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

开发马尔可夫链蒙特卡罗算法分析新方法，以促进 MCMC 在实践中的原则性使用

• 批准号: 1511945
• 负责人: James Hobert
• 金额: $ 20万
• 依托单位: University of Florida
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1511945&HistoricalAwards=false
• 关键词: Development; New; Approaches; Analysis; Markov

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.

马尔可夫链蒙特卡洛（MCMC）是一种基于概率的模拟技术，用于近似高维顽固性积分。在过去的二十年中，MCMC通过在遗传学，农业科学，计算机科学，物理学和经济学等多样化的各种学科中使用复杂的统计模型来彻底改变了科学计算。任何导致采用MCMC算法的更有效方法的新方法在无数科学领域都有无数的潜在应用。对于MCMC的用户来说，具有为产生估计构建错误界限的原则方法，并具有基本马尔可夫链的融合的理论保证。不幸的是，目前缺乏此类方法和保证。该研究项目旨在通过开发新方法来解决此问题，该方法将允许更有原则地应用MCMC。由于MCMC的使用变得如此普遍，因此该项目在该项目中开发的方法具有很大的潜力，可以从许多不同的科学角度改善社会的改善。通常直接构建一种从给定的可怕的后验概率分布中构建MCMC算法来进行采样。但是，MCMC的长期困难是确定应运行算法多长时间以产生有用的结果。做出这种确定的原则性方法都是基于MCMC估计器的渐近正态性的。不幸的是，建立必要的中央限制定理（CLT）的存在需要对基础马尔可夫链的详细分析。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极难应用。实际上，对于实践中使用的绝大多数MCMC算法，这些CLT是否存在。该项目涉及开发用于分析MCMC算法的复杂马尔可夫连锁店的新技术，以使其更容易建立CLT的存在。这是将要提出的两个主要思想：（1）使用总变化（TV）度量（概率分布）开发了用于分析蒙特卡洛·马尔可夫链的标准技术，但是现在越来越明显的是，Wasserstein Metric实际上比电视更自然要比分析MCMC统计应用中的Markov链的类型更为自然。这表明，与基于电视的马尔可夫链相比，在实践中，瓦斯斯坦距离代替电视距离的距离“回到绘图板”可能会导致新方法更有用。由于MCMC社区的普遍看法，与实际上的Markov Markov连锁店相关的Markov运营商很少，但最近的工作表明，与这些算法相比，许多Gibbs Samplers和数据增强算法（DA）算法实际上都具有紧凑的Markov操作员。用于分析与Gibbs采样器和DA算法相关的Markov运算符的光谱技术。