Efficiency of Markov chain Monte Carlo methods

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Metropolis-Hastings algorithms, an important class of MCMC algorithms, allow for data generation from highly complex distributions (the target distribution).  In applying Metropolis-Hastings algorithms, it is necessary to choose a proposal density; due to its simplicity, the normal density is a very popular choice.  In order to have some level of optimality in the performance of the algorithm, it then becomes necessary to select the variance of the normal distribution.  Results about the optimal scaling issue are available in the literature for high-dimensional target distributions with independent components.My goal is to improve the efficiency of Metropolis-Hastings algorithms by extending current optimal scaling results to the case where the components of the target distribution assume a nontrivial correlation structure.  This shall be achieved by first considering hierarchical target models, which are interesting in themselves due to their popularity in Bayesian statistics.  There also exist many variations of the Metropolis-Hastings algorithms that work beautifully in practice, but which have not been studied theoretically. One of them is the Delayed Rejection Metropolis-Hastings algorithm, which allow us to "freeze time" and adjust the proposal scaling upon the rejection of a move. I would like to study the weak convergence and optimal scaling theory of such variations.Recently, adaptive algorithms have become quite popular and I intend to use my optimal scaling results to derive an efficient adaptive method to be used by researchers and practitioners in various fields of application. I shall also study some applications of MCMC algorithms, specifically in statistical inference (to compare frequentist and Bayesian methods) and engineering (in a problem related to the control of air traffic).