Adaptive Markov Chain Monte Carlo methods

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This award is funded under the American Recovery and Reinvestment Act of2009 (Public Law 111-5).Markov Chain Monte Carlo (MCMC) is a flexible computational technique thathas proven very useful to many scientific disciplines and is the backboneof current implementations of Bayesian inference. Recent researchdevelopments suggest that the use of adaptive methods can make Monte Carloalgorithms considerably more effective. This research proposal has twomajor components. The first part will contribute to the development of alimit theory for adaptive MCMC algorithms. More Specifically, the PI willdevelop a resolvent-based martingale approximation technique to investigatethe central limit theorem and the asymptotic variance estimation forvarious adaptive MCMC algorithms. The second part of this research activitywill develop a new MCMC algorithm for the Bayesian analysis of statisticalmodels with intractable normalizing constants, a topic that currently posesmajor computational challenges. This part of the research is driven by theprotein design problem in computational biology but the same problem alsofrequently occurs in many other statistical models including Markov randomfields, Markov point processes.Markov Chain Monte Carlo is a well-established Monte Carlo technique forsampling probability distributions. The method is used widely to solvesubstantive problems in many areas of applications. It is especially usefulin situations with high-dimensional data, which is increasingly common inscience and engineering research as well as applications. Thus, theresearch developments from this project can have a significant impact onmethods for doing statistical inference in these areas. Through itstheoretical component, this research project will advance the generalunderstanding and strengthen the use of adaptive Monte Carlo methods inpractice. In the course of developing the asymptotic theory, the PI willdevelop original extensions to some well-established probabilistic tools.The research is also proposing a new algorithm to tackle one of the mostchallenging current problem in Monte Carlo simulation; sampling from theposterior distribution of statistical models with intractable normalizingconstants.