Speeding up MCMC by Delayed Acceptance and Data Subsampling

The complexity of the Metropolis–Hastings (MH) algorithm arises from the requirement of a likelihood evaluation for the full dataset in each iteration. One solution has been proposed to speed up the algorithm by a delayed acceptance approach where the acceptance decision proceeds in two stages. In the first stage, an estimate of the likelihood based on a random subsample determines if it is likely that the draw will be accepted and, if so, the second stage uses the full data likelihood to decide upon final acceptance. Evaluating the full data likelihood is thus avoided for draws that are unlikely to be accepted. We propose a more precise likelihood estimator that incorporates auxiliary information about the full data likelihood while only operating on a sparse set of the data. We prove that the resulting delayed acceptance MH is more efficient. The caveat of this approach is that the full dataset needs to be evaluated in the second stage. We therefore propose to substitute this evaluation by an estimate and construct a state-dependent approximation thereof to use in the first stage. This results in an algorithm that (i) can use a smaller subsample m by leveraging on recent advances in Pseudo-Marginal MH (PMMH) and (ii) is provably within O(m^{? 2}) of the true posterior.