  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.
