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

频率论视角下贝叶斯统计马尔可夫链蒙特卡罗方法的大样本分析

• 批准号: 2112887
• 负责人: Qian Qin
• 金额: $ 19.94万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-15 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2112887&HistoricalAwards=false
• 关键词: Large; Sample; Analysis; Markov; Chain

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.

这个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未知参数的不确定性，例如一个大国的平均家庭收入，或者两个人口群体之间的平均预期寿命差异。估计程序基于一个数据集，通常是通过调查或实验从基本人口中收集的样本。MCMC被广泛认为是高质量估计的一个非常强大的工具，但对于它在大样本情况下的可靠性还知之甚少。本项目旨在研究大样本环境下各种MCMC算法的数学特性。该研究的结果有望促进对MCMC算法性能的理解，这些算法适用于经济学、生物学和天文学等领域的现代数据集。参与本项目研究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，本项目侧重于MCMC算法，用于探索贝叶斯统计模型中的后验分布。从频率论的角度来看，与贝叶斯模型相关联的数据集被假设为从底层分布生成。因此，MCMC算法的马尔可夫转移核可以被视为统计量，即，可观察的随机元素，与经典的向量值统计没有什么不同，例如，样本平均值。就像任何统计数据一样，当数据集的样本量增加时，了解MCMC转换内核的渐进行为非常重要。本项目旨在利用经典大样本理论和马尔可夫链理论的技术，为该问题开发一般理论。将特别注意数据增强算法，这是一个广泛的类实际相关的MCMC算法，表现出丰富的和有意义的大样本属性。该项目还将研究MCMC算法在大样本范围内的混合时间，这对MCMC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。

项目成果

Convergence rates of two-component MCMC samplers

双分量 MCMC 采样器的收敛率

• DOI: 10.3150/21-bej1369
• 发表时间: 2022
• 期刊: Bernoulli
• 影响因子: 1.5
• 作者: Qin, Qian;Jones, Galin L.
• 通讯作者: Jones, Galin L.