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。它们用于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，通用算法，如Metropolis和Langevin算法，由于易于实现而非常受欢迎。然而，众所周知，选择最好的提案分发并不是一件容易的事。如果做不到这一点，可能会导致低效的算法和误导性的结果。我的研究计划将集中在MCMC算法的理论和实现上，通常是在无界欧几里得状态空间上，由贝叶斯推理中遇到的分布所驱动。我提出的研究计划的三个主要目标是：1。定量地限定收敛速率。从离散状态空间到一般状态空间，我将通过分解定理的应用来推广和结合可用的技术。这项工作将对在MCMC中工作的理论家有直接价值，并有望统一估计和优化算法收敛速度的方法；2. 当目标密度维数趋近于无穷大时，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，它是指以（或接近）当前状态为中心的增量分布的方法，并为更广泛的目标和提议分布证明弱收敛结果。这项工作将从现有理论中扩展和完善该领域所有从业者的缩放指南；3. 对科学文献中广泛使用的一些现有采样算法（如MCMC、自适应MCMC等）提出修改建议并研究实际实现问题。这包括适应性的稳定性，这些算法的收敛性和效率，以及从业者提出的其他问题。这一基础性工作将丰富MCMC方法和相关扩展的理论。

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.