Efficiency and computability of MCMC with Langevin, Hamiltonian, and other matrix-splitting proposals

TL;DR

This paper analyzes the efficiency of Metropolis-Hastings algorithms using AR(1) process proposals, including Langevin and Hamiltonian dynamics, extending previous work to non-Gaussian targets through matrix splitting techniques.

Contribution

It extends matrix splitting analysis to non-Gaussian targets by examining AR(1) process proposals within Metropolis-Hastings algorithms.

Findings

Provides a framework for analyzing MCMC efficiency with non-Gaussian targets.

Extends previous Gaussian-focused analysis to broader classes of distributions.

Offers insights into the computational performance of Langevin and Hamiltonian proposals.

Abstract

We analyse computational efficiency of Metropolis-Hastings algorithms with AR(1) process proposals. These proposals include, as a subclass, discretized Langevin diffusion (e.g. MALA) and discretized Hamiltonian dynamics (e.g. HMC). By including the effect of Metropolis-Hastings we extend earlier work by Fox and Parker, who used matrix splitting techniques to analyse the performance and improve efficiency of AR(1) processes for targeting Gaussian distributions. Our research enables analysis of MCMC methods that draw samples from non-Gaussian target distributions by using AR(1) process proposals in Metropolis-Hastings algorithms, by analysing the matrix splitting of the precision matrix for a local Gaussian approximation of the non-Gaussian target.