发表机构
Departamento de Matemáticas, Universidad Carlos III de Madrid（马德里卡洛斯三世大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出一种基于Metropolis-Hastings公式对称性的通用方法，研究高维Metropolised MCMC采样算法的缩放性质，统一并推广了随机游走Metropolis、MALA等算法的已知结果，并得到多种提议机制的新最优缩放结果。
AI 中文摘要
我们提出了一种简单而通用的方法来研究随着维度增加，Metropolised MCMC采样算法的缩放性质。该研究最终依赖于Metropolis-Hastings公式的对称性。我们的结果包括随机游走Metropolis、MALA和其他算法的许多已知结果作为特例。此外，它们以简单的方式为各种提议机制提供了新的最优缩放结果，包括隐式提议和借助微分方程积分器生成的提议。该分析适用于目标分布是给定（不一定是一元）分布的乘积的情况，也适用于乘积中不同项被不同缩放的情况。我们展示了如何构建基于梯度的类似MALA的提议，其中提议的方差随着维度$d$的增加可以取为$O(1/d^\mu)$，其中$\mu>0$可以任意小，相比之下随机游走Metropolis的$\mu=1$，MALA的$\mu=1/3$。
英文摘要
We 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 MALA-like proposals where the variance of the proposal as the dimension $d$ increases may be taken as $O(1/d^μ)$, with $μ>0$ arbitrarily small, to be compared with the values $μ= 1$ for Random Walk Metropolis and $μ=1/3$ for MALA.
Comments25 pages, 3 figures
