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Control Variates for Markov Chain Monte Carlo Variance Reduction

Final Report Summary - CVMCMC (Control Variates for Markov Chain Monte Carlo Variance Reduction)

I. Objectives.
Research in numerous scientific fields, including many of the fundamental research frontiers in science and engineering, has produced large empirical data sets with highly complex structure. There, the search for efficient ways of detecting and evaluating relevant information is currently one of the dominant problems, and the use of sophisticated statistical methods has been advanced as a necessary and central part of the analysis. In particular, recent advances in Markov chain Monte Carlo (MCMC) methods have revolutionized statistical analysis, vastly increasing its impact. The ability of MCMC algorithms to simulate from high-dimensional and potentially awkward distributions has made MCMC a major reason for – and an indispensable part of – the spectacular spread of statistical modeling to virtually every quantitative scientific area.
But in many applications, MCMC fails. On certain complex, high-dimensional problems, existing MCMC algorithms take too long to converge. And generally, the slow convergence of MCMC methods is one of the main limitations – perhaps the main limitation – of their applicability in statistics.
The main objective of this project is to develop a new family of efficient methodologies based on the method of control variates, for overcoming this obstacle in the context of statistical estimation. Our effort toward achieving this objective naturally brakes down into three separate directions, as described next.

II. Summary of work performed.
After a preliminary phase during which a detailed literature survey on the larger topic of variance reduction in MCMC estimation was conducted, the first major task was the development of the necessary theory: A firm mathematical foundation was created for the application of control variates to Markov chains in general. In the second phase, the resulting theoretical insights led to the introduction of new generic methodologies: For the large class of so-called conjugate Gibbs samplers, we developed a clear, detailed and provably effective method for the direct use of control variates in the estimation process. Then, for the complementary class of Metropolis-Hastings samplers, we proposed several variants of the earlier basic methodology, which can be applied to a large number of the most common MCMC algorithms used in Bayesian inference studies.
The third part of the work was devoted to applications: We examined a large array of scientific and engineering problems of active research interest across the range of disciplines that require intensive use of statistical computing via MCMC, including genetics, health studies, environmental epidemiology, animal development and ecology, demography, statistical physics, computer science, signal processing, econometrics, astronomy and neuroscience. In each case we studied how our methodology needs to be tailored in order to be maximally effective, and in all cases the empirical results we obtained showed that this methodology is very effective in producing significantly more reliable statistical conclusions. Indeed, in some cases the difference was dramatic enough to suggest that certain tasks that up to now may have been considered impossible, may in fact now become possible.
Our findings are described in detail in a series of papers (see [1,2,3,4] below) as well as in the Ph.D. thesis of Zoi Tsourti, a graduate student at the Athens University of Economics and Business, jointly supervised by Kontoyiannis and Dellaportas. We have given more than 15 seminar and conference talks in leading institutions around the world, and in September of 2009 we organized a small, focused workshop on the topics of this project, which was attended by several of the world’s leading experts.

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