Multidimensional examples of the Metropolis algorithm

TL;DR

This paper investigates the convergence speed of the Metropolis algorithm in approximating distributions on multi-dimensional cubes, focusing on spectral gap bounds for fixed dimensions, especially two-dimensional cases.

Contribution

It provides explicit bounds on the spectral gap of the Metropolis algorithm in multidimensional settings using path techniques, enhancing understanding of convergence rates.

Findings

Derived upper and lower bounds on spectral gap $bb$

Applied path techniques to analyze convergence

Focused on two-dimensional case for detailed analysis

Abstract

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, $\lambda$, 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 $\lambda$ can be obtained via appropriate application of path techniques.