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，用于逼近高维难解积分。在过去的二十年里，MCMC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，导致更有效地使用MCMC算法有无数的潜在应用在无数的科学领域。对于MCMC的用户来说，重要的是要有原则的方法来构造结果估计的误差界，并有理论保证底层马尔可夫链的收敛性。不幸的是，目前缺乏这样的方法和保证。本研究项目旨在通过开发新的方法来解决这一问题，该方法将允许更有原则地应用MCMC。由于MCMC的使用已经变得如此广泛，在这个项目中开发的方法有很大的潜力，可以从许多不同的科学领域为社会的改善做出贡献。从一个给定的难以处理的后验概率分布中构造一个MCMC算法通常是很简单的。然而，MCMC长期存在的一个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（clt）的存在性需要对底层的马尔可夫链进行详细的分析。更糟糕的是，目前可用于分析蒙特卡洛马尔可夫链的方法在实践中非常难以应用。事实上，对于实践中使用的绝大多数MCMC算法来说，这些clt是否存在是未知的。该项目涉及开发分析复杂马尔可夫链的新技术，如MCMC算法的基础，目的是使建立clt的存在更容易。本文将探讨两个主要思想：(1)分析蒙特卡洛马尔可夫链的标准技术是使用总变差（TV）度量（在概率分布之间）开发的，但现在越来越清楚的是，对于分析MCMC统计应用中出现的马尔可夫链类型，Wasserstein度量实际上比TV更自然。这表明，用沃瑟斯坦距离代替电视距离“回到绘图板”可能会产生比基于电视的新方法在实践中更有用的新方法。(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，并取得了巨大的成功，但这种方法并不经常用于作为MCMC统计应用基础的马尔可夫链（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，即与实际相关的蒙特卡洛马尔可夫链相关的马尔可夫算子很少是紧凑的。然而，最近的研究表明，许多吉布斯采样器和数据增强（DA）算法实际上确实具有紧凑的马尔可夫算子。此外，将光谱技术应用于这些算子通常比标准分析简单得多。这就要求开发新的、通用的、用于分析与吉布斯采样器和DA算法相关的马尔可夫算子的光谱技术。

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.