Efficiencies of MCMC and nonparametric estimation methods

在本研究中，我将集中在几个主题，有关马尔可夫链蒙特卡罗（MCMC）算法和非参数估计方法。 1.高维MCMC算法的最优尺度化：该领域的先前分析主要集中在研究各种通用随机游走大都会算法的扩散极限，其中目标分布通常满足一些限制性条件。这项研究的目的是扩展现有的结果，更复杂的目标分布和建议密度的算法。这将导致更多的执行准则。 2. MCMC在凝聚态物理中的应用：蠕动量子蒙特卡罗（RQMC）算法是一种“近似”的Metropolis-Hastings算法，已经成为凝聚态物理中一种强大的计算工具。然而，在实践中，一旦维数达到数千，该算法就缓慢地收敛到目标分布，并且其未能给出可靠的估计是有据可查的。本研究的目的是将MCMC技术与现有的RQMC相结合，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了纵向分析。我们使用的方法是完全贝叶斯，后验抽样MCMC方法。 4.非参数估计方法：众所周知，许多现有的非参数估计方法的分布，分位数和回归的效率是相对较低的分布的尾部，特别是重尾分布。在本研究中，我们将研究各种加权经验分布函数族，并使用它们来开发新的估计方法，并研究它们的理论性质。这些方法将在文献中现有的方法进行比较，并应用到现实生活中的问题。

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.