Improving the efficiency of Markov chain Monte Carlo

马尔可夫链蒙特卡罗(MCMC)算法在科学界得到了广泛的应用。它们被用来从统计和应用概率模型产生的复杂目标分布中产生近似样本。特别是，Metropolis和Langevin算法等通用算法由于易于实现而非常流行。然而，众所周知，选择最好的提案分布并不是一件容易的事情。如果做不到这一点，可能会导致算法效率低下并产生误导性结果。我的研究项目将专注于MCMC算法的理论和实现，通常是在无界欧几里德状态空间上，受分布的激励，例如在贝叶斯推理中遇到的那些分布。我提出的研究计划的三个主要目标是：1.定量地界定收敛速度。我将通过应用分解定理来概括和组合可用的技术，从离散状态空间到一般状态空间。这项工作将对在MCMC工作的理论家具有直接价值，并有望统一估计和优化算法收敛速度的方法；2.当目标密度的维度接近无穷大时，通过适当缩放建议分布来优化收敛速度。我将继续关注局部MCMC算法，泛指增量分布以当前状态为中心(或接近当前状态)的方法，并证明更广泛类别的目标分布和建议分布的弱收敛结果。这项工作将从现有理论中为该领域的所有从业者扩展和提炼缩放指南；3.对一些现有采样算法(如MCMC、自适应MCMC等)提出修改建议并研究实际实施问题。在科学文献中被广泛使用。这包括算法的适应性稳定性、收敛速度和效率，以及实践者所关心的其他问题。这项具有基础性的工作将丰富MCMC方法及其相关扩展的理论。

Markov chain Monte Carlo (MCMC) algorithms are widely used in the scientific community. They are used to generate approximate samples from complicated target distributions resulting from statistical and applied probability models. In particular, generic algorithms such as Metropolis and Langevin algorithms are very popular because of the ease of implementations. However, it is well known that choosing the best proposal distribution is not an easy task. Failing to do that can result in an inefficient algorithm and misleading results. My research program will focus on the theory and implementation of MCMC algorithms, typically on unbounded Euclidean state spaces, motivated by distributions such as those encountered in Bayesian inference. The three main objectives of my proposed research program are:  1. Bounding the convergence rates quantitatively. I shall generalize and combine available techniques through the application of decomposition theorems, from discrete state spaces, to general state spaces. This work will have direct value for theoreticians working in MCMC and will hopefully unify approaches for estimating and optimizing rates of convergence of algorithms;  2. Optimizing the convergence rates by proper scaling of the proposal distribution, when the dimension of the target density approaches infinity. I shall continue to focus on local MCMC algorithms, loosely referring to methods which have increment distributions centered at (or close to) the current state, and on proving weak convergence results for a wider class of target and proposal distributions. This work will extend and refine the scaling guidelines from available theory for all practitioners in the area;  3. Suggest modifications and study practical implementation issues on some existing sampling algorithms (e.g. MCMC, adaptive MCMC, etc.) widely used in the scientific literature. This includes stabilities of adaptations, convergence and efficiencies of these algorithms, and other problems motivated by practitioners.  This work, which is foundational, will enrich the theories of MCMC methods and related extensions.