Optimal Scaling of Random-Walk Metropolis Algorithms on General Target Distributions

Abstract

One main limitation of the existing optimal scaling results for Metropolis--Hastings algorithms is that the assumptions on the target distribution are unrealistic. In this paper, we consider optimal scaling of random-walk Metropolis algorithms on general target distributions in high dimensions arising from practical MCMC models from Bayesian statistics. For optimal scaling by maximizing expected squared jumping distance (ESJD), we show the asymptotically optimal acceptance rate $0.234$ can be obtained under general realistic sufficient conditions on the target distribution. The new sufficient conditions are easy to be verified and may hold for some general classes of MCMC models arising from Bayesian statistics applications, which substantially generalize the product i.i.d. condition required in most existing literature of optimal scaling. Furthermore, we show one-dimensional diffusion limits can be obtained under slightly stronger conditions, which still allow dependent coordinates of the target distribution. We also connect the new diffusion limit results to complexity bounds of Metropolis algorithms in high dimensions.

arXiv comment45 pages

References in corpus (14)

• MCMC using Hamiltonian dynamics
• Renewal theory and computable convergence rates for geometrically ergodic Markov chains
• On the efficiency of pseudo-marginal random walk Metropolis algorithms
• Optimal scalings for local Metropolis--Hastings chains on nonproduct targets in high dimensions
• The Random Walk Metropolis: Linking Theory and Practice Through a Case Study
• Diffusion limits of the random walk Metropolis algorithm in high dimensions
• Conditions for rapid mixing of parallel and simulated tempering on multimodal distributions
• Optimal scaling for partially updating MCMC algorithms
• Optimal scaling of the random walk Metropolis on elliptically symmetric unimodal targets
• MCMC-Based Inference in the Era of Big Data: A Fundamental Analysis of the Convergence Complexity of High-Dimensional Chains
• Optimal scaling for the transient phase of Metropolis Hastings algorithms: The longtime behavior
• Complexity Results for MCMC derived from Quantitative Bounds
• Diffusion Limit For The Random Walk Metropolis Algorithm Out Of stationarity
• Optimal scaling of the Random Walk Metropolis algorithm under Lp mean differentiability