  MCMC algorithms such as Metropolis-Hastings algorithms are slowed down by the
computation of complex target distributions as exemplified by huge datasets. We
offer in this paper a useful generalisation of the Delayed Acceptance approach,
devised to reduce the computational costs of such algorithms by a simple and
universal divide-and-conquer strategy. The idea behind the generic acceleration
is to divide the acceptance step into several parts, aiming at a major
reduction in computing time that out-ranks the corresponding reduction in
acceptance probability. Each of the components can be sequentially compared
with a uniform variate, the first rejection signalling that the proposed value
is considered no further. We develop moreover theoretical bounds for the
variance of associated estimators with respect to the variance of the standard
Metropolis-Hastings and detail some results on optimal scaling and general
optimisation of the procedure. We illustrate those accelerating features on a
series of examples

Computation