Estimating the spectral gap of a trace-class Markov operator

The utility of a Markov chain Monte Carlo algorithm is, in large part, determined by the size of the spectral gap of the corresponding Markov operator. However, calculating (and even approximating) the spectral gaps of practical Monte Carlo Markov chains in statistics has proven to be an extremely difficult and often insurmountable task, especially when these chains move on continuous state spaces. In this paper, a method for accurate estimation of the spectral gap is developed for general state space Markov chains whose operators are non-negative and trace-class. The method is based on the fact that the second largest eigenvalue (and hence the spectral gap) of such operators can be bounded above and below by simple functions of the power sums of the eigenvalues. These power sums often have nice integral representations. A classical Monte Carlo method is proposed to estimate these integrals, and a simple sufficient condition for finite variance is provided. This leads to asymptotically valid confidence intervals for the second largest eigenvalue (and the spectral gap) of the Markov operator. In contrast with previously existing techniques, our method is not based on a near-stationary version of the Markov chain, which, paradoxically, cannot be obtained in a principled manner without bounds on the spectral gap. On the other hand, it can be quite expensive from a computational standpoint. The efficiency of the method is studied both theoretically and empirically

General Design Bayesian Generalized Linear Mixed Models

Linear mixed models are able to handle an extraordinary range of complications in regression-type analyses. Their most common use is to account for within-subject correlation in longitudinal data analysis. They are also the standard vehicle for smoothing spatial count data. However, when treated in full generality, mixed models can also handle spline-type smoothing and closely approximate kriging. This allows for nonparametric regression models (e.g., additive models and varying coefficient models) to be handled within the mixed model framework. The key is to allow the random effects design matrix to have general structure; hence our label general design. For continuous response data, particularly when Gaussianity of the response is reasonably assumed, computation is now quite mature and supported by the R, SAS and S-PLUS packages. Such is not the case for binary and count responses, where generalized linear mixed models (GLMMs) are required, but are hindered by the presence of intractable multivariate integrals. Software known to us supports special cases of the GLMM (e.g., PROC NLMIXED in SAS or glmmML in R) or relies on the sometimes crude Laplace-type approximation of integrals (e.g., the SAS macro glimmix or glmmPQL in R). This paper describes the fitting of general design generalized linear mixed models. A Bayesian approach is taken and Markov chain Monte Carlo (MCMC) is used for estimation and inference. In this generalized setting, MCMC requires sampling from nonstandard distributions. In this article, we demonstrate that the MCMC package WinBUGS facilitates sound fitting of general design Bayesian generalized linear mixed models in practice.Comment: Published at http://dx.doi.org/10.1214/088342306000000015 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Bayesian Approach of Joint Models of Longitudinal Outcomes and Informative Time

Longitudinal studies are commonly encountered in a variety of research areas in which the scientific interest is in the pattern of change in a response variable over time. In longitudinal data analyses, a number of methods have been proposed. Most of the traditional longitudinal methods assume that the independent variables are the same across all subjects. It is commonly assumed that time intervals for collecting outcomes are predetermined and have no information regarding the measured variables. However, in practice, researchers might occasionally have irregular time intervals and informative time, which violate the above assumptions. Hence, if traditional statistical methods are used for this situation, the results would be biased. The joint models of longitudinal outcomes and informative time are used as a solution to the above violations by using joint probability distributions, incorporating the relationships between outcomes and time. The joint models are designed to handle outcome distributions from a normal distribution with informative time following an exponential distribution. Several studies used the maximum likelihood parameter estimates of the joint model. This study, however, presented an alternative method for parameters estimation, based on a Bayesian approach, with respect to joint models of longitudinal outcomes and informative time. Using a Bayesian approach permitted the inclusion of knowledge of the observed data within the analysis through the prior distribution of unknown parameters. In this dissertation, the prior distribution adopted three scenarios: (1) the prior distributions of all unknown parameters are noninformative prior, which will set to be vague but proper prior: Normal(0, 1e6). (2) The prior distributions of all unknown parameters are informative prior, which will be set to be normal for unrestricted parameters, and inverse gamma (IG) priors for positive parameters such as the variance σ2. (3) A combination of two above scenarios, so the prior distributions of some unknown parameters are noninformative, and the others are informative. The procedure for estimating the model parameters was developed via a Markov chain Monte Carlo method using the Metropolis-Hastings algorithm. The key idea was to construct the likelihood function, specify the prior information, and then calculate the posterior distribution. Simulated observations were generated by the MCMC technique from the posterior distribution. Thus, the primary purpose of this study was to find Bayesian estimates for the unknown parameters in the joint model, with the assumptions of a normal distribution for the outcome process and an exponential distribution for informative time. The properties and merits of the proposed procedure were illustrated employing a simulation study through a written R program and OpenBUGS