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

马尔可夫链蒙特卡罗(MCMC)方法已经成为现代统计学实践中最重要的方法之一，因为它们在各种统计环境中提供了直接的计算方法，如贝叶斯参数估计、估计参数的不确定性量化和模型拟合，同时允许模型规格中的不确定性。尽管被广泛使用，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。该项目将开发新的估计器，用于量化各种MCMC抽样设置中的不确定性。除了作出技术贡献外，该项目还将导致开发实用方法和开放源码软件包，使从业人员能够更准确地量化MCMC估计中的不确定性，并更有效地利用计算资源。该项目整合了多个领域的活跃研究课题，包括统计机器学习、MCMC和非参数统计，因此将为研究生提供在这些重要统计领域进行培训的机会。在这个项目中，我们结合了MCMC抽样和形状约束估计领域的思想，提出了一种新的用于MCMC抽样不确定性量化的非参数估计器。在这样做的过程中，研究人员的目标是推进与马尔可夫链中的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。基于形状约束推理的马尔可夫链蒙特卡罗(MCMC)渐近方差估计器将被开发出来，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iid输入的离散支持函数的非参数最小二乘估计器。这些发现将被用来建立新估计量的理论性质，包括一致性、收敛速度和偏差-方差权衡特征。将为蒙特卡罗方法开发新的方差减少方法，并将开发高效的算法来计算形状约束估计器，并在开源软件包中实施。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。

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.