Markov chain Monte Carlo (MCMC) has several disadvantages when compared to classical Monte Carlo methods. In particular, an important issue that every practitioner faces when implementing MCMC is when to stop the computation. Typically, a mixture of experience and ad hoc methods is employed to make this decision. Thus one is forced to wonder about the quality of the inference. The investigator studies sequential fixed-width methods that allow construction of an interval estimator for the quantity of interest. The interval estimator describes the confidence in the point estimate. The investigator uses this to study the development of valid stopping rules when the MCMC computation is aimed at estimating general quantities of the target distribution. These methods require the Markov chain to converge at a geometric rate which in turn implies there is a limiting distribution of the point estimate in the settings of interest. Thus the investigator studies the convergence rates of Markov chains encountered in two broad classes of Bayesian models.
MCMC methods have become a standard technique in the toolbox of applied statisticians (and many scientists in other disciplines). Indeed, it is not much of an overstatement to say that it has revolutionized applied statistics, especially that of the Bayesian variety. Unfortunately, MCMC methods are not always used carefully leading to dubious claims in the literature. In particular, there has been little effort to include measures of uncertainty in inferential conclusions. Rigorously addressing the issue of stopping rules in terms of these measures of uncertainty enhances infrastructure for research and education by providing statisticians and other scientists valid techniques for using MCMC to make inference in their research setting.
Markov chain Monte Carlo is a key computational tool for doing statistical inference based on practically relevant models for real-world phenomena. The importance of Markov chain Monte Carlo is easily reecognized as it has has found application in fields as divergent as economics, lingustics, medical imaging and fisheries science, among many others. Unfortunately, Markov chain Monte Carlo is difficult to implement rigorously. Much of this research focused on methods for ensuring that the implementation of Markov chain Monte Carlo can be trusted. In particular, we considered the theoretical properties of the computational algorithms and developed software to implement our findings. This will allow practitioners from a wide range of disciplines to implement Markov chain Monte Carlo methods safely and rigorously. Our main interdisciplinary contribution has been to the development of a model for investigating brain structure and function in subjects at high risk for Allzheimer's disease. Markov chain Monte Carlo was required for understanding the model and developing our conclusions based on a given fMRI data set. The last part of the project involved investigating the properties of statistical models which have been developed for so-called objective Bayesian inference. The models often encountered are quite complicated due to a desire to provide reasonable representations of real-world processes.
