Metropolis-Hastings algorithms with autoregressive proposals, and a few examples

TL;DR

This paper analyzes the efficiency of Metropolis-Hastings algorithms with stochastic AR(1) process proposals, including Langevin and Hamiltonian dynamics, deriving formulas for acceptance rates and jump sizes, and optimizing parameters for improved sampling performance.

Contribution

It extends existing analysis of AR(1) proposals to more general Gaussian targets and includes the effect of Metropolis-Hastings, providing new insights into optimal parameter choices and efficiency.

Findings

Derived formulas for acceptance rate and jump size for AR(1) proposals.

Extended analysis to Gaussian targets with off-diagonal covariance.

Identified optimal parameters for Hybrid Monte Carlo, including integration time and mass matrix.

Abstract

We analyse computational efficiency of Metropolis-Hastings algorithms with stochastic AR(1) process proposals. These proposals include, as a subclass, discretized Langevin diffusion (e.g. MALA) and discretized Hamiltonian dynamics (e.g. HMC). We derive expressions for the expected acceptance rate and expected jump size for MCMC methods with general stochastic AR(1) process proposals for the case where the target distribution is absolutely continuous with respect to a Gaussian and the covariance of the Gaussian is allowed to have off-diagonal terms. This allows us to extend what is known about several MCMC methods as well as determining the efficiency of new MCMC methods of this type. In the special case of Hybrid Monte Carlo, we can determine the optimal integration time and the effect of the choice of mass matrix. By including the effect of Metropolis-Hastings we also extend results by Fox and Parker, who used matrix splitting techniques to analyse the performance and improve efficiency of stochastic AR(1) processes for sampling from Gaussian distributions.