Metropolis-Hastings 算法是 MCMC 中一种流行的采样方法. 该文章主要研究了如何选取底过程 P 来加速 Z+ 上的 Metropolis-Hastings 过程 P! 的收敛速度. 我们首先在非常广的框架下完全刻画了底图 Γ(P ) 和Metropolis-Hastings 图 Γ(P!) 的差异. 之后我们具体研究了两种底过程: 星型过程和随机游动. 对于星型过程, 通过矩方法我们发现, 通过适当地选取星型底过程, 几何 (强) 遍历总是可以达到的. 对于随机游动, 我们首先证明了 P! 的几何遍历性完全由目标测度 μ 所决定并且给出了一个充要条件来完全刻画保证几何遍历的 μ 的特性. 然后, 通过局部最大化狄氏型 E(f, f), 我们得出了提高收敛速度的底过程的显式表达. 最后, 我们提出了可操作的方法来选取随机游动底过程的参数.

Metropolis-Hastings algorithm is a popular way of sampling in MCMC. This article mainly investigates how to choose the base process P to accelerate the convergent speed of the Metropolis-Hastings process P! on Z+. We first completely characterize discrepancies between the base graph Γ(P ) and the Metropolis-Hastings graph Γ(P!) in a general setting. Then we specifically investigate two types of base processes: star-like processes and random walks. For star-like processes, the method of moments is applied to find that geometrical (strong) ergodicity is always feasible by choosing proper star-like base processes. For random walks, we first prove that the geometrical ergodicity of P! is determined by its target measure μ, and provide a sufficient and necessary condition to completely capture the characteristic of μ to guarantee the geometrical ergodicity. Next, by locally maximizing the Dirichlet form E(f,f), we obtain the explicit expression of the base process for higher convergent speed. Finally, a practical procedure to choose parameters of random walks base process is presented.