🤖 AI Summary
This study investigates the optimal scaling of high-dimensional Metropolised MCMC algorithms, focusing on how to adjust proposal distributions with increasing dimensionality to maintain sampling efficiency. Building upon the symmetry of the Metropolis–Hastings algorithm and high-dimensional asymptotic analysis, the authors develop a unified framework applicable to a broad class of target distributions and proposal mechanisms. The approach not only recovers classical results—such as the $O(1/d)$ variance scaling for Random Walk Metropolis (RWM) and $O(1/d^{1/3})$ for Metropolis-Adjusted Langevin Algorithm (MALA)—but also derives a novel class of gradient-driven MALA proposals with an optimal scaling law: their variance can be set to $O(1/d^\mu)$ for arbitrarily small $\mu > 0$, substantially outperforming existing methods. The theoretical analysis integrates non-product target measures and proposal distributions generated by implicit integrators of differential equations, demonstrating enhanced adaptability to dimensionality.
📝 Abstract
We present a simple, yet general approach to study the scaling properties as the dimensionality of Metropolised MCMC sampling algorithms increases. The study relies ultimately on the symmetry of the Metropolis-Hastings formula. Our findings contain, as particular cases, many known results for the Random Walk Metropolis, MALA and other algorithms. In addition, they provide, in an easy way, new optimal scaling results for a variety of proposal mechanisms, including implicit proposals and proposals generated with the help of differential equation integrators. The analysis applies to targets that are products of a given, not necessarily univariate distribution, and also to cases where the different terms in the product are scaled differently. We show how to construct gradient-based MALA-like proposals where the variance of the proposal as the dimension $d$ increases may be taken as $O(1/d^μ)$, with $μ>0$ arbitrarily small, to be compared with the values $μ= 1$ for Random Walk Metropolis and $μ=1/3$ for MALA.