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.     

Statistical Certification of Software Systems

Common software release procedures based on statistical techniques try to optimize the trade-off between further testing costs and costs due to remaining errors. We propose new software release procedures where the aim is to certify with a certain confidence level that the software does not contain errors. The underlying model is a discrete time model similar to the geometric Moranda model. The decisions are based on a mix of classical and Bayesian approaches to sequential testing and do not require any assumption on the initial number of errors.     

Bayesian Estimation of the Parameters of Bivariate Exponential Distributions

In this article, we develop an empirical Bayesian approach for the Bayesian estimation of parameters in four bivariate exponential (BVE) distributions. We have opted for gamma distribution as a prior for the parameters of the model in which the hyper parameters have been estimated based on the method of moments and maximum likelihood estimates (MLEs). A simulation study was conducted to compute empirical Bayesian estimates of the parameters and their standard errors. We use moment estimators or MLEs to estimate the hyper parameters of the prior distributions. Furthermore, we compare the posterior mode of parameters obtained by different prior distributions and the Bayesian estimates based on gamma priors are very close to the true values as compared to improper priors. We use MCMC method to obtain the posterior mean and compared the same using the improper priors and the classical estimates, MLEs.     

Bayesian model averaging in longitudinal regression models with AR(1) errors with application to a myopia data set

We propose a new iterative algorithm, called model walking algorithm, to the Bayesian model averaging method on the longitudinal regression models with AR(1) random errors within subjects. The Markov chain Monte Carlo method together with the model walking algorithm are employed. The proposed method is successfully applied to predict the progression rates on a myopia intervention trial in children.     

Bayesian generalized varying coefficient models for longitudinal proportional data with errors-in-covariates

This paper is motivated from a neurophysiological study of muscle fatigue, in which biomedical researchers are interested in understanding the time-dependent relationships of handgrip force and electromyography measures. A varying coefficient model is appealing here to investigate the dynamic pattern in the longitudinal data. The response variable in the study is continuous but bounded on the standard unit interval (0, 1) over time, while the longitudinal covariates are contaminated with measurement errors. We propose a generalization of varying coefficient models for the longitudinal proportional data with errors-in-covariates. We describe two estimation methods with penalized splines, which are formalized under a Bayesian inferential perspective. The first method is an adaptation of the popular regression calibration approach. The second method is based on a joint likelihood under the hierarchical Bayesian model. A simulation study is conducted to evaluate the efficacy of the proposed methods under different scenarios. The analysis of the neurophysiological data is presented to demonstrate the use of the methods.     

Bayesian validation assessment of multivariate computational models

Multivariate model validation is a complex decision-making problem involving comparison of multiple correlated quantities, based upon the available information and prior knowledge. This paper presents a Bayesian risk-based decision method for validation assessment of multivariate predictive models under uncertainty. A generalized likelihood ratio is derived as a quantitative validation metric based on Bayes’ theorem and Gaussian distribution assumption of errors between validation data and model prediction. The multivariate model is then assessed based on the comparison of the likelihood ratio with a Bayesian decision threshold, a function of the decision costs and prior of each hypothesis. The probability density function of the likelihood ratio is constructed using the statistics of multiple response quantities and Monte Carlo simulation. The proposed methodology is implemented in the validation of a transient heat conduction model, using a multivariate data set from experiments. The Bayesian methodology provides a quantitative approach to facilitate rational decisions in multivariate model assessment under uncertainty.     

Selection of Multivariate Stochastic Volatility Models via Bayesian Stochastic Search

We propose a Bayesian stochastic search approach to selecting restrictions on multivariate regression models where the errors exhibit deterministic or stochastic conditional volatilities. We develop a Markov chain Monte Carlo (MCMC) algorithm that generates posterior restrictions on the regression coefficients and Cholesky decompositions of the covariance matrix of the errors. Numerical simulations with artificially generated data show that the proposed method is effective in selecting the data-generating model restrictions and improving the forecasting performance of the model. Applying the method to daily foreign exchange rate data, we conduct stochastic search on a VAR model with stochastic conditional volatilities.     

Methods for Constructing Uncertainty Intervals for Queries of Bayesian Nets

In this paper the issue of finding uncertainty intervals for queries in a Bayesian Network is reconsidered. The investigation focuses on Bayesian Nets with discrete nodes and finite populations. An earlier asymptotic approach is compared with a simulation‐based approach, together with further alternatives, one based on a single sample of the Bayesian Net of a particular finite population size, and another which uses expected population sizes together with exact probabilities. We conclude that a query of a Bayesian Net should be expressed as a probability embedded in an uncertainty interval. Based on an investigation of two Bayesian Net structures, the preferred method is the simulation method. However, both the single sample method and the expected sample size methods may be useful and are simpler to compute. Any method at all is more useful than none, when assessing a Bayesian Net under development, or when drawing conclusions from an ‘expert’ system.     

Semiparametric Bayesian inference for regression models

This paper presents a method for Bayesian inference for the regression parameters in a linear model with independent and identically distributed errors that does not require the specification of a parametric family of densities for the error distribution. This method first selects a nonparametric kernel density estimate of the error distribution which is unimodal and based on the least-squares residuals. Once the error distribution is selected, the Metropolis algorithm is used to obtain the marginal posterior distribution of the regression parameters. The methodology is illustrated with data sets, and its performance relative to standard Bayesian techniques is evaluated using simulation results.     

Bayesian modeling of autoregressive partial linear models with scale mixture of normal errors

Normality and independence of error terms are typical assumptions for partial linear models. However, these assumptions may be unrealistic in many fields, such as economics, finance and biostatistics. In this paper, a Bayesian analysis for partial linear model with first-order autoregressive errors belonging to the class of the scale mixtures of normal distributions is studied in detail. The proposed model provides a useful generalization of the symmetrical linear regression model with independent errors, since the distribution of the error term covers both correlated and thick-tailed distributions, and has a convenient hierarchical representation allowing easy implementation of a Markov chain Monte Carlo scheme. In order to examine the robustness of the model against outlying and influential observations, a Bayesian case deletion influence diagnostics based on the Kullback–Leibler (K–L) divergence is presented. The proposed method is applied to monthly and daily returns of two Chilean companies.     

A Bayesian Analysis of Autoregressive Models with Exogenous Variables and Power-Transformed and Threshold GARCH Errors

Consider a class of autoregressive models with exogenous variables and power transformed and threshold GARCH (ARX-PTTGARCH) errors, which is a natural generalization of the standard and special GARCH model. We propose a Bayesian method to show that combining Gibbs sampler and Metropolis-Hastings algorithm to give a Bayesian analysis can be applied to estimate parameters of ARX-PTTGARCH models with success.     

Choosing summary statistics by least angle regression for approximate Bayesian computation

Bayesian statistical inference relies on the posterior distribution. Depending on the model, the posterior can be more or less difficult to derive. In recent years, there has been a lot of interest in complex settings where the likelihood is analytically intractable. In such situations, approximate Bayesian computation (ABC) provides an attractive way of carrying out Bayesian inference. For obtaining reliable posterior estimates however, it is important to keep the approximation errors small in ABC. The choice of an appropriate set of summary statistics plays a crucial role in this effort. Here, we report the development of a new algorithm that is based on least angle regression for choosing summary statistics. In two population genetic examples, the performance of the new algorithm is better than a previously proposed approach that uses partial least squares.     

Bayesian composite quantile regression for linear mixed-effects models

Longitudinal data are commonly modeled with the normal mixed-effects models. Most modeling methods are based on traditional mean regression, which results in non robust estimation when suffering extreme values or outliers. Median regression is also not a best choice to estimation especially for non normal errors. Compared to conventional modeling methods, composite quantile regression can provide robust estimation results even for non normal errors. In this paper, based on a so-called pseudo composite asymmetric Laplace distribution (PCALD), we develop a Bayesian treatment to composite quantile regression for mixed-effects models. Furthermore, with the location-scale mixture representation of the PCALD, we establish a Bayesian hierarchical model and achieve the posterior inference of all unknown parameters and latent variables using Markov Chain Monte Carlo (MCMC) method. Finally, this newly developed procedure is illustrated by some Monte Carlo simulations and a case analysis of HIV/AIDS clinical data set.     

Model selection in spline nonparametric regression

A Bayesian approach is presented for model selection in nonparametric regression with Gaussian errors and in binary nonparametric regression. A smoothness prior is assumed for each component of the model and the posterior probabilities of the candidate models are approximated using the Bayesian information criterion. We study the model selection method by simulation and show that it has excellent frequentist properties and gives improved estimates of the regression surface. All the computations are carried out efficiently using the Gibbs sampler.     

Combining Economic Forecasts

A method for combining forecasts may or may not account for dependence and differing precision among forecasts. In this article we test a variety of such methods in the context of combining forecasts of GNP from four major econometric models. The methods include one in which forecasting errors are jointly normally distributed and several variants of this model as well as some simpler procedures and a Bayesian approach with a prior distribution based on exchangeability of forecasters. The results indicate that a simple average, the normal model with an independence assumption, and the Bayesian model perform better than the other approaches that are studied here.     

Bayesian Estimators for Small Area Models when Auxiliary Information is Measured with Error

Small area estimators in linear models are typically expressed as a convex combination of direct estimators and synthetic estimators from a suitable model. When auxiliary information used in the model is measured with error, a new estimator, accounting for the measurement error in the covariates, has been proposed in the literature. Recently, for area‐level model, Ybarra & Lohr (Biometrika, 95, 2008, 919) suggested a suitable modification to the estimates of small area means based on Fay & Herriot (J. Am. Stat. Assoc., 74, 1979, 269) model where some of the covariates are measured with error. They used a frequentist approach based on the method of moments. Adopting a Bayesian approach, we propose to rewrite the measurement error model as a hierarchical model; we use improper non‐informative priors on the model parameters and show, under a mild condition, that the joint posterior distribution is proper and the marginal posterior distributions of the model parameters have finite variances. We conduct a simulation study exploring different scenarios. The Bayesian predictors we propose show smaller empirical mean squared errors than the frequentist predictors of Ybarra & Lohr (Biometrika, 95, 2008, 919), and they seem also to be more stable in terms of variability and bias. We apply the proposed methodology to two real examples.     

A Bayesian approach to assessing population bioequivalence in a 2 ×2 crossover design

A Bayesian testing procedure is proposed for assessment of the bioequivalence in both mean and variance, which ensures population bioequivalence under the normality assumption. We derive the joint posterior distribution of the means and variances in a standard 2 ×2 crossover experimental design and propose a Bayesian testing procedure for bioequivalence based on a Markov chain Monte Carlo method. The proposed method is applied to a real data set.     

Bayesian spectral analysis models for quantile regression with Dirichlet process mixtures

This paper presents a Bayesian analysis of partially linear additive models for quantile regression. We develop a semiparametric Bayesian approach to quantile regression models using a spectral representation of the nonparametric regression functions and the Dirichlet process (DP) mixture for error distribution. We also consider Bayesian variable selection procedures for both parametric and nonparametric components in a partially linear additive model structure based on the Bayesian shrinkage priors via a stochastic search algorithm. Based on the proposed Bayesian semiparametric additive quantile regression model referred to as BSAQ, the Bayesian inference is considered for estimation and model selection. For the posterior computation, we design a simple and efficient Gibbs sampler based on a location-scale mixture of exponential and normal distributions for an asymmetric Laplace distribution, which facilitates the commonly used collapsed Gibbs sampling algorithms for the DP mixture models. Additionally, we discuss the asymptotic property of the sempiparametric quantile regression model in terms of consistency of posterior distribution. Simulation studies and real data application examples illustrate the proposed method and compare it with Bayesian quantile regression methods in the literature.     

Bayesian Random Segmentation Models to Identify Shared Copy Number Aberrations for Array CGH Data

Array-based comparative genomic hybridization (aCGH) is a high-resolution high-throughput technique for studying the genetic basis of cancer. The resulting data consists of log fluorescence ratios as a function of the genomic DNA location and provides a cytogenetic representation of the relative DNA copy number variation. Analysis of such data typically involves estimation of the underlying copy number state at each location and segmenting regions of DNA with similar copy number states. Most current methods proceed by modeling a single sample/array at a time, and thus fail to borrow strength across multiple samples to infer shared regions of copy number aberrations. We propose a hierarchical Bayesian random segmentation approach for modeling aCGH data that utilizes information across arrays from a common population to yield segments of shared copy number changes. These changes characterize the underlying population and allow us to compare different population aCGH profiles to assess which regions of the genome have differential alterations. Our method, referred to as BDSAcgh (Bayesian Detection of Shared Aberrations in aCGH), is based on a unified Bayesian hierarchical model that allows us to obtain probabilities of alteration states as well as probabilities of differential alteration that correspond to local false discovery rates. We evaluate the operating characteristics of our method via simulations and an application using a lung cancer aCGH data set.     

On Variational Bayes Estimation and Variational Information Criteria for Linear Regression Models

Variational Bayes (VB) estimation is a fast alternative to Markov Chain Monte Carlo for performing approximate Baesian inference. This procedure can be an efficient and effective means of analyzing large datasets. However, VB estimation is often criticised, typically on empirical grounds, for being unable to produce valid statistical inferences. In this article we refute this criticism for one of the simplest models where Bayesian inference is not analytically tractable, that is, the Bayesian linear model (for a particular choice of priors). We prove that under mild regularity conditions, VB based estimators enjoy some desirable frequentist properties such as consistency and can be used to obtain asymptotically valid standard errors. In addition to these results we introduce two VB information criteria: the variational Akaike information criterion and the variational Bayesian information criterion. We show that variational Akaike information criterion is asymptotically equivalent to the frequentist Akaike information criterion and that the variational Bayesian information criterion is first order equivalent to the Bayesian information criterion in linear regression. These results motivate the potential use of the variational information criteria for more complex models. We support our theoretical results with numerical examples.