Large Sample Analysis of Markov Chain Monte Carlo Methods in Bayesian Statistics From a Frequentist Perspective

该项目涉及马尔可夫链蒙特卡罗（MCMC），这是一类广泛用于模拟复杂概率分布的计算机算法。在统计学领域，概率分布经常被用来模拟人们希望估计的未知参数的不确定性，例如，一个大国的平均家庭收入，或两个人口群体之间的平均预期寿命的差异。估计过程基于数据集，通常是通过调查或实验从潜在人群中收集的样本。MCMC被广泛认为是一种非常强大的高质量估计工具，但当样本大量时，对其可靠性的了解还不够。本项目旨在研究各种MCMC算法在大样本设置下的数学性质。研究结果有望促进对MCMC算法性能的理解，这些算法应用于经济学、生物学和天文学等领域的现代数据集。参与该项目的本科生和研究生将接受概率论、数学和应用统计方面的培训。更具体地说，该项目侧重于用于探索贝叶斯统计模型中的后验分布的MCMC算法。从频率论的角度来看，与贝叶斯模型相关的数据集假定是从底层分布生成的。因此，MCMC算法的马尔可夫转移核可以看作是一个统计量，即可观察的随机元素，与经典的向量值统计量（如样本均值）没有什么不同。就像任何统计量一样，当数据集的样本量增加时，理解MCMC转移核的渐近行为是很重要的。本项目旨在利用经典大样本理论与马尔可夫链理论相结合的技术，为该问题发展一般理论。将特别关注数据增强算法，这是一大类实际相关的MCMC算法，表现出丰富和有意义的大样本特性。该项目还将涉及研究MCMC算法在大样本状态下的混合时间，这对基于MCMC的推理在实践中的有效性至关重要。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project concerns Markov Chain Monte Carlo (MCMC), which is a class of computer algorithms that are widely used to simulate complicated probability distributions. In the field of statistics, probability distributions are often used to model uncertainty about unknown parameters that one wishes to estimate, for example, the average household income of a large nation, or the difference in average life expectancy between two demographic groups. The estimation procedure is based on a data set, which is usually a sample collected from an underlying population through a survey or experiment. MCMC is widely regarded as an extremely powerful tool for high-quality estimation, but not enough is known about its reliability when the sample is massive. This project aims to study the mathematical properties of various MCMC algorithms in large sample settings. Results of the research are expected to advance understanding in the performance of MCMC algorithms that are applied to modern data sets in fields such as economics, biology, and astronomy. Undergraduate and graduate students involved in the research efforts of this project will receive training in probability theory as well as mathematical and applied statistics.More specifically, this project focuses on MCMC algorithms that are used to explore posterior distributions in Bayesian statistical models. From a frequentist perspective, the data set associated with a Bayesian model is assumed to be generated from an underlying distribution. The Markov transition kernel of an MCMC algorithm can thus be regarded as a statistic, i.e., observable random element, no different from a classical vector-valued statistic, e.g., a sample mean. Just like with any statistic, it is important to understand the asymptotic behavior of an MCMC transition kernel when the sample size of the data set grows. This project aims to develop general theories for the problem using techniques from classical large sample theory in conjunction with those from Markov chain theory. Special attention will be given to data augmentation algorithms, which are a broad class of practically relevant MCMC algorithms that exhibit rich and meaningful large sample properties. The project will also involve studying the mixing times of MCMC algorithms in the large sample regime, which is crucial to the effectiveness of MCMC-based inference in practice.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.