Optimal scaling of Metropolis algorithms: Heading toward general target distributions

Abstract

Abstract:The authors provide an overview of optimal scaling results for the Metropolis algorithm with Gaussian proposal distribution. They address in more depth the case of high-dimensional target distributions formed of independent, but not identically distributed components. They attempt to give an intuitive explanation as to why the well-known optimal acceptance rate of 0.234 is not always suitable. They show how to find the asymptotically optimal acceptance rate when needed, and they explain why it is sometimes necessary to turn to inhomogeneous proposal distributions. Their results are illustrated with a simple example. Echelonnage optimal de l'algorithme Metropolis : en route vers des lois cibles généralesRésumé : Les auteurs font un survol des résultats sur l'échelonnage optimal de l'algorithme Metropolis avec loi instrumentale gaussienne. Ils s'intéressent de plus près au cas des lois cibles multivariéesà composantes indépendantes mais non identiquement distribuées. Ils tentent d'expliquer en termes intuitifs pourquoi le taux d'acceptation optimal bien connu de 0.234 ne convient pas toujours. Ils montrent comment déterminer au besoin le taux d'acceptation asymptotiquement optimal et ils expliquent pourquoi il est parfois indispensable d'avoir recoursà des lois instrumentales non homogènes. Leurs résultats sont illustrés dans un cas simple.

On the efficiency of pseudo-marginal random walk Metropolis algorithms

“…However, these i.i.d. assumptions have been relaxed in various directions [5,6,8,9,23], and we believe that our diffusion limit Theorem 2 could also be extended to similar settings at the cost of considerably less transparent proofs.…”

“…Small proposed jumps lead to high acceptance rates but little movement across the state space, whereas large proposed jumps lead to low acceptance rates and again to inefficient exploration of the state space. The problem of choosing the optimal scale of the RWM proposal has been tackled for various shapes of target (e.g., [5,6,8,10,26,28,31,33]) and has led to the following rule of thumb: choose the scale so that the acceptance rate is approximately 0.234. Although nearly all of the theoretical results are based upon limiting arguments in high dimension, the rule of thumb appears to be applicable even in relatively low dimensions (e.g., [32]).…”

Optimal Proposal Distributions and Adaptive MCMC

“…In a related direction, Bédard (2007Bédard ( , 2008Bédard ( , 2006; see also Bédard and Rosenthal, 2008) considered the case where the target distribution π has independent coordinates with vastly different scalings (i.e., different powers of d as d → ∞). She proved that if each individual component is dominated by the sum of all components, then the optimal acceptance rate of 0.234 still holds.…”

Some of Statistics Canada’s Contributions to Survey Methodology

“…A number of authors have attempted to weaken and generalize the original strong assumptions; see e.g., Bédard (2007Bédard ( , 2008, Bédard and Rosenthal (2008), Beskos et al (2009), andSherlock and. Corresponding results have been developed for Langevin MCMC algorithms (Roberts and Rosenthal, 1998), and for simulated tempering algorithms (Atchadé et al, 2011;Roberts and Rosenthal, 2013).…”