N. Brosse, A. Durmus, S. Meyn, E. Moulines, S. Samsonov
{"title":"Diffusion Approximations and Control Variates for MCMC","authors":"N. Brosse, A. Durmus, S. Meyn, E. Moulines, S. Samsonov","doi":"10.1134/s0965542524700167","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>A new method is introduced for the construction of control variates to reduce the variance of additive functionals of Markov Chain Monte Carlo (MCMC) samplers. These control variates are obtained by minimizing the asymptotic variance associated with the Langevin diffusion over a family of functions. To motivate our approach, we then show that the asymptotic variance of some well-known MCMC algorithms, including the Random Walk Metropolis and the (Metropolis) Unadjusted/Adjusted Langevin Algorithm, are well approximated by that of the Langevin diffusion. We finally theoretically justify the use of a class of linear control variates we introduce. In particular, we show that the variance of the resulting estimators is smaller, for a given computational complexity, than the standard Monte Carlo estimator. Several examples of Bayesian inference problems support our findings showing, in some cases, very significant reduction of the variance.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computational Mathematics and Mathematical Physics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1134/s0965542524700167","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
A new method is introduced for the construction of control variates to reduce the variance of additive functionals of Markov Chain Monte Carlo (MCMC) samplers. These control variates are obtained by minimizing the asymptotic variance associated with the Langevin diffusion over a family of functions. To motivate our approach, we then show that the asymptotic variance of some well-known MCMC algorithms, including the Random Walk Metropolis and the (Metropolis) Unadjusted/Adjusted Langevin Algorithm, are well approximated by that of the Langevin diffusion. We finally theoretically justify the use of a class of linear control variates we introduce. In particular, we show that the variance of the resulting estimators is smaller, for a given computational complexity, than the standard Monte Carlo estimator. Several examples of Bayesian inference problems support our findings showing, in some cases, very significant reduction of the variance.
期刊介绍：
Computational Mathematics and Mathematical Physics is a monthly journal published in collaboration with the Russian Academy of Sciences. The journal includes reviews and original papers on computational mathematics, computational methods of mathematical physics, informatics, and other mathematical sciences. The journal welcomes reviews and original articles from all countries in the English or Russian language.
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
