Gibbs sampling for Bayesian non-conjugate and hierarchical models by using auxiliary variables

We demonstrate the use of auxiliary (or latent) variables for sampling non-standard densities which arise in the context of the Bayesian analysis of non-conjugate and hierarchical models by using a Gibbs sampler. Their strategic use can result in a Gibbs sampler having easily sampled full conditionals. We propose such a procedure to simplify or speed up the Markov chain Monte Carlo algorithm. The strength of this approach lies in its generality and its ease of implementation. The aim of the paper, therefore, is to provide an alternative sampling algorithm to rejection-based methods and other sampling approaches such as the Metropolis–Hastings algorithm.     

Behaviour of the Gibbs sampler when conditional distributions are potentially incompatible

The Gibbs sampler has been used extensively in the statistics literature. It relies on iteratively sampling from a set of compatible conditional distributions and the sampler is known to converge to a unique invariant joint distribution. However, the Gibbs sampler behaves rather differently when the conditional distributions are not compatible. Such applications have seen increasing use in areas such as multiple imputation. In this paper, we demonstrate that what a Gibbs sampler converges to is a function of the order of the sampling scheme. Besides providing the mathematical background of this behaviour, we also explain how that happens through a thorough analysis of the examples.     

Sequential importance sampling for nonparametric Bayes models: The next generation

There are two generations of Gibbs sampling methods for semiparametric models involving the Dirichlet process. The first generation suffered from a severe drawback: the locations of the clusters, or groups of parameters, could essentially become fixed, moving only rarely. Two strategies that have been proposed to create the second generation of Gibbs samplers are integration and appending a second stage to the Gibbs sampler wherein the cluster locations are moved. We show that these same strategies are easily implemented for the sequential importance sampler, and that the first strategy dramatically improves results. As in the case of Gibbs sampling, these strategies are applicable to a much wider class of models. They are shown to provide more uniform importance sampling weights and lead to additional Rao-Blackwellization of estimators.     

A more efficient Gibbs sampler estimation using steady-state simulation: applications to public health studies

Markov chain Monte Carlo methods, in particular, the Gibbs sampler, are widely used algorithms both in application and theoretical works in the classical and Bayesian paradigms. However, these algorithms are often computer intensive. Samawi et al. [Steady-state ranked Gibbs sampler. J. Stat. Comput. Simul. 2012;82(8), 1223–1238. doi:10.1080/00949655.2011.575378] demonstrate through theory and simulation that the dependent steady-state Gibbs sampler is more efficient and accurate in model parameter estimation than the original Gibbs sampler. This paper proposes the independent steady-state Gibbs sampler (ISSGS) approach 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 multidimensional problems.     

An iterative Monte Carlo method for nonconjugate Bayesian analysis

The Gibbs sampler has been proposed as a general method for Bayesian calculation in Gelfand and Smith (1990). However, the predominance of experience to date resides in applications assuming conjugacy where implementation is reasonably straightforward. This paper describes a tailored approximate rejection method approach for implementation of the Gibbs sampler when nonconjugate structure is present. Several challenging applications are presented for illustration.     

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.     

Steady-state ranked Gibbs sampler

Gibbs sampler as a computer-intensive algorithm is an important statistical tool both in application and in theoretical work. This algorithm, in many cases, is time-consuming; this paper extends the concept of using the steady-state ranked simulated sampling approach, utilized in Monte Carlo methods by Samawi [On the approximation of multiple integrals using steady state ranked simulated sampling, 2010, submitted for publication], to improve the well-known Gibbs sampling algorithm. It is demonstrated that this approach provides unbiased estimators, in the case of estimating the means and the distribution function, and substantially improves the performance of the Gibbs sampling algorithm and convergence, which results in a significant reduction in the costs and time required to attain a certain level of accuracy. Similar to Casella and George [Explaining the Gibbs sampler, Am. Statist. 46(3) (1992), pp. 167–174], we provide some analytical properties in simple cases and compare the performance of our method using the same illustrations.     

Nonlinear mixed‐effect models with nonignorably missing covariates

Nonlinear mixed‐effect models are often used in the analysis of longitudinal data. However, it sometimes happens that missing values for some of the model covariates are not purely random. Motivated by an application to HTV viral dynamics, where this situation occurs, the author considers likelihood inference for this type of problem. His approach involves a Monte Carlo EM algorithm, along with a Gibbs sampler and rejection/importance sampling methods. A concrete application is provided.     

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.     

Sampling the Dirichlet Mixture Model with Slices

We provide a new approach to the sampling of the well known mixture of Dirichlet process model. Recent attention has focused on retention of the random distribution function in the model, but sampling algorithms have then suffered from the countably infinite representation these distributions have. The key to the algorithm detailed in this article, which also keeps the random distribution functions, is the introduction of a latent variable which allows a finite number, which is known, of objects to be sampled within each iteration of a Gibbs sampler.     

On implementation of the Gibbs sampler for estimating the accuracy of multiple diagnostic tests

Implementation of the Gibbs sampler for estimating the accuracy of multiple binary diagnostic tests in one population has been investigated. This method, proposed by Joseph, Gyorkos and Coupal, makes use of a Bayesian approach and is used in the absence of a gold standard to estimate the prevalence, the sensitivity and specificity of medical diagnostic tests. The expressions that allow this method to be implemented for an arbitrary number of tests are given. By using the convergence diagnostics procedure of Raftery and Lewis, the relation between the number of iterations of Gibbs sampling and the precision of the estimated quantiles of the posterior distributions is derived. An example concerning a data set of gastro-esophageal reflux disease patients collected to evaluate the accuracy of the water siphon test compared with 24 h pH-monitoring, endoscopy and histology tests is presented. The main message that emerges from our analysis is that implementation of the Gibbs sampler to estimate the parameters of multiple binary diagnostic tests can be critical and convergence diagnostic is advised for this method. The factors which affect the convergence of the chains to the posterior distributions and those that influence the precision of their quantiles are analyzed.     

A new strategy for speeding Markov chain Monte Carlo algorithms

Markov chain Monte Carlo (MCMC) methods have become popular as a basis for drawing inference from complex statistical models. Two common difficulties with MCMC algorithms are slow mixing and long run-times, which are frequently closely related. Mixing over the entire state space can often be aided by careful tuning of the chain's transition kernel. In order to preserve the algorithm's stationary distribution, however, care must be taken when updating a chain's transition kernel based on that same chain's history. In this paper we introduce a technique that allows the transition kernel of the Gibbs sampler to be updated at user specified intervals, while preserving the chain's stationary distribution. This technique seems to be beneficial both in increasing efficiency of the resulting estimates (via Rao-Blackwellization) and in reducing the run-time. A reinterpretation of the modified Gibbs sampling scheme introduced in terms of auxiliary samples allows its extension to the more general Metropolis-Hastings framework. The strategies we develop are particularly helpful when calculation of the full conditional (for a Gibbs algorithm) or of the proposal distribution (for a Metropolis-Hastings algorithm) is computationally expensive. Partial financial support from FAR 2002-3, University of Insubria is gratefully acknowledged.     

Optimal direction Gibbs sampler for truncated multivariate normal distributions

Generalized Gibbs samplers simulate from any direction, not necessarily limited to the coordinate directions of the parameters of the objective function. We study how to optimally choose such directions in a random scan Gibbs sampler setting. We consider that optimal directions will be those that minimize the Kullback–Leibler divergence of two Markov chain Monte Carlo steps. Two distributions over direction are proposed for the multivariate Normal objective function. The resulting algorithms are used to simulate from a truncated multivariate Normal distribution, and the performance of our algorithms is compared with the performance of two algorithms based on the Gibbs sampler.     

Analysis of the Gibbs sampler for a model related to James-Stein estimators

We analyse a hierarchical Bayes model which is related to the usual empirical Bayes formulation of James-Stein estimators. We consider running a Gibbs sampler on this model. Using previous results about convergence rates of Markov chains, we provide rigorous, numerical, reasonable bounds on the running time of the Gibbs sampler, for a suitable range of prior distributions. We apply these results to baseball data from Efron and Morris (1975). For a different range of prior distributions, we prove that the Gibbs sampler will fail to converge, and use this information to prove that in this case the associated posterior distribution is non-normalizable.     

Variable selection in quantile regression via Gibbs sampling

Due to computational challenges and non-availability of conjugate prior distributions, Bayesian variable selection in quantile regression models is often a difficult task. In this paper, we address these two issues for quantile regression models. In particular, we develop an informative stochastic search variable selection (ISSVS) for quantile regression models that introduces an informative prior distribution. We adopt prior structures which incorporate historical data into the current data by quantifying them with a suitable prior distribution on the model parameters. This allows ISSVS to search more efficiently in the model space and choose the more likely models. In addition, a Gibbs sampler is derived to facilitate the computation of the posterior probabilities. A major advantage of ISSVS is that it avoids instability in the posterior estimates for the Gibbs sampler as well as convergence problems that may arise from choosing vague priors. Finally, the proposed methods are illustrated with both simulation and real data.     

市场风险的高效率加速算法研究

在计算投资组合市场风险时,采用高效率重要性抽样技术来处理大规模、高维度和稀有事件问题可以提高计算的速度和效率。在对投资组合损失进行Delta-Gamma近似的基础上,通过利用辅助分布变换函数,将求解抽样参数的最小抽样方差问题转化为一个非线性的广义最小二乘问题;在指数族抽样核的假设下,进一步将问题转化为迭代线性回归问题,从而简化了计算;通过德尔塔对冲和指数对冲投资组合的模拟算例验证了所提出方法的有效性。     

Efficient Markov chain Monte Carlo with incomplete multinomial data

We propose a block Gibbs sampling scheme for incomplete multinomial data. We show that the new approach facilitates maximal blocking, thereby reducing serial dependency and speeding up the convergence of the Gibbs sampler. We compare the efficiency of the new method with the standard, non-block Gibbs sampler via a number of numerical examples.     

A simulation approach to convergence rates for Markov chain Monte Carlo algorithms

Markov chain Monte Carlo (MCMC) methods, including the Gibbs sampler and the Metropolis–Hastings algorithm, are very commonly used in Bayesian statistics for sampling from complicated, high-dimensional posterior distributions. A continuing source of uncertainty is how long such a sampler must be run in order to converge approximately to its target stationary distribution. A method has previously been developed to compute rigorous theoretical upper bounds on the number of iterations required to achieve a specified degree of convergence in total variation distance by verifying drift and minorization conditions. We propose the use of auxiliary simulations to estimate the numerical values needed in this theorem. Our simulation method makes it possible to compute quantitative convergence bounds for models for which the requisite analytical computations would be prohibitively difficult or impossible. On the other hand, although our method appears to perform well in our example problems, it cannot provide the guarantees offered by analytical proof.     

A bayesian approach to dynamic tobit models

This paper develops a posterior simulation method for a dynamic Tobit model. The major obstacle rooted in such a problem lies in high dimensional integrals, induced by dependence among censored observations, in the likelihood function. The primary contribution of this study is to develop a practical and efficient sampling scheme for the conditional posterior distributions of the censored (i.e., unobserved) data, so that the Gibbs sampler with the data augmentation algorithm is successfully applied. The substantial differences between this approach and some existing methods are highlighted. The proposed simulation method is investigated by means of a Monte Carlo study and applied to a regression model of Japanese exports of passenger cars to the U.S. subject to a non-tariff trade barrier.     

A bayesian approach to dynamic tobit models

This paper develops a posterior simulation method for a dynamic Tobit model. The major obstacle rooted in such a problem lies in high dimensional integrals, induced by dependence among censored observations, in the likelihood function. The primary contribution of this study is to develop a practical and efficient sampling scheme for the conditional posterior distributions of the censored (i.e., unobserved) data, so that the Gibbs sampler with the data augmentation algorithm is successfully applied. The substantial differences between this approach and some existing methods are highlighted. The proposed simulation method is investigated by means of a Monte Carlo study and applied to a regression model of Japanese exports of passenger cars to the U.S. subject to a non-tariff trade barrier.