Development of New Approaches for Analysis of Markov Chain Monte Carlo Algorithms to Facilitate Principled Use of MCMC in Practice

马尔可夫链蒙特卡罗（MCMC）是一种基于概率的模拟技术，用于近似高维棘手的积分。MCMC在过去的二十年里通过在遗传学、农业科学、计算机科学、物理学和经济学等众多学科中使用复杂的统计模型，彻底改变了科学计算。任何新的方法，导致更有效的方式来使用MCMC算法在无数的科学领域有无数的潜在应用。对于MCMC的用户来说，重要的是要有原则的方法来构建所得到的估计的误差界，并有理论上的保证收敛的基础马尔可夫链。不幸的是，目前缺乏这种方法和保障。该研究项目旨在通过开发新的方法来解决这个问题，该方法将允许MCMC更有原则的应用。由于MCMC的使用已经变得如此广泛，因此本项目中开发的方法具有很大的潜力，可以从科学的许多不同角落为社会的进步做出贡献。然而，MCMC的一个长期存在的困难是确定一个算法应该运行多长时间才能产生有用的结果。作出这种决定的原则性方法都是基于MCMC估计量的渐近正态性。不幸的是，建立必要的中心极限定理（CLTs）的存在需要详细分析的基础马尔可夫链。更糟糕的是，目前用于分析蒙特卡罗马尔可夫链的方法在实践中非常难以应用。事实上，对于绝大多数实际使用的MCMC算法，这些CLT是否存在是未知的。该项目涉及分析复杂马尔可夫链的新技术的开发，如MCMC算法的基础，着眼于使其更容易建立CLT的存在。这是两个主要的想法，将追求：（1）标准技术分析蒙特卡罗马尔可夫链的开发使用总变差（TV）度量（概率分布之间），但现在越来越清楚，沃瑟斯坦度量实际上是更自然的比电视分析类型的马尔可夫链出现在MCMC的统计应用。这表明，用Wasserstein距离代替TV距离“回到绘图板”可能会导致新的方法，这些方法在实践中比基于TV的方法更有用。(2)具有可数状态空间的马尔可夫链通常通过谱技术进行分析，但这种方法并不经常用于MCMC统计应用的基础马尔可夫链（通常具有不可数状态空间）。这（至少）部分是由于MCMC社区的普遍看法，即与实际相关的Monte Carlo Markov链相关的Markov算子很少是紧凑的。然而，最近的工作表明，许多吉布斯采样器和数据增强（DA）算法，事实上，有紧凑的马尔可夫算子。此外，谱技术的应用，这些运营商往往比标准的分析简单得多。这就要求发展新的，一般的，谱技术的分析与吉布斯采样器和DA算法的马尔可夫算子。

Markov Chain Monte Carlo (MCMC) is a probability-based simulation technique that is used to approximate high-dimensional intractable integrals. MCMC has revolutionized scientific computing in the last two decades by enabling the use of intricate statistical models in a vast array of disciplines as diverse as genetics, agricultural science, computer science, physics, and economics. Any new methodology that leads to more effective ways to employ MCMC algorithms has countless potential applications in myriad scientific fields. It is vital for users of MCMC to have principled methods for constructing error bounds for the resulting estimates, and to have theoretical guarantees of convergence for the underlying Markov chains. Unfortunately, such methods and guarantees are currently lacking. This research project aims to address this problem by developing new methodology that will allow for more principled application of MCMC. Because the use of MCMC has become so widespread, there is great potential for the methods developed in this project to contribute to the improvement of society from many different corners of science.It is typically straightforward to construct an MCMC algorithm for sampling from a given intractable posterior probability distribution. However, a long-standing difficulty with MCMC is in determining how long an algorithm should be run to produce useful results. The principled approaches to making such a determination are all predicated on the asymptotic normality of the MCMC estimators. Unfortunately, establishing the existence of the requisite central limit theorems (CLTs) requires a detailed analysis of the underlying Markov chain. Worse yet, the methods that are currently available for analyzing Monte Carlo Markov chains are extremely difficult to apply in practice. In fact, for the vast majority of MCMC algorithms that are used in practice, it is unknown whether these CLTs exist. This project concerns the development of new techniques for analyzing complex Markov chains, like those that underlie MCMC algorithms, with an eye towards making it easier to establish the existence of CLTs. The are two main ideas that will be pursued: (1) The standard techniques for analyzing Monte Carlo Markov chains were developed using the total variation (TV) metric (between probability distributions), but it is now becoming clear that the Wasserstein metric is actually much more natural than TV for analyzing the types of Markov chains that arise in statistical applications of MCMC. This suggests that going "back to the drawing board'' with Wasserstein distance in place of TV distance may lead to new methods that are far more useful in practice than those based on TV. (2) Markov chains with countable state spaces are routinely analyzed with great success via spectral techniques, but this approach is not often used for the Markov chains that underlie statistical applications of MCMC (which usually have uncountable state spaces). This is (at least) partly due to a general perception in the MCMC community that very few of the Markov operators associated with practically relevant Monte Carlo Markov chains are compact. However, recent work suggests that many Gibbs samplers and data augmentation (DA) algorithms do, in fact, have compact Markov operators. Furthermore, application of the spectral techniques to these operators is often much simpler than the standard analysis. This calls for the development of new, general, spectral techniques for the analysis of Markov operators associated with Gibbs samplers and DA algorithms.