Efficiency of Markov chain Monte Carlo methods

马尔可夫链蒙特卡罗方法的效率

• 批准号: 346215-2007
• 负责人: Bédard, Mylène
• 金额: $ 0.87万
• 依托单位: Université de Montréal
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2007
• 资助国家: 加拿大
• 起止时间: 2007-01-01 至 2008-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=363206
• 关键词: Efficiency; Markov; chain; Monte; Carlo

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).

Metropolis-Hastings算法是MCMC算法的一个重要分支，它可以从高度复杂的分布中生成数据在应用Metropolis-Hastings算法时，需要选择一个建议密度;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优性，然后，有必要选择正态分布的方差。关于最优尺度问题的结果在文献中可用于具有独立分量的高维目标分布。我的目标是提高大都会的效率-Hastings算法，将当前的最优标度结果扩展到目标分布的分量假设非平凡相关结构的情况。这将首先考虑分层目标模型，这是有趣的，因为它们在贝叶斯统计中很受欢迎。还有许多Metropolis-Hastings算法的变体，它们在实践中工作得很好，但尚未在理论上进行研究。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提案比例。我想研究这种变化的弱收敛和最佳尺度理论，最近，自适应算法已经变得相当流行，我打算使用我的最佳尺度结果来推导出一个有效的自适应方法，供研究人员和实践者在各个应用领域使用。我还将研究MCMC算法的一些应用，特别是在统计推断（比较频率论和贝叶斯方法）和工程（在与空中交通控制有关的问题中）。