Robustness of Bayesian D-optimal design for the logistic mixed model against misspecification of autocorrelation

In medicine and health sciences mixed effects models are often used to study time-structured data. Optimal designs for such studies have been shown useful to improve the precision of the estimators of the parameters. However, optimal designs for such studies are often derived under the assumption of a zero autocorrelation between the errors, especially for binary data. Ignoring or misspecifying the autocorrelation in the design stage can result in loss of efficiency. This paper addresses robustness of Bayesian D-optimal designs for the logistic mixed effects model for longitudinal data with a linear or quadratic time effect against incorrect specification of the autocorrelation. To find the Bayesian D-optimal allocations of time points for different values of the autocorrelation, under different priors for the fixed effects and different covariance structures of the random effects, a scalar function of the approximate variance–covariance matrix of the fixed effects is optimized. Two approximations are compared; one based on a first order penalized quasi likelihood (PQL1) and one based on an extended version of the generalized estimating equations (GEE). The results show that Bayesian D-optimal allocations of time points are robust against misspecification of the autocorrelation and are approximately equally spaced. Moreover, PQL1 and extended GEE give essentially the same Bayesian D-optimal allocation of time points for a given subject-to-measurement cost ratio. Furthermore, Bayesian optimal designs are hardly affected either by the choice of a covariance structure or by the choice of a prior distribution.     

����ά�ɹ��̲���Ӧ�������˻�����Ŀ͹۱�Ҷ˹����

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.     

Posterior propriety in nonparametric mixed effects model

It is well known that spline smoothing estimator relates to the Bayesian estimate under partially informative normal prior.In this paper,we derive the conditions for the propriety of the posterior in the nonparametric mixed efects model under this class of partially informative normal prior for fxed efect with inverse gamma priors on the variance components and hierarchical priors for covariance matrix of random efect,then we explore the Gibbs sampling procedure.     

Bayesian inference and life testing plans for generalized exponential distribution

Recently generalized exponential distribution has received considerable attentions. In this paper, we deal with the Bayesian inference of the unknown parameters of the progressively censored generalized exponential distribution. It is assumed that the scale and the shape parameters have independent gamma priors. The Bayes estimates of the unknown parameters cannot be obtained in the closed form. Lindley’s approximation and importance sampling technique have been suggested to compute the approximate Bayes estimates. Markov Chain Monte Carlo method has been used to compute the approximate Bayes estimates and also to construct the highest posterior density credible intervals. We also provide different criteria to compare two different sampling schemes and hence to find the optimal sampling schemes. It is observed that finding the optimum censoring procedure is a computationally expensive process. And we have recommended to use the sub-optimal censoring procedure, which can be obtained very easily. Monte Carlo simulations are performed to compare the performances of the different methods and one data analysis has been performed for illustrative purposes. This work was partially supported by a grant from the Department of Science and Technology, Government of India     

Sparse Variational Analysis of Linear Mixed Models for Large Data Sets

It is increasingly common to be faced with longitudinal or multi-level data sets that have large numbers of predictors and/or a large sample size. Current methods of fitting and inference for mixed effects models tend to perform poorly in such settings. When there are many variables, it is appealing to allow uncertainty in subset selection and to obtain a sparse characterization of the data. Bayesian methods are available to address these goals using Markov chain Monte Carlo (MCMC), but MCMC is very computationally expensive and can be infeasible in large p and/or large n problems. As a fast approximate Bayes solution, we recommend a novel approximation to the posterior relying on variational methods. Variational methods are used to approximate the posterior of the parameters in a decomposition of the variance components, with priors chosen to obtain a sparse solution that allows selection of random effects. The method is evaluated through a simulation study, and applied to an epidemiological application.     

Estimation of the multivariate normal precision and covariance matrices in a star-shape model

In this paper, we introduce the star-shape models, where the precision matrix Ω (the inverse of the covariance matrix) is structured by the special conditional independence. We want to estimate the precision matrix under entropy loss and symmetric loss. We show that the maximal likelihood estimator (MLE) of the precision matrix is biased. Based on the MLE, an unbiased estimate is obtained. We consider a type of Cholesky decomposition of Ω, in the sense that Ω=Ψ′Ψ, where Ψ is a lower triangular matrix with positive diagonal elements. A special group , which is a subgroup of the group consisting all lower triangular matrices, is introduced. General forms of equivariant estimates of the covariance matrix and precision matrix are obtained. The invariant Haar measures on , the reference prior, and the Jeffreys prior of Ψ are also discussed. We also introduce a class of priors of Ψ, which includes all the priors described above. The posterior properties are discussed and the closed forms of Bayesian estimators are derived under either the entropy loss or the symmetric loss. We also show that the best equivariant estimators with respect to is the special case of Bayesian estimators. Consequently, the MLE of the precision matrix is inadmissible under either entropy or symmetric loss. The closed form of risks of equivariant estimators are obtained. Some numerical results are given for illustration. The project is supported by the National Science Foundation grants DMS-9972598, SES-0095919, and SES-0351523, and a grant from Federal Aid in Wildlife Restoration Project W-13-R through Missouri Department of Conservation.     

Reference Bayesian analysis of inverse Gaussian degradation process

In this paper, objective Bayesian method is applied to analyze degradation model based on the inverse Gaussian process. Noninformative priors (Jefferys prior and two reference priors) for model parameters are obtained and their properties are discussed. Moreover, we propose a class of modified reference priors to remedy weaknesses of the usual reference priors and show that the modified reference priors not only have proper posterior distributions but also have probability matching properties for model parameters. Gibbs sampling algorithms for Bayesian inference based on the Jefferys prior and the modified reference priors are studied. Simulations are conducted to compare the objective Bayesian estimates with the maximum likelihood estimates and subjective Bayesian estimates and shows better performance of the objective method than the other two estimates especially for the case of small sample size. Finally, two real data examples are analyzed for illustration.     

Estimation of a Multivariate Normal Covariance Matrix with Staircase Pattern Data

In this paper, we study the problem of estimating a multivariate normal covariance matrix with staircase pattern data. Two kinds of parameterizations in terms of the covariance matrix are used. One is Cholesky decomposition and another is Bartlett decomposition. Based on Cholesky decomposition of the covariance matrix, the closed form of the maximum likelihood estimator (MLE) of the covariance matrix is given. Using Bayesian method, we prove that the best equivariant estimator of the covariance matrix with respect to the special group related to Cholesky decomposition uniquely exists under the Stein loss. Consequently, the MLE of the covariance matrix is inadmissible under the Stein loss. Our method can also be applied to other invariant loss functions like the entropy loss and the symmetric loss. In addition, based on Bartlett decomposition of the covariance matrix, the Jeffreys prior and the reference prior of the covariance matrix with staircase pattern data are also obtained. Our reference prior is different from Berger and Yang’s reference prior. Interestingly, the Jeffreys prior with staircase pattern data is the same as that with complete data. The posterior properties are also investigated. Some simulation results are given for illustration.     

Geometric Ergodicity of Gibbs and Block Gibbs Samplers for a Hierarchical Random Effects Model

We consider fixed scan Gibbs and block Gibbs samplers for a Bayesian hierarchical random effects model with proper conjugate priors. A drift condition given in Meyn and Tweedie (1993, Chapter 15) is used to show that these Markov chains are geometrically ergodic. Showing that a Gibbs sampler is geometrically ergodic is the first step toward establishing central limit theorems, which can be used to approximate the error associated with Monte Carlo estimates of posterior quantities of interest. Thus, our results will be of practical interest to researchers using these Gibbs samplers for Bayesian data analysis.     

A general divergence criterion for prior selection

The paper revisits the problem of selection of priors for regular one-parameter family of distributions. The goal is to find some “objective” or “default” prior by approximate maximization of the distance between the prior and the posterior under a general divergence criterion as introduced by Amari (Ann Stat 10:357–387, 1982) and Cressie and Read (J R Stat Soc Ser B 46:440–464, 1984). The maximization is based on an asymptotic expansion of this distance. The Kullback–Leibler, Bhattacharyya–Hellinger and Chi-square divergence are special cases of this general divergence criterion. It is shown that with the exception of one particular case, namely the Chi-square divergence, the general divergence criterion yields Jeffreys’ prior. For the Chi-square divergence, we obtain a prior different from that of Jeffreys and also from that of Clarke and Sun (Sankhya Ser A 59:215–231, 1997).     

Priors for Bayesian adaptive spline smoothing

Adaptive smoothing has been proposed for curve-fitting problems where the underlying function is spatially inhomogeneous. Two Bayesian adaptive smoothing models, Bayesian adaptive smoothing splines on a lattice and Bayesian adaptive P-splines, are studied in this paper. Estimation is fully Bayesian and carried out by efficient Gibbs sampling. Choice of prior is critical in any Bayesian non-parametric regression method. We use objective priors on the first level parameters where feasible, specifically independent Jeffreys priors (right Haar priors) on the implied base linear model and error variance, and we derive sufficient conditions on higher level components to ensure that the posterior is proper. Through simulation, we demonstrate that the common practice of approximating improper priors by proper but diffuse priors may lead to invalid inference, and we show how appropriate choices of proper but only weakly informative priors yields satisfactory inference.     

Objective Bayesian analysis for competing risks model with Wiener degradation phenomena and catastrophic failures

In this paper, the objective Bayesian method is applied to investigate the competing risks model involving both catastrophic and degradation failures. By modeling soft failure as the Wiener degradation process, and hard failures as a Weibull distribution, we obtain the noninformative priors (Jefferys prior and two reference priors) for the parameters. Moreover, we show that their posterior distributions have good properties and we propose Gibbs sampling algorithms for the Bayesian inference based on the Jefferys prior and two reference priors. Some simulation studies are conducted to illustrate the superiority of objective Bayesian method. Finally, we apply our methods to two real data examples and compare the objective Bayesian estimates with the other estimates.     

Joint Bayesian Analysis of Factor Scores and Structural Parameters in the Factor Analysis Model

A Bayesian approach is developed to assess the factor analysis model. Joint Bayesian estimates of the factor scores and the structural parameters in the covariance structure are obtained simultaneously. The basic idea is to treat the latent factor scores as missing data and augment them with the observed data in generating a sequence of random observations from the posterior distributions by the Gibbs sampler. Then, the Bayesian estimates are taken as the sample means of these random observations. Expressions for implementing the algorithm are derived and some statistical properties of the estimates are presented. Some aspects of the algorithm are illustrated by a real example and the performance of the Bayesian procedure is studied using simulation.     

Bayesian estimation of system reliability in Brownian stress-strength models

A stress-strength system fails as soon as the applied stress,X, is at least as much as the strength,Y, of the system. Stress and strength are time-varying in many real-life systems but typical statistical models for stress-strength systems are static. In this article, the stress and strength processes are dynamically modeled as Brownian motions. The resulting stress-strength system is then governed by a time-homogeneous Markov process with an absorption barrier at O. Conjugate as well as non-informative priors are developed for the model parameters and Markov chain sampling methods are used for posterior inference of the reliability of the stress-strength system. A generalization of this model is described next where the different stress-strength systems are assumed to be exchangeable. The proposed Bayesian analyses are illustrated in two examples where we obtain posterior estimates as well as perform model checking by cross-validation.     

Probability matching priors for predicting a dependent variable with application to regression models

In a Bayesian setup, we consider the problem of predicting a dependent variable given an independent variable and past observations on the two variables. An asymptotic formula for the relevant posterior predictive density is worked out. Considering posterior quantiles and highest predictive density regions, we then characterize priors that ensure approximate frequentist validity of Bayesian prediction in the above setting. Application to regression models is also discussed.     

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.     

Bayesian priors in sequential binomial design

The status of sequential analysis in Bayesian inference is revisited. The information on the experimental design, including the stopping rule, is one part of the evidence, prior to the sampling. Consequently this information must be incorporated in the prior distribution. This approach allows to relax the likelihood principle when appropriate. It is illustrated in the case of successive Binomial trials. Using Jeffreys' rule, a prior based on the Fisher information and conditional on the design characteristics is derived. The corrected Jeffreys prior, which involves a new distribution called Beta-J, extends the classical Jeffreys priors for the Binomial and Pascal sampling models to more general stopping rules. As an illustration, we show that the correction induced on the posterior is proportional to the bias induced by the stopping rule on the maximum likelihood estimator. To cite this article: P. Bunouf, B. Lecoutre, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Bayesian shrinkage prediction for the regression problem

We consider Bayesian shrinkage predictions for the Normal regression problem under the frequentist Kullback-Leibler risk function.Firstly, we consider the multivariate Normal model with an unknown mean and a known covariance. While the unknown mean is fixed, the covariance of future samples can be different from that of training samples. We show that the Bayesian predictive distribution based on the uniform prior is dominated by that based on a class of priors if the prior distributions for the covariance and future covariance matrices are rotation invariant.Then, we consider a class of priors for the mean parameters depending on the future covariance matrix. With such a prior, we can construct a Bayesian predictive distribution dominating that based on the uniform prior.Lastly, applying this result to the prediction of response variables in the Normal linear regression model, we show that there exists a Bayesian predictive distribution dominating that based on the uniform prior. Minimaxity of these Bayesian predictions follows from these results.     

Triply fuzzy function approximation for hierarchical Bayesian inference

We prove that three independent fuzzy systems can uniformly approximate Bayesian posterior probability density functions by approximating the prior and likelihood probability densities as well as the hyperprior probability densities that underly the priors. This triply fuzzy function approximation extends the recent theorem for uniformly approximating the posterior density by approximating just the prior and likelihood densities. This approximation allows users to state priors and hyper-priors in words or rules as well as to adapt them from sample data. A fuzzy system with just two rules can exactly represent common closed-form probability densities so long as they are bounded. The function approximators can also be neural networks or any other type of uniform function approximator. Iterative fuzzy Bayesian inference can lead to rule explosion. We prove that conjugacy in the if-part set functions for prior, hyperprior, and likelihood fuzzy approximators reduces rule explosion. We also prove that a type of semi-conjugacy of if-part set functions for those fuzzy approximators results in fewer parameters in the fuzzy posterior approximator.     

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.