992) vorgeschlagenen Reference Priors in linearen gemischten Modellen angewandt. Trotz der bekannten Schwierigkeiten gibt es hauptsächlich anhand von Beispielen mehr und mehr Anhaltspunkte, dass die Reference Priors aus der Sicht von Bayesians zu brauchbaren Antworten führen. Es wurde auch untersucht, ob diese Reference Priors den Probability-Matching Kriterien genügen. Die Theorie und die Ergebnisse sind an einem Beispiel mit 879 Datensätzen mit Absatzgewichten von Schafen, die von 17 Böcken abstammen, verifiziert worden. Die wichtigen Aspekte sind mittels Monte Carlo Simulation untersucht worden. [source]

Bayesian Hypothesis Testing: a Reference Approach

INTERNATIONAL STATISTICAL REVIEW, Issue 3 2002
José M. Bernardo
Summary For any probability model M={p(x|,, ,), ,,,, ,,,} assumed to describe the probabilistic behaviour of data x,X, it is argued that testing whether or not the available data are compatible with the hypothesis H0={,=,0} is best considered as a formal decision problem on whether to use (a0), or not to use (a0), the simpler probability model (or null model) M0={p(x|,0, ,), ,,,}, where the loss difference L(a0, ,, ,) ,L(a0, ,, ,) is proportional to the amount of information ,(,0, ,), which would be lost if the simplified model M0 were used as a proxy for the assumed model M. For any prior distribution ,(,, ,), the appropriate normative solution is obtained by rejecting the null model M0 whenever the corresponding posterior expectation ,,,(,0, ,, ,),(,, ,|x)d,d, is sufficiently large. Specification of a subjective prior is always difficult, and often polemical, in scientific communication. Information theory may be used to specify a prior, the reference prior, which only depends on the assumed model M, and mathematically describes a situation where no prior information is available about the quantity of interest. The reference posterior expectation, d(,0, x) =,,,(,|x)d,, of the amount of information ,(,0, ,, ,) which could be lost if the null model were used, provides an attractive nonnegative test function, the intrinsic statistic, which is invariant under reparametrization. The intrinsic statistic d(,0, x) is measured in units of information, and it is easily calibrated (for any sample size and any dimensionality) in terms of some average log-likelihood ratios. The corresponding Bayes decision rule, the Bayesian reference criterion (BRC), indicates that the null model M0 should only be rejected if the posterior expected loss of information from using the simplified model M0 is too large or, equivalently, if the associated expected average log-likelihood ratio is large enough. The BRC criterion provides a general reference Bayesian solution to hypothesis testing which does not assume a probability mass concentrated on M0 and, hence, it is immune to Lindley's paradox. The theory is illustrated within the context of multivariate normal data, where it is shown to avoid Rao's paradox on the inconsistency between univariate and multivariate frequentist hypothesis testing. Résumé Pour un modèle probabiliste M={p(x|,, ,) ,,,, ,,,} censé décrire le comportement probabiliste de données x,X, nous soutenons que tester si les données sont compatibles avec une hypothèse H0={,=,0 doit être considéré comme un problème décisionnel concernant l'usage du modèle M0={p(x|,0, ,) ,,,}, avec une fonction de coût qui mesure la quantité d'information qui peut être perdue si le modèle simplifiéM0 est utilisé comme approximation du véritable modèle M. Le coût moyen, calculé par rapport à une loi a priori de référence idoine fournit une statistique de test pertinente, la statistique intrinsèque d(,0, x), invariante par reparamétrisation. La statistique intrinsèque d(,0, x) est mesurée en unités d'information, et sa calibrage, qui est independante de la taille de léchantillon et de la dimension du paramètre, ne dépend pas de sa distribution à l'échantillonage. La règle de Bayes correspondante, le critère de Bayes de référence (BRC), indique que H0 doit seulement êetre rejeté si le coût a posteriori moyen de la perte d'information à utiliser le modèle simplifiéM0 est trop grande. Le critère BRC fournit une solution bayésienne générale et objective pour les tests d'hypothèses précises qui ne réclame pas une masse de Dirac concentrée sur M0. Par conséquent, elle échappe au paradoxe de Lindley. Cette théorie est illustrée dans le contexte de variables normales multivariées, et on montre qu'elle évite le paradoxe de Rao sur l'inconsistence existant entre tests univariés et multivariés. [source]

Reference and probability-matching priors in Bayesian analysis of mixed linear models

JOURNAL OF ANIMAL BREEDING AND GENETICS, Issue 5 2002
A. L. Pretorius
Summary Determination of reasonable non-informative priors in multiparameter problems is not easy; common non-informative priors, such as Jeffrey's prior, can have features that have an unexpectedly dramatic effect on the posterior. In recognition of this problem Berger and Bernardo (Bayesian Statistics IV. Oxford University Press, Oxford, UK, pp. 35,70, 1992), proposed the Reference Prior approach to the development of non-informative priors. In the present paper the reference priors of Berger and Bernardo (1992) are derived for the mixed linear model. In spite of these difficulties, there is growing evidence, mainly through examples that reference priors provide ,sensible' answers from a Bayesian point of view. We also examine whether the reference priors satisfy the probability-matching criterion. The theory and results are applied to a real problem consisting of 879 weaning weight records, from the progeny of 17 sires. These important aspects are explored via Monte Carlo simulations. Zusammenfassung Reference und Probability-Matching Priors in Bayesian Analysen von gemischten linearen Modellen Die Festlegung von vernünftigen Non-Informative Priors in Multi-Paramter Analysen ist nicht leicht; gewöhnliche Non-Informative Priors, wie beispielsweise Jeffrey's Priors, können unerwartete dramatische Effekte auf die Lösungen haben. Zur Lösung dieses Problems schlagen Berger and Bernardo (Bayesian Statistics IV. Oxford University Press, Oxford, UK, pp. 35,70, 1992) den Reference Prior Ansatz zur Entwicklung von Non-Informative Priors vor. In der vorliegenden Untersuchung werden die von Berger and Bernardo (1992) vorgeschlagenen Reference Priors in linearen gemischten Modellen angewandt. Trotz der bekannten Schwierigkeiten gibt es hauptsächlich anhand von Beispielen mehr und mehr Anhaltspunkte, dass die Reference Priors aus der Sicht von Bayesians zu brauchbaren Antworten führen. Es wurde auch untersucht, ob diese Reference Priors den Probability-Matching Kriterien genügen. Die Theorie und die Ergebnisse sind an einem Beispiel mit 879 Datensätzen mit Absatzgewichten von Schafen, die von 17 Böcken abstammen, verifiziert worden. Die wichtigen Aspekte sind mittels Monte Carlo Simulation untersucht worden. [source]

Default Bayesian Priors for Regression Models with First-Order Autoregressive Residuals

JOURNAL OF TIME SERIES ANALYSIS, Issue 3 2003
Malay Ghosh
Abstract. The objective of this paper is to develop default priors when the parameter of interest is the autocorrelation coefficient in normal regression models with first-order autoregressive residuals. Jeffreys' prior as well as reference priors are found. These priors are compared in the light of how accurately the coverage probabilities of Bayesian credible intervals match the corresponding frequentist coverage probabilities. It is found that the reference priors have a definite edge over Jeffreys' prior in this respect. Also, the credible intervals based on these reference priors seem superior to similar intervals based on certain divergence measures. [source]