  Markov chain Monte Carlo (MCMC) algorithms are generally regarded as the gold
standard technique for Bayesian inference. They are theoretically
well-understood and conceptually simple to apply in practice. The drawback of
MCMC is that in general performing exact inference requires all of the data to
be processed at each iteration of the algorithm. For large data sets, the
computational cost of MCMC can be prohibitive, which has led to recent
developments in scalable Monte Carlo algorithms that have a significantly lower
computational cost than standard MCMC. In this paper, we focus on a particular
class of scalable Monte Carlo algorithms, stochastic gradient Markov chain
Monte Carlo (SGMCMC) which utilises data subsampling techniques to reduce the
per-iteration cost of MCMC. We provide an introduction to some popular SGMCMC
algorithms and review the supporting theoretical results, as well as comparing
the efficiency of SGMCMC algorithms against MCMC on benchmark examples. The
supporting R code is available online.
