[Paper Review] Optimal Scaling of Random-Walk Metropolis Algorithms on General Target Distributions

Jun Yang, Gareth O. Roberts+1 more|arXiv (Cornell University)|Apr 27, 2019

Markov Chains and Monte Carlo Methods57 references4 citations

TL;DR

This paper establishes optimal scaling for random-walk Metropolis algorithms on general high-dimensional target distributions in Bayesian statistics, showing that the asymptotically optimal acceptance rate of 0.234 is achievable under realistic, verifiable conditions that generalize beyond the restrictive i.i.d. product assumptions of prior work. The results are derived using diffusion limit approximations and apply to models with dependent coordinates and non-Gaussian targets.

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.

Motivation & Objective

• To extend optimal scaling theory for random-walk Metropolis algorithms beyond the restrictive i.i.d. product target assumption.
• To identify general, verifiable sufficient conditions under which the optimal acceptance rate of 0.234 holds for high-dimensional target distributions.
• To derive one-dimensional diffusion limits for RWM under slightly stronger but still realistic conditions on dependent target distributions.
• To connect the diffusion limit results to complexity bounds for MCMC algorithms in high dimensions.
• To validate the theory on practical Bayesian models with non-i.i.d. structures, such as hierarchical and graphical models.

Proposed method

• Derives optimal scaling by maximizing the expected squared jumping distance (ESJD) under general target distributions.
• Introduces new sufficient conditions (A1–A5) that are easy to verify and allow for dependent coordinates in the target distribution.
• Establishes one-dimensional diffusion limits under slightly stronger versions of the conditions, enabling asymptotic analysis of convergence speed.
• Uses a sequence of typical sets and concentration inequalities to control tail probabilities and ensure convergence of the diffusion limit.
• Applies the framework to hierarchical and graphical models, verifying conditions via conditional distributions and log-density gradients.
• Employs probabilistic bounds and moment conditions (e.g., m=5) to ensure summability of tail probabilities and almost sure convergence.

Experimental results

Research questions

• RQ1 Under what general conditions on high-dimensional target distributions does the optimal acceptance rate of 0.234 hold for random-walk Metropolis algorithms?
• RQ2 Can one-dimensional diffusion limits be established for RWM when the target distribution has dependent coordinates?
• RQ3 How do the new sufficient conditions compare to the classical i.i.d. product assumption in terms of applicability to real Bayesian models?
• RQ4 What is the relationship between the diffusion limit and the computational complexity of RWM in high dimensions?
• RQ5 Can the theoretical framework be applied to practical Bayesian hierarchical models with non-i.i.d. structure?

Key findings

• The asymptotically optimal acceptance rate of 0.234 for random-walk Metropolis is achieved under general, verifiable conditions that do not require i.i.d. or product structure.
• The sufficient conditions (A1–A5) are easy to verify and hold for several classes of Bayesian models, including hierarchical and graphical models with dependent coordinates.
• One-dimensional diffusion limits are established under slightly stronger but still realistic conditions, allowing for asymptotic analysis of convergence speed.
• The results generalize prior work by removing the restrictive i.i.d. assumption while maintaining the same optimal acceptance rate.
• The framework applies to models with non-Gaussian targets and non-Gaussian proposals, extending beyond Gaussian reference measures.
• Theoretical results are validated on a hierarchical normal variance component model, where all required conditions are verified using conditional distributions and moment bounds.