[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85650-en":3,"doc-seo-85650-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},85650,4398048949847,"Eliana","https://ap-avatar.wpscdn.com/avatar/400002536579ef2da7f?_k=1778318612642679267",8,"Research & Report","Optimal Scaling of MCMC Algorithms: the Hamiltonian Approach","We present a general method to analyze how Metropolised Markov chain Monte Carlo sampling algorithms scale as target dimensionality increases. The approach uses symmetries from the Hamiltonian formulation and the symmetry underlying the Metropolis–Hastings acceptance rule. Known results are recovered for Random Walk Metropolis, MALA, and related proposals. New optimal scaling laws are derived for multiple proposal mechanisms, including implicit proposals and proposals from differential-equation integrators, with extensions to product targets whose components may be scaled differently, and gradient-based MALA-like schemes achieving proposal variance O(1/d^µ).","arXiv :2607 .00586v2 [ stat .CO] 13 Jul 2026  \nOptimal scaling of MCMC algorithms: the Hamiltonian approach  \nP. Dobson  \nMaxwell Institute for Mathematical Sciences and Mathematics Department Heriot-Watt University, Edinburgh, EH14 4AS, UK  \nJ. M. Sanz-Serna  \nDepartamento de Matem´aticas, Universidad Carlos III de Madrid  \nAvenida Universidad 30, 28911 Legan´es, Madrid  \nK. C. Zygalakis  \nMaxwell Institute for Mathematical Sciences and School of Mathematics University of Edinburgh, Peter Guthrie Tait Rd, EH9 3FD, Edinburgh  \nAbstract  \nWe present a simple, yet general approach to study the scaling properties as the dimensionality of Metropolised MCMC sampling algorithms increases. The study relies on the symmetries of the Hamiltonian formalism and ultimately on the symmetry of the Metropolis-Hastings formula. Our findings contain, as particular cases, many known results for the Random Walk Metropolis, MALA and other algorithms. In addition, they provide, in an easy way, new optimal scaling results for a variety of proposal mechanisms, including implicit proposals and proposals generated with the help of differential equation integrators. The analysis applies to targets that are products of a given, not necessarily univariate distribution, and also to cases where the different terms in the product are scaled differently. We show how to construct gradient-based MALAlike proposals where the variance of the proposal as the dimension d increases may be taken as O(1/dµ ), with µ > 0 arbitrarily small, to be compared with the values µ = 1 for Random Walk Metropolis and µ = 1/3 for MALA.  \nMSC 2020 subject classifications: Primary 60J22; secondary 65C05 .  \nKeywords: Markov chain Monte Carlo; Metropolis–Hastings; optimal scaling; Hamiltonian dynamics; Langevin algorithms.  \n1 Introduction  \nThis paper presents a unified approach to the investigation of the scaling properties of different MCMC sampling algorithms. Hundreds or perhaps thousands of MCMC algorithms based on Metropolisation have been suggested to sample from probability densities  \nπ (q) ∝ exp(−V(q)), q ∈ Rm. (1)  \nTwo of the best known proposals are given by the Random Walk Metropolis (RWM) formula  \nq⋆ = q +√δp, p ∼ N (0, Im), (2)  \nand the MALA formula [23]  \nq⋆ = q − δ2∇V(q) +√δp, p ∼ N (0, Im) . (3)  \nThe latter provides a consistent discretisation of the Langevin equation  \n1  \ndq (t) = − ∇V(q(t)) dt + dW (t) (4)  \n2 ,  \nwhich preserves the target (1) .  \nA criterion to choose between all the different algorithms is to study how their parameters have to be varied as the dimensionality of the target increases. For targets consisting of d independent copies of a univariate distribution, it was proved in the pioneering contribution [20] that the variance of the RWM proposal should be scaled as δ d = ℓ2 /d, with a constant ℓ, to ensure that, as d → ∞ , the acceptance probability converges to a nontrivial limit not equal to zero or one. For MALA [21], the scaling is less demanding: δ d = ℓ2 /d1/3 . The best result available for a MALA-like sampler corresponds to fMALA [13] where δ d = ℓ2 /d1/5, but this comes at the cost of calculating higher order derivatives of the potential V. These studies also identify optimal values of the acceptance probability, so that the algorithms are most efficient if ℓ is tuned to achieve those optimal values. It is remarkable that, at least in the product of identical copies scenario, the optimal values of the acceptance probability may be proved to be independent of the target. The literature on optimal scaling is by now substantial; a selection of useful references include [22 , 6 , 3 , 13 , 1 , 18 , 19 , 17 , 28 , 29 , 25 , 2 , 15 , 16] among others.  \nIn this paper we provide a simple, yet general approach to deriving scaling results that, on the one hand, makes it possible to recover the results in [20 , 21 , 6 , 13] and, on the other, provides, inan easy way, optimal scaling results for many other proposals. These include implicit proposals th","cbCaihgrcWXvxVYO","https://ap.wps.com/l/cbCaihgrcWXvxVYO","pdf",343672,1,25,"English","en",105,"# Introduction\n## Optimal scaling background\n## Hamiltonian approach overview\n## Implicit and integrator-based proposals","[{\"question\":\"What problem does the paper address in MCMC scaling?\",\"answer\":\"It studies how the scaling properties of Metropolised MCMC algorithms change as the dimensionality of the target distribution increases.\"},{\"question\":\"How does the paper derive its scaling results?\",\"answer\":\"It exploits symmetries from the Hamiltonian formalism and the symmetry structure of the Metropolis–Hastings formula.\"},{\"question\":\"Which kinds of proposal mechanisms are covered beyond standard RWM and MALA?\",\"answer\":\"The framework yields optimal scaling results for many proposals, including implicit proposals that require solving nonlinear systems and proposals generated using differential-equation integrators from Hamiltonian numerical methods.\"}]",1784205354,63,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"optimal-scaling-of-mcmc-algorithms-the-hamiltonian-approach","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/optimal-scaling-of-mcmc-algorithms-the-hamiltonian-approach/85650/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does the paper address in MCMC scaling?","Question",{"text":75,"@type":76},"It studies how the scaling properties of Metropolised MCMC algorithms change as the dimensionality of the target distribution increases.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the paper derive its scaling results?",{"text":80,"@type":76},"It exploits symmetries from the Hamiltonian formalism and the symmetry structure of the Metropolis–Hastings formula.",{"name":82,"@type":73,"acceptedAnswer":83},"Which kinds of proposal mechanisms are covered beyond standard RWM and MALA?",{"text":84,"@type":76},"The framework yields optimal scaling results for many proposals, including implicit proposals that require solving nonlinear systems and proposals generated using differential-equation integrators from Hamiltonian numerical methods.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]