On the choice of a noninformative prior for Bayesian inference of discretized normal observations

We consider the task of Bayesian inference of the mean of normal observations when the available data have been discretized and when no prior knowledge about the mean and the variance exists. An application is presented which illustrates that the discretization of the data should not be ignored when their variability is of the order of the discretization step. We show that the standard (noninformative) prior for location-scale family distributions is no longer appropriate. We work out the reference prior of Berger and Bernardo, which leads to different and more reasonable results. However, for this prior the posterior also shows some non-desirable properties. We argue that this is due to the inherent difficulty of the considered problem, which also affects other methods of inference. We therefore complement our analysis by an empirical Bayes approach. While such proceeding overcomes the disadvantages of the standard and reference priors and appears to provide a reasonable inference, it may raise conceptual concerns. We conclude that it is difficult to provide a widely accepted prior for the considered problem.     

Bayesian Analysis for Random Coefficient Regression Models Using Noninformative Priors

We apply Bayesian approach, through noninformative priors, to analyze a Random Coefficient Regression (RCR) model. The Fisher information matrix, the Jeffreys prior and reference priors are derived for this model. Then, we prove that the corresponding posteriors are proper when the number of full rank design matrices are greater than or equal to twice the number of regression coefficient parameters plus 1 and that the posterior means for all parameters exist if one more additional full rank design matrix is available. A hybrid Markov chain sampling scheme is developed for computing the Bayesian estimators for parameters of interest. A small-scale simulation study is conducted for comparing the performance of different noninformative priors. A real data example is also provided and the data are analyzed by a non-Bayesian method as well as Bayesian methods with noninformative priors.     

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.     

Reliability analysis for Weibull distribution with homogeneous heavily censored data based on Bayesian and least-squares methods

The reliability for Weibull distribution with homogeneous heavily censored data is analyzed in this study. The universal model of heavily censored data and existing methods, including maximum likelihood, least-squares, E-Bayesian estimation, and hierarchical Bayesian methods, are introduced. An improved method is proposed based on Bayesian inference and least-squares method. In this method, the Bayes estimations of failure probabilities are focused on for all the samples. The conjugate prior distribution of failure probability is set, and an optimization model is developed by maximizing the information entropy of prior distribution to determine the hyper-parameters. By integrating the likelihood function, the posterior distribution of failure probability is then derived to yield the Bayes estimation of failure probability. The estimations of reliability parameters are obtained by fitting distribution curve using least-squares method. The four existing methods are compared with the proposed method in terms of applicability, precision, efficiency, robustness, and simplicity. Specifically, the closed form expressions concerning E-Bayesian estimation and hierarchical Bayesian methods are derived and used. The comparisons demonstrate that the improved method is superior. Finally, three illustrative examples are presented to show the application of the proposed method.     

A Bayesian inference approach to identify a Robin coefficient in one-dimensional parabolic problems

This paper investigates a nonlinear inverse problem associated with the heat conduction problem of identifying a Robin coefficient from boundary temperature measurement. A Bayesian inference approach is presented for the solution of this problem. The prior modeling is achieved via the Markov random field (MRF). The use of a hierarchical Bayesian method for automatic selection of the regularization parameter in the function estimation inverse problem is discussed. The Markov chain Monte Carlo (MCMC) algorithm is used to explore the posterior state space. Numerical results indicate that MRF provides an effective prior regularization, and the Bayesian inference approach can provide accurate estimates as well as uncertainty quantification to the solution of the inverse problem.     

The Conditional-Potts Clustering Model

This article presents a Bayesian kernel-based clustering method. The associated model arises as an embedding of the Potts density for class membership probabilities into an extended Bayesian model for joint data and class membership probabilities. The method may be seen as a principled extension of the super-paramagnetic clustering. The model depends on two parameters: the temperature and the kernel bandwidth. The clustering is obtained from the posterior marginal adjacency membership probabilities and does not depend on any particular value of the parameters. We elicit an informative prior based on random graph theory and kernel density estimation. A stochastic population Monte Carlo algorithm, based on parallel runs of the Wang–Landau algorithm, is developed to estimate the posterior adjacency membership probabilities and the parameter posterior. The convergence of the algorithm is also established. The method is applied to the whole human proteome to uncover human genes that share common evolutionary history. Our experiments and application show that good clustering results are obtained at many different values of the temperature and bandwidth parameters. Hence, instead of focusing on finding adequate values of the parameters, we advocate making clustering inference based on the study of the distribution of the posterior adjacency membership probabilities. This article has online supplementary material.     

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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.     

Bayesian Variable Selection for Median Regression

When the data has heavy tail feature or contains outliers, conventional variable selection methods based on penalized least squares or likelihood functions perform poorly. Based on Bayesian inference method, we study the Bayesian variable selection problem for median linear models. The Bayesian estimation method is proposed by using Bayesian model selection theory and Bayesian estimation method through selecting the Spike and Slab prior for regression coefficients, and the effective posterior Gibbs sampling procedure is also given. Extensive numerical simulations and Boston house price data analysis are used to illustrate the effectiveness of the proposed method.     

Bayesian estimation of a covariance matrix with flexible prior specification

Bayesian analysis for a covariance structure has been in use for decades. The commonly adopted Bayesian setup involves the conjugate inverse Wishart prior specification for the covariance matrix. Here we depart from this approach and adopt a novel prior specification by considering a multivariate normal prior for the elements of the matrix logarithm of the covariance structure. This specification allows for a richer class of prior distributions for the covariance structure with respect to strength of beliefs in prior location hyperparameters and the added ability to model potential correlation amongst the covariance structure. We provide three computational methods for calculating the posterior moment of the covariance matrix. The moments of interest are calculated based upon computational results via Importance sampling, Laplacian approximation and Markov Chain Monte Carlo/Metropolis–Hastings techniques. As a particular application of the proposed technique we investigate educational test score data from the project talent data set.     

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.     

Frequentist and Bayesian measures of confidence via multiscale bootstrap for testing three regions

A new computation method of frequentist p values and Bayesian posterior probabilities based on the bootstrap probability is discussed for the multivariate normal model with unknown expectation parameter vector. The null hypothesis is represented as an arbitrary-shaped region of the parameter vector. We introduce new functional forms for the scaling-law of bootstrap probability so that the multiscale bootstrap method, which was designed for a one-sided test, can also compute confidence measures of a two-sided test, extending applicability to a wider class of hypotheses. Parameter estimation for the scaling-law is improved by the two-step multiscale bootstrap and also by including higher order terms. Model selection is important not only as a motivating application of our method, but also as an essential ingredient in the method. A compromise between frequentist and Bayesian is attempted by showing that the Bayesian posterior probability with a noninformative prior is interpreted as a frequentist p value of “zero-sided” test.     

Bayesian analysis of Birnbaum–Saunders distribution via the generalized ratio-of-uniforms method

This paper deals with the Bayesian inference for the parameters of the Birnbaum–Saunders distribution. We adopt the inverse-gamma priors for the shape and scale parameters because the continuous conjugate joint prior distribution does not exist and the reference prior (or independent Jeffreys’ prior) results in an improper posterior distribution. We propose an efficient sampling algorithm via the generalized ratio-of-uniforms method to compute the Bayesian estimates and the credible intervals. One appealing advantage of the proposed procedure over other sampling techniques is that it efficiently generates independent samples from the required posterior distribution. Simulation studies are conducted to investigate the behavior of the proposed method, and two real-data applications are analyzed for illustrative purposes.     

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??When the data has heavy tail feature or contains outliers, conventional variable selection methods based on penalized least squares or likelihood functions perform poorly. Based on Bayesian inference method, we study the Bayesian variable selection problem for median linear models. The Bayesian estimation method is proposed by using Bayesian model selection theory and Bayesian estimation method through selecting the Spike and Slab prior for regression coefficients, and the effective posterior Gibbs sampling procedure is also given. Extensive numerical simulations and Boston house price data analysis are used to illustrate the effectiveness of the proposed method.     

广义变系数模型的Bayesian B样条估计

提出了广义变系数模型函数系数的一种新的估计方法．我们用B样条函数逼近函数系数,不具体选择节点的个数,而是节点个数取均匀的无信息先验,样条函数系数取正态先验,用Bayesian模型平均的方法估计各个函数系数．这种估计方法一个主要特点是允许各个函数系数所需节点个数的后验分布不同,因此允许不同函数系数使用不同的光滑参数．另外,本文还给出了Bayesian B样条估计的计算方法,并通过模拟例子,说明广义变系数模型的函数系数可以由Bayesian B样条估计方法得到很好的估计．     

Semiparametric Bayesian Regression via Potts Model

We consider Bayesian nonparametric regression through random partition models. Our approach involves the construction of a covariate-dependent prior distribution on partitions of individuals. Our goal is to use covariate information to improve predictive inference. To do so, we propose a prior on partitions based on the Potts clustering model associated with the observed covariates. This drives by covariate proximity both the formation of clusters, and the prior predictive distribution. The resulting prior model is flexible enough to support many different types of likelihood models. We focus the discussion on nonparametric regression. Implementation details are discussed for the specific case of multivariate multiple linear regression. The proposed model performs well in terms of model fitting and prediction when compared to other alternative nonparametric regression approaches. We illustrate the methodology with an application to the health status of nations at the turn of the 21st century. Supplementary materials are available online.     

Properties of Prior and Posterior Distributions for Multivariate Categorical Response Data Models

In this article, we model multivariate categorical (binary and ordinal) response data using a very rich class of scale mixture of multivariate normal (SMMVN) link functions to accommodate heavy tailed distributions. We consider both noninformative as well as informative prior distributions for SMMVN-link models. The notation of informative prior elicitation is based on available similar historical studies. The main objectives of this article are (i) to derive theoretical properties of noninformative and informative priors as well as the resulting posteriors and (ii) to develop an efficient Markov chain Monte Carlo algorithm to sample from the resulting posterior distribution. A real data example from prostate cancer studies is used to illustrate the proposed methodologies.     

Bayesian hierarchical linear mixed models for additive smoothing splines

Bayesian hierarchical models have been used for smoothing splines, thin-plate splines, and L-splines. In analyzing high dimensional data sets, additive models and backfitting methods are often used. A full Bayesian analysis for such models may include a large number of random effects, many of which are not intuitive, so researchers typically use noninformative improper or nearly improper priors. We investigate propriety of the posterior for these cases. Our findings extend known results for normal linear mixed models to certain cases with Bayesian additive smoothing spline models. Supported by National Science Foundation grant SES-0351523 and by National Institutes of Health grants R01-CA100760 and R01-MH071418.     

Modelling non-stationary dynamic gene regulatory processes with the BGM model

Recently, a Bayesian network model for inferring non-stationary regulatory processes from gene expression time series has been proposed. The Bayesian Gaussian Mixture (BGM) Bayesian network model divides the data into disjunct compartments (data subsets) by a free allocation model, and infers network structures, which are kept fixed for all compartments. Fixing the network structure allows for some information sharing among compartments, and each compartment is modelled separately and independently with the Gaussian BGe scoring metric for Bayesian networks. The BGM model can equally be applied to both static (steady-state) and dynamic (time series) gene expression data. However, it is this flexibility that renders its application to time series data suboptimal. To improve the performance of the BGM model on time series data we propose a revised approach in which the free allocation of data points is replaced by a changepoint process so as to take the temporal structure into account. The practical inference follows the Bayesian paradigm and approximately samples the network, the number of compartments and the changepoint locations from the posterior distribution with Markov chain Monte Carlo (MCMC). Our empirical results show that the proposed modification leads to a more efficient inference tool for analysing gene expression time series.     

Computation of reference Bayesian inference for variance components in longitudinal studies

Generalized linear mixed models (GLMMs) have been applied widely in the analysis of longitudinal data. This model confers two important advantages, namely, the flexibility to include random effects and the ability to make inference about complex covariances. In practice, however, the inference of variance components can be a difficult task due to the complexity of the model itself and the dimensionality of the covariance matrix of random effects. Here we first discuss for GLMMs the relation between Bayesian posterior estimates and penalized quasi-likelihood (PQL) estimates, based on the generalization of Harville’s result for general linear models. Next, we perform fully Bayesian analyses for the random covariance matrix using three different reference priors, two with Jeffreys’ priors derived from approximate likelihoods and one with the approximate uniform shrinkage prior. Computations are carried out via the combination of asymptotic approximations and Markov chain Monte Carlo methods. Under the criterion of the squared Euclidean norm, we compare the performances of Bayesian estimates of variance components with that of PQL estimates when the responses are non-normal, and with that of the restricted maximum likelihood (REML) estimates when data are assumed normal. Three applications and simulations of binary, normal, and count responses with multiple random effects and of small sample sizes are illustrated. The analyses examine the differences in estimation performance when the covariance structure is complex, and demonstrate the equivalence between PQL and the posterior modes when the former can be derived. The results also show that the Bayesian approach, particularly under the approximate Jeffreys’ priors, outperforms other procedures.     

A semiparametric Bayesian joint model for multiple mixed-type outcomes: an application to acute myocardial infarction

We propose a Bayesian semiparametric regression model to represent mixed-type multiple outcomes concerning patients affected by Acute Myocardial Infarction. Our approach is motivated by data coming from the ST-Elevation Myocardial Infarction (STEMI) Archive, a multi-center observational prospective clinical study planned as part of the Strategic Program of Lombardy, Italy. We specifically consider a joint model for a variable measuring treatment time and in-hospital and 60-day survival indicators. One of our main motivations is to understand how the various hospitals differ in terms of the variety of information collected as part of the study. To do so we postulate a semiparametric random effects model that incorporates dependence on a location indicator that is used to explicitly differentiate among hospitals in or outside the city of Milano. The model is based on the two parameter Poisson-Dirichlet prior, also known as the Pitman-Yor process prior. We discuss the resulting posterior inference, including sensitivity analysis, and a comparison with the particular sub-model arising when a Dirichlet process prior is assumed.