Accelerating adaptation in the adaptive Metropolis–Hastings random walk algorithm

The 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.