  Speeding up Markov Chain Monte Carlo (MCMC) for datasets with many
observations by data subsampling has recently received considerable attention.
A pseudo-marginal MCMC method is proposed that estimates the likelihood by data
subsampling using a block-Poisson estimator. The estimator is a product of
Poisson estimators, allowing us to update a single block of subsample
indicators in each MCMC iteration so that a desired correlation is achieved
between the logs of successive likelihood estimates. This is important since
pseudo-marginal MCMC with positively correlated likelihood estimates can use
substantially smaller subsamples without adversely affecting the sampling
efficiency. The block-Poisson estimator is unbiased but not necessarily
positive, so the algorithm runs the MCMC on the absolute value of the
likelihood estimator and uses an importance sampling correction to obtain
consistent estimates of the posterior mean of any function of the parameters.
Our article derives guidelines to select the optimal tuning parameters for our
method and shows that it compares very favourably to regular MCMC without
subsampling, and to two other recently proposed exact subsampling approaches in
the literature.
