Efficiencies of MCMC and nonparametric estimation methods

MCMC 和非参数估计方法的效率

• 批准号: 293260-2012
• 负责人: Yuen, WaiKong
• 金额: $ 0.87万
• 依托单位: Brock University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=571423
• 关键词: Efficiencies; MCMC; nonparametric; estimation; methods

In this research, I will focus on several topics concerning Markov chain Monte Carlo (MCMC) algorithms and nonparametric estimation methods. 1. Optimal scaling of high-dimensional MCMC algorithms: Previous analysis in this field has mostly focused upon studying the diffusion limits of various generic random walk Metropolis algorithms where the target distribution typically satisfies some restrictive conditions. This research aims to extend existing results to algorithms with more complex target distributions and proposal densities. This will lead to more implementation guidelines. 2. Applications of MCMC in the area of condensed matter physics: The reptation quantum Monte Carlo (RQMC) algorithm, which is an "approximate" Metropolis-Hastings algorithm, has been a powerful computational tool in condensed matter physics. In practice, however, this algorithm converges slowly to the target distribution once the dimension reaches the thousands, and its failure to give reliable estimates is well documented. The goal of this research is to develop new algorithms by combining various MCMC techniques with the existing RQMC, and study other applications of MCMC algorithms in condensed matter physics. 3. Applications of Bayesian methods and MCMC algorithms in applied health sciences: we perform a longitudinal analysis on the impact of lipids change over time on the risk of coronary heart disease. The approach we use is fully Bayesian, with posterior sampling done by MCMC methods. 4. Nonparametric estimation methods: It is well-known that the efficiencies of many existing nonparametric estimation methods for distribution, quantile and regression are relatively low on the tails of the distribution, particularly for heavy-tailed distributions. In this research, we will study various families of weighted empirical distribution functions, and use them to develop new estimation methods and study their theoretical properties. These methods will be compared to existing methods in the literature and applied to real-life problems.

在本研究中，我将重点关注有关马尔可夫链蒙特卡罗（MCMC）算法和非参数估计方法的几个主题。 1.高维MCMC算法的优化缩放：该领域之前的分析主要集中在研究各种通用随机游走Metropolis算法的扩散极限，其中目标分布通常满足一些限制条件。这项研究旨在将现有结果扩展到具有更复杂目标分布和提议密度的算法。这将导致更多的实施指南。 2. MCMC在凝聚态物理领域的应用： 蠕动量子蒙特卡罗（RQMC）算法是一种“近似”Metropolis-Hastings算法，已成为凝聚态物理领域强大的计算工具。然而，在实践中，一旦维度达到数千，该算法就会缓慢地收敛到目标分布，并且它无法给出可靠的估计是有据可查的。本研究的目标是将各种MCMC技术与现有的RQMC相结合来开发新的算法，并研究MCMC算法在凝聚态物理中的其他应用。 3.贝叶斯方法和MCMC算法在应用健康科学中的应用：我们对血脂随时间变化对冠心病风险的影响进行纵向分析。我们使用的方法完全是贝叶斯方法，通过 MCMC 方法完成后采样。 4.非参数估计方法：众所周知，现有的许多分布、分位数和回归的非参数估计方法在分布的尾部，特别是对于重尾分布，效率相对较低。在本研究中，我们将研究各种加权经验分布函数族，并使用它们开发新的估计方法并研究它们的理论特性。这些方法将与文献中的现有方法进行比较，并应用于现实生活中的问题。