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，使科学计算发生了革命性的变化。任何一种新的方法都会导致更有效的方法来使用MCMC算法，在无数的科学领域中都有无数的潜在应用。对于MCMC的用户来说，重要的是要有原则性的方法来为结果估计构建误差界，并在理论上保证基本的马尔可夫链的收敛。不幸的是，目前缺乏这样的方法和保障。本研究项目旨在通过开发新的方法来解决这一问题，以便更有原则地应用MCMC。由于MCMC的使用已经变得如此广泛，本项目中开发的方法具有巨大的潜力，可以从科学的许多不同角落为社会进步做出贡献。从给定的难于处理的后验概率分布构造MCMC算法通常是简单的。然而，MCMC的一个长期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。作出这种判断的原则性方法都是基于MCMC估计量的渐近正态。不幸的是，建立必要的中心极限定理(CLT)的存在需要对潜在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罗马尔可夫链的方法在实践中应用极其困难。事实上，对于实际使用的绝大多数MCMC算法来说，这些CLT是否存在都是未知的。这个项目涉及分析复杂马尔可夫链的新技术的发展，比如那些作为MCMC算法基础的技术，着眼于使建立CLT的存在变得更容易。(1)用于分析蒙特卡罗马尔可夫链的标准技术是使用(概率分布之间的)全变差(TV)度量来开发的，但现在变得明显的是，对于分析在MCMC的统计应用中出现的马尔可夫链的类型，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“重新开始”，可能会产生比基于电视的方法更有用的新方法。(2)具有可数状态空间的马尔可夫链通常是通过谱技术来分析的，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社区中的一种普遍看法，即与实际相关的蒙特卡罗马尔可夫链相关的马尔可夫算子中很少是紧凑的。然而，最近的工作表明，许多Gibbs采样器和数据增强(DA)算法实际上确实具有紧致马尔可夫算子。此外，将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、通用的谱技术，用于分析与Gibbs采样器和DA算法相关的马尔可夫算子。

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.