Efficiency of delayed-acceptance random walk Metropolis algorithms

Delayed-acceptance Metropolis-Hastings and delayed-acceptance pseudo-marginal Metropolis-Hastings algorithms can be applied when it is computationally expensive to calculate the true posterior or an unbiased stochastic approximation thereof, but a computationally cheap deterministic approximation is available. An initial accept-reject stage uses the cheap approximation for computing the Metropolis-Hastings ratio; proposals which are accepted at this stage are then subjected to a further accept-reject step which corrects for the error in the approximation. Since the expensive posterior, or the approximation thereof, is only evaluated for proposals which are accepted at the first stage, the cost of the algorithm is reduced and larger scalings may be used. We focus on the random walk Metropolis (RWM) and consider the delayed-acceptance RWM and the delayed-acceptance pseudo-marginal RWM. We provide a framework for incorporating relatively general deterministic approximations into the theoretical analysis of high-dimensional targets. Justified by diffusion approximation arguments, we derive expressions for the limiting efficiency and acceptance rates in high-dimensional settings. These theoretical insights are finally leveraged to formulate practical guidelines for the efficient tuning of the algorithms. The robustness of these guidelines and predicted properties are verified against simulation studies, all of which are strictly outside of the domain of validity of our limit results.