Extensions of the conjugate prior through the Kullback-Leibler separators

The conjugate prior for the exponential family, referred to also as the natural conjugate prior, is represented in terms of the Kullback-Leibler separator. This representation permits us to extend the conjugate prior to that for a general family of sampling distributions. Further, by replacing the Kullback-Leibler separator with its dual form, we define another form of a prior, which will be called the mean conjugate prior. Various results on duality between the two conjugate priors are shown. Implications of this approach include richer families of prior distributions induced by a sampling distribution and the empirical Bayes estimation of a high-dimensional mean parameter.     

Posterior mode estimation for the generalized linear model

Posterior mode estimators are proposed, which arise from simply expressed prior opinion about expected outcomes, roughly as follows: a conjugate family of prior distributions is determined by a given variance function. Using a conjugate prior, a posterior mode estimator and its estimated (co-)variances are obtained through conventional maximum likelihood computations, by means of small alterations to the observed outcomes and/or to the modelled variance function. Within the conjugate family, for purposes of inference about the regression vector, a reference prior is proposed for a given choice of linear design of the canonical link. The resulting approximate reference inferences approximate the Bayesian inferences which arise from a minimally informative reference prior. A set of subjective prior upper and lower percentage points for the expected outcomes can be used to determine a conjugate family member. Alternatively, a set of subjective prior means and standard deviations determines a member. The subfamily of priors determinable by percentage points either includes or approximates the proposed reference prior.The research of the first-named author was funded in part by a Natural Sciences and Engineering Research Council grant.The second named author gratefully acknowledges the support of the National Science Foundation, grant #DMS-8901494 and of the Kansas Geological Survey where he visited during the term of the majority of this research.     

Weakly Informative Reparameterizations for Location-Scale Mixtures

While mixtures of Gaussian distributions have been studied for more than a century, the construction of a reference Bayesian analysis of those models remains unsolved, with a general prohibition of improper priors due to the ill-posed nature of such statistical objects. This difficulty is usually bypassed by an empirical Bayes resolution. By creating a new parameterization centered on the mean and possibly the variance of the mixture distribution itself, we manage to develop here a weakly informative prior for a wide class of mixtures with an arbitrary number of components. We demonstrate that some posterior distributions associated with this prior and a minimal sample size are proper. We provide Markov chain Monte Carlo (MCMC) implementations that exhibit the expected exchangeability. We only study here the univariate case, the extension to multivariate location-scale mixtures being currently under study. An R package called Ultimixt is associated with this article. Supplementary material for this article is available online.     

Bayesian analysis of finite mixture models of distributions from exponential families

This paper deals with the Bayesian analysis of finite mixture models with a fixed number of component distributions from natural exponential families with quadratic variance function (NEF-QVF). A unified Bayesian framework addressing the two main difficulties in this context is presented, i.e., the prior distribution choice and the parameter unidentifiability problem. In order to deal with the first issue, conjugate prior distributions are used. An algorithm to calculate the parameters in the prior distribution to obtain the least informative one into the class of conjugate distributions is developed. Regarding the second issue, a general algorithm to solve the label-switching problem is presented. These techniques are easily applied in practice as it is shown with an illustrative example.     

Bayesian inference in spherical linear models: robustness and conjugate analysis

The early work of Zellner on the multivariate Student-t linear model has been extended to Bayesian inference for linear models with dependent non-normal error terms, particularly through various papers by Osiewalski, Steel and coworkers. This article provides a full Bayesian analysis for a spherical linear model. The density generator of the spherical distribution is here allowed to depend both on the precision parameter φ and on the regression coefficients β. Another distinctive aspect of this paper is that proper priors for the precision parameter are discussed.The normal-chi-squared family of prior distributions is extended to a new class, which allows the posterior analysis to be carried out analytically. On the other hand, a direct joint modelling of the data vector and of the parameters leads to conjugate distributions for the regression and the precision parameters, both individually and jointly. It is shown that some model specifications lead to Bayes estimators that do not depend on the choice of the density generator, in agreement with previous results obtained in the literature under different assumptions. Finally, the distribution theory developed to tackle the main problem is useful on its own right.     

Posterior Simulation with Priors Specified on Functionals

Abstract

Many Bayesian analyses use Markov chain Monte Carlo (MCMC) techniques. MCMC techniques work fastest (per iteration) when the prior distribution of the parameters is chosen conveniently, such as a conjugate prior. However, this is sometimes at odds with the prior desired by the investigator. We describe two motivating examples where nonconjugate priors are preferred. One is a Dirichlet process where it is difficult to implement alternative, nonconjugate priors. We develop a method that allows computation to be done with a convenient prior but adjusts the equilibrium distribution of the Markov chain to be the posterior distribution from the desired prior. In addition to allowing more freedom in choosing prior distributions, the method enables the investigator to perform quick sensitivity analyses, even in nonparametric settings.     

Power-Conditional-Expected Priors: Using g-Priors With Random Imaginary Data for Variable Selection

The Zellner's g-prior and its recent hierarchical extensions are the most popular default prior choices in the Bayesian variable selection context. These prior setups can be expressed as power-priors with fixed set of imaginary data. In this article, we borrow ideas from the power-expected-posterior (PEP) priors to introduce, under the g-prior approach, an extra hierarchical level that accounts for the imaginary data uncertainty. For normal regression variable selection problems, the resulting power-conditional-expected-posterior (PCEP) prior is a conjugate normal-inverse gamma prior that provides a consistent variable selection procedure and gives support to more parsimonious models than the ones supported using the g-prior and the hyper-g prior for finite samples. Detailed illustrations and comparisons of the variable selection procedures using the proposed method, the g-prior, and the hyper-g prior are provided using both simulated and real data examples. Supplementary materials for this article are available online.     

A semiparametric Bernstein–von Mises theorem for Gaussian process priors

This paper is a contribution to the Bayesian theory of semiparametric estimation. We are interested in the so-called Bernstein–von Mises theorem, in a semiparametric framework where the unknown quantity is (θ, f), with θ the parameter of interest and f an infinite-dimensional nuisance parameter. Two theorems are established, one in the case with no loss of information and one in the information loss case with Gaussian process priors. The general theory is applied to three specific models: the estimation of the center of symmetry of a symmetric function in Gaussian white noise, a time-discrete functional data analysis model and Cox’s proportional hazards model. In all cases, the range of application of the theorems is investigated by using a family of Gaussian priors parametrized by a continuous parameter.     

Objective Bayesian analysis of the Frechet stress–strength model

Several reference priors and a general form of matching priors are derived for a stress–strength system, and it is concluded that none of the reference priors is a matching prior. The study shows that the matching prior performs better than Jeffreys prior and reference priors in meeting the target coverage probabilities.     

Generalized beta prior models on fraction defective in reliability test planning

In many reliability analyses, the probability of obtaining a defective unit in a production process should not be considered constant even though the process is stable and in control. Engineering experience or previous data of similar or related products may often be used in the proper selection of a prior model to describe the random fluctuations in the fraction defective. A generalized beta family of priors, several maximum entropy priors and other prior models are considered for this purpose. In order to determine the acceptability of a product based on the lifelengths of some test units, failure-censored reliability sampling plans for location-scale distributions using average producer and consumer risks are designed. Our procedure allows the practitioners to incorporate a restricted parameter space into the reliability analysis, and it is reasonably insensitive to small disturbances in the prior information. Impartial priors are used to reflect prior neutrality between the producer and the consumer when a consensus on the elicited prior model is required. Nonetheless, our approach also enables the producer and the consumer to assume their own prior distributions. The use of substantial prior information can, in many cases, significantly reduce the amount of testing required. However, the main advantage of utilizing a prior model for the fraction defective is not necessarily reduced sample size but improved assessment of the true sampling risks. An example involving shifted exponential lifetimes is considered to illustrate the results.     

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Reference分析最早由Bernardo(1979)提出的, Berger和Bernardo(1992a)做了进一步的发展. 而Berger 等(2001)提出了一个获得精确reference先验的方法, 它已经成为获取无信息先验的最成功的方法之一. 本文基于Berger等(2001)所提出的的算法, 研究了具有一般协方差结构的增长曲线模型的reference先验. 同时, 给出了相应结果的一些应用.     

The Sensitivity of the Number of Clusters in a Gaussian Mixture Model to Prior Distributions

One of the main advantages of Bayesian approaches is that they offer principled methods of inference in models of varying dimensionality and of models of infinite dimensionality. What is less widely appreciated is how the model inference is sensitive to prior distributions and therefore how priors should be set for real problems. In this paper prior sensitivity is considered with respect to the problem of inference in Gaussian mixture models. Two distinct Bayesian approaches have been proposed. The first is to use Bayesian model selection based upon the marginal likelihood; the second is to use an infinite mixture model which ‘side steps’ model selection. Explanations for the prior sensitivity are given in order to give practitioners guidance in setting prior distributions. In particular the use of conditionally conjugate prior distributions instead of purely conjugate prior distributions are advocated as a method for investigating prior sensitivity of the mean and variance individually.     

Bayes empirical Bayes estimation for discrete exponential families

Bayes-empiric Bayes estimation of the parameter of certain one parameter discrete exponential families based on orthogonal polynomials on an interval (a, b) is introduced. The resulting estimator is shown to be asymptotically optimal. The application of this method to three special distributions, the binomial, Poisson and negative binomial, is discussed.The first author was supported by NSF grant DCR-8504620.     

Bayesian reference analysis for Gaussian Markov random fields

Gaussian Markov random fields (GMRF) are important families of distributions for the modeling of spatial data and have been extensively used in different areas of spatial statistics such as disease mapping, image analysis and remote sensing. GMRFs have been used for the modeling of spatial data, both as models for the sampling distribution of the observed data and as models for the prior of latent processes/random effects; we consider mainly the former use of GMRFs. We study a large class of GMRF models that includes several models previously proposed in the literature. An objective Bayesian analysis is presented for the parameters of the above class of GMRFs, where explicit expressions for the Jeffreys (two versions) and reference priors are derived, and for each of these priors results on posterior propriety of the model parameters are established. We describe a simple MCMC algorithm for sampling from the posterior distribution of the model parameters, and study frequentist properties of the Bayesian inferences resulting from the use of these automatic priors. Finally, we illustrate the use of the proposed GMRF model and reference prior for studying the spatial variability of lip cancer cases in the districts of Scotland over the period 1975-1980.     

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.     

Noninformative priors for the ratio of the shape parameters of two Weibull distributions

The Weibull distribution is one of the most widely used lifetime distributions in reliability engineering. Here, the noninformative priors for the ratio of the shape parameters of two Weibull models are introduced. The first criterion used is the asymptotic matching of the coverage probabilities of Bayesian credible intervals with the corresponding frequentist coverage probabilities. We develop the probability matching priors for the ratio of the shape parameters using the following matching criteria: quantile matching, matching of the distribution function, highest posterior density matching, and matching via inversion of the test statistics. We obtain one particular prior that meets all the matching criteria. Next, we derive the reference priors for different groups of ordering. Our findings show that some of the reference priors satisfy a first-order matching criterion and the one-at-a-time reference prior is a second-order matching prior. Lastly, we perform a simulation study and provide a real-world example.     

Bayesian analysis for step‐stress accelerated life testing under progressive interval censoring

A step‐stress accelerated life testing model is considered for progressive type‐I censored experiments when the tested items are not monitored continuously but inspected at prespecified time points, producing thus grouped data. The underlying lifetime distributions belong to a general scale family of distributions. The points of stress‐level change are simultaneously inspection points as well while there is the option of assigning additional inspection points in between the stress‐level change points. In a Bayesian framework, the posterior distributions of the parameters of the model are derived for characteristic choices of prior distributions, as conjugate‐like and normal priors; vague or noninformative. The developed approach is illustrated on a simulated example and on a real data set, both known from the literature. The results are compared to previous analyses; frequentist or Bayes.     

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Step-stress accelerated degradation test (SSADT) is a useful tool for assessing the lifetime distribution of highly reliable products when the available test items are very few. In this paper, we discuss multiple-steps step-stress accelerated degradation models based on Wiener process, and we apply the objective Bayesian method for such analytically intractable models to obtain the noninformative priors (Jefferys prior and two Reference priors). Moreover, we show that their posterior distributions are proper, and we propose Gibbs sampling algorithms for the Bayesian inference based on the Jefferys prior and two Reference priors. Finally, we present some simulation studies to compare the objective Bayesian estimates with the other Bayesian estimate and the maximum likelihood estimates (MLEs). Simulation results demonstrate the superiority of objective Bayesian analysis method.     

Objective Bayesian analysis for CAR models

Objective priors, especially reference priors, have been studied extensively for spatial data in the last decade. In this paper, we study objective priors for a CAR model. In particular, the properties of the reference prior and the corresponding posterior are studied. Furthermore, we show that the frequentist coverage probabilities of posterior credible intervals depend only on the spatial dependence parameter $\rho $ , and not on the regression coefficient or the error variance. Based on the simulation study for comparing the reference and Jeffreys priors, the performance of two reference priors is similar and better than the Jeffreys priors. One spatial dataset is used for illustration.     

Sequential order statistics with an order statistics prior

In the model of sequential order statistics, prior distributions are considered for the model parameters, which, for example, describe increasing load put on remaining components. Gamma priors are examined as well as priors out of a class of extended truncated Erlang distributions (ETED), which is introduced along with some properties. The choice of independent priors in both set-ups leads to respective independent, conjugate posterior distributions for the model parameters of sequential order statistics. Since, in practical applications, the model parameters will often be increasingly ordered, a multivariate prior is applied being the joint distribution of common ETED-order statistics. Whatever baseline distribution of the sequential order statistics is chosen, the joint posterior distribution turns out to be a Weinman multivariate exponential distribution. Posterior moments are given explicitly, and HPD credible sets for the model parameters are stated.