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 inference for double SARMA models

In this paper, we present a Bayesian analysis of double seasonal autoregressive moving average models. We first consider the problem of estimating unknown lagged errors in the moving average part using non linear least squares method, and then using natural conjugate and Jeffreys’ priors we approximate the marginal posterior distributions to be multivariate t and gamma distributions for the model coefficients and precision, respectively. We evaluate the proposed Bayesian methodology using simulation study, and apply to real-world hourly electricity load data sets.     

Bayesian mismeasurement t-models for censored responses

This paper aims at introducing a Bayesian robust error-in-variable regression model in which the dependent variable is censored. We extend previous works by assuming a multivariate t distribution for jointly modelling the behaviour of the errors and the latent explanatory variable. Inference is done under the Bayesian paradigm. We use a data augmentation approach and develop a Markov chain Monte Carlo algorithm to sample from the posterior distributions. We run a Monte Carlo study to evaluate the efficiency of the posterior estimators in different settings. We compare the proposed model to three other models previously discussed in the literature. As a by-product we also provide a Bayesian analysis of the t-tobit model. We fit all four models to analyse the 2001 Medical Expenditure Panel Survey data.     

Bayesian identification of double seasonal autoregressive time series models

ABSTRACT

Seasonal autoregressive (SAR) models have been modified and extended to model high frequency time series characterized by exhibiting double seasonal patterns. Some researchers have introduced Bayesian inference for double seasonal autoregressive (DSAR) models; however, none has tackled the problem of Bayesian identification of DSAR models. Therefore, in order to fill this gap, we present a Bayesian methodology to identify the order of DSAR models. Assuming the model errors are normally distributed and using three priors, i.e. natural conjugate, g, and Jeffreys’ priors, on the model parameters, we derive the joint posterior mass function of the model order in a closed-form. Accordingly, the posterior mass function can be investigated and the best order of DSAR model is chosen as a value with the highest posterior probability for the time series being analyzed. We evaluate the proposed Bayesian methodology using simulation study, and we then apply it to real-world hourly internet amount of traffic dataset.     

Bayesian Computations for Random Environment Models

This paper deals with the analysis of reliability data from a Bayesian perspective for Random Environment (RE) models. We give an overview of current literature on RE models. We also study the computational problems associated with the implementations of RE models in a Bayesian setting. Then, we present the Markov Chain Monte Carlo technique to solve such problems. These problems arise in posterior and predictive analysis and their relevant quantities such as mean, variance, and median. The suggested methodology is incorporated with an illustration.     

Hierarchical Bayesian models for predicting spatially correlated curves

Functional data analysis has emerged as a new area of statistical research with a wide range of applications. In this paper, we propose novel models based on wavelets for spatially correlated functional data. These models enable one to regularize curves observed over space and predict curves at unobserved sites. We compare the performance of these Bayesian models with several priors on the wavelet coefficients using the posterior predictive criterion. The proposed models are illustrated in the analysis of porosity data.     

Bayesian Analysis in Regression Models Using Pseudo-Likelihoods

This article deals with the issue of using a suitable pseudo-likelihood, instead of an integrated likelihood, when performing Bayesian inference about a scalar parameter of interest in the presence of nuisance parameters. The proposed approach has the advantages of avoiding the elicitation on the nuisance parameters and the computation of multidimensional integrals. Moreover, it is particularly useful when it is difficult, or even impractical, to write the full likelihood function.

We focus on Bayesian inference about a scalar regression coefficient in various regression models. First, in the context of non-normal regression-scale models, we give a theroetical result showing that there is no loss of information about the parameter of interest when using a posterior distribution derived from a pseudo-likelihood instead of the correct posterior distribution. Second, we present non trivial applications with high-dimensional, or even infinite-dimensional, nuisance parameters in the context of nonlinear normal heteroscedastic regression models, and of models for binary outcomes and count data, accounting also for possibile overdispersion. In all these situtations, we show that non Bayesian methods for eliminating nuisance parameters can be usefully incorporated into a one-parameter Bayesian analysis.     

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.     

Semiparametric Bayesian hierarchical models for heterogeneous population in nonlinear mixed effect model: application to gastric emptying studies

Gastric emptying studies are frequently used in medical research, both human and animal, when evaluating the effectiveness and determining the unintended side-effects of new and existing medications, diets, and procedures or interventions. It is essential that gastric emptying data be appropriately summarized before making comparisons between study groups of interest and to allow study the comparisons. Since gastric emptying data have a nonlinear emptying curve and are longitudinal data, nonlinear mixed effect (NLME) models can accommodate both the variation among measurements within individuals and the individual-to-individual variation. However, the NLME model requires strong assumptions that are often not satisfied in real applications that involve a relatively small number of subjects, have heterogeneous measurement errors, or have large variation among subjects. Therefore, we propose three semiparametric Bayesian NLMEs constructed with Dirichlet process priors, which automatically cluster sub-populations and estimate heterogeneous measurement errors. To compare three semiparametric models with the parametric model we propose a penalized posterior Bayes factor. We compare the performance of our semiparametric hierarchical Bayesian approaches with that of the parametric Bayesian hierarchical approach. Simulation results suggest that our semiparametric approaches are more robust and flexible. Our gastric emptying studies from equine medicine are used to demonstrate the advantage of our approaches.     

Posterior Analysis for Normalized Random Measures with Independent Increments

Abstract.  One of the main research areas in Bayesian Nonparametrics is the proposal and study of priors which generalize the Dirichlet process. In this paper, we provide a comprehensive Bayesian non-parametric analysis of random probabilities which are obtained by normalizing random measures with independent increments (NRMI). Special cases of these priors have already shown to be useful for statistical applications such as mixture models and species sampling problems. However, in order to fully exploit these priors, the derivation of the posterior distribution of NRMIs is crucial: here we achieve this goal and, indeed, provide explicit and tractable expressions suitable for practical implementation. The posterior distribution of an NRMI turns out to be a mixture with respect to the distribution of a specific latent variable. The analysis is completed by the derivation of the corresponding predictive distributions and by a thorough investigation of the marginal structure. These results allow to derive a generalized Blackwell–MacQueen sampling scheme, which is then adapted to cover also mixture models driven by general NRMIs.     

Bayesian inference for joint location and scale nonlinear models with skew-normal errors

A regression model with skew-normal errors provides a useful extension for ordinary normal regression models when the dataset under consideration involves asymmetric outcomes. In this article, we explore the use of Markov Chain Monte Carlo (MCMC) methods to develop a Bayesian analysis for joint location and scale nonlinear models with skew-normal errors, which relax the normality assumption and include the normal one as a special case. The main advantage of these class of distributions is that they have a nice hierarchical representation that allows the implementation of MCMC methods to simulate samples from the joint posterior distribution. Finally, simulation studies and a real example are used to illustrate the proposed methodology.     

Bayesian model selection in linear mixed effects models with autoregressive(p) errors using mixture priors

In this article, we apply the Bayesian approach to the linear mixed effect models with autoregressive(p) random errors under mixture priors obtained with the Markov chain Monte Carlo (MCMC) method. The mixture structure of a point mass and continuous distribution can help to select the variables in fixed and random effects models from the posterior sample generated using the MCMC method. Bayesian prediction of future observations is also one of the major concerns. To get the best model, we consider the commonly used highest posterior probability model and the median posterior probability model. As a result, both criteria tend to be needed to choose the best model from the entire simulation study. In terms of predictive accuracy, a real example confirms that the proposed method provides accurate results.     

A Semi-parametric Bayesian Analysis of Survival Data Based on Lévy-driven Processes

In the presence of covariate information, the proportional hazards model is one of the most popular models. In this paper, in a Bayesian nonparametric framework, we use a Markov (Lévy-driven) process to model the baseline hazard rate. Previous Bayesian nonparametric models have been based on neutral to the right processes, which have a number of drawbacks, such as discreteness of the cumulative hazard function. We allow the covariates to be time dependent functions and develop a full posterior analysis via substitution sampling. A detailed illustration is presented.     

Nonparametric Bayesian survival analysis using mixtures of Weibull distributions

Bayesian nonparametric methods have been applied to survival analysis problems since the emergence of the area of Bayesian nonparametrics. However, the use of the flexible class of Dirichlet process mixture models has been rather limited in this context. This is, arguably, to a large extent, due to the standard way of fitting such models that precludes full posterior inference for many functionals of interest in survival analysis applications. To overcome this difficulty, we provide a computational approach to obtain the posterior distribution of general functionals of a Dirichlet process mixture. We model the survival distribution employing a flexible Dirichlet process mixture, with a Weibull kernel, that yields rich inference for several important functionals. In the process, a method for hazard function estimation emerges. Methods for simulation-based model fitting, in the presence of censoring, and for prior specification are provided. We illustrate the modeling approach with simulated and real data.     

Fast Bayesian variable screenings for binary response regressions with small sample size

Screening procedures play an important role in data analysis, especially in high-throughput biological studies where the datasets consist of more covariates than independent subjects. In this article, a Bayesian screening procedure is introduced for the binary response models with logit and probit links. In contrast to many screening rules based on marginal information involving one or a few covariates, the proposed Bayesian procedure simultaneously models all covariates and uses closed-form screening statistics. Specifically, we use the posterior means of the regression coefficients as screening statistics; by imposing a generalized g-prior on the regression coefficients, we derive the analytical form of their posterior means and compute the screening statistics without Markov chain Monte Carlo implementation. We evaluate the utility of the proposed Bayesian screening method using simulations and real data analysis. When the sample size is small, the simulation results suggest improved performance with comparable computational cost.     

Interval-censored data with misclassification: a Bayesian approach

Survival data involving silent events are often subject to interval censoring (the event is known to occur within a time interval) and classification errors if a test with no perfect sensitivity and specificity is applied. Considering the nature of this data plays an important role in estimating the time distribution until the occurrence of the event. In this context, we incorporate validation subsets into the parametric proportional hazard model, and show that this additional data, combined with Bayesian inference, compensate the lack of knowledge about test sensitivity and specificity improving the parameter estimates. The proposed model is evaluated through simulation studies, and Bayesian analysis is conducted within a Gibbs sampling procedure. The posterior estimates obtained under validation subset models present lower bias and standard deviation compared to the scenario with no validation subset or the model that assumes perfect sensitivity and specificity. Finally, we illustrate the usefulness of the new methodology with an analysis of real data about HIV acquisition in female sex workers that have been discussed in the literature.     

A bayesian analysis of trend determination in economic time series

In this paper we provide a comprehensive Bayesian posterior analysis of trend determination in general autoregressive models. Multiple lag autoregressive models with fitted drifts and time trends as well as models that allow for certain types of structural change in the deterministic components are considered. We utilize a modified information matrix-based prior that accommodates stochastic nonstationarity, takes into account the interactions between long-run and short-run dynamics and controls the degree of stochastic nonstationarity permitted. We derive analytic posterior densities for all of the trend determining parameters via the Laplace approximation to multivariate integrals. We also address the sampling properties of our posteriors under alternative data generating processes by simulation methods. We apply our Bayesian techniques to the Nelson-Plosser macroeconomic data and various stock price and dividend data. Contrary to DeJong and Whiteman (1989a,b,c), we do not find that the data overwhelmingly favor the existence of deterministic trends over stochastic trends. In addition, we find evidence supporting Perron's (1989) view that some of the Nelson and Plosser data are best construed as trend stationary with a change in the trend function occurring at 1929.     

Rate estimation in partially observed Markov jump processes with measurement errors

We present a simulation methodology for Bayesian estimation of rate parameters in Markov jump processes arising for example in stochastic kinetic models. To handle the problem of missing components and measurement errors in observed data, we embed the Markov jump process into the framework of a general state space model. We do not use diffusion approximations. Markov chain Monte Carlo and particle filter type algorithms are introduced which allow sampling from the posterior distribution of the rate parameters and the Markov jump process also in data-poor scenarios. The algorithms are illustrated by applying them to rate estimation in a model for prokaryotic auto-regulation and the stochastic Oregonator, respectively.     

A bayesian analysis of trend determination in economic time series

In this paper we provide a comprehensive Bayesian posterior analysis of trend determination in general autoregressive models. Multiple lag autoregressive models with fitted drifts and time trends as well as models that allow for certain types of structural change in the deterministic components are considered. We utilize a modified information matrix-based prior that accommodates stochastic nonstationarity, takes into account the interactions between long-run and short-run dynamics and controls the degree of stochastic nonstationarity permitted. We derive analytic posterior densities for all of the trend determining parameters via the Laplace approximation to multivariate integrals. We also address the sampling properties of our posteriors under alternative data generating processes by simulation methods. We apply our Bayesian techniques to the Nelson-Plosser macroeconomic data and various stock price and dividend data. Contrary to DeJong and Whiteman (1989a,b,c), we do not find that the data overwhelmingly favor the existence of deterministic trends over stochastic trends. In addition, we find evidence supporting Perron's (1989) view that some of the Nelson and Plosser data are best construed as trend stationary with a change in the trend function occurring at 1929.     

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.