论文标题
大都市算法的多维示例
Multidimensional examples of the Metropolis algorithm
论文作者
论文摘要
考虑通过使用带有网格大小$ 1/n $和大都市算法的方形离散化和大都市算法的方格离散化，考虑了立方体$ [0,1]^n $上给定概率分布的问题。在这里，尺寸$ n $是固定的，我们大部分都集中在案例$ n = 2 $上。为了了解这种过程的收敛速度，需要控制相关有限马尔可夫链的光谱差距，$λ$，以及它如何依赖于参数$ n $。在这项工作中，我们研究了基本示例，这些示例可以通过适当应用路径技术来获得$λ$的良好上限和下限。
Consider the problem of approximating a given probability distribution on the cube $[0,1]^n$ via the use of a square lattice discretization with mesh-size $1/N$ and the Metropolis algorithm. Here the dimension $n$ is fixed and we focus for the most part on the case $n=2$. In order to understand the speed of convergence of such a procedure, one needs to control the spectral gap, $λ$, of the associated finite Markov chain, and how it depends on the parameter $N$. In this work, we study basic examples for which good upper-bounds and lower-bounds on $λ$ can be obtained via appropriate application of path techniques.
