提出哈密顿框架下的MCMC最优缩放方法，统一分析多种采样算法性能。
Optimal scaling of MCMC algorithms: the Hamiltonian approach
- 基于哈密顿对称性与MH公式对称性，建立通用缩放分析框架。
- 证明可实现渐进方差与维度呈O(1/d^μ)关系，μ可任意接近0。
- 适用于多变量目标分布及不同缩放比例，适合优化复杂高维采样。
我们提出一种简洁而通用的方法，研究马尔可夫链蒙特卡洛（MCMC）采样算法在维度增加时的缩放特性。该方法基于哈密顿形式的对称性，最终依赖于梅特罗波利斯-哈斯金斯（Metropolis-Hastings）公式的对称性。所得结果涵盖随机游走梅特罗波利斯（RWM）、MALA及其他算法的已知结论，并以简便方式推导出多种提议机制（包括隐式提议和微分方程积分器生成的提议）的最优缩放结果。分析适用于乘积形式的目标分布，且各因子可分别缩放。我们展示了如何构造基于梯度的类似MALA的提议，其方差随维度d增长可取O(1/d^μ)，其中μ>0可任意小；相比之下，随机游走梅特罗波利斯为μ=1，而MALA为μ=1/3。
原文摘要 · Abstract (English)
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.
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