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

本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。在统计领域，概率分布经常被用来模拟人们想要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差异。估算程序基于数据集，该数据集通常是通过调查或实验从基础人群中收集的样本。MCMC被广泛认为是一种非常强大的高质量估计工具，但当样本数量较大时，人们对其可靠性知之甚少。本项目旨在研究各种MCMC算法在大样本环境下的数学性质。这项研究的结果有望增进对MCMC算法性能的理解，这些算法应用于经济、生物和天文学等领域的现代数据集。参与该项目研究工作的本科生和研究生将接受概率论以及数学和应用统计学的培训。更具体地说，该项目重点介绍用于探索贝叶斯统计模型中的后验分布的MCMC算法。从频数的角度来看，假设与贝叶斯模型相关联的数据集是从基础分布中生成的。因此，MCMC算法的马尔可夫转移核可以被视为与经典向量值统计量(例如，样本平均值)没有区别的统计量，即可观测的随机元素。就像任何统计量一样，当数据集的样本量增加时，了解MCMC转移核的渐近行为是很重要的。这个项目的目的是使用经典大样本理论和马尔可夫链理论的技术来开发这个问题的一般理论。将特别关注数据增强算法，这是一类广泛的实际相关的MCMC算法，显示出丰富和有意义的大样本特性。该项目还将研究MCMC算法在大样本制度下的混合时间，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NSF的法定使命，并通过使用基金会的智力优势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.