Efficiency of Markov chain Monte Carlo methods

Metropolis-Hastings算法是一类重要的MCMC算法，允许从高度复杂的分布(目标分布)生成数据。由于其简单性，正态密度是一种非常流行的选择。为了在算法的性能中具有某种程度的最佳性，则有必要选择正态分布的方差。在具有独立分量的高维目标分布的文献中有关于最优缩放问题的文献。我的目标是通过将当前的最优缩放结果扩展到目标分布的分量具有非平凡的相关结构的情况来提高Metropolis-Hastings算法的效率。这应该通过首先考虑分层目标模型来实现，它们本身很有趣，因为它们在贝叶斯统计中很受欢迎。也存在许多Metropolis-Hastings算法的变体，它们在实践中工作得很好，但还没有从理论上进行研究。其中之一是延迟拒绝Metropolis-Hastings算法，该算法允许我们冻结时间，并在拒绝移动时调整建议的比例。最近，自适应算法已经变得非常流行，我打算利用我的最优尺度结果来推导出一种有效的自适应方法，供研究人员和实践者在不同的应用领域使用。我还将研究MCMC算法的一些应用，特别是在统计推断(比较频率法和贝叶斯法)和工程学(在与空中交通控制有关的问题中)。

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).