Gibbs sampling method for the Bayesian adaptive elastic net

This article considers the adaptive elastic net estimator for regularized mean regression from a Bayesian perspective. Representing the Laplace distribution as a mixture of Bartlett–Fejer kernels with a Gamma mixing density, a Gibbs sampling algorithm for the adaptive elastic net is developed. By introducing slice variables, it is shown that the mixture representation provides a Gibbs sampler that can be accomplished by sampling from either truncated normal or truncated Gamma distribution. The proposed method is illustrated using several simulation studies and analyzing a real dataset. Both simulation studies and real data analysis indicate that the proposed approach performs well.     

Inference for nonconjugate Bayesian Models using the Gibbs sampler

A Bayesian approach to modeling a rich class of nonconjugate problems is presented. An adaptive Monte Carlo integration technique known as the Gibbs sampler is proposed as a mechanism for implementing a conceptually and computationally simple solution in such a framework. The result is a general strategy for obtaining marginal posterior densities under changing specification of the model error densities and related prior densities. We illustrate the approach in a nonlinear regression setting, comparing the merits of three candidate error distributions.     

Bayesian prediction for small areas using sur models

Sample surveys are usually designed and analyzed to produce estimates for larger areas and/or populations. Nevertheless, sample sizes are often not large enough to give adequate precision for small area estimates of interest. To circumvent such difficulties, borrowing strength from related small areas via modeling becomes essential. In line with this, we propose a hierarchical multivariate Bayes prediction method for small area estimation based on the seemingly unrelated regressions (SUR) model. The performance of the proposed method was evaluated through simulation studies.     

Gibbs sampling methods for Bayesian quantile regression

This paper considers quantile regression models using an asymmetric Laplace distribution from a Bayesian point of view. We develop a simple and efficient Gibbs sampling algorithm for fitting the quantile regression model based on a location-scale mixture representation of the asymmetric Laplace distribution. It is shown that the resulting Gibbs sampler can be accomplished by sampling from either normal or generalized inverse Gaussian distribution. We also discuss some possible extensions of our approach, including the incorporation of a scale parameter, the use of double exponential prior, and a Bayesian analysis of Tobit quantile regression. The proposed methods are illustrated by both simulated and real data.     

Gibbs sampling for marginal posterior expectations

In earlier work (Gelfand and Smith, 1990 and Gelfand et al, 1990) a sampling based approach using the Gibbs sampler was offered as a means for developing marginal posterior densities for a wide range of Bayesian problems several of which were previously inaccessible. Our purpose here is two-fold. First we flesh out the implementation of this approach for calculation of arbitrary expectations of interest. Secondly we offer comparison with perhaps the most prominent approach for calculating posterior expectations, analytic approximation involving application of the LaPlace method. Several illustrative examples are discussed as well. Clear advantages for the sampling based approach emerge.     

Bayesian Model Choice in Exponential Survival Models

ABSTRACT

Log-linear models for the distribution on a contingency table are represented as the intersection of only two kinds of log-linear models. One assuming that a certain group of the variables, if conditioned on all other variables, has a jointly independent distribution and another one assuming that a certain group of the variables, if conditioned on all other variables, has no highest order interaction. The subsets entering into these models are uniquely determined by the original log-linear model. This canonical representation suggests considering joint conditional independence and conditional no highest order association as the elementary building blocks of log-linear models.     

On MCMC sampling in hierarchical longitudinal models

Markov chain Monte Carlo (MCMC) algorithms have revolutionized Bayesian practice. In their simplest form (i.e., when parameters are updated one at a time) they are, however, often slow to converge when applied to high-dimensional statistical models. A remedy for this problem is to block the parameters into groups, which are then updated simultaneously using either a Gibbs or Metropolis-Hastings step. In this paper we construct several (partially and fully blocked) MCMC algorithms for minimizing the autocorrelation in MCMC samples arising from important classes of longitudinal data models. We exploit an identity used by Chib (1995) in the context of Bayes factor computation to show how the parameters in a general linear mixed model may be updated in a single block, improving convergence and producing essentially independent draws from the posterior of the parameters of interest. We also investigate the value of blocking in non-Gaussian mixed models, as well as in a class of binary response data longitudinal models. We illustrate the approaches in detail with three real-data examples.     

Bayesian and fiducial inference for the inverse gaussian distribution via Gibbs sampler

This paper presents a kernel estimation of the distribution of the scale parameter of the inverse Gaussian distribution under type II censoring together with the distribution of the remaining time. Estimation is carried out via the Gibbs sampling algorithm combined with a missing data approach. Estimates and confidence intervals for the parameters of interest are also presented.     

Bayesian latent variable models for clustered mixed outcomes

A general framework is proposed for modelling clustered mixed outcomes. A mixture of generalized linear models is used to describe the joint distribution of a set of underlying variables, and an arbitrary function relates the underlying variables to be observed outcomes. The model accommodates multilevel data structures, general covariate effects and distinct link functions and error distributions for each underlying variable. Within the framework proposed, novel models are developed for clustered multiple binary, unordered categorical and joint discrete and continuous outcomes. A Markov chain Monte Carlo sampling algorithm is described for estimating the posterior distributions of the parameters and latent variables. Because of the flexibility of the modelling framework and estimation procedure, extensions to ordered categorical outcomes and more complex data structures are straightforward. The methods are illustrated by using data from a reproductive toxicity study.     

Multivariate partially linear single-index models: Bayesian analysis

Partially linear single-index models play important roles in advanced non-/semi-parametric statistics due to their generality and flexibility. We generalise these models from univariate response to multivariate responses. A Bayesian method with free-knot spline is used to analyse the proposed models, including the estimation and the prediction, and a Metropolis-within-Gibbs sampler is provided for posterior exploration. We also utilise the partially collapsed idea in our algorithm to speed up the convergence. The proposed models and methods of analysis are demonstrated by simulation studies and are applied to a real data set.     

Comparison of Bayesian models for production efficiency

In this paper, we use Markov Chain Monte Carlo (MCMC) methods in order to estimate and compare stochastic production frontier models from a Bayesian perspective. We consider a number of competing models in terms of different production functions and the distribution of the asymmetric error term. All MCMC simulations are done using the package JAGS (Just Another Gibbs Sampler), a clone of the classic BUGS package which works closely with the R package where all the statistical computations and graphics are done.     

Bayesian hierarchical models for linear networks

The purpose of this study is to highlight dangerous motorways via estimating the intensity of accidents and study its pattern across the UK motorway network. Two methods have been developed to achieve this aim. First, the motorway-specific intensity is estimated by using a homogeneous Poisson process. The heterogeneity across motorways is incorporated using two-level hierarchical models. The data structure is multilevel since each motorway consists of junctions that are joined by grouped segments. In the second method, the segment-specific intensity is estimated. The homogeneous Poisson process is used to model accident data within grouped segments but heterogeneity across grouped segments is incorporated using three-level hierarchical models. A Bayesian method via Markov Chain Monte Carlo is used to estimate the unknown parameters in the models and the sensitivity to the choice of priors is assessed. The performance of the proposed models is evaluated by a simulation study and an application to traffic accidents in 2016 on the UK motorway network. The deviance information criterion (DIC) and the widely applicable information criterion (WAIC) are employed to choose between models.     

Nonparametric binary regression using a Gaussian process prior

The article describes a nonparametric Bayesian approach to estimating the regression function for binary response data measured with multiple covariates. A multiparameter Gaussian process, after some transformation, is used as a prior on the regression function. Such a prior does not require any assumptions like monotonicity or additivity of the covariate effects. However, additivity, if desired, may be imposed through the selection of appropriate parameters of the prior. By introducing some latent variables, the conditional distributions in the posterior may be shown to be conjugate, and thus an efficient Gibbs sampler to compute the posterior distribution may be developed. A hierarchical scheme to construct a prior around a parametric family is described. A robustification technique to protect the resulting Bayes estimator against miscoded observations is also designed. A detailed simulation study is conducted to investigate the performance of the proposed methods. We also analyze some real data using the methods developed in this article.     

Bayesian LASSO-Regularized quantile regression for linear regression models with autoregressive errors

Quantile regression (QR) is a natural alternative for depicting the impact of covariates on the conditional distributions of a outcome variable instead of the mean. In this paper, we investigate Bayesian regularized QR for the linear models with autoregressive errors. LASSO-penalized type priors are forced on regression coefficients and autoregressive parameters of the model. Gibbs sampler algorithm is employed to draw the full posterior distributions of unknown parameters. Finally, the proposed procedures are illustrated by some simulation studies and applied to a real data analysis of the electricity consumption.     

Steady-state Gibbs sampler estimation for lung cancer data

This paper is based on the application of a Bayesian model to a clinical trial study to determine a more effective treatment to lower mortality rates and consequently to increase survival times among patients with lung cancer. In this study, Qian et al. [13 J. Qian, D.K. Stangl, and S. George, A Weibull model for survival data: Using prediction to decide when to stop a clinical trial, in Bayesian Biostatistics, D. Berry and D. Stangl, eds., Marcel Dekker, New York, 1996, pp. 187–205. [Google Scholar]] strived to determine if a Weibull survival model can be used to decide whether to stop a clinical trial. The traditional Gibbs sampler was used to estimate the model parameters. This paper proposes to use the independent steady-state Gibbs sampling (ISSGS) approach, introduced by Dunbar et al. [3 M. Dunbar, H.M. Samawi, R. Vogel, and L. Yu, A more efficient Gibbs sampler estimation using steady state simulation: Application to public health studies, J. Stat. Simul. Comput. 10.1080/00949655.2013.770857.[Taylor &; Francis Online] , [Google Scholar]], to improve the original Gibbs sampler in multidimensional problems. It is demonstrated that ISSGS provides accuracy with unbiased estimation and improves the performance and convergence of the Gibbs sampler in this application.     

Bayesian quantile regression for hierarchical linear models

The paper proposes a Bayesian quantile regression method for hierarchical linear models. Existing approaches of hierarchical linear quantile regression models are scarce and most of them were not from the perspective of Bayesian thoughts, which is important for hierarchical models. In this paper, based on Bayesian theories and Markov Chain Monte Carlo methods, we introduce Asymmetric Laplace distributed errors to simulate joint posterior distributions of population parameters and across-unit parameters and then derive their posterior quantile inferences. We run a simulation as the proposed method to examine the effects on parameters induced by units and quantile levels; the method is also applied to study the relationship between Chinese rural residents' family annual income and their cultivated areas. Both the simulation and real data analysis indicate that the method is effective and accurate.     

Bayesian hierarchical joint modeling using skew-normal/independent distributions

The multiple longitudinal outcomes collected in many clinical trials are often analyzed by multilevel item response theory (MLIRT) models. The normality assumption for the continuous outcomes in the MLIRT models can be violated due to skewness and/or outliers. Moreover, patients’ follow-up may be stopped by some terminal events (e.g., death or dropout), which are dependent on the multiple longitudinal outcomes. We proposed a joint modeling framework based on the MLIRT model to account for three data features: skewness, outliers, and dependent censoring. Our method development was motivated by a clinical study for Parkinson’s disease.     

Bayesian Tobit quantile regression using g-prior distribution with ridge parameter

A Bayesian approach is proposed for coefficient estimation in the Tobit quantile regression model. The proposed approach is based on placing a g-prior distribution depends on the quantile level on the regression coefficients. The prior is generalized by introducing a ridge parameter to address important challenges that may arise with censored data, such as multicollinearity and overfitting problems. Then, a stochastic search variable selection approach is proposed for Tobit quantile regression model based on g-prior. An expression for the hyperparameter g is proposed to calibrate the modified g-prior with a ridge parameter to the corresponding g-prior. Some possible extensions of the proposed approach are discussed, including the continuous and binary responses in quantile regression. The methods are illustrated using several simulation studies and a microarray study. The simulation studies and the microarray study indicate that the proposed approach performs well.     

Bayesian methods for contingency tables using Gibbs sampling

Cell counts in contingency tables can be smoothed using loglinear models. Recently, sampling-based methods such as Markov chain Monte Carlo (MCMC) have been introduced, making it possible to sample from posterior distributions. The novelty of the approach presented here is that all conditional distributions can be specified directly, so that straight-forward Gibbs sampling is possible. Thus, the model is constructed in a way that makes burn-in and checking convergence a relatively minor issue. The emphasis of this paper is on smoothing cell counts in contingency tables, and not so much on estimation of regression parameters. Therefore, the prior distribution consists of two stages. We rely on a normal nonconjugate prior at the first stage, and a vague prior for hyperparameters at the second stage. The smoothed counts tend to compromise between the observed data and a log-linear model. The methods are demonstrated with a sparse data table taken from a multi-center clinical trial. The research for the first author was supported by Brain Pool program of the Korean Federation of Science and Technology Societies. The research for the second author was partially supported by KOSEF through Statistical Research Center for Complex Systems at Seoul National University.