Bayesian approach to errors-in-variables in count data regression models with departures from normality and overdispersion

In most practical applications, the quality of count data is often compromised due to errors-in-variables (EIVs). In this paper, we apply Bayesian approach to reduce bias in estimating the parameters of count data regression models that have mismeasured independent variables. Furthermore, the exposure model is misspecified with a flexible distribution, hence our approach remains robust against any departures from normality in its true underlying exposure distribution. The proposed method is also useful in realistic situations as the variance of EIVs is estimated instead of assumed as known, in contrast with other methods of correcting bias especially in count data EIVs regression models. We conduct simulation studies on synthetic data sets using Markov chain Monte Carlo simulation techniques to investigate the performance of our approach. Our findings show that the flexible Bayesian approach is able to estimate the values of the true regression parameters consistently and accurately.     

Bayesian Tobit quantile regression with single-index models

Based on the Bayesian framework of utilizing a Gaussian prior for the univariate nonparametric link function and an asymmetric Laplace distribution (ALD) for the residuals, we develop a Bayesian treatment for the Tobit quantile single-index regression model (TQSIM). With the location-scale mixture representation of the ALD, the posterior inferences of the latent variables and other parameters are achieved via the Markov Chain Monte Carlo computation method. TQSIM broadens the scope of applicability of the Tobit models by accommodating nonlinearity in the data. The proposed method is illustrated by two simulation examples and a labour supply dataset.     

Parallel Markov chain Monte Carlo for Bayesian hierarchical models with big data,in two stages

Due to the escalating growth of big data sets in recent years, new Bayesian Markov chain Monte Carlo (MCMC) parallel computing methods have been developed. These methods partition large data sets by observations into subsets. However, for Bayesian nested hierarchical models, typically only a few parameters are common for the full data set, with most parameters being group specific. Thus, parallel Bayesian MCMC methods that take into account the structure of the model and split the full data set by groups rather than by observations are a more natural approach for analysis. Here, we adapt and extend a recently introduced two-stage Bayesian hierarchical modeling approach, and we partition complete data sets by groups. In stage 1, the group-specific parameters are estimated independently in parallel. The stage 1 posteriors are used as proposal distributions in stage 2, where the target distribution is the full model. Using three-level and four-level models, we show in both simulation and real data studies that results of our method agree closely with the full data analysis, with greatly increased MCMC efficiency and greatly reduced computation times. The advantages of our method versus existing parallel MCMC computing methods are also described.     

A hierarchical Bayesian regression model for the uncertain functional constraint using screened scale mixtures of Gaussian distributions

This paper considers a hierarchical Bayesian analysis of regression models using a class of Gaussian scale mixtures. This class provides a robust alternative to the common use of the Gaussian distribution as a prior distribution in particular for estimating the regression function subject to uncertainty about the constraint. For this purpose, we use a family of rectangular screened multivariate scale mixtures of Gaussian distribution as a prior for the regression function, which is flexible enough to reflect the degrees of uncertainty about the functional constraint. Specifically, we propose a hierarchical Bayesian regression model for the constrained regression function with uncertainty on the basis of three stages of a prior hierarchy with Gaussian scale mixtures, referred to as a hierarchical screened scale mixture of Gaussian regression models (HSMGRM). We describe distributional properties of HSMGRM and an efficient Markov chain Monte Carlo algorithm for posterior inference, and apply the proposed model to real applications with constrained regression models subject to uncertainty.     

Estimating catch at age from market sampling data by using a Bayesian hierarchical model

Summary.  The paper develops a Bayesian hierarchical model for estimating the catch at age of cod landed in Norway. The model includes covariate effects such as season and gear, and can also account for the within-boat correlation. The hierarchical structure allows us to account properly for the uncertainty in the estimates.     

Bayesian generalized fused lasso modeling via NEG distribution

The fused lasso penalizes a loss function by the L₁ norm for both the regression coefficients and their successive differences to encourage sparsity of both. In this paper, we propose a Bayesian generalized fused lasso modeling based on a normal-exponential-gamma (NEG) prior distribution. The NEG prior is assumed into the difference of successive regression coefficients. The proposed method enables us to construct a more versatile sparse model than the ordinary fused lasso using a flexible regularization term. Simulation studies and real data analyses show that the proposed method has superior performance to the ordinary fused lasso.     

Bayesian quantile regression for longitudinal data models

In this paper, we discuss a fully Bayesian quantile inference using Markov Chain Monte Carlo (MCMC) method for longitudinal data models with random effects. Under the assumption of error term subject to asymmetric Laplace distribution, we establish a hierarchical Bayesian model and obtain the posterior distribution of unknown parameters at τ-th level. We overcome the current computational limitations using two approaches. One is the general MCMC technique with Metropolis–Hastings algorithm and another is the Gibbs sampling from the full conditional distribution. These two methods outperform the traditional frequentist methods under a wide array of simulated data models and are flexible enough to easily accommodate changes in the number of random effects and in their assumed distribution. We apply the Gibbs sampling method to analyse a mouse growth data and some different conclusions from those in the literatures are obtained.     

Bayesian expectile regression with asymmetric normal distribution

In this paper, we adopt the Bayesian approach to expectile regression employing a likelihood function that is based on an asymmetric normal distribution. We demonstrate that improper uniform priors for the unknown model parameters yield a proper joint posterior. Three simulated data sets were generated to evaluate the proposed method which show that Bayesian expectile regression performs well and has different characteristics comparing with Bayesian quantile regression. We also apply this approach into two real data analysis.     

Parallelizing MCMC for Bayesian spatiotemporal geostatistical models

When MCMC methods for Bayesian spatiotemporal modeling are applied to large geostatistical problems, challenges arise as a consequence of memory requirements, computing costs, and convergence monitoring. This article describes the parallelization of a reparametrized and marginalized posterior sampling (RAMPS) algorithm, which is carefully designed to generate posterior samples efficiently. The algorithm is implemented using the Parallel Linear Algebra Package (PLAPACK). The scalability of the algorithm is investigated via simulation experiments that are implemented using a cluster with 25 processors. The usefulness of the method is illustrated with an application to sulfur dioxide concentration data from the Air Quality System database of the U.S. Environmental Protection Agency.     

Bayesian parsimonious covariance estimation for hierarchical linear mixed models

We consider a non-centered parameterization of the standard random-effects model, which is based on the Cholesky decomposition of the variance-covariance matrix. The regression type structure of the non-centered parameterization allows us to use Bayesian variable selection methods for covariance selection. We search for a parsimonious variance-covariance matrix by identifying the non-zero elements of the Cholesky factors. With this method we are able to learn from the data for each effect whether it is random or not, and whether covariances among random effects are zero. An application in marketing shows a substantial reduction of the number of free elements in the variance-covariance matrix.     

Bayesian analysis of two overdispersed poisson regression models

Shookri and Consul (1989) and Scollnik (1995) have previously considered the Bayesian analysis of an overdispersed generalized Poisson model. Scollnik (1995) also considered the Bayesian analysis of an ordinary Poisson and over-dispersed generalized Poisson mixture model. In this paper, we discuss the Bayesian analysis of these models when they are utilised in a regression context. Markov chain Monte Carlo methods are utilised, and an illustrative analysis is provided.     

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 analyses of an exponential-Poisson and related zero augmented type models

We consider several alternatives to the continuous exponential-Poisson distribution in order to accommodate the occurrence of zeros. Three of these are modifications of the exponential-Poisson model. One of these remains a fully continuous model. The other models we consider are all semi-continuous models, each with a discrete point mass at zero and a continuous density on the positive values. All of the models are applied to two environmental data sets concerning precipitation, and their Bayesian analyses using MCMC are discussed. This discussion covers convergence of the MCMC simulations and model selection procedures and considerations.     

A Bayesian hierarchical model for photometric red shifts

Summary.  The Sloan digital sky survey is an extremely large astronomical survey that is conducted with the intention of mapping more than a quarter of the sky. Among the data that it is generating are spectroscopic and photometric measurements, both containing information about the red shift of galaxies. The former are precise and easy to interpret but expensive to gather; the latter are far cheaper but correspondingly more difficult to interpret. Recently, Csabai and co-workers have described various calibration techniques aiming to predict red shift from photometric measurements. We investigate what a structured Bayesian approach to the problem can add. In particular, we are interested in providing uncertainty bounds that are associated with the underlying red shifts and the classifications of the galaxies. We find that quite a generic statistical modelling approach, using for the most part standard model ingredients, can compete with much more specific custom-made and highly tuned techniques that are already available in the astronomical literature.     

Bayesian variable selection in Poisson change-point regression analysis

In this article, we develop a Bayesian variable selection method that concerns selection of covariates in the Poisson change-point regression model with both discrete and continuous candidate covariates. Ranging from a null model with no selected covariates to a full model including all covariates, the Bayesian variable selection method searches the entire model space, estimates posterior inclusion probabilities of covariates, and obtains model averaged estimates on coefficients to covariates, while simultaneously estimating a time-varying baseline rate due to change-points. For posterior computation, the Metropolis-Hastings within partially collapsed Gibbs sampler is developed to efficiently fit the Poisson change-point regression model with variable selection. We illustrate the proposed method using simulated and real datasets.     

On cross‐validation of Bayesian models

The authors examine several aspects of cross‐validation for Bayesian models. In particular, they propose a computational scheme which does not require a separate posterior sample for each training sample.     

Hierarchical Bayesian meta-analysis models for cross-platform microarray studies

The development of new technologies to measure gene expression has been calling for statistical methods to integrate findings across multiple-platform studies. A common goal of microarray analysis is to identify genes with differential expression between two conditions, such as treatment versus control. Here, we introduce a hierarchical Bayesian meta-analysis model to pool gene expression studies from different microarray platforms: spotted DNA arrays and short oligonucleotide arrays. The studies have different array design layouts, each with multiple sources of data replication, including repeated experiments, slides and probes. Our model produces the gene-specific posterior probability of differential expression, which is the basis for inference. In simulations combining two and five independent studies, our meta-analysis model outperformed separate analyses for three commonly used comparison measures; it also showed improved receiver operating characteristic curves. When combining spotted DNA and CombiMatrix short oligonucleotide array studies of Geobacter sulfurreducens, our meta-analysis model discovered more genes for fixed thresholds of posterior probability of differential expression and Bayesian false discovery than individual study analyses. We also examine an alternative model and compare models using the deviance information criterion.     

Bayesian outlier analysis in binary regression

We propose alternative approaches to analyze residuals in binary regression models based on random effect components. Our preferred model does not depend upon any tuning parameter, being completely automatic. Although the focus is mainly on accommodation of outliers, the proposed methodology is also able to detect them. Our approach consists of evaluating the posterior distribution of random effects included in the linear predictor. The evaluation of the posterior distributions of interest involves cumbersome integration, which is easily dealt with through stochastic simulation methods. We also discuss different specifications of prior distributions for the random effects. The potential of these strategies is compared in a real data set. The main finding is that the inclusion of extra variability accommodates the outliers, improving the adjustment of the model substantially, besides correctly indicating the possible outliers.