Abstract
SummaryThe Metropolis–Hastings random walk algorithm remains popular with practitioners due to the wide variety of situations in which it can be successfully applied and the extreme ease with which it can be implemented. Adaptive versions of the algorithm use information from the early iterations of the Markov chain to improve the efficiency of the proposal. The aim of this paper is to reduce the number of iterations needed to adapt the proposal to the target, which is particularly important when the likelihood is time‐consuming to evaluate. First, the accelerated shaping algorithm is a generalisation of both the adaptive proposal and adaptive Metropolis algorithms. It is designed to remove, from the estimate of the covariance matrix of the target, misleading information from the start of the chain. Second, the accelerated scaling algorithm rapidly changes the scale of the proposal to achieve a target acceptance rate. The usefulness of these approaches is illustrated with a range of examples.
Highlights
The Metropolis–Hastings random walk (MHRW) algorithm (Metropolis et al 1953; Hastings 1970) is a Markov Chain Monte Carlo (MCMC) algorithm that has an enduring popularity with practitioners due to the ease of implementation and the wide variety of circumstances in which it is applicable
A simple two-dimensional example shows that removing as well as adding observations to the estimation of the covariance matrix speeds up the time taken to obtain a reasonable estimate when the chain is not started close to the posterior mode
The example target is a multivariate normal with mean (0, 200) and covariance matrix = [50, − 40; −40, 50]. This is an elliptical ridge with strong negative correlation
Summary
The Metropolis–Hastings random walk (MHRW) algorithm (Metropolis et al 1953; Hastings 1970) is a Markov Chain Monte Carlo (MCMC) algorithm that has an enduring popularity with practitioners due to the ease of implementation and the wide variety of circumstances in which it is applicable. Landmark papers have shown that, for a d dimensional target with covariance matrix , in a range of circumstances the optimal proposal covariance matrix is 2.382 =d , leading to an optimal acceptance rate of 0.234 (Gelman, Roberts & Gilks 1996; Roberts, Gelman & Gilks 1997; Roberts & Rosenthal 2001). This acceptance rate has been proved to be optimal for some other MCMC proposals (Lee & Neal 2018). Accelerating the speed at which adaptations occur during the burn-in phase of the MHRW algorithm is the subject of this paper
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