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和模型拟合，同时允许模型规格的不确定性。尽管广泛使用，但量化 MCMC 模拟估计的不确定性仍然存在实际和理论上的困难。该项目将开发新颖的估计器，用于量化各种 MCMC 采样设置中的不确定性。除了做出技术贡献外，该项目还将开发实用方法和开源软件包，使从业者能够更准确地量化 MCMC 估计中的不确定性，并更有效地利用计算资源。该项目整合了统计机器学习、MCMC 和非参数统计等多个领域的活跃研究主题，因此将为在这些重要的统计学领域培训研究生提供机会。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，提出了用于 MCMC 采样中不确定性量化的新型非参数估计器。在此过程中，研究人员的目标是推进与马尔可夫链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量的理解。将开发基于形状约束推理的新型马尔可夫链蒙特卡罗（MCMC）渐近方差估计器，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独立同分布输入的离散支持的函数的非参数最小二乘估计器。这些发现将用于建立新估计量的理论特性，包括一致性、收敛速度和偏差-方差权衡特征。将为蒙特卡洛方法开发新的方差减少方法，并将开发计算形状约束估计量的有效算法并在开源软件包中实现。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准。