主 题：Stratification and Optimal Resampling for Sequential Monte Carlo

主讲人：清华大学邓柯副教授

主持人：统计学院林华珍教授

时间：2022年8月4日（周四）上午9:30-10:30

直播平台及会议ID：腾讯会议，ID: 717-802-666

主办单位：统计研究中心和统计学院 科研处

主讲人简介：

邓柯博士2008年获得北京大学统计学博士学位，同年进入哈佛大学统计系从事研究工作，历任博士后、副研究员。2013年加入清华大学丘成桐数学科学中心任助理教授。2015年6月，清华大学成立统计学研究中心，统筹清华大学统计学学科的建设，邓柯博士作为创始成员加入统计学研究中心并担任副主任；2016年12月晋升副教授，随后2019年晋升长聘副教授。邓柯博士的研究兴趣包括统计建模、统计计算、生物信息、文本分析、计算机网络透视等领域。邓柯博士的一系列论文发表在国际统计学顶级杂志 Journal of Royal Statistics Association (Series B), Journal of American Statistics Association, Annuals of Applied Statistics 和 Statistics in Medicine, 以及其他领域的顶级杂志如 Nature Communications, PNAS, IEEE Transaction on Signal Processing, Bioinformatics, Plos Computational Biology 等。他2008年获得中国概率统计学会颁发的“钟家庆优秀论文奖”，2014年当选中国数学会概率统计学会第十届理事会理事，2015年当选中国医疗保健国际交流促进会医学数据与医学计量分会常务委员，2017年当选中国现场统计研究会计算统计分会首任理事长、中国现场统计研究会环境与资源分会常务理事，2018年当选国际计算统计学会亚太地区分会委员。他还获得了“2016科学中国人年度人物”的荣誉称号。2020年荣获世界华人数学家国际联盟颁发的“华人数学家最佳论文奖”。

内容提要：

Sequential Monte Carlo algorithms are widely accepted as powerful computational tools for making inference with dynamical systems. A key step in sequential Monte Carlo is resampling, which plays the role of steering the algorithm towards the future dynamics. Several strategies have been used in practice, including multinomial resampling, residual resampling, optimal resampling, stratified resampling and optimal transport resampling. In one-dimensional cases, we show that optimal transport resampling is equivalent to stratified resampling on the sorted particles, and both strategies minimize the resampling variance as well as the expected squared energy distance between the original and resampled empirical distributions. For general d-dimensional cases, we show that if the particles are first sorted using the Hilbert curve, the variance of stratified resampling is O(m−(1+2/d)), an improvement over the best previously known rate of O(m−(1+1/d)), where m is the number of resampled particles. We show that this improved rate is optimal for ordered stratified resampling schemes, as conjectured in Gerber et al. (2019). We also present an almost-sure bound on the Wasserstein distance between the original and Hilbert-curve-resampled empirical distributions. In light of these results, we show that for dimension d > 1 the mean square error of sequential quasi-Monte Carlo with n particles can be O(n−1−4/{d(d+4)}) if Hilbert curve resampling is used and a specific low-discrepancy set is chosen. To our knowledge, this is the first known convergence rate lower than o(n−1).