Probability Matching Priors Research Articles

Probability matching priors for predicting unobservable random effects with application to ANOVA models

We characterize priors that ensure approximate frequentist validity of posterior quantiles of unobservable random effects. Application to analysis of variance (ANOVA) models is explored.

Noninformative priors, credible sets and Bayesian hypothesis testing for the intraclass model

The objective of this paper is twofold. The first is to find a Bayesian credible interval for the intraclass correlation coefficient in symmetric normal models based on some “default” or “noninformative” priors. Probability matching priors and reference priors along with Jeffreys’ prior are considered, and the one-at-a-time reference prior emerges as the “optimal” according to several criteria. The second objective is to compare two nested models such as the intraclass and independence models using the distance or divergence between the two as the basis of comparison. A suitable criterion for this is the “power divergence measure” as introduced by Cressie and Read (J. Roy. Statist. Soc. Ser. B 46 (1984) 440). Such a measure includes the two Kullback-Leibler divergence measures and the Hellinger divergence measure as special cases. For the specific problem, the power divergence measure turns out to be a function solely of ρ, the intraclass correlation coefficient. Also, this function is increasing for ρ>0, is decreasing for ρ<0, and the minimum is attained at ρ=0. Thus the model comparison problem in this case amounts to testing the hypothesis H 0 : ρ=0 . Due to the duality between hypothesis tests and set estimation, the hypothesis testing problem can also be solved by solving a corresponding set estimation problem. The present paper develops Bayesian methods based on the Kullback-Leibler and Hellinger divergence measures, rejecting H 0 : ρ=0 when the specified divergence measure exceeds some number d. This number d is so chosen that the resulting credible interval for the divergence measure has specified coverage probability 1− α. The length of such an interval is compared with the (i) equal two-tailed credible interval and (ii) the HPD credible interval for ρ with the same coverage probability which can also be inverted into acceptance regions of H 0 : ρ=0 . An example is considered where the HPD interval based on the one-at-a-time reference prior turns out to be the shortest credible interval having the same coverage probability.

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

SummaryDetermination 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.

Second-order probability matching priors for a parametric function with application to Bayesian tolerance limits

Conditions are obtained for the approximate frequentist validity of the posterior quantiles of any smooth parametric function. An application to Bayesian tolerance limits is indicated.

Some New Results on Probability Matching Priors

Some New Results on Probability Matching Priors

Some new Results on Probability Matching Priors

The paper has three components. First, for a realvalued parameter of interest orthogonal (Cox and Reid, 1987) to the nuisance parameter vector, we find a necessary and sufficient condition for the equivalence of second order quantile matching priors and highest posterior density regions matching priors within the class of first order quantile matching priors. Examples are presented to illustrate the result. Second, we develop a quantile matching prior in a normal hierarchical Bayesian model. This prior turns out to be different from the one proposed earlier by Morris (1983). Third, we obtain an exact matching result when the objective is prediction of a real-valued random variable from a location family of distributions. AMS (2000) Subject Classification: 62F15, 62F25, 62E20

Noninformative Priors for Multivariate Linear Calibration

This paper derives a class of first order probability matching priors and a complete catalog of the reference priors for the general multivariate linear calibration problem. In an important special case, a complete characterization of first order probability matching priors is given, and a fairly general class of second order probability matching priors is also provided. Orthogonal transformations (1987, D. R. Cox and N. Reid, J. Roy. Statist. Soc. Ser. B49, 1–18) are found to facilitate the derivations. It turns out that under orthogonal parameterization, reference priors (including Jeffreys' prior) are first order probability matching priors for unidimensional multivariate linear calibration. Also, in univariate linear calibration, the prior of W. G. Hunter and W. F. Lamboy (1981, Technometrics23, 323–350) is a second order probability matching prior.

Miscellanea. Probability matching priors for non-regular cases

We study probability matching priors in non-regular cases. Both one-parameter and multi-parameter cases are considered. The resulting priors are shown to satisfy certain differential equations. Several examples are discussed. It is observed that the reference prior in the sense of Bernardo is in many cases also probability matching. As a necessary tool for finding such priors, asymptotic expansion of posterior distributions for multi-parameter non-regular cases is also obtained.

On a property of probability matching priors: matching the alternative coverage probabilities

SUMMARY In frequentist inference based on confidence sets, both the true coverage and the probability for a confidence set to include an alternative value of the parameter of interest are important. Thus if probability matching priors also match such alternative coverage probabilites there is perhaps a stronger justification for calling them noninformative. Considering contiguous alternatives, we obtain the relevant necessary and sufficient conditions. In particular, with n denoting the sample size, it is seen that probability matching priors up to o(n-2) also match the alternative coverage probabilities up to that order while this is not necessarily the case with probability matching priors up to o(n-').

On the property of posteriors for dispersion models

Jorgensen (The Theory of Exponential Dispersion Models and Analysis of Deviance, Instituto de Mathematica Pura e Aplicada, Rio de Janerio (1992)) introduced a general class of probability models, and named the same as dispersion models. Garvan and Ghosh (1997, Biometrika, 84, 976-982) developed first- and second-order probability matching priors for two parameter dispersion models. We examine in this paper the propriety of posteriors under these priors, and other noninformative priors including Jeffreys' and reference priors. Also, we illustrate the Bayesian methods with real and simulated data.

Probability matching priors: a review

In recent years, extensive work has been done concerning the derivation of noninformative prior distributions assuring approximate frequentist validity of Bayesian inferences. This paper provides a review of matching priors obtained via quantiles andvia the distribution function. Various matching criteria are described and discussed. Emphasis is laid on a proposal of designing priors matching the true coverage probability as well as the false coverage probabilities of contiguous alternatives with the respective Bayesian counterparts. The review is not primarily meant to be a comprehensive account on the area, but, rather, to convey the main underlying ideas and point out the relationships between matching priors and other noninformative priors, such as the Jeifreys’ and the reference priors.

Miscellanea. Noninformative priors for dispersion models

SUMMARY The present paper provides a Bayesian analysis of two-parameter dispersion models under noninformative priors. The focus is on first- and second-order probability matching priors. Other noninformative priors, including Jeffreys' and reference priors, are examined in the light of these criteria. For certain models some unique second-order probability matching priors are found which are different from either Jeffreys' or reference priors.

Miscellanea. Second-order probability matching priors

SUMMARY The paper considers priors obtained by ensuring approximate frequentist validity of (a) posterior quantiles, and of (b) the posterior distribution function. It is seen that, at the second order of approximation, the two approaches do not necessarily lead to identical conclusions. Examples are given to illustrate this. The role of invariance in the context of probability matching is also discussed.

A Note on the Probability Difference Between Matching Priors Based on Posterior Quantiles and on Inversion of Conditional Likelihood Ratio Statistics

A necessary and sufficient condition is given so that the second order probability matching priors based on posterior quantiles will be the priors under which the Bayesian and frequentist Bartlett corrected conditional likelihood ratio statistics differ by 0(l). Several examples are given to illustrate the idea.

On priors providing frequentist validity of Bayesian inference for multiple parametric functions

SUMMARY We characterise priors which match, up to O(n- 1), the posterior joint cumulative distribution function of multiple parametric functions with the corresponding frequentist cumulative distribution function. This work extends and unifies the work of Ghosh & Mukerjee (1993) and Datta & Ghosh (1995a) on the topic of probability-matching priors. A set of necessary and sufficient conditions is obtained for the above characterisation. Some of these conditions depend only on the parametric functions and not on the prior. Examples are given where the joint probability matching is possible and where it is not possible.