This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
Markov Chain Monte Carlo (MCMC) is a flexible computational technique that has proven very useful to many scientific disciplines and is the backbone of current implementations of Bayesian inference. Recent research developments suggest that the use of adaptive methods can make Monte Carlo algorithms considerably more effective. This research proposal has two major components. The first part will contribute to the development of a limit theory for adaptive MCMC algorithms. More Specifically, the PI will develop a resolvent-based martingale approximation technique to investigate the central limit theorem and the asymptotic variance estimation for various adaptive MCMC algorithms. The second part of this research activity will develop a new MCMC algorithm for the Bayesian analysis of statistical models with intractable normalizing constants, a topic that currently poses major computational challenges. This part of the research is driven by the protein design problem in computational biology but the same problem also frequently occurs in many other statistical models including Markov random fields, Markov point processes.
Markov Chain Monte Carlo is a well-established Monte Carlo technique for sampling probability distributions. The method is used widely to solve substantive problems in many areas of applications. It is especially useful in situations with high-dimensional data, which is increasingly common in science and engineering research as well as applications. Thus, the research developments from this project can have a significant impact on methods for doing statistical inference in these areas. Through its theoretical component, this research project will advance the general understanding and strengthen the use of adaptive Monte Carlo methods in practice. In the course of developing the asymptotic theory, the PI will develop original extensions to some well-established probabilistic tools. The research is also proposing a new algorithm to tackle one of the most challenging current problem in Monte Carlo simulation; sampling from the posterior distribution of statistical models with intractable normalizing constants.
