Sujit K. Sahu

A new class of multivariate skew distributions with applications to bayesian regression models

The authors develop a new class of distributions by introducing skewness in multivariate ellip- tically symmetric distributions. The class, which is obtained by using transformation and conditioning, contains many standard families including the multivariate skew-normal and distributions. The authors obtain analytical forms of the densities and study distributional properties. They give practical applica- tions in Bayesian regression models and results on the existence of the posterior distributions and moments under improper priors for the regression coefficients. They illustrate their methods using practical examples.

Updating Schemes, Correlation Structure, Blocking and Parameterization for the Gibbs Sampler

In this paper many convergence issues concerning the implementation of the Gibbs sampler are investigated. Exact computable rates of convergence for Gaussian target distributions are obtained. Different random and non-random updating strategies and blocking combinations are compared using the rates. The effect of dimensionality and correlation structure on the convergence rates are studied. Some examples are considered to demonstrate the results. For a Gaussian image analysis problem several updating strategies are described and compared. For problems in Bayesian linear models several possible parameterizations are analysed in terms of their convergence rates characterizing the optimal choice.

Adaptive Markov Chain Monte Carlo through Regeneration

Abstract Markov chain Monte Carlo (MCMC) is used for evaluating expectations of functions of interest under a target distribution π. This is done by calculating averages over the sample path of a Markov chain having π as its stationary distribution. For computational efficiency, the Markov chain should be rapidly mixing. This sometimes can be achieved only by careful design of the transition kernel of the chain, on the basis of a detailed preliminary exploratory analysis of π. An alternative approach might be to allow the transition kernel to adapt whenever new features of π are encountered during the MCMC run. However, if such adaptation occurs infinitely often, then the stationary distribution of the chain may be disturbed. We describe a framework, based on the concept of Markov chain regeneration, which allows adaptation to occur infinitely often but does not disturb the stationary distribution of the chain or the consistency of sample path averages.

Identifiability, Improper Priors, and Gibbs Sampling for Generalized Linear Models

Markov chain Monte Carlo algorithms are widely used in the fitting of generalized linear models (GLMs). Such model fitting is somewhat of an art form, requiring suitable trickery and tuning to obtain results in which one can have confidence. A wide range of practical issues arise. The focus here is on parameter identifiability and posterior propriety. In particular, we clarify that nonidentifiability arises for usual GLMs and discuss its implications for simulation-based model fitting. Because often some part of the prior specification is vague, we consider whether the resulting posterior is proper, providing rather general and easily checked results for GLMs. We also show that if a Gibbs sampler is run with an improper posterior, then it may be possible to use the output to obtain meaningful inference for certain model unknowns.

A Weibull Regression Model with Gamma Frailties for Multivariate Survival Data

Frequently in the analysis of survival data, survival times within the same group are correlated due to unobserved co-variates. One way these co-variates can be included in the model is as frailties. These frailty random block effects generate dependency between the survival times of the individuals which are conditionally independent given the frailty. Using a conditional proportional hazards model, in conjunction with the frailty, a whole new family of models is introduced. By considering a gamma frailty model, often the issue is to find an appropriate model for the baseline hazard function. In this paper a flexible baseline hazard model based on a correlated prior process is proposed and is compared with a standard Weibull model. Several model diagnostics methods are developed and model comparison is made using recently developed Bayesian model selection criteria. The above methodologies are applied to the McGilchrist and Aisbett (1991) kidney infection data and the analysis is performed using Markov Chain Monte Carlo methods.

High Resolution Space-Time Ozone Modeling for Assessing Trends.

This article proposes a space–time model for daily 8-hour maximum ozone levels to provide input for regulatory activities: detection, evaluation, and analysis of spatial patterns and temporal trend in ozone summaries. The model is applied to the analysis of data from the state of Ohio that contains a mix of urban, suburban, and rural ozone monitoring sites. The proposed space–time model is autoregressive and incorporates the most important meteorological variables observed at a collection of ozone monitoring sites as well as at several weather stations where ozone levels have not been observed. This misalignment is handled through spatial modeling. In so doing we adopt a computationally convenient approach based on the successive daily increments in meteorological variables. The resulting hierarchical model is specified within a Bayesian framework and is fitted using Markov chain Monte Carlo techniques. Full inference with regard to model unknowns as well as for predictions in time and space, evaluation of annual summaries, and assessment of trends are presented.

Spatio-temporal modeling of fine particulate matter

Studies indicate that even short-term exposure to high concentrations of fine atmospheric particulate matter (PM2.5) can lead to long-term health effects. In this article, we propose a random effects model for PM2.5 concentrations. In particular, we anticipate urban/rural differences with regard to both mean levels and variability. Hence we introduce two random effects components, one for rural or background levels and the other as a supplement for urban areas. These are specified in the form of spatio-temporal processes. Weighting these processes through a population density surface results in nonstationarity in space. We analyze daily PM2.5 concentrations in three midwestern U.S. states for the year 2001. A fully Bayesian model is implemented, using MCMC techniques, which enables full inference with regard to process unknowns as well as predictions in time and space.

Bayesian Estimation and Model Choice in Item Response Models

Item response models are essential tools for analyzing results from many educational and psychological tests. Such models are used to quantify the probability of correct response as a function of unobserved examinee ability and other parameters explaining the difficulty and the discriminatory power of the questions in the test. Some of these models also incorporate a threshold parameter for the probability of the correct response to account for the effect of guessing the correct answer in multiple choice type tests. In this article we consider fitting of such models using the Gibbs sampler. A data augmentation method to analyze a normal-ogive model incorporating a threshold guessing parameter is introduced and compared with a Metropolis-Hastings sampling method. The proposed method is an order of magnitude more efficient than the existing method. Another objective of this paper is to develop Bayesian model choice techniques for model discrimination. A predictive approach based on a variant of the Bayes factor is used and compared with another decision theoretic method which minimizes an expected loss function on the predictive space. A classical model choice technique based on a modified likelihood ratio test statistic is shown as one component of the second criterion. As a consequence the Bayesian methods proposed in this paper are contrasted with the classical approach based on the likelihood ratio test. Several examples are given to illustrate the methods.

On Markov Chain Monte Carlo Acceleration

Abstract Markov chain Monte Carlo (MCMC) methods are currently enjoying a surge of interest within the statistical community. The goal of this work is to formalize and support two distinct adaptive strategies that typically accelerate the convergence of an MCMC algorithm. One approach is through resampling; the other incorporates adaptive switching of the transition kernel. Support is both by analytic arguments and simulation study. Application is envisioned in low-dimensional but nontrivial problems. Two pathological illustrations are presented. Connections with reparameterization are discussed as well as possible difficulties with infinitely often adaptation.

On convergence of the EM algorithmand the Gibbs sampler

In this article we investigate the relationship between the EM algorithm and the Gibbs sampler. We show that the approximate rate of convergence of the Gibbs sampler by Gaussian approximation is equal to that of the corresponding EM-type algorithm. This helps in implementing either of the algorithms as improvement strategies for one algorithm can be directly transported to the other. In particular, by running the EM algorithm we know approximately how many iterations are needed for convergence of the Gibbs sampler. We also obtain a result that under certain conditions, the EM algorithm used for finding the maximum likelihood estimates can be slower to converge than the corresponding Gibbs sampler for Bayesian inference. We illustrate our results in a number of realistic examples all based on the generalized linear mixed models.