  The Metropolis-Hastings (MH) algorithm is one of the most widely used Markov
Chain Monte Carlo schemes for generating samples from Bayesian posterior
distributions. The algorithm is asymptotically exact, flexible and easy to
implement. However, in the context of Bayesian inference for large datasets,
evaluating the likelihood on the full data for thousands of iterations until
convergence can be prohibitively expensive. This paper introduces a new
subsample MH algorithm that satisfies detailed balance with respect to the
target posterior and utilises control variates to enable exact, efficient
Bayesian inference on datasets with large numbers of observations. Through
theoretical results, simulation experiments and real-world applications on
certain generalised linear models, we demonstrate that our method requires
substantially smaller subsamples and is computationally more efficient than the
standard MH algorithm and other exact subsample MH algorithms.
