Optimal Scaling of MCMC Beyond Metropolis

TL;DR

This paper derives optimal scaling rules for a broad class of acceptance functions in MCMC, including Barker's algorithm, showing how to tune proposal distributions effectively beyond traditional Metropolis methods.

Contribution

It extends optimal scaling results to general acceptance functions like Barker's, providing new guidelines for tuning proposals in complex MCMC scenarios.

Findings

Optimal proposal variance for Barker's algorithm differs from Metropolis-Hastings.

Tuning to the optimal acceptance probability is effective even when the variance is unknown.

Numerical simulations support the theoretical optimal scaling results.

Abstract

The problem of optimally scaling the proposal distribution in a Markov chain Monte Carlo algorithm is critical to the quality of the generated samples. Much work has gone into obtaining such results for various Metropolis-Hastings (MH) algorithms. Recently, acceptance probabilities other than MH are being employed in problems with intractable target distributions. There is little resource available on tuning the Gaussian proposal distributions for this situation. We obtain optimal scaling results for a general class of acceptance functions, which includes Barker's and Lazy-MH. In particular, optimal values for the Barker's algorithm are derived and found to be significantly different from that obtained for the MH algorithm. Our theoretical conclusions are supported by numerical simulations indicating that when the optimal proposal variance is unknown, tuning to the optimal acceptance probability remains an effective strategy.