  The pseudo-marginal (PM) approach is increasingly used for Bayesian inference
in statistical models, where the likelihood is intractable but can be estimated
unbiasedly. %Examples include random effect models, state-space models and data
subsampling in big-data settings. Deligiannidis et al. (2016) show how the PM
approach can be made much more efficient by correlating the underlying Monte
Carlo (MC) random numbers used to form the estimate of the likelihood at the
current and proposed values of the unknown parameters. Their approach greatly
speeds up the standard PM algorithm, as it requires a much smaller number of
samples or particles to form the optimal likelihood estimate. Our paper
presents an alternative implementation of the correlated PM approach, called
the block PM, which divides the underlying random numbers into blocks so that
the likelihood estimates for the proposed and current values of the parameters
only differ by the random numbers in one block. We show that this
implementation of the correlated PM can be much more efficient for some
specific problems than the implementation in Deligiannidis et al. (2016); for
example when the likelihood is estimated by subsampling or the likelihood is a
product of terms each of which is given by an integral which can be estimated
unbiasedly by randomised quasi-Monte Carlo. Our article provides methodology
and guidelines for efficiently implementing the block PM. A second advantage of
the the block PM is that it provides a direct way to control the correlation
between the logarithms of the estimates of the likelihood at the current and
proposed values of the parameters than the implementation in Deligiannidis et
al. (2016). We obtain methods and guidelines for selecting the optimal number
of samples based on idealized but realistic assumptions.
