{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,26]],"date-time":"2026-09-26T09:15:30Z","timestamp":1790414130592,"version":"4.1.0"},"reference-count":53,"publisher":"Oxford University Press (OUP)","issue":"1","license":[{"start":{"date-parts":[[2011,11,3]],"date-time":"2011-11-03T00:00:00Z","timestamp":1320278400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"funder":[{"name":"Marie Curie International Outgoing Fellowship","award":["PIOF-GA-2009-235837"],"award-info":[{"award-number":["PIOF-GA-2009-235837"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2012,1,1]]},"abstract":"<jats:title>Summary<\/jats:title><jats:p>A general methodology is introduced for the construction and effective application of control variates to estimation problems involving data from reversible Markov chain Monte Carlo samplers. We propose the use of a specific class of functions as control variates, and we introduce a new consistent estimator for the values of the coefficients of the optimal linear combination of these functions. For a specific Markov chain Monte Carlo scenario, the form and proposed construction of the control variates is shown to provide an exact solution of the associated Poisson equation. This implies that the estimation variance in this case (in the central limit theorem regime) is exactly zero. The new estimator is derived from a novel, finite dimensional, explicit representation for the optimal coefficients. The resulting variance reduction methodology is primarily (though certainly not exclusively) applicable when the simulated data are generated by a random-scan Gibbs sampler. Markov chain Monte Carlo examples of Bayesian inference problems demonstrate that the corresponding reduction in the estimation variance is significant, and that in some cases it can be quite dramatic. Extensions of this methodology are discussed and simulation examples are presented illustrating the utility of the methods proposed. All methodological and asymptotic arguments are rigorously justified under essentially minimal conditions.<\/jats:p>","DOI":"10.1111\/j.1467-9868.2011.01000.x","type":"journal-article","created":{"date-parts":[[2011,11,3]],"date-time":"2011-11-03T17:47:24Z","timestamp":1320342444000},"page":"133-161","source":"Crossref","is-referenced-by-count":33,"title":["Control Variates for Estimation Based on Reversible Markov Chain Monte Carlo Samplers"],"prefix":"10.1093","volume":"74","author":[{"given":"Petros","family":"Dellaportas","sequence":"first","affiliation":[{"name":"Athens University of Economics and Business , Greece"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ioannis","family":"Kontoyiannis","sequence":"additional","affiliation":[{"name":"Athens University of Economics and Business , Greece"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2011,11,3]]},"reference":[{"key":"2023031108151014700_","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1145\/174153.174154","article-title":"Variance reduction through smoothing and control variates for Markov Chain simulations","volume":"3","author":"Andrad\u00f2ttir","year":"1993","journal-title":"ACM Trans. Modlng Comput. Simuln"},{"key":"2023031108151014700_","doi-asserted-by":"crossref","first-page":"4682","DOI":"10.1103\/PhysRevLett.83.4682","article-title":"Zero-variance principle for monte carlo algorithms","volume":"83","author":"Assaraf","year":"1999","journal-title":"Phys. Rev. Lett."},{"key":"2023031108151014700_","first-page":"3","article-title":"Improving on the independent Metropolis-Hastings algorithm","volume":"15","author":"Atchad\u00e9","year":"2005","journal-title":"Statist. Sin."},{"key":"2023031108151014700_","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1017\/S0269964800001674","article-title":"Improving stochastic relaxation for Gaussian random fields","volume":"4","author":"Barone","year":"1989","journal-title":"Probab. Engng Inform. Sci."},{"key":"2023031108151014700_","doi-asserted-by":"crossref","first-page":"846","DOI":"10.1080\/03610911003650375","article-title":"A new multinomial model and a zero variance estimation","volume":"39","author":"Dalla Valle","year":"2010","journal-title":"Communs Statist. Simuln Computn"},{"key":"2023031108151014700_","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1111\/1467-9868.00179","article-title":"Gibbs sampling for Bayesian non-conjugate and hierarchical models by using auxiliary variables","volume":"61","author":"Damien","year":"1999","journal-title":"J. R. Statist. Soc."},{"key":"2023031108151014700_","author":"Dellaportas","year":"2009"},{"key":"2023031108151014700_","volume-title":"Manuscript","author":"Dellaportas","year":"2010"},{"key":"2023031108151014700_","first-page":"1","volume-title":"Spatial Statistics and Computational Methods","author":"Dellaportas","year":"2003"},{"key":"2023031108151014700_","doi-asserted-by":"crossref","first-page":"938","DOI":"10.1239\/jap\/1261670681","article-title":"Does waste recycling really improve the multi-proposal Metropolis-Hastings algorithm?: an analysis based on control variates","volume":"46","author":"Delmas","year":"2009","journal-title":"J. Appl. Probab."},{"key":"2023031108151014700_","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1307\/mmj\/1030132713","article-title":"Analysis of systematic scan Metropolis algorithms using Iwahori-Hecke algebra techniques","volume":"48","author":"Diaconis","year":"2000","journal-title":"Mich. Math. J."},{"key":"2023031108151014700_","doi-asserted-by":"crossref","first-page":"178","DOI":"10.1198\/106186006X96908","article-title":"Output assessment for Monte Carlo simulations via the score statistic","volume":"15","author":"Fan","year":"2006","journal-title":"J. Computnl Graph. Statist."},{"key":"2023031108151014700_","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1239\/jap\/1044476831","article-title":"On the geometric ergodicity of hybrid samplers","volume":"40","author":"Fort","year":"2003","journal-title":"J. Appl. Probab."},{"key":"2023031108151014700_","doi-asserted-by":"crossref","first-page":"513","DOI":"10.1016\/S0167-9473(02)00211-6","article-title":"Space-varying regression models: specifications and simulation","volume":"42","author":"Gamerman","year":"2003","journal-title":"Computnl Statist. Data Anal."},{"key":"2023031108151014700_","doi-asserted-by":"crossref","first-page":"972","DOI":"10.1080\/01621459.1990.10474968","article-title":"Illustration of Bayesian inference in normal data models using Gibbs sampling","volume":"85","author":"Gelfand","year":"1990","journal-title":"J. Am. Statist. Ass."},{"key":"2023031108151014700_","doi-asserted-by":"crossref","first-page":"398","DOI":"10.1080\/01621459.1990.10476213","article-title":"Sampling-based approaches to calculating marginal densities","volume":"85","author":"Gelfand","year":"1990","journal-title":"J. Am. Statist. Ass."},{"key":"2023031108151014700_","volume-title":"Computational Statistics","author":"Givens","year":"2005"},{"key":"2023031108151014700_","volume-title":"Monte Carlo Methods in Financial Engineering","author":"Glasserman","year":"2004"},{"key":"2023031108151014700_","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1007\/978-3-642-56046-0_3","volume-title":"Monte Carlo and Quasi-Monte Carlo Methods, 2000 (Hong Kong)","author":"Glynn","year":"2002"},{"key":"2023031108151014700_","doi-asserted-by":"crossref","first-page":"142","DOI":"10.1007\/978-1-4612-2920-9_10","article-title":"Metropolis methods, Gaussian proposals, and antithetic variables","volume":"74","author":"Green","year":"1992","journal-title":"Lect. Notes Statist."},{"key":"2023031108151014700_","doi-asserted-by":"crossref","first-page":"400","DOI":"10.1111\/j.1467-9469.2008.00601.x","article-title":"Control variates for the Metropolis-Hastings algorithm","volume":"35","author":"Hammer","year":"2008","journal-title":"Scand. J. Statist."},{"key":"2023031108151014700_","volume-title":"PhD Thesis","author":"Henderson","year":"1997"},{"key":"2023031108151014700_","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1287\/moor.27.2.253.329","article-title":"Approximating martingales for variance reduction in Markov process simulation","volume":"27","author":"Henderson","year":"2002","journal-title":"Math. Oper. Res."},{"key":"2023031108151014700_","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1023\/A:1022197004856","article-title":"Performance evaluation and policy selection in multiclass networks","volume":"13","author":"Henderson","year":"2003","journal-title":"Discr. Event Dyn. Syst."},{"key":"2023031108151014700_","doi-asserted-by":"crossref","first-page":"859","DOI":"10.1239\/jap\/1091543430","article-title":"Adaptive simulation using perfect control variates","volume":"41","author":"Henderson","year":"2004","journal-title":"J. Appl. Probab."},{"key":"2023031108151014700_","doi-asserted-by":"crossref","first-page":"414","DOI":"10.1006\/jmva.1998.1778","article-title":"Geometric ergodicity of Gibbs and block Gibbs samplers for a hierarchical random effects model","volume":"67","author":"Hobert","year":"1998","journal-title":"J. Multiv. Anal."},{"key":"2023031108151014700_","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1016\/S0304-4149(99)00082-4","article-title":"Geometric ergodicity of Metropolis algorithms","volume":"85","author":"Jarner","year":"2000","journal-title":"Stochast. Process. Appl."},{"key":"2023031108151014700_","doi-asserted-by":"crossref","first-page":"508","DOI":"10.1287\/moor.1070.0251","article-title":"Adaptive control variates for finite-horizon simulation","volume":"32","author":"Kim","year":"2007","journal-title":"Math. Oper. Res."},{"key":"2023031108151014700_","doi-asserted-by":"crossref","first-page":"304","DOI":"10.1214\/aoap\/1042765670","article-title":"Spectral theory and limit theorems for geometrically ergodic Markov processes","volume":"13","author":"Kontoyiannis","year":"2003","journal-title":"Ann. Appl. Probab."},{"key":"2023031108151014700_","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1214\/EJP.v10-231","article-title":"Large deviation asymptotics and the spectral theory of multiplicatively regular Markov processes","volume":"10","author":"Kontoyiannis","year":"2005","journal-title":"Electron. J. Probab."},{"key":"2023031108151014700_","doi-asserted-by":"crossref","first-page":"322","DOI":"10.1287\/mnsc.27.3.322","article-title":"A perspective on the use of control variables to increase the efficiency of Monte Carlo simulations","volume":"27","author":"Lavenberg","year":"1981","journal-title":"Mangmnt Sci."},{"key":"2023031108151014700_","volume-title":"Simulation Modeling and Analysis","author":"Law","year":"1982"},{"key":"2023031108151014700_","volume-title":"Monte Carlo Strategies in Scientific Computing","author":"Liu","year":"2001"},{"key":"2023031108151014700_","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1080\/10618600.1998.10474772","article-title":"Estimating mixture of Dirichlet process models","volume":"7","author":"MacEachern","year":"1998","journal-title":"J. Computnl Graph. Statist."},{"key":"2023031108151014700_","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1093\/oso\/9780198504856.003.0018","volume-title":"Bayesian Statistics 6","author":"Mengersen","year":"1999"},{"key":"2023031108151014700_","doi-asserted-by":"crossref","first-page":"310","DOI":"10.1214\/105051605000000737","article-title":"Large deviation asymptotics and control variates for simulating large functions","volume":"16","author":"Meyn","year":"2006","journal-title":"Ann. Appl. Probab."},{"key":"2023031108151014700_","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511804410","volume-title":"Control Techniques for Complex Networks","author":"Meyn","year":"2007"},{"key":"2023031108151014700_","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511626630","volume-title":"Markov Chains and Stochastic Stability","author":"Meyn","year":"2009"},{"key":"2023031108151014700_","article-title":"Zero variance Markov chain Monte Carlo for Bayesian estimators","author":"Mira","year":"2010","journal-title":"Technical Report"},{"key":"2023031108151014700_","volume-title":"Technical Report 2003\/29","author":"Mira","year":"2003"},{"key":"2023031108151014700_","volume-title":"Technical Report","author":"M\u00fcller","year":"1993"},{"key":"2023031108151014700_","doi-asserted-by":"crossref","first-page":"855","DOI":"10.1287\/mnsc.1040.0246","article-title":"Stochastic simulation research in management science","volume":"50","author":"Nelson","year":"2004","journal-title":"Mangmnt Sci."},{"key":"2023031108151014700_","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511526237","volume-title":"General Irreducible Markov Chains and Nonnegative Operators","author":"Nummelin","year":"1984"},{"key":"2023031108151014700_","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1023\/A:1008926514119","article-title":"Riemann sums for MCMC estimation and convergence monitoring","volume":"11","author":"Philippe","year":"2001","journal-title":"Statist. Comput."},{"key":"2023031108151014700_","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1111\/j.1751-5823.2006.tb00170.x","article-title":"Comparison of sampling schemes for dynamic linear models","volume":"74","author":"Reis","year":"2006","journal-title":"Int. Statist. Rev."},{"key":"2023031108151014700_","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-4145-2","volume-title":"Monte Carlo Statistical Methods","author":"Robert","year":"2004"},{"key":"2023031108151014700_","doi-asserted-by":"crossref","first-page":"5","DOI":"10.2307\/3315667","article-title":"Markov-chain Monte Carlo: some practical implications of theoretical results (with discussion)","volume":"26","author":"Roberts","year":"1998","journal-title":"Can. J. Statist."},{"key":"2023031108151014700_","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1214\/154957804100000024","article-title":"General state space Markov chains and MCMC algorithms","volume":"1","author":"Roberts","year":"2004","journal-title":"Probab. Surv."},{"key":"2023031108151014700_","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1198\/jcgs.2009.06134","article-title":"Examples of adaptive MCMC","volume":"18","author":"Roberts","year":"2009","journal-title":"J. Computnl Graph. Statist."},{"key":"2023031108151014700_","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1111\/1467-9868.00070","article-title":"Updating schemes, correlation structure, blocking and parameterization for the Gibbs sampler","volume":"59","author":"Roberts","year":"1997","journal-title":"J. R. Statist. Soc."},{"key":"2023031108151014700_","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1093\/biomet\/83.1.95","article-title":"Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms","volume":"83","author":"Roberts","year":"1996","journal-title":"Biometrika"},{"key":"2023031108151014700_","first-page":"1","volume-title":"Stein\u2019s Method: Expository Lectures and Applications","author":"Stein","year":"2004"},{"key":"2023031108151014700_","first-page":"1701","article-title":"Markov chains for exploring posterior distributions (with discussion)","volume":"22","author":"Tierney","year":"1994","journal-title":"Ann. Statist."}],"container-title":["Journal of the Royal Statistical Society Series B: Statistical Methodology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1111%2Fj.1467-9868.2011.01000.x","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/74\/1\/133\/49509063\/jrsssb_74_1_133.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/74\/1\/133\/49509063\/jrsssb_74_1_133.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,3,13]],"date-time":"2025-03-13T19:31:03Z","timestamp":1741894263000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jrsssb\/article\/74\/1\/133\/7074993"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,11,3]]},"references-count":53,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2011,11,3]]},"published-print":{"date-parts":[[2012,1,1]]}},"URL":"https:\/\/doi.org\/10.1111\/j.1467-9868.2011.01000.x","relation":{},"ISSN":["1369-7412","1467-9868"],"issn-type":[{"value":"1369-7412","type":"print"},{"value":"1467-9868","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2012,1]]},"published":{"date-parts":[[2011,11,3]]}}}