Weak convergence of Metropolis algorithms for non-i.i.d. target distributions

TL;DR

This paper investigates the asymptotic behavior of Metropolis algorithms for high-dimensional, non-i.i.d. target distributions, proposing a method to optimize proposal scaling and confirming the optimal acceptance rate of 0.234.

Contribution

It introduces a new approach to determine proposal scaling in high dimensions and proves an asymptotic diffusion theorem for non-i.i.d. targets.

Findings

Optimal acceptance rate is 0.234 under certain conditions.

Proposed scaling method improves algorithm efficiency.

Proven asymptotic diffusion behavior for the algorithms.

Abstract

In this paper, we shall optimize the efficiency of Metropolis algorithms for multidimensional target distributions with scaling terms possibly depending on the dimension. We propose a method for determining the appropriate form for the scaling of the proposal distribution as a function of the dimension, which leads to the proof of an asymptotic diffusion theorem. We show that when there does not exist any component with a scaling term significantly smaller than the others, the asymptotically optimal acceptance rate is the well-known 0.234.