{"id":1003,"date":"2025-12-23T16:09:41","date_gmt":"2025-12-23T16:09:41","guid":{"rendered":"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/?p=1003"},"modified":"2026-09-30T15:00:08","modified_gmt":"2026-09-30T15:00:08","slug":"decoupling-for-markov-chains","status":"publish","type":"post","link":"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/decoupling-for-markov-chains\/","title":{"rendered":"Decoupling for Markov Chains"},"content":{"rendered":"<p><strong>How can we rigorously quantify Monte Carlo error and assess convergence in modern MCMC methods such as the No-U-Turn Sampler?<\/strong><\/p>\n<p>This question motivates <a href=\"https:\/\/arxiv.org\/abs\/2512.19351\">new joint work<\/a> with <a href=\"https:\/\/www.columbia.edu\/~vhd1\/\">Victor de la Pe\u00f1a<\/a> (Columbia), which introduces a decoupling-based perspective for Markov chains with direct applications to MCMC.<\/p>\n<p><strong>Core idea.<\/strong><\/p>\n<p>Essentially all MCMC algorithms &#8212; from Gibbs samplers to Hamiltonian Monte Carlo and NUTS &#8212; can be written as deterministic update maps driven by i.i.d. auxiliary random variables.\u00a0 While this representation is well known, we show that it can be exploited in a systematic and largely unexplored way.<\/p>\n<p>The paper introduces a <strong>tangent-decoupling framework for Markov chains<\/strong>.\u00a0Alongside the original chain, we construct a companion process by re-running the same update map along the realized trajectory, but with fresh auxiliary randomness injected at each step. The resulting tangent-decoupled sequence<\/p>\n<ul>\n<li class=\"p1\">is straightforward to simulate in tandem with the original chain,<\/li>\n<li class=\"p1\">is conditionally independent given the realized backbone trajectory, and<\/li>\n<li class=\"p1\">is generally non-Markovian, yet remains ergodically correct.<\/li>\n<\/ul>\n<p class=\"p1\"><b>Main results.<\/b><b><\/b><\/p>\n<p class=\"p1\">Two theoretical consequences of this construction are particularly striking.<\/p>\n<p class=\"p1\">First, we establish <i>almost sure consistency of empirical averages<\/i> computed from the tangent-decoupled sequence, despite its lack of Markovian structure. Under mild conditions, these averages converge to the correct target expectation, yielding a structurally simple and broadly applicable estimator.<\/p>\n<p class=\"p1\">Second, we prove a <i>sharp, nonasymptotic variance inequality<\/i>: for any square-integrable observable and any finite sample size, the variance of the standard MCMC estimator is bounded above by twice the variance of the tangent-decoupled estimator. This bound is assumption-free: it does not rely on reversibility, spectral gaps, or mixing-time arguments; and leads directly to principled uncertainty quantification for MCMC output.<\/p>\n<p><strong>Why the connection is interesting mathematically<\/strong><\/p>\n<p>Decoupling theory is a classical tool in probability, with deep connections to martingale theory, empirical processes, and the study of dependent random variables. At its core, decoupling provides a principled way to compare dependent processes with suitably constructed independent counterparts.<\/p>\n<p>What is novel here is that this abstract theory is brought into direct contact with the concrete structure of Markov chains. By identifying a natural tangent-decoupled companion process, the work shows how general decoupling principles can be realized in an explicit, algorithmically meaningful setting.<\/p>\n<p class=\"p1\">Beyond the specific results, this perspective suggests a new way to think about convergence diagnostics and error assessment in MCMC algorithms, yielding tools for error assessment that remain valid even when classical Markov-chain assumptions are unavailable or hard to verify.<\/p>\n<blockquote><p>I am very grateful to Victor for this collaboration. The project was sparked by a question he raised after an applied probability seminar I gave at Columbia, a reminder of the well-known adage that <i>good questions are often the engine of good mathematics<\/i>, and highlighting the role of seminars not only as venues for dissemination, but as catalysts for new ideas and collaborations.<\/p><\/blockquote>\n<p>The full paper is available here:\u00a0<a href=\"https:\/\/arxiv.org\/abs\/2512.19351\">https:\/\/arxiv.org\/abs\/2512.19351<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>How can we rigorously quantify Monte Carlo error and assess convergence in modern MCMC methods such as the No-U-Turn Sampler? This question motivates new joint work with Victor de la &hellip; <a href=\"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/decoupling-for-markov-chains\/\" class=\"\">Read More<\/a><\/p>\n","protected":false},"author":2614,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[8],"tags":[],"class_list":["post-1003","post","type-post","status-publish","format-standard","hentry","category-preprints"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v23.5 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Decoupling for Markov Chains - Nawaf Bou-Rabee<\/title>\n<meta name=\"description\" content=\"Quantifying Monte Carlo error and assessing convergence in modern MCMC methods such as the No-U-Turn Sampler, through decoupling for Markov chains.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/decoupling-for-markov-chains\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Decoupling for Markov Chains - 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