Austerity in MCMC Land: Cutting the Metropolis-Hastings Budget

Markov chain Monte Carlo (MCMC) sampling has been the main workhorse of Bayesian computation since the 1990s. A canonical MCMC algorithm proposes samples from a distribution q and then accepts or rejects these proposals with a certain probability given by the Metropolis-Hastings (MH) formula [Metropolis et al., 1953, Hastings, 1970]. For each proposed sample, the MH rule needs to examine the likelihood of all dataitems. When the number of data-cases is large this is an awful lot of computation for one bit of information, namely whether to accept or reject a proposal. In today's Big Data world, we need to rethink our Bayesian inference algorithms. Standard MCMC methods do not meet the Big Data challenge for the reason described above. Researchers have made some progress in terms of making MCMC more efficient, mostly by focusing on parallelization. Very few question the algorithm itself: is the standard MCMC paradigm really optimally efficient in achieving its goals?