A shape-constrained approach for non-parametric variance estimation for Markov Chains

马尔可夫链非参数方差估计的形状约束方法

• 批准号: 2311141
• 负责人: Hyebin Song
• 金额: $ 25.68万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2311141&HistoricalAwards=false
• 关键词: shape; constrained; approach; non; parametric

Markov chain Monte Carlo (MCMC) methods have become one of the most important methods in modern statistics practice, as they provide straightforward computational approaches in a wide variety of statistical settings, such as Bayesian parameter estimation, uncertainty quantification for the estimated parameters, and model fitting while allowing uncertainties in model specifications. Despite widespread use, practical and theoretical difficulties remain for quantifying the uncertainty of estimates from MCMC simulations. This project will develop novel estimators for quantifying uncertainties in various MCMC sampling settings. In addition to making technical contributions, the project will result in the development of practical methods and open-source software packages that will enable practitioners to quantify uncertainty in MCMC estimates more accurately and make more efficient use of computational resources. This project integrates active research topics from multiple areas including statistical machine learning, MCMC, and nonparametric statistics, and therefore will provide an opportunity to train graduate students in these important areas of statistics. In this project, we combine ideas from the fields of MCMC sampling and shape-constrained estimation to propose novel non-parametric estimators for uncertainty quantification in MCMC sampling. In doing so, the investigators aim to advance various aspects of statistical inference related to variance estimation in Markov chains, and to improve understanding of shape-constrained estimators. Novel asymptotic variance estimators for Markov chain Monte Carlo (MCMC) based on shape-constrained inference will be developed, which will aid in uncertainty quantification for computer simulations based on MCMC. Additionally, the investigators will develop new technical tools to analyze non-parametric least squares estimators for functions with discrete supports with non-iid inputs. These findings will be used to establish the theoretical properties, including consistency, convergence rate, and bias-variance tradeoff characterizations, of the new estimators. New variance reduction methods will be developed for Monte Carlo methods, and efficient algorithms for computing shape-constrained estimators will be developed and implemented in open-source software packages.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.

马尔可夫链蒙特卡罗（MCMC）方法已成为现代统计实践中最重要的方法之一，因为它们在各种统计设置中提供了直接的计算方法，例如贝叶斯参数估计，估计参数的不确定性量化以及模型拟合，同时允许模型规格中的不确定性。尽管广泛使用，但在量化MCMC模拟估计的不确定性方面仍然存在实际和理论上的困难。该项目将开发新的估算器，用于量化各种MCMC采样设置中的不确定性。除了做出技术贡献之外，该项目还将导致实用方法和开源软件包的开发，这将使从业者能够更准确地量化MCMC评估中的不确定性，并更有效地利用计算资源。该项目整合了包括统计机器学习、MCMC和非参数统计在内的多个领域的活跃研究课题，因此将为培养这些重要统计领域的研究生提供机会。在这个项目中，我们结合了MCMC采样和形状约束估计领域的思想，提出了一种新的用于MCMC采样不确定性量化的非参数估计。在此过程中，研究人员旨在推进与马尔可夫链中方差估计相关的统计推断的各个方面，并提高对形状约束估计器的理解。提出了一种基于形状约束推理的马尔可夫链蒙特卡罗（MCMC）渐近方差估计方法，这将有助于基于MCMC的计算机模拟的不确定性量化。此外，研究人员将开发新的技术工具来分析具有非id输入的离散支持的函数的非参数最小二乘估计。这些发现将用于建立新的估计器的理论性质，包括一致性、收敛率和偏方差权衡特征。新的方差减少方法将被开发用于蒙特卡罗方法，有效的计算形状约束估计的算法将被开发并在开源软件包中实现。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。