{"title":"可逆 MCMC 方法的效率:基本推导及对复合方法的应用","authors":"Radford M. Neal, Jeffrey S. Rosenthal","doi":"10.1017/jpr.2024.48","DOIUrl":null,"url":null,"abstract":"We review criteria for comparing the efficiency of Markov chain Monte Carlo (MCMC) methods with respect to the asymptotic variance of estimates of expectations of functions of state, and show how such criteria can justify ways of combining improvements to MCMC methods. We say that a chain on a finite state space with transition matrix <jats:italic>P</jats:italic> efficiency-dominates one with transition matrix <jats:italic>Q</jats:italic> if for every function of state it has lower (or equal) asymptotic variance. We give elementary proofs of some previous results regarding efficiency dominance, leading to a self-contained demonstration that a reversible chain with transition matrix <jats:italic>P</jats:italic> efficiency-dominates a reversible chain with transition matrix <jats:italic>Q</jats:italic> if and only if none of the eigenvalues of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000482_inline1.png\"/> <jats:tex-math> $Q-P$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> are negative. This allows us to conclude that modifying a reversible MCMC method to improve its efficiency will also improve the efficiency of a method that randomly chooses either this or some other reversible method, and to conclude that improving the efficiency of a reversible update for one component of state (as in Gibbs sampling) will improve the overall efficiency of a reversible method that combines this and other updates. It also explains how antithetic MCMC can be more efficient than independent and identically distributed sampling. We also establish conditions that can guarantee that a method is not efficiency-dominated by any other method.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Efficiency of reversible MCMC methods: elementary derivations and applications to composite methods\",\"authors\":\"Radford M. Neal, Jeffrey S. Rosenthal\",\"doi\":\"10.1017/jpr.2024.48\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We review criteria for comparing the efficiency of Markov chain Monte Carlo (MCMC) methods with respect to the asymptotic variance of estimates of expectations of functions of state, and show how such criteria can justify ways of combining improvements to MCMC methods. 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引用次数: 0
摘要
我们回顾了比较马尔可夫链蒙特卡罗(MCMC)方法在状态函数期望估计值渐近方差方面的效率的标准,并说明了这些标准如何证明将改进 MCMC 方法结合起来的方法是合理的。如果对每个状态函数而言,具有过渡矩阵 P 的有限状态空间链的渐近方差较小(或相等),我们就认为该链的效率优于具有过渡矩阵 Q 的有限状态空间链。我们给出了以前关于效率优势的一些结果的基本证明,从而得出一个自足的论证:当且仅当 $Q-P$ 的特征值都不是负数时,具有过渡矩阵 P 的可逆链效率优势于具有过渡矩阵 Q 的可逆链。这让我们得出结论:修改一种可逆 MCMC 方法以提高其效率,也会提高随机选择这种或其他可逆方法的方法的效率;还让我们得出结论:提高状态的一个组成部分的可逆更新(如吉布斯采样)的效率,会提高结合这种更新和其他更新的可逆方法的整体效率。这也解释了反可逆 MCMC 如何比独立同分布采样更高效。我们还建立了一些条件,以保证一种方法的效率不会被任何其他方法所支配。
Efficiency of reversible MCMC methods: elementary derivations and applications to composite methods
We review criteria for comparing the efficiency of Markov chain Monte Carlo (MCMC) methods with respect to the asymptotic variance of estimates of expectations of functions of state, and show how such criteria can justify ways of combining improvements to MCMC methods. We say that a chain on a finite state space with transition matrix P efficiency-dominates one with transition matrix Q if for every function of state it has lower (or equal) asymptotic variance. We give elementary proofs of some previous results regarding efficiency dominance, leading to a self-contained demonstration that a reversible chain with transition matrix P efficiency-dominates a reversible chain with transition matrix Q if and only if none of the eigenvalues of $Q-P$ are negative. This allows us to conclude that modifying a reversible MCMC method to improve its efficiency will also improve the efficiency of a method that randomly chooses either this or some other reversible method, and to conclude that improving the efficiency of a reversible update for one component of state (as in Gibbs sampling) will improve the overall efficiency of a reversible method that combines this and other updates. It also explains how antithetic MCMC can be more efficient than independent and identically distributed sampling. We also establish conditions that can guarantee that a method is not efficiency-dominated by any other method.
期刊介绍:
Journal of Applied Probability is the oldest journal devoted to the publication of research in the field of applied probability. It is an international journal published by the Applied Probability Trust, and it serves as a companion publication to the Advances in Applied Probability. Its wide audience includes leading researchers across the entire spectrum of applied probability, including biosciences applications, operations research, telecommunications, computer science, engineering, epidemiology, financial mathematics, the physical and social sciences, and any field where stochastic modeling is used.
A submission to Applied Probability represents a submission that may, at the Editor-in-Chief’s discretion, appear in either the Journal of Applied Probability or the Advances in Applied Probability. Typically, shorter papers appear in the Journal, with longer contributions appearing in the Advances.