走向现代数学学术报告 - 何志坚教授（No. 746)

题目：Unbiased Markov chain quasi-Monte Carlo for Gibbs samplers 

报告人：何志坚 教授（华南理工大学） 

时间：2024年11月17日 10:00 

地点：东海岸校区-D实209 

摘要：In statistical analysis, Monte Carlo (MC) stands as a classical numerical integration method. When encountering challenging sample problem, Markov chain Monte Carlo (MCMC) is a commonly employed method. However, the MCMC estimator is biased after a fixed number of iterations. Unbiased MCMC, an advancement achieved through coupling techniques, addresses this bias issue in MCMC. It allows us to run many short chains in parallel. Quasi-Monte Carlo (QMC), known for its high order of convergence, is an alternative of MC. By incorporating the idea of QMC into MCMC, Markov chain quasi-Monte Carlo (MCQMC) effectively reduces the variance of MCMC, especially in Gibbs samplers. This work presents a novel approach that integrates unbiased MCMC with MCQMC, called as an unbiased MCQMC method. This method renders unbiased estimators while improving the rate of convergence significantly. Numerical experiments demonstrate that for Gibbs sampling, unbiased MCQMC with a sample size of N yields a faster root mean square error (RMSE) rate than the O(N^{-1/2}) rate of unbiased MCMC, toward an RMSE rate of O(N^{-1}) for low-dimensional problems. Surprisingly, in a challenging problem of 1049-dimensional P\'olya Gamma Gibbs sampler, the RMSE can still be reduced by several times for moderate sample sizes. In the setting of parallelization, unbiased MCQMC also performs better than unbiased MCMC, even running with short chains. This is joint work with Jiarui Du. 

报告人简介：何志坚，华南理工大学数学学院教授、博导、副院长，广东省计算数学学会副理事长，广东省现场统计学会理事。于2015年7月在清华大学获得理学博士学位。研究兴趣为随机计算方法与不确定性量化，特别是拟蒙特卡罗方法的理论和应用研究。目前发表SCI论文近20篇，其中十余篇发表在统计学和计算科学的国际著名刊物Journal of the Royal Statistical Society: Series B，SIAM Journal on Numerical Analysis，SIAM Journal on Scientific Computing，Mathematics of Computation。博士论文获得新世界数学奖（ICCM毕业论文奖）银奖。主持一项国家级青年人才计划项目、两项国家自然科学基金项目以及四项省部级项目。