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

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.